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Resonant Thermoelectric Nanophotonics: Applications in Spectral and Thermal Sensing
Citation
Mauser, Kelly Ann Weekley
(2019)
Resonant Thermoelectric Nanophotonics: Applications in Spectral and Thermal Sensing.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/0HJF-X691.
Abstract
Plasmon excitation enables extreme light confinement at the nanoscale, localizing energy in subwavelength volumes and thus can enable increased absorption in photovoltaic or photoconductive detectors. Nonetheless, plasmon decay also results in energy transfer to the lattice as heat which is detrimental to photovoltaic detector performance. However, heat generation in resonant subwavelength nanostructures also represents an energy source for voltage generation, as we demonstrate in the first part of this thesis via design of resonant thermoelectric (TE) plasmonic absorbers for optical detection. Though TEs have been used to observe resonantly coupled surface plasmon polaritons in noble-metal thin films and microelectrodes, they have not been employed previously as resonant absorbers in functional TE nanophotonic structures.
We demonstrate nanostructures composed of TE thermocouple junctions using established TE materials – chromel/alumel and bismuth telluride/antimony telluride – but patterned so as to support guided mode resonances with sharp absorption profiles, and which thus generate large thermal gradients upon optical excitation and localized heat generation in the TE material. Unlike previous TE absorbers, our structures feature tunable narrowband absorption and measured single junction responsivities 4 times higher than the most similar (albeit broadband) graphene structures, with potential for much higher responsivities in thermopile architectures. For bismuth telluride – antimony telluride single thermocouple structures, we measure a maximum responsivity of 38 V/W, referenced to incident illumination power. We also find that the small heat capacity of optically resonant TE nanowires enables a fast, 3 kHz temporal response, 10-100 times faster than conventional TE detectors. We show that TE nanophotonic structures are tunable from the visible to the MIR, with small structure sizes of 50 microns x 100 micons. Our nanophotonic TE structures are suspended on thin membranes to reduce substrate heat losses and improve thermal isolation between TE structures arranged in arrays suitable for imaging or spectroscopy. Whereas photoconductive and photovoltaic detectors are typically insensitive to sub-bandgap radiation, nanophotonic TEs can be designed to be sensitive to any specific wavelength dictated by nanoscale geometry, without bandgap wavelength cutoff limitations. From the point of view of imaging and spectroscopy, they enable integration of filter and photodetector functions into a single structure. Other thermoelectric nanophotonic motifs are also explored.
Generating localized, high electric field intensity in nanophotonic and plasmonic devices has many applications, from enhancing chemical reaction rates, to thermal radiation steering, to chemical sensing, and to photovoltaics. Along with a strongly localized electric field comes a temperature rise in non-lossless photonic materials, which can affect reaction rate, photovoltaic efficiency, or other properties of the system. Measuring temperature rises in nanophotonic structures is difficult, and methods commonly employed suffer from various limitations, such as low spatial resolution (Fourier transform infrared microscopy), bulky and expensive setups (scanning thermal microscopy), intrusive methods that interfere with nanophotonic structures (Pt resistive thermometry), or the need for specialized materials (temperature dependent photoluminescence).
In the second part of this thesis, we overcome these limitations with the first-ever demonstration of temperature measurements of nanophotonic structures by employing both room temperature noise thermometry and the thermoelectric effect under ambient conditions without external probes by utilizing the properties of the materials that make up the nanophotonic structure itself. We have previously estimated the Δ T in a nanophotonic device using the thermoelectric effect, but could not determine the absolute temperature of the system. In the application we will discuss, the absolute electron temperature of the nanophotonic material itself is measured. Because Johnson-Nyquist noise is material independent and is a fundamental measure of absolute temperature, there is theoretically no need for calibration as in the case of resistive thermometry. To measure the temperature rise of a nanophotonic resonant region remotely, the Seebeck coefficient of the material is first carefully measured using noise thermometry, then the thermoelectric voltage generated in the nanophotonic materials themselves is measured from electrical leads spanning the resonantly excited region. To accomplish this, we have developed a metrology technique capable of simultaneously measuring electrical noise at two locations on the nanophotonic structure as well as the electrical potential between the two points, under chopped laser illumination that heats the structure via nanophotonic absorption, thus providing drift-corrected light on/off temperature information.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Nanophotonics; Thermoelectrics; Noise Thermometry; Thermoelectric Nanophotonics
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Atwater, Harry Albert
Group:
Kavli Nanoscience Institute
Thesis Committee:
Schwab, Keith C. (chair)
Painter, Oskar J.
Minnich, Austin J.
Atwater, Harry Albert
Defense Date:
19 April 2019
Record Number:
CaltechTHESIS:06062019-191307495
Persistent URL:
DOI:
10.7907/0HJF-X691
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DOI
Article adapted for thesis.
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Author
ORCID
Mauser, Kelly Ann Weekley
0000-0001-9903-8559
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No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
11700
Collection:
CaltechTHESIS
Deposited By:
Kelly Mauser
Deposited On:
07 Jun 2019 22:10
Last Modified:
08 Nov 2023 00:12
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Resonant Thermoelectric Nanophotonics: Applications
in Spectral and Thermal Sensing

Thesis by

Kelly Ann Weekley Mauser

In Partial Fulfillment of the Requirements for the
Degree of
Doctor of Philosophy in Applied Physics

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2019
Defended April 19, 2019

Kelly Ann Weekley Mauser
ORCID: 0000-0001-9903-8559

iii

ACKNOWLEDGEMENTS
I would like to thank Prof. Harry Atwater for the scientific support and advice he
has provided, and for welcoming me into his group. I would also like to thank Prof.
Keith Schwab for allowing me to use his lab and for all the advice he has given
me. Prof. Painter and Prof. Minnich (and their students) have also been helpful
resources at various points in my time at Caltech.
Many people have helped me accomplish the work completed in this thesis. On the
fabrication side, those who have given advice or who have been willing to discuss
my many problems are Max, Dvin, Seyoon, Phil, Hal, Victor, Mike, Michelle, and
Dagny, among others. Additionally, the KNI staff have been invaluable, and have
made the KNI a great place to work, so thank you Guy, Alex, Matt, Nathan, Melissa,
Bert, Barry, and Steven. I would especially like to thank Alex and Dvin (my sputter
buddies) for being very accommodating. On the simulation side, Seyoon taught
me a great deal, and Phil was excellent at keeping the computers up and running.
I would especially like to thank Keith for teaching me everything about noise and
noise thermometry and all kinds of different measurements, and for Constantine
(Doug) Sideris for answering all of my circuit questions. Team safety has been a
fun team to be a part of. Thanks Rebecca, Cris, Haley, Kelly 2.0, and Eowyn. I
wish you the best in continuing to fight the unwinnable war.
I would like to thank Seyoon Kim (my very best friend), for being a constant source
of support, and for a lot of fun times. Not only is he a great friend, but he is a
great scientist and has very helpfully and directly told me I was wrong a lot (he was
usually right).
I would like to thank Jacob, Ellen, Max, Charlie, and Cedric, who helped me survive
Caltech classes and are great friends and programming resources. Keeping me sane
outside of lab, I would like to thank Anya, Yury, Ruzan, Nick, Dylan, Andrew, Nate,
Fred, Soichi, Ryan (I guess), Roberta, Emily, and Cora. I would also like to thank
the amazing administrators in our department, including Jonathan, Angie, Christy,
Jennifer, Tiffany, Connie, Lyann, and Liz. They have been so helpful over the years
and always so kind.
I would also like to thank Nathan for being a great partner and travel buddy during
grad school. I couldn’t have done this without his support.
Finally, and most importantly, I would like to thank Mom, Dad, and Paul for always

iv
supporting me through the good and the bad.

ABSTRACT
Plasmon excitation enables extreme light confinement at the nanoscale, localizing
energy in subwavelength volumes and thus can enable increased absorption in
photovoltaic or photoconductive detectors. Nonetheless, plasmon decay also results
in energy transfer to the lattice as heat which is detrimental to photovoltaic detector
performance. However, heat generation in resonant subwavelength nanostructures
also represents an energy source for voltage generation, as we demonstrate in the first
part of this thesis via design of resonant thermoelectric (TE) plasmonic absorbers
for optical detection. Though TEs have been used to observe resonantly coupled
surface plasmon polaritons in noble-metal thin films and microelectrodes, they have
not been employed previously as resonant absorbers in functional TE nanophotonic
structures.
We demonstrate nanostructures composed of TE thermocouple junctions using established TE materials – chromel/alumel and bismuth telluride/antimony telluride
– but patterned so as to support guided mode resonances with sharp absorption
profiles, and which thus generate large thermal gradients upon optical excitation
and localized heat generation in the TE material. Unlike previous TE absorbers,
our structures feature tunable narrowband absorption and measured single junction
responsivities 4 times higher than the most similar (albeit broadband) graphene
structures, with potential for much higher responsivities in thermopile architectures.
For bismuth telluride – antimony telluride single thermocouple structures, we measure a maximum responsivity of 38 V/W, referenced to incident illumination power.
We also find that the small heat capacity of optically resonant TE nanowires enables
a fast, 3 kHz temporal response, 10-100 times faster than conventional TE detectors.
We show that TE nanophotonic structures are tunable from the visible to the MIR,
with small structure sizes of 50 microns x 100 micons. Our nanophotonic TE structures are suspended on thin membranes to reduce substrate heat losses and improve
thermal isolation between TE structures arranged in arrays suitable for imaging or
spectroscopy. Whereas photoconductive and photovoltaic detectors are typically
insensitive to sub-bandgap radiation, nanophotonic TEs can be designed to be sensitive to any specific wavelength dictated by nanoscale geometry, without bandgap
wavelength cutoff limitations. From the point of view of imaging and spectroscopy,
they enable integration of filter and photodetector functions into a single structure.
Other thermoelectric nanophotonic motifs are also explored.

vi
Generating localized, high electric field intensity in nanophotonic and plasmonic
devices has many applications, from enhancing chemical reaction rates, to thermal
radiation steering, to chemical sensing, and to photovoltaics. Along with a strongly
localized electric field comes a temperature rise in non-lossless photonic materials,
which can affect reaction rate, photovoltaic efficiency, or other properties of the
system. Measuring temperature rises in nanophotonic structures is difficult, and
methods commonly employed suffer from various limitations, such as low spatial
resolution (Fourier transform infrared microscopy), bulky and expensive setups
(scanning thermal microscopy), intrusive methods that interfere with nanophotonic
structures (Pt resistive thermometry), or the need for specialized materials (temperature dependent photoluminescence).
In the second part of this thesis, we overcome these limitations with the first-ever
demonstration of temperature measurements of nanophotonic structures by employing both room temperature noise thermometry and the thermoelectric effect under
ambient conditions without external probes by utilizing the properties of the materials that make up the nanophotonic structure itself. We have previously estimated
the ∆ T in a nanophotonic device using the thermoelectric effect, but could not determine the absolute temperature of the system. In the application we will discuss,
the absolute electron temperature of the nanophotonic material itself is measured.
Because Johnson-Nyquist noise is material independent and is a fundamental measure of absolute temperature, there is theoretically no need for calibration as in the
case of resistive thermometry. To measure the temperature rise of a nanophotonic
resonant region remotely, the Seebeck coefficient of the material is first carefully
measured using noise thermometry, then the thermoelectric voltage generated in
the nanophotonic materials themselves is measured from electrical leads spanning
the resonantly excited region. To accomplish this, we have developed a metrology
technique capable of simultaneously measuring electrical noise at two locations
on the nanophotonic structure as well as the electrical potential between the two
points, under chopped laser illumination that heats the structure via nanophotonic
absorption, thus providing drift-corrected light on/off temperature information.

vii

PUBLISHED CONTENT AND CONTRIBUTIONS

Mauser, Kelly W et al. (2017). “Resonant thermoelectric nanophotonics”. In: Nature
nanotechnology 12.8, p. 770. doi: https://doi.org/10.1038/nnano.2017.
87.
K.A.W.M. participated in the conception of the project, fabricated the structures,
performed the measurements and simulations, analyzed the data, and participated
in the writing of the manuscript.

viii

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . vii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Nanoscale light-matter interactions . . . . . . . . . . . . . . . . . . 1
1.2 Thermoelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Noise thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Scope of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Chapter II: Design considerations and simulations of nanophotonic devices . 16
2.1 Design considerations in nanophotonic thermoelectric devices . . . . 16
2.2 Scale of nanophotonic vs. thermal simulations . . . . . . . . . . . . 17
2.3 Thermal time constant of devices . . . . . . . . . . . . . . . . . . . 17
2.4 Simulation material values . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Perfect absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Split ring resonator perfect absorber . . . . . . . . . . . . . . . . . . 22
2.7 Plasmonic bowtie antenna . . . . . . . . . . . . . . . . . . . . . . . 25
2.8 Guided mode resonator thermocouple . . . . . . . . . . . . . . . . . 29
2.9 Guided mode resonator thermopile . . . . . . . . . . . . . . . . . . 48
2.10 Analysis and summary of simulation findings and guidance for future
designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter III: Hyperspectral detector application . . . . . . . . . . . . . . . . . 52
3.1 Background of hyperspectral detectors . . . . . . . . . . . . . . . . 52
3.2 Fabrication and materials analysis . . . . . . . . . . . . . . . . . . . 53
3.3 Measurement results . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 Best possible performance with state-of-the-art materials for current
design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5 Why D∗ is not a valid FOM in our structures and other thermal detectors 70
3.6 Comparison with bolometers . . . . . . . . . . . . . . . . . . . . . 73
3.7 Outlook, improving performance, and potential applications . . . . . 76
Chapter IV: Nanoscale temperature measurements with noise thermometry
and the thermoelectric effect - Low noise design . . . . . . . . . . . . . . 78
4.1 Nanophotonic temperature measurement review . . . . . . . . . . . 78
4.2 Measurement plan . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Design considerations for circuit board . . . . . . . . . . . . . . . . 81
Chapter V: Noise thermometry results . . . . . . . . . . . . . . . . . . . . . 104

ix
5.1 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2 Noise thermometry circuit measurement verification . . . . . . . . . 111
5.3 Noise measurements under chopped illumination . . . . . . . . . . . 114
5.4 Outlook and future work . . . . . . . . . . . . . . . . . . . . . . . . 122
Chapter VI: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.1 Potential for nanophotonic thermoelectric detectors . . . . . . . . . . 126
6.2 Potential for noise thermometry in nanophotonic devices . . . . . . . 127
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Appendix A: FFTW C program for data analysis . . . . . . . . . . . . . . . . 139
Appendix B: Python GUIs for instrument control . . . . . . . . . . . . . . . 217

LIST OF ILLUSTRATIONS

Number
Page
1.1 Simplified version of thermoelectric operation for p-type (top left),
n-type (top right), and a thermocouple (bottom). . . . . . . . . . . . 5
1.2 Equivalent circuit of a single "pulse" of an electron in a resistor. . . . 12
2.1 Artistic rendition of perfect absorber structure, rendered in POV-ray. . 19
2.2 Dielectric function of bismuth telluride in visible and near-IR wavelengths measured using J.A. Woollam Co. VASE and IR-VASE
MARK II ellipsometers and analyzed with WVASE software.. . . . . 19
2.3 Dielectric function of bismuth telluride in IR wavelengths. . . . . . . 20
2.4 Dielectric function of antimony telluride in visible to near-IR wavelengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Dielectric function of antimony telluride in IR wavelengths. . . . . . 21
2.6 Lumerical simulation of normalized electric field magnitude of the
perfect absorber structure described in the text, on resonance at 1,648
nm (left), and as a function of wavelength (right). Scale bar is 300
nm. Credit to Dr. Seyoon Kim for nicely plotting the simulated data
on the left. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

xi
2.7

Perfect absorber and split ring resonator thermal simulation design.
(Top left) Diagram of thermal simulation design for a perfect absorber
and a split ring resonator thermal simulation corresponding to data
in Figures 2.9, 2.15 and Figures 2.8, 2.14, 2.10, 2.16, 2.13. The split
ring resonator design has a 50 nm thick SiNx membrane with 60 nm
of thermoelectric material above it, which serves as a backreflector
in the optical design. The perfect absorber has a 100 nm thick SiNx
membrane and 100 nm thick TE materials. It uses a 50 nm thick
layer of gold as the backreflector in the 10 µm by 10 µm center
absorber patch. The 200 nm gold above the silica increases thermal
conduction of heat from the cold end of the device (i.e. acts as a heat
sink), and any other thermally conductive material would serve this
purpose well. Thermal simulations involved a volumetric heat influx
into the centrally located split ring resonator or cylinder absorbers in
the array (this assumes absorption near the edges of the arrays would
be worse). (Top right) Thermal simulation of perfect absorber with a
total absorbed power of 23 µW. The split ring resonator thermal profile
was similar (see Figure 2.13). (Bottom left) Thermal simulations of
the perfect absorber design. The perfect absorber array covers a 10
µm by 10 µm square in the center of a suspended, TE junction on top
of a SiNx membrane. The “cold” ends of the TE materials sit on a
100 nm SiNx /200 nm Au/20 µm SiO2 substrate. The Au layer acts as
a high thermal conductivity heat sink to conduct heat away from the
“cold” ends of the TE materials. Both simulations were performed
in vacuum. The scale bars are 10 µm, and the absorbed power is 23 µW. 22
2.8 Temperature differences between the hot and cold sides of the thermoelectric materials as a function of power absorbed for the perfect
absorber structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.9 Simulated responsivities as a function of incident angle of the perfect
absorber structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.10 Simulated noise equivalent power as a function of incident angle of
the perfect absorber structure. . . . . . . . . . . . . . . . . . . . . . 24
2.11 Artistic rendition of split ring resonator design, rendered in POV-ray. 24

xii
2.12

2.13

2.14

2.15
2.16
2.17
2.18

2.19

2.20
2.21
2.22

Lumerical simulation of normalized electric field magnitude of the
split ring resonator structure described in the text, on resonance at
1,710 nm (left), and as a function of wavelength (right). Scale bar is
50 nm. Credit to Dr. Seyoon Kim for nicely plotting the simulated
data on the left. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thermal simulation for split ring resonator perfect absorbers utilizing
a thin membrane to lower thermal heat loss to the substrate. Total
absorbed power is 23 µW. Large scale bar is 10 µm. . . . . . . . . . .
Temperature differences between the hot and cold sides of the thermoelectric materials as a function of power absorbed for the split ring
resonator absorber structure. . . . . . . . . . . . . . . . . . . . . . .
Simulated responsivities as a function of incident angle of the split
ring resonator absorber structure. . . . . . . . . . . . . . . . . . . .
Simulated noise equivalent power as a function of incident angle of
the split ring resonator absorber structure. . . . . . . . . . . . . . . .
Artistic rendition of the plasmonic bowtie absorber, rendered in POVray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lumerical simulation of normalized electric field magnitude of the
bowtie absorber structure described in the text, on resonance at 660
nm (left), and absorption cross-section as a function of wavelength
(right) for the wire alone (blue) and the wire with the bowtie (orange).
Scale bar is 200 nm. Credit to Dr. Seyoon Kim for these simulation
results and for nicely plotting the simulated data on the left. . . . . .
(Top left) Thermal simulations of teh resonant bowtie antenna, performed in vacuum. Scale bar is 500 nm and the absorbed power
is 7.9 µW. (Top right) Thermal simulation of bowtie structure with
absorbed power of 7.9 µW. Maximum temperature is 330 K, but a
smaller scale was used to show temperature differences better. Scale
bar is 20 µm and inset scale bar is 500 nm. (Bottom left) The same
thermal simulation as in the top right is shown, but with the full temperature scale bar. The outline of the bowtie is given in blue dotted
line. Scale bar is 500 nm. . . . . . . . . . . . . . . . . . . . . . . .
Temperature difference between hot and cold spot of thermoelectric
wires in the resonant bowtie antenna design. . . . . . . . . . . . . .
Responsivity of the resonant bowtie antenna design. . . . . . . . . .
Noise equivalent power of the resonant bowtie antenna design. . . . .

26
27
27
28

29
30
30
31

xiii
2.23

2.24

2.25

2.26

2.27

2.28

Conceptual design of GMR resonant thermoelectric structure. The
junction of the thermoelectric wires absorbs incident, spatially uniform illumination and is absorbed, heating the junction and producing
a thermoelectric voltage. Light illuminating the pads will be reflected,
creating the necessary temperature gradient. . . . . . . . . . . . . . .
Diagram of interaction of bright (broad resonance) and dark (narrow
resonance) modes in the production of Fano lineshapes. The bright
mode is the Fabry-Perot resonance of light with k-vector parallel to
incident illumination. Analysis from (Gallinet and O. J. F. Martin,
2011a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
a, The Fano formula (Equation 2.7) is fit to our simulation data for
a region near the resonance. textbfb, The extracted bright mode
from a is shown in more detail, and is compared with the normalized
magnitude of the electric field at the top of a 50 nm of SiO2 /100 nm
of SiN x waveguide without wires due to Fabry-Perot resonances. c,
Bright mode and Fano formula fits to simulated and experimental
data. d The simulation and experimental bright modes from c along
with the Fabry-Perot resonance for a waveguide-only structure. . . .
A comparison of absorption spectra of different wire materials in
our guided mode resonant structure composed of 40 nm high, 68 nm
wide, pitch of 488 nm wires on a waveguide of 50 nm SiO2 /100 nm
SiN x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Full wave simulations for guided mode resonance structure with dimensions of 40 nm high, 68 nm wide, pitch of 488 nm wires on a
waveguide of 50 nm SiO2 /100 nm SiN x , with Sb2 Te3 wires (see Figure 2.4 for dielectric function). a, At peak absorption, and b, at the
minimum absorption. a, b, Electric field distributions normalized to
incident electric field. c, d, Power absorption density is calculated
by Pabs = 21 ω 00 |E| 2 , and is normalized by P0 , the incident power
divided by the wire volume. . . . . . . . . . . . . . . . . . . . . . .
(Left) Guided mode resonance structure geometry. Calculated absorption of (Top right) 60 nm wide, and (Bottom right) 100 nm wide
wires with thicknesses of 40 nm and varying pitch on suspended 50
nm SiO2 /100 nm SiN x waveguides. . . . . . . . . . . . . . . . . . .

xiv
2.29

2.30

2.31

2.32

2.33

Experimental absorption (black dotted), simulated absorption corresponding to the experimental dimensions (blue), and simulated absorption corresponding to fitted and scaled absorption spectra (red)
for varying wire pitches and widths on a 45 nm SiO2 /100 nm SiN x
waveguide (see Table 2.2 for dimensions and parameters). Offnormal angle of illumination causes the smaller peak to the left of
the larger absorption peak to form (see Figure 2.30). . . . . . . . . .
Dependence of absorption spectra on incident illumination angle and
wire thickness. a, Measured absorption spectra for different angles
of incidence. An objective with numerical aperture 0.14 was used,
giving an angular spread even at normal incidence, producing the
side peak at normal (0°) incident angle. b, Full wave simulations
of the incident illumination angle dependence of 40 nm tall, 67 nm
wide Sb2 Te3 wires with a pitch of 488 nm on a 50 nm SiO2 on 100
nm SiN x waveguide. Even at 1 degree offset, the single peak splits
into two. c, The dependence of wire thickness on absorption spectra,
with pitch of 488 nm. The absorption asymptotes to its maximum
value for wire heights around 40 nm. . . . . . . . . . . . . . . . . .
(Top) Wavelength versus pitch absorption plot in the visible regime
for 40 nm thick Sb2 Te3 wires, on a 50 nm SiO2 /100 nm SiN x suspended membrane. (Center) Absorption spectra for 50 nm thick, 300
nm wide Sb2 Te3 wires on a 300 nm SiO2 /500 nm SiN x suspended
membrane. (Bottom) Absorption spectra in the mid-IR for 50 nm
thick, 1.5 µm wide Bi2 Te3 wires on a 500 nm SiO2 /500 nm SiN x
suspended membrane. All calculations use either Sb2 Te3 or Bi2 Te3
as the wire material (see Figures 2.2, 2.3,2.4, 2.5 for dielectric functions). Simulations were performed first coarsely with Lumerical
FDTD (Lumerical, n.d.), then were refined by Seyoon Kim with finer
pitch steps using an RCWA method based on (Moharam et al., 1995).
Theoretical absorption (blue) and temperature difference between the
center of the wire and edge of the pad (orange) for a structure with
40 nm tall by 100 nm wide Sb2 Te3 wires spaced 488 nm apart. . . .
Electric field profile normalized to incident electric field of a periodic
structure at peak absorption (1 µm scale bar). Highest |E| occurs in
the wires, leading to absorption, while the pads largely reflect light
creating the necessary temperature gradient. . . . . . . . . . . . . .

xv
2.34

Power density absorbed along a line cut through the simulation in
Figure 2.33. Asymmetry arises from half of the device being Sb2 Te3
and the other half being Bi2 Te3 . Power density is normalized to
incident power divided by thermoelectric structure volume. . . . . .
2.35 A thermal simulation of the Bi2 Te3 - Sb2 Te3 structure at peak absorption with input power of 20 µW. Scale bar is 500 µm, inset scale
bar is 50 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.36 False color SEM of a fabricated p/n thermoelectric structure, with
Au contacts (20 µm scale bar). Inset is the junction between Bi2 Te3
- Sb2 Te3 wires (1 µm scale bar). . . . . . . . . . . . . . . . . . . .
2.37 Thermopile and wire length analysis. a, Absorption spectra used in
simulations for b,c,e,f. b, Responsivity for a structure with 50 µm
long wires and 27 µm long pads, as a function of number of wires in
a thermopile configuration. The entire structure is illuminated and
responsivity is calculated relative to power striking the wire area. The
pads are assumed to have a 20% absorption, independent of wavelength. c, Noise equivalent power (NEP) for the thermopiles in b,
assuming Johnson noise as the noise spectral density using simulated
average temperatures. d, Temperature difference between the edge of
the pad and the center of the wires versus power density for different
wire lengths. Pad sizes remain constant with dimensions of 50 µm by
27 µm by 50 nm. e, Responsivity as a function of wavelength for the
absorption spectrum in a, for different wire lengths, relative to power
illuminating the entire structure. f, Noise equivalent power corresponding to the responsivity in e, for different wire lengths. Noise
spectral density is theoretical Johnson noise using simulated average
temperatures for the structures. Simulation details are given in the
text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.38 Conceptual design of hyperspectral pixel. Each thermoelectric structure in the pixel has a different wire pitch, causing each structure to
have an absorption peak that corresponds to a different wavelength
(Figure 2.31). When light of unknown wavelength content illuminates the pixel, voltages will be produced in each structure depending
on their specific absorption profiles. Deconvolution of these voltage
signals through an algorithm can enable identification of the unknown
wavelengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvi
2.39

2.40

3.1

3.2
3.3

3.4

Artistic rendition of a thermopiled guided mode resonance structure, when p-type and n-type thermoelectrics are alternated in series.
Image rendered in POV-ray. . . . . . . . . . . . . . . . . . . . . . .
Full wave simulations of the thermopiled guided mode resonance
structure, with electric field |E| normalized by incident electric filed
|E0 |. Excitation wavelength is 631 nm (peak absorption for this
geometry). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
XRD data of 100 nm of Bi2 Te3 (left) and 50 nm Sb2 Te3 (right) show
very little crystallinity, as sputtered in experiments. Two-dimensional
diffraction image frames were collected with frame centers set to 20,
40, 60 and 80 degrees in 2Θ, from right to left, and then merged.
Credit to Slobodan Mitrovic for data and analysis. . . . . . . . . . .
XPS survey scans. Sb2 Te3 (top) and Bi2 Te3 (bottom) samples. Credit
to Slobodan Mitrovic for data and analysis. . . . . . . . . . . . . . .
Compositional analysis. Detailed XPS data and fits for bismuth telluride peaks (a-b) and antimony telluride peaks (c-d) for our samples.
a, Three components are visible in Bi 4f levels: the major components are Bi2 O3 , and Bi2 Te3 (157.1 eV and spin-orbit pair at + 5.3
eV), with a small amount of elemental bismuth (156.6 eV). b, Te
3d level in bismuth-telluride is mostly TeO2 and Bi2 Te3 (582.3 eV,
SO-splitting of 10.4 eV). c, Sb 3d levels show that most of the surface
of antimony-telluride is oxidized (Sb2 O3 ), much more so than the
bismuth-telluride, with a measurable Sb2 Te3 component (538 eV). d,
Te 3d levels in antimony-telluride show the telluride, an oxide and
elemental Te peaks. Credit to Slobodan Mitrovic for data and analysis.
(Top) Absorption (0° and 5°) or 1 – transmission (10°) for 0°, 5°, and
10° (±1° error) incident illumination on a Bi2 Te3 - Sb2 Te3 structure
described in the text with wire dimensions of 40 nm thick × 130 nm
wide × 50 µm long. (Bottom) Responsivity for unfocused, spatially
uniform illumination of the entire structure (including the pads, Figure 3.5d) with a 120 µm by 100 µm spot size at 0°, 5°, and 10° (±1°
error) off-normal incidence. . . . . . . . . . . . . . . . . . . . . . .

55
56

xvii
3.5

Focused versus unfocused, spatially uniform light responsivity and
noise characteristics. Focused (blue, 60 µm by 5 µm spot size, e)
and unfocused, spatially uniform (orange, 120 µm by 100 µm spot
size, d) illumination incident on Bi2 Te3 -Sb2 Te3 structures at given
angles off normal incidence, with ±1° error. A 5× objective with
numerical aperture 0.14 was used for both the focused and spatially
uniform illumination data collection. a, The input power used to
calculated responsivity in the case of uniform illumination was only
the power that illuminated the wires (a 50 × 50 µm2 area). The
spatially uniform illumination spot was 120 µm by 100 µm, and
completely covered the wires and pads of the structure. Noise spectral
density, b, was measured under the power spectrum shown in Figure
3.6. Higher noise spectral density in focused light was likely due
to back currents from uneven heating in the structure, discussed
further in the text. c, Noise equivalent power was found to be lower
for spatially uniform illumination than for focused illumination, due
to higher responsivity values combined with lower noise values for
spatially uniform illumination. d, Black circle illustrates uniform
illumination of structure. e, Focused illumination used in a-c, Figure
3.8. f, Focused illumination in Figure 3.7. . . . . . . . . . . . . . .
3.6 Laser power illuminating the wire region as a function of wavelength
for focused illumination (blue) and uniform illumination (orange). .
3.7 Maximum responsivity found for a structure when only the junction
is illuminated (60 um by 5 um spot size, Figure 3.5f). . . . . . . . .
3.8 Thermoelectric voltage (TEV) dependence on incident power for a
Bi2 Te3 - Sb2 Te3 structure at 0° (±1° error) off normal angle under
focused illumination (see Figure 3.5e, and Figure 3.5a for focused
responsivity spectrum). The temperature scale on the right axis
corresponds to ∆T between the hot wire junctions and cold pad edges
based on average measured Seebeck coefficients. We estimate that
1,000 µV would give a temperature range of a 2.8 to 3.4 K temperature
rise, based on the range of Seebeck coefficients of our materials
measured. Error bars are sample standard deviation of measurements.

62
63
63

xviii
3.9

Time response of a Bi2 Te3 - Sb2 Te3 structure. The time constant
fit line (red) plotted over the data from our thermoelectric detector
(green) is measured as 155.13 s ± 3.06 s, corresponding to a 10%90% rise time of 337 s. The response of a Si photodiode at the same
chopper speed is shown in blue. . . . . . . . . . . . . . . . . . . . .
3.10 Noise spectral density (NSD) and noise equivalent power (NEP) as a
function of wavelength corresponding the data shown in Figure 3.4
(bottom). All data were taken under polarized illumination with the
E-field perpendicular to the wires. . . . . . . . . . . . . . . . . . .
3.11 Chromel-alumel structure results. Thermoelectric potential (TEV)
and absorption results for a chromel-alumel structure with the same
dimensions as that of the Bi2 Te3 - Sb2 Te3 structures. The structure is
under 70.92 W illumination, or 30.4 W/cm2 incident power density.
Data points are taken as the focused beam is moved across the junction
of the detector (over a 400 µm2 area). All data points are averaged
together for a given wavelength. . . . . . . . . . . . . . . . . . . . .
3.12 a-c, Simulated absorption spectra for wires of the dimensions of
the experimental Bi2 Te3 -Sb2 Te3 structures at 1, 5, and 10 degrees
off normal incidence. d-l, Responsivity calculated from thermal
simulations using absorption to guide power input at different angles
off normal incidence. 20% absorption in the pads was assumed
independent of wavelength. Simulated absorption from a-c, thermal
properties and Seebeck coefficient of state-of-the-art thermoelectric
materials, given in the text, were used in d-f. Simulated absorption
from a-c, thermal properties given in the previous chapter and average
Seebeck coefficient of our materials (242 µV/K for Sb2Te3 and -84
µV/K for Bi2Te3, see Supplementary Note 8) was used in g-i to
calculate responsivity. Experimental absorption from Figure ??(top),
thermal properties from the previous chapter and average Seebeck
coefficient of our materials (above) was used in j-l to calculated
responsivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Measurement plan outline. The inset shows the wire junctions. The
laser broadly illuminates the wires, and the junction should be the
hottest point in the structure. In principle, because of the optical
properties of this structure discussed in earlier chapters, the entire
structure could be illuminated by the laser. . . . . . . . . . . . . . .

xix
4.2
4.3
4.4

4.5

4.6

4.7

4.8
4.9

First
stage
of
the
circuit.
Magnitude of (blue) and real part of (red) the impedance at the noninverting input of the 1st stage of the opamp shown in Fig. 4.2. . . . .
Spice simulated voltage and phase versus time (top) and frequency
(bottom) seen at the input of the opamp. The signal is a square wave
at 55 Hz with 1 V amplitude generated across the source resistor,
which is 3300 Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison between the power spectral density (PSD) of AD797
(top) and OPA827 (bottom) for an Rs of 3300 Ω. E2s is the noise
power density from Rs , E2total is the sum of all noise power sources,
E2n1 is the voltage noise power of the opamp, E2i1 is the current noise
power of the opamp acting on |Zs |, E R 21 is the voltage noise power
of R1 , E R 22 is the voltage noise power of R2 , and E R 22i is the current
noise power of the opamp current noise acting on R2 (ignoring any
stray capacitances that may contribute). . . . . . . . . . . . . . . . .
Comparison between the power spectral density (PSD) of AD797
(top) and OPA827 (bottom) for an Rs of 10 kΩ. E2s is the noise power
density from Rs , E2total is the sum of all noise power sources, E2n1 is
the voltage noise power of the opamp, E2i1 is the current noise power
of the opamp acting on |Zs |, E R 21 is the voltage noise power of R1 ,
E R 22 is the voltage noise power of R2 , and E R 22i is the current noise
power of the opamp current noise acting on R2 (ignoring any stray
capacitances that may contribute). . . . . . . . . . . . . . . . . . . .
Comparison between the power spectral density (PSD) of AD797
(top) and OPA827 (bottom) for an Rs of 500 Ω. E2s is the noise power
density from Rs , E2total is the sum of all noise power sources, E2n1 is
the voltage noise power of the opamp, E2i1 is the current noise power
of the opamp acting on |Zs |, E R 21 is the voltage noise power of R1 ,
E R 22 is the voltage noise power of R2 , and E R 22i is the current noise
power of the opamp current noise acting on R2 (ignoring any stray
capacitances that may contribute). . . . . . . . . . . . . . . . . . . .
Variation of output noise spectral density with current noise on a
2-stage OPA827 opamp circuit for Rs = 3300Ω. . . . . . . . . . . . .
Variation of output noise spectral density with source resistance and
current noise on a OPA827 2-stage opamp circuit. . . . . . . . . . .

90
91
91

xx
4.10
4.11
4.12

4.13
4.14

4.15

Variation of output noise spectral density with source resistance and
current noise on a OPA827 2-stage opamp circuit. . . . . . . . . . .
2-stage circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power spectral density versus frequency for each individual contribution to the output noise. Here, Rs is 3.3 kΩ. E2s is the noise power
density from Rs , E2total is the sum of all noise power sources, E2n1 is
the voltage noise power of the first stage opamp, E2i1 is the current
noise power of the first stage opamp acting on |Zs |, E R 21 is the voltage
noise power of R1 , E R 22 is the voltage noise power of R2 , E R 22i is the
current noise power of the first stage opamp current noise acting on
R2 (ignoring any stray capacitances that may contribute), E2Zout is the
noise power density from the output impedance of the OPA827 (20
Ω), E2n2 is the voltage noise power of the second stage opamp, E2i2 is
the current noise power of the second stage opamp acting on |Zout | (the
output impendance of the first stage opamp), E R 24 is the voltage noise
power of R4 , E R 25 is the voltage noise power of R5 , E R 25i is the current
noise power of the opamp current noise acting on R5 (ignoring any
stray capacitances that may contribute), and E2DAQ is the noise from
the DAQ we used in our measurements (National Instruments USB
6366). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Total noise thermometry circuit. . . . . . . . . . . . . . . . . . . . .
Measurement of thermoelectric sample under 639 nm, 43 Hz chopped
illumination with 590 µW of absorbed power. The thermoelectric voltage of the bismuth telluride close to the wires is shown in
turquoise, the thermoelectric voltage of the bismuth telluride far from
the wires is shown in red, the thermoelectric voltage of the platinum
close to the wires is shown in green, and the chopper voltage signal
(smoothed digitally with a low pass filter) is shown in blue. . . . . .
Measurement from DAQ, including gain and circuit noise (plotted
with different axis), with Rs = 3.3kΩ metal film resistor. . . . . . . .

92
93

94
95

96
97

xxi
4.16

4.17
4.18

4.19

4.20
4.21
5.1

5.2
5.3

Second-order bandpass filter for injecting resistance-measuring current while filtering high and low frequency noise produced by the
signal generator (the sine reference output of a SRS 830 lock-in amplifier, in this case). Here, Ra is 1 kΩ, and Rb , Rc , Rd , and Re are
1 MΩ each. Ca , Cb , Cc , and Cd are 100 nF, 1 nF, 0.1 nF, and 1
nF, respectively. The other resistor and capacitor values are given
previously in the text. . . . . . . . . . . . . . . . . . . . . . . . . . 98
NSD spectrum with resistance signal. . . . . . . . . . . . . . . . . . 99
Heat generation in inamps in early circuit after 5 minutes of operation. Hot spots near the bottom and left side of the image were due to
reflections off metal surfaces, and did not correspond to actual temperatures (these "hot spots" disappeared when pictured from different
angles, while the inamp hot spots remained). . . . . . . . . . . . . . 100
NSD as a function of absorbed laser power in thermoelectric device.
This data collection took several hours, and the laser power was swept
from high power to low power. The approximate temperature drift in
the measurement corresponds to about 60 K, which we attribute to an
increase in current noise from the inamp due to heating, as dicussed
in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Separate box for batteries. . . . . . . . . . . . . . . . . . . . . . . . 102
Final circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Sequence for analyzing noise data, as described in Steps 2-4 in the
text. Note that the signal wave is not a square wave, as is depicted here
in a) for simplicity. See Figure 4.14 for measured signal shape. b)
shows the deletion of transition regions from the chopper and signal
traces. c) depicts joining the "light on" and "light off" data into long
data arrays. d) shows a cartoon PSD trace from the "light on" and
"light off" data showing the expected increase in noise for the "light
on" data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Measured gain versus frequency on our 2-stage opamp circuit. Note
the similarity to theory in Figure 4.4. . . . . . . . . . . . . . . . . . 110
A sine wave with piece-wise discontinuous phase. . . . . . . . . . . 111

xxii
5.4

5.5

5.6

5.7

5.8

5.9

5.10

The styrofoam box is holding the noise thermometry circuit. The
small blue box with orange twisted shielded wires is connected to the
surface mount RTDs on the circuit is a DAQ from Omega described
above for logging temperature data. The power supply on the left is
connected to the resistive heater in the box. The instrumetn in the
foreground is the NI-USB 6366 DAQ AC-coupled to BNC cables
connected to the output of the 2nd stage opamp circuit. . . . . . . . . 112
The inside of the cooler. The left side shows the suspended resistive cooler. The circuit is in the shielded metal box labeled "Boxy
McBoxFace". The green wire is a grounding wire for the shielding. . 113
The inside of the shielding box. The syrofoam divider is to further
thermally isolate the sample from opamp heating. The samples,
on the right, are metal film resistors. The cables connected to the
circuit, from left to right, are as follows: shielded SMA cable for -5V
power supply from the voltage regular box; the matching +5 V power
supply; the orange cable is shielded twisted wires taking 4-wire RTD
measurements from the SMT RTD; the next 4 cables are shielded
BNC cables carrying the opamp output to the USB 6366 DAQ; and
the far right cable is another SMT RTD cable. . . . . . . . . . . . . . 114
Test of the circuit with metal film resistors, with values of 6.8 kΩ
(top) and 10 kΩ (bottom). The channel refers to which op amp on the
circuit did the measurement. The legend shows the slope of a best fit
line to the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Test of the circuit with metal film resistors, with values of 3.3 kΩ
(top) and 4.7 kΩ (bottom). The channel refers to which op amp on
the circuit did the measurement. The legend shows the slope of a best
fit line to the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Test of the circuit with metal film resistors, with values of 3.3 kΩ
(top) and 4.7 kΩ (bottom). These measurements were taken about a
week after those shown in Figure 5.8. The channel refers to which op
amp on the circuit did the measurement. The legend shows the slope
of a best fit line to the data. . . . . . . . . . . . . . . . . . . . . . . . 117
Test of the circuit with metal film resistors, with values of 1.5 kΩ
(top) and 2.2 kΩ (bottom). The channel refers to which op amp on
the circuit did the measurement. The legend shows the slope of a best
fit line to the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

xxiii
5.11
5.12

5.13

5.14

5.15
5.16
5.17
5.18

B.1
B.2

B.3

Plot of only channel 2 in Figure 5.8 (bottom), with resistance value
of 4.7 kΩ. The dotted line is the best fit line. . . . . . . . . . . . . . 119
Plot of system temperature, with resistors of values 3.3 kΩ (top) and
4.7 kΩ (bottom). The channel refers to which op amp on the circuit
did the measurement. . . . . . . . . . . . . . . . . . . . . . . . . . 120
Plot of system temperature, with resistors of values 6.8 kΩ (top) and
10 kΩ (bottom). The channel refers to which op amp on the circuit
did the measurement. . . . . . . . . . . . . . . . . . . . . . . . . . 121
Precision of temperature measurements as a function of sampling
time, for a 2 MHz sampling rate. The theoretical line (orange) is
given by Dicke’s equation, and the measurements (blue) are for a
1500 Ω source resistor. δT is the standard deviation of 10 measurements of NSD for the given measurement time, without incorporating
a temperature offset correction. The on-board RTD reported a temperature of 298.2 K for the measurements. . . . . . . . . . . . . . . . 122
Shown is PSD under laser illumination without accounting for sample
resistance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Shown is resistance of sample as a function of absorbed power. . . . 123
Shown is the measured Temperature of the sample as a function of
absorbed power, correcting for sample resistance. . . . . . . . . . . . 124
Shown is the measured Seebeck coefficient (left) for our bismuth
telluride, using the data from Figure 5.17 and the measured thermoelectric voltage (right). . . . . . . . . . . . . . . . . . . . . . . . . . 124
Settings tab for noise measurements. . . . . . . . . . . . . . . . . . . 217
Tab in GUI which shows the NSD for 4 channels. The display on
the right averages the NSD bewteen values set in the settings tab.
The theoretical R can be set in the settings tab, but the theoretical
NSD prediction is only valid for very simple one-inamp systems, but
this could be easily modified. The channel 4 spectrum is showing
noise peaks from the proximity of the surface mount RTD; we had
to remove the connection to this RTD for measurements due to these
peaks. Physically moving the channel 4 signal traces farther from the
SMT RTD would likely fix this problem. . . . . . . . . . . . . . . . 218
Tab in GUI which shows the noise spectral density average for 2 channels, a voltage measurement between them (DC), and temperatures
from the RTDs on the circuit as a function of time. . . . . . . . . . . 219

xxiv
B.4
B.5
B.6
B.7

Data for channel 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Data for channel 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Data for channel 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Data for channel 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

xxv

LIST OF TABLES
Number
Page
2.1 Fitting parameters for Equation 2.7 for Figure 2.25. 95% confidence
intervals are given. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Comparison of experimental dimensions and illumination angle (Pitchs ,
Widths , Θs ) with best-fit simulation dimensions, illumination angle
(Pitch f , Width f , Θ f ) and scaling factor corresponding to Figure 2.29
in the main text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Measured Seebeck coefficient of 6 different samples of thermoelectric
material. To clarify the sign convention, Sb2 Te3 here is p-type and
Bi2 Te3 is n-type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Literature values of Seebeck coefficient and resistivity. . . . . . . . . 59

Chapter 1

INTRODUCTION
Loss of incident photons to heat is often undesirable in light-detection or energy
conversion technologies. For instance, in photovoltaic systems, thermalization of
above-bandgap electrons to the band edge is lost power, and thermalization of
valence band electrons into the conduction band leads to decreased sensitivity via
excess noise in photodiode-based detectors. This energy lost to heat, however, can
be harnessed as a detection mechanism. Using thermoelectric materials, which
convert heat into electrical power, with the help of nanophotonics to capture more
light in a smaller volume and thereby decrease response times, and by measuring the
energy of the electrons directly via noise thermometry, a number of new detectors
and applications can be imagined.
The following chapter is an introduction to all of the necessary concepts in this thesis,
including nanophotonics, thermoelectricity, and noise thermometry. Applications
combining these various topics will appear later in the thesis.
1.1

Nanoscale light-matter interactions

Light interacting with sub-wavelength geometries behaves differently than with
macroscale objects. While the same Maxwell’s equations apply in both cases,
simpler approximations can be used to solve macroscale optics problems, such as
ray tracing. On the other hand, nanoscale structures with dimensions on the order
of the wavelength of incident light, cannot use such simple laws, and Maxwell’s
equations must be directly solved to determine the behavior of light within each
structure. Maxwell’s equations are written as follows:
∂B(r, t)
∇ × E(r, t) = −
∂t
∇ × H(r, t) =

∂D(r, t)
+ j(r, t),
∂t

∇ · D(r, t) = ρ(r, t),

∇ · B(r, t) = 0,

where E is electric field, D is electric displacement, H is magnetic field, B is
magnetic inductance, j is current density, and ρ is charge density. The relationship
between electric field and electric displacement, and between the magnetic field and
magnetic induction is
D(r, t) = ε0 E(r, t) + P(r, t),

H(r, t) =

B(r, t) − M(r, t),
µ0

where P is the polarization, M is the magnetization, ε0 is the permittivity of free
space, and µ0 is the permeability of free space.
These equations are often intractable analytically for a given nanoscale structure
and must be solved numerically. Occasionally, simplified models can be used to
approximate nanophotonic behavior, such as coupled mode theory (Haus and W.
Huang, 1991).
Interactions of light with subwavelength structures
Interesting effects can arise when light interacts with subwavelength particles. For
example, a macroscale piece of gold appears yellow under a white light source, but if
this piece of gold were separated into 30 nm spheres and suspended in a transparent
liquid, the gold would appear red (Njoki et al., 2007). Prior to modern computers,
calculations done with light used some variation of ray-tracing to determine light
trajectories for structures much larger than the wavelength of light.
Only the very simplest "nanophotonic" structures could be designed without a
computer. One example is the Salisbury screen (Salisbury, 1952), first designed to
absorb radar. It consists of a dielectric of thickness equal to 1/4 of the wavelength
of interest (when traveling in the dielectric), sandwiched between a backreflector
and a thin absorbing layer. This design can absorb close to 100% of light of
a specific wavelength. With today’s computing power, Maxwell’s equations can
be solved numerically, opening up a wide range of previously unsolvable (or too
time-consuming to solve) sub-wavelength geometries for study.
The material composing the subwavelength structure as well as the incident wavelength of light can have an enormous impact on the outcome of the light-matter
interaction. For instance, light with energy close to a resonant lattice vibration

energy (phonon) or a resonant electronic oscillation energy (plasmon) will interact
with the material differently than a photon with energy much greater or less than
these features. Materials have intrinsic (bulk) electron oscillation energies, which
can be shifted or modified by geometry; the edges of a nanostructure can introduce
boundary conditions which support different oscillating modes of electrons. Metals,
due to the large populations of freely moving electrons in their conduction band,
are especially affected by structure size in this way. Semiconductors have far fewer
electrons in their conduction bands and have additional interactions with light in
interband transitions between the valence and conduction band. Semiconductorlight interactions close to or above the band edge are often more complicated than
metal-light interactions to intuit or model in simple ways. Insulators (which can
be thought of as semiconductors with very large bandgaps), can have phononic
interactions with lower energy light (similar to the semiconductors and metals) or
electronic interactions with very high energy light. In essence, different materials
in the same nanophotonic geometry can behave in radically different ways when
exposed to light.
Metals can be used in plasmonic applications to focus light in very small mode
volumes, but suffer from high loss (which can be beneficial in some applications
(X. Huang et al., 2008)). Insulators are less adept at concentrating light into small
volumes, but have very low loss and have found uses in metalens applications (Khorasaninejad et al., 2016). Semiconductors (and semimetals) find uses related to their
bandgap. Either used as an insulator below their band edge (i.e. in silicon photonics for low-loss waveguides), or their bandgap is used to segregate and transport
electrons (photovoltaics, thermoelectrics).
Not only does the atomic composition of the material play a role, but the crystal structure is also important. For example, VO2 has an insulating phase below
68°C and metallic phase above, yielding correspondingly different optical properties (which has recently found uses in optical switching applications, e.g. (Briggs,
Pryce, and Harry A. Atwater, 2010)). Silicon photovoltaics perform remarkably
better when in very crystalline form, due to higher carrier lifetimes (i.e. electrons
scatter and recombine at grain boundaries) (Imaizumi et al., 1997).
1.2

Thermoelectrics

The thermoelectric effect was first discovered by Thomas Seebeck around 1821
(Zhang and Zhao, 2015). The uses of thermoelectrics have been limited, how-

ever, due to the difficulty of increasing the efficiency above 7% (Vining, 2009).
Thermoelectrics have found uses in niche applications, however, including many
space-based applications. A radioactive thermoelectric generator is being used to
power the Mars Curiosity rover as well as other deep space probes which cannot
rely solely on solar power due to either dust storms or their distance from the sun.
Additionally, thermoelectrics are much less sensitive to radiation damage (H. Wang
and Leonard, 2017), whereas silicon solar cells have efficiency degradation when
exposed to radiation (Treble, 1962) . In fact, thermoelectric performance can actually be enhanced by crystal defects, due to phonon scattering at crystal boundaries
and corresponding lowering of thermal conductivity (Koumoto and Mori, 2013).
As additional evidence of the robustness of thermoelectrics against radiation, thermoelectric materials are in practice attached adjacent to a decaying radioactive core
in radioactive thermoelectric generators.
The basic operating principle behind thermoelectric emf generation is if one end
of a material is heated, the majority carriers will "diffuse" toward the cold end of
the material (the minority carriers will also "diffuse" toward the cold end, but the
majority carrier "diffusion" produces a larger effect), generating a voltage difference
that is proportional to the temperature difference by the Seebeck coefficient, α. α is
material dependent and temperature dependent (Zhang and Zhao, 2015). See Figure
1.1 for a heavily simplified explanation of thermoelectric operation, including the
operature of a thermocouple.
If instead of heating one end, a current is passed through the thermoelectric element,
the temperature will fall at one end of the device and rise at the other end with polarity
depending on the sign of the Seebeck coefficient. This is called the Peltier effect.
The next section will discuss the thermoelectric effect theory in more detail.
Thermoelectric effect theory
The thermoelectric effect can be derived from the electronic Boltzmann transport
equation, and is the analog of thermal conductivity in the phononic Boltzmann
transport equation. In reality, calculations rarely predict the Seebeck coefficient of
materials accurately. This is in part due to imperfect crystal structure in experiment
and the effect of grain boundaries, among other factors. The Seebeck coefficient
can be highly dependent on deposition method and stoichiometry, (see Table 3.2).
A brief theory of the thermoelectric effect follows, from (W. Wang and Z. M. Wang,
2014).

Figure 1.1: Simplified version of thermoelectric operation for p-type (top left),
n-type (top right), and a thermocouple (bottom).

The electrons in a material obey Fermi-Dirac statistics with a distribution function,
f0 of

f0 =

e((k)−µ)/k b T + 1

(1.1)

where (k) is the electron energy, µ is the chemical potential, or the Fermi potential
at 0 K, k B is the Boltzmann constant, and T is the absolute temperature in Kelvin.
In a material, this distribution function becomes dependant on position, r, wave
number (which depends on the band structure), k, and time, t. In equilibrium, this
function is

f (r,k) =

e((k)−µ(r))/k b T(r) + 1

(1.2)

We can find the time dependence of this distribution function by taking the time
derivative and using the chain rule,

df
dr 1
dp ∂ f
∂f
= ∇r f ·
+ ∇k f +
= v · ∇r f + ∇k f · Fa +
= 0,
dt
dt ~
dt
∂t
∂t

(1.3)

where this equation equals zero because the total number of states is constant in
time. In the above equation, Fa is the set of applied external and internal (e.g.
impurities, phonons, etc...). In the right-most form of the equation above, the last
term is due to internal fields and is the scattering term, and the first two terms are
due to external forces, or are drift terms. This gives, in the time independant case
where ∂ f /∂t = 0,

∂f
∂f
∂f
= 0.
∂t
∂t dri f t
∂t scattering

(1.4)

While the drift term is classical and deterministic, the scattering term is quantum
mechanical and non-deterministic, and is the difficult term to deal with as it requires
scattering theory to solve. One method of dealing with this scattering term is by
using the relaxation time approximation, where it is assumed that the change in the
distribution function with time is proportional to the difference between the current
distribution state and f0 , the equilibrium distribution state, with proportionality
constant 1/τ(k), where the relaxation time τ is how fast the system returns to
equilibrium. This approximation assumes the system is not far from the equilibrium
state. This approximation gives

∂f
∂t

=−
scattering

f − f0
τ(k)

(1.5)

Using the time-independant form of the Boltzmann Equation where the relaxation
time approximation is used and assuming spatial homogeneity and Fa = eE, the
Boltzmann equation can be analytically solved as

f (k) ≈ f0 (k) − τ(k)E · ∇k f .

(1.6)

If we are interested in linear E-field phenomena, then we can replace f on the right
hand side with f0 , then taylor expand about k, giving

f (k) ≈ f0 (k) − τ(k)E · ∇k f0 ≈ f0 k − τ(k)E .

(1.7)

In a similar way, taking away the spatial homogeneity condition, the Boltzmann
equation can be linearized as

f (r,k) − f0 (r,k)
f (1) (r,k)
≡−
v · ∇r f0 + E · k f0 = −
τ(k)
τ(k)

(1.8)

where f (1) is a first order term in a Taylor expansion. Because we know the form
of f0 , the gradients in the above expression can be simplified and rewritten (see (W.
Wang and Z. M. Wang, 2014)), so we find

f (1) (r,k) = −

∂ f0
τ(k)v · [∇(eφ + µ) + ( − µ)∇lnT],
∂

(1.9)

where E = −∇φ, where φ is the electrostatic potential. Next, we look a the current
density, j,

j=
4pi 3

v f (r,k)d 3 k.

(1.10)

Substituting in f (1) , and assuming an isotropic medium (no r dependence), and
introducing the density of states D(),

µ
− eL21 ∇lnT,
j = eL11 eE − T ∇

(1.11)

∂ f0 j i
τ D()d
∂

(1.12)

where

Li j =
3m∗
and the energy flux is given by

∫

jQ =
4pi 3

µ
(k)v f (r,k)d 3 k = eL2 1 eE − T ∇
− L31 ∇lnT .

(1.13)

From L, we can extract the electrical conctivity, σ = e2 L11 . If j = 0 (no currents)
and there is no concentration gradient (∇µ = 0), then from Equation 1.11,

L21 − µL11
∇T,
eL11T

(1.14)

L21 − µL11
eL11T

(1.15)

i.e. the Seebeck coefficient, α, is

α=

If we put the electrical current j in Equation 1.13, we find

jQ = Πj − κe ∇T,

(1.16)

where the Peltier coefficient is (again with no gradient in chemical potential),

Π=

L21
= αT,
eL11

(1.17)

and the electron thermal conductivity is

2 − eL L
L21
31 11
κe =
eL11

(1.18)

For a metal or degenerate semiconductor, we can find using additional terms in the
Fermi integral, that

α=

π 2 k B2 T
∂D()
3eF
∂ =F

(1.19)

The Seebeck coefficient for metals is typically a few µV/K. For a semiconductor
with assumed τ() = τ0 λ ,

Nc
α = kBe
+ λ − ln

(1.20)

where n, Nc are carrier concentration terms. This equation gives general guidelines
for semiconductor Seebeck coefficients, but can be drastically different from experimental data, especially with multiple bandgaps near the Fermi level (W. Wang and
Z. M. Wang, 2014; Koumoto and Mori, 2013). Seebeck coefficients of semiconductors can be as high as 100’s of µV/K.
The thermoelectric effect can also be derived with thermodynamic arguments with
the first and second Kelvin relations (see http://www.eng.fsu.edu/ dommelen/quantum/style_a/nt_pelt.html for example).
In the limiting case where no current is flowing, only the Seebeck coefficient is
needed to determine voltage generation. When the system is isothermal (with good
heat sinking, for instance) and only current is flowing, the Peltier coefficient is the
relevant quantity. When current is flowing and the system is not isothermal, both j
and jQ are coupled and need to be solved together.
The Seebeck coefficient is just one contribution to thermoelectric efficiency. The
figure of merit for thermoelectric efficiency is

zT =

α2 σT

(1.21)

where "good" thermoelectric materials have a zT of around 1 or higher, corresponding to about 4-5% efficiency or higher (Zhang and Zhao, 2015).
1.3

Noise thermometry

The presence of the random motion of electrons in circuits was first suggested by
Albert Einstein in his 1906 paper on Brownian Motion (Einstein, 1905). Johnson
was the first to measure the presence of this noise (John B Johnson, 1927; John
Bertrand Johnson, 1928), and a year later Nyquist developed the theory to explain
Johnson’s findings (Nyquist, 1928). Nyquist’s theorem was later generalized into
the fluctuation dissipation theorem (Rep Kubo, 1966), which governs a wide range
of processes.

10
The first noise thermometer, unknowingly, was created by Dicke (Dicke, 1946) in
1946 to measure thermal radiation at microwave frequencies. A general overview
of noise can be found in (White et al., 1996). More recently, noise thermometry was
used in observing the quanta of thermal conductance (Keith Schwab et al., 2000),
and is being used by NIST to redefine the Boltzmann constant (Qu et al., 2013).
Noise is often measured in noise spectral density (NSD), in units of V/ Hz or
A/ Hz. Another important measure is the power spectral density (PSD), which is
the square of the NSD. There are a number of different noise sources, including
Johnson noise from resistive elements, in which N SD = 4Rk BT, where R is
resistance, k B is the Boltzmann constant, and T is temperature in Kelvin. Shot noise
is given by N SD = 2qI, where q is the electron charge, and I is the current. Both
Johnson and shot noise appear as a flat background at all frequencies. 1/f noise is
another source of noise in semiconductor devices, which occurs at lower frequencies
with a 1/f β shape, where β is component dependent. Other noise sources, such as
avalanche noise, exist, but in noise thermometry, Johnson noise is the noise source
of interest and what we will focus on in the next section.
Theory of Johnson-Nyquist noise
The derivation in this section is summarized from (Buckingham, 1983), and is an
alternative derivation of Johnson noise than that foundin Nyquist’s original paper.
Noise in electronic device is stochastic, and assumed to be small fluctuations about
the mean, so "small signal" theory can be used. In other words, if an electron
randomly moves away from the equilibrium state, on average, its next move will
be back toward equilibrium. Noise is also statistically stationary, meaning that the
statistical properties are independent of epoch where measured. Shot noise and
Johnson noise (which will be derived), follow a normal distribution about the mean.
These noise sources can also be approximated well by a chain of uncorrelated delta
functions, called an impulse process.
The Fourier transform is used in measuring the power spectral density of noise. In
order for a function to be Fourier transformable, its absolute value integrated over
all of time must be finite. For a noise signal, this is not the case, as the noise lasts
for all time. But while the total energy is infinite, the power, or energy per time
is finite, so Fourier analysis can be conducted as long as a section of the signal is
analyzed, i.e. the signal, f (t), is 0 at all times except for − T2 < t < T2 , where T is
finite. If we look at the inverse Fourier transform of f (t),

f (t) =
2π

∫ ∞

F(ω)eiωt dω,

(1.22)

−∞

we see that the kernel, eiωt is a harmonic wave, so F(iω) must be an amplitude
spectrum. By Plancherel’s theorem (assuming f (t) is real over the interval [− T2 , T2 ]
and zero everywhere else),

∫ ∞

[ f (t)] dt =
2π
−∞

∫ ∞

|F(ω)| 2 dω.

(1.23)

Both sides of this equation equal the total energy in f (t), and therefore |F(ω)| 2 is
the energy density (with units energy/Hz), and dividing by T will give the average
power density (with units power/Hz). Thus, the average power spectral density is
given by

2|F(ω)| 2
T→∞

Sx (ω) = lim

(1.24)

As an aside, which can be read about further in (Buckingham, 1983), the PSD
and autocorrelation function of a process are related by the Wiener-Khintchine
theorem. The electronic and thermal autocorrelation functions together with their
cross-correlation function can be used to obtain the kinetic coefficients of electronic
thermal conductivity, electrical resistivity, and the Seebeck coefficient through
Green-Kubo theory (Ryogo Kubo, Yokota, and Nakajima, 1957; Garrity, 2011).
This appears to be an experimentally understudied method of measuring Seebeck
coefficient via equilibrium measurements, which should be further explored.
Continuing with the summarized derivation from (Buckingham, 1983), thermal and
shot noise can be modeled as random pulse trains,

f (t) =

an g(t − tn ),

(1.25)

n=1

where g(t) is the pulse shape function, an is the amplitude of the nth pulse, tn is the
time of the nth pulse, and N is the number of pulses in the time interval T. tn is
Poisson distributed since the events are independent, so the expected value is

f (t) = va

∫ ∞
g(t)dt,

(1.26)

−∞

where v is the mean number of pulses per second and a is the mean amplitude of
the pulses. The Fourier transform of f (t) is

F(ω) = G(ω)

an e−iωtn .

(1.27)

n=1

From Equation 1.24, we can find the PSD of f (t) as

2|G(ω)| 2 Õ Õ
Sx (ω) = lim
an am e−iω(tn −tm ) .
T→∞
n=1 m=1

(1.28)

Only the n = m terms survive assuming the an are distributed symmetrically about
zero, yielding (with v = N/T),

Sx (ω) = 2va2 |G(ω)| 2,

(1.29)

which is Carson’s theorem. We now apply this to a resistor with cross-sectional
area A and length L. An electron travel length between collisions in the resistor is
l f . It is assumed that when an electron is "pulsed" out of equilibrium, it will relax
to equilibrium. When the pulse is applied, it can be imaged that an instantaneous
capacitor made of two charged plates with charge density ±q/A separate by length
l f form, then come back together. The equivalent circuit is shown in Figure 1.2,

Figure 1.2: Equivalent circuit of a single "pulse" of an electron in a resistor.

13
where R f and C f are the resistance and capacitance between the two sheets, R is
the resistance of the total resistor, and qδ(t) is the "pulse". The voltage, vn (t) seen
at the nodes can be found with the RC circuit equation

where C =

dvn (t)
vn (t)
=−
+ qδ(t),
dt

(1.30)

Cf lv
L . Fourier transforming both sides yields

Vn (ω) =

ql f (R/L)
(1 + iωτ)

(1.31)

where τ = RC, which is very small in this case. From Carson’s theorem (Equation
1.29), then

Sv (ω) =

2vq2 (R/L)2 l 2f
(1 + ω2 τ 2 )

4k BT R
(1 + ω2 τ 2 )

(1.32)

where k B is the Boltzmann constant and T is temperature, and the right hand term
comes after making substitutions for v, R, and µ described in (Buckingham, 1983),
where this entire derivation is summarized from. In resistors, τ is negligible, so we
find the familiar equation for PSD,

Sv (ω) = PSD = 4Rk BT .

(1.33)

The noise spectral density (NSD) with units V/ Hz is then

Sv (ω) = N SD =

4Rk BT .

(1.34)

This description is valid for metallic-like materials for kT >> ~ω, where ~ is
Planck’s constant divided by 2π and ω is the signal frequency. Nyquist (Nyquist,
1928) noted that when ~ω becomes comparable or larger than kT, kT should be
replaced and the zero-point energy term should be added, yielding

~ω
Sv (ω) = 4R ~ω + ~ω
e kT − 1

(1.35)

In this thesis, we are in the classical regime and will not need to use Equation 1.35.

14
Precision of noise thermometry
The temperature resolution we can achieve is dictated by the Dicke radiometer
relation(Dicke, 1946), which is given by
T + Tcirq
∆T = √
∆ft

(1.36)

where ∆T is the temperature resolution, T is the temperature of the resistor in Kelvin,
Tcirq is the temperature of the circuit, ∆ f is the measurement bandwidth in Hz, and
t is the measurement time, in seconds.
1.4

Scope of thesis

This thesis explores performance and properties of nanophotonic structures composed of thermoelectric materials, both theoretically and experimentally. The chapters are organized as follows:
• Chapter II – Design considerations and simulations of nanophotonic thermoelectric devices
This section will give an overview of what should be considered when designing thermoelectric nanophotonic structures in order to maximizing performance. We will discuss how the simulations must be developed, as well
as the impact of materials choices on performance, and the interplay between
responsivity and thermal time constant. Simulations of various nanophotonic
thermoelectric devices for sensing applications will be presented.
• Chapter III – Hyperspectral detector application
One of the devices described in Chapter II will be explored experimentally in
this section for its potential as a hyperspectral detector.
• Chapter IV – Nanoscale temperature measurements with noise thermometry
In this chapter, the theoretical considerations will be introduced for performing noise measurements on thermoelectric nanophotonic devices, including
circuit design and component selection.
• Chapter V – Noise thermometry results
The data analysis technique, as well as noise thermometer circuit tests and
preliminary results will be introduced in this section.

15
• Chapter VI – Conclusion
This thesis will be summarized and an outlook will be given for thermoelectric nanophotonic device performance for both spectral and thermometry
applications.
• Appendices
The appendices provide code for data analysis using the FFTW package in C,
as well as description of the Python generated GUIs for instrument control in
data collection.

16
Chapter 2

DESIGN CONSIDERATIONS AND SIMULATIONS OF
NANOPHOTONIC DEVICES
2.1

Design considerations in nanophotonic thermoelectric devices

Plasmon excitation enables extreme light confinement at the nanoscale, localizing
energy in subwavelength volumes, and thus can enable increased absorption in
photovoltaic or photoconductive detectors (Harry A Atwater and Polman, 2010).
Nonetheless, plasmon decay also results in energy transfer to the lattice as heat
which is detrimental to photovoltaic detector performance (Skoplaki and Palyvos,
2009). However, heat generation in resonant subwavelength nanostructures also represents an energy source for voltage generation, as we demonstrate here via design of
resonant thermoelectric plasmonic absorbers for optical detection. Though thermoelectrics have been used to observe resonantly coupled surface plasmon polaritons
in noble-metal thin films and microelectrodes (Innes and Sambles, 1985; Weeber
et al., 2011) and have been explored theoretically for generation of ultrafast intense
magnetic pulses in a dual-metal split ring resonator (Tsiatmas et al., 2013), they
have not been employed as resonant absorbers in functional thermoelectric nanophotonic structures. Previously, non-narrowband photodetection has been demonstrated
through the photothermoelectric effect in gated graphene structures(Cai et al., 2014;
Xu et al., 2009) and the laser heating of nanoantennas and micropatterned materials
(Russer et al., 2015; Szakmany et al., 2015; A. S. Gawarikar, R. P. Shea, and J. J.
Talghader, 2013b; F Völklein and A Wiegand, 1990; Hsu et al., 2015), all shown
to be promising for infrared to THz broadband detection. Typical responsivities
of the graphene structures are around 10 V/W for infrared and THz detectors, relative to incident (not absorbed) power, with a time response ranging from 23 ms
to nearly 10 ps. Responsivities of non-graphene detectors range from 10’s of V/W
to nearly 7,000 V/W (A. S. Gawarikar, R. P. Shea, and J. J. Talghader, 2013b) for
thermopiles made of many thermocouples of up to mm sizes. The response time of
these structures range from 10’s to 100’s of ms, though ns response times have been
predicted8 for nanoantenna structures. High-figure-of-merit thermoelectrics have
been investigated as solar power generators, but the light absorption process was
entirely separate from the thermoelectric functionality and relied on black carbon
absorbers (Kraemer et al., 2011) or solar concentrators (Amatya and Ram, 2010).

17
2.2

Scale of nanophotonic vs. thermal simulations

In our applications, nanophotonic electromagnetic fields extend over the sub-nanometer
to hundreds of nanometers scale, whereas temperature gradients occur on the 10’s of
microns scale. In simulations, this creates difficulties with scales and mesh sizes, as
the mesh sizes needed for accurate nanophotonic simulations are impractical to use
in thermal simulations due to memory constraints. So while coupled physics could
be used in COMSOL to solve for both the thermal properties and optical properties simultaneously, this was computationally prohibitive in our applications, as the
membranes in some simulations, in order to have proper thermal boundary conditions, were over 1 mm2 . Additionally, nanophotonic light absorption occurred on a
much smaller time scale than the thermal processes, so the nanophotonic absorption
could be treated as a steady-state power input into the thermal simulations. We assumed that the optical and other properties of materials did not change significantly
over the small temperature ranges in our simulations, although if larger temperatures
are generated or materials with significantly temperature-dependent optical properties are used, it would be necessary to incorporate these temperature-dependent
changes into the model.
2.3

Thermal time constant of devices

The thermal time constant of nanophotonic devices depends primarily on the size
of the structure being heated as well as the thermal conductivity of the material and
its surroundings. The thermal time constant, τ, for convective cooling or heating of
a simple geometry can be taken as

τ=

ρcpV
hAs

(2.1)

where ρ is density, cp is heat capacity, V is volume, h is the heat transfer coefficient
between the object and its surroundings, and As is the surface area. If examining
conductive instead of convective heating or cooling, the thermal conductivity would
replace h to first approximation in the denominator of the time constant equation,
as it plays a similar role. We can see that the larger structures will have a larger
time constant, while higher h (or thermal conductivity) will decrease the time
constant. Therefore, a small, high thermal conductivity structure will have fast
response time. On the other hand, high thermal conductivity will decrease the
maximum temperature achievable by a nanophotonic thermoelectric device under
illumination, which will decrease responsivity (sensitivity). Therefore, there is a

18
tradeoff between responsivity and response time which must be tailored for the
specific application.
2.4

Simulation material values

Starting with nanophotonic (EM) and thermal simulations, we can determine the
performance characteristics of thermoelectric nanophotonic devices. In the sections below, several different nanophotonic designs coupled to different thermal
management designs will be explored.
In this chapter, the following values are used in simulations. Heat capacity of
Bi2 Te3 (Gorbachuk and Sidorko, 2004) was taken as 158 J/kg·K, density (Nolas,
Sharp, and Goldsmid, 2001) as 7,859 kg/m3 , surface emissivity as 0.34, and thermal
conductivity (D. M. Rowe and Bhandari, 1995) as 2.05 W/m·K. SiN x heat capacity
(Greenwood and Earnshaw, 1984) was taken as 800 J/kg·K, density as 3,185 kg/m3 ,
surface emissivity as 0.9, and thermal conductivity (Ftouni et al., 2015) as 2.5
W/m·K. SiO2 heat capacity (Greenwood and Earnshaw, 1984) was taken as 703
J/kg·K, density as 2,196 kg/m3 , surface emissivity as 0.9, and thermal conductivity
(Yamane et al., 2002) as 1.38 W/m·K. Au heat capacity was taken as 129 J/kg·K,
density as 19300 kg/m3 , surface emissivity as 0.025, and thermal conductivity as
317 W/m·K.
2.5

Perfect absorber

Nanophotonic design
To increase the sensitivity of a detector, it is desirable to absorb close to 100%
of the incident light. While a carbon black absorber can do this, they are often
large (10s or 100s of microns thick) and thus, slow due to their large heat capacity.
Nanophotonic structures can be used to make close to 100% absorbing structures
within a wavelength range, with a much smaller footprint than carbon black. The
perfect absorber design we explored was modeled off a design from the literature
(Liu et al., 2010a), and is shown artistically in Figure 2.1. An additional benefit
of these structures is the insensitivity to incident angle of illumination, as will be
shown. The cylinder height was 20 nm, the radius 176 nm, and the pitch 600 nm.
The dielectric gap was 30 nm thick SiO2 with index from Palik (Palik, 1997a).
The backreflector was 50 nm gold with index data from Johnson and Christy (P. B.
Johnson and Christy, 1972), beneath which was 100 nm of bismuth telluride (see
Figure 2.2 for measured dielectric function). The magnitude of the electric field
at the resonant wavelength (at peak absorption) of 1,648 nm is shown in Figure

19
2.6 (left). As a function of wavelength, the absorption of the structure is shown in
Figure 2.6 (right).

Figure 2.1: Artistic rendition of perfect absorber structure, rendered in POV-ray.

Figure 2.2: Dielectric function of bismuth telluride in visible and near-IR wavelengths measured using J.A. Woollam Co. VASE and IR-VASE MARK II ellipsometers and analyzed with WVASE software..

Thermal design
In the thermal simulations, a patch of the perfect absorber materials was placed in
the center of the 100 nm thick thermoelectric junction on top of a 10 µm wide, 100
nm thick SiN x membrane suspended between two large blocks of SiO2 , shown in
Supplementary Fig. 12a and Supplementary Fig. 10j. Sandwiched between the

Figure 2.3: Dielectric function of bismuth telluride in IR wavelengths.

Figure 2.4: Dielectric function of antimony telluride in visible to near-IR wavelengths.

large SiO2 blocks and the “cold” end of the SiN x membrane is a 200 nm thick
layer of Au. Because Au has a large thermal conductivity, this layer acts as a heat
sink for the “cold” end of the thermoelectric materials, enhancing the temperature
gradient across the thermoelectric materials. The simulations neglected convection,
i.e. were simulated a vacuum environment.
Using these thermal simulations, we can plot the difference in temperature between

Figure 2.5: Dielectric function of antimony telluride in IR wavelengths.

Figure 2.6: Lumerical simulation of normalized electric field magnitude of the
perfect absorber structure described in the text, on resonance at 1,648 nm (left), and
as a function of wavelength (right). Scale bar is 300 nm. Credit to Dr. Seyoon Kim
for nicely plotting the simulated data on the left.

the hot and cold edges of the bismuth telluride material as a function of power
absorbed by the perfect absorber structure, as shown in Figure 2.8. The simulated
responsivity as a function of wavelength is shown in Figure 2.9, and Figure 2.10
shows noise equivalent power as a function of wavelength for this structure. Noise
equivalent power is caluculated by dividing the simulated responsivity by the theoretical Johnson noise of the structure at room temperature, using the geometry and
measured resistivity of our materials to compute resistance.

Figure 2.7: Perfect absorber and split ring resonator thermal simulation design.
(Top left) Diagram of thermal simulation design for a perfect absorber and a split
ring resonator thermal simulation corresponding to data in Figures 2.9, 2.15 and
Figures 2.8, 2.14, 2.10, 2.16, 2.13. The split ring resonator design has a 50 nm
thick SiNx membrane with 60 nm of thermoelectric material above it, which serves
as a backreflector in the optical design. The perfect absorber has a 100 nm thick
SiNx membrane and 100 nm thick TE materials. It uses a 50 nm thick layer of
gold as the backreflector in the 10 µm by 10 µm center absorber patch. The 200
nm gold above the silica increases thermal conduction of heat from the cold end of
the device (i.e. acts as a heat sink), and any other thermally conductive material
would serve this purpose well. Thermal simulations involved a volumetric heat
influx into the centrally located split ring resonator or cylinder absorbers in the
array (this assumes absorption near the edges of the arrays would be worse). (Top
right) Thermal simulation of perfect absorber with a total absorbed power of 23
µW. The split ring resonator thermal profile was similar (see Figure 2.13). (Bottom
left) Thermal simulations of the perfect absorber design. The perfect absorber array
covers a 10 µm by 10 µm square in the center of a suspended, TE junction on top
of a SiNx membrane. The “cold” ends of the TE materials sit on a 100 nm SiNx
/200 nm Au/20 µm SiO2 substrate. The Au layer acts as a high thermal conductivity
heat sink to conduct heat away from the “cold” ends of the TE materials. Both
simulations were performed in vacuum. The scale bars are 10 µm, and the absorbed
power is 23 µW.
2.6

Split ring resonator perfect absorber

Nanophotonic design
The split ring resonator can also be a perfect absorber, angle-insensitive design if
the backreflector is a low-loss metal in the wavelength range of interest (such as

Figure 2.8: Temperature differences between the hot and cold sides of the thermoelectric materials as a function of power absorbed for the perfect absorber structure.

Figure 2.9: Simulated responsivities as a function of incident angle of the perfect
absorber structure.
gold in the near-IR). The split ring resonator perfect absorber (shown artistically in
Figure 2.11) consists of two gold rings, the outer ring with outer radius 70 nm and
inner radius 60 nm, with a split 15 nm wide. The inner ring had outer radius of
40 nm and inner radius of 30 nm, with split 20 nm wide. Both rings were 20 nm
tall and had optical constants from Johnson and Christy (P. B. Johnson and Christy,

Figure 2.10: Simulated noise equivalent power as a function of incident angle of the
perfect absorber structure.
1972). From center to center, the distance between each split ring resonator in the
array is 240 nm. The dielectric layer was 40 nm of Palik (Palik, 1997a) SiO2 . The
backreflector layer was made of antimony telluride. The thermal simulations were
performed in a similar manner as the perfect absorber structures described above,
but with a 60 nm thick thermoelectric layer and 50 nm thick SiN x membrane below.

Figure 2.11: Artistic rendition of split ring resonator design, rendered in POV-ray.
The magnitude of the electric field at one resonance can be found in Figure 2.12

25
(left), and the absorption as a function of wavelength and incident angle can be
found in Figure 2.12 (right).

Figure 2.12: Lumerical simulation of normalized electric field magnitude of the
split ring resonator structure described in the text, on resonance at 1,710 nm (left),
and as a function of wavelength (right). Scale bar is 50 nm. Credit to Dr. Seyoon
Kim for nicely plotting the simulated data on the left.

Thermal design
The thermal design for this structure is similar to the perfect absorber structure above,
but with a thinner membrane, which leads to a greater temperature difference, shown
in Figure 2.13. We can plot the temperature difference between the hot and cold
ends of the thermoelectric elements in Figure 2.14. The simulated responsivity is
shown in Figure 2.15, and the simulated noise equivalent power is shown in Figure
2.16.
2.7

Plasmonic bowtie antenna

Nanophotonic design
Plasmonic structures can focus certain light wavelengths into mode volumes much
smaller than their free-space wavelength. One example is the bowtie structure
artistically rendered in Figure 2.17. The bowtie structure simulated had an internal
angle of 90 degrees and was cut from a circle of radius 500 nm and thickness 80
nm. The gap between the “wings” of the bowtie was 30 nm. The wire was 20 nm
wide and 20 nm tall and made of one material, bismuth telluride, due to memory
and symmetry constraints. The bowtie was gold with index of refraction from Palik

Figure 2.13: Thermal simulation for split ring resonator perfect absorbers utilizing
a thin membrane to lower thermal heat loss to the substrate. Total absorbed power
is 23 µW. Large scale bar is 10 µm.

Figure 2.14: Temperature differences between the hot and cold sides of the thermoelectric materials as a function of power absorbed for the split ring resonator
absorber structure.
(Palik, 1997a). The substrate was an infinitely thick slab of SiO2 with index of
1.455.
The magnitude of the electric field at one resonance can be found in Figure 2.18
(left), and the absorption cross section as a function of wavelength and incident
angle can be found in Figure 2.18 (right).

Figure 2.15: Simulated responsivities as a function of incident angle of the split
ring resonator absorber structure.

Figure 2.16: Simulated noise equivalent power as a function of incident angle of the
split ring resonator absorber structure.

Thermal design
The thermal design for this structure consisted of placing a 20 by 20 nm thermoelectric wire junction at the plasmonic focusing point of the bowtie antenna. A

Figure 2.17: Artistic rendition of the plasmonic bowtie absorber, rendered in POVray.

Figure 2.18: Lumerical simulation of normalized electric field magnitude of the
bowtie absorber structure described in the text, on resonance at 660 nm (left), and
absorption cross-section as a function of wavelength (right) for the wire alone (blue)
and the wire with the bowtie (orange). Scale bar is 200 nm. Credit to Dr. Seyoon
Kim for these simulation results and for nicely plotting the simulated data on the
left.
COMSOL simulation of the bowtie is shown with several different temperature
scales in Figure 2.19.
The temperature difference between the hot spot at the center of the bowtie antenna
and the edge of the thermoelectric wires is shown in Figure 2.20. Figure 2.21 shows
the responsivity of this structure, and 2.22 shows the noise equivalent power.

Figure 2.19: (Top left) Thermal simulations of teh resonant bowtie antenna, performed in vacuum. Scale bar is 500 nm and the absorbed power is 7.9 µW. (Top right)
Thermal simulation of bowtie structure with absorbed power of 7.9 µW. Maximum
temperature is 330 K, but a smaller scale was used to show temperature differences
better. Scale bar is 20 µm and inset scale bar is 500 nm. (Bottom left) The same
thermal simulation as in the top right is shown, but with the full temperature scale
bar. The outline of the bowtie is given in blue dotted line. Scale bar is 500 nm.

2.8

Guided mode resonator thermocouple

The guided mode resonance thermocouple explored in this section was fabricated
and experimentally tested in the next chapter. Here we present an in-depth analysis
of the nanophotonic structure.
Nanophotonic design
This section is broken up into two parts. The first is concerned solely with why the
given guided mode resonance structure absorbs the wavelength of light it does, and
the second part looks at how this guided mode resonance design can be incorported
with thermoelectrics, to generate a necessary temperature gradient. An artistic
rendition of the thermocouple structure is shown in Figure 2.23. Figure 2.23 shows

Figure 2.20: Temperature difference between hot and cold spot of thermoelectric
wires in the resonant bowtie antenna design.

Figure 2.21: Responsivity of the resonant bowtie antenna design.

a schematic of our experimental structure, a guided mode resonance wire array,
with wire dimensions of 40 nm × 100 nm × 50 µm, in which TM polarized, normal
incident, unfocused optical radiation is coupled into a waveguide mode via a periodic
thermoelectric wire array that serves as a light absorber.

Figure 2.22: Noise equivalent power of the resonant bowtie antenna design.

Figure 2.23: Conceptual design of GMR resonant thermoelectric structure. The
junction of the thermoelectric wires absorbs incident, spatially uniform illumination
and is absorbed, heating the junction and producing a thermoelectric voltage. Light
illuminating the pads will be reflected, creating the necessary temperature gradient.

Guided mode resonance with Fano analysis
Nanophotonic thermoelectric structures must concentrate the electric field in the
thermoelectric material to maximize absorption. Our guided mode resonance structures achieve this via Fano interference (Fano, 1961) of a waveguide mode and a
Fabry-Perot resonance in the waveguide, which will be described in detail. The
resonant wavelength of this waveguide mode is predicted quite well by the grating
coupler equation for normally incident light, assuming infinitely narrow gratings,

32
2π/d = β, where d is the grating pitch and β is the propagation constant of the
two-layer slab waveguide. Small deviations from the grating coupler equation predictions are due to waveguide mode interactions with Fabry-Perot resonances.
Fano lineshapes are produced when a continuum of states interacts with discrete
or narrow modes near the same energy, and appear in electronic circuits, nanophotonics, and atomic spectra (Gallinet, 2012). In our specific case, we have a broad,
Fabry-Perot resonance in our waveguide layers acting as the continuum background
(radiative bright mode), and a narrow, waveguide mode (nonradiative dark mode)
interacting with it. The effect is developed thoroughly in work by Gallinet et al.
(Gallinet and O. J. F. Martin, 2011a; Gallinet, 2012; Gallinet and O. J. F. Martin,
2011b), which will be summarized here.

Figure 2.24: Diagram of interaction of bright (broad resonance) and dark (narrow resonance) modes in the production of Fano lineshapes. The bright mode is
the Fabry-Perot resonance of light with k-vector parallel to incident illumination.
Analysis from (Gallinet and O. J. F. Martin, 2011a).

Figure 2.25: a, The Fano formula (Equation 2.7) is fit to our simulation data for a
region near the resonance. textbfb, The extracted bright mode from a is shown in
more detail, and is compared with the normalized magnitude of the electric field
at the top of a 50 nm of SiO2 /100 nm of SiN x waveguide without wires due to
Fabry-Perot resonances. c, Bright mode and Fano formula fits to simulated and
experimental data. d The simulation and experimental bright modes from c along
with the Fabry-Perot resonance for a waveguide-only structure.

From the interaction between the bright (continuum) and dark (waveguide) modes,
as Figure 2.24a outlines, we get a new resonance of the combined system, at a
position equal to

ωa = ωd2 + ωd ∆,

(2.2)

where ωd is the resonant frequency of the dark mode and ∆ is the shift away from
this frequency due to coupling with the bright mode calculated explicitly in Gallinet

34
(Gallinet, 2012). The shape of the new resonance is asymmetric about the new
resonance, ωa2 , and is given by

ω −ω2a
+q +b

2
σa (ω) =

ω −ω2a

(2.3)

where q is an asymmetry term, Γ is a width term equal to 2ωaWa (Wa is a width) for
ωa >> Wa , and b is a modulated damping term (Gallinet, 2012). The asymmetry
of this dark mode comes from a rapid phase asymmetry of π across the resonance
from the original dark mode, interfering with the symmetric phase difference in
the bright mode across the resonance. On one side of the resonance, these bright
and dark modes destructively interfere, and on the other side of the resonance they
constructively interfere. The location of destructive interference on either side of
the resonance depends on the sign of the phase difference between the dark and
bright mode resonances, along with whether the loss is real or imaginary. This is
expressed in the asymmetry term in Equation 2.3, q, where

q=±

ωd2 − ωb2

,
2Γb 1 + ΓΓci

(2.4)

where Γb is equal to 2ωbWb (Wb is a width) for ωb >> Wb , Γi is intrinsic loss, and
Γc is coupling loss. b, the modulation damping parameter, is equal to

2
b=

Γi
Γc

Γi
2 = Γ + Γ
Γi

(2.5)

1 + Γc

The bright mode Fano formula, on the other hand, is pseudo-Lorentzian and therefore
symmetric, given by

σb (ω) =

a2
ω2 −ω2 2

Γb

(2.6)

35
From Gallinet (Gallinet, 2012), we find that the total optical response of the system
comes from multiplying the Fano formulas of the bright mode with the dark mode
modified by the bright mode, or

σtot (ω) = σb (ω)σa (ω).

(2.7)

Fitting our resonance to this formula, the frequency of the bright mode resonance
can be extracted to determine its source, whether it be from plasma resonances of
the wires, incident radiation, or Fabry-Perot resonances in the waveguide. Figure
2.25a shows a full wave simulation of Sb2 Te3 wires that are 60 nm wide, 40 nm
thick, with a pitch of 431 nm on a 50 nm SiO2 /100 nm SiN x suspended waveguide,
that is fit to the Fano formula (Equation 2.7). Shown are the simulation (dotted
orange), the fit of the combined Fano formula (black, Equation 2.7) between 440
nm and 650 nm, and the extracted Fano formula of the bright mode (blue, Equation
2.6). All fitting parameters for Figures 2.25a-d are located in Table 2.1.
The extracted bright mode from Figure 2.25a is shown in more detail in Figure
2.25b, and is compared with the normalized magnitude of the electric field at the
top of a 50 nm of SiO2 /100 nm of SiN x waveguide without wires due to FabryPerot resonances. The Fabry-Perot resonance (orange) is calculated via full-wave
simulation as the normalized magnitude of the electric field at a point on the surface
of a bare waveguide structure. The bright mode fit (blue) is described in Figure
2.25a. The shift in the Fabry-Perot peak is associated with the contributions of
the wires to the effective index of the entire photonic crystal structure. The minor
misalignment of the bright mode and Fabry-Perot peak is likely caused by the grating
itself altering the location of the bright mode, as the effective index of the photonic
crystal made up of the grating plus the waveguide will be different than the index of
the waveguide alone.
Using this method to compare our experimental and simulated data, we can compare
differences quantitatively. Figure 2.25c shows such a comparison: the experiment
versus simulation for 45 nm SiO2 /100 nm SiN x suspended waveguide with Sb2 Te3
wires. Wire thickness is 40 nm, width is 89 nm, and pitch is 511 nm. Shown are the
full wave simulation for this structure (solid blue), the total Fano function fit (dotted
magenta) to the full wave simulation with parameters given in Table 2.1, and the
bright mode (dotted blue) extracted from the total Fano fit function, corresponding to
the Fabry-Perot resonance. Also shown are the measured absorption (solid orange),

36
the fitted total Fano function to the measured absorption (dotted yellow), and the
extracted bright mode profile from the measured extinction (dashed purple). Figure
2.25d plots the simulation and experimental bright modes from Figure 2.25c along
with the Fabry-Perot resonance for a waveguide-only structure. Extracted bright
mode for the experimental (dotted orange) and simulated (dotted blue) data from
2.25c, with the simulated electric field magnitude (dotted yellow) at the surface of
the waveguide (without the wires) due to the Fabry-Perot resonance. All curves
are normalized to their maximum value in the given wavelength range for ease of
comparing resonance peaks. The experimental, simulated, and Fabry-Perot peaks
align reasonably well.
If we compare the various values of the fitting parameters from Table 2.1, we
note that b, the modulated damping term, is higher in the experiment than in the
simulation. From Equation 2.5, we can see that this indicates the intrinsic loss has a
larger influence (or the coupling loss has a lesser influence) in the experiment than
in the simulation. This could be attributed to a, the bright mode amplitude, having
a higher magnitude in the simulation than in the experiment.
Figure
Figures
2.25a,b
Figures
2.25c,d simulation
Figures
2.25c,d experiment

a (unitless)
0.13038±
0.000704
0.58557±
0.00514

ωa (eV)
2.2174±
0.00014
1.9725±
0.000991

Wa (eV)
0.020862±
0.000133
0.041887±
0.000941

ωb (eV)
2.4448±
0.0067
2.383± 0.0233

Wb (eV)
0.48832±
0.00726
0.38067±
0.0203

q (unitless)
-1.6986±
0.00943
-1.2762±
0.0227

b (unitless)
0.82025±
0.0388
0.53022±
0.0578

0.27396±
0.00968

1.9637±
0.00144

0.032595±
0.00151

2.3473±
0.0563

0.48954±
0.0991

-2.185± 0.11

0.6564± 0.412

Table 2.1: Fitting parameters for Equation 2.7 for Figure 2.25. 95% confidence
intervals are given.
Absorption
curve
ii
iii
iv
vi
vii

Pitchs (nm)

Widths (nm)

Θs (deg)

Pitch f (nm)

Width f (nm)

Θ f (deg)

567
566
511
509
511
452
452

97
91
119
98
89
131
101

0.5
0.5
0.5
0.5
0.5

560
560
507
507
507
455
455

97
90
102
87
82
102
87

0.5
0.5
0.6
0.6
0.6

Scaling (unitless)
0.71
0.66
1.06
0.87
0.76
0.99
0.87

Table 2.2: Comparison of experimental dimensions and illumination angle (Pitchs ,
Widths , Θs ) with best-fit simulation dimensions, illumination angle (Pitch f , Width f ,
Θ f ) and scaling factor corresponding to Figure 2.29 in the main text.
A wide range of materials with varying Seebeck coefficients including Al, Cr,
and Sb2 Te3 give rise to guided mode resonances with very similar peak heights,
positions, and widths, as shown in Figure 2.26. Sb2 Te3 and Cr exhibit a large

37
extinction coefficient k at the waveguide resonance wavelength and are plasmonic
(’<0) in this wavelength range. By contrast Al has a more negative value of ’
in this region and has a narrower resonant linewidth, whereas Au and Cu have
resonances that are spectrally shifted in wavelength due to interband transitions or
plasmon resonances that couple to the waveguide mode, causing a Rabi splitting of
the modes (Christ et al., 2004).

Figure 2.26: A comparison of absorption spectra of different wire materials in our
guided mode resonant structure composed of 40 nm high, 68 nm wide, pitch of 488
nm wires on a waveguide of 50 nm SiO2 /100 nm SiN x .
Cross-sections of Sb2 Te3 wire guided mode resonance structures are shown in Figure
2.27. Figure 2.27a,c correspond to the absorption maximum wavelength, and Figure
2.27b,d correspond to the absorption minimum just to the left of the maximum, as
shown in Fig. 2a (Sb2 Te3 ). For this geometry, the absorption maximum occurs
at 611 nm, and the absorption minimum occurs at 596 nm. Figure 2.27a shows
the electric field surrounding the wires at the maximum absorption wavelength,
resulting from a constructive interference of the waveguide mode and the FabryPerot resonance. The large electric field magnitude in the wire corresponds to
high power absorption on resonance, shown in Figure 2.27c, whereas Figure 2.27b
illustrates the off-resonance electric field, at an absorption minimum, shown in
Figure 2.27d.
Thermoelectric nanophotonic structures supporting guided mode resonances exhibit

Figure 2.27: Full wave simulations for guided mode resonance structure with dimensions of 40 nm high, 68 nm wide, pitch of 488 nm wires on a waveguide of 50 nm
SiO2 /100 nm SiN x , with Sb2 Te3 wires (see Figure 2.4 for dielectric function). a, At
peak absorption, and b, at the minimum absorption. a, b, Electric field distributions
normalized to incident electric field. c, d, Power absorption density is calculated by
Pabs = 21 ω 00 |E| 2 , and is normalized by P0 , the incident power divided by the wire
volume.
tunable narrowband absorption over a wide wavelength range by variation of wire
array geometrical parameters. We can tune the absorption resonance over the entire
visible spectrum at constant waveguide thickness (50 nm SiO2 , 100 nm SiN x )
by varying the wire array pitch (Figure 2.28). Figure 2.29 shows experimental
absorption (black dotted, procedure described further in the next chapter), simulated
absorption (blue), and simulated best-fit (red). The peak positions in our experiment
closely match those predicted by simulations. The best-fit simulation (red) was
achieved by fitting the experimental data with altered wire dimensions in simulations
(fitting parameters in Table 2.2). Fitting experimental and simulation spectra to a
Fano shape (Gallinet and O. J. Martin, 2011) for one wire pitch (Figure 2.25c, Table
2.1), we found that the experimental spectrum exhibited larger damping caused
by losses in the wires, which altered the absorption spectrum shape. Pitchs and
Widths of our fabricated structures were found via SEM imaging and used in
simulations for absorption spectra (red line in Figure 2.29). θ s was a best fit incident
illumination angle from simulation. Pitch f , Width f , and f were dimensions and
incident illumination angle used in a simulation to best fit the experimental data.
The best-fit simulation was multiplied by a scaling factor to further fit.

Figure 2.28: (Left) Guided mode resonance structure geometry. Calculated absorption of (Top right) 60 nm wide, and (Bottom right) 100 nm wide wires with
thicknesses of 40 nm and varying pitch on suspended 50 nm SiO2 /100 nm SiN x
waveguides.

The absorption maximum can be tuned across several hundred nanometers of wavelength for a given waveguide thickness. Figure 2.31 shows wavelength versus
wire pitch for three different SiO2 /SiN x waveguide thicknesses which display pitchtunable, narrowband absorption maximum in three different wavelength regimes.
Using thicker waveguide layers, Figure 2.31 (center, bottom) show absorption peaks
beyond the detection limit of Si photodetectors, which is around 1.1 µm. In principle, the only limitation in infrared tunability for these detectors is the phonon
absorption band in SiO2 (and SiN x ) at around 8-11 µm (Cataldo et al., 2012; Palik,
1997b).
Guided mode resonance incorporated into a thermocouple device
Figure 2.23 show a schematic of our experimental structure, a guided mode resonance wire array, with wire dimensions of 40 nm × 100 nm × 50 µm, in which TM

Figure 2.29: Experimental absorption (black dotted), simulated absorption corresponding to the experimental dimensions (blue), and simulated absorption corresponding to fitted and scaled absorption spectra (red) for varying wire pitches and
widths on a 45 nm SiO2 /100 nm SiN x waveguide (see Table 2.2 for dimensions and
parameters). Off-normal angle of illumination causes the smaller peak to the left of
the larger absorption peak to form (see Figure 2.30).

polarized, normal incident, unfocused optical radiation is coupled into a waveguide
mode via a periodic thermoelectric wire array that serves as a light absorber with
spectra of the shape shown in Figure 2.32 (blue). Optical power is generated at the
thermoelectric junction from absorption in the wires, while the ends of the thermoelectric wires terminate in a broad pad of the same thermoelectric material which
reflects most incident light and remains cooler. The resulting temperature difference
between the center and edge of the structure is shown Figure 2.32 (orange). Figure
2.33 shows a full wave simulation illustrating the difference in absorption between
the pads and wires under unfocused, spatially-uniform illumination. Figure 2.34
shows the difference in power absorbed along a line cut through the length of the
simulation in Figure 2.33, which leads to a temperature gradient and results in a

Figure 2.30: Dependence of absorption spectra on incident illumination angle and
wire thickness. a, Measured absorption spectra for different angles of incidence.
An objective with numerical aperture 0.14 was used, giving an angular spread even
at normal incidence, producing the side peak at normal (0°) incident angle. b,
Full wave simulations of the incident illumination angle dependence of 40 nm tall,
67 nm wide Sb2 Te3 wires with a pitch of 488 nm on a 50 nm SiO2 on 100 nm
SiN x waveguide. Even at 1 degree offset, the single peak splits into two. c, The
dependence of wire thickness on absorption spectra, with pitch of 488 nm. The
absorption asymptotes to its maximum value for wire heights around 40 nm.

thermoelectric voltage (TEV). Our nanophotonic thermoelectric structures on thermally insulating membrane substrates have dimensions large enough that bulk heat
transport equations can be used (i.e. no ballistic or quantized thermal conductance).
To maximize responsivity, we seek to maximize the TEV, which is proportional
to the Seebeck coefficient, , and the temperature difference, ∆T, between cold and
hot ends of the material, i.e., T EV = α∆T. α is primarily dependent on material
and deposition methods, while nanostructuring has been shown to alter the Seebeck
coefficient to some degree (W. Wang and Z. M. Wang, 2014; Koumoto and Mori,
2013). ∆T can be increased through five primary design approaches. First, high

Figure 2.31: (Top) Wavelength versus pitch absorption plot in the visible regime for
40 nm thick Sb2 Te3 wires, on a 50 nm SiO2 /100 nm SiN x suspended membrane.
(Center) Absorption spectra for 50 nm thick, 300 nm wide Sb2 Te3 wires on a 300
nm SiO2 /500 nm SiN x suspended membrane. (Bottom) Absorption spectra in the
mid-IR for 50 nm thick, 1.5 µm wide Bi2 Te3 wires on a 500 nm SiO2 /500 nm SiN x
suspended membrane. All calculations use either Sb2 Te3 or Bi2 Te3 as the wire
material (see Figures 2.2, 2.3,2.4, 2.5 for dielectric functions). Simulations were
performed first coarsely with Lumerical FDTD (Lumerical, n.d.), then were refined
by Seyoon Kim with finer pitch steps using an RCWA method based on (Moharam
et al., 1995).

light absorption in the desired “hot region” is essential. Secondly, low energy loss
via radiation (i.e. low emissivity) in the “hot region” is desirable, with higher
emissivity in the “cold region”. Thirdly, low conduction through the interface is
preferred, via suspending the thermoelectric “hot region” or having high thermal
interface resistance. Fourth, as with any thermoelectric device, a low thermal conductivity is necessary to maintain a high temperature gradient, achieved by material

43
selection, nanostructuring, or by choice of deposition methods. Finally, low convective losses to the surrounding gas ambient in the “hot region” are preferred, and
can be achieved by operation of the thermoelectric structure in vacuum (although
the loss of convection in the “cold region” could be detrimental to a temperature
gradient and should be carefully considered).
As an example of a thermoelectric plasmonic nanostructure, we consider a periodic
array of wires composed of thermoelectric materials on a thin, suspended, electrically insulating, low thermal conductivity substrate. Using the electromagnetic
power absorption simulations as inputs, we can simulate the temperature profiles
in our structures; an example is shown in Figure 2.35. Temperature difference ∆T
as a function of wire length is shown in Supplementary Fig. 1d. Longer wires
produce a larger temperature difference for a given power density, but will have a
larger resistance, increasing the Johnson noise and therefore increasing noise equivalent power (NEP), shown in Figure 2.37f. Additionally, smaller structure sizes are
preferable, e.g., for camera pixel applications, motivating us to choose a wire length
of 50 µm, which shows reasonable responsivity for the chosen power densities and
yields a low NEP. A wire array/substrate heterostructure supporting guided mode
resonances in an n/p-type TE junction is shown in Figure 2.36. The absorption
resonance can be spectrally shifted by several hundred nm by varying the wire array
period. Thus, a periodic tiling of wire array pixels each with a different period
and resonance frequency could function as a thermoelectric hyperspectral detector,
shown conceptually in Figure 2.38.

Figure 2.32: Theoretical absorption (blue) and temperature difference between the
center of the wire and edge of the pad (orange) for a structure with 40 nm tall by
100 nm wide Sb2 Te3 wires spaced 488 nm apart.

Figure 2.33: Electric field profile normalized to incident electric field of a periodic
structure at peak absorption (1 µm scale bar). Highest |E| occurs in the wires,
leading to absorption, while the pads largely reflect light creating the necessary
temperature gradient.

Figure 2.34: Power density absorbed along a line cut through the simulation in
Figure 2.33. Asymmetry arises from half of the device being Sb2 Te3 and the
other half being Bi2 Te3 . Power density is normalized to incident power divided by
thermoelectric structure volume.
Thermal design
The theoretical temperature spectrum found in Figure 2.32 for the guided mode
resonance wire structure was constructed as follows. Absorption data were taken
from 2D periodic electromagnetic simulations and combined with experimentally
measured power. The combined data was used as a volumetric input power source
in a thermal simulation. Simulation sizes were large enough that changes in fixed
boundary temperatures did not affect central temperature profiles. Due to memory

Figure 2.35: A thermal simulation of the Bi2 Te3 - Sb2 Te3 structure at peak absorption
with input power of 20 µW. Scale bar is 500 µm, inset scale bar is 50 µm.

Figure 2.36: False color SEM of a fabricated p/n thermoelectric structure, with Au
contacts (20 µm scale bar). Inset is the junction between Bi2 Te3 - Sb2 Te3 wires (1
µm scale bar).

Figure 2.37: Thermopile and wire length analysis. a, Absorption spectra used in
simulations for b,c,e,f. b, Responsivity for a structure with 50 µm long wires and
27 µm long pads, as a function of number of wires in a thermopile configuration.
The entire structure is illuminated and responsivity is calculated relative to power
striking the wire area. The pads are assumed to have a 20% absorption, independent
of wavelength. c, Noise equivalent power (NEP) for the thermopiles in b, assuming
Johnson noise as the noise spectral density using simulated average temperatures. d,
Temperature difference between the edge of the pad and the center of the wires versus
power density for different wire lengths. Pad sizes remain constant with dimensions
of 50 µm by 27 µm by 50 nm. e, Responsivity as a function of wavelength for the
absorption spectrum in a, for different wire lengths, relative to power illuminating
the entire structure. f, Noise equivalent power corresponding to the responsivity
in e, for different wire lengths. Noise spectral density is theoretical Johnson noise
using simulated average temperatures for the structures. Simulation details are given
in the text.

Figure 2.38: Conceptual design of hyperspectral pixel. Each thermoelectric structure in the pixel has a different wire pitch, causing each structure to have an absorption peak that corresponds to a different wavelength (Figure 2.31). When light
of unknown wavelength content illuminates the pixel, voltages will be produced in
each structure depending on their specific absorption profiles. Deconvolution of
these voltage signals through an algorithm can enable identification of the unknown
wavelengths.

constraints, symmetry boundary conditions were used, and so simulations used one
material, bismuth telluride, instead of both bismuth telluride and antimony telluride.
Of course, while our thermal simulations were able to explain experimental results
reasonably well (compare Figure 3.4 and Figure 3.12j-l), it is known that small crystal
grain sizes can decrease thermal conductivity in thin film BiTe-based materials
(Takashiri et al., 2008). From our XRD data (Figure 3.1), we can see that the grain
structures must not be large. We recognize the bulk values of thermal conductivity
used in our thermoelectric materials in the simulations may be different than that of
our thin-film materials. Dimensions in simulations were the same as those of the
fabricated sample: 40 nm tall wires and pads, 100 wires total, 50 µm long (25 µm
for each thermoelectric material), with a pitch of 520 nm and wire width of 130 nm.

48
The thermoelectric pad lengths were 27 µm and as wide as the wire array. The Au
contacts were 6 µm wide and 70 nm tall, and overlapped the pads by 5 µm.
2.9

Guided mode resonator thermopile

Nanophotonic design
Thermopiling would further increase the responsivity of the guided mode resonance
design. Figure 2.39 shows an artist’s rendition of this idea. The nanophotonic
response would be nearly identical to the thermocouple case, as the pads would
still be wide enough to act like bulk materials and reflect light (Figure 2.39). The
absorption for the thermopiled structure wires is shown in Figure 2.37a.

Figure 2.39: Artistic rendition of a thermopiled guided mode resonance structure,
when p-type and n-type thermoelectrics are alternated in series. Image rendered in
POV-ray.

Thermal design
Simulations done in Figure 2.37 were performed in a similar manner to those described for the guided mode resonance wire structures above. The results of the wire
simulations were also used in Figures 2.37b,c to approximated outputs of thermopile

Figure 2.40: Full wave simulations of the thermopiled guided mode resonance
structure, with electric field |E| normalized by incident electric filed |E0 |. Excitation
wavelength is 631 nm (peak absorption for this geometry).

configurations. We find that increasing power density will linearly increase the difference in temperature between the “cold” edge of the thermoelectric pads and the
junction at the center of the wires (Figure 2.37d). Relative to power impingent upon
the entire structure, the responsivity of wires increases with length until 90 µm, then
slightly decreases at 110 µm (Figure 2.37e, average of measured Seebeck coefficients
of –84 µV/K (Bi2 Te3 ) and 242 µV/K (Sb2 Te3 ) used). We can explain the decrease
in responsivity at 110 µm in the following way: while there is more area available
to absorb light, and therefore more total power absorbed, this absorbed power is
not localized on the thermoelectric junctions as much as the absorbed power in the
smaller wire structures is. Because pad lengths are held constant, the larger wire
area will simply heat up the entire structure, including the edges of the pads more.
The maximum heat of the wires will increase, but the temperature of the “cold” ends
will also increase, creating a smaller overall ∆T, and therefore a smaller voltage.
Increasing the length of the pads would allow responsivity to increase with wire
length for wires longer than 110 µm. Using theoretical Johnson noise as our noise
spectral density and using the measured resistivity of our materials and simulated
temperature rises, we find that because of their overall higher resistances, longer
wires will have higher noise equivalent powers. We chose to study structures with
50 µm wires as they give a high responsivity with lower noise equivalent power.

50
2.10

Analysis and summary of simulation findings and guidance for future
designs

Noise considerations
The perfect absorber and split ring resonator designs studied in this chapter theoretically could have higher detectivities than the guided mode resonance wire structures
described and tested in the later part of the chapter and the next chapter. For the perfect absorber (Figure 2.1) and split ring resonator absorbers (Figure 2.11), we have
designed a thermal management scheme to provide motivation for future research.
The nanophotonic absorber arrays were placed in the center of a thin thermoelectric
junction on top of a thin membrane of SiN x , described further in Figures 2.7. The
simulation was done without any assumed thermal convection in the surrounding
ambient. The supporting edges of the thermoelectric ‘bridge’ structure depicted in
Figure 2.7 were placed on a laminate of SiN x on a thin Au film on a thick layer
of SiO2 . The high thermal conductivity of the thin Au layer greatly lowered the
“cold side” temperature of the thermoelectric material to near room temperature,
compared to when this layer was not present. This allowed us to increase the
thermal gradient within the thermoelectric materials, creating higher responsivities
than in our resonant wire structures, from 180 to 390 V/W (Figures 2.9, 2.15).
Using the measured resistivities for our structures, (2.75×10−2 ± 7.2×10−5 Ω·cm
for Bi2 Te3 and 4.27×10−1 ± 1.1×10−3 Ω·cm for Sb2 Te3 , discussed later) and the
dimensions described in Figure 2.7, the resistance of the thermoelectric element
would be around 91 kΩ in the perfect absorber, and 152 k in the split ring resonator,
giving a theoretical Johnson noise of 38 nV/Hz1/2 and 50 nV/Hz1/2 , respectively.
Thus, the minimum noise equivalent power (NEP) would be 0.2 nW/Hz1/2 for the
perfect absorber and 0.13 nW/Hz1/2 for the split ring resonator absorber, shown in
Figures 2.10, 2.16. This is 10 to 20 times less than our measured and simulated
NEP minima of our structures, which is significant.
Resonant thermoelectric nanophotonics versus conventional thermoelectric detectors
The distinction between a conventional (c.f. (F. Völklein and A. Wiegand, 1990))
thermopile and a resonant thermopile (Figure 2.39) lies in the total heat capacity of
the hot end of the structure. A smaller heat capacity (i.e. smaller structure) will generally have both higher responsivity and smaller time constant for the same amount
of incident power absorbed. Resonant absorbing structures can absorb a large fraction of light in small volumes because their absorption cross-section is much larger

51
than their geometric cross-section, unlike conventional absorbing layers such as
black paint. Furthermore, layers like black paint, which rely on single-pass material
absorption, need to be thicker than the skin depth of the incident light, whereas
this is not a requirement for resonant structures which rely on multiple-passes of
light. If the thermopile detector takes advantage of absorption from multiple passes
of light with non-thermoelectric absorbers (A. S. Gawarikar, R. P. Shea, and J. J.
Talghader, 2013a), the heat capacity for this structure will still be higher than if
the thermoelectric materials themselves were the absorbers, and still gives a 10’s
of milliseconds time constant, as there is simply more material if an absorber plus
a thermoelectric element is needed. Thus, if equal amounts of light are absorbed,
thermoelectric structures which themselves absorb (especially suspended, isolated
thermoelectric absorbers) have the potential for higher responsivity with lower time
constant than devices with non-thermoelectric absorbing structures, because of the
smaller volume and therefore heat capacity of the materials involved.
Take as an example the thermopile structure in (F. Völklein and A. Wiegand, 1990).
This structure relies on an absorbing layer on the order of microns thick and 500
µm in diameter inside a circle of thermopiles. If we have the guided mode resonant
thermoelectric thermopile structure shown in Figure 2.39 that uses thermoelectric
material as the absorber, and assume the thermoelectric materials absorb an equivalent amount as the membrane, then the maximum temperature will occur in the
thermoelectric wires themselves, but with much less material than in the membrane
in (F. Völklein and A. Wiegand, 1990). This will lead to a higher responsivity
and lower time constant in the resonant thermoelectric structure, although the noise
spectral density may be higher due to larger resistance in the structure. Our present
design has an absorption maximum of about 60% in the thermoelectric wires, but
by altering the design slightly to form a Salisbury screen (Fante and McCormack,
1988), 100% absorption can be reached in the thermoelectric materials.

52
Chapter 3

HYPERSPECTRAL DETECTOR APPLICATION
We propose and demonstrate here nanostructures composed of thermoelectric thermocouple junctions using established thermoelectric materials – Bi2 Te3 /Sb2 Te3 –
but patterned so as to support guided mode resonances with spectrally sharp absorption profiles. Spatially localized absorption in resonant thermoelectric nanophotonic
structures results in localized heating of the thermoelectric material, generating large
thermal gradients under unfocused optical excitation. We find that the small heat
capacity of optically resonant thermoelectric nanowires enables a fast, 337 µs temporal response, 10-100 times faster than conventional thermoelectric detectors. We
show that TE nanophotonic structures are tunable from the visible to the mid-IR,
with small structure sizes of 50 µm by 110 µm. Whereas photoconductive and photovoltaic detectors are typically broadband (with exceptions noted, e.g. (Mokkapati
et al., 2015; Chang et al., 2007; Wipiejewski, Panzlaff, and Ebeling, 1992)) and are
insensitive to sub-bandgap radiation, nanophotonic thermoelectrics can be designed
to be sensitive to any specific wavelength dictated by nanoscale geometry, without
bandgap wavelength cutoff limitations or need for cooling. From the point of view
of imaging and spectroscopy, they enable integration of filter and photodetector
functions into a single structure.
3.1

Background of hyperspectral detectors

Hyperspectral detectors detect light with much finer wavelength resolution than RGB
detectors. Originally developed by NASA and JPL (Goetz, 2009) for space-based
geology analysis and military applications, hyperspectral detectors have found uses
from art conservation (Fischer and Kakoulli, 2006), to wetland vegetation identification (Adam, Mutanga, and Rugege, 2010), to food safety (Gowen et al., 2007).
Additionally, hyperspectral imaging has found applications in medical imaging (Lu
and Fei, 2014), as different tissues (such as healthy and cancerous) absorb and
emit different wavelengths of light which RGB imaging may not have fine enough
wavelength resolution to elucidate.
Hyperspectral imaging is more complicated than RGB imaging. First, some form
of narrow-band filter must be used to descriminate between different wavelengths
of light. This decreases the incident power of light, so detectors must be have a

53
good signal to noise ratio. Photovoltaic-based materials are also band gap limited,
so a single material, like Si, has a wavelength cutoff of 1100 nm. Thus, a visible
to IR hyperspectral detector would need several different materials to detect the
entire spectrum, and the small bandgap materials would need cooling to maximize
sensitivity.
3.2

Fabrication and materials analysis

Fabrication
The thermoelectric hyperspectral detectors were fabricated as follows. On top of
the waveguide layer of 100 nm thick SiN x membrane (Norcada NX7150C), the 45
or 50 nm SiO2 spacer layer was deposited via PECVD (Oxford Instruments System
100 PECVD) at 350°C.
The structures were written via electron beam lithography (Raith EBPG 5000+)
in a series of aligned writes, followed by deposition and liftoff. In order to spin
resist on the membrane, a sort of "stilts" were constructed to allow airflow under
the membrane, using a method designed by Seyoon Kim. If the membrane were
directly glued to a substrate, when put under vacuum, the air bubble trapped below
the membrane will burst the membrane if not allowed to escape. These "stilts" were
composed of two thin silicon chips that were first glued to a larger silicon wafer,
and the membrane was placed across these thin silicon chips so that air could flow
below the membrane. The "stilts" were attached the the larger silicon wafer by
spinning PMMA 495 A8 at 1100 RPM for one minute, placing the "stilts" on the
wafer, and baking at 180°C for 5 minutes. Then PMMA 495 A8 was spun on top
of the "stilts", again at 1100 RPM for 1 minute. The membrane was then placed on
the "stilts", and baked at 180°C for 5 minutes to adhere. Next, a bilayer resist was
spun on the membrane to aid in liftoff. PMMA 495 A4 was spun at 4,000 RPM
for one minute, then baked at 180°C for 5 minutes, followed by PMMA 950 A2
spun at 4,000 RPM for one minute, then baked at 180°C for 5 minutes. The idea
behind a bilayer resist is that the lower resist, the PMMA 495 A4, will develop move
than the top layer under an electron beam, creating an overhang. This overhang
will prevent deposited material from adhering to the sidewalls of the resist, which
prevents liftoff. This process works very well for directional deposition, such as
electon beam evaporation or thermal evaporation with a long throw distance (i.e.
material impinges primarily normal to the surface). This directionality is due first
to, geometry, due to the long distance from source to sample, and second to low
base pressure in the chamber (10−5 - 10−9 Torr generally), which means evaporated

54
material has zero or few collisions between evaporation and landing on the sample
due to the long mean free path of atoms at these low pressures. Sputtering, on the
other hand, is very non-directional. Sputtering is very effective at evenly coating
surfaces for several reasons. First, the throw distance in sputterers is generally must
be much shorter than in electron beam or thermal evaporators. This is due to the
higher pressure during deposition needed to sustain a plasma (around 10−3 - 10−2
Torr), so the mean free path of particles is much shorter, and cannot travel for before
being redirected by a collision with another particle or gas particle. These frequent
collisions additionally allow materials to slowly coat the inside of resist overhangs,
making liftoff more difficult, although not impossible.
Alignment markers were 20 by 20 micron squares composed of an electron beam
evaporation (Kurt J Lesker Labline or CHA Industries Mark 40) deposited 5 nm Ti
adhesion layer (0.5 Å/s) and 55-70 nm of Au (1-1.5 Å/s), and lifted off in Acetone.
40 nm of Bi2 Te3 and Sb2 Te3 were magnetron sputter deposited (Kurt J Lesker) with
40 W RF power (US Guns). 40 nm of alumel and chromel were magnetron sputter
deposited with 500 W DC power (AJA Orion). Contacts were deposited in the same
way as alignment markers, but with closer to 100 nm of Au. In this experiment, Ti
was used as an adhesion layer to the thermoelectric materials (Ni was used as an
adhesion layer for cloverleaf Hall measurement samples for Bi2 Te3 ).
The liftoff process could be done fast or slow. The fast process involved submerging
the sample in acetone on a 70°C hot plate. After approximately 5-15 minutes,
depending on the deposition method used (the Lesker Labline took 5 minutes, the
CHA took closer to 15 minutes), the edges of the chip were scraped to aid the liftoff
process. This was done when the sample surface started to look "wrinkly". After
about 5 more minutes, acetone was forcibly sprayed on the sample (avoiding directly
sqirting the membrane) to help loosen the remaining material. The material was
generally lifted off after about 30 minutes. Occasionally, letting the sample soak
for several days would aid in lifting off more material, but this usually was not the
case. The slow liftoff process involved allowing the sample to sit covered in room
temperature acetone overnight. Liftoff success was extremely variable, especially
with cosputtered and thermally evaporated thermoelectric materials, explored in
later chapters. Bismuth-heavy materials lifted off much worse than tellurium-heavy
materials. This could be due to surface tension differences.
Fabricated dimensions of Figure 2.36 are 100 nm wide, 40 nm thick, wires with a
470 nm period, fabricated on a 145 nm thick freestanding dielectric slab waveguide

55
composed of 45 nm SiO2 and 100 nm SiN x layers.
Bismuth telluride and antimony telluride compositional and structural analysis
and materials characterization

Figure 3.1: XRD data of 100 nm of Bi2 Te3 (left) and 50 nm Sb2 Te3 (right) show
very little crystallinity, as sputtered in experiments. Two-dimensional diffraction
image frames were collected with frame centers set to 20, 40, 60 and 80 degrees in
2Θ, from right to left, and then merged. Credit to Slobodan Mitrovic for data and
analysis.

Figure 3.1 shows XRD data on thin films of Bi2 Te3 and Sb2 Te3 , deposited from a
stoichiometric target as described in the previous section. We notice that in both
cases the data show signatures of nanocrystallinity or perhaps even amorphous
structure in the case of antimony telluride.
Two-dimensional XRD data were collected with a Bruker Discover D8 system, with
a Vantec 500 detector and Cu Kα x-ray line produced by a microfocused IµS source,
in a θ − 2θ measurement. Figure 3.1 shows results on thin films of Bi2 Te3 and
Sb2 Te3 .
X-ray photoemission spectroscopy (XPS) was performed on a Kratos Nova (Kratos
Analytical Instruments), with monochromatized x-rays at 1486.6 eV and using a
delay-line detector at a take-off angle of 35 degrees. The pressure during measurement was better than 5x10−9 Torr, and the data were collected at 15 mA and 15 kV
from an area of about 0.32 mm2 . Survey scans were collected at pass energy 160,
and high-resolution scans at pass energy 10. Figure 3.2 shows XPS survey spectra
for our Bi2 Te3 and Sb2 Te3 thin films, and due to surface sensitivity of the technique,
these represent only the top few nanometers of the sample.
Apart from expected surface oxidation and hydrocarbons from air, there are no
other contaminants present that could affect the Seebeck coefficient. The stoichiometry of bismuth telluride and antimony telluride greatly affects the Seebeck coefficient(Horne, 1959; Bottner et al., 2004; Silva, Kaviany, and Uher, 2005). Bismuth

Figure 3.2: XPS survey scans. Sb2 Te3 (top) and Bi2 Te3 (bottom) samples. Credit
to Slobodan Mitrovic for data and analysis.

telluride is particularly sensitive: a small variation in atomic percent composition is
capable of changing the carrier type from electrons to holes(Horne, 1959).
Using quantitative analysis based on Te 3d and Bi 4f levels, shown in Figure 3.3, we
determined that the composition of our bismuth telluride was 42.5%:57.5%, Bi:Te
for surface relative concentrations. This corresponds to a wt% of about 53.7% for
the bismuth.
The XPS measured composition of our 50 nm antimony telluride film was determined from Sb 3d 3/2 and Te 3d 3/2 peak areas (as identified to belong to the
compound), and indicates a composition of 32%:68% Sb:Te. A large amount of
antimony on the surface had oxidized.

Figure 3.3: Compositional analysis. Detailed XPS data and fits for bismuth telluride
peaks (a-b) and antimony telluride peaks (c-d) for our samples. a, Three components
are visible in Bi 4f levels: the major components are Bi2 O3 , and Bi2 Te3 (157.1 eV and
spin-orbit pair at + 5.3 eV), with a small amount of elemental bismuth (156.6 eV). b,
Te 3d level in bismuth-telluride is mostly TeO2 and Bi2 Te3 (582.3 eV, SO-splitting
of 10.4 eV). c, Sb 3d levels show that most of the surface of antimony-telluride
is oxidized (Sb2 O3 ), much more so than the bismuth-telluride, with a measurable
Sb2 Te3 component (538 eV). d, Te 3d levels in antimony-telluride show the telluride,
an oxide and elemental Te peaks. Credit to Slobodan Mitrovic for data and analysis.

Seebeck measurements at room temperature were performed based on a thin film
Seebeck measurement technique described in the literature (Singh and Shakouri,
2009). The sample thermoelectric materials were deposited on 500 nm layer of
SiO2 on a 1 cm by 1 cm silicon chip. The thermoelectric materials were of the
same thickness of the structures used in this paper. Copper blocks were placed
with thermally conductive paste on the hot and cold sides of a thermoelectric
heating stage with the thin-film sample straddling the copper blocks, attached with
thermally conductive paste. Thin K-type thermocouple wire junctions were placed
on the surface of the thermoelectric material on the hot and cold ends, using physical
pressure. Soldering or using thermally conductive paste to connect the wires to the
samples was not recommended (Singh and Shakouri, 2009), and we measured large
hysteresis when using thermally conductive paste. A voltmeter (Keithley 6430 sub-

58
femtoamp remote source meter) was attached to the chromel end termination of each
thermocouple before it was passed into a temperature meter, providing temperature
and voltage data from the exact same location on the sample. Stage temperature
was measured using a build-on thermocouple attached to the stage, which roughly
verified the sample temperature measurements were in the correct range. Next, the
Seebeck coefficient of the chromel wire used as half of the thermocouple/electrode
was measured in a similar manner to subtract out its contribution to the thin film
voltage measured. The thin film samples were not measured in vacuum, which could
contribute error to the temperature measurements, as convection could slightly cool
thermocouples used in the measurements relative to the sample surface. Data from
several specimens of each material were collected, and is displayed in Table 3.1.
Variability among specimens deposited at the same time in the same chamber can
be accounted for by distribution of substrate positions within the deposition system.
There was no substrate rotation during the deposition, which leads to small thickness
variations or unequal substrate temperatures amongst our samples. Amorphous thin
films of Sb2 Te3 have shown a variation of 100 µV/K when deposited under the same
conditions (Baily and Emin, 2006), indicating that our small sample-to-sample
variation is not unique.
Sample
Sb2 Te3 – 1
Sb2 Te3 – 2
Sb2 Te3 – 3
Bi2 Te3 – 1
Bi2 Te3 – 2
Bi2 Te3 – 3

Seebeck coefficient (µV/K)
264 ±16.6
214 ±14.6
247 ±15.9
-81 ±3.1
-77 ±3.9
-95 ±5.2

Table 3.1: Measured Seebeck coefficient of 6 different samples of thermoelectric
material. To clarify the sign convention, Sb2 Te3 here is p-type and Bi2 Te3 is n-type.
Seebeck coefficient is theoretically determined primarily by crystal band structure.
For quintuple layers (unit of Te-Bi-Te-Bi-Te or Te-Sb-Te-Sb-Te) of bismuth telluride
and antimony telluride, it has been calculated that band structures will be altered
for less than 12 quintuple layers, due to opening of a bandgap attributed to topological surface-states, but this gap will close with more than 12 quintuple layers and
approaches the bulk value (Sung et al., 2014). One quintuple layer is approximately
1.1 nm (Sung et al., 2014), putting our materials at roughly 36 quintuple layers,
albeit disordered in our case. Therefore, we would not expect surface states to
significantly alter the bandstructure, and therefore, the Seebeck coefficient in our

59
Citation

Material

(Horne, 1959)

Bi2 Te3

(Bottner et al., 2004)

Bi2 Te3

(Silva, Kaviany, and Bi2 Te3
Uher, 2005)
(Zou, D. Rowe, and Bi2 Te3
Williams, 2002)
(Silva, Kaviany, and Sb2 Te3
Uher, 2005)
(Pinisetty et al., 2011) Sb2 Te3

(Baily
2006)

and

Emin, Sb2 Te3

(Shi et al., 2008)

Sb2 Te3

(Zou, D. Rowe, and Sb2 Te3
Williams, 2002)

Resistivity
(Ωcm)
2.7 × 10− 4 to
1.71 × 10− 3
2.1 × 10− 3

Seebeck Coef- Details
ficient (µV/K)
+160 to -145
bulk with oxide impurities
-50
10
microns,
sputtered
8 × 10− 3
-50
coevaporated,
1 micron
1.29 × 10− 3 to -143 to -228
coevaporated,
2.6 × 10− 3
coevaporated,
700 nm
4 × 10− 3
+140
coevaporated,
1 micron
+70 to + 365
100 and 400
nm diameters.
100 nm amorphous gave α
+230 µV/K
1.11
+700, +800
amorphous,
cosputtered, 1
micron
0.91
+125
hydrothermal
synthesis
1.04 × 10− 3 to +140 to +171
coevaporated,
4.90 × 10− 3
700 nm

Table 3.2: Literature values of Seebeck coefficient and resistivity.

130 nm by 40 nm wires from the Seebeck coefficient in our 40 nm thick films. Our
thermal simulations combined with our experimentally measured voltage support
this theory. Furthermore, in a study on thin film Sb2 Te3 , it was found that while
resistivity increased with decreasing film thickness (from 790 nm to 160 nm), the
film thickness had little effect on the Seebeck coefficient (Rajagopalan and Ghosh,
1963), which approaches our thickness value of 40 nm.
Bi2 Te3 and Sb2 Te3 have been extensively studied and characterized. As with other
thermoelectric materials, the Seebeck coefficient has been found to depend heavily
on deposition method and deposition temperature, as well as post-deposition treatment, such as annealing (Takashiri et al., 2008; D.-H. Kim and Lee, 2006; Rashid,
Cho, and Chung, 2013). Our Seebeck coefficients fall within the range of those
found in the literature, a very small subset of which is shown in Table 3.2. The
ranges of Seebeck coefficient for Bi2 Te3 in this table range from +160 µV/K to -228
µV/K, depending on deposition method, percent Te, and oxide content. Our measured Bi2 Te3 samples lie well within this range. Comparing our measured values

60
to any of the values in the literature is difficult, as we were not able to determine
the substrate temperature during deposition. As sputterers have different working
distances and different deposition powers can be used, the substrate heating would
be different in each sputterer or evaporator unless substrate heating controls are used.
Sb2 Te3 in Table 3.2 ranges in Seebeck coefficient from +70 µV/K to +800 µV/K.
Of note, noncrystaline Sb2 Te3 (Ref. 7) has a higher Seebeck coefficient, 700-800
µV/K. Our samples are nearly noncrystaline, based on our XRD data (Figure 3.1).
Also, in this paper, two different samples deposited in the same way had a difference in Seebeck coefficient of about 100 µV/K, showing the variability in Seebeck
coefficients among similar thin film samples.
The resistivities of our materials were found by measuring cloverleaf samples in a DC
hall measurement system with short time constants, and showed very small error. We
found a resistivity of 2.75×10−2 7.2×10−5 Ω·cm for Bi2 Te3 and 4.27×10−1 1.1×10−3
Ω·cm for Sb2 Te3 . A subset of literature resistivity values, shown in Table 3.2, vary
between 1.71 × 10−3 Ω·cm to 8.71 × 10−3 Ω·cm. Our resistivity is higher than these
values, but our films are much thinner and surface oxidation could play a significant
role in decreasing conductivity. A subset of literature values (Table 3.2) show
1.11 Ω·cm to 1.04 × 10−3 Ω·cm resistivities for various thin film thicknesses and
deposition methods for antimony telluride. Our values lie within the literature range,
closer to the values of amorphous films. Additionally, it was found that resistivity
increases with decreasing film thickness in Sb2 Te3 (Rajagopalan and Ghosh, 1963).
As our films are thinner than many in the literature, a higher resistivity, similar to
ours, is expected.
We expect resistivity to be the same in our structures as in our wires, as the crystal
grain sizes are small in both. Using our measured resistivity, the power factors for
Sb2 Te3 are 1.1 × 10−5 ˘1.6 × 10−5 W/K2 ·m and for Bi2 Te3 2.2 × 10−5 − 3.3 × 10−5
W/K2 ·m. Using the literature values of thermal conductivity used in our thermal
simulations (see previous section), this would give us zT values in the 10−3 - 10−4
range, indicating that our materials could be further optimized.
3.3

Measurement results

Measured absorption and responsivity
This section summarizes the measurements for our thermoelectric plasmonic guided
mode resonance structures, with measured absorption (1-transmission-reflection) at
normal (0°±1°), 5°±1° off-normal, and 10°±1° off-normal incidence (1-transmission

Figure 3.4: (Top) Absorption (0° and 5°) or 1 – transmission (10°) for 0°, 5°, and
10° (±1° error) incident illumination on a Bi2 Te3 - Sb2 Te3 structure described in the
text with wire dimensions of 40 nm thick × 130 nm wide × 50 µm long. (Bottom)
Responsivity for unfocused, spatially uniform illumination of the entire structure
(including the pads, Figure 3.5d) with a 120 µm by 100 µm spot size at 0°, 5°, and
10° (±1° error) off-normal incidence.

Figure 3.5: Focused versus unfocused, spatially uniform light responsivity and
noise characteristics. Focused (blue, 60 µm by 5 µm spot size, e) and unfocused,
spatially uniform (orange, 120 µm by 100 µm spot size, d) illumination incident
on Bi2 Te3 -Sb2 Te3 structures at given angles off normal incidence, with ±1° error.
A 5× objective with numerical aperture 0.14 was used for both the focused and
spatially uniform illumination data collection. a, The input power used to calculated
responsivity in the case of uniform illumination was only the power that illuminated
the wires (a 50 × 50 µm2 area). The spatially uniform illumination spot was 120
µm by 100 µm, and completely covered the wires and pads of the structure. Noise
spectral density, b, was measured under the power spectrum shown in Figure 3.6.
Higher noise spectral density in focused light was likely due to back currents from
uneven heating in the structure, discussed further in the text. c, Noise equivalent
power was found to be lower for spatially uniform illumination than for focused
illumination, due to higher responsivity values combined with lower noise values
for spatially uniform illumination. d, Black circle illustrates uniform illumination of
structure. e, Focused illumination used in a-c, Figure 3.8. f, Focused illumination
in Figure 3.7.

shown for this case), shown in Figure 3.4 (top). Figure 3.4 (bottom) depicts the
responsivity, relative to power illuminating the wire region, of a Bi2 Te3 - Sb2 Te3
structure completely and uniformly illuminated (pads and wires). In Figure 3.7, a

Figure 3.6: Laser power illuminating the wire region as a function of wavelength
for focused illumination (blue) and uniform illumination (orange).

Figure 3.7: Maximum responsivity found for a structure when only the junction is
illuminated (60 um by 5 um spot size, Figure 3.5f).

long, narrow beam was focused on the junction of all wires at 5°±1° off-normal
incidence and represents the maximum responsivity found. The responsivity is
noisier due to the sensitivity of the sample to the position of the light at the junction. Comparison of illumination configurations and alumel-chromel structure data
is discussed in Figures 3.5,3.11 and in a later section. While the ratio of maximum to minimum responsivity is not large in our structures, the ratio is nearly the
same as the maximum to minimum absorption ratio, suggesting that the absorption
spectra largely dictates responsivity spectral shape, as demonstrated in simulations
of responsivity in guided mode resonance structures in Figure 3.12. Therefore, a
spectrum with a larger maximum to minimum absorption ratio would have a larger

Figure 3.8: Thermoelectric voltage (TEV) dependence on incident power for a
Bi2 Te3 - Sb2 Te3 structure at 0° (±1° error) off normal angle under focused illumination (see Figure 3.5e, and Figure 3.5a for focused responsivity spectrum). The
temperature scale on the right axis corresponds to ∆T between the hot wire junctions
and cold pad edges based on average measured Seebeck coefficients. We estimate
that 1,000 µV would give a temperature range of a 2.8 to 3.4 K temperature rise,
based on the range of Seebeck coefficients of our materials measured. Error bars
are sample standard deviation of measurements.

Figure 3.9: Time response of a Bi2 Te3 - Sb2 Te3 structure. The time constant fit line
(red) plotted over the data from our thermoelectric detector (green) is measured as
155.13 s ± 3.06 s, corresponding to a 10%-90% rise time of 337 s. The response of
a Si photodiode at the same chopper speed is shown in blue.

maximum to minimum responsivity ratio.
Comparison with thermal simulations and response time
We found the voltage to be linearly dependent on incident power, as shown in Figure
3.8 for a Bi2 Te3 - Sb2 Te3 structure under focused illumination at 5°±1° off-normal

Figure 3.10: Noise spectral density (NSD) and noise equivalent power (NEP) as a
function of wavelength corresponding the data shown in Figure 3.4 (bottom). All
data were taken under polarized illumination with the E-field perpendicular to the
wires.
incidence. The weighted root mean squared error values were 0.58 µV, 0.45 µV,
1.05 µV, 0.82 µV, and 0.74 µV for our first order polynomial fit for illumination

Figure 3.11: Chromel-alumel structure results. Thermoelectric potential (TEV) and
absorption results for a chromel-alumel structure with the same dimensions as that
of the Bi2 Te3 - Sb2 Te3 structures. The structure is under 70.92 W illumination, or
30.4 W/cm2 incident power density. Data points are taken as the focused beam is
moved across the junction of the detector (over a 400 µm2 area). All data points are
averaged together for a given wavelength.

wavelengths of 700 nm, 675 nm, 650 nm, 625 nm, and 600 nm, respectively. These
results strongly suggest a linear dependence of TEV on incident power, which is
supported by simulation (see Figure 2.37d). The temperature scale in Figure 3.8
is based on a measured average Seebeck coefficient (at room temperature) of 242
µV/K for Sb2 Te3 and –84 µV/K for Bi2 Te3 (see Fabrication and Materials Analysis
section for details). This indicates a maximum temperature gradient ∆T of nearly
3 K, under illumination. We find a similar temperature gradient created in thermal
simulations, shown in Figure 2.35. Note that the relevant ∆T is between the edge
of the thermoelectric pads and the wire junctions, not the wire junctions and the
simulation edge.
Measurements of the response time under chopped illumination yielded time constants of 155.13 s ± 3.06 s and 153.56 s ± 2.50 s during heat up and cool down,
respectively (see Figure 3.9). This corresponds to a 10%-90% rise time of 341 s,
or almost 3 kHz, which is a fast enough response for many detection and imaging
applications.

67
Noise performance
Figure 3.10 (top) shows the noise spectral density (NSD) from the detector for an input power spectra shown in Figure 3.6. The resistance of our device is approximately
113 kΩ, giving a theoretical Johnson noise at room temperature of approximately 42
nV/Hz1/2 . Noise density detected above this level is attributed to temperature rise
and unaccounted for stray capacitances or other noise sources (see next chapter for
more discussion on noise measurement setups). Johnson noise provides the largest
contribution to the NSD. Thus noise density could be decreased by lower device
resistance through structural engineering or material selection. The noise equivalent
power (NEP) is shown in Figure 3.10 (bottom). This corresponds to a detectivity
of around 1 × 108 Hz1/2 /W, for a maximum of roughly 8 × 108 Hz1/2 /W if using
our maximum responsivity measurement in Figure 3.7 (see section in introduction
discussing why D∗ is not appropriate for most thermal detectors). In a comparison
between responsivity, NSD, and NEP for focused and spatially uniform illumination conditions as a function of incident angle (shown in Figure 3.5 and discussed
further in the next section), we found that the responsivity measured under spatially
uniform illumination more closely matched the absorption spectra shape, and the
uniform illumination had lower NEP and NSD. It is possible the higher NSD in the
focused illumination case arises from shot noise or capacitor Johnson noise from
back currents due to uneven heating of the thermoelectric junctions.
Comparison of illumination configuration: effect on responsivity and noise
Figure 3.5 compares responsivity and noise characteristics for a Bi2 Te3 - Sb2 Te3
structure under focused illumination (a 60 µm × 5 µm spot size illuminating a
few single wires in the center of the array, Figure 3.5e) and spatially uniform
illumination (a 120 µm × 100 µm spot size which completely uniformly illuminated
the entire structure, including the pads, Figure 3.5d) over a range of angles off
normal incidence, with ±1° error, and with electric field polarized perpendicular to
the wires. An objective with numerical aperture 0.14 was used for all measurements.
Responsivity of the spatially uniform illumination was relative to the power incident
on the area of the wires only. The unfocused, spatially uniform beam leads to a higher
responsivity at nearly every wavelength than that of the focused beam, and exhibited
a lineshape which more closely followed the absorption lineshape in Figure 3.4 (top),
and has similar shape and magnitude to our simulated values shown in Figure 3.12gi (simulated absorption and thermal simulation) and Figure 3.12j-l (experimental
absorption and thermal simulation). We believe the discrepancy between absorption

68
and responsivity lineshape under focused illumination arises from uneven heating
of the wire array. Because our wires are very optically lossy (see the non-negligible
imaginary dielectric constant in Figures 2.2, 2.4, 2.3, 2.5), light diffracted into a
waveguide mode by light focused on the center wires will not propagate far, and
therefore wires on the edge of the structure will not absorb (or heat up) as much
as wires in the center of the array. The wires with minimal temperature gradient
provide a leakage pathway for thermoelectric currents, reducing responsivity and
altering responsivity spectra. Spatially uniform illumination will heat the wires
more equally, largely eliminating these effects.
Leakage currents from uneven heating could cause the larger noise spectral density
(NSD) under focused illumination, seen in Figure 3.5b. While this NSD is convoluted with the input laser power (shown in Figure 3.6) at each wavelength, attempts
to normalize NSD to Johnson noise of a temperature rise for a given power input
(using knowledge that temperature rise is linear with power input, shown in Figure
3.8) did not yield a flat NSD spectra. Therefore, excess NSD must be due to noise
sources which do not vary as T1/2 , as Johnson noise does. Shot noise, on the other
hand, is proportional to the magnitude of the currents in a structure, and is a possible
noise source arising from these back currents if there is generation/recombination
within the semiconductor materials. The NEP is lower for nearly every wavelength
for the uniform illumination, and gives detectivity values in the 1-3×108 Hz1/2 /W
range.
Comparison with measured alumel-chromel device
Thermoelectric potential (TEV) and absorption results for a chromel - alumel structure with the same dimensions as that of the bismuth telluride structures is shown
in Figure 3.11. The structure is shown under 70.92 W illumination, or 30.4 W/cm2
incident power density. The focused beam is raster scanned across the wire junction
region of the detector, and at each location, a data point is taken (over a 400 µm2
area). All data points are averaged together.
3.4

Best possible performance with state-of-the-art materials for current design

Large responsivity and low noise are needed to have high detectivity. Larger
responsivity arises from a higher Seebeck coefficient, α, and a larger temperature
gradient (therefore, smaller thermal conductivity, k). Lower Johnson noise (the
primary noise source in our structure, shown in Figure 3.10 (top)), comes from lower

69
resistance, and therefore lower resistivity, ρ. Thus, detectivity can be improved with
a higher thermoelectric figure of merit, zT = αρkT , material. At room temperature,
one of the best p-type materials is a BiSbTe alloy which has a zT of 1.2 at room
temperature (Poudel, Hao, Ma, Lan, Minnich, Yu, X. Yan, D. Wang, Muto, Vashaee,
et al., 2008b). One of the best performing n-type materials is a PbSeTe quantum dot
superlattice material with a zT of 1.3-1.6 at room temperature (T. C. Harman et al.,
2002). The resistivity of the high zT materials are 1.71 x10-3 Ωcm (n-type) and
8.00 x10-4 Ωcm (p-type), the thermal conductivities are 0.58 W/m·K (n-type) and
1.1 W/m·K (p-type), and the Seebeck coefficients are -219 µV/K (n-type) and 185
µV/K (p-type). While we did not measure the thermal conductivity of our materials,
assuming bulk values (D. M. Rowe and Bhandari, 1995) of thermal conductivity for
our Bi2 Te3 ( 2.05 W/m·K) and Sb2 Te3 ( 3.54 W/m·K), we can see that the overall
thermal conductivity with the high zT materials would be decreased by a factor
of 4. While our thermal simulations were able to explain experimental results
reasonably well (compare Figures 3.12g-l and Figure 3.4 (bottom)), it is known that
small crystal grain sizes can decrease thermal conductivity in thin film BiTe-based
materials(Takashiri et al., 2008). From our XPS data (Figure 3.1), we can see that
the grain sizes must be small. This indicates that our thermal conductivity may not
be greatly improved by using the state-of-the-art materials above. In results from
thermal simulations we can see that using state-of-the-art n-type material, shown
in Figures 3.12d-f, versus our n-type Bi2 Te3 material, shown in Figures 3.12g-i,
we get 40% increase in responsivity. Our structures depend both on the thermal
conductivity of the thermoelectric material, and the thermal conductivity of the
substrate. Fortunately, suspended low-stress SiN x membranes(Ftouni et al., 2015)
and SiO2 can have thermal conductivities lower than or similar to the thermoelectric
materials we study. To remove the thermal conductivity contribution from the
substrate, the substrate could be etched from beneath the wires, although this would
affect the optical absorption characteristics.
The Seebeck coefficient for the large zT n-type material are larger than ours by a
factor of 2.6 (-219 µV/K versus an average of -84 µV/K), and the Seebeck coefficients
for the large zT p-type material were smaller than our materials by a factor of 1.3 (185
µV/K versus an average of 242 µV/K). This gives an overall Seebeck coefficient of
404 µV/K for the optimal materials, versus 326 µV/K for our materials, or a factor
of 1.24 increase in Seebeck coefficient. The resistivity of the large zT n-type
material decreases by a factor of 16 over our n-type material resistivity (1.71×10-3
Ω·cm versus 2.75×10-2 Ω·cm), and the resistivity of the large zT p-type material

70
decreases by a factor of 534 over our p-type material resistivity (8.00×10-4 Ω·cm
versus 4.27×10-1 Ω·cm). We can treat our system as two resistors in series, one
resistor composed of n-type material and the other composed of p-type material.
Then, assuming a symmetric system, the total resistance is proportional to the sum
of the resistivities of the two materials. The sum of the resistivities of the Bi2 Te3 and
Sb2 Te3 materials in our paper is 4.55×10-1 Ω·cm, and the sum of the resistivities
of the state-of-the-art materials is 2.51×10-3 Ω·cm. Thus, resistance decreases by a
factor of about 181 from our materials to the state-of-the-art. While nanostructuring
could change the material properties of the high zT materials, an approximation for
the expected detectivity increase can still be found.
Based on our simulations, a factor of 4 decrease in thermal conductivity in our thermoelectric materials will produce a factor of 1.4 increase in temperature difference
between the hot and cold end of our device. Coupled with a factor of 1.24 increase
in Seebeck coefficient, we can expect the responsivity to increase by a factor of 1.7
using the state-of-the-art thermoelectric materials. Noise spectral density depends
on the square root of resistance (assuming Johnson noise dominates), so decreasing
resistance by a factor of 181 will lead to a decrease in noise by about a factor of 13.
Therefore, the detectivity can be increased by around 1.7 × 13 = 22 times by using
state-of-the-art thermoelectric materials.
3.5

Why D∗ is not a valid FOM in our structures and other thermal detectors

As described in (Datskos and Lavrik, 2003), specific detectivity (D∗ ) was a figure
of merit designed for comparing quantum detectors, independent of detector size.
Quantum detectors have detectivity which scales with the square root of the detector
area. As we explain below, detectivity for our structure does not scale with the
square root of the detector area, but with a more complicated function of the width
of detector and length of the legs. Thus, our size-independent detectivity figure of
merit would have different units than that of quantum detectors, so we hesitate to
present this value. Page 351 of (Datskos and Lavrik, 2003) states:
“It should be noted that the definition of specific detectivity, D*, was
originally proposed for quantum detectors, in which the noise power is
always proportional to the detector area and noise signal (V or I) is
proportional to the square root of the area. However, the noise in
thermal IR detectors does not always obey this scaling trend. In fact,
neither temperature fluctuations nor thermo-mechanical noise (see the

Figure 3.12: a-c, Simulated absorption spectra for wires of the dimensions of the
experimental Bi2 Te3 -Sb2 Te3 structures at 1, 5, and 10 degrees off normal incidence.
d-l, Responsivity calculated from thermal simulations using absorption to guide
power input at different angles off normal incidence. 20% absorption in the pads
was assumed independent of wavelength. Simulated absorption from a-c, thermal
properties and Seebeck coefficient of state-of-the-art thermoelectric materials, given
in the text, were used in d-f. Simulated absorption from a-c, thermal properties
given in the previous chapter and average Seebeck coefficient of our materials (242
µV/K for Sb2Te3 and -84 µV/K for Bi2Te3, see Supplementary Note 8) was used in
g-i to calculate responsivity. Experimental absorption from Figure ??(top), thermal
properties from the previous chapter and average Seebeck coefficient of our materials
(above) was used in j-l to calculated responsivity.

next section) scales up with the detector area. Therefore, D* should
be very cautiously interpreted when applied to thermal IR detector.
In fact, D* tends to overestimate the performance of larger absorbing
area thermal detectors and underestimates the performance of smaller
ones.”

72
We therefore have deemed it appropriate to include only NEP and D calculations,
for which (Datskos and Lavrik, 2003), p. 351 remarks:
“The parameter NEP is generally sufficient to evaluate and compare
the performance of single (spot) IR detectors by predicting the minimum
power.”
By definition, D∗ = A · D, where D∗ is selective detectivity, A is detector area,
and D is detectivity. Detectivity is defined as D = N E1 P , where N E P, or noise
equivalent power, is defined as N E P = N SD
r , where r is responsivity, or the output
voltage over the input power, and NSD is the noise spectral density, with a Johnson
noise floor of N SD Johnson = 4Rk bT, where R is the structure resistance, k b is the
Boltzmann constant, and T is temperature in Kelvin. In the following calculations,
we ignore shot noise from emf-induced currents, and examine the minimum D that
can be found assuming only Johnson noise. This is a reasonable approximation,
given that our experimental N SD is very close to the Johnson noise limit (Figure
3.10 (top) and caption), indicating Johnson noise is the dominant noise source. The
temperature differences between the pad edges and wire centers should be similar
between a thermopiled structure (Figure 2.39), and our non-thermopiled structure
(Figure 2.23), indicating we can study r, D, and D∗ only as a function of resistance
and dimension.
Using the relations above, we find

r1 (l)
=q
D=
N SD
4R
n b

r1 (l),

(3.1)

for a guided mode resonance wire structure shown in Figure 2.23 with n wires in a
parallel (non-thermopile) configuration, where R is the resistance of a single wire
(so Rtotal = Rn ), and r1 (l) is the responsivity of a single wire. We assume that r1 (l),
for wires of fixed cross-sectional area and with fixed, uniform thermal conductivity,
is some function of the wire length (See Figure 2.37e, for example). Assuming the
majority of resistance comes from the wires (not the pads), and the cross-sectional
area of the wires are constant and resistivity is the same for all wires, R ∼ l, where l
is the length of the wire. Thus, D ∼ nl r1 (l). The total detector area of the structure
(counting the wire area only) is approximately A = nl p, where p is the pitch of the
wires. Then,

D ∼

nl p ·

npr1 (l)
r1 (l)
r1 (l) =
p1/2
p1/2

(3.2)

where W is structure width. W and r1 (l) are independent of one another and will not
cancel in general. Therefore, D* will not give a detection figure of merit independent
of area and is not a good metric for our structures. Other thermal detectors have
similar problems using D∗ as a figure of merit(Datskos and Lavrik, 2003).
For a thermopiled device of n wires, where the wires are in series, i.e. total resistance
Rtotal = nR, the detectivity (DT P ) is

nr1 (l)
nr1 (l)
rT P
=√
∼ √
DT P =
N SD
4nRk bT
nR

r1 (l) ∼

r1 (l),

(3.3)

which is the same as detectivity for a thermocouple (Equation 3.1), again neglecting
contributions of the pads to resistance. Since the detector area is the same for
thermopiled and non-thermopiled structures, then D∗ will also be the same.
As discussed in (Datskos and Lavrik, 2003), there exists other figures of merit for
thermal detectors. For instance, noise equivalent temperature difference (NETD)
is common in bolometric detectors, or other IR focal plane arrays. NETD is
defined as "the temperature of a target above (or below) the background temperature
that produces a signal in the detector equal to the rms detector noise." (Datskos
and Lavrik, 2003). This figure of merit is useful as room-temperature blackbody
radiation is noise in IR detectors, unlike in visible detectors. Another figure of merit
is the minimum resolvable temperature difference (MRTD) which characterizes the
spatial resolution and temperature sensitivity of IR detectors. Another figure of
merit, D**, normalizes the detectivity by the focal ratio.
3.6

Comparison with bolometers

There exists an enormous number of different light-detecting technologies, from
semiconductor photodiodes, to photomultiplier tubes, to Golay cells, to photoreceptor cells in our retina, just to list a small fraction of available detectors. Each
technology has its advantages and disadvantages, and when choosing a detector,
a number of factors must be taken into account, including the wavelengths and
bandwidth of interest, noise equivalent power or other relevant detection efficiency

74
figure of merit, the sensitivity range (the ability to withstand Watts of laser power
versus detect at the single photon level), the speed, the suitability of the detector
to its environment (radiation, temperature, mechanical vibrations), the lifetime of
the detector, the power consumption, the weight, the need for cooling, the cost or
availability of materials, the fabrication difficulties, etc... No detector is ideal in
all situations. For instance, silicon photodiodes work well for visible light at room
temperature at the nano or micro Watt level and are inexpensive, but do not work well
in the infrared regime below their bandgap, or at high temperatures including under
kWatt illumination, and will degrade under radiation. Photomultiplier tubes and
avalanche photodiodes can work well for detecting single photons, but conversely
can be permanently damaged when exposed to room light levels.
Without a specific application in mind, comparison between technologies is akin to
asking the question, "In a fight, would a shark or a polar bear win?", as clearly, the
environment of the fight matters. Fifty meters underwater, a shark is the clear winner.
On land, a polar bear would be victorious. Similarly, comparing a photomulitiplier
tube for single photon detection to a thermoelectric detector designed for kWatt
detection doesn’t make sense without a specific application in mind, as they are well
suited for completely different applications.
Choosing a similar technology, we will compare our thermoelectric nanophotonic
structure with a bolometer based on the criterion listed above. A bolometer is
composed of a material which has a well characterized resistance change with
temperature, so that when incident light impinges upon the material, the temperature rises and the corresponding resistance change is measured by monitoring the
voltage change of a small current passing through the resistor. Both bolometric
and thermoelectric light detectors are most sensitive when size is minimized and a
large thermal resistance exists between the hot region and reservoir (or "cold side"
of the thermoelectric element). The wavelength selectivity for both technologies
is determined in similar ways, via geometry, or primarily non-material-dependent
properties (in contrast to photodiodes, for instance, which have wavelength selectivity dictated by bandgap). Materials selection of the sensing region is important
in both cases, however. For more sensitive bolometers, materials with higher temperature coefficient of resistance is desired. Thermoelectrics require materials with
high Seebeck coefficients. Thermopiles operate most efficiently with two high zT
materials of opposite sign Seebeck coefficients, but this is not strictly necessary. A
high zT thermoelectric material could be paired with a low zT material with lower

75
resistance, such as a metal, which would decrease Johnson noise, potentially at the
expense of lower temperature drop due to the higher thermal conductivity of metals. Both bolometers and thermocouples additionally require materials with stable
characteristics to prevent drift in accuracy over time. The dominant noise source in
each application should be Johnson noise from the resistive character of the sensing
elements, as well as 1/f noise at low frequencies. Noise from amplifiers will also
contribute, and bolometers will have an additional noise source from the source of
injected current to extract voltage. In both cases, decreasing the temperature of the
device will decrease Johnson noise, but care must be taken that the sensing materials
still operate in lower temperature regimes. For instance, the Seebeck coefficient of
bismuth telluride drops at lower temperatures, and another thermoelectric material
must be used.
In terms of power requirements, from a theoretical perspective, thermoelectrics
should require less power. This is due to the thermoelectric materials producing
a voltage via the thermoelectric emf, which can be on the order of hundreds of
microvolts per Kelvin. This signal may or may not need to be amplified to be
sensed. Bolometers, on the other hand, require external work done in the form of
a supplied current in order to convert resistance changes into a measurable voltage
signal. Likely a small current must be used to prevent significant Joule heating, so
an amplifier also must be used to detect the signal. Therefore, a thermopile detector
should have lower voltage requirements. In reality, often thermopile detectors
have an internal silicon diode which monitors the room or reservoir temperature
(as thermoelectrics can only give information about temperature differences, not
absolute temperature). The silicon diode has a resistance change with temperature,
so a current must be supplied to monitor this as well, but in principle this does
not need to be continuously monitored, but checked at intervals, depending on the
required parameters.
Bolometers have been shown to detect single photons with near unity efficiency
(Lita, Miller, and Nam, 2008). In principle, assuming 100% light absorption, thermopiles should be able to sense single photons with some non-negligible efficiency,
dependent upon the efficiency of low temperature thermoelectric materials at the
low temperatures necessary for single photon detection via Johnson noise reduction.
Additionally, the thermoelectric effect relies only on the electron temperature, not
the phonon temperature, so at lower temperatures where electrons are more easily
decoupled from phonons, perhaps more efficient detection can be achieved, similar

76
to hot electron bolometers. More research is needed to determine the experimental
and theoretical sensitivity limit of thermoelectric nanophotonics.
3.7

Outlook, improving performance, and potential applications

The responsivity and detectivity of these structures could be increased through
thermopiling, optimizing the thermoelectric materials, measuring in vacuum to
eliminate convective loss, or suspending the wires to eliminate conductive losses to
the substrate. Focusing on material optimization alone as an example, responsivity
will increase with a higher Seebeck coefficient and lower thermal conductivity (k).
The noise floor can be decreased with lower resistivity (ρ). Therefore, detectivity
can be increased using a material with a larger thermoelectric figure of merit,
zT = (α2T)/ρk. For example, high room temperature zT n and p-type materials,
such as a p-type BiSbTe alloy (Poudel, Hao, Ma, Lan, Minnich, Yu, X. Yan, D.
Wang, Muto, Vashaee, et al., 2008a) with room temperature zT = 1.2, and ntype PbSeTe-based superlattice structure (T. Harman et al., 2002) with zT = 1.6
can be used. Alone, the increased Seebeck coefficient of these materials (∼25%
combined increase over our structure) would increase responsivity and detectivity by
roughly 25%. Using these state-of-the-art thermoelectric materials in our structure,
would lead to a factor of 1.7 and 22 overall increase in responsivity and detectivity,
respectively (see earlier section).
Thermopiling would further boost device responsivity, shown schematically in Figure 2.39 with simulated responsivity as a function of number of wires in the thermopile shown in Figure 2.37b. As we observed in our guided mode resonance structures, focusing light using a far field lens at all thermoelectric junctions maximized
responsivity (Figure 3.7). Light can also be focused onto a thermoelectric junction
by using plasmonic nanophotonic structures (Coppens et al., 2013a) designed to
maximize the electric field inside the thermoelectric material, as illustrated by the
plasmonic bow-tie antenna shown in Figures 2.17, 2.18, 2.19, 2.21. Guided mode
resonance structures are highly angle sensitive, whereas relatively angle-insensitive
performance can be achieved using e.g., ‘perfect absorber’ antenna structures (Alaee
et al., 2012; Aydin et al., 2011; K. Chen, Adato, and Altug, 2012; Liu et al., 2010b)
or split-ring resonators (Landy et al., 2008) that excite a thin thermoelectric junction
like those shown in Figures 2.1, 2.6, 2.7, 2.9 and Figures 2.11, 2.12, 2.15, respectively. The perfect absorbing structures and split-ring resonator absorbers also
exhibit 10-20 times lower noise equivalent power than the guided mode resonance
wire structures, as discussed.

77
While conventional photodiodes exhibit higher detectivity and response times in the
visible regime, resonant thermoelectric light detecting structures have two primary
advantageous features. First, thermoelectric resonant structures are band-gap insensitive and have shown potential as room temperature infrared light detectors, as an
alternative to super-cooled photodiodes or bolometers. Second, as we have shown,
resonant thermoelectric structures can have response times 100 times faster than
previously reported thermoelectric detectors made from high zT materials arising
from the smaller heat capacity of resonant thermoelectric structures resulting from
their large absorption cross-section (see previous section). Additionally, these structures combine responsivity with wavelength selectivity, enabling easier fabrication.
It may be possible to design very compact resonant thermoelectric structures that
exhibit sufficiently large thermal gradients over short distances (1-5 µm) such as the
one illustrated in Figure 2.19, which may make it possible to shrink thermoelectric
sensors to a scale more comparable to conventional camera pixel sizes of 10 µm2 .
Using nanophotonic designs to better focus the electric field on an as-small-aspossible section of the thermoelectric junction (Figure 2.18) could improve performance by maximizing the temperature difference between the hot and cold end of
the thermoelectric elements. Suspending the junction to minimize heat conducted
away by the substrate, combined with cooling the “cold” ends of the thermoelectric
materials by putting high thermal conductivity materials near the “cold” regions
(Figure 2.7), would increase responsivity by increasing the temperature difference
within the thermoelectric structures. Shrinking devices will additionally decrease
Johnson noise in resonant thermoelectric structures, thereby decreasing NEP. We
note that thermal design of parallel-connected thermoelectric junctions should minimize uneven junction heating, which can cause internal currents that waste input
energy. In general, careful consideration of matching the optical power absorbed
to the thermal impedance will be required to optimize thermoelectric nanophotonic
structure performance.

78
Chapter 4

NANOSCALE TEMPERATURE MEASUREMENTS WITH
NOISE THERMOMETRY AND THE THERMOELECTRIC
EFFECT - LOW NOISE DESIGN
4.1

Nanophotonic temperature measurement review

Measuring the temperature of nanostructures is important for a number of applications. There has been an increase in interest in determining the temperature of
nanophotonic structures under illumination, specifically those illuminated structures
which increase chemical reaction rates. It is desirable to elucidate the exact mechanisms for the rate increase, such as temperature rises versus hot electron generation
(Golubev et al., 2018). Additionally, measuring temperature in nanoscale structures
or gaps is of interest for thermophotovoltaic devices (Lenert et al., 2014), thermal
emitters (Yeng et al., 2012), or in near-field heat transfer studies and applications
(Yang, Narayanaswamy, and G. Chen, 2005). Other applications lie in measuring the heat capacity or thermal conductivity on the nanoscale, which has proven
challenging. Finally, one may want to study electronic property changes under
nanophotonic heating or illumination, which requires temperature measurements
(Schoenlein et al., 1987).
The challenge is finding a way to measure the temperature of nanophotonic structures without affecting their nanophotonic properties. A very large and diverse
number of creative techniques exist for measuring the temperature of nanophotonic
structures. Raman thermometry (e.g. (Davis et al., 1993)) determines the temperature of a structure by examining shifts in Raman peaks or by comparing the
magnitude of Stokes and anti-Stokes peaks. Thermoreflectance (e.g. (Antonelli
et al., 2006)) measurements use changes in reflection of light from a surface under
different thermal conditions to extract temperature. Fourier transform infrared microscopy (FTIR) uses a measurement of the blackbody spectrum of a material to
determine temperature. By measuring infrared wavelengths, however, the spot size
is diffraction limited and can be quite large, on the order of 10’s of microns. The
diffraction limit can be overcome by coupling an FTIR to an atomic force microscope
(AFM) to make Photothermal infrared (PTIR) measurements (Katzenmeyer et al.,
2015; Hammiche et al., 1999). A different AFM-based technique is scanning ther-

79
mal microscopy (SThM) (K. Kim et al., 2012), which has a thermocouple junction at
the tip of an AFM probe. This can be used for nanophotonic heating measurements
(Grajower et al., 2015). The drawback of this measurement technique is that the
probe is approximately 10 nm from the surface and can potentially interfere with the
photonics of the system. Another method is temperature-dependent photoluminescence measurements (Coppens et al., 2013b), where the substrate a nanostructure
sits on has temperature-dependent photoluminescence. Resistive thermometry is
also a temperature measurement method, whereby either the resistance of a material
is carefully measured with temperature (Herzog, Knight, and Natelson, 2014), or a
platinum resistive thermometer is placed on the material, and the resistance of this
is measured to extract temperature. A number of other interesting methods exist,
including but by no means limited to (Guillaume Baffou et al., 2012; Carlson, Khan,
and Richardson, 2011; G Baffou et al., 2009; Ward et al., 2011; Rosencwaig et al.,
1985).
The major drawback of many of these methods is the need for either an expensive
microscope or laser setup, and/or the need for excessive calibration for each device (as in the case of resistive thermometry). Additionally, the temperature must
be extrapolated from a measurement method which introduces a number of error
sources, such as laser power. We propose noise thermometry, a primary temperature measurement method, (see introduction for noise thermometry overview), in
conjunction with the thermoelectric effect as an alternative measurement technique
which only requires electrical leads to the nanophotonic device, does not involve
coupling and aligning an extra laser, has minimal calibration, and measures the
temperature of the electrons themselves.
4.2

Measurement plan

The aim of this project is to measure the temperature at the hot spot of a nanophotonic
device. Others have used noise thermometry in a graphene bolometer application
(Efetov et al., 2018), but they required a thermal model to determine the temperature
of the hot spot in the device relative to the edges (∆T), which increases uncertainty.
In our method, we directly measure the hot-spot absolute temperature. Figure
4.1 shows an SEM of a typical structure we extract the temperature of. In order to
determine the temperature at the "hot spot" of the device, or the junction of the wires,
two steps must be performed. First, the Seebeck coefficient of the Bi2 Te3 must be
determined. From an earlier chapter, we know that Seebeck coefficient of Bi2 Te3 can
vary from -228 to +160 µV/K depending on deposition conditions, so estimating the

80
Seebeck coefficient from the literature will almost certainly yield the wrong result,
possibly including the wrong sign. To measure the Seebeck coefficient, we measure
the noise temperature across each lead on the Bi2 Te3 device, and measure the voltage
between each set of leads. We know ∆V = α∆T, where α is the Seebeck coefficient,
so by measuring ∆V and ∆T, we can extract α. With α, we can proceed to step
2. Here, we measure the noise temperature at the lead set on the Bi2 Te3 closest to
the hot spot, and the voltage difference between these leads and the platinum leads
shown in the picture. Thus, with ∆V and α (we assume the Seebeck coefficient of
Pt is much smaller than that of Bi2 Te3 and can be neglected (Moore and Graves,
1973)), we can extract ∆Thot , the temperature difference between the center of the
wires and the Bi2 Te3 lead closest to the hot spot. The noise temperature gives us
the absolute temperature at the leads closest to the hot spot on the Bi2 Te3 side, and
this temperature plus ∆Thot gives the absolute temperature of the hot spot.

Figure 4.1: Measurement plan outline. The inset shows the wire junctions. The
laser broadly illuminates the wires, and the junction should be the hottest point in the
structure. In principle, because of the optical properties of this structure discussed
in earlier chapters, the entire structure could be illuminated by the laser.
This method gives the added benefit of being able to monitor the temperature
changes across the device due to thermoelectric currents in the device. In addition
to performing noise thermometry across the device leads, we also monitor the

81
resistance in the same locations using a bandpass filter and lock-in at frequencies
outside of the noise thermometry measurement window. This allows us to extract
more precisely the change in noise due to change in temperature by eliminating
noise changes due to changes in resistance.
This method requires collection of data from at least three leads at the same time.
The amplifing circuit will be discussed in the next chapter. The DAQ we used at first
was the National Instruments USB 4431 with 4 channels of simultaneous 24-bit 100
kSamples/s sampling. This DAQ was used for noise measurements in the previous
chapter; however, we found that with the slower data rate, to collect enough data
to get small enough error bars on our temperature measurements, we had to collect
continuously (with breaks to save data due to memory constraints) for about an hour.
During this time, the temperature in the room or on the circuit board could drift by
several Kelvin, hurting our temperature resolution. We upgraded to the 8-channel
National Instruments USB 6366, which has 2 MSamples/s sampling rate with 16-bit
resolution. Temperature resolution will be discussed in the next chapter.
The USB 6366 only has DC coupling, so AC coupling for each channel (in order
to get better resolution in case of DC offsets), composed of a high pass filter in a
Pomona box consisting of a 100 kΩ resistor and a 10 µF capacitor. Shorting the
DAQ, we measured a noise of approximately 38 nV/ Hz.
4.3

Fabrication

Fabrication of this device was similar to the device studied above. The same PECVD
and electron beam lithography systems were used. Instead of Sb2 Te3 , Pt was used
deposited with an electron beam evaporator, with a 15 nm Ni adhesion layer, as Ni
makes an ohmic contact to Bi2 Te3 . Bi2 Te3 was cosputtered this time, with 24 W RF
power for Bi and 40 W RF power for Te, sputtered at 3 mTorr in Ar. Leads to the
samples were composed of 15 nm of Ni (for Ohmic contact) and 120 nm of Au.
4.4

Design considerations for circuit board

In designing the circuit board, several important factors had to be considered.
First, the choice of operational amplifier depended on the source resistance, or
the resistance of the thermoelectric nanostructure we were measuring the noise
temperature of. Second, the thermal heating of the circuit board will cause drift
in the noise of the opamps over time. These two points will be discussed futher
in the following sections. Much of the following information is sourced from
(Motchenbacher and Connelly, 1992) and (Kay, 2012). An alternative method for

82
noise measurement of high resistance sources is to use an inductor (Fong and KC
Schwab, 2012; Efetov et al., 2018). We opted not to, as our source resistances were
low enough to not have significant rolloff from stray capacitances. Higher fidelity
noise measurements can be obtained via techniques NIST is exploring (e.g. (Qu
et al., 2013)).
Opamp considerations
There are two important noise figures in opamps: voltage noise, and current noise.
Voltage noise arrises from various resistances within the opamp, and the current
noise arises from shot noise in the transistors (see the the noise theory section in
the introduction for more general information). For a source impedance, Zs , which
includes the source resistance along with any coupling capacitors or inductors, the
equation for noise power relative to the ouput of the opamp (not including the data
acquisition device noise) is
E02 = 4k bTs Re[Zs ]
+ en2

R2
+1
R1

+ (i n |Zs |)

R2
+1
R1

+ 4k b R2TR2 + (i n |R2 |) + 4k b R1TR1

R2
R1

(4.1)

where the first term is the Johnson noise of the source impedance (note that this
term only depends on the real part of the source impedence) multiplied by the gain
of the non-inverting amplifier, the second term is the noise in the source impedance
generated by the current noise of the opamp (note that this term depends on the
magnitude of the impedance, which is not stated in most opamp data sheets), the
third term is the voltage noise of the opamp, the fourth term is the Johnson noise of
the larger gain resistor, the fifth term is from the noise current of the opamp through
the larger gain resistor, and the last term is the Johnson noise of the smaller gain
resistor, multiplied by the gain of the inverting input. Note that in the noise sources
from the gain resistors, the real part of the impedance is used, with the exception
of the fifth term which uses the magnitude of the impedance (useful if you have a
capacitor in parallel, for instance).
Clearly, the noise of a complicated circuit can be difficult to calculate, which is
why keeping a circuit simple is important. Using eq. 4.1, we can determine the
best opamp for our source resistance. Broadly, we want to have the noise from
our source, or the first term in Equation 4.1 dominate the other noise sources. If
our source resistance is large, en can be overcome, but i n begins to play a role,

83
and the second term can become quite large. If our source resistance is small, en
is more important, as the second term will be quite small. Unfortunately, there
is not, to our knowledge, an opamp with low enough voltage and current noise to
be used for all applications. For small source resistance, the AD797 and OPA211
from Analog Devices and Texas Instruments, respectively, have excellent voltage
noise, near 1 nV/ Hz. However, their current noises are relatively large, around
2 pA/ Hz, making them unsuitable for applications with source resistances larger
than approximately 2kΩ. The AD745 and OPA827, again from Analog Devices and
Texas Instruments, respectively, have fairly low voltage noises (2.9 nV/ Hz and 3.8
/ Hz, respectively, though not nearly as low as the AD797 and OPA211, but have
extremely low current noise (6.9 fA/ Hz and 2.2 fA/ Hz, respectively). Thus, for
low source resistances, the en of these opamps will dominate, but for high source
resistances, the en will be negligible, as will the i n . Because the resistances of our
sources are several thousand kΩ’s, we chose the OPA827. The added benefit of the
OPA827 over the AD745 are its well-characterized noise specs and its stability and
lack of need for external compensation capacitors. The AD745 has lower voltage
noise, but has a more complicated and harder to characterize noise profile because
of the added external capacitance.
While the ideal circuit configuration would be to have Rs grounded on one side
and connected directly to the non-inverting input on the other side, this was not
possible in our circuit. For one, directly grounding one side of Rs would short
circuit any thermoelectric voltage generated under illumination. Adding a large
resistor between Rs and ground (shown as Rg in Fig. 4.2) would solve the problem
of preserving generated thermoelectric voltage, however, this large resistor Johnson
noise would dominate noise from Rs . To correct for this, C2 is added parallel to
Rg in Fig. 4.2. This "looks" like a short circuit to ground at higher frequencies
where we measure noise, but acts like an open circuit at lower frequencies where we
chop the laser, allowing us to extract the thermoelectric voltage while still seeing
the noise. R3 and C1 in Fig. 4.2 AC couple to the non-inverting input opamp, which
serves two purposes. The first is to avoid DC offsets which drive the opamp close
to the rails (a +/- 5 V supply is used for thermal management reasons discussed in
the next section). The second is to avoid interference with the other opamps also
connected to the thermoelectric device, which we observed early on in the circuit
design. At high frequencies, the real part of the Thevanin equivalent impedance at
the non-inverting input of the opamp comes primarily from Rs , so Rg contributes a
negligible amount to the noise. The equivalent impedance of our source, shown in

84
Fig. 4.2 in the red box, can be written in complex form as

Zs = 1

R3 + −

(4.2)

2πC1 f + 1 +2iπC f +Rs
Rg

where f is frequency in Hertz and the resistance and capacitance values are defined
in Fig. 4.2. In our final design, Rg = 10 MΩ, C2 = 100 nF, C1 = 1 nF, R3 = 1 MΩ,
R1 = 33.2 Ω, R2 = 300 Ω, and Rs varies around 1-5 kΩ.

Figure 4.2: First stage of the circuit. R
A plot of the real part and the magnitude of Zs is shown in Fig. 4.2, for Rs = 3.3 kΩ.
At low frequencies, C1 acts like an effective open circuit, so the only current path
available is through R3 , which is 1 MΩ. At high frequencies, past the high-pass filter
cutoff, the capacitors act like shorts so Rs is the primary source of impedance. Thus,
past approximately 30 kHz, all of the Johnson noise will be from our thermoelectric
structure, and not the circuit components.
We need to extract both the high frequency data for temperature measurements
and the low frequency data for thermoelectric voltage measurements. This circuit
with the given values allows us to get information for both. Data analysis will be
discussed in detail in the next section, but for a 50 Hz square wave with an amplitude
(1/2 V pp ) of 1 V, for a source resistor of 3300 Ω, we expect the signal at the input of
the first opamp to be that shown in Figure 4.4.
Returning to the comparison between various opamp types, Fig. 4.5 compares the
AD797 (top) with the OPA827 (bottom) for Rs = 3300 Ω. In these plots and the

Figure 4.3: Magnitude of (blue) and real part of (red) the impedance at the noninverting input of the 1st stage of the opamp shown in Fig. 4.2.

following plots, we ignore any amplifier gain rolloff at higher frequencies (at 10x
this will be fairly small for the two studied opamps), 1/f noise (should be negligible
at the high frequencies we are concerned with) and also the noise of the DAQ. The
DAQ noise we observed was generally on the order of 10’s of nV/ Hz, so a 10x
gain would not generally be sufficient to overcome this noise source. If gain is
raised to 100x, the amplifier rolloff at higher frequencies becomes significant for
these two opamps. Therefore, the best option is to use a 2-stage opamp system, both
with 10x gain, or whatever gain allows sufficient bandwidth for the thermometry
measurements. The OPA827 has higher voltage noise and lower current noise,
making it a better first stage amplifier for our system, while the AD797 has low
voltage noise and higher current noise, making it a better second stage opamp for
our system. We measure the noise generally above 100 kHz in our circuit, well
past where the capacitors contribute to the impedance of the source. Note that
the AD797 has not been tested for stability using the given gain resistors, though
the data sheet does indicate it should be stable. From experience working with
this opamp, pin 8, the decompensation and distortion neutralization pin, can be
critical for performance. These plots represent theory calculated using values from
the data sheet, not measurement, though the OPA827 is characterized further in
this section experimentally. In general, commercial opamps are used for signal
measurements, not noise measurements. Typically, filters are used to cut out noise
in the frequencies far from the signal of interest, so much effort is not spent on
creating an opamp of ideal noise characteristics at a large frequency band. We have

Figure 4.4: Spice simulated voltage and phase versus time (top) and frequency
(bottom) seen at the input of the opamp. The signal is a square wave at 55 Hz with
1 V amplitude generated across the source resistor, which is 3300 Ω.

found that opamps with external compensation pins are more difficult to stabilize
across a broad frequency range and get close to ideal noise conditions. Opamps,
like the OPA827, require minimal external compensation tweaking as they have
no compensation pins to tweak, and are therefore self-compensated and behave
generally well. Thus, we used the OPA827 for both the first and second stage opamp
in our circuit, as the noise variation from opamp to opamp seemed to be be much
smaller than the other tested opamps, and we found it easier to work with, though
the AD797 would be the better choice if an additional iteration of the circuit were
to be made.
One important point in Fig. 4.5 is that while the current noise power is a negligible

Figure 4.5: Comparison between the power spectral density (PSD) of AD797 (top)
and OPA827 (bottom) for an Rs of 3300 Ω. E2s is the noise power density from Rs ,
E2total is the sum of all noise power sources, E2n1 is the voltage noise power of the
opamp, E2i1 is the current noise power of the opamp acting on |Zs |, E R 21 is the voltage
noise power of R1 , E R 22 is the voltage noise power of R2 , and E R 22i is the current noise
power of the opamp current noise acting on R2 (ignoring any stray capacitances that
may contribute).

contribution to the total noise for the OPA827, it is nearly equal to the source
voltage noise in the AD797. Therefore, the OPA827 is a much better choice for
this resistance, as the source resistance makes up the largest component of the total

88
noise. This effect is exaggerated further if the source resistance is increased to 10
kΩ, as shown in Fig. 4.6. In this case, for the AD797 (top of Fig. 4.6, the opamp
current noise power is clearly the dominant noise source, compared to the OPA827
(bottom of Fig. 4.6), where the current noise power is orders of magnitude lower
than the sample noise power.
Conversely, if our source resistance is smaller, the AD797 is a better choice. Figure
4.7 shows the power spectral density contributions for a source resistance of 500
Ω as measured by the AD797 (top) and the OPA827 (bottom). The opamp voltage
noise power is higher than the noise power of the source resistance, so this opamp
would not be suitable for this noise measurement. The AD797, despite having 3
orders of magnitude higher opamp current noise than the OPA827, has low enough
Johnson current noise above about 200 kHz to render this opamp usable for this
noise measurement application. Furthermore, increasing the values of the coupling
capacitors could reduce the minimum measurable frequency in this application with
the AD797.
In JFET amplifiers (JFET transistors are found in most low-noise amplifiers sold
today), the current noise doubles for every 10-20 K temperature rise in the circuit
(see any JFET input opamp Data Sheet plot of input bias current versus T, such as the
OPA827). Therefore, it is important to first thermally manage the circuit, including
using the lowest possible power supply to the opamp, as much power dissipation
happens at or near the opamp. An easier and likely more effective solution, as heatsinking opamps can be difficult or not possible in small spaces, one should choose
a source resistance and opamp for which if the current noise doubles or triples, the
output noise will be unaffected. Figure 4.8 shows the voltage noise produced by
varying current noise for a number of different source resistance values (noise is
relative to output, and for a 2-stage circuit, which will be discussed shortly). The
OPA827, with 2.2 fA/ Hz current noise, would have a negligible contribution to the
overall noise of the circuit if multiplied by a factor of 10, which makes it ideal for use
in applications where the environment temperature cannot be carefully controlled.
Figures 4.9,4.10 show how the output noise changes as a function of current noise
and source resistor.
One important note is that instrumentation amplifiers must be very carefully selected
if used in noise measurements. Most have very high current noise, stemming from
internal input bias current compensation. While the input bias current of these
inamps can be small, there are sufficient internal currents carefully balancing these

Figure 4.6: Comparison between the power spectral density (PSD) of AD797 (top)
and OPA827 (bottom) for an Rs of 10 kΩ. E2s is the noise power density from Rs ,
E2total is the sum of all noise power sources, E2n1 is the voltage noise power of the
opamp, E2i1 is the current noise power of the opamp acting on |Zs |, E R 21 is the voltage
noise power of R1 , E R 22 is the voltage noise power of R2 , and E R 22i is the current noise
power of the opamp current noise acting on R2 (ignoring any stray capacitances that
may contribute).
output currents to minimize DC offset, that generate excessive current noise.
The source resistances across our thermoelectric devices are controllable via the

Figure 4.7: Comparison between the power spectral density (PSD) of AD797 (top)
and OPA827 (bottom) for an Rs of 500 Ω. E2s is the noise power density from Rs ,
E2total is the sum of all noise power sources, E2n1 is the voltage noise power of the
opamp, E2i1 is the current noise power of the opamp acting on |Zs |, E R 21 is the voltage
noise power of R1 , E R 22 is the voltage noise power of R2 , and E R 22i is the current noise
power of the opamp current noise acting on R2 (ignoring any stray capacitances that
may contribute).

material stoichiometry and thickness. While it would be most desirable to use a
low source resistance material, such as a metal like chromium, which is on the
lower end of thermal conductivity for metals and has a decent Seebeck coefficient,

Figure 4.8: Variation of output noise spectral density with current noise on a 2-stage
OPA827 opamp circuit for Rs = 3300Ω.

Figure 4.9: Variation of output noise spectral density with source resistance and
current noise on a OPA827 2-stage opamp circuit.

these materials have drawbacks. Because the size of the device is so small, the
resistance of the gold leads to the device (on the order of 50-100 Ω depending on
lead geometry) will dominated the resistance of the device, which will be less than
10 Ω for a metal. Having a resistance that is too high, such as 100 kΩ, is also
undesirable, due to the stray capacitances on the PCB board, as we have discovered.
About 5 pF is a typical stray capacitance value for our signal leads on our PCB
board without excess engineering to account for it, which when combined with a
large resistance, creates a low-pass filter, and can cut-off noise signal in the higher

Figure 4.10: Variation of output noise spectral density with source resistance and
current noise on a OPA827 2-stage opamp circuit.

frequencies. By controlling the stoichiometry and thicknesses of our materials, we
aimed for a resistance of around 4 kΩ, which yielded a theoretical Johnson noise
above the opamp voltage noise of the OPA827, but was not a large enough resistance
to be affected at high frequencies by stray capacitances.
As stated previously, a 2-stage opamp system is desirable in our case, as gain higher
than 10x is needed to overcome the DAQ noise, but the low-noise opamps available
to us do not have the gain-bandwidth product necessary to have greater than 10x
gain without significant amplifier gain rolloff at higher frequencies. Figure 4.11
shows our 2-stage amplifier circuit. It is critical for the first stage amplifier to have
low-noise characteristics, but the second stage amplifier has more relaxed noise
requirements, as its contribution to the output noise will be less by a factor of
the first stage gain than the first stage amplifier noise. Additionally, while a nonnegligible source resistance generates extra noise from the output current noise of
the first stage amplifier, the output impedance of most amplifiers is very low, so the
current noise of the second stage opamp will not generate much extra noise at the
output. The best amplifier to use at the second stage would be an amplifier with
a very large gain bandwidth product, such as the AD829, with a 600 - 750 MHz
gain bandwidth product. This would allow for a very large gain on the second stage.
In our experience, the AD829 was found to have high variation in noise between
different amplifiers, and was tricky to work with initially. For this reason, we used
the OPA827 as the second stage amplifier, although in a second generation device,
optimizing the compensation and layout of the OPA827 would be worth the time

93
spent. In Figure 4.11, R1 = R4 and R2 = R5 , which are given above.

Figure 4.11: 2-stage circuit.
Figure 4.12 shows the power spectral density referred to the output of a 2-stage
opamp circuit shown in Figure 4.11, with the values of the circuit elements defined
previously, and R1 = R4 and R2 = R5 , giving each stage approximately 10x gain
each. The total output noise of this circuit is given by

E02 =

2
2
R2
2 R2
+ 1 + (i n1 |Zs |)
+1
4k bTs Re[Zs ]
R1
R1
2 #
2
2
R2
R5
+ 1 + 4k b R2TR2 + (i n1 |R2 |) + 4k b R1TR1
+1
+ en1
R1
R1
R4
2
2
R5
2 R5
+ 4k bT1 Re[Z1 ]
+ 1 + (i n2 |Z1 |)
+1
R4
R4
2
2
R5
R5
+ en2
+ 1 + 4k b R2TR5 + (i n2 |R5 |) + 4k b R1TR4
+ N SD2DAQ,
R4
R4
(4.3)

where Z1 is the output impedance of the first stage opamp, i nx and enx refer to the
current noise and voltage noise, respectively, for the first stage opamp (x = 1) and
second stage opamp (x = 2), and N SDDAQ is the Johnson noise of the DAQ input.
It is important to keep in mind that while it is useful to examine each noise thermometry circuit on its own, the entire circuit is shown in Figure 4.13. The resistor
between each noise thermometry stage, we assume to be much larger than the Rs for
each 2-stage opamp circuit, so the circuit should not interfere.

Figure 4.12: Power spectral density versus frequency for each individual contribution to the output noise. Here, Rs is 3.3 kΩ. E2s is the noise power density from Rs ,
E2total is the sum of all noise power sources, E2n1 is the voltage noise power of the first
stage opamp, E2i1 is the current noise power of the first stage opamp acting on |Zs |,
E R 21 is the voltage noise power of R1 , E R 22 is the voltage noise power of R2 , E R 22i is
the current noise power of the first stage opamp current noise acting on R2 (ignoring
any stray capacitances that may contribute), E2Zout is the noise power density from
the output impedance of the OPA827 (20 Ω), E2n2 is the voltage noise power of the
second stage opamp, E2i2 is the current noise power of the second stage opamp acting
on |Zout | (the output impendance of the first stage opamp), E R 24 is the voltage noise
power of R4 , E R 25 is the voltage noise power of R5 , E R 25i is the current noise power
of the opamp current noise acting on R5 (ignoring any stray capacitances that may
contribute), and E2DAQ is the noise from the DAQ we used in our measurements
(National Instruments USB 6366).

A note on capacitors: the knowledge in the literature or internet about noise levels
of various capacitor types is varied and sometimes contradictory. We observed
no noticeable difference when using mica versus ceramic (low and high quality)
versus polypropylene film in our circuit, at least at the higher frequencies we were
interested in. Presence of excess flux on the circuit board also did not contribute
noticeably to noise levels. Shielding, however, is extremely important to avoid high
frequency signals from the environment. A grounded metal box surrounding the
circuit sufficed.
Figure 4.14 shows the raw voltage data collected from the DAQ under chopped
illumination of a thermoelectric sample. Note its similarity in shape to Figure 4.4.

Figure 4.13: Total noise thermometry circuit.

Figure 4.14: Measurement of thermoelectric sample under 639 nm, 43 Hz chopped
illumination with 590 µW of absorbed power. The thermoelectric voltage of the
bismuth telluride close to the wires is shown in turquoise, the thermoelectric voltage
of the bismuth telluride far from the wires is shown in red, the thermoelectric voltage
of the platinum close to the wires is shown in green, and the chopper voltage signal
(smoothed digitally with a low pass filter) is shown in blue.

Figure 4.15 shows data collected from our circuit, plotted with two different axes.
There is a small oscillation at lower frequencies, but as we are measuring the
frequencies between 100,000 and 900,000 Hz, this is tolerable.
Measuring resistance changes across leads
In order to account for noise changes due to resistance changes, we precisely measured resistance in our samples by inserting a small current at a certain frequency
and monitoring the resulting change in magnitude of the voltage it produced during
the course of noise measurements. Thermoelectric voltage measurements were performed below 100 Hz and noise measurements were performed above 100 kHz, so
the resistance measurements were performed between 18 - 22 kHz, as at these frequencies the source resistance noise dominates (Figure 4.12). Two SRS 830 lock-in
amplifiers were used as signal generators to produce a small current (nanoamps) at
the desired frequencies. The insertion point of this signal is shown in Figure 4.16,
along with the resistor and capacitor values in the bandpass filter. In designing
the filter, an emphasis was put on decreasing the lock-in noise contribution in the
100 kHz and higher frequency regime, which meant that the bandpass filter is not
centered on the resistance measurement frequencies of interest, but allows through

Figure 4.15: Measurement from DAQ, including gain and circuit noise (plotted with
different axis), with Rs = 3.3kΩ metal film resistor.
enough signal for our purposes. This filter could be optimized further.
Figure 4.17 shows the resulting NSD spectrum on our thermoelectric sample with
a resistance measurement signal. The change in magnitude of the resistance measurement with power is amplified by the opamps and analyzed well the noise data
as described in a later chapter.

Figure 4.16: Second-order bandpass filter for injecting resistance-measuring current
while filtering high and low frequency noise produced by the signal generator (the
sine reference output of a SRS 830 lock-in amplifier, in this case). Here, Ra is 1 kΩ,
and Rb , Rc , Rd , and Re are 1 MΩ each. Ca , Cb , Cc , and Cd are 100 nF, 1 nF, 0.1 nF,
and 1 nF, respectively. The other resistor and capacitor values are given previously
in the text.
Thermal design of board
As discussed earlier, temperature increases in opamps can increase current noise
significantly above datasheet values, which is undesirable for accurate noise thermometry measurements. We discovered this using an early version of our circuit
which used AD8421 instrumentation amplifiers and a ±15V power supply, a large
amount of heat was being dissapated in the inamps, due to the large power supply
and small signal outputs. Figure 4.18 shows an image of our circuit next to a IR
camera image, illustrating the temperature rise of about 10 K in the inamps after only
5 minutes of operation. At this point, we were using the NI-USB 4431 with 100 kHz

Figure 4.17: NSD spectrum with resistance signal.

data collection rate, so measurements took hours, which would have significantly
heated the circuit board and the sample, and also increase the inamp current noise
output.
One data set taken with this circuit is shown in Figure 4.19. Laser power was swept
from high to low power over the course of several hours, and the overall drift in
temperature extracted from the noise was about 60 K, which we attribute to addition
of current noise over time due to inamp heating. We later discovered, as we will
show, that the resistance changes may also have accounted for some of the drift in
noise with time, however, with this circuit, we did observe increased noise over time
scanning from both high to low and low to high laser powers, which brings us to
conclude that the circuit heating did have a non-negligible effect on this noise rise.
A number of changes were made to correct these issues. First, as the signals we
were looking at were small, we replaced the ±15 V power supply with ±5 V power
supplies, the minimum value for most amplifiers. This would decrease the power
dissipated across the amplifier. Because we lowered the power supply rails, we
had to decrease the AC coupling resistor in front of the inamp from 10 MΩ to
1 MΩ, as the input bias current offsets sometimes gave a DC input voltage that
exceeded the rails when amplified. This had negligible effect on noise. As we
were using the lowest current noise inamp available (200 pA/ Hz, we switched to
the opamp configuration shown in the preceeding section, as there existed opamps

100

Figure 4.18: Heat generation in inamps in early circuit after 5 minutes of operation.
Hot spots near the bottom and left side of the image were due to reflections off metal
surfaces, and did not correspond to actual temperatures (these "hot spots" disappeared when pictured from different angles, while the inamp hot spots remained).

with current noise so low that a 50 K increase in circuit board temperature would
still have no effect on the circuit board. Finally, we reduced the 9V batteries from
4 to 2 to decrease power dissipation across the voltage regulators, and moved the
batteries and voltage regulators to a separate box (Figure 4.20) so as to not affect
the circuit board temperature. We also installed 2 surface mount RTDs (resistive
thermometers) to monitor the circuit temperature, and connected these to a DAQ
(OM-USB-TEMP-AI from Omega) to collect circuit board temperature data during
noise collection. We installed DC fans to further cool the circuit, but these added
peaks to the noise spectra, so we eliminated them. We also redesigned the circuit to
keep the opamps as far as possible from the sample. All of these changes sufficed
to fix the heating problems.
The final circuit is shown in Figure 4.21.

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Figure 4.19: NSD as a function of absorbed laser power in thermoelectric device.
This data collection took several hours, and the laser power was swept from high
power to low power. The approximate temperature drift in the measurement corresponds to about 60 K, which we attribute to an increase in current noise from the
inamp due to heating, as dicussed in the text.

102

Figure 4.20: Separate box for batteries.

103

Figure 4.21: Final circuit.

104
Chapter 5

NOISE THERMOMETRY RESULTS
5.1

Data analysis

The data analysis process was generally split into two pieces. The first piece is determining the noise temperature under chopped illumination or without illumination,
and the second piece, if under chopped illumination, was to determine the magnitude of the thermoelectric voltage produced. Both will be discussed in subsequent
sections. Because of the large volume of data involved (billions of data points), C
was used with the FFTW (Frigo and S. Johnson, 2005) libraries, which was several
orders of magnitude faster than Python scripts (many hours with Python versus 20
minutes with C). Both C and Python scripts can be found in the appendices.
Determining noise temperature with Welch’s method
To determine the temperature from our measurements, the process is as follows (the
C analysis script can be found in the appendix A):
1. Read large binary file for each channel plus chopper signal.
2. Separate the data into "on" and "off" data using the chopper signal (this step
is skipped in a control run with no laser signal).
3. Generate a Welch’s periodogram from the "on" and "off" data separately.
4. Average together the high frequency data to get the power spectral density
(PSD).
5. Take the square root of the averaged data to get the noise spectral density
(NSD).
Note that NSD must be averaged in quadrature (i.e. the mean of the PSD is taken)
to avoid underestimating the noise. Figure 5.1 outlines Steps 2-4.
Step 2 can be accomplished with a computer-generated chopper signal if proper
synchronization between instruments and computers can be achieved, otherwise the
chopper and signal will drift relative to one another. In our case, we read in the

105

Figure 5.1: Sequence for analyzing noise data, as described in Steps 2-4 in the
text. Note that the signal wave is not a square wave, as is depicted here in a) for
simplicity. See Figure 4.14 for measured signal shape. b) shows the deletion of
transition regions from the chopper and signal traces. c) depicts joining the "light
on" and "light off" data into long data arrays. d) shows a cartoon PSD trace from the
"light on" and "light off" data showing the expected increase in noise for the "light
on" data.
chopper signal from the waveform generator or chopper from the DAQ and lowpass
filtered the signal in order to find transition regions (the sharp on/off points in the
chopper signal). The non-flat regions (when currents were still flowing) to the left
and right of the sharp spike were deleted, as the Nyquist noise theorem only applies
to systems in equilibrium. This non-equilibrium noise data could be interesting to
examine in the future, but was not the focus of the present study. The "on" data was
put into one data set, and the "off" data was put into a different data set. These were
plotted to make sure they looked relatively flat to ensure the transition regions had
been cut effectively.
The "on" and "off" data then went through a process to generate a Welch Periodogram
(Welch, 1967). A discussion of different methods of taking periodograms can be
found in (Stoica, Moses, et al., 2005). Welch’s method is one of the most widely
used. The process is as follows:
1. Partition the data into multiple segments of a fixed length. The segments have
a 50% overlap (this can be adjusted).
2. Each segment is multiplied by a window function (we chose to use a Blackman-

106
Harris window function, but found that the window function makes no noticeable difference in the results with the number of data points we had).
The windowing functions are less useful for measuring the power spectrum of
white noise, but must be selected carefully for measuring signal peaks (Harris,
1978).
3. The window-multiplied segment is then FFTed and properly normalized for
the windowing segment used (1/(sample rate x sum(window2 )).
4. All of the FFTed segments are first squared then averaged together, and then
the high frequency ranges are averaged (for us, between 100 and 900 kHz),
and divided by the number of frequency points to get the PSD. The square
root of the PSD at this point is the NSD.
5. The theoretical noise contributions from the circuit components can be subtracted from the PSD (always subtract power, not voltage) to get the noise
attributable to the source resistance.
6. If an offset noise from unknown sources is present, this is subtracted from the
PSD at this point, and reported.
Note that different programs normalize FFTs differently. For instance, FFTW
normalizes its FFTs differently than Mathematica. Additionally, if a 1-sided versus
2-sided FFT is generated by the program, the result must be multiplied by a factor
of 2. We strongly suggest generating a test file with generated noise to double
check code, as Welch’s method, implemented in slightly different ways, can be used
to measure PSD with units in V 2 /Hz or just V 2 . The Blackman-Harris window
function we used is

2π
2π
2π
n + a2 cos
2n − a3 cos
3n ,
w(n) = a0 − a1 cos

(5.1)

where n = 0, 1, 2, ..., N − 1, where N is the number of data points per segment,
a0 = 0.35875, a1 = 0.48829, a2 = 0.14128, and a3 = 0.01168.
C and the FFTW libraries (Frigo and S. Johnson, 2005) are ideal for this sort of data
analysis as the use of pointers makes the process less memory intensive. The FFTW
allows you to make plans for an array of a certain length accessed by a pointer,
which can measure the fastest method for doing a transform. This array can then be

107
overwritten with each successive segment data for fast processing. Generally, for
one temperature data point, 500 million data points were taken and saved, and C
was far superior at manipulating such large amounts of data.
Determining thermoelectric voltage
The following process was used to determine the amplitude of the thermoelectric
voltage produced in our devices:
1. Read in the binary file
2. FFT the data and find the magnitude of the signal at the chopper frequency
3. Multiply the magnitude of this signal by the gain of the circuit at this frequency
and the sine - to - square wave conversion factor to get the thermoelectric
voltage amplitude.
The FFT was taken using either the entire length of the data set (1e8 data points
per data set) or 226 = 67108864 data points if speed is an issue (FFTs are faster
if the number of data points is a power of 2). This number of points gave a very
high frequency resolution, and the magnitude of the FFT signal was taken to be the
maximum point within about 5 or 10 Hz of the chopper signal, as the chopper was
often a fraction of a Hz off the set value, attributable to the precision of the chopper.
To ensure the capacitors in the circuit had been charged and equilibrium temperature
was reached, the device was chopped slowly, at around 43 Hz. As we can see from
Figure 4.4, the AC coupling to the sample attenuates low frequency signals below
approximately 500 Hz. Therefore, finding the thermoelectric voltage amplitude is
not entirely straightforward to extract, which we will describe presently.
We start with several assumptions. First, that the incident laser light is approximately
a perfect square wave, and the induced heating in the circuit is approximately a perfect
square wave. From (Mauser et al., 2017), (or an earlier chapter), a similar device has
a response time of about 3 kHz, which is much greater than 43 Hz, so this should
be a safe assumption. Next, we know that any piecewise smooth periodic function
can be written as a Fourier series. A function is piecewise smooth on some interval
if the interval can be broken up into pieces such that in each piece the function is
continuous and its derivative is continuous (Haberman, 2004). A finite number of
jump discontinuities are allowed. From (Haberman, 2004), the Fourier series of
f (x) over the interval −L ≤ x ≤ L is defined to be the infinite series

108

nπx Õ
nπx
Fourier series = a0 +
an cos
bn sin
∼ f (x),
n=1
n=1

(5.2)

where the Fourier coefficients are given by

a0 =
2L

an =

∫ L

bn =

∫ L

∫ L
f (x)dx,

nπx
dx,

(5.4)

nπx
dx.

(5.5)

f (x)cos
−L

f (x)sin
−L

(5.3)

−L

We use ∼ f (x) as the infinite Fourier series doesn’t always converge and if it does,
it doesn’t always converge to f (x). For most physical problems, the Fourier series
should converge to f (x). Conveniently, this is the case for a square wave with a 50%
duty cycle, g(t), which has the following Fourier series

4A Õ 1
sinnωt,
g(t) =
π n=1,3,5,... n

(5.6)

where the wavelength is 2L and A is the amplitude (A = 12 Vpp ).
Further, from (Haberman, 2004), If f (x) is piecewise smooth, then the Fourier series
of a continuous function f (x) can be differentiated term by term if f (−L) = f (L),
which is true in the case of a periodic square wave. This is relevant, as our ACcoupled circuit is called an "RC differentiator circuit," where the voltage seen by the
opamp across R3 in Figure 4.2 can be found by first using Kirchoff’s voltage law

Vin =

+ I R3,
C1

(5.7)

109
where Vin is the voltage (relative to ground) just to the left of C1 in Figure 4.2, and
Q is the charge on C1 . We can differentiate this to

dI
dVin
dt C1
dt

(5.8)

We then set Vin (t) = g(t) in Equation 5.6 above, but for convenience, change to comÍ
1 inωt
plex notation so Vin (t) = 2A
and I(t) = n=...−3,−1,1,3,... Inn einωt .
n=...−3,−1,1,3,... n e
iπ
We assume In is complex. Then,

n=...−3,−1,1,3,...

Iniωeinωt +

In inωt 2A
ωeinωt .
C1 n=...−3,−1,1,3,... n
π n=...−3,−1,1,3,...
(5.9)

Due to the orthogonality of the exponents, we can drop the sums, and cancel the
exponents, leaving

2A
R3iω +
ω.
C1 n

(5.10)

Rearranging,

In =

2A
π ω

R3iω + C11 n

(5.11)

Thus, Voutn = Im[In R3 ] (since we only want the sine components), where Voutn is the
amplitude of the nth harmonic at the output, and is

Voutn =

2nωC1
A,
π(1 − (R3 ωC1 n)2 )

(5.12)

which shows that the amplitude of the input voltage can be theoretically determined
from the amplitude of the output voltage for each harmonic of the Fourier series,
if we solve for A. More succinctly, differentiation is a linear operation. Note that

110

Figure 5.2: Measured gain versus frequency on our 2-stage opamp circuit. Note the
similarity to theory in Figure 4.4.

Equation 5.9 actually has two infinite sums, due to differentiating a non-continuous
Fourier series. Clearly this is not physical, and a real function is continuous and not
a perfect square wave, so there will be a truncation of terms.
For greater accuracy, we measured Voutn as a function of frequency for fixed A to
determine this attenuation factor. We connected the external reference generator
of a lock-in (a sine wave) to a point in the circuit between Rs = 3300 Ω and C1 ,
and measured the output after both opamps with the lockin input. We swept the
frequency (see Appendix for code to connect to lockin), with results shown in Figure
5.2.
In our analysis code, we multiply the magnitude of the measured FFT signal at the
chopper frequency by this gain, and then by 2π
4 , to convert the sine wave amplitude
into a square wave amplitude (the 4 term). The factor of 2 is because we are
interested in the peak-to-peak amplitude, or 2A.
Determining resistance
The resistance changes were determined by measuring the magnitude of the voltage
peak from injection of a current between 18 and 21 kHz, as described in an earlier
section. To measure the magnitude of this peak, steps a) and b) depicted in Figure
5.1 were enacted on the data, and then each individual section of "on" and "off" data
was Fourier transformed and the magnitude at the frequency of interest was probed.
We did not proceed to step c) in Figure 5.1, as "squishing" together all of the on and

111
off data would create a wave discontinuous phase (Figure 5.3), the Fourier transform
of which would not give the desired amplitude. Fourier transforming each "on" and
"off" section of data individually gives less frequency resolution due to fewer data
points and therefore may underestimate the resistance peak, but this can be mitigated
by using no windowing function when Fourier transforming the data, which will
give poor frequency resolution, but good magnitude estimation. As we care more
about magnitude than frequency resolution, this is acceptable.

Figure 5.3: A sine wave with piece-wise discontinuous phase.

5.2

Noise thermometry circuit measurement verification

To test the noise thermometry circuit, we attached metal film resistors in place of
the sample. Metal film resistors have noise properties close to theoretical values.
The circuit was placed in a sealed styrofoam box (Figure 5.4) with a resistive heater
(Figure 5.5). A constant DC current was run through the resistive heater, and the
system was allowed to thermalize for about an hour, after which time 500 million
data points at 2 MS/s were collected from each of the 4 opamps systems in the
circuit. Then the DC current through the resistive heater was raised, and the system
thermalized for an hour, and the process was repeated. Figure 5.6 shows the inside
of the metal box in Figure 5.5.
To test the noise thermometry circuit, we first put 4 different 1% metal film resistors
in each of the four channels. In the first round of testing, we used 3.3 kΩ, 4.7 kΩ,
6.8 kΩ, and 10 kΩ each in a different channel. In the second, third, and fourth round
we rotated these resistors among each opamp. Figure 5.7 show 6.8 kΩ (top) and 10
kkΩ (bottom). Figure 5.8 shows 3.3 kΩ (top) and 4.7 kΩ (bottom).
In the next round, we used smaller resistances: 1.5 kΩ, 2.2 kΩ, 3.3 kΩ, and 4.7
kΩ. We redid the latter two resistances to examine the stability of our circuit after
about a week. Figure 5.9 shows 3.3 kΩ (top) and 4.7 kΩ (bottom), measured about
a week after the data taken in Figure 5.8. Figure 5.10 shows 1.5 kΩ (top) and 2.2
kΩ (bottom).

112

Figure 5.4: The styrofoam box is holding the noise thermometry circuit. The small
blue box with orange twisted shielded wires is connected to the surface mount RTDs
on the circuit is a DAQ from Omega described above for logging temperature data.
The power supply on the left is connected to the resistive heater in the box. The
instrumetn in the foreground is the NI-USB 6366 DAQ AC-coupled to BNC cables
connected to the output of the 2nd stage opamp circuit.

We notice several trends from the data. First, there is an offset from theory that is
apparent in the plots. This is attributable to unaccounted for noise sources, such
as errors in opamp data sheets. The slope of the theory and experiment is similar,
especially at higher resistances, leading us to believe our noise thermometry circuit
is capable of measuring temperature. Deviations from the theoretical slope could
be due to error in the 1% resistor values, as well as temperature measurement errors
from the on-board RTD. The RTD has accuracy of 0.15 K. The circuit was designed

113

Figure 5.5: The inside of the cooler. The left side shows the suspended resistive
cooler. The circuit is in the shielded metal box labeled "Boxy McBoxFace". The
green wire is a grounding wire for the shielding.

with channel 4 signal traces too close to the other on-board RTD (the one close
to the sample), so we were forced to leave it unhooked to not interfere with the
measurements. This RTD would have been a slightly more accurate measurement
of the resistor temperature, as it was several centimeters closer, in case the thermal
stability time of 1 hour was not long enough. Figure 5.11 shows an expanded view
of the measurements from one circuit.
System temperature and precision
The system temperature was determined using the following method. To approximate the system temperature, we used data from the previous section. We took the
PSD data from the output of the DAQ and divided out the gain. We then plotted a
best fit line to this data, and followed the data back to the point at which it crossed
the x-axis. The absolute value of this x-value is the system temperature. Figures
5.12, 5.13 shows this data for different resistor values.
The precision of the system was measured and compared to Dicke’s radiometry
equation, shown in Figure 5.14. The orange line is the prediction from Dicke’s
equation.

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Figure 5.6: The inside of the shielding box. The syrofoam divider is to further
thermally isolate the sample from opamp heating. The samples, on the right, are
metal film resistors. The cables connected to the circuit, from left to right, are as
follows: shielded SMA cable for -5V power supply from the voltage regular box;
the matching +5 V power supply; the orange cable is shielded twisted wires taking
4-wire RTD measurements from the SMT RTD; the next 4 cables are shielded BNC
cables carrying the opamp output to the USB 6366 DAQ; and the far right cable is
another SMT RTD cable.
5.3

Noise measurements under chopped illumination

The setup consists of either a broad-spectrum laser passed through a monocromator
to filter out the wavelength of interest or a diode laser, depending on the power
requirements. The light passes through a chopper and then a glass slide (∼10/90
beamsplitter), which is used to monitor the stability of the laser power with time.
The light passes through various alignment mirrors then through the sample to a
photodiode behind the sample, which monitors any shifts in the sample position
with time.
We first plot the PSD as a function of absorbed laser power for a 639 nm diode laser,
with electric field polarized perpendicular to the wires, and a chopper frequency
of 43 Hz. At this wavelength, the absorption in the wires was approximately 17%.

115

Figure 5.7: Test of the circuit with metal film resistors, with values of 6.8 kΩ (top)
and 10 kΩ (bottom). The channel refers to which op amp on the circuit did the
measurement. The legend shows the slope of a best fit line to the data.

Figure 5.15 shows PSD versus absorbed laser power assuming constant sample
resistance. Blue data points are when the light is blocked by the chopper, and red
data points are when the light is not blocked by the chopper. We see that the noise
appears to decrease when the light is off, which is unexpected, and can be explained

116

Figure 5.8: Test of the circuit with metal film resistors, with values of 3.3 kΩ (top)
and 4.7 kΩ (bottom). The channel refers to which op amp on the circuit did the
measurement. The legend shows the slope of a best fit line to the data.

either by decreasing temperature or decreasing resistance of the sample.
The resistance of the sample as a function of absorbed power is plotted in Figure
5.16. The dotted line and greed arrow indicate the order in which measurements were

117

Figure 5.9: Test of the circuit with metal film resistors, with values of 3.3 kΩ (top)
and 4.7 kΩ (bottom). These measurements were taken about a week after those
shown in Figure 5.8. The channel refers to which op amp on the circuit did the
measurement. The legend shows the slope of a best fit line to the data.

taken. This plot offers two important insights: first, that the resistance decreases with
increasing power absorption, and second, the resistance does not return to its original
value after illumination. Addressing the first point, we had expected that because the

118

Figure 5.10: Test of the circuit with metal film resistors, with values of 1.5 kΩ (top)
and 2.2 kΩ (bottom). The channel refers to which op amp on the circuit did the
measurement. The legend shows the slope of a best fit line to the data.
carrier concentration of bismuth telluride is so high (1019 −1021 /cm3 ), the resistance
would increase with temperature (assuming that illuminating the sample heats it),
similar to a metal. Instead, bismuth telluride acts more like a semi-conductor, in that

119

Figure 5.11: Plot of only channel 2 in Figure 5.8 (bottom), with resistance value of
4.7 kΩ. The dotted line is the best fit line.
temperature (or potentially photo-excited electrons) decrease resistance. As for the
second point, the resistance of bismuth telluride does not appear stable with time.
This could be due to the cyclic annealing of the material under chopped illumination
altering the crystal structure. This indicates that bismuth telluride would not be a
good candidate for resistive thermometry, where the temperature of a material is
determined by change in resistance.
Putting the noise and resistance data together, we find the temperature as a function
of absorbed power, shown in Figure 5.17, across two different leads. We assumed
a constant offset of noise (subtracted in quadrature), found by matching the onboard RTD temperature with the no-illumination data points, incorporating the
resistance corrections. For the leads closer to the wires or point of illumination,
the subtracted offset was 3.44 nV/ Hz, and for the leads farther from the wires or
point of illumination, the offset was 3.14 nV/ Hz. Incorporating resistance into

120

Figure 5.12: Plot of system temperature, with resistors of values 3.3 kΩ (top)
and 4.7 kΩ (bottom). The channel refers to which op amp on the circuit did the
measurement.
the noise measurements, we find that the temperature with the light off drops to
room temperature for every power, indicating that the thermal time constant of the
device is much faster than our chopping speed, which is expected based on results
from our hyperspectral light detecting structure from a previous chapter. Further,

121

Figure 5.13: Plot of system temperature, with resistors of values 6.8 kΩ (top)
and 10 kΩ (bottom). The channel refers to which op amp on the circuit did the
measurement.
the temperature increases approximately linearly with increasing absorbed power,
and reaches as much as 70 K above the no illumination condition, for the maximum
power we used.
We can calculate the Seebeck coefficient for our bismuth telluride by dividing the

122

Figure 5.14: Precision of temperature measurements as a function of sampling
time, for a 2 MHz sampling rate. The theoretical line (orange) is given by Dicke’s
equation, and the measurements (blue) are for a 1500 Ω source resistor. δT is the
standard deviation of 10 measurements of NSD for the given measurement time,
without incorporating a temperature offset correction. The on-board RTD reported
a temperature of 298.2 K for the measurements.

measured thermoelectric voltage (the voltage difference between each lead pair when
the light is on) by the temperature difference between each lead pair when the light
is on. The results are shown in Figure 5.18 (left). The bismuth telluride has a
Seebeck coefficient around -140 µV/K, making it n-type. We note that the Seebeck
coefficient is relatively constant with temperature, as is expected over our relatively
small temperature range. We find a Seebeck coefficient of zero at zero power, as the
voltage difference is zero.
5.4

Outlook and future work

We were able to design and build a circuit and data analysis program that worked
effectively as a primary temperature measurement system and was also able to
measure thermoelectric voltages, at relatively low cost. Experimentally, the circuit

123

Figure 5.15: Shown is PSD under laser illumination without accounting for sample
resistance.

Figure 5.16: Shown is resistance of sample as a function of absorbed power.

was an effective noise thermometer for metal film resistors. Results thus far show
successful measurements of temperature, voltage, and thereby, Seebeck coefficients
of nanophotonic-heated structures. Further work will use these findings in addition
to thermoelectric voltage measurements between the platinum and bismuth telluride
leads to measure the temperature rise remotely at the center of the wires, far from

124

Figure 5.17: Shown is the measured Temperature of the sample as a function of
absorbed power, correcting for sample resistance.

Figure 5.18: Shown is the measured Seebeck coefficient (left) for our bismuth
telluride, using the data from Figure 5.17 and the measured thermoelectric voltage
(right).

the leads.
Noise measurements combined with the thermoelectric effect show potential for non-

125
invasive measurements of temperature of nanostructures. While leads are required
to connect to the nanophotonic structure, this could be a better option that needing
to couple in a laser (Raman thermometry) or use a microscope (SThM, FTIR).
Additionally, because we have a low noise circuit, high-zT thermoelectric materials
don’t necessarily need to be used in measurements. The low voltages produced
by small Seebeck coefficients in gold, for instances, could be measured with our
circuit, as long as they are above the noise floor of about 4-5 nV. Alternative circuits
with different opamps could be designed to work more effectively for low resistance
sources, as described in earlier chapters.

126
Chapter 6

CONCLUSION
6.1

Potential for nanophotonic thermoelectric detectors

This work combines the often disparate fields of nanophotonics and thermoelectrics
by using high light-confinement to control temperature gradients in nanoscale
volumes to produce highly wavelength-dependent thermoelectric voltages from
spatially-uniform illumination. This wavelength dependence is geometrically tunable and enables a non-bandgap limited, uncooled, filterless two-material spectrometer from visible to infrared on one chip. Because of the small volume, a rise time 100
times faster than conventional thermoelectric detector was found. Thermopiling,
optimized thermoelectric materials, or thermal management via wire suspension
would improve responsivity, while shrinking absorber dimensions would decrease
response time. Extending this thermoelectric detection motif to other resonant
nanophotonics structures will be straightforward, as we briefly explored in our
work, and opens up a new, untested world of uncooled self-filtering light detecting
structures.
Designing thermoelectric nanophotonic detectors involves a vast parameter space,
involving not only nanophotonic design, but thermal design and thermoelectric
properties of materials used. For higher responsivity, larger temperature gradients
are required, which favors long and thin structures. For faster response time, smaller
structures with lower heat capacities win out, which makes device design tricky, as
these are contradictory requirements. One interesting area for these thermoelectric
nanophotonic detectors would be in the ultra-violet regime. Here, nanophotonic
structures naturally shrink to handle smaller wavelengths, which would decrease the
response time. Environments with higher background radiation, such as nuclear or
space applications, could also find uses for these detectors, as thermoelectrics are
less sensitive to radiation than photovoltaic detectors. An interesting study would
be on the optimization of thermoelectric nanophotonics for speed and responsivity,
and to understand if these have absolute limits.
More thermal modeling is needed to determine the absolute detectivity limit of these
thermoelectric detectors. This is limited in part by the approximately 7% efficiency
ceiling of currently usable thermoelectric materials. Nanophotonic devices have

127
been designed which can absorb nearly 100% of incident light, though there will
be efficiency losses through heat loss via radiation, conduction, and convection,
which need to be studied in more detail (a similar study for solar thermoelectrics
was performed in (Kraemer et al., 2011)). Studying cooled detectors, weighing the
benefits of lower noise with a different material selection also needs to be studied.
Another interesting detector idea would be to combine resistive detectors, or bolometers, with thermoelectrics. In principle, an increase in temperature caused by an
impinging photon changes the resistance of the bolometer. The addition of a thermoelectric material could give additional information about the location of the photon
hitting the detector (based on the sign of the voltage), or generally enhance the
signal by generating added thermoelectric voltage, which would act as additional or
decreased resistance depending on the location of the photon on the structure.
6.2

Potential for noise thermometry in nanophotonic devices

By measuring the energy of the electrons themselves within a material, we are able
to make primary thermometry measurements less invasively and with simpler connections to the sample than with other temperature measurement techniques. While
noise thermometry may not be the best thermometry technique for all nanophotonic
applications, it does show promise for very thin structures where resistive thermometry probes would be on the order of the size of the structure and would thus
effect the temperature profile. Additionally, the cooling results when coupling a
thermoelectric nanophotonic device to a capacitive circuit show promise for on-chip
cooling applications.
One interesting path to explore would be resistance-corrected noise thermometry,
or conversely, noise-corrected resistive thermometry. Resistive thermometry works
well for materials with well-defined resistance versus temperature curves, but not all
materials have this characteristic, as we observed in bismuth telluride, for instance,
which had a resistance that drifted with time. By either monitoring resistance
constantly during noise measurements, or by periodically using noise thermometry
to recalibrate a resistor used in noise thermometry, any material could be used for
thermometry, regardless of the stability of its resistance with time.
Noise measurements ave not been extensively used in nanophotonics, and there
are many avenues to explore. For instance, to avoid metallic leads from affecting
nanophotonic temperatures, capacitive coupling to nanophotonic structures could
be accomplished by separating the structures from metallic leads with a thin dielec-

128
tric layer, effectively forming a capacitor. At high enough frequencies, capacitors
have negligible, allowing for noise temperature extraction without additional noise.
Measuring the noise in interferometric measurements can reflect the actual motion
of electrons in a material, thereby allowing for noise temperature extraction. Similar methods could be used, for example, by diffracting light with subwavelength
structures and using the noise of the diffracted pattern to determine the temperature
of the nanophotonic structure.
Additionally, while noise in equilibrium systems is well studied and what we have
studied here, the "transition" noise, or the noise in the heating and cooling transition
regions we ignored, could have interesting information about material properties.
In addition, exploring the possibility of using coupling between large capacitors,
nanophotonic heating, and thermoelectric materials to generate periodic thermoelectric cooling would be an interesting direction to pursue theoretically and potentially
experimentally. Finally, using measurements of power spectral density and thereby
correlation functions to extract the Seebeck coefficient through Green-Kubo theory
of hard-to-measure materials, such as 2D materials, would be an interesting avenue
to explore.

129

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Yeng, Yi Xiang et al. (2012). “Enabling high-temperature nanophotonics for energy applications”. In: Proceedings of the National Academy of Sciences 109.7,
pp. 2280–2285.
Zhang, Xiao and Li-Dong Zhao (2015). “Thermoelectric materials: Energy conversion between heat and electricity”. In: Journal of Materiomics 1.2, pp. 92–
105. issn: 2352-8478. doi: https : / / doi . org / 10 . 1016 / j . jmat . 2015 .
01.001. url: http://www.sciencedirect.com/science/article/pii/
S2352847815000258.
Zou, Helin, D.M. Rowe, and S.G.K. Williams (2002). “Peltier effect in a coevaporated Sb2Te3(P)-Bi2Te3(N) thin film thermocouple”. In: Thin Solid Films
408.1, pp. 270–274. issn: 0040-6090. doi: https : / / doi . org / 10 . 1016 /
S0040 - 6090(02 ) 00077 - 9. url: http : / / www . sciencedirect . com /
science/article/pii/S0040609002000779.

139
Appendix A

FFTW C PROGRAM FOR DATA ANALYSIS
From www.fftw.org, "FFTW is a C subroutine library for computing the discrete
Fourier transform (DFT) in one or more dimensions, of arbitrary input size, and of
both real and complex data (as well as of even/odd data, i.e. the discrete cosine/sine
transforms or DCT/DST).". FFTW (Frigo and S. Johnson, 2005) stands for "Fastest
Fourier Transform in the West", and is free. We found Matlab to be impossibly
slow at analyzing data, and Python to take several hours with multiprocessing to
analyze data. We wrote this data analysis program in C using the FFTW library, and
it has cut down the data analysis time from 3-5 hours in Python to approximately 20
minutes in C. One of the components that was severely slowing the Python program
was simply opening the large binary files the data was stored in. C takes a fraction
of a second to access the data. While Python has built-in Welch functions, we had
to write our own in C, which is detailed in an earlier chapter of this thesis.
The FFTW program can be installed on most Linux computers by using the general
installation procedure. On CentOS, for example, type the following into a terminal:
sudo yum install fftw
followed by
sudo yum install fftw-devel
If using a Windows operating system, VirtualBox (with RedHat and CentOS) works
with fftw, as does Linux Subsystem for Windows, which is available in the Windows
Store. The Linux Subsystem is easier for accessing files shared between the Windows
and Linux system, but the VirtualBox is much more forgiving when trying new
techniques, as you can delete the Virtual Machine and make a new one if you
generate serious problems, which you can’t do with Linux Subsystem. In our
application, we required a Windows machine to take data with the NI USB 6366, as
National Instruments only created a USB driver for this DAQ for Windows machines,
and not Linux, as of this writing. The PXIe version of the 6366 has a Linux driver
as well as a Windows driver, but is less portable than the USB version, and cannot
as easily be transferred between computers.
We used a Python program with wrappers for C to communicate with and collect data

140
from the NI USB6366, using the Python package PyDAQmx, which allows Python
to interact with the DAQmx driver for certain National Instruments products. This
data was saved in binary files.
If installed the standard way, using a gcc compiler, the compile command is
gcc -Wall fftw_data_analysis_program.c -I$HOME/usr/include -L$HOME/
usr/lib -lfftw3 -lm -o fftw_executable_program
The C program we wrote took several arguments at the command line, a path to a
directory with the data to analyze, and the extension for the analyzed data text file,
among other inputs. We changed the .c program and recompiled if we were changing
the order of resistors in the circuit, or were using chopped data or unchopped data
(baseline or control). To run data analysis in parallel, threading can be used with
FFTW, though we chose to use bash scripting. An example script is below:

#!/bin/bash

for num in {0..10}

fftw_executable path_to_data _analyzed_data_$num other_args &

done

wait

GNUplot, a free Linux plotting software, was also installed for examining the data
to determine the transition region truncation. The following is the full data analysis
program. We apologize for un-optimized coding, but we appreciate that C is fast
enough to compensate for poor form. The code explained in detail in a previous
chapter.

/* Compile with

gcc -Wall program_name.c -I$HOME/usr/include -L$HOME/usr/lib -lfftw3
,→ -lm -o analysis_excecutable

Run with (for example)

* /analysis_excecutable /path_to_data_to_analyze
,→ _suffix_for_analyzed_file 1 1 1 1 right_freq_limit
,→ left_freq_limit 3 R1_freq R2_freq

141

//complex.h needs to be above fftw3.h for this program to work, as
,→ complex.h complex number formatting is used instead of fftw3.h
,→ . See fftw documentation for details.

#include

18
19

#define PI 3.14159265

#define KB 1.38064852E-23

21
22
23
24

//Windowing function - we chose blackmanharris - for noise
,→ measurements of very large data sets, the window choice does
,→ not (and should not) have an effect.

double blackmanharris(long n,long segment_length);

26
27

//Linear interpolation function

double linear_interpy(double x_new, double *x_base, double *y_base,
,→ double max_freqq);

29
30

//Finding complex impedence at a given frequency and source
,→ resistance

double find_magz(double freq_val, double rss);

32
33

//Calculate the noise from external sources.

double external_noise_calc(double gain_data, double daq, double r4,
,→ double temp, double in, double r3, double gainz2, double en,
,→ double zout,double r2, double r1, double gainz1, double rs );

35
36

//Plots index versus ydata in gnuplot - must have gnuplot installed

142
37

void plotplot(long data_len, fftw_complex *ydata);

38
39

//Plots index versus 2 ydatas in gnuplot

void plotmultiplot(long data_len,double *ydata1, double *ydata2);

41
42

//Plots loglog in gnuplot (doesn’t yet read in x data)

void plotlogplot(long data_len,double *ydata);

44
45

//Plots loglog in gnuplot the fft data for TEV

void plotlogplotfft(long data_len,long nyq,double *ydata);

47
48

//Plot 8 lines in loglog (for checking noise data components)

void plotlogplot8(long data_len,double *ydata1,double *ydata2,double
,→ *ydata3,double *ydata4,double *ydata5,double *ydata6,double *
,→ ydata7,double *ydata8);

50
51

//Plot many things

void plotlogplotmany(int num_rows,long data_len,double *total_meas,
,→ double **cirq_noise);

53
54

//Make 2D array (not for use with fftw arrays, not supported)

double** array2D(int xdim, int ydim);

56
57

//function to find amplitude of frequency value. fft_length is #
,→ samples/2 +1

void FindAmpAtFreq(double freq_of_interest, double fft_length, double
,→ nyquist_freq, double *fft_to_search, double *max_value,
,→ double *max_index);

59
60

int main(int argc, char **argv){

61
62
63

if( argc != 12){
printf("Exactly 11 arguments needed, path to data
,→ directory, save extension, 4 numbers of
,→ resistance in ohms, and right cut (number) and
,→ left cut (number), chopper freq, and CH2 R
,→ frequency and CH3 R frequency (ex. /home/
,→ kamauser/C_programs/20190308_heater_cals/

143
,→ Data_binary_20190308_cirq_1_2/ and cirq1_2 and
,→ 3300 4700 1500 2200 and 12000 2000 and 43 and
,→ 21630 18387).\n");
exit(-1);

65
66
67

//SET DATA ANALYSIS FORMAT (WITH/WITHOUT GAIN, ETC...)

,→ **********************************************************************************
,→
69
70

//Is it a flat spectrum (as in, unchopped, so a control or
,→ temperature calibration run)? 1 = yes, 0 = is chopped

int is_flat = 0;

72
73

//If is_flat = 0, set this equal to 0 (or problems may arise).
,→ Otherwise, if is_flat=1, this will use the TEV extractor to
,→ get the R value if you feed the chopper frequency as the R
,→ frequency. The answer is NOT multiplied by 2pi/4 (pi/4 for
,→ square wave correction, 2 for Vpp) as it is for is_flat
,→ =0.

int getRconst = 0;

75
76

//data path

char data_path[500];//

78
79

//Do you want raw data or data with gain versus freq corrected
,→ and circuit component noise contributions subtracted?

int raw_data = 0; //1 = raw data, 0 = data with above
,→ calculations

81
82

//If raw_data = 0, do you want to use theoretical gain (r2/r1
,→ +1)*(r4/r3+1) or measured gain?

int theo_gain = 1;//1 means using theoretical gain on all
,→ frequencies. If raw_data = 1, make theo_gain = 0.

84
85

//If tev_gain_meas = 1, then a separate gain file is used during
,→ tev analysis which overrides the raw_data and theo_gain for

144
,→ tev voltages. This is useful if you want theoretical gain
,→ at high frequencies, but can’t use it at low frequencies
,→ due to lowpass filtering, or if there is separate frequency
,→ -related attenuation analysis needed. If tev_gain_meas = 0,
,→ the same gain file is used for tev analysis as for noise
,→ analysis. (so set to 0 if is_flat=1 and getRconst=1 if the
,→ tev gain file doesn’t go to high enough frequencies).
int tev_gain_meas = 1;

86
87
88

//save data path

char filename_nsd_on[500];// =

char filename_nsd_off[500];// =

char filename_tev[500];// =

char filename_R_on[500];

char filename_R_off[500];

94
95
96

//MAKE SAVE FILE NAMES

,→ **********************************************************************************
,→
//Read in the arguments for the file path and data save

,→ extension.
99

char data_ext[100];

100

sscanf(argv[1], "%s", data_path);

101

sscanf(argv[2], "%s", data_ext);

102

snprintf(filename_nsd_on,500,"/path/NSD_on_data%s.txt",
,→ data_ext);
snprintf(filename_nsd_off,500,"/path/NSD_off_data%s.txt",

103

,→ data_ext);
104

snprintf(filename_tev,500,"/path/NSD_tev%s.txt",data_ext);

105

snprintf(filename_R_on,500,"/path/R_on%s.txt",data_ext);

106

snprintf(filename_R_off,500,"/path/R_off%s.txt",data_ext);

107

printf("path to data files: %s",filename_nsd_off);

108
109

//GAIN FILE LOCATIONS

110

,→ **********************************************************************************

145
,→
111
112

char gain1[500];

113

char gain2[500];

114

char gain3[500];

115

char gain4[500];

116

if(raw_data == 1){//Don’t change these - these give a gain of 1 ,→ and will give raw data if that is desired
snprintf(gain1,500,"/path/gainversusfreq_201811/ch1_gain_1.txt")

117

,→ ;
snprintf(gain2,500,"/path/C_programs/gainversusfreq_201811/

118

,→ ch2_gain_1.txt");
snprintf(gain3,500,"/path/C_programs/gainversusfreq_201811/

119

,→ ch3_gain_1.txt");
snprintf(gain4,500,"/path/C_programs/gainversusfreq_201811/

120

,→ ch4_gain_1.txt");
121

122

else if(raw_data ==0){//change these to the path where you have
,→ gain versus frequency data
snprintf(gain1,500,"/path/

123

,→ lockin_data_20190308_opamp4_3p3k_4p76mV_finest_ch1.csv");
snprintf(gain2,500,"/path/

124

,→ lockin_data_20190308_opamp4_3p3k_4p76mV_finest_ch2.csv");
snprintf(gain3,500,"/path/

125

,→ lockin_data_20190308_opamp4_3p3k_4p76mV_finest_ch3.csv");
snprintf(gain4,500,"/path/

126

,→ lockin_data_20190308_opamp4_3p3k_4p76mV_finest_ch4.csv");
127

128

else{

129

printf("ERROR: raw_data must be either 0 or 1\n");

130

exit(-1);

131

132
133

//TEV gain file locations - theoretical opamp gain is not valid at
,→ low frequencies because of low-pass filtering in circuit

134

char gain1_tev[500];//These must be declared outside of if
,→ function

135

char gain2_tev[500];

146
136

char gain3_tev[500];

137

char gain4_tev[500];

138

if(tev_gain_meas == 1){

139
140

snprintf(gain1_tev,500,"/path/

141

,→ lockin_data_20190308_opamp4_3p3k_4p76mV_finest_ch1.csv");
snprintf(gain2_tev,500,"/path/

142

,→ lockin_data_20190308_opamp4_3p3k_4p76mV_finest_ch2.csv");
snprintf(gain3_tev,500,"/path/

143

,→ lockin_data_20190308_opamp4_3p3k_4p76mV_finest_ch3.csv");
snprintf(gain4_tev,500,"/path/

144

,→ lockin_data_20190308_opamp4_3p3k_4p76mV_finest_ch4.csv");

145
146
147
148

//OTHER VARIABLES

149

,→ **********************************************************************************
,→
150
151
152

//Welch variables for later - we assume a 50% overlap

153

double segment_len = 2097152; //Length of welch segmentprint ,→ should be faster if a power of 2 (2^21 = 2097152, 2^15
,→ = 32768, 2^16 = 65536, 2^25 = 33554432, 2^26 =
,→ 67108864) This actually works. Don’t make larger than
,→ ~1/5 of data size. Smaller is better. Because split
,→ into on/off and ~1/2 of data thrown out, you could end
,→ up with segment_len>fft size

154

double segment_len_4fft = 100000000;//For taking an FFT to get
,→ thermoelectric voltage (we don’t have to FFT all the
,→ data)

155
156

double samp_rate = 2000000; //sample rate during measurement

157
158

double chop_rate = atof(argv[9]);//43.0; //chopper frequency,
,→ used to guide the TEV peak-finder in the fft, will

147
,→ search about +/-5 Hz for the peak
159
160

//Frequency to average between to get the average PSD

161

double freq_min = 100000; //White noise frequency minimum

162

double freq_max = 900000; //White noise frequency maximum

163

//This is for cutting the transition region, in units of index

164

,→ . The data should be plotted and this determined.
long right_cut = atol(argv[7]);//12000; //Number of values to

165

,→ the right of the transition cutoff to remove
long left_cut = atol(argv[8]);//2000; //Number of values to

166

,→ the left of the transition cutoff to remove
167

//Length of segment for determining resistance (should be

168

,→ approximately the size of the region analyzed for on/
,→ off
double r_segment_len = 0.5*(1.0/((double)chop_rate))*samp_rate

169

,→ - right_cut - left_cut;
170
171

//Frequencies used for measuring resistance

172

double ch2_r_freq = atof(argv[10]);

173

double ch3_r_freq = atof(argv[11]);

174
175
176

//PLOTTING FOR ERROR CHECKING

177

,→ **********************************************************************************
,→
178

int plot_data = 0; //Set 1 to see data when setting right_cut
,→ and left_cut

179

long ind2plot = 50000; //Number of values to plot

180

int plot_psd_cumul_array = 0;//set equal to one to see the psd
,→ cumulative array

181

int plot_separate_source = 0;//1 means each contribution to
,→ noise will be plotted (error checking)

182

int plot_errchk = 0;//for exhaustive error checking

183

int plot_fft = 0;

184

int plot_fft_R = 0;

148
185
186
187

//NUMBER OF CHANNELS AND SCANS

188

,→ **********************************************************************************
,→
int num_scans = 5;//Number of runs you wish to analyze within

189

,→ the data folder
int num_chans = 4;//NSD channels, not chopper channels

190
191
192
193

//RESISTANCE OF EACH CHANNEL AND NOISE CALCULATIONS

194

,→ **********************************************************************************
,→
195
196

double r_of_chans[4];//in order from ch1, ch2, ch3, ch4

197

sscanf(argv[3], "%lf", &r_of_chans[0]);//Put resistances from
,→ command line argumetns into r_of_chans array

198

sscanf(argv[4], "%lf", &r_of_chans[1]);

199

sscanf(argv[5], "%lf", &r_of_chans[2]);

200

sscanf(argv[6], "%lf", &r_of_chans[3]);

201
202
203

double r1 = 33.2;//smaller gain resistor first stage

204

double r2 = 300;//larger gain resistor first stage

205

double r3 = 33.2;//smaller gain resistor second stage

206

double r4 = 300;//larger gain resistor second stage

207

double zout = 20;//output impedance of first stage opamp (
,→ seriously negligible)

208

double temp = 298;//temperature of gain resistors

209

double en = 3.8E-9;//input voltage noise of amplifier (in V/
,→ sqrt(Hz))

210

double in = 2.2E-15;//current noise of amplifier (in A/sqrt(Hz
,→ ))

211

double daq = 38e-9;//voltage noise of DAQ (in V/sqrt(Hz))

212
213

//Calculations for extraction of noise from opamp, don’t

149
,→ change these unless configuration changes
214

double gainz1, gainz2;

215

gainz1 = r2/r1 + 1;//Gain of first stage amplifier

216

gainz2 = r4/r3 + 1;//Gain of second stage amplifier

217
218
219

//GAIN CORRECTION INTERPOLATIONS

220

,→ **********************************************************************************
,→
221

//Read in txtfiles for gain corrections

222

printf("reading in gain corrections\n");

223

FILE *ch1_gptr,*ch2_gptr,*ch3_gptr,*ch4_gptr;

224

long gain_file_size = 0;

225

double * gain_file_freq_data;

226

double *gain_ch1_data,*gain_ch2_data,*gain_ch3_data,*
,→ gain_ch4_data;

227

ch1_gptr = fopen(gain1,"r");

228

ch2_gptr = fopen(gain2,"r");

229

ch3_gptr = fopen(gain3,"r");

230

ch4_gptr = fopen(gain4,"r");

231

if(ch1_gptr == NULL){

232

printf("error reading gain files\n");

233

exit(0);

234

235

if(ch2_gptr == NULL){

236

printf("error reading gain files\n");

237

exit(0);

238

239

if(ch3_gptr == NULL){

240

printf("error reading gain files\n");

241

exit(0);

242

243

if(ch4_gptr == NULL){

244

printf("error reading gain files\n");

245

exit(0);

246

247

//find the number of lines

150
248

printf("finding number of lines in gain file\n");

249

char line[128];

250

while(fgets(line,sizeof(line),ch1_gptr)!=NULL){
gain_file_size++;

251
252

253
254

printf("size of gain_file_size: %ld\n",gain_file_size);

255

fseek(ch1_gptr,0,SEEK_SET);

256
257
258

//malloc space for gain arrays
gain_file_freq_data = (double *)malloc(sizeof(double)*
,→ gain_file_size);

259

gain_ch1_data = (double *)malloc(sizeof(double)*gain_file_size
,→ );

260

gain_ch2_data = (double *)malloc(sizeof(double)*gain_file_size
,→ );

261

gain_ch3_data = (double *)malloc(sizeof(double)*gain_file_size
,→ );

262

gain_ch4_data = (double *)malloc(sizeof(double)*gain_file_size
,→ );

263
264

//put the data into the arrays

265

int hm=0;

266

char str1[100];

267

char str2[100];

268

char str3[100];

269

char str4[100];

270

for(hm=0;hm

271

fgets(str1,sizeof(str1),ch1_gptr);

272

fgets(str2,sizeof(str2),ch2_gptr);

273

fgets(str3,sizeof(str3),ch3_gptr);

274

fgets(str4,sizeof(str4),ch4_gptr);

275

sscanf(str1,"%le,%le",&gain_file_freq_data[hm],&
,→ gain_ch1_data[hm]);

276

sscanf(str2,"%*e,%le",&gain_ch2_data[hm]);

277

sscanf(str3,"%*e,%le",&gain_ch3_data[hm]);

278

sscanf(str4,"%*e,%le",&gain_ch4_data[hm]);

279

151
280
281
282

int testint;

283

for(testint=0;testint<3;testint++){
printf("gain_file_freq_data = %e\n",gain_file_freq_data

284

,→ [testint]);
285

286
287

fclose(ch1_gptr);

288

fclose(ch2_gptr);

289

fclose(ch3_gptr);

290

fclose(ch4_gptr);

291
292

printf("read in gain corrections successfully\n");

293
294

printf("interpolating gain corrections for frequencies\n");

295

double *gain_ch1_data_interp;

296

double *gain_ch2_data_interp;

297

double *gain_ch3_data_interp;

298

double *gain_ch4_data_interp;

299

gain_ch1_data_interp = (double *)malloc(sizeof(double)*(
,→ segment_len/2+1));

300

gain_ch2_data_interp = (double *)malloc(sizeof(double)*(
,→ segment_len/2+1));

301

gain_ch3_data_interp = (double *)malloc(sizeof(double)*(
,→ segment_len/2+1));

302

gain_ch4_data_interp = (double *)malloc(sizeof(double)*(
,→ segment_len/2+1));

303

int freq_cntr2;

304

double freqq2;

305

if(theo_gain==0){

306

for(freq_cntr2 = 0;freq_cntr2<(segment_len/2 + 1);
,→ freq_cntr2++){

307

freqq2 = freq_cntr2*((samp_rate/2)/(segment_len
,→ /2 +1));//frequency needed

308

gain_ch1_data_interp[freq_cntr2] =
,→ linear_interpy(freqq2,gain_file_freq_data
,→ ,gain_ch1_data,samp_rate/2.0);

152
gain_ch2_data_interp[freq_cntr2] =

309

,→ linear_interpy(freqq2,gain_file_freq_data
,→ ,gain_ch2_data,samp_rate/2.0);
gain_ch3_data_interp[freq_cntr2] =

310

,→ linear_interpy(freqq2,gain_file_freq_data
,→ ,gain_ch3_data,samp_rate/2.0);
gain_ch4_data_interp[freq_cntr2] =

311

,→ linear_interpy(freqq2,gain_file_freq_data
,→ ,gain_ch4_data,samp_rate/2.0);

312
313

314

else if(theo_gain==1){//using theoretical gain
for(freq_cntr2 = 0;freq_cntr2<(segment_len/2 + 1);

315

,→ freq_cntr2++){
freqq2 = freq_cntr2*((samp_rate/2)/(segment_len

316

,→ /2 +1));//frequency needed
gain_ch1_data_interp[freq_cntr2] = gainz1*gainz2

317

,→ ;
gain_ch2_data_interp[freq_cntr2] = gainz1*gainz2

318

,→ ;
gain_ch3_data_interp[freq_cntr2] = gainz1*gainz2

319

,→ ;
gain_ch4_data_interp[freq_cntr2] = gainz1*gainz2

320

,→ ;

321
322

323

else{
printf("ERROR: theo_gain must be either 0 or 1\n");

324

exit(-1);

325
326
327

printf("interpolation done\n");

328
329

for(testint=0;testint<10;testint++){
printf("gain data ch1 line %d is: %f\n",testint,

330

,→ gain_ch1_data_interp[testint]);
331

332
333

//TEV GAIN CORRECTION INTERPOLATIONS (IF tev_gain_meas == 1)

153
334

,→ **********************************************************************************
,→
335

double * gain_file_freq_data_tev;//must be declared outside of
,→ if statement

336

double *gain_ch1_data_tev,*gain_ch2_data_tev,*
,→ gain_ch3_data_tev,*gain_ch4_data_tev;

337

if(tev_gain_meas == 1){

338

//Read in txtfiles for gain corrections

339

printf("reading in gain corrections\n");

340

FILE *ch1_gptr_tev,*ch2_gptr_tev,*ch3_gptr_tev,*ch4_gptr_tev;

341

long gain_file_size_tev = 0;

342
343

ch1_gptr_tev = fopen(gain1_tev,"r");

344

ch2_gptr_tev = fopen(gain2_tev,"r");

345

ch3_gptr_tev = fopen(gain3_tev,"r");

346

ch4_gptr_tev = fopen(gain4_tev,"r");

347

if(ch1_gptr_tev == NULL){

348

printf("error reading tev gain files\n");

349

exit(0);

350

351

if(ch2_gptr_tev == NULL){

352

printf("error reading tev gain files\n");

353

exit(0);

354

355

if(ch3_gptr_tev == NULL){

356

printf("error reading tev gain files\n");

357

exit(0);

358

359

if(ch4_gptr_tev == NULL){

360

printf("error reading tev gain files\n");

361

exit(0);

362

363

//find the number of lines

364

printf("finding number of lines in gain file\n");

365

char line[128];

366

while(fgets(line,sizeof(line),ch1_gptr_tev)!=NULL){

367

gain_file_size_tev++;

154
368

369
370

printf("size of gain_file_size: %ld\n",gain_file_size_tev);

371

fseek(ch1_gptr_tev,0,SEEK_SET);

372
373
374

//malloc space for gain arrays
gain_file_freq_data_tev = (double *)malloc(sizeof(double)*
,→ gain_file_size_tev);

375

gain_ch1_data_tev = (double *)malloc(sizeof(double)*
,→ gain_file_size_tev);

376

gain_ch2_data_tev = (double *)malloc(sizeof(double)*
,→ gain_file_size_tev);

377

gain_ch3_data_tev = (double *)malloc(sizeof(double)*
,→ gain_file_size_tev);

378

gain_ch4_data_tev = (double *)malloc(sizeof(double)*
,→ gain_file_size_tev);

379
380

//put the data into the arrays

381

int hm=0;

382

char str1[100];

383

char str2[100];

384

char str3[100];

385

char str4[100];

386

for(hm=0;hm

387

fgets(str1,sizeof(str1),ch1_gptr_tev);

388

fgets(str2,sizeof(str2),ch2_gptr_tev);

389

fgets(str3,sizeof(str3),ch3_gptr_tev);

390

fgets(str4,sizeof(str4),ch4_gptr_tev);

391

sscanf(str1,"%le,%le",&gain_file_freq_data_tev[hm],&
,→ gain_ch1_data_tev[hm]);

392

sscanf(str2,"%*e,%le",&gain_ch2_data_tev[hm]);

393

sscanf(str3,"%*e,%le",&gain_ch3_data_tev[hm]);

394

sscanf(str4,"%*e,%le",&gain_ch4_data_tev[hm]);

395

396
397
398

int testint;

399

for(testint=0;testint<3;testint++){

155
printf("gain_file_freq_data_tev = %e\n",

400

,→ gain_file_freq_data_tev[testint]);

401
402
403

fclose(ch1_gptr_tev);

404

fclose(ch2_gptr_tev);

405

fclose(ch3_gptr_tev);

406

fclose(ch4_gptr_tev);

407

printf("read in TEV gain corrections successfully\n");

408
409

410

else{
printf("no separate TEV gain corrections used\n");

411

412
413
414

//GAIN-CORRECTED NOISE OF CIRCUIT COMPONENTS

415

,→ **********************************************************************************
,→
416
417

//Calculate the other sources of noise in the system.

418
419

int freq_cntr = 0;

420
421

//NOTE ABOUT THE E1 VOLTAGE NOISE: gain corrected gain resistor 1 (
,→ smaller one) voltage noise (gain of the inverting node that r1
,→ sees is actually (r2/r1), not (r2/r1 +1). Now, we cannot
,→ exactly measure the gain rolloff from this input, but we
,→ assume it is the fractionally same rolloff as in the other
,→ input. So if the + input has a gain of 90% of (r2/r1+1), we
,→ assume the - input has a gain of 90% of (r2/r1). Therefore,
,→ gain-corrected e1 will be x, where (gain_correction_measured/(
,→ r2/r1+1) = x/(r2/r1)), or x = (r2/r1)*gain_correction_measured
,→ /gainz

422

double *ch1_indep_noise_pwr;//gain corrected opamp current
,→ noise acting on source impedence magnitude

423

double *ch2_indep_noise_pwr;

424

double *ch3_indep_noise_pwr;

156
425

double *ch4_indep_noise_pwr;

426
427

ch1_indep_noise_pwr = (double *)malloc(sizeof(double)*(
,→ segment_len/2+1));

428

ch2_indep_noise_pwr = (double *)malloc(sizeof(double)*(
,→ segment_len/2+1));

429

ch3_indep_noise_pwr = (double *)malloc(sizeof(double)*(
,→ segment_len/2+1));

430

ch4_indep_noise_pwr = (double *)malloc(sizeof(double)*(
,→ segment_len/2+1));

431
432

double g1;

433

double g2;

434
435

for(freq_cntr = 0;freq_cntr<(segment_len/2 + 1);freq_cntr++){
,→ //number of frequencies dictated by welch segment
,→ length

436
437

if(raw_data == 1){
ch1_indep_noise_pwr[freq_cntr] = 0;//no

438

,→ subtraction of circuit component noise
ch2_indep_noise_pwr[freq_cntr] = 0;//no

439

,→ subtraction of circuit component noise
ch3_indep_noise_pwr[freq_cntr] = 0;//no

440

,→ subtraction of circuit component noise
ch4_indep_noise_pwr[freq_cntr] = 0;//no

441

,→ subtraction of circuit component noise
442

443

else if(raw_data == 0){

444

//freqq = freq_cntr*(samp_rate/2)/(segment_len/2
,→ +1);//The frequency of the future
,→ calculated power spectrum

445
446

// all sources of noise to be subtracted from
,→ measured PSD. 4*Rs*kb*T*G^2 = (measured
,→ PSD) - (stuff), where (stuff) =
,→ chx_indep_noise_pwr. This is not
,→ extremely computationally efficient, but

157
,→ easier to error check.
447
448

//ch1

449

ch1_indep_noise_pwr[freq_cntr] =
,→ external_noise_calc(gain_ch1_data_interp[
,→ freq_cntr],daq, r4, temp, in, r3, gainz2,
,→ en, zout, r2, r1, gainz1, r_of_chans[0]
,→ );

450

//ch2

451

ch2_indep_noise_pwr[freq_cntr] =
,→ external_noise_calc(gain_ch2_data_interp[
,→ freq_cntr],daq, r4, temp, in, r3, gainz2,
,→ en, zout, r2, r1, gainz1, r_of_chans[1]
,→ );

452

//ch3

453

ch3_indep_noise_pwr[freq_cntr] =
,→ external_noise_calc(gain_ch3_data_interp[
,→ freq_cntr],daq, r4, temp, in, r3, gainz2,
,→ en, zout, r2, r1, gainz1, r_of_chans[2]
,→ );

454

//ch4

455

ch4_indep_noise_pwr[freq_cntr] =
,→ external_noise_calc(gain_ch4_data_interp[
,→ freq_cntr],daq, r4, temp, in, r3, gainz2,
,→ en, zout, r2, r1, gainz1, r_of_chans[3]
,→ );

456
457
458

459

else{
printf("ERROR: raw_data must be either 0 or 1\n"

460

,→ );
exit(-1);

461

462
463
464
465
466

158
467

//INITIALIZE NSD AND TEV DATA FILES

468

,→ **********************************************************************************
,→
469

//Initialize the files for NSD and TEV data

470

FILE *fp_on;

471

FILE *fp_off;

472

FILE *fp_tev;

473

FILE *fp_Ron;

474

FILE *fp_Roff;

475

fp_on = fopen(filename_nsd_on,"w+");

476

fp_off = fopen(filename_nsd_off,"w+");

477

fp_tev = fopen(filename_tev,"w+");

478

fp_Ron = fopen(filename_R_on,"w+");

479

fp_Roff = fopen(filename_R_off,"w+");

480

if(fp_on == NULL){

481

printf("trouble creating on data file\n");

482

exit(-1);

483

484
485

if(fp_off == NULL){

486

printf("trouble creating off data file\n");

487

exit(-1);

488

489
490

if(fp_off == NULL){

491

printf("trouble creating tev data file\n");

492

exit(-1);

493

494
495

if(fp_Ron == NULL){

496

printf("trouble creating R on data file\n");

497

exit(-1);

498

499
500

if(fp_Roff == NULL){

501

printf("trouble creating R off data file\n");

502

exit(-1);

159

503
504
505
506

fclose(fp_on);

507

fclose(fp_off);

508

fclose(fp_tev);

509

fclose(fp_Ron);

510

fclose(fp_Roff);

511
512

printf("starting\n");

513
514
515

//INITIALIZING A BUNCH OF STUFF

516

,→ **********************************************************************************
,→
517
518

char ch_data_filename[100];

519

FILE *ch_file_ptr;

520

long filelen;//length of file, long is an int that can be very
,→ large

521

int array_len;//length of array

522

double *NSYNC; //data goes into this array

523

int *on_starts;

524

on_starts = (int *)malloc(1*sizeof(int));

525

int *off_starts;

526

off_starts = (int *)malloc(1*sizeof(int));

527

//printf("made the new mallocs\n");

528

double sub_val;

529

int jt;

530

int ons = 0;//index for location of on starts

531

int offs = 0;//index for location of off starts

532

double *data4plot;

533

int vally;

534

fftw_complex * ch_on_data;

535

fftw_complex * ch_off_data;

536

int k;

537

//int z;

160
538

int ind_on;

539

int ind_off;

540

int cntr2;

541

int zz;

542

double max_psd;

543

double min_psd;

544
545

//Spacing of fft frequencies

546

double freq_spacing = (samp_rate/2.0)/(segment_len/2.0+1.0);

547

int freq_min_ind = (int)(freq_min/freq_spacing);

548

int freq_max_ind = (int)(freq_max/freq_spacing);

549
550

double avg_PSD;

551

double avg_NSD;

552
553
554

//INITIALIZING FFTW PLANS AND ARRAYS

555

,→ **********************************************************************************
,→
556
557

fftw_plan plan_segment;

558

//Getting all of the Welch variables ready

559

fftw_complex *segment_data_array;//length of each welch
,→ segment length the fft

560

segment_data_array = (fftw_complex *)fftw_malloc((segment_len)
,→ *sizeof(fftw_complex));

561
562

//Getting all of the thermoelectric voltage variables ready

563

int tev_ind=0;

564

double max_tev;

565

int max_tev_ind;

566

fftw_plan plan_segment_tev;

567

fftw_complex *segment_tev_array;

568

fftw_complex *segment_tev_fft_array;

569

double *segment_tev_fft_array_norm;

570

if(is_flat==0 || getRconst == 1){

571

//Getting all of the thermoelectric voltage variables

161
,→ ready
segment_tev_array = (fftw_complex *)fftw_malloc((

572

,→ segment_len_4fft)*sizeof(fftw_complex));
segment_tev_fft_array = (fftw_complex *)fftw_malloc(

573

,→ sizeof(fftw_complex)*(segment_len_4fft));
574

printf("Starting TEV fftw plan\n");

575

plan_segment_tev = fftw_plan_dft_1d(segment_len_4fft,
,→ segment_tev_array,segment_tev_fft_array,
,→ FFTW_FORWARD,FFTW_ESTIMATE);//

576

//change to

FFTW_MEASURE eventually

577

printf("Finished TEV fftw plan\n");

578

segment_tev_fft_array_norm = (double *)malloc(sizeof(
,→ double)*segment_len_4fft);

579

580
581

//PSD data array

582

double *psd_cumulative_array;

583

psd_cumulative_array = (double *)calloc((segment_len/2+1),
,→ sizeof(double));//calloc initializes array to zero

584
585

int segment_counter = 0; //number of segments

586
587

fftw_complex *segment_fft_array;//the array the welch segment
,→ will be ffted into

588
589

segment_fft_array = (fftw_complex *)fftw_malloc(sizeof(
,→ fftw_complex)*(segment_len));

590
591

printf("planning the segment\n");

592

plan_segment = fftw_plan_dft_1d(segment_len,
,→ segment_data_array,segment_fft_array,FFTW_FORWARD,
,→ FFTW_ESTIMATE);//change to FFTW_MEASURE eventually

593
594
595

//The fftw initialization for extracting the resistance
,→ signals

596

int r_cntr;

597

fftw_plan plan_resistance;

162
598

fftw_complex *resistance_array_time;

599

fftw_complex *resistance_array_fftd;

600

resistance_array_time = (fftw_complex *)fftw_malloc((

601

,→ r_segment_len)*sizeof(fftw_complex));
resistance_array_fftd = (fftw_complex *)fftw_malloc((

602

,→ r_segment_len)*sizeof(fftw_complex));
603
604

double *resistance_array_fftd_norm;

605

resistance_array_fftd_norm = (double *)malloc((r_segment_len/2.0 +
,→ 1.0)*sizeof(double));

606

plan_resistance = fftw_plan_dft_1d(r_segment_len,

607

,→ resistance_array_time, resistance_array_fftd,
,→ FFTW_FORWARD,FFTW_ESTIMATE);
608

double *r_array_4avg_on;//holds the R data of each individual

609

,→ segment, and is averaged at the end
double *r_array_4avg_off;

610

r_array_4avg_on = (double *)malloc(sizeof(double)*3000);//this

611

,→ will be realloced later
612

r_array_4avg_off = (double *)malloc(sizeof(double)*3000);

613

int numRon = 0;

614

int numRoff = 0;

615
616

//variables for storing the amplitude of the resistance signal

617

double max_valueR;

618

double max_indexR;

619

double R_avg_on=0;

620

double R_avg_off=0;

621
622
623

//BLACKMANHARRIS WINDOW AND NORMALIZATION FACTOR

624

,→ **********************************************************************************
,→
625
626

//The blackmanharris PSD window normalization

163
627

//Make the blackmanharris window array

628

double *blackmanharris_array;

629

blackmanharris_array = (double *)malloc(sizeof(double)*(
,→ segment_len));

630

for(zz=0;zzblackmanharris_array[zz] = blackmanharris(zz,

631

,→ segment_len);
632

633
634

double window_norm = 0.0;

635

printf("making the window norm\n");

636

for(zz=0;zzwindow_norm = window_norm + blackmanharris_array[zz]*

637

,→ blackmanharris_array[zz];//for fft
638

639
640

double window_norm_correct = 0.0;

641

printf("making the window norm\n");

642

for(zz=0;zzwindow_norm_correct = window_norm_correct +

643

,→ blackmanharris_array[zz]*blackmanharris_array[zz
,→ ];
644

645
646

window_norm_correct = window_norm_correct*samp_rate;

647
648
649

//int zz;

650

int on_cntr = 0;

651

int off_cntr = 0;

652

long cntr=0;

653

int cum_data=0;

654

int segs_left=1;

655

int is_on_ch = 0;

656

int scan_num;

657
658
659

164
660

//BEGIN DATA READ IN

661

,→ **********************************************************************************
,→
662
663
664

//looping over the number of scans at each power or wavelength (a for
,→ loop inside this one loops over the channels)

665

for(scan_num = 0;scan_num

666
667

//If a flat scan, slightly different route:

668

if(is_flat == 1){
sprintf(ch_data_filename,"%sch1_scan%d.bin",

669

,→ data_path,scan_num+1);
printf("Data is not chopped\n");

670
671

672

else if(is_flat == 0){

673

//Open the chopper data
sprintf(ch_data_filename,"%sch5_scan%d.bin",

674

,→ data_path,scan_num+1);
printf("Data is chopped\n");

675
676

677

else{

678

printf("Incorrect value for is_flat\n");

679

exit(0);

680

681
682

//Open file

683

ch_file_ptr = fopen(ch_data_filename,"rb");

684

if(ch_file_ptr == NULL){

685

printf("Problem opening first data file\n");

686

exit(-1);

687

688

printf("opened the first data file\n");

689

//Read the binary file data contents into an array
,→ ready for fft

690

//initialize array for first scan. All arrays must be
,→ the same size.

165
691
692

if(scan_num == 0){//just do this stuff the first time,
,→ since all arrays are the same length

693

fseek(ch_file_ptr, 0, SEEK_END); //Go to end of
,→ file

694

printf("found end of file\n");

695

filelen = ftell(ch_file_ptr); //say what
,→ position is the end

696

rewind(ch_file_ptr); //Put pointer back to
,→ beginning of file

697

printf("filelen %ld\n",filelen);

698

printf("about to malloc\n");

699

array_len = filelen/8; //assumes data is doubles
,→ , so 8 bytes each

700

NSYNC = (double *)malloc((filelen+1)*sizeof(
,→ double));//may need to do filelen+1, don’
,→ t know if binary doubles have a null
,→ character, don’t think so

701

printf("mallocked\n");

702
703

//Again, the different route for a flat scan

704

if(is_flat ==1){
ch_off_data = (fftw_complex*)malloc(

705

,→ sizeof(fftw_complex)*array_len);
//ch_on_data will be unused

706
707

708

else if(is_flat == 0){

709

//Determine amount of space for the "on" and "
,→ off" data vectors. -> Overallocating so
,→ we don’t run out of space (we know it
,→ must be length of total data/2 +1 period
,→ maximum, so we will do that, and can cut
,→ later

710

ch_on_data = (fftw_complex*)fftw_malloc(
,→ sizeof(fftw_complex)*((array_len/2)
,→ +300000)); //200000 is number of
,→ points in 1 period of 10Hz with 2e6
,→ S/s

166
ch_off_data = (fftw_complex*)fftw_malloc(

711

,→ sizeof(fftw_complex)*((array_len/2)
,→ +300000));
712

713

else{

714

printf("error with is_flat value\n");

715

exit(0);

716

717
718
719

fread(NSYNC,filelen,1,ch_file_ptr);

720

fclose(ch_file_ptr);

721

printf("read in data successfully\n");

722
723
724

//CONVERT CHOPPER DATA TO +1 AND -1

725

,→ **********************************************************************************
,→
726
727

if(is_flat == 0){
//find mean of data (or first 1/3 of data) so
,→ the "on" will be >0 and the "off" will be
,→ <0

728

double mean_val = 0.0;

729

int tj;

730

for(tj=0;tjmean_val = mean_val + NSYNC[tj];

731
732

733

mean_val = mean_val/((array_len));

734

printf("mean_val is: %f\n",mean_val);

735
736

//find the on/off regions from the chopper
,→ signal. Make everything +/-1, then take
,→ the difference. Note: position 2-position
,→ 1 = position 0. So essentially I am
,→ cutting off the first 2 parts of the
,→ array.

737

167
738

ons = 0;

739

offs = 0;

740

for(jt = 1;jt,→ from 1, not 0

741

//1 ahead

742

if(jt == 1){//changed 20190407
if((NSYNC[jt-1]-mean_val)<0.0 && (

743

,→ NSYNC[jt]-mean_val)<0.0){//
,→ changed 20190407 to sample the
,→ point and the next point, works
,→ for noisy data
NSYNC[jt-1]=-1.0;

744
745

746

else if((NSYNC[jt-1]-mean_val)>0.0 &&
,→ (NSYNC[jt]-mean_val)>0.0){
NSYNC[jt-1]=1.0;

747
748

749

else{
NSYNC[jt-1]=0.0;

750

751
752

753
754

//2 ahead

755

//4/7/2019 - smooths slightly noisy
,→ chopper data (assuming problem
,→ spike only lasts one data point)

756
757

//case 1: random spike above zero from "
,→ light off" (low HIGH low)

758

if((NSYNC[jt]-mean_val)>0.0 && (NSYNC[jt
,→ -1]-mean_val)<0.0 && (NSYNC[jt+1],→ mean_val)<0.0){
NSYNC[jt]=-1.0;//Is a random high

759

,→ spike when the light is off
//printf("case 1\n");

760
761

762

//case 2: random spike below zero from "
,→ light on" (high LOW high)

168
763

else if((NSYNC[jt]-mean_val)<0.0 && (
,→ NSYNC[jt-1]-mean_val)>0.0 && (NSYNC
,→ [jt+1]-mean_val)>0.0){
NSYNC[jt]=1.0;//Is a random low

764

,→ spike when the light is off
//printf("case 2\n");

765
766

767

//case 3: data point right before "on" (
,→ low LOW high)

768

else if((NSYNC[jt]-mean_val)<0.0 && (
,→ NSYNC[jt-1]-mean_val)<0.0 && (NSYNC
,→ [jt+1]-mean_val)>0.0){
NSYNC[jt]=-1.0;//right before step

769

,→ up
770

771

//case 4: data point right after "on" (
,→ low HIGH high)

772

else if((NSYNC[jt]-mean_val)>0.0 && (
,→ NSYNC[jt-1]-mean_val)<0.0 && (NSYNC
,→ [jt+1]-mean_val)>0.0){
NSYNC[jt]=1.0;//right after step

773

,→ up
774

775

//case 5: data point right before "off" (
,→ high HIGH low)

776

else if((NSYNC[jt]-mean_val)>0.0 && (
,→ NSYNC[jt-1]-mean_val)>0.0 && (NSYNC
,→ [jt+1]-mean_val)<0.0){
NSYNC[jt]=1.0;//right before step

777

,→ down
778

779

//case 6: data point right after "off" (
,→ high LOW low)

780

else if((NSYNC[jt]-mean_val)<0.0 && (
,→ NSYNC[jt-1]-mean_val)>0.0 && (NSYNC
,→ [jt+1]-mean_val)<0.0){

781

NSYNC[jt]=-1.0;//right after step
,→ up

169
782

783

//case 7: data point in the low region (
,→ low LOW low)

784

else if((NSYNC[jt]-mean_val)<0.0 && (
,→ NSYNC[jt-1]-mean_val)<0.0 && (NSYNC
,→ [jt+1]-mean_val)<0.0){
NSYNC[jt]=-1.0;//low

785
786

787

//case 8: data point in the low region (
,→ low LOW low)

788

else if((NSYNC[jt]-mean_val)>0.0 && (
,→ NSYNC[jt-1]-mean_val)>0.0 && (NSYNC
,→ [jt+1]-mean_val)>0.0){
NSYNC[jt]=1.0;//low

789
790

791

else{
NSYNC[jt]=NSYNC[jt-1];//if none of

792

,→ the above, like it equals
,→ zero, just make it the same
,→ as the point before
793

794

//Where you find the indices of break
,→ points

795
796

sub_val = NSYNC[jt]-NSYNC[jt-1];

797
798

if(sub_val>0.0){//change 4/7/2019 from
,→ 0.0 to 1.1
on_starts = (int *)realloc(

799

,→ on_starts,(ons+1)*sizeof(int
,→ ));
on_starts[ons] = jt;//changed

800

,→ 4/7/2019 from jt-1 to jt,
,→ should make no difference
ons++;

801
802

803

else if(sub_val<0.0){

804

off_starts = (int *)realloc(

170
,→ off_starts,(offs+1)*sizeof(
,→ int));
if(off_starts == NULL){

805

printf("memory allocation

806

,→ failed\n");
exit(1);

807
808

809

off_starts[offs] = jt;//changed
,→ 4/7/2019 from jt-1 to jt
offs++;

810
811

812

else{
//printf("is zero\n");

813

814
815

816

817
818

printf("ons = %d, offs = %d\n",ons,offs);

819
820
821
822

//PLOTTING PERFECT CHOPPER PLUS DATA CHANNEL 3

823

,→ **********************************************************************************
,→
824

//Plotting the transition location

825

if(plot_data == 1){

826
827

sprintf(ch_data_filename,"%sch3_scan1.bin",
,→ data_path);

828

ch_file_ptr = fopen(ch_data_filename,"rb");

829
830
831

data4plot = (double *)malloc(ind2plot*sizeof(
,→ double));

832

fread(data4plot,ind2plot*8,1,ch_file_ptr); //8
,→ assumes the data is a double, so 8 bytes
,→ each

171
fclose(ch_file_ptr);

833
834
835

for(vally=0;vally<10;vally++)

836

printf("data is ch3, ch4: %f, %f\n",

837

,→ data4plot[vally],NSYNC[vally]);
838
839

plotmultiplot(ind2plot,NSYNC,data4plot);

840

exit(1);

841
842
843
844

//LOOP OVER EACH CHANNEL (k)

845

,→ **********************************************************************************
,→
846

//Looping over the channels

847

for(k=0; k

848
849

is_on_ch = 0;

850

on_cntr = 0;

851

off_cntr = 0;

852

cntr = 0;

853

cum_data = 0;

854

segs_left = 1;

855
856

//read in data - containing file into NSYNC
,→ array

857

sprintf(ch_data_filename,"%sch%d_scan%d.bin",
,→ data_path,k+1,scan_num+1);

858

ch_file_ptr = fopen(ch_data_filename,"rb");

859

if(ch_file_ptr == NULL){
printf("error reading files channel loop\

860

,→ n");
exit(0);

861
862

863
864

fread(NSYNC,filelen,1,ch_file_ptr);

172
fclose(ch_file_ptr);

865
866
867

//THERMOELECTRIC VOLTAGE ANALYSIS

868

,→ **********************************************************************************
,→
869

//Here we find the thermoelectric voltage, if
,→ not a control spectrum

870

if(is_flat == 0 || getRconst == 1){

871
872

//fft NSYNC - make a plan beforehand and
,→ a place to put the fft

873

for(tev_ind = 0;tev_ind,→ tev_ind++){
segment_tev_array[tev_ind] = NSYNC

874

,→ [tev_ind];
875

876
877

fftw_execute(plan_segment_tev);//fft data
,→ in segment_tev_fft_array

878
879

//properly normalize (is no window, so
,→ just divide by segment_len_4fft, x
,→ by 2 because 2-sided).

880

for(tev_ind = 0;tev_ind,→ tev_ind++){
segment_tev_fft_array_norm[tev_ind

881

,→ ] = 2.0*sqrt(creal(
,→ segment_tev_fft_array[
,→ tev_ind])*creal(
,→ segment_tev_fft_array[
,→ tev_ind]) + cimag(
,→ segment_tev_fft_array[
,→ tev_ind])*cimag(
,→ segment_tev_fft_array[
,→ tev_ind]))/((double)(
,→ segment_len_4fft));
882

173
883
884

//Plot the fft if desired

885

if(plot_fft == 1){
plotlogplotfft(segment_len_4fft

886

,→ /2+1,samp_rate/2,
,→ segment_tev_fft_array_norm);
printf("press ENTER to continue\n"

887

,→ );
getchar();

888
889

890
891

//ind_chop_rate = chop_rate/((samp_rate
,→ /2.0)/(segment_len_4fft/2.0+1.0));

892

max_tev = 0.0;//initialize to zero to
,→ find max voltage peak

893

max_tev_ind = 0;//find index of peak

894

for(tev_ind = (int)((chop_rate-10)/((
,→ samp_rate/2.0)/(segment_len_4fft
,→ /2.0+1.0)));tev_ind<(int)((
,→ chop_rate+10)/((samp_rate/2.0)/(
,→ segment_len_4fft/2.0+1.0)));tev_ind
,→ ++){//search for the peak +/- 5 Hz
,→ around the chopper rate (in case
,→ the frequency was actually 43.1 Hz
,→ instead of 43 Hz, just to get
,→ better accuracy)
if(segment_tev_fft_array_norm[

895

,→ tev_ind]>max_tev){
max_tev =

896

,→ segment_tev_fft_array_norm
,→ [tev_ind];//find max
,→ voltage in this range
max_tev_ind = tev_ind;

897

898
899

900

printf("Peak voltage is %e V at %f Hz\n",
,→ max_tev,max_tev_ind*((samp_rate
,→ /2.0)/(segment_len_4fft/2.0+1.0)));

174
901

//multiply the peak by 2*pi/4 (for TEV

902

,→ extraction, not R extraction) - the
,→ 2 is because we want the peak to
,→ peak
903

if(is_flat==0){//only for TEV extraction, not R extraction

904

printf("max_tev before alteration is %e\
,→ n",max_tev);
max_tev = max_tev * 2.0 * PI / 4.0;

905
906

907
908

//analyze differently if using separate gain for TEV

909

if(tev_gain_meas == 0){

910
911

//multiply by the gain of the amplitude
,→ at the frequency of interest
if(k==0){//if chan 1

912

max_tev = max_tev / linear_interpy

913

,→ (max_tev_ind*((samp_rate
,→ /2.0)/(segment_len_4fft
,→ /2.0+1.0)),
,→ gain_file_freq_data,
,→ gain_ch1_data, samp_rate
,→ /2.0);
914

915

else if(k == 1){//if chan 2
max_tev = max_tev / linear_interpy

916

,→ (max_tev_ind*((samp_rate
,→ /2.0)/(segment_len_4fft
,→ /2.0+1.0)),
,→ gain_file_freq_data,
,→ gain_ch2_data, samp_rate
,→ /2.0);
917

918

else if(k == 2){//if chan 3

919

max_tev = max_tev / linear_interpy
,→ (max_tev_ind*((samp_rate
,→ /2.0)/(segment_len_4fft

175
,→ /2.0+1.0)),
,→ gain_file_freq_data,
,→ gain_ch3_data, samp_rate
,→ /2.0);
920

921

else if(k == 3){//if chan 4
max_tev = max_tev / linear_interpy

922

,→ (max_tev_ind*((samp_rate
,→ /2.0)/(segment_len_4fft
,→ /2.0+1.0)),
,→ gain_file_freq_data,
,→ gain_ch4_data, samp_rate
,→ /2.0);
923

924

else{
printf("TEV extraction - too many

925

,→ channels, you can only have
,→ 4 signal channels +
,→ chopper signal at this time
,→ .\n");
exit(0);

926

927
928

929

else if(tev_gain_meas == 1){

930
931

//multiply by the gain of the amplitude
,→ at the frequency of interest
if(k==0){//if chan 1

932

max_tev = max_tev / linear_interpy

933

,→ (max_tev_ind*((samp_rate
,→ /2.0)/(segment_len_4fft
,→ /2.0+1.0)),
,→ gain_file_freq_data_tev,
,→ gain_ch1_data_tev,
,→ samp_rate/2.0);
934

935

else if(k == 1){//if chan 2

936

max_tev = max_tev / linear_interpy

176
,→ (max_tev_ind*((samp_rate
,→ /2.0)/(segment_len_4fft
,→ /2.0+1.0)),
,→ gain_file_freq_data_tev,
,→ gain_ch2_data_tev,
,→ samp_rate/2.0);
937

938

else if(k == 2){//if chan 3
max_tev = max_tev / linear_interpy

939

,→ (max_tev_ind*((samp_rate
,→ /2.0)/(segment_len_4fft
,→ /2.0+1.0)),
,→ gain_file_freq_data_tev,
,→ gain_ch3_data_tev,
,→ samp_rate/2.0);
940

941

else if(k == 3){//if chan 4
max_tev = max_tev / linear_interpy

942

,→ (max_tev_ind*((samp_rate
,→ /2.0)/(segment_len_4fft
,→ /2.0+1.0)),
,→ gain_file_freq_data_tev,
,→ gain_ch4_data_tev,
,→ samp_rate/2.0);
943

944

else{
printf("TEV extraction - too many

945

,→ channels, you can only have
,→ 4 signal channels +
,→ chopper signal at this time
,→ .\n");
exit(0);

946

947
948

949

else{

950

printf("tev_gain_meas must be 0 or 1, failed during TEV
,→ analysis step\n");

951

exit(0);

177

952

//save this data in a file which also

953

,→ must be made before and terminated.
954

fp_tev = fopen(filename_tev,"a");

955

fprintf(fp_tev,"%le\t",max_tev);

956

fclose(fp_tev);

957

958
959
960
961

//SEPARATE ON AND OFF DATA

962

,→ **********************************************************************************
,→
963

//Here I separate the on and off data into the
,→ different arrays to fft, if not a control
,→ spectrum

964

if(is_flat == 0){

965
966

for(ind_on = 0;ind_on < ons;ind_on++){//
,→ Loop over all the on transitions

967

//Basically determine which index
,→ follows

968

for(cntr2 = 0;cntr2<(fabs(
,→ off_starts[5]-on_starts[5]),→ right_cut-left_cut);cntr2++)
,→ {//Will cut correctly +/- a
,→ few spots on each side

969

ch_on_data[on_cntr] = NSYNC[
,→ on_starts[ind_on]+
,→ right_cut+cntr2];

970

on_cntr++;

971

//R array, only for channels
,→ 2 and 3

972
973

if(k==1 || k==2){
if(cntr2<
,→ r_segment_len)
,→ {//We don’t

178
,→ want to
,→ overflow the
,→ array. If it
,→ underflows,
,→ there will be
,→ zeros at the
,→ end. Will look
,→ into this
,→ futher.
resistance_array_time

974

,→ [cntr2]
,→ =
,→ NSYNC[
,→ on_starts
,→ [ind_on
,→ ]+
,→ right_cut
,→ +cntr2
,→ ];

975

976
977

978

//printf("cntr2 is %d\n",cntr2);

979

//printf("r_segment_len is %f\n",
,→ r_segment_len);

980
981

//Only for channels 2 and 3 do we
,→ look for the resistance peak

982
983

if(k==1 || k==2){
//Here we fft the resistance
,→ array and find the
,→ peak at the correct
,→ resistance

984

fftw_execute(plan_resistance
,→ );

985
986

//Properly normalize the fft
,→ (2*absolute value/

179
,→ sample num). .
987

for(r_cntr=0;r_cntr<(
,→ r_segment_len/2.0)
,→ +1.0;r_cntr++){
resistance_array_fftd_norm

988

,→ [r_cntr] =
,→ 2.0*sqrt(creal
,→ (
,→ resistance_array_fftd
,→ [r_cntr])*
,→ creal(
,→ resistance_array_fftd
,→ [r_cntr]) +
,→ cimag(
,→ resistance_array_fftd
,→ [r_cntr])*
,→ cimag(
,→ resistance_array_fftd
,→ [r_cntr]))/((
,→ double)(
,→ r_segment_len)
,→ );
989

990
991

//Plot the fft if desired

992

if(plot_fft_R == 1){
plotlogplotfft(

993

,→ r_segment_len
,→ /2+1,samp_rate
,→ /2,
,→ resistance_array_fftd_norm
,→ );
printf("R fft plot,

994

,→ press ENTER to
,→ continue\n");
getchar();

995
996
997

180
998

//Now we find the peak of
,→ the resistance signal
,→ . (for ch2 or ch3)

999

if(k==1){
FindAmpAtFreq(ch2_r_freq

1000

,→ , r_segment_len
,→ /2.0+1.0,
,→ samp_rate/2.0,
,→ resistance_array_fftd_norm
,→ , &max_valueR, &
,→ max_indexR);
1001

1002

else if(k==2){
FindAmpAtFreq(ch3_r_freq

1003

,→ , r_segment_len
,→ /2.0+1.0,
,→ samp_rate/2.0,
,→ resistance_array_fftd_norm
,→ , &max_valueR, &
,→ max_indexR);
1004

1005

//printf("Peak R voltage (on
,→ ) is %e V at %f Hz\n
,→ ",max_valueR,
,→ max_indexR*((
,→ samp_rate/2.0)/(
,→ r_segment_len
,→ /2.0+1.0)));

1006
1007

//Save this data in an array
,→ which will be
,→ averaged later

1008

if(numRon>3000){//reallocate
,→ length if needed

1009

r_array_4avg_on = (
,→ double *)realloc(
,→ r_array_4avg_on,(
,→ numRoff+1)*sizeof

181
,→ (double));
1010

1011

r_array_4avg_on[ind_on] =
,→ max_valueR;
numRon++;

1012

1013
1014

1015
1016

for(ind_off = 0;ind_off < offs;ind_off++)
,→ {

1017

for(cntr2 = 0;cntr2<(fabs(
,→ off_starts[5]-on_starts[5]),→ right_cut-left_cut);cntr2++)
,→ {

1018

ch_off_data[off_cntr] =
,→ NSYNC[off_starts[
,→ ind_off]+right_cut+
,→ cntr2];

1019

off_cntr++;

1020
1021
1022

if(k==1 || k==2){
if(cntr2<
,→ r_segment_len)
,→ {//We don’t
,→ want to
,→ overflow the
,→ array. If it
,→ underflows,
,→ there will be
,→ zeros at the
,→ end. Will look
,→ into this
,→ futher.

1023

resistance_array_time
,→ [cntr2]
,→ =
,→ NSYNC[
,→ off_starts

182
,→ [
,→ ind_off
,→ ]+
,→ right_cut
,→ +cntr2
,→ ];

1024

1025

1026
1027

//printf("cntr2 is %d\n",cntr2);

1028

//printf("r_segment_len is %f\n",
,→ r_segment_len);

1029
1030
1031

//LOOK FOR RESISTANCE PEAK

1032

,→ **********************************************************************************
,→
1033

//Only for channels 2 and 3 do we
,→ look for the resistance peak

1034
1035

if(k==1 || k==2){
//Here we fft the resistance
,→ array and find the
,→ peak at the correct
,→ resistance

1036

fftw_execute(plan_resistance
,→ );

1037
1038

//Properly normalize the fft
,→ (2*absolute value/
,→ sample num). We are
,→ putting it back into
,→ the fftd array, just
,→ the first half. So
,→ only the first half
,→ of the array should
,→ be used from this
,→ point on.

183
1039

for(r_cntr=0;r_cntr<(
,→ r_segment_len/2.0)
,→ +1.0;r_cntr++){
resistance_array_fftd_norm

1040

1042
1043

//Plot the fft if desired

1044

if(plot_fft_R == 1){
plotlogplotfft(

1045

,→ r_segment_len
,→ /2+1,samp_rate
,→ /2,
,→ resistance_array_fftd_norm
,→ );
printf("R fft plot,

1046

,→ press ENTER to
,→ continue\n");
getchar();

1047
1048

1049
1050

//Now we find the peak of

184
,→ the resistance signal
,→ . (for ch2 or ch3)
1051

if(k==1){
FindAmpAtFreq(ch2_r_freq

1052

,→ , r_segment_len
,→ /2.0+1.0,
,→ samp_rate/2.0,
,→ resistance_array_fftd_norm
,→ , &max_valueR, &
,→ max_indexR);
1053

1054

else if(k==2){
FindAmpAtFreq(ch3_r_freq

1055

,→ , r_segment_len
,→ /2.0+1.0,
,→ samp_rate/2.0,
,→ resistance_array_fftd_norm
,→ , &max_valueR, &
,→ max_indexR);
1056

1057

//printf("Peak R voltage (
,→ off) is %e V at %f Hz
,→ \n",max_valueR,
,→ max_indexR*((
,→ samp_rate/2.0)/(
,→ r_segment_len
,→ /2.0+1.0)));

1058
1059

//Save this data in an array
,→ which will be
,→ averaged later

1060

if(numRoff>3000){//
,→ reallocate length if
,→ needed

1061

r_array_4avg_off = (
,→ double *)realloc(
,→ r_array_4avg_off
,→ ,(numRoff+1)*

185
,→ sizeof(double));
1062

1063

r_array_4avg_off[ind_off] =
,→ max_valueR;
numRoff++;

1064

1065
1066

1067
1068
1069

if(k==1 || k==2){
//Here we average the Ron/off data
,→ and save in file.

1070

//The on data

1071

for(r_cntr=0;r_cntr,→ ++){
R_avg_on = R_avg_on +

1072

,→ r_array_4avg_on[
,→ r_cntr]/((double)
,→ numRon);
1073

1074

//The off data

1075

for(r_cntr=0;r_cntr,→ ++){
R_avg_off = R_avg_off +

1076

,→ r_array_4avg_off[
,→ r_cntr]/((double)
,→ numRoff);
1077

1078
1079

//Save the data

1080

fp_Ron = fopen(filename_R_on,"a");

1081

fprintf(fp_Ron,"%le\t",R_avg_on);

1082

fclose(fp_Ron);

1083
1084

fp_Roff = fopen(filename_R_off,"a"
,→ );

1085

fprintf(fp_Roff,"%le\t",R_avg_off)
,→ ;

1086

fclose(fp_Roff);

186
1087
1088

//Reset values

1089

numRoff = 0;

1090

numRon = 0;

1091

R_avg_on = 0;

1092

R_avg_off = 0;

1093

1094
1095

printf("split the data into on and off
,→ portions\n");

1096
1097

if(plot_errchk==1){
printf("plotting first 1 values of

1098

,→ on and off data\n");
1099

int b;

1100

for(b=0;b<10;b++){
printf("on data: %f, off

1101

,→ data: %f\n",creal(
,→ ch_on_data[b]),creal(
,→ ch_off_data[b]));
1102

1103

printf("plotting on and off data
,→ for errorchecking\n");

1104

printf("plotting on data\n");

1105

plotplot(ind2plot,ch_on_data);

1106

printf("press ENTER to continue\n"
,→ );

1107

getchar();

1108

printf("plotting off data\n");

1109

plotplot(ind2plot,ch_off_data);

1110

printf("press ENTER to continue\n"
,→ );
getchar();

1111
1112

1113

printf("on_cntr = %d, off_cntr = %d\n",
,→ on_cntr,off_cntr);

1114
1115

if(on_cntrprintf("segment_len too large,

187
,→ larger than the on data.\n")
,→ ;
exit(0);

1116
1117

1118

if(off_cntrprintf("segment_len too large,

1119

,→ larger than the off data.\n"
,→ );
exit(0);

1120

1121
1122

1123
1124
1125

else{//for flat sprectrum, all data goes into

1126

,→ ch_off_data
for(ind_off = 0; ind_off < array_len;

1127

,→ ind_off++){
ch_off_data[ind_off] = NSYNC[

1128

,→ ind_off];

1129

1130
1131
1132
1133

//LOOP OVER THE ON AND OFF DATA

1134

,→ **********************************************************************************
,→
1135

for(is_on_ch = 0;is_on_ch < 2;){//We first do

1136

,→ the ffts of the on data, then the off
,→ data. 0 is off, 1 is on.
1137
1138
1139

//EXTRACT THE RESISTANCE SIGNAL

1140

,→ **********************************************************************************
,→

188
1141

//if(r_segment_len < off_cntr){
//printf("r_segment_len is longer

1142

,→ than the number in off array
,→ , for R measurements\n");
//exit(-1);

1143
1144

//}

1145

//for(r_cntr = 0; r_cntr< r_segment_len;
,→ r_cntr++){
////read the on or off data into

1146

,→ the array for ffting
//if(is_on_ch == 0){//read off

1147

,→ data
//resistance_array_time[

1148

,→ r_cntr] = ch_off_data
,→ [r_cntr];
1149

//}

1150

//else if(is_on_ch == 1){//read on
,→ data
//resistance_array_time[

1151

,→ r_cntr] = ch_on_data[
,→ r_cntr];
1152

//}

1153

//else{
//printf("problem with

1154

,→ number of is_on_ch,
,→ should be only 0 or
,→ 1\n");
//exit(1);

1155

//}

1156
1157

//}

1158

////perform the fft, puts the fft’d
,→ resistance_array_time into
,→ resistance_array_fftd

1159

//fftw_execute(plan_resistance);

1160
1161

////Problem: we can’t split up a signal,
,→ patch it together, and get accurate
,→ signal data for a single signal.

189
1162
1163
1164

//}

1165
1166
1167

//SPLIT DATA INTO SEGMENTS AND APPLY WINDOW FUNCTION
printf("is_on_ch is: %d (0 is off, 1 is
,→ on)\n",is_on_ch);

1168

segment_counter = 0;

1169

segs_left = 1;

1170

cum_data = 0;//reinitialize the
,→ cumulative data counter

1171

while(segs_left == 1){

1172
1173

//Putting the data into the segment
,→ allocation, multiplied by
,→ blackmanharris

1174

for(cntr=0;cntr,→ ++){

1175

//printf("ch_data_array %f\n
,→ ", ch_data_array[
,→ cum_data]);

1176

if(is_on_ch == 0){//read off
,→ data
segment_data_array[

1177

,→ cntr] =
,→ ch_off_data[
,→ cum_data]*
,→ blackmanharris_array
,→ [cntr];
1178

1179

else if(is_on_ch == 1){//
,→ read on data

1180

segment_data_array[
,→ cntr] =
,→ ch_on_data[
,→ cum_data]*
,→ blackmanharris_array
,→ [cntr];

190
1181

1182

else{
printf("problem with

1183

,→ number of
,→ is_on_ch,
,→ should be only
,→ 0 or 1\n");
exit(1);

1184

1185
1186

cum_data = cum_data+1;

1187
1188

1189
1190
1191
1192

//DO THE FFT FOR NOISE CALCULATIONS

1193

,→ **********************************************************************************
,→
1194
1195

//printf("executing segment fft %d

1196

,→ \n",cum_data);
fftw_execute(plan_segment);

1197
1198
1199
1200

//DIRECTLY PLOT THE NSD AND GET AVERAGE IF ERROR CHECKING

1201

,→ **********************************************************************************
,→
1202
1203

//error checking fft

1204

if(plot_errchk == 1 &&
,→ segment_counter==0){

1205

printf("plotting fft\n");

1206

double *fft_mag;//for
,→ plotting

1207

fft_mag = (double*)malloc(

191
,→ sizeof(double)*(
,→ segment_len/2+1));
1208

double avg_nsd_test=0.0;

1209

int bb;

1210

for(bb=0;bb<(segment_len
,→ /2+1);bb++){

1211

fft_mag[bb] = (2.0*

1212

,→ creal(
,→ segment_fft_array
,→ [bb])*2.0*
,→ creal(
,→ segment_fft_array
,→ [bb]) + 2.0*
,→ cimag(
,→ segment_fft_array
,→ [bb])*2.0*
,→ cimag(
,→ segment_fft_array
,→ [bb]))/((
,→ window_norm_correct
,→ ));//actually
,→ the psd
,→ currently,
,→ this is a
,→ convenient
,→ place to plot
,→ NSD, FFT, PSD
,→ for error
,→ checking
avg_nsd_test =

1213

,→ avg_nsd_test +
,→ fft_mag[bb];
1214

1215

avg_nsd_test = avg_nsd_test
,→ /(samp_rate/2);

1216

printf("avg_nsd_test = %e\n"
,→ ,sqrt(avg_nsd_test));

192
plotlogplot(segment_len/2+1,

1217

,→ fft_mag);
1218

printf("press ENTER to

1219

,→ continue.\n");
1220

getchar();

1221

free(fft_mag);

1222
1223
1224
1225

//DETERMINE IF THERE ARE SEGMENTS LEFT TO FFT

1226

,→ **********************************************************************************
,→
1227
1228

if(is_flat == 0){
if(is_on_ch == 0){

1229

if(off_cntr-(cum_data

1230

,→ -(int)
,→ segment_len/2)
,→ < (int)
,→ segment_len){
segs_left = 0;

1231

,→
,→
,→ }

1232

else if(is_on_ch == 1){

1233

if(on_cntr-(cum_data

1234

,→ -(int)
,→ segment_len/2)
,→ < (int)
,→ segment_len){
segs_left = 0;

1235

1236

1237
1238

1239

else{//if not chopped

193
if(array_len-(cum_data-(int)

1240

,→ segment_len/2) < (int
,→ )segment_len){//Could
,→ simplify by setting
,→ off_cntr = array_len
,→ earlier in the code,
,→ but this is probably
,→ easier to debug later
segs_left = 0;

1241

1242

1243
1244
1245

//FIND THE POWER SPECTRUM AND ADD TO THE CUMULATIVE ARRAY AND
,→ NORMALIZE

1246

,→ **********************************************************************************
,→
1247
1248
1249
1250

//do the welch and normalize here
,→ and cumulative

1251

for(cntr=0;cntr<(int)segment_len
,→ /2+1;cntr++){
psd_cumulative_array[cntr] =

1252

,→ psd_cumulative_array
,→ [cntr] + 2.0*(creal(
,→ segment_fft_array[
,→ cntr])*creal(
,→ segment_fft_array[
,→ cntr]) + cimag(
,→ segment_fft_array[
,→ cntr])*cimag(
,→ segment_fft_array[
,→ cntr]))/
,→ window_norm_correct;
1253
1254

194
1255

//printf("psd_cumulative_array

1256

,→ [200] = %0.9f\n",
,→ psd_cumulative_array[200]);
segment_counter = segment_counter

1257

,→ + 1;
//printf("segment counter is: %d\n

1258

,→ ",segment_counter);
cum_data = cum_data - (int)

1259

,→ segment_len/2; //This is the
,→ 0.5 overlap of Welch’s
,→ method.
//printf("cumulative data = %d\n",

1260

,→ cum_data);
1261
1262

1263

//last step of averaging all of the PSD
,→ data - divide by number of segments

1264

max_psd = 0.0;

1265

min_psd = 100.0;

1266
1267

for(cntr=0;cntr<(int)segment_len/2+1;cntr
,→ ++){

1268

psd_cumulative_array[cntr] =
,→ psd_cumulative_array[cntr]/
,→ segment_counter;

1269

if(psd_cumulative_array[cntr]<
,→ min_psd){//find min of psd
min_psd =

1270

,→ psd_cumulative_array[
,→ cntr];
1271

1272

if(psd_cumulative_array[cntr]>
,→ max_psd){//find max of psd
max_psd =

1273

,→ psd_cumulative_array[
,→ cntr];
1274

195
1275

1276
1277

printf("min of psd_cumulative_array is:

1278

,→ %0.10e\n",min_psd);
printf("max of psd_cumulative_array is:

1279

,→ %0.10e\n",max_psd);
1280
1281
1282
1283

//PLOTTING SEPARATE NOISE SOURCES

1284

,→ **********************************************************************************
,→
1285
1286
1287

//plotting all noise sources separately

1288

if(plot_separate_source == 1){

1289

int ggg=0;

1290

double **cirq_comp_noise;

1291

cirq_comp_noise = array2D(14, (int
,→ )(segment_len/2+1));//malloc
,→ a 2d array

1292
1293
1294

double *ch_gc;

1295

ch_gc = (double *)malloc(sizeof(
,→ double)*(segment_len/2+1));

1296
1297

double rrss;//rs

1298

if(k==0){//ch1

1299

printf("plotting channel 1\n
,→ ");

1300

int xy;

1301

for(xy=0;xy<(segment_len
,→ /2+1);xy++){

1302

ch_gc[xy] =
,→ gain_ch1_data_interp

196
,→ [xy];
//printf("ch_gc: %f\n

1303

,→ ",ch_gc[xy]);
1304

1305

rrss = r_of_chans[0];

1306
1307
1308

1309

if(k==1){//ch2
printf("plotting channel 2\n

1310

,→ ");
1311

int xy;

1312

for(xy=0;xy<(segment_len
,→ /2+1);xy++){
ch_gc[xy] =

1313

,→ gain_ch2_data_interp
,→ [xy];
1314

1315

rrss = r_of_chans[1];

1316

1317

if(k==2){//ch3
printf("plotting channel 3\n

1318

,→ ");
1319

int xy;

1320

for(xy=0;xy<(segment_len
,→ /2+1);xy++){
ch_gc[xy] =

1321

,→ gain_ch3_data_interp
,→ [xy];
1322

1323

rrss = r_of_chans[2];

1324

1325

if(k==3){//ch4

1326

printf("plotting channel 4\n
,→ ");

1327

int xy;

1328

for(xy=0;xy<(segment_len
,→ /2+1);xy++){

197
ch_gc[xy] =

1329

,→ gain_ch4_data_interp
,→ [xy];
1330

1331

rrss = r_of_chans[3];

1332

1333

//double freqy;

1334

int hh;

1335

for(ggg=0;ggg<(segment_len/2+1);
,→ ggg++){

1336

g1 = sqrt(ch_gc[ggg]);

1337

g2 = g1;

1338

//frequency -> 0th row

1339

cirq_comp_noise[0][ggg] =
,→ ggg*(samp_rate/2.0)/(
,→ segment_len/2.0+1.0);

1340

//daq -> 1st row

1341

cirq_comp_noise[1][ggg] =
,→ daq*daq;

1342

//r4 V -> 2nd row

1343

cirq_comp_noise[2][ggg] =
,→ 4.0*r4*KB*temp;

1344

//r4 I -> 3rd row

1345

cirq_comp_noise[3][ggg] = (
,→ in*r4)*(in*r4);

1346

//r3 V -> 4th row

1347

cirq_comp_noise[4][ggg] =
,→ 4.0*r3*KB*temp*(r4/r3
,→ )*(r4/r3)*(g2/gainz2)
,→ *(g2/gainz2);

1348

//en 2nd stage -> 5th row

1349

cirq_comp_noise[5][ggg] = en
,→ *en*g2*g2;

1350

//in 2nd stage -> 6th row

1351

cirq_comp_noise[6][ggg] = (
,→ in*zout)*(in*zout)*g2
,→ *g2;

1352

//r2 V -> 7th row

198
1353

cirq_comp_noise[7][ggg] =
,→ 4.0*r2*KB*temp*g2*g2;

1354

//r2 I -> 8th row

1355

cirq_comp_noise[8][ggg] = (
,→ in*r2)*(in*r2)*g2*g2;

1356

//r1 V -> 9th row

1357

cirq_comp_noise[9][ggg] =
,→ 4.0*r1*KB*temp*(r2/r1
,→ )*(r2/r1)*(g1/gainz1)
,→ *(g1/gainz1)*g2*g2;

1358

//en 1st stage -> 10th row

1359

cirq_comp_noise[10][ggg] =
,→ en*en*g1*g1*g2*g2;

1360

//in 1st stage -> 11th row

1361

cirq_comp_noise[11][ggg] = (
,→ in*find_magz(
,→ cirq_comp_noise[0][
,→ ggg],rrss))*(in*
,→ find_magz(
,→ cirq_comp_noise[0][
,→ ggg],rrss))*g1*g1*g2*
,→ g2;

1362

//sample V (theory) -> 12th
,→ row

1363

cirq_comp_noise[12][ggg] =
,→ 4.0*KB*rrss*temp*g1*
,→ g1*g2*g2;

1364

//total (theory) -> 13th row

1365

cirq_comp_noise[13][ggg] =
,→ 0.0;

1366

for(hh=1;hh<13;hh++){//skip
,→ row 0, the frequency
,→ row

1367

cirq_comp_noise[13][
,→ ggg] =
,→ cirq_comp_noise
,→ [13][ggg] +
,→ cirq_comp_noise

199
,→ [hh][ggg];

1368
1369
1370
1371

1372
1373
1374

// + + + + + + + + + + ;

1375
1376

plotlogplotmany(14,segment_len
,→ /2+1,psd_cumulative_array,
,→ cirq_comp_noise);

1377
1378

printf("Press ENTER to continue\n"
,→ );

1379

getchar();

1380
1381

//Here plot the signals relative
,→ to input (divide out the
,→ gain)

1382
1383

double *
,→ psd_cumulative_array_rel2in;

1384

psd_cumulative_array_rel2in = (
,→ double *)malloc(sizeof(
,→ double)*(segment_len/2.0
,→ +1.0));

1385
1386
1387

for(ggg=0;ggg<(segment_len/2+1);
,→ ggg++){

1388

g1 = sqrt(ch_gc[ggg]);

1389

g2 = g1;

1390
1391

//daq -> 1st row

1392

cirq_comp_noise[1][ggg] =
,→ cirq_comp_noise[1][
,→ ggg]/(g1*g1*g2*g2);

200
1393

//r4 V -> 2nd row

1394

cirq_comp_noise[2][ggg] =
,→ cirq_comp_noise[2][
,→ ggg]/(g1*g1*g2*g2);

1395

//r4 I -> 3rd row

1396

cirq_comp_noise[3][ggg] =
,→ cirq_comp_noise[3][
,→ ggg]/(g1*g1*g2*g2);

1397

//r3 V -> 4th row

1398

cirq_comp_noise[4][ggg] =
,→ cirq_comp_noise[4][
,→ ggg]/(g1*g1*g2*g2);

1399

//en 2nd stage -> 5th row

1400

cirq_comp_noise[5][ggg] =
,→ cirq_comp_noise[5][
,→ ggg]/(g1*g1*g2*g2);

1401

//in 2nd stage -> 6th row

1402

cirq_comp_noise[6][ggg] =
,→ cirq_comp_noise[6][
,→ ggg]/(g1*g1*g2*g2);

1403

//r2 V -> 7th row

1404

cirq_comp_noise[7][ggg] =
,→ cirq_comp_noise[7][
,→ ggg]/(g1*g1*g2*g2);

1405

//r2 I -> 8th row

1406

cirq_comp_noise[8][ggg] =
,→ cirq_comp_noise[8][
,→ ggg]/(g1*g1*g2*g2);

1407

//r1 V -> 9th row

1408

cirq_comp_noise[9][ggg] =
,→ cirq_comp_noise[9][
,→ ggg]/(g1*g1*g2*g2);

1409

//en 1st stage -> 10th row

1410

cirq_comp_noise[10][ggg] =
,→ cirq_comp_noise[10][
,→ ggg]/(g1*g1*g2*g2);

1411

//in 1st stage -> 11th row

1412

cirq_comp_noise[11][ggg] =

201
,→ cirq_comp_noise[11][
,→ ggg]/(g1*g1*g2*g2);
//sample V (theory) -> 12th

1413

,→ row
cirq_comp_noise[12][ggg] =

1414

,→ cirq_comp_noise[12][
,→ ggg]/(g1*g1*g2*g2);
1415

//total (theory) -> 13th row

1416

cirq_comp_noise[13][ggg] =
,→ 0.0;
for(hh=1;hh<13;hh++){//skip

1417

,→ row 0, the frequency
,→ row
cirq_comp_noise[13][

1418

,→ ggg] =
,→ cirq_comp_noise
,→ [13][ggg] +
,→ cirq_comp_noise
,→ [hh][ggg];
1419

1420

psd_cumulative_array_rel2in[
,→ ggg] =
,→ psd_cumulative_array[
,→ ggg]/(g1*g1*g2*g2);

1421

1422

printf("Plotting relative to input
,→ .\n");

1423

plotlogplotmany(14,segment_len
,→ /2+1,
,→ psd_cumulative_array_rel2in,
,→ cirq_comp_noise);

1424

printf("Press ENTER to continue\n"
,→ );

1425

getchar();

1426

free(psd_cumulative_array_rel2in);

1427

free(ch_gc);

1428
1429

free(cirq_comp_noise);

202
1430
1431

1432
1433

//Here is where the gain versus frequency
,→ is taken into account. 4*Rs*kb*T*
,→ gain^2 = (PSD measured) - (circuit
,→ noise power), so 4*Rs*kb*T = (
,→ PSD_measured - circuit_noise_power)
,→ /gain^2

1434
1435

for(cntr=0;cntrif(k==0){//if it’s ch1
psd_cumulative_array[cntr] =

1436

,→ (
,→ psd_cumulative_array[
,→ cntr],→ ch1_indep_noise_pwr[
,→ cntr])/(
,→ gain_ch1_data_interp[
,→ cntr]*
,→ gain_ch1_data_interp[
,→ cntr]);
1437

1438

else if(k==1){//if its ch2
psd_cumulative_array[cntr] =

1439

,→ (
,→ psd_cumulative_array[
,→ cntr],→ ch2_indep_noise_pwr[
,→ cntr])/(
,→ gain_ch2_data_interp[
,→ cntr]*
,→ gain_ch2_data_interp[
,→ cntr]);
1440

1441

else if(k==2){//if its ch3

1442

psd_cumulative_array[cntr] =
,→ (
,→ psd_cumulative_array[

203
,→ cntr],→ ch3_indep_noise_pwr[
,→ cntr])/(
,→ gain_ch3_data_interp[
,→ cntr]*
,→ gain_ch3_data_interp[
,→ cntr]);
1443

1444

else if(k==3){//if its ch3
psd_cumulative_array[cntr] =

1445

,→ (
,→ psd_cumulative_array[
,→ cntr],→ ch4_indep_noise_pwr[
,→ cntr])/(
,→ gain_ch4_data_interp[
,→ cntr]*
,→ gain_ch4_data_interp[
,→ cntr]);
1446

1447

else{
printf("gain correction not

1448

,→ set up for more than
,→ 4 channels\n");
exit(0);

1449

1450
1451

1452
1453

printf("min of psd_cumulative_array after
,→ gain correction is: %0.10e\n",
,→ min_psd);

1454

printf("max of psd_cumulative_array after
,→ gain correction is: %0.10e\n",
,→ max_psd);

1455
1456
1457
1458

//plotting psd averaged

204
1459

if(plot_psd_cumul_array == 1){
plotlogplot(segment_len/2+1,

1460

,→ psd_cumulative_array);
printf("Press ENTER to continue\n"

1461

,→ );
getchar();

1462
1463

1464
1465

//average the PSD here and get the NSD as
,→ well

1466

avg_PSD = 0;

1467

avg_NSD = 0;

1468

for(cntr = freq_min_ind;cntr,→ +1;cntr++){
avg_PSD = avg_PSD +

1469

,→ psd_cumulative_array[cntr]/(
,→ freq_max_ind-freq_min_ind);
1470
1471

1472
1473

avg_NSD = sqrt(avg_PSD);

1474

printf("writing to file\n");

1475

if(is_on_ch == 0){

1476

fp_off = fopen(filename_nsd_off,"a

1477

,→ ");
1478

fprintf(fp_off,"%le\t",avg_NSD);

1479

fclose(fp_off);

1480
1481

1482

else if(is_on_ch == 1){
fp_on = fopen(filename_nsd_on,"a")

1483

,→ ;
1484

fprintf(fp_on,"%le\t",avg_NSD);

1485

fclose(fp_on);

1486

1487

printf("finished writing to file\n");

1488

printf("avg PSD is %le\n",avg_PSD);

205
printf("avg NSD is %le\n",avg_NSD);

1489
1490
1491

//SET psd_cumulative_array BACK TO ZERO

1492

,→ **********************************************************************************
,→
for(cntr = 0;cntr<(segment_len/2.0 + 1);

1493

,→ cntr++){
psd_cumulative_array[cntr] = 0.0;

1494

1495
1496

//If using chopped data, you need the on

1497

,→ part as well
1498

if(is_flat == 0){

1499

is_on_ch++;

1500

1501

else{//don’t need the on part here.
is_on_ch = is_on_ch + 2;

1502

1503
1504
1505

1506
1507
1508
1509

printf("it worked! congrats!\n");

1510
1511
1512
1513

1514

//add a newline to the text files to represent a new
,→ scan

1515
1516

fp_off = fopen(filename_nsd_off,"a");

1517

fprintf(fp_off,"\n");

1518

fclose(fp_off);

1519
1520

if(is_flat == 0 || getRconst == 1){//need to save the
,→ on data if chopped, and the tev data

206
1521

fp_on = fopen(filename_nsd_on,"a");

1522

fp_tev = fopen(filename_tev,"a");

1523

fp_Ron = fopen(filename_R_on,"a");

1524

fp_Roff = fopen(filename_R_off,"a");

1525
1526

fprintf(fp_on,"\n");

1527

fprintf(fp_tev,"\n");

1528

fprintf(fp_Ron,"\n");

1529

fprintf(fp_Roff,"\n");

1530
1531

fclose(fp_on);

1532

fclose(fp_tev);

1533

fclose(fp_Ron);

1534

fclose(fp_Roff);

1535
1536
1537
1538

1539

//fftw_destroy_plan(plan);

1540

free(NSYNC);

1541

free(on_starts);

1542

free(off_starts);

1543

free(ch1_indep_noise_pwr);

1544

free(ch2_indep_noise_pwr);

1545

free(ch3_indep_noise_pwr);

1546

free(gain_ch1_data_interp);

1547

free(gain_ch2_data_interp);

1548

free(gain_ch3_data_interp);

1549
1550
1551

//fftw_free(ch_fft_array);

1552

if(is_flat == 0){

1553

fftw_free(ch_on_data);

1554

free(segment_tev_array);

1555

fftw_destroy_plan(plan_segment_tev);

1556

fftw_free(segment_tev_fft_array);

1557

fftw_destroy_plan(plan_resistance);

1558

fftw_free(resistance_array_time);

207
fftw_free(resistance_array_fftd);

1559

1560
1561

fftw_free(ch_off_data);

1562
1563
1564

fftw_destroy_plan(plan_segment);

1565

free(segment_data_array);

1566

fftw_free(segment_fft_array);

1567

//freeing uninterpolated gain data as it is no longer needed

1568

free(gain_ch1_data);

1569

free(gain_ch2_data);

1570

free(gain_ch3_data);

1571

free(gain_ch4_data);

1572

free(gain_file_freq_data);

1573

if(tev_gain_meas == 1){

1574
1575

free(gain_ch1_data_tev);

1576

free(gain_ch2_data_tev);

1577

free(gain_ch3_data_tev);

1578

free(gain_ch4_data_tev);
free(gain_file_freq_data_tev);

1579

1580
1581
1582
1583

return 0;

1584
1585
1586
1587
1588
1589
1590
1591

1592
1593

//Windowing function for Welch’s PSD method - blackmanharris. This
,→ takes in the length of the welch segment, the place of the
,→ number in the welch segment (n), and returns a fraction to
,→ attenuate the number by to make a good blackmanharris window.

208
,→ So it returns a single number.
1594

void plotplot(long data_len,fftw_complex *ydata){

1595

FILE *pipefft = popen("gnuplot -persist","w");

1596

fprintf(pipefft, "set xlabel ’Index’\n");

1597

fprintf(pipefft, "set ylabel ’Value x 1e6’\n");

1598

fprintf(pipefft, "plot ’-’ with lines\n");

1599
1600

int kk;

1601

for(kk=0;kkfprintf(pipefft, "%d %f\n",kk,creal(ydata[kk]));//

1602

,→ *1000000);
1603

printf("kitty\n");

1604

fprintf(pipefft, "e\n");

1605

fprintf(pipefft, "refresh\n");

1606

pclose(pipefft);

1607
1608

1609
1610

void plotmultiplot(long data_len,double *ydata1, double *ydata2){
FILE *pipefft = popen("gnuplot -persist","w");

1611
1612
1613

fprintf(pipefft, "set xlabel ’Index’\n");

1614

fprintf(pipefft, "set ylabel ’Value’\n");

1615

fprintf(pipefft, "plot ’-’\n");

1616
1617
1618

int kk;

1619

for(kk=0;kk

1620

fprintf(pipefft, "%d %f\n",kk,ydata1[kk]);

1621

fprintf(pipefft, "%d %f\n",kk,ydata2[kk]);

1622

1623
1624
1625

printf("kitty\n");

1626

fprintf(pipefft, "e\n");

1627

fprintf(pipefft, "refresh\n");

1628

pclose(pipefft);

1629

209
1630
1631

void plotlogplot(long data_len,double *ydata){

1632

FILE *pipefft = popen("gnuplot -persist","w");

1633

fprintf(pipefft, "set logscale xy 10\n");

1634

fprintf(pipefft, "set xlabel ’Frequency (Hz)’\n");

1635

fprintf(pipefft, "set ylabel ’NSD (V/sqrt(Hz))’\n");

1636
1637

fprintf(pipefft, "plot ’-’ with linespoints\n");

1638

int kk;

1639

for(kk=0;kkfprintf(pipefft, "%f %f\n",kk*(1000000.0/(data_len)),

1640

,→ ydata[kk]*1.0e15);
1641

1642
1643
1644

printf("kitty\n");

1645

fprintf(pipefft, "e\n");

1646

fprintf(pipefft, "refresh\n");

1647

pclose(pipefft);

1648

1649
1650

void plotlogplotfft(long data_len,long nyq,double *ydata){

1651

FILE *pipefft = popen("gnuplot -persist","w");

1652

fprintf(pipefft, "set logscale xy 10\n");

1653

fprintf(pipefft, "set xlabel ’Frequency (Hz)’\n");

1654

fprintf(pipefft, "set ylabel ’FFT (V)’\n");

1655
1656

fprintf(pipefft, "plot ’-’ with linespoints\n");

1657

int kk;

1658

printf("plotting every 1 data points\n");

1659

for(kk=0;kkfprintf(pipefft, "%f %f\n",kk*((double)nyq/((double)

1660

,→ data_len)),ydata[kk]*1.0e6);
1661

kk=kk+1;

1662
1663

1664
1665

printf("kitty\n");

210
1666

fprintf(pipefft, "e\n");

1667

fprintf(pipefft, "refresh\n");

1668

pclose(pipefft);

1669

1670
1671

void plotlogplot8(long data_len,double *ydata1,double *ydata2,double
,→ *ydata3,double *ydata4,double *ydata5,double *ydata6,double *
,→ ydata7,double *ydata8){

1672
1673

//I was unable to generate a plot with multiple lines of
,→ different color using just a c script. The data must be
,→ saved to a text file first

1674
1675

FILE *plotptr = fopen("data2plot.txt","w");

1676

int h;

1677

for(h=0;hfprintf(plotptr,"%f %f %f %f %f %f %f %f %f %f\n",h

1678

,→ *(1000000.0/(data_len)),ydata1[h]*1.0e10,ydata2[
,→ h]*1.0e10,ydata3[h]*1.0e10,ydata4[h]*1.0e10,
,→ ydata5[h]*1.0e10,ydata6[h]*1.0e10,ydata7[h]*1.0
,→ e10,ydata8[h]*1.0e10,(1.0e10)*(ydata2[h]+ydata3[
,→ h]+ydata4[h]+ydata5[h]+ydata6[h]+ydata7[h]+
,→ ydata8[h]));
1679

1680
1681

fclose(plotptr);

1682
1683

FILE *pipefft = popen("gnuplot -persist","w");

1684

fprintf(pipefft, "set logscale xy 10\n");

1685

fprintf(pipefft, "set xlabel ’Frequency (Hz)’\n");

1686

fprintf(pipefft, "set ylabel ’PSD (V^2/Hz)’\n");

1687
1688

fprintf(pipefft,"set datafile separator whitespace\n");

1689

fprintf(pipefft, "plot ’data2plot.txt’ using 1:2 title ’
,→ Measured PSD’ with lines, ’’ using 1:3 title ’En^2’
,→ with linespoints, ’’ using 1:4 title ’E1^2’, ’’ using
,→ 1:5 title ’E2^2’, ’’ using 1:6 title ’(In|Zs|^2)’, ’’
,→ using 1:7 title ’DAQ^2’, ’’ using 1:8 title ’(InR2)^2’,

211
,→ ’’ using 1:9 title ’Es^2’, ’’ using 1:10 with lines lc
,→ rgb ’black’ title ’theory all’\n");// with linespoints
,→ \n");
1690
1691
1692

printf("kitty\n");

1693

//fprintf(pipefft, "e\n");

1694

fprintf(pipefft, "refresh\n");

1695

pclose(pipefft);

1696

printf("closed the pipe\n");

1697

1698
1699
1700

void plotlogplotmany(int num_rows,long data_len,double *total_meas,
,→ double **cirq_noise){

1701

//I was unable to generate a plot with multiple lines of
,→ different color using just a c script. The data must be
,→ saved to a text file first

1702
1703

FILE *plotptr = fopen("data2plot.txt","w");

1704

int cl;

1705

int rw;

1706

for(cl=0;clfprintf(plotptr,"%f ",cirq_noise[0][cl]);//frequency (x

1707

,→ -axis)
for(rw=1;rw

1708

fprintf(plotptr,"%f ",cirq_noise[rw][cl]*1.0e20)

1709

,→ ;
1710

1711

fprintf(plotptr,"%f\n",total_meas[cl]*1.0e20);//
,→ measured data

1712

1713
1714

fclose(plotptr);

1715
1716

FILE *pipefft = popen("gnuplot -persist","w");

1717

fprintf(pipefft, "set logscale xy 10\n");

1718

fprintf(pipefft, "set xlabel ’Frequency (Hz)’\n");

212
fprintf(pipefft, "set ylabel ’PSD (1e20 x V^2/Hz)’\n");

1719
1720
1721

fprintf(pipefft,"set datafile separator whitespace\n");

1722

fprintf(pipefft, "plot ’data2plot.txt’ using 1:2 title ’DAQ’
,→ with lines, ’’ using 1:3 title ’R4 V’ with linespoints,
,→ ’’ using 1:4 title ’R4 I’, ’’ using 1:5 title ’R3 V’,
,→ ’’ using 1:6 title ’en 2nd’, ’’ using 1:7 title ’in 2nd
,→ ’, ’’ using 1:8 title ’R2 V’, ’’ using 1:9 title ’R2 I
,→ ’, ’’ using 1:10 with lines lc rgb ’black’ title ’R1 V
,→ ’, ’’ using 1:11 title ’en 1st’, ’’ using 1:12 title ’
,→ in 1st’, ’’ using 1:13 title ’Rs (theory)’, ’’ using
,→ 1:14 title ’total (theory)’, ’’ using 1:15 title ’total
,→ (meas)’\n");

1723

printf("kitty\n");

1724

fprintf(pipefft, "refresh\n");

1725

pclose(pipefft);

1726

printf("closed the pipe\n");

1727
1728

1729
1730
1731

//Function for calculating the complex impedence as a function of
,→ frequency. Inputs are the frequency of interest and the rs
,→ source resistance, output is the magnitude of the complex
,→ impedence for that frequency. Note that the C and R’s besides
,→ Rs are fixed, so for a different circuit this funciton will
,→ need to be modified.

1732

double find_magz(double freq_val, double rss){

1733

double magz; //magnitude of z that will be returned

1734

double complex rg = 10e6; //resistor to ground

1735

double complex c2_z = -I/(freq_val*100e-9);//complex impedence
,→ of the capacitor parallel to rg

1736

rss = (double complex)rss;//convert to complex number (not
,→ sure if this is necessary)

1737

double complex c1_z = -I/(freq_val*1e-9);//complex impedence
,→ of AC coupling capacitor to DAQ

1738

double complex r3 = 1e6;//resistor to ground between c1 and
,→ DAQ

213
1739
1740

//computing the impedence

1741

double complex z1 = (1/(1/rg + 1/c2_z));//thevenin equivalent

1742

double complex z2 = z1 + rss + c1_z;//thevenin equivalent

1743

double complex z3 = 1/(1/z2 + 1/r3);//final thevanin
,→ equivalent
magz = sqrt(creal(z3)*creal(z3) + cimag(z3)*cimag(z3));//

1744

,→ getting the norm of the impedence
return magz;

1745
1746

1747
1748

//Interpolation function - returns interpolated y value for inputed x
,→ value. x_new is the x value you have and want the y value for
,→ , and x_base and y_base are the data you have to interpolate
,→ with.

1749

double linear_interpy(double x_new, double *x_base, double *y_base,
,→ double max_freqq){

1750

double y_new;

1751

int cntry=0;

1752

//taking care of special cases

1753

if(x_new<=x_base[0]){
y_new = y_base[0];

1754
1755

1756

else if(x_new > max_freqq){

1757

printf("interpolation frequency too high\n");

1758

exit(0);

1759

1760

else{

1761

while(x_new>x_base[cntry]){//find the location of the
,→ value in x_base x_new is just larger than
cntry++;

1762
1763
1764

1765

cntry = cntry-1;//back up one

1766

//y=mx+b. m=(y_base(n+1)-y_base(n))/(x_base(n+1)-x_base
,→ (n)). b = y_base(n)-m*x_base(n). y_new=m*x_new+b
,→ . Algebraically simplified.

1767

y_new = y_base[cntry] + ((y_base[cntry+1]-y_base[cntry

214
,→ ])/(x_base[cntry+1]-x_base[cntry]))*(x_new,→ x_base[cntry]);
1768

1769

return y_new;

1770

1771
1772

//Calculate the noise to be subtracted from the measured noise. May
,→ be better to make inputs global variables.

1773

1774
1775

double indep_noise_pwr, g1, g2;

1776

g1 = sqrt(gain_data);

1777

g2 = g1;//Assume both stages have the same gain and rolloff (
,→ if rolloff used)

1778

indep_noise_pwr = daq*daq + 4.0*r4*KB*temp + (in*r4)*(in*r4) +

1779

,→ 4.0*r3*KB*temp*(r4/r3)*(r4/r3)*(g2/gainz2)*(g2/gainz2)
,→ + en*en*g2*g2 + (in*zout)*(in*zout)*g2*g2 + 4.0*r2*KB*
,→ temp*g2*g2 + (in*r2)*(in*r2)*g2*g2 + 4.0*r1*KB*temp*(r2
,→ /r1)*(r2/r1)*(g1/gainz1)*(g1/gainz1)*g2*g2 + en*en*g1*
,→ g1*g2*g2 + (in*rs)*(in*rs)*g1*g1*g2*g2;
1780

return indep_noise_pwr;

1781
1782

1783
1784

//FFT window function (4-term blackmanharris)

1785

double blackmanharris(long n,long segment_length){

1786

double result;

1787

double a0,a1,a2,a3;

1788

a0 = 0.35875;

1789

a1 = 0.48829;

1790

a2 = 0.14128;

1791

a3 = 0.01168;

1792

result = a0-a1*cos(2.0*M_PI*(double)n/((double)segment_length
,→ -1.0))+a2*cos(4.0*M_PI*(double)n/((double)
,→ segment_length-1.0))-a3*cos(6.0*M_PI*(double)n/((double

215
,→ )segment_length-1.0));
return result;

1793
1794

1795
1796

//function to make 2D array

1797

double** array2D(int xdim, int ydim) {

1798

double** array2D_return;

1799

array2D_return = (double**) malloc(xdim*sizeof(double*));

1800

int z;

1801

for (z = 0; z < xdim; z++)
array2D_return[z] = (double*) malloc(ydim*sizeof(double

1802

,→ ));
1803

return array2D_return;

1804
1805

1806
1807

//function to find amplitude of frequency value. fft_length is #
,→ samples/2 +1

1808

void FindAmpAtFreq(double freq_of_interest, double fft_length, double
,→ nyquist_freq, double *fft_to_search, double *max_value,
,→ double *max_index){

1809

double max_tev = 0.0;//initialize to zero to find max voltage
,→ peak

1810

int max_tev_ind = 0;//find index of peak

1811

int tev_ind;

1812

for(tev_ind = (int)((freq_of_interest-5)/(nyquist_freq/
,→ fft_length));tev_ind<(int)((freq_of_interest+300)/(
,→ nyquist_freq/fft_length));tev_ind++){//search for the
,→ peak +/- 5 Hz around the chopper rate (in case the
,→ frequency was actually 43.1 Hz instead of 43 Hz, just
,→ to get better accuracy)
if(fft_to_search[tev_ind]>max_tev){

1813

max_tev = fft_to_search[tev_ind];//find max

1814

,→ voltage in this range
max_tev_ind = tev_ind;

1815

1816
1817

1818

//Put the max values into the pointers which store these

216
,→ values
1819

*max_value = max_tev;

1820

*max_index = max_tev_ind;

1821

//printf("max_value should be: %f\n",max_tev);

1822

//printf("max_index should be: %d\n",max_tev_ind);

1823

217
Appendix B

PYTHON GUIS FOR INSTRUMENT CONTROL
We created GUIs in python using the tkinter package (see Figures in this Appendix),
which is free and comes with most Python installations. It allows for data collection
from the USB 6366 (and with minor modifications, can take data from the USB
4431). Plotting is fairly slow, and efficient data collection is not built into this GUI,
so it is best for ensuring there are no peaks in the noise spectra and if the measured
noise is close to the theoretical noise.

Figure B.1: Settings tab for noise measurements.

As an aside, the metal film resistors in the circuit are, from channel 1 to channel 4,
1.5 kΩ, 2.2 kΩ, 3.3 kΩ, and 4.7 kΩ, respectively. The theoretical noise, using the
circuit described in the thesis, we expect to see measured noise of approximately
636, 722, 839, and 967 nV/sqrtHz, respectively. The error seen between theory
and measurement is then between 7-14%, which is acceptable (Kay, 2012). For our
measurements, as earlier chapters describe, the noise percentage off from theory
was constant. For example, by the error stated above, we mean that channel 1 was
consistently too high by about 7%, not that the measurement was within theory by

218
± 7%. Thus, our measurements were very precise, but less accurate, indicating that
using a constant offset to extract noise was acceptable. The RTD data and DAQ data
was collected simultaneously via multiprocessing.

Figure B.2: Tab in GUI which shows the NSD for 4 channels. The display on the
right averages the NSD bewteen values set in the settings tab. The theoretical R
can be set in the settings tab, but the theoretical NSD prediction is only valid for
very simple one-inamp systems, but this could be easily modified. The channel 4
spectrum is showing noise peaks from the proximity of the surface mount RTD; we
had to remove the connection to this RTD for measurements due to these peaks.
Physically moving the channel 4 signal traces farther from the SMT RTD would
likely fix this problem.
The code for the GUI and data collection programs for the USB6366 and OM-USBTEMP-AI is available upon request.

219

Figure B.3: Tab in GUI which shows the noise spectral density average for 2
channels, a voltage measurement between them (DC), and temperatures from the
RTDs on the circuit as a function of time.

Figure B.4: Data for channel 1.

220

Figure B.5: Data for channel 2.

Figure B.6: Data for channel 3.

221

Figure B.7: Data for channel 4.

INDEX
bibliography, 128
figures, 5, 12, 19–33, 37–49, 55–57, 61–66, 71, 80, 84–87, 89–103, 105, 110–124,
217–221
noise spectral density NSD, 10
power spectral density PSD, 10, 11
Seebeck coefficient α, 4, 11
tables, 36, 58, 59

222