Self-Heating of HEMT Low-Noise Amplifiers in Liquid Cryogenic Environments and the Limits of Microwave Noise Performance - Calt

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Self-Heating of HEMT Low-Noise Amplifiers in Liquid Cryogenic Environments and the Limits of Microwave Noise Performance
Citation
Ardizzi, Anthony Joseph
(2022)
Self-Heating of HEMT Low-Noise Amplifiers in Liquid Cryogenic Environments and the Limits of Microwave Noise Performance.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/9va8-tc64.
Abstract
Detection and processing of microwave signals is of substantial scientific importance in fields ranging from radio astronomy to quantum computing. An essential component of the signal processing chain is the microwave amplifier, which adds gain to the signal so that it may be processed by subsequent microwave components. However, the amplifier itself adds its own internally generated noise into the measurement chain. As a result, amplifiers which add a minimal amount of noise are crucial to any high precision measurement scheme. A device which is commonly employed for this task is the high-electron-mobility transistor (HEMT) amplifier. Understanding the fundamental limits of the microwave noise performance of HEMT amplifiers is highly desirable. Noise temperatures in these devices as low as 3 times the quantum limit have been observed in the last decade, but the lack of understanding of the origin of the excess noise has hindered further improvements. Noise in HEMTs is attributed to a generator at the output, known as drain noise; and a generator at the input, which is attributed to thermal noise of the gate. At cryogenic temperatures of ~4 K, thermal noise is predicted to be negligible. However, a plateau in noise temperature has been observed at physical temperatures below ~20 K, with a negligible improvement in noise performance upon further cooling.
The primary noise mechanism responsible for this plateau is believed to be ohmic heating of the HEMT structure induced by current in the active device channel, a process known as self-heating. At room temperature the ambient thermal noise dominates the amplifier’s overall noise performance, but at the cryogenic temperatures required to achieve low-noise performance the self-heating effect produces thermal noise at the input of the HEMT gate which contributes significantly to the total noise. A potential mechanism to mitigate self-heating is to provide an additional thermal dissipation path for the Joule heating in the channel. However, given the sub-micron length scales and buried gate structure of HEMTs, thermal management is challenging. The primary heat conduction pathway, that of phonons travelling through the bulk HEMT substrate, decreases rapidly in magnitude at cryogenic temperatures. An alternative option is to submerge the HEMT in a cryogenic fluid, thereby presenting an alternate thermal conduction route through the HEMT surface into the fluid. This technique, while commonly employed in cryogenic thermal management of superconducting magnets, has not been investigated for HEMTs.
In this work we explore the use of liquid cryogenic cooling to directly mitigate the effect of HEMT self-heating. We test in particular the effectiveness of cooling using superfluid helium-4, which has the highest known thermal conductivity of any known substance. We report a systematic experimental investigation of the noise performance of a cryogenic packaged two-stage HEMT low-noise amplifier over a wide range of biases in a 4.0 - 5.5 GHz frequency band, with the device immersed in a variety of cryogenic baths including helium-4 vapor, liquid helium-4, superfluid helium-4, and vacuum. We present the details of the experimental apparatus which was constructed to perform microwave noise measurements of the low-noise amplifier when submerged in a liquid cryogen environment. We interpret our results using a small-signal model of the amplifier and compare our findings with the predictions of a phonon radiation model of heat dissipation. We find that liquid cryogenic cooling is unable to mitigate the thermal noise associated with self-heating. Considering this finding, we examine the implications for the lower bounds of cryogenic noise performance in HEMTs by incorporating the effects of self-heating into the existing noise modelling of HEMT amplifiers. Our analysis supports the general design principle for cryogenic HEMTs of maximizing gain at the lowest possible power.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
electronic noise, self-heating, low-noise amplifiers, HEMTs, superfluid helium
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Minnich, Austin J. (co-advisor)
Schwab, Keith C. (co-advisor)
Thesis Committee:
Nadj-Perge, Stevan (chair)
Minnich, Austin J.
Readhead, Anthony C. S.
Schwab, Keith C.
Defense Date:
26 May 2022
Funders:
Funding Agency
Grant Number
NSF
1911220
Projects:
Experimental characterization of HEMT self-heating in liquid cryogen environments
Record Number:
CaltechTHESIS:05272022-180844058
Persistent URL:
DOI:
10.7907/9va8-tc64
Related URLs:
URL
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Description
arXiv
Article adapted for sections of this thesis
ORCID:
Author
ORCID
Ardizzi, Anthony Joseph
0000-0001-8667-1208
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No commercial reproduction, distribution, display or performance rights in this work are provided.
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14638
Collection:
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Deposited By:
Anthony Ardizzi
Deposited On:
08 Jun 2022 15:11
Last Modified:
20 Feb 2025 21:11
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Self-heating of HEMT low-noise amplifiers in liquid
cryogenic environments and the limits of microwave
noise performance

Thesis by

Anthony J. Ardizzi

In Partial Fulfillment of the Requirements for the
Degree of
Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2022
Defended May 26, 2022

ii

Anthony J. Ardizzi
ORCID: 0000-0001-8667-1208

iii

ACKNOWLEDGMENTS
I sincerely thank both my advisors, Austin Minnich and Keith Schwab, for their
immeasurable patience, support, and instruction. Without them I would not be
where I am today, and I am forever indebted to each of them. I also extend my
gratitude to all of my peers and collaborators during my time at Caltech, especially
Jeff Botimer, Bekari Gabritchidze, Iretomiwa Esho, Alex Choi, Jacob Kooi, Kieran
Cleary, and Greg MacCabe.
I also thank my mother, father, and my entire family for their unconditional love and
support, and for granting me such a privileged and fortunate life where graduating
with a PhD from Caltech could become a reality.
And to my lovely bride-to-be, words cannot express both my love and gratefulness
for supporting me over the last several years, and for putting up with all my long
hours in the lab.

iv

ABSTRACT
Detection and processing of microwave signals is of substantial scientific importance
in fields ranging from radio astronomy to quantum computing. An essential component of the signal processing chain is the microwave amplifier, which adds gain to the
signal so that it may be processed by subsequent microwave components. However,
the amplifier itself adds its own internally generated noise into the measurement
chain. As a result, amplifiers which add a minimal amount of noise are crucial to
any high precision measurement scheme. A device which is commonly employed for
this task is the high-electron-mobility transistor (HEMT) amplifier. Understanding
the fundamental limits of the microwave noise performance of HEMT amplifiers is
highly desirable. Noise temperatures in these devices as low as 3 times the quantum
limit have been observed in the last decade, but the lack of understanding of the
origin of the excess noise has hindered further improvements. Noise in HEMTs is
attributed to a generator at the output, known as drain noise; and a generator at the
input, which is attributed to thermal noise of the gate. At cryogenic temperatures of
∼4 K, thermal noise is predicted to be negligible. However, a plateau in noise temperature has been observed at physical temperatures below ∼20 K, with a negligible
improvement in noise performance upon further cooling.
The primary noise mechanism responsible for this plateau is believed to be ohmic
heating of the HEMT structure induced by current in the active device channel, a
process known as self-heating. At room temperature the ambient thermal noise dominates the amplifier’s overall noise performance, but at the cryogenic temperatures
required to achieve low-noise performance the self-heating effect produces thermal
noise at the input of the HEMT gate which contributes significantly to the total
noise. A potential mechanism to mitigate self-heating is to provide an additional
thermal dissipation path for the Joule heating in the channel. However, given the
sub-micron length scales and buried gate structure of HEMTs, thermal management
is challenging. The primary heat conduction pathway, that of phonons travelling
through the bulk HEMT substrate, decreases rapidly in magnitude at cryogenic temperatures. An alternative option is to submerge the HEMT in a cryogenic fluid,
thereby presenting an alternate thermal conduction route through the HEMT surface into the fluid. This technique, while commonly employed in cryogenic thermal
management of superconducting magnets, has not been investigated for HEMTs.
In this work we explore the use of liquid cryogenic cooling to directly mitigate

the effect of HEMT self-heating. We test in particular the effectiveness of cooling
using superfluid helium-4, which has the highest known thermal conductivity of
any known substance. We report a systematic experimental investigation of the
noise performance of a cryogenic packaged two-stage HEMT low-noise amplifier
over a wide range of biases in a 4.0–5.5 GHz frequency band, with the device
immersed in a variety of cryogenic baths including helium-4 vapor, liquid helium4, superfluid helium-4, and vacuum. We present the details of the experimental
apparatus which was constructed to perform microwave noise measurements of the
low-noise amplifier when submerged in a liquid cryogen environment. We interpret
our results using a small-signal model of the amplifier and compare our findings
with the predictions of a phonon radiation model of heat dissipation. We find that
liquid cryogenic cooling is unable to mitigate the thermal noise associated with
self-heating. Considering this finding, we examine the implications for the lower
bounds of cryogenic noise performance in HEMTs by incorporating the effects of
self-heating into the existing noise modelling of HEMT amplifiers. Our analysis
supports the general design principle for cryogenic HEMTs of maximizing gain at
the lowest possible power.

vi

PUBLISHED CONTENT AND CONTRIBUTIONS

1 A. J. Ardizzi, A. Y. Choi, B. Gabritchidze, J. Kooi, K. A. Cleary, A. C. Readhead,

and A. J. Minnich, “Self-heating of cryogenic hemt amplifiers and the limits of
microwave noise performance”, arXiv:2205.03975 (2022),
Contributions: A.J.A. designed and constructed the experimental apparatus,
wrote the LabVIEW and MATLAB scripts, acquired and analyzed the data,
performed the small-signal model fitting, extended the amplifier noise modelling
to include the effects of self-heating, and wrote the manuscript.

vii

CONTENTS

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . vi
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 HEMT overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Helium-4 overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Superfluid helium-4 heat transport . . . . . . . . . . . . . . . . . . . 11
1.4 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Chapter II: Experimental methods: liquid cryogenic cooling of HEMT amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1 Background theory for cold attenuator Y-factor measurements . . . . 20
2.2 Experimental apparatus overview . . . . . . . . . . . . . . . . . . . 26
2.3 Cryogenic engineering . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Device characterization and calibrations at room temperature . . . . 34
2.5 Calibrations in liquid cryogen environments . . . . . . . . . . . . . 41
2.6 LNA noise and gain measurements . . . . . . . . . . . . . . . . . . 48
Chapter III: Measurement results, device modeling, and interpretations . . . . 51
3.1 Microwave noise temperature versus bias and frequency . . . . . . . 51
3.2 Interpretation using small-signal modeling . . . . . . . . . . . . . . 55
3.3 Comparison with a phonon radiation model . . . . . . . . . . . . . . 58
3.4 Noise temperature dependence on cryogenic environment . . . . . . 62
3.5 Measurement uncertainty analysis . . . . . . . . . . . . . . . . . . . 62
3.6 Limits on thermal conductance at the helium-gate interface . . . . . 67
3.7 Implications for noise performance of cryogenic HEMTs . . . . . . . 67
Chapter IV: Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . 70
4.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Appendix A: Correction algorithm for a changing noise source impedance . . 85

viii

LIST OF FIGURES

Number
Page
1.1 (a) Epitaxial layer profile of the OMMIC 70 nm GaAs mHEMT. (b)
Schematic of the energy band structure along the dotted-line slice in
(a), demonstrating how the two-dimensional electron (green region)
is populated from the band bending provided by doping of the InAlAs
barrier layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Schematic representations of a noisy amplifier. (a) Noise is interpreted as originating in the amplifier. (b) Noise is interpreted as
originating in a 50 Ω resistor across the amplifier’s input. (c) Noise
is interpreted as originating in both the ‘gate’ noise generator at the
amplifier’s input and ‘drain’ noise generator at the amplifier’s output.
1.3 Equivalent circuit of a FET. The intrinsic elements are enclosed by
dotted lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Dispersion curve of He II emphasizing both the phonon and roton
dissipation regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 The normal fluid and superfluid fractions comprising a given volume
of He II versus temperature. Above 𝑇𝜆 the fluid is entirely in the
normal state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Change in liquid level height versus time of a bath of He II contained
in a partially filled quarter-wave resonator connected through a small
channel to another volume after a step in pressure is applied at time
𝑡 = 0 s. The data is shown here for He II temperatures of 𝑇 = 0.9 K
(blue line) and 𝑇 = 1.1 K (orange line). The model is also shown for
𝑇 = 0.9 K (dash-dotted black line) and 𝑇 = 1.1 K (dashed black line). 15
1.7 Schematic representation of phonon scattering at the interface between two dissimilar media, in this case between a hot solid and
a cool liquid, which resembles the scattering of light between two
media of dissimilar indices of refraction. . . . . . . . . . . . . . . . 16
2.1 Diagram of measured noise power versus noise source noise temperature in the Y-factor measurement scheme. . . . . . . . . . . . . . . 22

ix
2.2

Schematic of the cold attenuator Y-factor measurement chain. From
left to right the components are: Noise source with hot (cold) noise
temperature 𝑇H (𝑇C ), input coaxial cable with loss 𝐿 1 and physical
temperature 𝑇𝐿 1 , attenuator with loss 𝐿 2 and physical temperature
𝑇𝐿 2 , DUT with gain 𝐺 and input-referred noise temperature 𝑇e , output
coaxial cable with loss 𝐿 3 and physical temperature 𝑇𝐿 3 , and backend
noise power detector with input-referred noise temperature 𝑇BE . . . .
2.3 (a) Schematic of the measurement apparatus inside a liquid helium-4
dewar, including the following components: (1) 15 dB ENR 2-18
GHz solid state SMA packaged noise diode biased at 28 V through a
MOSFET amplifier circuit (not shown); (2) input and output silverplated stainless steel SMA coaxial cables each 1.3 m in length; (3)
20 dB packaged cryogenic chip attenuator with factory calibrated
DT-670-SD diodes mounted directly on the attenuator substrate; (4)
packaged WBA46A LNA; (5) backend noise power detector. (b)
Representative raw Y-factor data versus time. The diode detector DC
offset voltage 𝑉0 (black lines), hot voltage 𝑉H (orange lines), and cold
voltage 𝑉C (blue lines) are all shown. . . . . . . . . . . . . . . . . .
2.4 Image of the vacuum apparatus mounted to a liquid helium dewar
with the following components: (1) pressure transducer; (2) leaktight cryogenic pressure-relief valve; (3) bellows valve connecting
to an ultra high purity helium-4 gas supply; (4) pumping line connection consisting of a bellows valve for flow-rate control in series
with a normally-closed electromagnetically actuated block valve; (5)
vertically mounted straight connector tube; (6) heating tape. The
helium flow path induced by pumping is shown as a blue dashed
line, and frosting of condensation on the exterior walls of the vacuum
components along this path can be seen. . . . . . . . . . . . . . . . .
2.5 Image of the hermetic breakout flange with the following labeled components: (1) conical reducing adapter; (2) CF flange with hermetic
breakouts; (3) DC cable; (4) input and output inner-outer DC blocks
terminating the coaxial cable leads; (5) SMA test cables connected
to the VNA (not shown). The reference plane for all microwave
measurements was the male connector of the inner-outer DC blocks.

23

27

29

30

2.6

Image of the copper mounting stage with the following labeled components: (1) dipstick; (2) input and output coaxial cables; (3) Lake
Shore DT-670-CU temperature diodes; (4) packaged 20 dB cold attenuator; (5) DUT. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Schematic of the liquid helium-4 dewar pumping model which includes the pumping speed 𝑆, the hydrodynamic tubing conductance
Ctube , the pump inlet pressure and temperature 𝑃𝑖 and 𝑇𝑖 , the external
heating 𝑃ext on the liquid bath and the temperature dependence of
the saturated vapor pressure 𝑃vap , liquid helium-4 density 𝜌4He , heat
capacity 𝐶4He , and the enthalpy of vaporization Δ 𝐻vap . . . . . . . .
2.8 (a) Liquid bath temperature versus time. The pumping model solution
for 𝑇L is shown for pumping speeds of 60 m3 h−1 (dash-dotted gray
line), 30 m3 h−1 (dash-dotted beige line), 15 m3 h−1 (dash-dotted
purple line), and 5 m3 h−1 (dash-dotted green line). The measured
bath temperature during a dewar cooldown is also shown (solid blue
line). (b) Remaining liquid volume versus time. The pumping model
solution for 𝑉L is shown for the same pumping speeds as in (a). . . .
2.9 (a) 𝐼DS versus 𝑉DS measured at room temperature with the reversebias of the first transistor stage ranging from 𝑉GS = −8 V to 𝑉GS = −2
V in 0.5 V steps (black lines from bottom to top) with the second
transistor stage pinched off at -8 V. (b) 𝐼DS versus 𝑉DS measured at
room temperature with the reverse-bias of the second transistor stage
ranging from 𝑉GS = −8 V to 𝑉GS = −2 V in 0.5 V steps (black lines
from bottom to top) with the first transistor stage pinched off at -8 V.
An asymmetry in the IV curves between the two stages can be seen.
(c) Noise temperature (left axis, blue circles), |𝑆21 | (right axis, red
line), and |𝑆11 | (right axis, magenta line) versus microwave frequency
with the device biased at its low-noise bias of 𝑃DC = 37.1 mW mm− 1
(𝑉DS = 0.85 V, 𝐼DS = 43.7 mA, 𝑉GS1 = −4.5 V, 𝑉GS2 = −4.9 V) at
room temperature. Noise temperature was measured using a NFA,
and S-pars were measured using a VNA. . . . . . . . . . . . . . . .

30

32

33

35

xi
2.10

Schematic of the room temperature backend noise power detector
consisting of (1) Pasternak PE8327 isolator, (2) Minicircuits ZX6083LN-S+ low-noise amplifiers, (3) Minicircuits filters with 3-6 GHz
bandwidth, (4) Miteq AMF-3B-04000800-25-25P medium power
amplifier, (5) RF switch for calibration, (6) 0 – 20 dB variable attenuator, (7) microwave power splitter, (8) Reactel cavity filter with
5 GHz center frequency and 20 MHz bandwidth, (9) Micro Lambda
MLFM-42008 20 MHz bandwidth tunable YIG filter, (10) Pasternak PE8224 inner-outer DC blocks, (11) Herotek DT4080 tunnel
diode detectors, (12) SRS560 low-noise preamps, (13) National Instruments NI6259-USB DAQ. Also shown is the Agilent 33210A
arbitrary waveform generator (AWG) used to pulse the microwave
switch MOSFET biasing circuit, as well as the DAQ terminals connecting to the AWG port used to bias the noise source (not shown),
temperature diode voltage outputs from the temperature controller,
and liquid level sensor voltage output from the American Magnetics
1700 liquid level instrument, enabling synchronous measurement of
each component with the noise power data. The losses of the SMA
cabling and attenuator pads are not shown. . . . . . . . . . . . . . . 37
2.11 |𝑆21 | of the fixed 5 GHz filter channel (black line) and the YIG filter
channel with center frequency set to 4.5 GHz (blue line), 5.0 GHz
(black line), and 5.5 GHz (red line) versus frequency. The filters
exhibit better than ±0.5 dB flatness over their 20 MHz bandwidth.
The −100 dB noise floor is set by the VNA noise floor. . . . . . . . . 38
2.12 Diode detector output voltage versus incident microwave power on
the diode detector for the fixed 5 GHz filter channel (blue circles)
and the YIG channel with center frequency set to 5 GHz (magenta
circles). (a) Data shown for incident microwave power ranging from
0 𝜇W to 10 𝜇W (−20 dBm). A linear fit to the data below 4 𝜇W
(−24 dBm) is also shown (solid lines). A nonlinear response can be
seen above 5 𝜇W (-23 dBm). (b) Data shown for incident microwave
power ranging from −44 dBm to −22 dBm, plotted logarithmically.
A linear fit to the full range of data is also shown (solid lines) . . . . 39

xii
2.13

Allan deviation versus integration time for the fixed 5 GHz filter
channel (blue dots) and the YIG channel with center frequency set to 5
GHz (magenta dots) with −100 dBm incident microwave power on the
detectors’ inputs. A minimum is observed at approximately 1 s, which
was used as the recalibration period for subsequent measurements.
Another minimum is seen at 500 s, suggesting the presence of a noise
source other than 1/f noise. . . . . . . . . . . . . . . . . . . . . . . .
2.14 (a) Backend detector noise temperature versus frequency. Error bars
reflect the uncertainty in the temperature of the cable connecting the
50 Ω load to the backend. (b) Noise source ENR versus frequency.
Error bars reflect the error propagated from uncertainty in the backend
noise temperature. (c) Diode detector output voltage sampled at
𝑓s = 600 kHz with the noise source connected directly to the input of
the backend detector and pulsed at 𝑓ENR = 50 kHz. . . . . . . . . . .
2.15 (a) Total loss of input and output coaxial cables versus frequency
measured at 300 K (red line), 4.2 K (magenta line), and 1.6 K (blue
line) with a commercial VNA. (b) Lumped physical coaxial cable
temperature versus frequency obtained from Y-factor measurements
(magenta circles) and from a heat conduction model (black line).
Error bars represent an estimate of the total uncertainty including
systematic errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.16 (a) Schematic of the coaxial cable temperature model showing a slice
of length dx along the cable in contact with the gas environment. (b)
Coaxial cable temperature profile (blue line) versus height along the
cable. The height of the liquid surface is also shown (vertical dashed
black line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

41

43

46

xiii
2.17

2.18
2.19
3.1

3.2

(a) Cable temperature (blue line), (b) cable loss (red line), and (c)
stage temperature (orange line) measured over the lifetime of the first
calibration dewar. Cable temperature and cable loss were measured
using the Y-factor method, and stage temperature was measured by
the temperature diode housed in the attenuator. Recalibration using a
VNA was performed at 𝑡 = 7 hours and 𝑡 = 62 hours, reflected in the
discontinuities in acquired data. (d) Cable temperature (blue line)
and cable loss (red line) versus physical stage temperature, generated
by fitting a smoothing spline to the time series data of the cable
temperature and loss plotted versus stage temperature, respectively,
during the warming phase of the calibration measurements. These
curves were used as the calibration curves for subsequent warming
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Attenuator loss versus frequency at room temperature (red line), 4.2
K (magenta line) and 1.6 K (blue line). . . . . . . . . . . . . . . . .
Thickness of a He II film versus height above the bulk fluid. . . . . .
Noise temperature (left axis) and gain (right axis) versus bias measured at a fixed frequency of 𝑓 = 4.55 GHz in the following cryogenic
environments: 1.6 K He II (blue triangles), 4.2 K He I (cyan triangles), 8.0 K vapor (green diamonds), 19.7 K vapor (green squares),
and 33.8 K vacuum (purple circles). The bias was varied by changing the drain-source voltage for a fixed gate-source voltage of (a)
𝑉GS = −2.2 V, (b) 𝑉GS = −2.8 V, and (c) 𝑉GS = −3.2 V. For each
fixed 𝑉GS the low-noise bias shifts by less than 10 mW mm−1 and
the noise temperature at the low-noise bias changes by less than 1 K
across all physical temperatures. . . . . . . . . . . . . . . . . . . . .
Noise temperature (left axis) and gain (right axis) versus drain-source
current measured at a fixed frequency of 𝑓 = 4.55 GHz in the same
cryogenic environments as in Fig. 3.1. The bias was varied by changing the gate-source voltage for a fixed drain-source voltage output by
PS=1.0 V and (b) 𝑉 PS=1.4 V. . . . . . . . . .
the power supply of (a) 𝑉DS
DS

47
48
50

53

54

xiv
3.3 (a) Noise temperature (left axis) and gain (right axis) versus frequency, measured at the device’s low-noise bias of 𝑃DC = 24.5 mW
mm−1 (𝑉DS = 0.56 V, 𝐼DS = 43.9 mA mm−1 , 𝑉GS = −2.7 V) in the
following cryogenic environments: 1.6 K He II (blue triangles), 4.2
K He I (cyan triangles), 8.2 K vapor (green diamonds), 20.1 K vapor
(green squares), and 35.9 K vacuum (purple circles). Only the gain
under He II conditions is shown for clarity since the gain varies by
less than 0.5 dB across all temperatures. The small-signal model
fits for each dataset is also shown (solid lines). (b) Noise temperature (left axis) and gain (right axis) versus frequency measured at
biases of 𝑃DC = 24.5 mW mm−1 (magenta circles; 𝑉DS = 0.56 V,
𝐼DS = 43.9 mA mm−1 , 𝑉GS = −2.7 V), 𝑃DC = 79.5 mW mm−1 (dark
blue circles; 𝑉DS = 1.0 V, 𝐼DS = 79.5 mA mm−1 , 𝑉GS = −2.7 V), and
𝑃DC = 120 mW mm−1 (red circles; 𝑉DS = 1.2 V, 𝐼DS = 100.0 mA
mm−1 , 𝑉GS = −2.7 V) with the DUT submerged in He II at 1.6 K. To
vary the bias, the gate-source voltage was held constant at 𝑉GS = −2.7
V while the drain-source voltage 𝑉DS was varied. The small-signal
model fits (solid lines) are also shown. Where omitted in both (a)
and (b), the vertical error bars are equal to the height of the symbols.
3.4 High-resolution micrograph image of the packaged device including
the input matching network and MMIC. The inset shows a zoom of
the MMIC fabricated by OMMIC. . . . . . . . . . . . . . . . . . . .
3.5 Schematic of the transistor model made using Microwave Office. . . .
3.6 (a) Extracted gate temperature versus physical temperature at the
device’s low-noise bias of 𝑃DC = 24.5 mW mm−1 . Symbols indicate
extracted values and represent the same conditions as in Fig. 3.3(a),
along with extracted values in 4.2 K He I (cyan triangles) and 8.1 K
vapor (green diamonds). The radiation model is also shown (dashdotted black line). (b) Extracted gate temperature versus bias at
1.6 K physical temperature (blue triangles). The radiation model is
also shown (dash-dotted black line). In both (a) and (b) the error
bars were generated by determining the range of gate temperatures
that accounted for the uncertainty in the frequency-dependent noise
temperature data. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

57
58

59

xv
3.7 (a) Extracted drain temperature versus physical temperature generated from the same extraction process as in Fig. 3.6(a). A linear fit is
also shown (dashed black line). (b) Extracted drain temperature versus bias generated from the same extraction process as in Fig. 3.6(b).
In both (a) and (b) the error bars were generated by determining the
range of drain temperatures that accounted for the uncertainty in the
frequency-dependent noise temperature data. . . . . . . . . . . . . .
3.8 (a) Noise temperature (left axis, blue line) and physical temperature
(right axis, black line) versus time in an evaporating He II bath
sampled at 𝑓ENR = 10 Hz and digitally filtered at 1 Hz, taken at a fixed
bias 𝑃DC = 80 mW mm−1 (𝑉DS = 1.0 V, 𝐼DS = 80 mA mm−1 , 𝑉GS =
−2.8 V) and frequency 𝑓 = 4.55 GHz. The sharp kink in the physical
temperature at time 𝑡 = 0 minutes, interpreted as the time at which
superfluid is no longer present on the attenuator and device, is not
reflected in the noise temperature. (b) Noise temperature (black line)
versus physical temperature obtained from the transient data shown
in (a). Symbols show independently measured noise temperatures
representing the same bath conditions as in Fig. 3.3(a), and the same
bias conditions as in (a). The presence of liquid cryogens does not
affect the noise temperature within the measurement uncertainty. . . .
3.9 Gain (top plot) and noise temperature (bottom plot) versus frequency
measured with the device biased at its low-noise bias of 𝑃DC =
24.5 mW (𝑉DS = 0.56 V, 𝐼DS = 43.9 mA mm−1 , 𝑉GS = −2.7 V).
The approximately quadratic shape of the noise temperature curve is
determined by the IMN. . . . . . . . . . . . . . . . . . . . . . . . .
3.10 (a) Modeled 𝑇min versus bias, shown for a fixed gate temperature
𝑇g = 20 K (dashed green line) and for a gate temperature with bias
dependence determined by a radiation model (solid blue line). The
radiation model predicts a gate temperature below 20 K for powers
below 40 mW mm−1 , which is reflected in 𝑇min . (b) Modeled 𝑇50
noise temperature versus bias, shown for 𝑇d = 500 K (dashed red
line), 𝑇d = 200 K (dash-dotted gold line), and 𝑇d = 20 K (solid pink
line). Both the minimum 𝑇50 and the power required to achieve this
minimum decrease with decreasing 𝑇d . . . . . . . . . . . . . . . . .

60

61

63

68

xvi
A.1

Equivalent noise measurement scheme representations involving a
noise source, an intermediate 2-port network, and a detector where
(a) reflections are attributed to the noise source output plane and
detector input plane, (b) the noise source and detector are treated
as ideal and virtual components carry the associated S-pars, and (c)
the noise source and detector are treated as ideal and all S-pars are
cascaded into a single element. . . . . . . . . . . . . . . . . . . . . . 85
A.2 Noise temperature of packaged HEMT amplifier versus frequency
measured at room temperature with a commerical NFA (blue circles)
and with the noise measurement setup described in Section 2.2 using
the uncorrected Eq. (2.6) (red circles) and the corrected Eq. (A.6)
(green circles). The agreement between the two measurements improves upon correction. All measurements were performed at the
same bias. Error bars reflect the overall measurement uncertainty,
including the impedance mismatch in the uncorrected case. . . . . . . 88

xvii

LIST OF TABLES
Number
Page
2.1 Table of parameters used in solving the coaxial cable thermal model. 45
3.1 Table of parameters used to extract 𝑇𝑒 , along with their associated
uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Chapter 1

INTRODUCTION
In this thesis we present a systematic investigation of the noise performance of
a packaged high-electron-mobility transistor (HEMT) amplifier when immersed
in various liquid cryogenic environments, including the quantum liquid superfluid
helium-4. In this chapter we begin by providing a brief history of semiconductor devices leading to the development of HEMT amplifier technology, which we describe
in greater depth. We then present an overview of helium-4, focusing particularly
on superfluid helium-4, before finishing by detailing the heat transport properties of
superfluid helium-4 and its uses as a cryogenic coolant.
1.1

HEMT overview

Electronic devices constructed from semiconductor materials are a hallmark of
modern technology. First discovered in the 19th century by Thomas Seebeck [1]
and Michael Faraday [2], it wasn’t until the discovery of the electron, the advent
of quantum mechanics, and the development of solid-state physics theory by scientists such as Johan Koenigsberger [3] and Ferdinand Bloch [4] in the early 20th
century that semiconductor physics was more rigorously understood. The unique
electronic properties of semiconductors, which possess resitivities spanning 11 orders of magnitude at room temperature [5, 6] that can be both precisely engineered
through fabrication techniques and electrically tuned in situ at frequencies of up to
several terahertz [7], naturally lend themselves to being excellent electrical switches
and valves, controlling the flow of electrical current in an analogous manner to the
control of liquid flow using hydrodynamic valves. The first semiconductor devices
were point-contact rectifiers which were used as early radio wave detectors [8]. In
1947 the famous Bell Labs group consisting of William Shockley, John Bardeen,
and Walter Brattain demonstrated successful operation of a bipolar transistor for the
first time [9], for which they were awarded a Nobel Prize. Over the coming decades,
semiconductor transistor technology exploded into widespread use and helped drive
the development of computer chips, with modern chips containing tens of billions of
metal–oxide–semiconductor field-effect transistor (MOSFET) switches [10]. Very
rough estimates place the total number of MOSFET switches produced globally
every year to be over 1021 as of 2020, a number which has grown exponentially

since the early 1970s [11].
The particular distinguishing feature of semiconductors when compared to pure insulators is their relatively smaller energy gap (bandgap), which are energies at which
electron states cannot exist in the material’s energy-momentum band structure. This
permits electrons in semiconductors to more readily cross the bandgap from the
valence band into the conduction band than in insulators [5, 6]. The conduction
band can be made even more accessible by introducing lattice impurities into a semiconductor through a process known as doping, which creates allowed energy bands
inside the bandgap near the bandgap edge and provides even more readily ionizable
electron states. Thus through either appropriate selection of undoped semiconductors, referred to as "intrinsic" semiconductors, or through the engineering of the
dopant density in doped semiconductors, referred to as "extrinsic" semiconductors,
the electrical transport parameters most critical for high-speed device performance
such as electron mobility (𝜇 𝑒 ) which characterizes the average speed of electrons in
a semiconductor, peak velocity (𝜈 𝑝 ) which is the peak charge carrier velocity for a
semiconductor subjected to strong electric fields, and the effective mass of charge
carriers (𝑚 ∗ ) can all be adjusted [12].
The most commonly used semiconductor material by far is silicon (Si) due to its
relatively low production costs, superior rectifying properties and high-temperature
workability when compared with its immediate competitor germanium (Ge) [13].
However, through decades of research into high-performance devices it was found
that III-V semiconductors such as gallium arsenide (GaAs) and more recently indium
phosphide (InP) possess far superior transport properties at high frequencies [12,
14, 15]. These materials are thus used to make analog amplifier devices with high
operating frequencies and excellent low-noise properties, known as high-electronmobility transistor (HEMT) amplifiers, the primary subject of study in this work.
The technology of HEMTs evolved from the metal-semiconductor field-effect transistor (MESFET) technologies of the 1960s and 1970s, first through research at IBM
where exceptional transport properties were demonstrated [16], and then through
research at laboratories located throughout the world such as Bell Laboratories [17,
18], the National Radio Astronomy Observatory in Charlottesville, VA [19–22], the
Thomson-CSF laboratory in France [23], Fujitsu Labs in Japan [24], the University
of Massachusetts [25], the Jet Propulsion Laboratory in Pasadena, CA [26–28], the
California Institute of Technology in Pasadena, CA [29, 30], Chalmers University of
Technology in Sweden [31–33], the University of Manchester [34], and Fraunhofer

IAF in Germany [35], to name just a few.
Conduction in GaAs MESFETs occurs through a Si doped GaAs conduction channel.
The use of an extrinsic semiconductor for the channel material causes a significant
reduction in electron mobility and an increase in overall noise when compared with
intrinsic semiconductors due to electron scattering off of the dopant impurities. The
improvement of the HEMT over the MESFET was to instead use an intrinsic narrow
bandgap channel layer, generally indium gallium arsenide (InGaAs) of varying
indium contents, with free carriers being supplied by the diffusion of electrons from
a doped donor layer [36]. Drastically improved electron mobilities are observed in
HEMTs over MESFETS, with typical increases of over 50% at room temperature
(𝜇 𝑒 ≃ 4000 cm2 V−1 s−1 in MESFETs versus 𝜇 𝑒 ≃ 8500 cm2 V−1 s−1 in HEMTS)
and over a factor of 10 improvement at liquid nitrogen temperatures (𝜇 𝑒 ≃ 6000 cm2
V−1 s−1 in MESFETs versus 𝜇 𝑒 ≃ 80000 cm2 V−1 s−1 in HEMTS) [12].
Figure 1.1(a) shows an example of an epitaxial layer structure of a metamorphic
HEMT grown on a GaAs substrate, and reflects the transistors used in the device
presented in this thesis. We discuss the stack from top to bottom. Coating the
entire device is a silicon nitride (SiN) passivation layer which improves stability by
preventing the semiconductor surfaces from being oxidized in uncontrolled ways,
both immediately after fabrication and in the long-term [37]. The gate metal is
of the double-mushroom style [38] and forms a Schottky contact with the widebandgap indium aluminum arsenide (InAlAs) barrier layer. This barrier layer is
delta doped with silicon, which is the doping of only a thin layer of semiconductor
material performed by growth-interrupted impurity deposition [39], and supplies the
electrons which ultimately conduct in the undoped narrow-bandgap InGaAs channel.
Conduction through the channel occurs in the two-dimensional electron gas which
forms below the interface of the barrier and channel. Between the channel layer
and the GaAs substrate, an InAlAs metamorphic buffer layer is employed which
gradually transitions the indium content to match the different lattice constants of
the channel and substrate, minimizing the presence of defects associated with lattice
mismatch which degrade device performance [40, 41]. Source and drain contacts
are made from Si doped InGaAs to yield small contact resistances. The general
convention is to refer to devices grown on GaAs substrates colloquially as GaAs
HEMTs, and HEMTs grown on InP substrates colloquially as InP HEMTs, with each
designation implying the use of an InGaAs channel, a convention which we follow.
The band structure of the HEMT is shown schematically in Fig. 1.1(b). The delta

doping, which is the doping of only a thin layer of semiconductor material performed
by growth-interrupted impurity deposition [39], causes bending of the InAlAs bands
so that when a sufficiently high gate-source bias 𝑉GS is applied electrons are able to
diffuse into the quantum well created at the heterojunction of the InAlAs-InGaAs
interface, with the wider bandgap of the barrier layer preventing electrons from
conducting in the doped InAlAs layer. A two-dimensional electron gas is confined
here, represented by the green shaded area, which acts as the conduction channel in
the device. The InGaAs conduction energy 𝐸𝐶 , valence energy 𝐸𝑉 , and the Fermi
energy 𝐸 𝐹 are also shown.
HEMT technology is now used ubiquitously in measurement schemes that require
a microwave-frequency low-noise amplifier (LNA), with modern devices exhibiting
truly exceptional noise properties. An amplifier’s noise contribution is typically
characterized by its input referred noise temperature 𝑇e , which is the temperature
that a matched resistor connected at a noiseless amplifier’s input would have if it
were to contribute the same magnitude of noise as the noisy amplifier, as shown
schematically in Figs. 1.2(a) and 1.2(b). Modern HEMTs exhibit noise temperatures
as low as 3–5 times larger than the absolute lower limit set by quantum mechanics
(the quantum limit) at operational frequencies of 2–50 GHz, and 10–100 times larger
than the quantum limit at frequencies of 50–1000 GHz, with worsening performance
at higher frequencies [42–45].
With such favorable noise properties, HEMTs are a key component of high precision
measurements across diverse fields in science and engineering such as radio astronomy [14, 44], deep-space communication [26], and quantum computing [46–50]. In
radio astronomy applications, HEMT LNAs cooled to 10–20 K typically serve as the
first stage of amplification, with superconducting mixers placed at the LNA input to
downconvert frequencies above 100 GHz [44] . In quantum computing applications,
particularly in superconducting qubit architectures, HEMT LNAs act as the second
stage of amplification after a first stage of quantum-limited amplification typically
performed by a superconducting parametric amplifier [48]. Since in general the
manufacturing and operation of HEMT LNAs are far simpler than for parametric amplifiers which require fabrication techniques akin to that of superconducting
qubits as well as an additional microwave frequency pump tone for biasing [51],
and since double and triple stage HEMT LNAs typically exhibit much higher gain
than a superconducting parametric amplifier, developing a quantum-limited HEMT
LNA whose operating power is sufficiently low for it to be mounted directly on

Fig. 1.1: (a) Epitaxial layer profile of the OMMIC 70 nm GaAs mHEMT. (b)
Schematic of the energy band structure along the dotted-line slice in (a), demonstrating how the two-dimensional electron (green region) is populated from the band
bending provided by doping of the InAlAs barrier layer.

Fig. 1.2: Schematic representations of a noisy amplifier. (a) Noise is interpreted as
originating in the amplifier. (b) Noise is interpreted as originating in a 50 Ω resistor
across the amplifier’s input. (c) Noise is interpreted as originating in both the ‘gate’
noise generator at the amplifier’s input and ‘drain’ noise generator at the amplifier’s
output.
the base stage of a dilution refrigerator would constitute a significant advancement
to the development of superconducting qubits and ultimately, it is hoped, a fully
fault-tolerant quantum computer based on this technology [49, 52].
Modelling the noise behavior of HEMT amplifiers is typically done using the
Pospieszalski model of field-effect transistors (FETs) [53]. A schematic of the
equivalent circuit in this representation is shown in Fig. 1.3. The elements of the
intrinsic FET are the intrinsic resistance 𝑟 i which is assumed to generate noise at
the gate temperature 𝑇g , the drain conductance 𝑔ds which is assumed to generate
noise at the drain temperature 𝑇d , the transconductance 𝑔m which represents the
transduction of voltage fluctuations in the gate to current fluctuations in the channel,
the gate-source capacitance 𝐶gs , and the gate-drain capacitance 𝐶gd . A simplified
schematic showing only the noise generators associated with 𝑇g and 𝑇d , with the
remaining parameters carried by the frequency dependence of the amplifier gain

Fig. 1.3: Equivalent circuit of a FET. The intrinsic elements are enclosed by dotted
lines.
𝐺, is presented in Fig. 1.2(c). The extrinsic resistances associated with the ohmic
contact resistances are the gate resistance 𝑅G , the drain resistance 𝑅D , and the source
resistance 𝑅S , which in this work we assume are all thermalized with the gate temperature 𝑇g . We restate here the minimum noise temperature 𝑇min of the intrinsic
FET written using the formulation of the Pospiesalski noise model [22]:
𝑇min  2

𝑓 p
𝑔ds𝑇d𝑟 i𝑇g
𝑓T

(1.1)

where 𝑓T = 𝑔m (2𝜋𝐶gs ) −1 is the intrinsic FET’s cut-off frequency, and the simplified
expression presented here is valid in the limit 𝑓
𝑓T . Typical figures of merit
for low-noise HEMTs involve the noise parameters comprising Eq. (1.1), with
representative state-of-the-art cryogenic devices exhibiting values of 𝑓T & 500
GHz, 𝑔m & 1400 mS mm−1 , 𝑔ds . 20 mS, 𝑇d . 200 K, 𝑟𝑖 . 1 Ω, and 𝑇g . 20 K [15,
38]. Since the Pospieszalski model is primarily a phenomenological model which
relies on fitting to empirical data in order to determine these constituent parameters,
other theories are required to understand the physical origins and limits of HEMT
noise.
At present, the drain temperature 𝑇d lacks an accepted physical origin, with several
theories, generally related to the fluctuations of hot electrons in the HEMT channel,
having been proposed. Early attempts at modelling drain noise attributed it to
the spontaneous generation of dipole layers which drift through the channel to the
drain contact [54]. A more testable theory was later made attributing drain noise

to suppressed shot noise, inspired by the fact that the observed drain noise current
spectral density is proportional to the current, as is the case for pure shot noise
[55, 56]. Most recently, drain noise has been attributed to real-space transfer of hot
electrons which thermionically emit across the interface between the channel and
barrier layers causing a significant fraction of electrons to conduct through the lower
mobility barrier layer [57]. The value of 𝑇d is typically much larger than the ambient
device temperature and ranges from 𝑇d = 100 K at liquid helium temperatures for
state-of-the-art InP devices [49] to 𝑇d > 10000 K at room temperature for GaAs
devices [30], supporting the belief that drain noise is not thermal in origin. Further
experimentation to test these theories [58] and develop devices with ever-lower
drain noise through, for example, different indium channel contents [49] is an area
of ongoing research. In this work, we simply take 𝑇d as a fitting parameter without
specific regard to its physical origin.
The gate temperature 𝑇g , on the other hand, is widely accepted to be equal to the
physical lattice temperature of the device down to cryogenic physical temperatures
[22, 59], and the noise is assumed to be purely thermal in origin. Specifically, the
gate noise is thought to originate primarily from ohmic losses associated with the
gate metal which imprints its voltage fluctuations on the channel conductance, as
well as possibly from losses in the small region of the channel closest to the source
lead before the electrons are accelerated by the strong electric fields supplied by
the gate and travel at their peak saturation velocity [55]. The parasitic resistances
associated with the gate and source leads are also assumed to be equal to the
ambient temperature. In this interpretation of gate noise, cryogenic cooling leads to
improvements in the noise performance of the HEMT in part by decreasing the gate
temperature and hence its thermal noise contribution, as well as by improving the
intrinsic transport properties of the HEMT semiconductor material.
Cryogenic cooling of HEMT devices thus leads to a monotonic decrease in noise
temperature with decreasing physical temperature. However, this trend has been
observed to plateau below physical temperatures of 20 – 40 K (see Fig. 10 of
[60], Fig. 1 of [28], and Fig. 2 of [61], for example). The physical origin of
this plateau has been attributed to heating of the intrinsic gate resistance caused
by power dissipated in the active channel, referred to as “self-heating”. In Ref.
[28], Monte Carlo numerical simulations were performed to solve the Boltzmann
transport equation inside an InP HEMT, which determined that the tens of milliwatts
of power dissipation required for device biasing induced a region of elevated lattice

temperature localized inside the conduction channel. This localized temperature
rise was found to saturate below ambient physical temperatures of approximately 30
K, persisting down to arbitrarily low temperatures. This channel heating is believed
to induce heating of both the intrinsic gate resistance and the parasitic resistances
associated with the gate and source leads.
A more intuitive understanding of self-heating can be understood through a black
body phonon radiation model [45]. In this model, heat conduction channels across
an interface are parameterized by occupation of available phonon modes, which at
low temperatures can be predicted by the Debye model. The result is analogous to
the Stefan-Boltzmann law for black body photon radiation, and it predicts a heat flux
𝑞 ∝ 𝑇 4 at low temperatures. Similar to the results of the Monte Carlo simulation
method, this model predicts that at cryogenic temperatures heat transport away
from the heated channel can no longer occur, and the gate temperature 𝑇g saturates
at approximately 20 K at physical temperatures below 20 K. These predictions
were experimentally confirmed using a Schottky thermometry method based on the
temperature-dependent diode characteristics of the HEMT gate [62]. Although heat
dissipation in this experiment occurred directly in the HEMT gate metal, which
was forward-biased to perform the thermometry measurements, as opposed to in the
channel which is the case for normal HEMT amplifier operation, the confirmation of
reduced heat transport at cryogenic temperatures is valid nonetheless. The predicted
plateau in the gate temperature is thought to be responsible for the corresponding
plateau in overall noise temperature.
To achieve improved device performance, mitigating the effect of self-heating is
desirable. One approach is to design devices with optimal operating biases at sufficiently low powers that the effect of self-heating becomes negligible, an approach
which has been actively pursued [31, 49]. Another approach is to provide additional
heat conduction modes to the heated region. Thermal management of the gate in
modern devices with sub-micron gate lengths and a buried gate structure is challenging, and there is no straightforward way to improve thermal conduction via solely
solid-state methods. The approach taken here, which is the primary topic of investigation in this thesis, is to submerge the HEMT in a cryogenic fluid, in particular
superfluid helium-4, a quantum fluid with the highest known thermal conductivity [63]. Such an approach is routinely used for cryogenic thermal management
of superconducting magnets [64] and is actively employed in high-energy physics
experiments [65–67]. However, the effectiveness of liquid cryogens to mitigate

10
self-heating in HEMTs has not yet been experimentally evaluated. The following
sections in this chapter provide the background information on superfluid helium-4
required to understand the details of the heat transport mechanism proposed here.
1.2

Helium-4 overview

Helium, the second chemical element on the periodic table, forms an inert, colorless,
odorless, non-toxic, monatomic gas at room temperature and barometric pressure.
Seemingly uninteresting under such conditions, at sufficiently low temperatures
helium becomes arguably the most interesting of all pure chemical elements. We
list just a few of helium’s unique properties, focusing in particular on the more
abundant isotope helium-4: (a) after hydrogen, it is the second lightest and second
most abundant element; (b) along with neon, it is the only element for which no
known compounds exist; (c) since its zero-point energy is larger than its binding
energy, it is the only element which does not freeze at any temperature under its own
vapor pressure, transitioning to the liquid phase at 4.23 K in which it persists down
to 0 K; (d) it is the only element which undergoes a second order phase transition
and condenses into a quantum liquid at sufficiently low temperature, known as the
“lambda point” due to the shape of the helium-4 specific heat versus temperature
curve around 𝑇𝜆 = 2.17 K [68], under its own vapor pressure; (e) in this quantum
liquid phase, it possesses the highest known thermal conductivity of any known
substance [69].
The properties of liquid helium-4, which we denote as He II at temperatures 𝑇 < 𝑇𝜆
where the helium-4 is a quantum liquid or “superfluid”, and He I at temperatures
𝑇 > 𝑇𝜆 where helium-4 is entirely a normal liquid, comprise an entire field of study in
their own right, particularly in the superfluid regime [70–74]. The first observation
of quantum fluidity was made in 1937 independently by Pyotr Kapitza in Russia
[75] (to whom the Nobel Prize in physics was awarded), and John Allen and Don
Misener at Cambridge [76, 77], which led to major advances in the understanding
of low-temperature physics and spawned the fields of superconductivity in solidstate physics and superfluidity in condensed-matter physics of liquids and ultracold
atomic gases. The complicated dynamics of superfluids such as quantized vortices
[78], phase-slippage [79], and Josephson effects [80] have been studied extensively
over the course of the twentieth century [81]. This has led to the development of
several He II based technologies in recent years such as matter-wave interferometric
devices [82, 83] used to measure the Earth’s rotation, and superfluid optomechanical
devices [84–86] which have been proposed as tabletop gravity wave detectors [87,

11
88] and dark-matter detectors [89, 90].
For our purposes, we focus on the study of heat transport and the use of liquid
helium as a cryogenic liquid coolant [63, 91]. Helium is used extensively in modern
cryogenic laboratories in a variety of different refrigeration systems, including gasphase cryocoolers (base temperatures of 20 K) [92, 93], helium-4 refrigerators
(base temperatures of 1 K), helium-3 refrigerators (base temperatures of 0.3
K), and dilution refrigerators (base temperatures of 0.02 K) [94–97]. Each of
these technologies uses their cooling power to convectively cool several mounting
“stages”, which are generally machined sections of oxygen-free high-conductivity
(OFHC) copper to which the components requiring cryogenic cooling are thermally
strapped.
A more direct cooling method using helium is to submerge an object to be cooled
directly in a bath of liquid helium. This method has the advantage of allowing the
helium to directly extract heat which cannot readily conduct through the body of
the object and is therefore localized to a particular region of the object’s surface,
but has the disadvantage of being more costly since in general a larger volume of
liquid helium is required. Furthermore, since evaporatively cooling a bath of liquid
helium-4 down to the superfluid state generally requires conversion of approximately
50% of the liquid into the gas state [96], if this vapor is not recaptured then a
significant quantity of helium can be wasted. In some situations, this convective
cooling method is required. Direct cooling using liquid helium is routinely used
to cool high-field superconducting magnets [64]. Liquid He I is used ubiquitously
in modern magnetic resonance imaging (MRI) devices [98] and in fusion reactor
research [99–101]. Liquid He II is used in high-energy physics experiments such
as the Large-Hadron Collider [65–67]. In the following section, we provide a
more detailed account of the physical mechanisms which determine the transport
properties of He II.
1.3

Superfluid helium-4 heat transport

The basis for understanding the transport properties of He II is the two-fluid model.
First postulated by Laszlo Tisza in 1938 [102], a more rigorous quantum hydrodynamical account was given by Lev Landau in 1941 [103]. In this description,
the fluid properties are comprised of a linear combination of two components, the
normal fluid component with density 𝜌𝑛 and the superfluid component with density
𝜌 𝑠 , with the overall fluid density given by the sum 𝜌 = 𝜌 𝑠 + 𝜌𝑛 . Landau showed

12
that the assumption of viscous-less fluid flow of the superfluid component leads to
a velocity field vs which obeys the Euler equation for ideal fluid flow, carrying with
it zero entropy. We restate Euler’s equation of motion here in the form derived in
Ref. [68]:
Dvs
𝜌𝑠
𝜌𝑠
= − ∇𝑃 + 𝜌 𝑠 𝜎∇𝑇
(1.2)
D𝑡
where ∇𝑃 is the pressure gradient, ∇𝑇 is the temperature gradient, and 𝜌 is the total
entropy per unit mass.
The normal fluid component responsible for entropy transport arises from the elementary excitations seen in the dispersion curve of energy 𝐸 versus momentum
𝑘, with conservation laws predicting a gas of elementary thermal excitations which
behave as an ordinary liquid governed by a Navier-Stokes type equation [68]:
𝜌𝑛

𝜌𝑛
Dvn
= − ∇𝑃 − 𝜌 𝑠 𝜎∇𝑇 + 𝜂𝑛 ∇2 vn
D𝑡

(1.3)

where 𝜂𝑛 is the normal fluid viscosity. In Fig. 1.4 we plot a smoothed spline to
compiled energy-momentum dispersion data from neutron scattering experiments
taken from Ref. [104], emphasizing both the linear dispersion regime near 𝑘 = 0
associated with phonons which move at the speed of sound in helium, and the
local minimum at higher energy associated with excitations called rotons. In 1947
Bogolyubov showed that a very similar dispersion for He II could be derived by
treating it as a weakly-interacting Bose gas with a macroscopically occupied lowest
energy state and excitations arising from the interactions [105]. In 1954 Feynamn
further demonstrated the strong connection between the quantum hydrodynamic
treatment by Landau and the interacting Bose condensate picture of Bogolyubov
by applying his path-integral formulation to prove that the dispersion curve derived
by Landau was a necessary consequence of an interacting Bose condensate [106].
Today, the understanding that He II is a Bose condensate described by the same
physics as ultracold atomic gas condensates is well established.
The internal consistency and overall applicability of the two-fluid model has been
confirmed by several classes of experiments, including the flow of He II in channels
[107, 108], the flow of He II films [109–111] for which the question of whether He II
film thickness depends on the motion of fluid in the film is still an open question, the
He II fountain effect [112] wherein the interdependence of heat and mass flow in He
II is most evident, and the propagation of sound waves [113] which have been used
to make precise determinations of the thermophysical properties of He II. Arguably
the most striking two-fluid model demonstration was performed by Andronikashvili

13

Fig. 1.4: Dispersion curve of He II emphasizing both the phonon and roton dissipation regimes.
in 1946 [114], where a stack of thin metal disks was suspended by a torsional fiber in
He II and driven to oscillate. A measurement of the disks’ oscillation frequency then
gave a direct measurement of the viscosity of the liquid, and from this the normal
fluid fraction 𝜌𝑛 /𝜌 responsible for the viscosity 𝜂 was inferred as a function of liquid
temperature. The normal fluid fraction was found to obey a 𝑇 5.6 dependence, in
approximate agreement with the predictions of the excitation spectrum determined
by the calculations of Landau and Feynman. In Fig. 1.5 we plot a smoothed spline
to compiled fluid fraction data from Ref. [104]. The sharp decrease in superfluid
fraction above 1 K can be attributed to the increased population of roton excitation
modes. Below 1 K, the superfluid fraction comprises over 99% of the total fluid.
We can understand the properties of steady state bulk He II heat transfer in the
context of the two-fluid model. Two loss mechanisms contribute to the limits of
bulk heat transfer. One is the mutual friction between the two fluid components,
and the other is the viscous nature of the normal fluid component. For conductive
heat transport in large channels such as those used for superconducting magnets and
particle accelerators, the latter effect of mutual friction tends to dominate [115]. In
this regime, the heat conductivity function exhibits a maximum at 𝑇 = 1.9 K near
the saturated vapor pressure (SVP), although typical large-scale heat exchangers use
sub-cooled He II at 1.9 K and 1 atm for two reasons. The first is simply to eliminate
risks associated with air in-leaks. The second is that in saturated He II, the bulk
liquid can only tolerate temperature excursions up to the local saturation point fixed

14

Fig. 1.5: The normal fluid and superfluid fractions comprising a given volume of
He II versus temperature. Above 𝑇𝜆 the fluid is entirely in the normal state.
by the pressure in the bulk liquid before boiling occurs. In sub-cooled He II a fixed
pressure of, for example, 1 atm allows for temperature excursions all the way up to
𝑇𝜆 , offering a form of cryogenic stabilization [116]. In the experiments presented
in this work, we consider only saturated He II due primarily to the fact that it is
straightforward to achieve directly via evaporative cooling.
The second loss mechanism, that of turbulence, must be included when considering
flow through small channels. Both the superfluid and normal fluid components
experience turbulent flow above a critical velocity. In the superfluid, the critical
velocity 𝑣 𝑐 is associated with the minimum energy above which excitations such as
rotons and quantized vortices can form and enable dissipative interaction between
the fluid and its surroundings [81]. Experimentally, 𝑣 𝑐 has been observed to vary
roughly as 𝑑 −1/4 where 𝑑 is the characteristic diameter of the transport channel
[117]. In the normal fluid the critical velocity is associated with the onset of classical
turbulence above Reynolds numbers of approximately 1200 in rigid channels [63].
Figure 1.6 shows both data and modelling of the motion of He II between two
volumes connected by a small channel from an unpublished experiment performed
by this author. The fluid height in one of the volumes, which was a partially filled
5 GHz quarter-wave tin-plated copper resonator with a quality factor of 3700, a
height of 2 cm, and a volume of 175 mm3 whose frequency change was used to
measure the fluid height, is plotted as a time series for different values of bulk liquid
temperature at SVP. At time 𝑡 = 0 seconds a pressure was applied by a cylindrical

15

Fig. 1.6: Change in liquid level height versus time of a bath of He II contained in a
partially filled quarter-wave resonator connected through a small channel to another
volume after a step in pressure is applied at time 𝑡 = 0 s. The data is shown here for
He II temperatures of 𝑇 = 0.9 K (blue line) and 𝑇 = 1.1 K (orange line). The model
is also shown for 𝑇 = 0.9 K (dash-dotted black line) and 𝑇 = 1.1 K (dashed black
line).
capacitor which formed the container for the other volume of fluid and coupled to
the liquid through weak dielectric constant of He II to induce mass flow. Before
𝑡 = 3 s, the fluid was constrained by the critical velocity in the channel before the
build-up of a temperature gradient caused a counter-flow of mass, exemplifying
the interdependence of mass and heat flow in He II. The lag between temperature
gradient and mass flow induced mass oscillations between the two volumes, seen
in the data at 𝑇 = 0.9 K. Above approximately 𝑇 = 1 K the oscillations become
over-damped. Modelling was performed by numerically solving a modified version
of Eq. (1.2) while including the limiting critical velocity value of 𝑣 𝑐 . We note the
resemblance to the film-flow trace shown in shown in Fig. 5 of Ref. [118].
In modelling such a system it was necessary to consider how heat was transported
not only through the bulk He II, but between the liquid and its containing walls.
Careful consideration of this interfacial heat transfer is of critical importance to the
work presented in this thesis, where we attempt to use He II to cool the HEMT

16

Fig. 1.7: Schematic representation of phonon scattering at the interface between two
dissimilar media, in this case between a hot solid and a cool liquid, which resembles
the scattering of light between two media of dissimilar indices of refraction.
gate surface. Fundamentally, the thermal boundary resistance arises due to the
discontinuity in the thermal environment experienced by the heat-carrying phonons
across the boundary. The effect was first observed by Pyotr Kapitza in 1941 [119]
and the interfacial conductance is generally referred to as the Kapitza conductance
ℎ 𝐾 . The first physical predictions of ℎ 𝐾 explained the interfacial heat conduction
in terms of the phonon radiation on each side of the boundary as predicted by
the Debye model, referred to as the "phonon radiation limit" [120, 121]. This
description, while capturing the observed ℎ 𝐾 ∝ 𝑇 3 dependence, over-predicts the
observed values of ℎ 𝐾 by more than a factor of 10 because it allows for more heat
transfer channels than are physically possible. An advancement to this model was
made by Khalatnikov in 1965 [122] and is known as “acoustic mismatch theory.”
While fundamentally similar to the phonon radiation limit, acoustic mismatch theory
attempts to account for the properties of the materials on both sides of the interface.
In particular it accounts for the finite reflection coefficient in a similar manner as
in photon reflection between two dissimilar optical media, as shown schematically
in Fig. 1.7. Particular to heat transfer between a solid surface and He II, it also
accounts for the fact that transverse phonon modes cannot exist in the liquid. While
believed to be the correct physical interpretation, acoustic mismatch theory tends to
under-predict the measured ℎ 𝐾 by a factor of nearly 20. Furthermore, neither the
phonon radiation limit nor acoustic mismatch theory correctly predict the observed
ℎ 𝐾 ∝ 𝑇𝐷−1 dependence on the solid Debye temperature 𝑇𝐷 , with the former predicting
ℎ 𝐾 ∝ 𝑇𝐷−2 and the latter predicting ℎ 𝐾 ∝ 𝑇𝐷−3 .
Further complications arise in the case where the solid surface is sufficiently hot

17
such that the local He II in its vicinity exceeds 𝑇𝜆 . This heat transfer regime is
known as the "film-boiling" regime where several phases of helium can coexist and
contribute to the film-boiling heat conductivity ℎ 𝑓 𝑏 . Given the elevated HEMT
gate temperatures of ∼ 20 K predicted to occur in the experiments presented in this
thesis, if the conductive heat flux due to phonon radiation is insufficient to reduce the
He II temperature in the vicinity of the gate below 𝑇𝜆 then film-boiling heat transfer
will necessarily occur. There are three cases two consider. The first is where the
local pressure at the heated surface remains below the SVP at 𝑇𝜆 , referred to as the
saturation boiling condition. In this case both He II liquid and helium vapor coexist
in the vicinity of the surface. The second is where the local pressure exceeds the
SVP at 𝑇𝜆 . If the heat flux does not exceed the critical heat flux in He I, then a film
of He I will form in which heat transfer occurs via nucleate boiling, and bulk heat
conduction occurs in the He II. Otherwise, there will be a coexistence of He II, He
I, and vapor at the heat transfer interface. The third case occurs in sub-cooled He
II, where again a coexistence of three phases can occur [63].
The film-boiling heat transfer regime, even for the simplest case of only a two-phase
coexistence of He II and vapor, remains the least well understood. The simplest
physical model, presented in Ref. [63], assumes a stable vapor film of constant
thickness covering the heat transfer surface, where heat transported through the
vapor film occurs by thermal conduction only. The variation in ℎ 𝑓 𝑏 with diameter
of a heated wire predicted by this model tends to agree with experiment; see for
example Fig. 7.42 of Ref. [63]. An alternative to this model presented in Ref.
[123] is to assume that the heat flux through the surface of the vapor film is equal
to the bulk He II heat conduction, and that the vapor film thickness increases to
limit the heat flux, a trend which is also borne out by experiment [124]. Yet
another alternative theory suggested in Ref. [125] uses molecular kinetic theory and
treats film boiling as a non-equilibrium process involving heat and mass transfer at
the vapor-He II interface. This theory predicts a minimum heat flux necessary to
establish a stable vapor film which is less than the critical heat flux predicted by
assuming a temperature excursion above 𝑇𝜆 , as was done previously. One way to
reconcile these two pictures is to treat the latter as the peak heat flux required to
establish a vapor film, and the former as the "recovery" heat flux which must be
subceeded to return to the non-boiling regime [126]. In all cases of film-boiling
heat transfer, the heat transfer coefficient ℎ 𝑓 𝑏 is typically 10–100 less than ℎ 𝐾 ,
although ℎ 𝑓 𝑏 is strongly dependent on a number of parameters such as heat surface
configuration, bath temperature and pressure, and immersion depth.

18
We now make some preliminary estimates of the heat flow 𝑄¤ = ℎ𝐴Δ𝑇 that is
expected between a submerged HEMT gate surface and a He II bath, where ℎ
is the heat transfer coefficient, 𝐴 is the heat transfer surface area, and Δ𝑇 is the
temperature difference across the interface. We take the heat transfer surface area
to be approximately that of the gate head in our device, 1 𝜇m × 200 𝜇m. In the
regime of the phonon radiation limit, we take ℎ 𝐾 = 5.7 kW m−2 K−1 which is
the measured value of the Kapitza conductance for SiO2 (taken from the compiled
data in Table 2 of Ref. [121]) and Δ𝑇 = 0.5 K as a representative value of the
temperature difference in the case of non-boiling heat transfer. Using these values,
we estimate 𝑄¤ 𝐾 = 0.6 𝜇W. In the film-boiling regime, we take ℎ 𝑓 𝑏 = 1 kW m−2
K−1 as a representative value for heat transfer between He II and flat surfaces
(taken from the compiled data in Table 7.5 of Ref. [63]), and Δ𝑇 = 10 K as a
representative value of the temperature difference if the He II is able to measurably
decreases the surface temperature. Using these values, we estimate 𝑄¤ 𝑓 𝑏 = 2 𝜇W.
Although both estimates 𝑄¤ 𝐾 and 𝑄¤ 𝑓 𝑏 predict heat flows which are several orders
of magnitude below the milliwatts of dissipated heat required to optimally bias the
device, we note that considerable uncertainty exists in these predictions owing to the
complexity of the heat transfer process in both the film-boiling and phonon radiation
regimes and its dependence on the surface conditions, as well as the appropriate
choice for the effective area of heat transfer given the mushroom-shaped HEMT
gate head and the SiN passivation layer covering the device. We therefore expect
our estimates to give at best an order-of-magnitude indication of the heat flux. We
also note that the observed increase in heat transfer coefficients with decreasing
heat transfer surface size (see Fig. 11 from Ref. [124], for example) lends support
that observing a cooling effect in small HEMT geometries may be possible, and
perform the experiment designed to conclusively test these predictions, the primary
experimental topic of this thesis.
1.4

Outline of thesis

In Chapter 2 we present the experimental apparatus which was constructed to perform microwave noise characterization of a packaged HEMT amplifier submerged
in saturated He II, and describe the calibration procedures involved. In Chapter 3
we present the results of the noise measurements and interpret them using a smallsignal model of the device. We compare these results with the predictions of a
phonon radiation heat conduction model. We find that liquid cryogenic cooling
is unable to mitigate self-heating in HEMTs, and discuss the implications of this

19
finding in the context of prior studies of He II heat transport in both the film-boiling
and non-boiling regimes. We also examine the implications for the limits of noise
performance in modern HEMT amplifiers using both the Pospiezalski noise model
and a phenomenological model of noisy linear amplifiers. Finally, in Chapter 4
we summarize the central findings of our work and identify potential directions for
future related research.

20
Chapter 2

EXPERIMENTAL METHODS: LIQUID CRYOGENIC COOLING
OF HEMT AMPLIFIERS
Microwave low-noise amplifiers based on III-V semiconductor high electron mobility transistor technology are a key component of high precision measurements
across diverse fields in science and engineering. In this chapter we establish the
approach taken in our experiment to mitigate self-heating in HEMTs using liquid
cryogenic cooling, in particular He II. We begin with background information on
the cold attenuator Y-factor measurement technique used here, before presenting
the experimental apparatus design and construction including both the cryogenic
vacuum apparatus and the microwave measurement components. We then detail
all calibration measurements, both at room temperature and in liquid cryogenic
environments, and finish by describing the noise measurement procedure for our
device under test (DUT), a common-source two-stage packaged amplifier (model
WBA46A, designed by Ahmed Akgiray and detailed further in Ch. 5.1 of Ref. [38])
comprised of OMMIC D007IH metamorphic HEMTs [35], each with a 70 nm gate
length and a 4 finger 200 𝜇m width (4f200) double-mushroom gate structure consisting of an InGaAs-InAlAs-InGaAs-InAlAs epitaxial stack on a semi-insulating
GaAs substrate with each stage biased nominally identically, using the cold attenuator Y-factor method. An input matching network (IMN) was employed to match
the optimal transistor impedance to the 50 Ω impedance of our measurement system
over a 4–5.5 GHz bandwidth at cryogenic temperatures. We refer to the device under
test using the terms “device”, “amplifier”, “DUT”, and “LNA” interchangeably for
the remainder of this work.
2.1

Background theory for cold attenuator Y-factor measurements

Although there are many techniques available to measure noise, by far the most
straightforward and commonly used method to measure the noise temperature of a
noisy amplifier at cryogenic temperatures is the Y-factor method [127–129]. In this
scheme, a noise source with a known noise temperature in the "hot" (noise source
turned on) and "cold" (noise source turned off) states is connected to the input of the
amplifier, pulsed on and off, and the output noise power in the hot and cold states is

21
measured. The Y-factor is then defined as:
𝑌=

𝑃H
𝑃C

(2.1)

where 𝑃H (𝑃C ) is the measured noise power with the noise source switched on (off).
Figure 2.1 shows the measured noise powers 𝑃H (noise source switched on with
noise temperature 𝑇H ) and 𝑃C (noise source switched off with noise temperature
𝑇C ), plotted as black dots. Also plotted is the line that passes through these two
points in the idealized scenario where the DUT has a linear power gain 𝐺, all
connectors are lossless, and only the amplifier and noise source contribute to the
measured noise power. We have also used the definition of noise temperature
𝑇𝑁 = 𝑃 𝑁 𝐵−1 𝑘 B−1 , where 𝑃 𝑁 is the noise power, 𝐵 is the measurement bandwidth
and 𝑘 B is the Boltzmann constant. By knowing the slope and y-intercept of the
established line along with the definition of the Y-factor, we can readily find the gain
𝐺 and noise temperature 𝑇e of the amplifier as:
𝑃H − 𝑃C
𝐵𝑘 B (𝑇H − 𝑇C )
𝑇H − 𝑌𝑇C
𝑇e =
𝑌 −1
𝐺=

(2.2)
(2.3)

In any real experiment, additional components will contribute to the measured
noise. Figure 2.2 shows a schematic of a Y-factor measurement which includes two
lossy input components with respective losses 𝐿 1 and 𝐿 2 and physical temperatures
𝑇𝐿 1 and 𝑇𝐿 2 , the DUT with gain 𝐺 and noise temperature 𝑇e , one lossy output
component with loss 𝐿 3 and physical temperature 𝑇𝐿 3 , and a noise power detector
which we refer to as the "backend" detector, with noise temperature 𝑇BE . The first
input component and output component represent coaxial cables connecting to the
noise source and backend detector. The second input component represents an
attenuator with an intentionally higher loss than the input coaxial cable by a factor
of approximately 100. For cryogenic noise measurements, this attenuator is useful
in that its temperature can be more easily manipulated and measured in order to
dominate over the relative noise contribution of the coaxial cables. The use of
such an attenuator is known as the "cold attenuator" Y-factor method [130]. In this
scheme, again using the definition of noise power in terms of noise temperature, one

22

Fig. 2.1: Diagram of measured noise power versus noise source noise temperature
in the Y-factor measurement scheme.
can write the measured hot and cold noise powers as:
𝐺 (𝐿 1 − 1)
𝐺 (𝐿 2 − 1)
(𝐿 3 − 1)
𝑃H
= 𝑇H
+ 𝑇𝐿 1
+ 𝑇𝐿 2
+ 𝑇𝑒
+ 𝑇𝐿 3
+ 𝑇BE
𝐵𝑘 B
𝐿1 𝐿2 𝐿3
𝐿1 𝐿2 𝐿3
𝐿2 𝐿3
𝐿3
𝐿3

(2.4)

𝑃C
𝐺 (𝐿 1 − 1)
𝐺 (𝐿 2 − 1)
(𝐿 3 − 1)
= 𝑇C
+ 𝑇𝐿 1
+ 𝑇𝐿 2
+ 𝑇𝑒
+ 𝑇𝐿 3
+ 𝑇BE
𝐵𝑘 B
𝐿1 𝐿2 𝐿3
𝐿1 𝐿2 𝐿3
𝐿2 𝐿3
𝐿3
𝐿3

(2.5)
where each term in Eqs. (2.4) and (2.5) represents the noise power added by successive elements in the measurement chain shown schematically in Fig. 2.2. We
have also used the fact that the input-referred noise temperature 𝑇in of a matched
attenuator with loss 𝐿 and physical temperature 𝑇𝐿 is given by 𝑇in = (𝐿 − 1)𝑇𝐿 as
derived in Ref. [128].
By plugging Eqs. (2.4) and (2.5) into Eq. (2.1) and solving for 𝑇e we arrive at the

23

Fig. 2.2: Schematic of the cold attenuator Y-factor measurement chain. From left
to right the components are: Noise source with hot (cold) noise temperature 𝑇H
(𝑇C ), input coaxial cable with loss 𝐿 1 and physical temperature 𝑇𝐿 1 , attenuator with
loss 𝐿 2 and physical temperature 𝑇𝐿 2 , DUT with gain 𝐺 and input-referred noise
temperature 𝑇e , output coaxial cable with loss 𝐿 3 and physical temperature 𝑇𝐿 3 , and
backend noise power detector with input-referred noise temperature 𝑇BE .
following expression for the DUT noise temperature:
𝑇0 𝐸
𝑇coax (𝐿 3 − 1) 𝑇BE
−𝑇C −𝑇coax (𝐿 1 −1)−𝑇𝐿 2 (𝐿 2 −1)𝐿 1 −
(2.6)
𝑇e =
𝐿1 𝐿2 𝑌 − 1
𝐺 full 𝐿 3
𝐺 full
where we have defined 𝐸 = (𝑇H − 𝑇C )𝑇0−1 to be the excess noise ration (ENR)
of the noise source where 𝑇0 = 290𝐾, and we have assumed that the input and
output coaxial cable temperatures are equal so that 𝑇coax = 𝑇𝐿 1 = 𝑇𝐿 3 . We have also
defined 𝐺 full = 𝐺 𝐿 1−1 𝐿 2−1 𝐿 3−1 to be the the total gain from the input plane of the
input coaxial cable to the output plane of the output coaxial cable, which is a useful
change of variables since 𝐺 full is the only gain which can be directly measured in this
configuration without disconnecting any coaxial components. We can also subtract
Eq. (2.4) from Eq. (2.5) to find an expression for the DUT gain:
𝐺=

𝐿 coax 𝐿 2 (𝑃H − 𝑃C )
𝐵𝑘 B𝑇0 𝐸

(2.7)

where we have defined 𝐿 coax = 𝐿 1 𝐿 3 to be the total loss of the coaxial cables.
In order to make gain and loss measurements with the highest possible precision,
it was necessary to use a Vector Network Analyzer (VNA). These instruments are
capable of fully characterizing multi-port networks by determining their scattering
matrix (S-matrix) S, consisting of scattering parameters (S-pars) 𝑆𝑖 𝑗 = 𝑉𝑖− /𝑉 𝑗+ , by
driving port 𝑗 with an incident wave of voltage 𝑉 𝑗+ and measuring the scattered wave
at port 𝑖 of voltage 𝑉𝑖− . In particular, a two-port network can be fully characterized by
the parameters 𝑆11 , 𝑆12 , 𝑆21 , and 𝑆22 . Of special interest for our purposes is that the
power gain 𝐺 (loss 𝐿) of a two-port device is given by 𝐺 = |𝑆21 | 2 (𝐿 = |𝑆21 | −2 ), and
the complex reflection coefficient between the device and a component connected
to the input of the device is given by Γ = 𝑆11 . A more complete treatment of
microwave network analysis can be found in Chapter 4 of Ref. [131]. Unless

24
specified otherwise, all loss and gain measurements shown in this work were taken
using a Rhode & Schwarz RSZVA50 VNA calibrated with a Maury 8050CK20
SOLT calibration kit, calibrated at most 24 hours before each measurement. The
uncertainty in any gain or loss measurement using this instrument was estimated to
be ±0.01 dB, given by the magnitude of the variation in the measured gain or loss
versus frequency of the calibration cable immediately after calibration.
With this in mind, we outline a method of transferring the high-precision calibration
of the VNA over to any noise power detector. We first note from Eq. (2.7) that
so long as the quantity (𝐵𝑘 B𝑇0 𝐸) −1 is constant (which is a reasonable assumption
over the timescale of a Y-factor measurement using stable filters and a stable noise
source), then the ratio 𝐺 full (𝑃H − 𝑃C ) −1 is also constant. Thus by fixing the gain at
some arbitrary calibration value 𝐺 cal
full , which can be measured with the VNA, and
cal and 𝑃 cal with the
then measuring the corresponding hot and cold noise powers 𝑃H
noise power detector, any other gain 𝐺 full can be determined by measuring only 𝑃H
and 𝑃C and using the following equation, derived by from Eq. (2.7) and the above
considerations:
𝑃H − 𝑃C cal
𝐺 𝐿 coax 𝐿 2
(2.8)
𝐺 = cal
𝑃H − 𝑃Ccal full
One subtlety to note is that in the derivations of 𝑇e and 𝐺, it was assumed that
all components are perfectly impedance matched so that the reflection coefficients
Γ𝑖 𝑗 = (𝑍𝑖 − 𝑍 𝑗 )(𝑍𝑖 + 𝑍 𝑗 ) −1 between components 𝑖 and 𝑗 with complex impedances
𝑍𝑖 and 𝑍 𝑗 are zero. In any real experiment, all microwave components are generally
manufactured with a consistent nominal impedance 𝑍0 (𝑍0 = 50 Ω in our experiment) so that the mismatch between components is small enough that negligible
error is introduced. The exceptions are the low-noise amplifiers themselves which
typically have a relatively poor impedance match over a given bandwidth, and the
noise source which has an impedance that changes between the hot and cold states.
If the noise source impedance changes significantly relative to the impedance of the
component connected to its output, the error introduced into the propagated power
will be significant. If the S-pars of all components including the noise source in
the hot and cold states are known, then this error can be corrected. The correction
algorithm is derived and demonstrated in Appendix A. Unless otherwise stated, this
correction was not used for data shown in this work since the impedance match
between the noise source and input coaxial cable was found to be sufficiently good
so as to introduce negligible uncertainty to the measurements.

25
Another subtlety of note is the distinction between representation of units in their
linear form (such as V, W, etc.) and in their logarithmic form (such as dB, dBV,
dBm, etc.). Unless otherwise stated, all units are assumed to be in their linear
representation and any logarithmic representation uses the "power ratio" definition
𝑃dB = 10 log10 ( 𝑃𝑃0 ) so that 𝑉dB = 20 log( 𝑉𝑉0 ). Furthermore, for error propagation
purposes it is useful to be able to convert uncertainties between their linear and
logarithmic representations. For a quantity 𝑋 with uncertainty Δ 𝑋 with logarithmic
representation 𝑋dB and logarithmic error Δ 𝑋dB this can be done using the following
equations:
𝑋dB

Δ 𝑋dB

Δ 𝑋 = 10 10 (10 10 − 1)
Δ𝑋
Δ 𝑋dB = 10 log10 1 +

(2.9)
(2.10)

Finally, it is also useful to write the measured Y-factor explicitly as a time-series in
terms of the measured voltages 𝑉H,𝑘 and 𝑉C,𝑘 output by the backend detector at time
index 𝑘. We assume the tunnel diode detectors linearly transduced the noise power
with some power-to-voltage gain KD and with a zero-power offset voltage 𝑉0,𝑘 so that
𝑉H,𝑘 = KD 𝑃H,𝑘 +𝑉0,𝑘 and 𝑉C,𝑘 = KD 𝑃C,𝑘 +𝑉0,𝑘 at time index k, which is the expected
behavior for microwave power tunnel diode detectors operating in the square-law
regime [132, 133]. The voltage data was analog filtered and over-sampled at 𝑓s to
avoid aliasing artifacts. We emphasize here that there is a trade-off between the faster
response time offered by tunnel diode detectors versus the improved stability offered
by commercial power sensors. We enhanced the stability of our diode detectors by
re-measuring the zero-power offset voltage at a frequency 𝑓0
𝑓ENR for a duration
𝑡0 = 𝐷 𝑓ENR to correct for DC offset drifts, where 𝐷 was the duty cycle of the
recalibration pulse. The noise source was pulsed at a frequency 𝑓ENR , and each
−1 .
half-pulse of hot (cold) voltage data was integrated for its duration 𝑡 ENR = 0.5 𝑓ENR
The integrated zero-power offset was subtracted from each integrated hot (cold)
0 (𝑉 0 ) for each pulse 𝑝. This
half-pulse to give hot (cold) offset-corrected data 𝑉H,𝑝
C,𝑝
procedure yielded Y-factor data effectively sampled at 𝑓ENR with 𝑡0 seconds of data
skipped every calibration cycle. The expression for the measured Y-factor for pulse
𝑝 considering the above specifications is:
 𝑘 HC, 𝑝
 𝑘
Í0
−1
−1
𝑁HC
𝑉H,𝑘 − 𝑁0
𝑉0,𝑘
𝑉H,𝑝
𝑘=𝑘 H, 𝑝
𝑘=0
(2.11)
𝑌𝑝 =
 𝑘 C, 𝑝
 𝑘
 = 0
𝑉C,𝑝
Í0
−1
−1
𝑁HC
𝑉C,𝑘 − 𝑁0
𝑉0,𝑘
𝑘=𝑘 HC, 𝑝

𝑘=0

26
where 𝑁0 = 𝑡0 𝑓s is the number of sampled points in each calibration pulse, 𝑁HC =
𝑓s 𝑡ENR is the number of sampled points in each half-pulse, 𝑘 H,𝑝 = (𝑘 0 + 1) + 2( 𝑝 −
1)𝑁HC is the time index at the start of the 𝑝th hot pulse, 𝑘 HC = 𝑘 H,𝑝 + 𝑁HC is the
time index at the center of the 𝑝th pulse when the noise source switches from hot to
cold, and 𝑘 C,𝑝 = 𝑘 H,𝑝 + 2𝑁HC is the time index at the end of the 𝑝th cold pulse.
2.2

Experimental apparatus overview

In order to measure the noise of the HEMT LNA when submerged in He II, a
microwave noise characterization apparatus was designed and constructed around
the principle of dipping the LNA directly into the liquid bath of a commercially
available liquid helium transport dewar, and evaporatively cooling this bath to reach
temperatures below the helium-4 lambda point. A schematic of the experimental
design which is an adaption of the cold attenuator Y-factor method with the inclusion
of a liquid helium bath surrounding the DUT and attenuator is shown in Fig. 2.3(a).
The apparatus consisted of: a 15 dB ENR 2-18 GHz solid state packaged noise diode
to generate noise power, stainless steel coaxial cabling to carry noise power between
the room temperature and cryogenic components, a 20 dB packaged chip attenuator
and LNA under test mounted to a copper mounting stage, and a room temperature
noise power detector. Further details of the cryogenic components and the noise
power detector are provided in Sections 2.3 and 2.4, respectively. Representative
output noise power voltage data is shown as a time series in Fig. 2.3(b) where
the data was analog-filtered at 1 kHz and over-sampled at 𝑓s = 1.1 kHz, and the
measurement parameters were 𝑓ENR = 10 Hz, 𝑓0 = 1 Hz, and 𝐷 = 0.1. The first and
last half-pulses of each cycle were discarded due to the 20 ms switching time of the
microwave switch used for recalibration, yielding 8 total pulses. Unless otherwise
indicated, these parameters were used for all subsequent measurements. For all
steady-state data the Y-factor was further averaged over a total measurement time
𝑡fin = 4 s.
2.3

Cryogenic engineering

Figure 2.4 shows an image of the mounted vacuum apparatus, which was helium
leak-tested at room temperature on an empty 60 L Cryofab liquid helium transport
dewar with a Pfeiffer SmartTest HLT 550 leak detector. A conical 2.5" Tri-Clamp to
KF-40 adapter was used to seal directly to the dewar. A 6-way KF-40 cross adapter
was used to attach the necessary vacuum fittings. These vacuum fittings consisted
of the following: (1) MKS 901P loadlock transducer to measure absolute pressure

27

Fig. 2.3: (a) Schematic of the measurement apparatus inside a liquid helium-4
dewar, including the following components: (1) 15 dB ENR 2-18 GHz solid state
SMA packaged noise diode biased at 28 V through a MOSFET amplifier circuit
(not shown); (2) input and output silver-plated stainless steel SMA coaxial cables
each 1.3 m in length; (3) 20 dB packaged cryogenic chip attenuator with factory
calibrated DT-670-SD diodes mounted directly on the attenuator substrate; (4)
packaged WBA46A LNA; (5) backend noise power detector. (b) Representative
raw Y-factor data versus time. The diode detector DC offset voltage 𝑉0 (black lines),
hot voltage 𝑉H (orange lines), and cold voltage 𝑉C (blue lines) are all shown.
from 1 atm down to ∼ 10−5 Torr; (2) leak-tight cryogenic 20 PSI pressure-relief
valve for safe release of helium pressure buildup in the event of a power failure to
the pumping system; (3) KurtLesker inline bellows valve connecting to an ultra high
purity helium-4 gas supply to control the back-filling rate of helium gas, with inlet
pressure measured by a Bourdon tube mechanical pressure gauge; (4) pumping line
connection consisting of a KurtLesker inline bellows valve for flow-rate control in
series with a normally-closed Agilent electromagnetically actuated block valve to
prevent back-filling of air in the event of power failure to the pumping system, which
was comprised of a Leybold DK-50 rotary piston vacuum pump (50 m3 h−1 pumping
speed) in parallel with an Anest Iwata ISP-500B dry scroll vacuum pump (35 m3 h−1
pumping speed) each outfitted with one-way valves on their outlet ports to further

28
protect from back-filling with air; (5) vertically mounted KF-40 straight connector
tube to extend the height of the vacuum fitting to match the length of the stainless
steel coaxial cables used in the experiment, which connected to two conical KF-40 to
KF-50 reducing adapters connected in series to provide enough space for stress-relief
loops in the coaxial cabling, and then to a KF-40 to 4-1/2" UHV CF flange which was
sealed with an OFHC copper gasket to a custom machined 4-1/2" UHV CF flange
electronics breakout fitting with hermetically sealed SubMiniature-A (SMA) and DC
feedthroughs; (6) rubber coated nickel-alloy heater tape with adjustable temperature
control, which was wrapped around the flanges that were in direct contact with
the helium pumping path, in order to maintain temperatures above the leak-tight
temperature tolerance of the O-rings during pumping. The conical reducing adapters
and custom hermetic breakout panel are shown in Fig. 2.5, with the 4" long input and
output coaxial cable leads connecting to SMA test cables via inner-outer DC blocks
which defined the reference plane for all subsequent microwave measurements, and a
24-wire DC cable connected to the vacuum feedthrough which enabled DC electrical
connections between room temperature electronics and cryogenic components inside
the dewar. The Tri-Clamp O-ring was made from PTFE which is rated leak-tight to
−100 ◦C, and all KF O-rings were made from silicone which is rated leak-tight to
−60 ◦C.
Inside the vacuum space, a thin-walled (1.250" outer diameter, 1.084" inner diameter) 304 stainless steel tube was used as a dipstick to submerge the DUT into the
helium-4 bath. Figure 2.6 shows the OFHC copper stage which was machined using
a TREE vertical milling machine in the Jim Hall Design and Prototyping Lab at
Caltech, and screw-mounted to the dipstick. The DUT, cold attenuator, two Lake
Shore DT-670-CU temperature diodes, and a heater (mounted behind the stage and
not shown) were screw-mounted to the stage using indium foil as an inter-facial
layer to promote thermalization between the components and the stage. An American Magnetics liquid helium level sensor calibrated for 4.2 K liquid was taped to
the interior wall of the dipstick using kapton tape, with the bottom of the sensor
sitting 1 cm above the top of the mounting stage. This sensor was primarily used
to ensure consistent starting heights across different liquid helium dewars. The
fluctuations in the readings of the temperature diodes mounted on the copper stage
were used to approximate the liquid level when the bath was in the superfluid state.
An improvement to this design would be to use a liquid level sensor calibrated to
the operating temperature of the superfluid bath, and to mount the sensor so that
the bottom of the sensor and the bottom of the mounting stage are level. Noise

29

Fig. 2.4: Image of the vacuum apparatus mounted to a liquid helium dewar with the
following components: (1) pressure transducer; (2) leak-tight cryogenic pressurerelief valve; (3) bellows valve connecting to an ultra high purity helium-4 gas supply;
(4) pumping line connection consisting of a bellows valve for flow-rate control in
series with a normally-closed electromagnetically actuated block valve; (5) vertically
mounted straight connector tube; (6) heating tape. The helium flow path induced by
pumping is shown as a blue dashed line, and frosting of condensation on the exterior
walls of the vacuum components along this path can be seen.
power was directed to and from the attenuator and DUT through 1.3 m long 0.141"
diameter low-loss silver-plated stainless steel SMA coaxial cables, which were connected to the hermetic SMA feedthrough via 0.3 m long copper-braided SMA flex
cables, each with one stress relief loop. The output cable was connected to the DUT
output through two SMA flex cables, which were included in the loss calibration
measurement of the cabling. DC signals were carried by phosphor-bronze cryogenic
wires manufactured by Lake Shore Cryotronics.
Since the experimental procedure required evaporatively cooling a relatively large

30

Fig. 2.5: Image of the hermetic breakout flange with the following labeled components: (1) conical reducing adapter; (2) CF flange with hermetic breakouts; (3)
DC cable; (4) input and output inner-outer DC blocks terminating the coaxial cable leads; (5) SMA test cables connected to the VNA (not shown). The reference
plane for all microwave measurements was the male connector of the inner-outer
DC blocks.
quantity of helium liquid, thermal modelling was performed to determine what
pumping speed was required so that the evaporative cooling power was sufficient to
overcome the heating power of the mounting apparatus, and to ensure that enough
liquid would remain to submerge the DUT for the length of time required for Yfactor measurements. Figure 2.7 shows a schematic of the model which includes

Fig. 2.6: Image of the copper mounting stage with the following labeled components: (1) dipstick; (2) input and output coaxial cables; (3) Lake Shore DT-670-CU
temperature diodes; (4) packaged 20 dB cold attenuator; (5) DUT.

31
the following parameters: the saturated vapor pressure 𝑃vap , the liquid helium-4
density 𝜌4He and heat capacity 𝐶4He , and the enthalpy (latent heat) of vaporization
Δ 𝐻vap , which all vary with the liquid bath temperature 𝑇L ; the effective pumping
speed 𝑆eff = Ctube 𝑆(𝑆 + 𝐶eff ) −1 given the pumping speed 𝑆 of the pump and the
¯ −1 where 𝑑 = 2.64 cm is the
hydrodynamic tubing conductance Ctube = 1350𝑑 4 𝑃𝑙
diameter of standard KF-25 tubing, 𝑙 = 1 m is the approximate length of tubing
used here, and 𝑃¯ ≈ 𝑃vap since we have Ctube
𝑆 for all pressures measured
in this experiment (down to 𝑃vap = 5.6 Torr at 𝑇L = 1.6 K); the pump inlet
pressure and temperature 𝑃𝑖 ≈ 𝑃vap and 𝑇𝑖 ≈ 300 K; and the external heating
𝑃ext ≈ 100 mW on the liquid bath carried by both the coaxial cabling and the
helium dewar walls, estimated by considering both the average evaporation rate of
a liquid helium transport dewar of 1 liter/day and the contributions from the coaxial
cabling connecting the 300 K breakout flange to the liquid bath. By using the fact
that both the total helium-4 mass and energy are conserved along with the above
specifications, we arrive at the following differential equations for the liquid bath
temperature 𝑇L and volume 𝑉L :
𝑃vap (𝑡)𝑆eff 𝑀4He
d𝑉L
=−
d𝑡
𝑅𝑇L (𝑡) 𝜌4He (𝑡)
𝑃vap (𝑡)𝑆eff Δ 𝐻vap (𝑡)
𝑀4He
d𝑇L
=−
− 𝑃ext
d𝑡
𝐶4He (𝑡) 𝜌4He (𝑡)𝑉L (𝑡)
𝑅𝑇L (𝑡)

(2.12)
(2.13)

where 𝑀4He = 4 g mol−1 is the molar mass of helium-4 and 𝑅 = 8.31 J (mol K)−1
is the gas constant. All temperature dependent helium-4 quantities were taken from
the compiled data in Ref. [104].
Equations (2.12) and (2.13) were solved using the ode45 differential equation solver
in MATLAB. Figure 2.8(a) shows the modelled bath temperature versus time for
various pumping speeds. The cooling speed from a starting bath temperature of 4.2
K down to 2.5 K happens relatively quickly, taking less than 3 hours for a modest
pumping speed of 15 m3 h−1 . The cooling slows drastically as the temperature
crosses the lambda point where the heat capacity increases, and then speeds up
again as the heat capacity comes back down. The critical pumping speed, below
which the evaporative cooling power is insufficient to ever achieve bath temperatures
below 𝑇𝜆 , is around 2 m3 h−1 , although pumping speeds of 25 m3 h−1 and above are
required in order to achieve stable base temperatures below 𝑇𝜆 in less than one day of
pumping. Furthermore, a high throughput pump is required to move the significant
mass of helium-4 vapor over the first several hours of pumping. This can be seen in
Fig. 2.8(b) which shows the volume of remaining liquid versus time, and suggests

32

Fig. 2.7: Schematic of the liquid helium-4 dewar pumping model which includes
the pumping speed 𝑆, the hydrodynamic tubing conductance Ctube , the pump inlet
pressure and temperature 𝑃𝑖 and 𝑇𝑖 , the external heating 𝑃ext on the liquid bath and
the temperature dependence of the saturated vapor pressure 𝑃vap , liquid helium-4
density 𝜌4He , heat capacity 𝐶4He , and the enthalpy of vaporization Δ 𝐻vap
that at least 20 L of liquid, which weighs 2.5 kg, must be removed. Since the rate
of liquid volume change decreases as the vapor pressure decreases with decreasing
temperature, Fig. 2.8(b) also suggests that even for relatively fast pumping speeds
the liquid volume will last for at least several days, an expectation which was indeed
borne out in the experiment.
Also shown in Fig. 2.8(a) is the measured bath temperature over the first 5 hours
of pumping during a cooldown. For the first 1.3 hours only the more powerful
Leybold DK-50 rotary piston pump was used to move the bulk of the helium, and
the pumping speed was throttled manually by adjusting the bellows valve in order
to minimize frosting of the O-rings while the helium vapor density was still high
enough to significantly cool the vacuum tubing, with adjustments of the throttling
seen as kinks in the cooling curve. After 1.3 hours the valve was fully opened, and
after another 20 minutes the Anest Iwata ISP-500B dry scroll pump was engaged, as
seen by the kink in the cooling curve at 1.6 hours. After this, the pumping occurred
at a rate of approximately 60 m3 h−1 , estimated by comparison with the modelled
curve, with the pumping rate decreasing with decreasing temperature. This pumping
speed is less than the highest rated pumping speed of the two pumps when added in
parallel (50 + 35 = 85 m3 h−1 ), which is expected when considering the reduction

33
in rated pumping speed associated with reduced pump inlet pressures.

Fig. 2.8: (a) Liquid bath temperature versus time. The pumping model solution
for 𝑇L is shown for pumping speeds of 60 m3 h−1 (dash-dotted gray line), 30 m3
h−1 (dash-dotted beige line), 15 m3 h−1 (dash-dotted purple line), and 5 m3 h−1
(dash-dotted green line). The measured bath temperature during a dewar cooldown
is also shown (solid blue line). (b) Remaining liquid volume versus time. The
pumping model solution for 𝑉L is shown for the same pumping speeds as in (a).
We conclude this section by noting several design considerations with regards to
pumping on the vacuum space of a cryogenic dewar. Firstly, it is critical to ensure
that all vacuum fittings are helium leak-tight over the full temperature range they
experience while pumping. In our design, the fast moving helium vapor from
the dewar space to the pump caused significant forced convective cooling of the
vacuum fittings in the pumping line, as evidenced by the frosting visible in Fig. 2.4.

34
Throttling of the pumping speed was performed to ensure all O-rings stayed within
their temperature tolerance. A possible improvement to this design would be to use
an inner lining tube along the pumping path to separate the flow of helium from
directly contacting the room temperature vacuum fittings, although due to space
limitations such a design is difficult to employ in narrow-neck dewars such as the
one used in this experiment. Second, a sufficiently strong pump must be employed
which can sustain pumping speeds above approximately 25 m3 h−1 at inlet pressures
above 10 Torr. It was found that a dry-scroll pump of the type commonly used in
low-temperature laboratories was insufficient for this task, necessitating the use of a
rotary piston pump or another pump of comparable or greater sustained throughput.
2.4

Device characterization and calibrations at room temperature

The operation of a transistor amplifier requires direct current (DC) voltage biasing
of both the drain terminal relative to the source terminal which we denote 𝑉DS and
the gate terminal relative to the source terminal which we denote 𝑉GS . A positive
𝑉DS is applied to drive a current 𝐼DS along the transistor channel from source to
drain, and since the GaAs HEMT transistors used in this work are depletion mode a
negative 𝑉GS is applied in order for a microwave frequency voltage signal applied to
the gate to modulate the conductance of the channel and induce gain. Device bias
voltages and currents were generated using Keithley 2400 SourceMeters.
Both DC and microwave characterization of the amplifier were performed at room
temperature in order to ensure reasonable device operation. Figures 2.9(a) and 2.9(b)
show the DC current-voltage (I-V) data taken over a range of 𝑉GS on the first and
second amplifier stages, respectively. In order to measure the I-V curve of an
individual transistor stage, the other stage was pinched off at 𝑉GS = −8 V. The
plotted 𝑉DS is corrected to account for the 44 Ω resistor in series with the drainsource bias line of each transistor as part of the protection circuitry, as are all
𝑉DS values reported hereafter. The current density 𝐼DS is normalized to the gate
periphery of 200 𝜇m. Figure 2.9(c) shows the 𝑆11 and 𝑆21 of the LNA measured
using the VNA as well as the noise temperature of the LNA measured using an
internally calibrated Agilent N8975A noise figure analyzer (NFA) and a N4000A
noise source, with the device biased at its low-noise bias of 𝑃DC = 37.1 mW mm−1
at room temperature. The device exhibits a noise temperature minimum of 𝑇e = 50𝐾
at a frequency of 4 GHz at room temperature. These characterization measurements
suggest a functioning device within the device design parameters, although the
differences between Fig. 2.9(a) and Fig. 2.9(b) suggest a slight asymmetry between

35
the two nominally identical stages.

Fig. 2.9: (a) 𝐼DS versus 𝑉DS measured at room temperature with the reverse-bias of
the first transistor stage ranging from 𝑉GS = −8 V to 𝑉GS = −2 V in 0.5 V steps
(black lines from bottom to top) with the second transistor stage pinched off at -8
V. (b) 𝐼DS versus 𝑉DS measured at room temperature with the reverse-bias of the
second transistor stage ranging from 𝑉GS = −8 V to 𝑉GS = −2 V in 0.5 V steps
(black lines from bottom to top) with the first transistor stage pinched off at -8 V.
An asymmetry in the IV curves between the two stages can be seen. (c) Noise
temperature (left axis, blue circles), |𝑆21 | (right axis, red line), and |𝑆11 | (right axis,
magenta line) versus microwave frequency with the device biased at its low-noise
bias of 𝑃DC = 37.1 mW mm− 1 (𝑉DS = 0.85 V, 𝐼DS = 43.7 mA, 𝑉GS1 = −4.5 V,
𝑉GS2 = −4.9 V) at room temperature. Noise temperature was measured using a
NFA, and S-pars were measured using a VNA.
In order to measure microwave noise power, a backend room temperature microwave
power detector was assembled. Figure 2.10 shows a schematic of the backend, which
consisted of the following components. Microwave power coupled to the input of the
detector first passed through a 3-–6 GHz microwave isolator to improve impedance

36
matching and minimize reflections. Two Minicircuits ZX60-83LN-S+ broadband
amplifiers were placed sequentially following the isolator. A microwave filter was
then used to limit the bandwidth of power amplified by the final gain stage, a Miteq
AMF-3B-04000800-25-25P power amplifier. Next, a microwave switch was used to
periodically switch between the signal path and a 50 Ω load, enabling recalibration
of the detector to correct for DC offset drifts. A variable attenuator was then
used to set the magnitude of noise power which was then split into two channels,
one with a temperature-controlled yttrium iron garnet (YIG) filter with adjustable
center frequency, and the other with a fixed Reactel 5 GHz bandpass cavity filter.
The latter channel was primarily used for diagnostic purposes. Microwave noise
power finally reached the Herotek DT4080 tunnel diode detectors, which linearly
transduced this power into a DC voltage. Inner-outer DC blocks were placed at the
diode detectors’ inputs to eliminate unwanted biasing. The final DC signals were
amplified and low-pass filtered by two SRS560 pre-amplifiers and then digitized
by a National Instruments NI6259-USB DAQ for further data processing. All data
acquisition was controlled via LabVIEW running on a Windows desktop computer,
and instrument control was generally performed using the GPIB protocol. All data
processing was performed in MATLAB.
Several diagnostic measurements were performed on the backend to ensure appropriate operation of the backend elements. Figure 2.11 shows the magnitude of the
|𝑆21 | versus frequency of each of the backend filter channels with the YIG filter set
at a few different center frequencies. Both filters exhibit reasonable flatness over
their 20 MHz bandwidth.
Figure 2.12(a) shows one calibration measurement of the voltage output by the
tunnel diode detectors versus incident microwave power, along with a linear fit to
the power range 0–4 𝜇W. The diodes exhibit good linearity in this range but deviate
from linearity above 5 𝜇W (−23 dBm). Figure 2.12(b) shows another calibration
measurement of diode detector voltage output versus incident microwave power
with more data taken at lower powers, plotted logarithmically along with a linear
fit. Linearity is observed here within measurement error down to −44 dBm (100
nW). The magnitude of incident microwave power was measured using a LadyBug
LB5940A TrueRMS power sensor which was connected to the diode detectors
via a calibrated microwave power splitter. We find power-to-voltage sensitivities
of 𝐾YIG = 1004 mV mW−1 and 𝐾fixed = 1111 mV mW−1 , in agreement with the
manufacturer reported minimum sensitivity of 900 mV mW−1 . A variable attenuator

37

Fig. 2.10: Schematic of the room temperature backend noise power detector consisting of (1) Pasternak PE8327 isolator, (2) Minicircuits ZX60-83LN-S+ low-noise
amplifiers, (3) Minicircuits filters with 3-6 GHz bandwidth, (4) Miteq AMF-3B04000800-25-25P medium power amplifier, (5) RF switch for calibration, (6) 0 –
20 dB variable attenuator, (7) microwave power splitter, (8) Reactel cavity filter
with 5 GHz center frequency and 20 MHz bandwidth, (9) Micro Lambda MLFM42008 20 MHz bandwidth tunable YIG filter, (10) Pasternak PE8224 inner-outer
DC blocks, (11) Herotek DT4080 tunnel diode detectors, (12) SRS560 low-noise
preamps, (13) National Instruments NI6259-USB DAQ. Also shown is the Agilent
33210A arbitrary waveform generator (AWG) used to pulse the microwave switch
MOSFET biasing circuit, as well as the DAQ terminals connecting to the AWG port
used to bias the noise source (not shown), temperature diode voltage outputs from
the temperature controller, and liquid level sensor voltage output from the American
Magnetics 1700 liquid level instrument, enabling synchronous measurement of each
component with the noise power data. The losses of the SMA cabling and attenuator
pads are not shown.
was used to ensure that the microwave power incident on the diode detectors was
between −40 dBm and −24 dBm for all subsequent measurements.
Figure 2.13 shows the Allan deviation of the output of the tunnel diode detectors
referred to their input versus integration time 𝜏, measured with an incident microwave power of −100 dBm which is five orders of magnitude below the detectors’

38

Fig. 2.11: |𝑆21 | of the fixed 5 GHz filter channel (black line) and the YIG filter
channel with center frequency set to 4.5 GHz (blue line), 5.0 GHz (black line), and
5.5 GHz (red line) versus frequency. The filters exhibit better than ±0.5 dB flatness
over their 20 MHz bandwidth. The −100 dB noise floor is set by the VNA noise
floor.
manufacturer-reported noise floor of −50 dBm. The voltage was sampled by the
DAQ at 500 Hz after being analog-filtered at 1000 Hz and amplified by a factor
of 1000 by the SRS560 preamps. This voltage gain was used for all subsequent
measurements. Both detectors exhibited a minimum in their Allan deviation of less
than 0.1 nW Hz−1/2 at an integration time of approximately 1 s, so a recalibration
period of 1 s was chosen for all subsequent measurements. Another minimum in
the Allan deviation was observed at 500 s, suggesting the presence of a noise source
other than 1/f noise at frequencies between roughly 1 Hz and 2 mHz, although such
long integration times are not relevant to our measurements.
In order to ultimately extract 𝑇e , the measurement components corresponding to
each term in Eq. (2.6) were calibrated. Figure 2.14(a) shows the calibration data
for the backend noise temperature 𝑇BE versus frequency. This measurement was
performed using a nitrogen cooled fixed load method, where the noise power of the
backend terminated with a 50 Ω load was first measured at room temperature, and
then measured again with the 50 Ω load submerged in a small liquid nitrogen bath at
77 K. Thermal insulation was placed over the bath to minimize vapor cooling of the
3.5" long stainless steel coaxial cable connecting the backend to the 50 Ω load. The
time between hot and cold measurements was 1 minute, which was the time it took to
dip the load into the nitrogen bath and allow its temperature to stabilize as indicated

39

Fig. 2.12: Diode detector output voltage versus incident microwave power on the
diode detector for the fixed 5 GHz filter channel (blue circles) and the YIG channel
with center frequency set to 5 GHz (magenta circles). (a) Data shown for incident
microwave power ranging from 0 𝜇W to 10 𝜇W (−20 dBm). A linear fit to the data
below 4 𝜇W (−24 dBm) is also shown (solid lines). A nonlinear response can be
seen above 5 𝜇W (-23 dBm). (b) Data shown for incident microwave power ranging
from −44 dBm to −22 dBm, plotted logarithmically. A linear fit to the full range of
data is also shown (solid lines)
by a plateauing of the measured noise power. The backend noise temperature was
found to be stable over at least a several-month timeframe. All backend amplifiers,
tunnel diode detectors, and DC pre-amplifiers were wrapped in thermal insulation
to mitigate drifts in gain and noise temperature.
Noise power was generated by a packaged 2 – 18 GHz solid state SMA noise diode
with a 15 dB excess noise ratio, which was biased using an Agilent 33220A arbitrary
waveform generator whose output was amplified by a MOSFET biasing circuit. The
circuit was formed by an Ohmite Arcol HS25 100 Ω power resistor in series with a
Vishay Siliconix IRFP140 power MOSFET and the noise diode in parallel, with the
MOSFET gate bias pulsed by the 3 V square wave output of the signal generator.
A schematic of an identical circuit which was used to pulse the microwave switch
is shown in Fig. 2.10 labeled as “Calibration switch.” Figure 2.14(b) shows the
calibration data for the noise source ENR versus frequency, measured by taking
Y-factor measurements with the noise source connected directly to the calibrated
backend detector. The inset shows a schematic of the MOSFET biasing circuit. The

40

Fig. 2.13: Allan deviation versus integration time for the fixed 5 GHz filter channel
(blue dots) and the YIG channel with center frequency set to 5 GHz (magenta dots)
with −100 dBm incident microwave power on the detectors’ inputs. A minimum
is observed at approximately 1 s, which was used as the recalibration period for
subsequent measurements. Another minimum is seen at 500 s, suggesting the
presence of a noise source other than 1/f noise.
ENR was extracted using the following equation:
𝐸=

𝑌 −1
(𝑇C + 𝑇BE )
𝑇0

(2.14)

The noise source chassis, which was wrapped in thermal insulation to promote thermal equilibration between the chassis and the internal noise diode, was monitored
at all times using a type T thermocouple, and was found to vary negligibly under all
experimental conditions. We took the chassis temperature to be equal to the internal
diode temperature 𝑇C .
The MOSFET biasing scheme allowed for Y-factor sampling as fast as hundreds
of kHz, limited by the RC time constant of the MOSFET and noise diode circuit.
Figure 2.14(c) shows the fastest Y-factor measurements recorded on this setup with
data sampled at 𝑓s = 600 kHz (the fastest stable sampling speed of the DAQ) and
the noise source pulsed at 𝑓ENR = 50 kHz, taken with the noise source connected
directly to the backend detector. The fact that no discernible rise time is observed
indicates that the the noise source circuit cutoff frequency is faster than 600 kHz.

41

Fig. 2.14: (a) Backend detector noise temperature versus frequency. Error bars
reflect the uncertainty in the temperature of the cable connecting the 50 Ω load to
the backend. (b) Noise source ENR versus frequency. Error bars reflect the error
propagated from uncertainty in the backend noise temperature. (c) Diode detector
output voltage sampled at 𝑓s = 600 kHz with the noise source connected directly to
the input of the backend detector and pulsed at 𝑓ENR = 50 kHz.
2.5

Calibrations in liquid cryogen environments

Two separate liquid helium-4 dewar baths were used for calibration of each cryogenic
component entering into Eq. (2.6). One dewar bath was used to calibrate the coaxial
cable losses 𝐿 1 and 𝐿 3 and temperature 𝑇coax , and then another dewar bath was used
to calibrate the attenuator loss 𝐿 2 .
The general procedure in each calibration dewar was as follows. The dipstick was
used to submerge the mounting stage 2 cm from the bottom of a fully filled 60 L
liquid helium-4 dewar at 4.2 K and ambient pressure. After waiting 30 minutes for

42
thermal equilibration, calibration measurements were taken. The dewar was then
sealed, and a vacuum pump was used to evaporatively cool the liquid into the He
II phase. A steady-state temperature of 1.6 K was reached after roughly 6 hours
of pumping, and further calibration measurements were taken. A heater was then
switched on for less than 2 hours to accelerate the boil-off rate of the remaining
liquid, and switched off when a spike in stage temperature was observed which
indicated that the liquid surface had dropped below the stage. Further calibration
was performed after turning off the heater as the stage was allowed to warm from 1.6
K under the ambient heating power of the measurement apparatus. The dewar was
then back-filled with helium-4 exchange gas at 1 psig to facilitate thermalization to
room temperature, at which point the dipstick was removed.
We now discuss the details of the measurements in each calibration dewar. In the first
calibration dewar, the attenuator and DUT shown in the configuration of Fig. 2.6
were replaced with a short through which was thermally anchored to the copper
stage. The total cable loss was measured, with the inner-outer DC block SMA
connectors used as the input and output reference planes. Figure 2.15(a) shows the
total loss 𝐿 coax of the coaxial cables measured at room temperature in air and dipped
in the first calibration dewar at 4.2 K and 1.6 K. To isolate the individual cable losses
𝐿 1 and 𝐿 3 , the losses of the input cable (up to the input of the attenuator) and output
cable (from the output of the DUT) were also measured independently at room
temperature, and their ratio was assumed to remain constant for all temperatures.
The lumped coaxial cable physical temperature 𝑇coax , which is the effective temperature at which the cables radiate their noise power, was measured directly using the
Y-factor method. Applying the same approach as in the derivation of Eq. (2.6), we
can write the measured hot and cold noise power of a single impedance matched
lossy element of loss 𝐿 and physical temperature 𝑇𝐿 as:
𝑇0 𝐸 + 𝑇C 𝑇𝐿 (𝐿 − 1)
+ 𝑇BE
𝑇C 𝑇𝐿 (𝐿 − 1)
𝑃C =
+ 𝑇BE

𝑃H =

(2.15)
(2.16)

taking the ratio 𝑌 = 𝑃𝑃HC and solving for 𝑇𝐿 gives:
𝑇0 𝐸
𝑇𝐿 =
− 𝑇C − 𝑇BE 𝐿
𝐿−1 𝑌 −1

(2.17)

Figure 2.15(b) shows the measured lumped cable temperature versus frequency with
the cables dipped in a 4.2 K He I bath, using Eq. (2.17). We assumed that both cables

43

Fig. 2.15: (a) Total loss of input and output coaxial cables versus frequency measured at 300 K (red line), 4.2 K (magenta line), and 1.6 K (blue line) with a
commercial VNA. (b) Lumped physical coaxial cable temperature versus frequency
obtained from Y-factor measurements (magenta circles) and from a heat conduction model (black line). Error bars represent an estimate of the total uncertainty
including systematic errors.
were at the same physical temperature. The 1.6 K He II calibration measurement is
omitted for clarity as it is within 10 K of the He I measurement.
To support the accuracy of this method, we developed a thermal model of the cables
used in our experiment. This model extends the work from Ref. [134] to include
the effect of heat transfer between the cables and the surrounding fluid. We consider
infinitesimal cross-sectional slices of the cable in contact with the surrounding bath.
Assuming steady-state where the net heat flux is zero we can write:
¤ + d𝑥) = 𝑄(𝑥)
𝑄(𝑥
+ d𝑄(𝑥)

(2.18)

where 𝑄(𝑥)
is the heat flux at height 𝑥 along the cable, and d𝑄(𝑥)
is the differential

44
convective heat flux along the slice of length d𝑥. Applying Fourier’s law along the
length of the cable and using the definition of convective heat transfer [135] we can
write:
d𝑇
𝑄(𝑥)
= 𝐶s (𝑥)
d𝑥 𝑥
d𝑄(𝑥)
= 𝑃𝐻g,l (𝑥) 𝑇 (𝑥) − 𝑇g,l (𝑥) d𝑥
𝐶s (𝑥) =

𝜅steel (𝑥) 𝐴steel + 𝜅 ptfe (𝑥) 𝐴ptfe
𝜅steel (𝑥)𝜅ptfe (𝑥) 𝐴steel 𝐴ptfe

(2.19)
(2.20)
(2.21)

where 𝐶s (𝑥) is the thermal conductance arising from the parallel stainless steel and
PTFE heat conduction channels in the cable, 𝑃 is the perimeter of the cable, 𝐻g,l (𝑥)
and 𝑇g,l (𝑥) are the convection coefficient and temperature of the surrounding gas
and liquid baths, respectively, and 𝜅steel (𝑥), 𝜅ptfe (𝑥), 𝐴steel and 𝐴ptfe are the thermal
conductivities and cross-sectional areas of the inner stainless steel conductor and
PTFE dielectric in the cable, respectively. The temperature dependence of the
thermal conductivities 𝜅 steel and 𝜅 ptfe were taken from compiled data from NIST
[136, 137]. Massaging the above equations and rearranging, we arrive at:
 2
d𝑇
d2𝑇 d𝐶s
(𝑇 (𝑥))
− 𝑃𝐻g,l (𝑥) 𝑇 (𝑥) − 𝑇g,l (𝑥) = 0
𝐶s (𝑇 (𝑥)) 2 +
d𝑇
d𝑥
d𝑥

(2.22)

A schematic of the model is shown in Fig. 2.16(a), where we have defined 𝑅steel =
(𝜅steel 𝐴steel ) −1 , 𝑅ptfe = (𝜅ptfe 𝐴ptfe ) −1 , and 𝑅g,l = (𝑃𝐻g,l d𝑥) −1 . Equation (2.22) was
solved numerically using the bvp4c routine in MATLAB with fixed boundary values
of 301 K and 4.2 K, and assuming a liquid surface height of 10 cm above the bottom
of the cable. Table 2.1 lists the remaining assumed parameter values, and Fig. 2.16(b)
shows the modelled coaxial cable temperature profile. Averaging over this curve
yields an effective lumped physical temperature of the stainless steel coaxial cable of
𝑇steel = 208𝐾. We now determine the effective lumped physical temperature of the
full cable by including the additional SMA cabling which connected the stainless
steel cables to the noise source and backend detector. We make the simplifying
assumptions that the entire stainless steel cable radiates its noise at 𝑇steel with a loss
𝐿 steel = 2.34 dB and that the additional cabling radiates entirely at 𝑇cab = 301 K
with a loss of 𝐿 cab = 0.65 dB. The additional cabling loss was measured directly,
and the stainless steel cable loss was found by subtracting 𝐿 cab from the total loss
measured at 4.2 K shown in Fig. 2.15(a), at 5 GHz. Cascading the noise from these

45
Parameter
𝜅steel
𝜅ptfe
𝐴steel
𝐴ptfe
𝐻g
𝐻l

Value
𝑇 dependent
𝑇 dependent
1.30 mm2
18.83 mm2
1.12 mm
30 Wm−2 K−1
15 kWm−2 K−1

Source
[136]
[137]
Measured
Measured
Measured
[138]
[63]

Table 2.1: Table of parameters used in solving the coaxial cable thermal model.
two cables we find:
𝐿 cab 𝐿 steel 𝑇cab (𝐿 cab − 1) 𝑇steel (𝐿 steel − 1)
𝑇coax =
𝐿 cab 𝐿 steel − 1
𝐿 cab 𝐿 steel
𝐿 steel

(2.23)

𝑇coax = 223.3K
The estimated lumped coaxial cable temperature from this model is plotted as a
horizontal line in Fig. 2.15(b).

The final calibration measurements taken in the first calibration dewar were the loss
𝐿 coax and temperature 𝑇coax of the cables as the stage was allowed to warm up after
the liquid level dropped below the level of the stage. These measurements were again
performed using the Y-factor measurement. Figures 2.17(a) and 2.17(b) shows the
continuous measurement of 𝑇coax and 𝐿 coax over the full lifetime of the calibration
dewar bath from the start of the cooldown until the stage reached 100 K during
warmup. Figure 2.17(c) shows the stage temperature over the same time period, as
measured by the temperature diode in the attenuator (all temperature diodes agreed
to within 1 % at all temperatures shown here). Figure 2.17(d) shows the calibration
curves for cable temperature and loss which were ultimately used. These curves
were generated by fitting a smoothing spline to the cable temperature and loss data
plotted versus stage temperature data during the warmup.
In the second calibration dewar, the short through was replaced by the cold attenuator,
which was a packaged 20 dB wideband chip attenuator. The mounted attenuator
is shown pictured in Fig. 2.6. The total loss 𝐿 tot = 𝐿 coax 𝐿 2 was measured using
the same procedure as in the first calibration dewar. The attenuator loss 𝐿 2 was
extracted by dividing 𝐿 tot by the previously measured 𝐿 coax . Figure 2.18 shows

46

Fig. 2.16: (a) Schematic of the coaxial cable temperature model showing a slice of
length dx along the cable in contact with the gas environment. (b) Coaxial cable
temperature profile (blue line) versus height along the cable. The height of the liquid
surface is also shown (vertical dashed black line).
the attenuator loss versus microwave frequency measured at room temperature, 4.2
K, and 1.6 K using the VNA. The attenuator loss was found to vary by less than
0.2 dB between all temperatures, which is consistent with other measurements of
similar chip attenuators [139]. A measurement of the temperature of the attenuator
using the Y-factor method was not possible in this case due to the larger loss of

47

Fig. 2.17: (a) Cable temperature (blue line), (b) cable loss (red line), and (c) stage
temperature (orange line) measured over the lifetime of the first calibration dewar.
Cable temperature and cable loss were measured using the Y-factor method, and
stage temperature was measured by the temperature diode housed in the attenuator.
Recalibration using a VNA was performed at 𝑡 = 7 hours and 𝑡 = 62 hours,
reflected in the discontinuities in acquired data. (d) Cable temperature (blue line)
and cable loss (red line) versus physical stage temperature, generated by fitting a
smoothing spline to the time series data of the cable temperature and loss plotted
versus stage temperature, respectively, during the warming phase of the calibration
measurements. These curves were used as the calibration curves for subsequent
warming data.
the attenuator. Instead, the temperature 𝑇𝐿 2 of the attenuator was measured by
a calibrated Lake Shore DT-670-SD temperature diode indium-bonded directly to

48
the attenuator chip substrate. The diode calibration curve was provided by Lake
Shore, and the saturated liquid temperature of He I at 4.23 K was used to correct
for DC offsets. We assumed that negligible temperature differences existed between
the attenuator, mounting stage, and DUT, and we therefore took the DUT physical
temperature to be 𝑇phys = 𝑇𝐿 2 . All temperature diodes were measured using a Lake
Shore 336 temperature controller, which converted the temperature to a voltage
which was measured by the DAQ synchronously with all Y-factor measurements.

Fig. 2.18: Attenuator loss versus frequency at room temperature (red line), 4.2 K
(magenta line) and 1.6 K (blue line).

2.6

LNA noise and gain measurements

With the calibration data obtained, the noise and gain of the DUT was measured using
two additional liquid helium-4 dewar baths at various frequencies, temperatures, and
biases. The gate and drain bias voltages 𝑉GS and 𝑉DS , which were nominally applied
equally to each individual transistor through , were varied to yield transistor drainsource current densities 𝐼DS from 35 mA mm−1 to 120 mA mm−1 , corresponding to
dissipated powers 𝑃DC = 𝑉DS 𝐼DS per transistor of 15 mW mm−1 to 150 mW mm−1 .
The relevant measured values from the calibration procedure described above were
used to extract 𝐺 and 𝑇e from Eqs. (2.6) and (2.8).
In the first measurement dewar, the device was mounted to the stage in the configu-

49
ration shown in Fig. 2.6, and the cooldown procedure followed that of the calibration
dewars described above. At each physical temperature, the DUT bias was varied
and Y-factor measurements were taken versus frequency by adjusting the YIG filter center frequency. In addition, the calibration measurement of 𝑉Hcal , 𝑉Ccal , and
𝐺 cal
full was performed, with the DUT bias chosen arbitrarily to be its low-noise bias.
Using these measurements, the device gain 𝐺 could be extracted from Y-factor
voltage measurements at any bias using Eq. (2.8) without requiring a separate VNA
measurement. This calibration was found to be stable over several days, and was
repeated each day before data acquisition.
In addition to measurements under specific liquid cryogen environments, continuous
Y-factor measurements were also taken as the He II bath was pumped away, yielding
noise data both before and after the DUT gate was submerged. The measurements
were performed at a fixed bias of 𝑃DC = 80 mW mm−1 and frequency of 𝑓 = 4.55
GHz. This frequency was chosen due to it being the optimum noise match frequency
as determined by the IMN of the DUT. The bias was chosen to be sufficiently high
that any self-heating mitigation would be readily observed without risking device
damage due to prolonged biasing.
The He II film creep effect [140] was expected to cause the entire mounting stage,
including the heated DUT region, to be coated in a superfluid film even after the
liquid bath surface dropped below the DUT height. To support this assumption, we
derive the expected saturated film thickness 𝑑 in terms of the height above the liquid
surface level ℎ. This can be done by expressing the chemical potential of the film as
𝜇film = 𝜇0 + 𝑚𝑔ℎ − 𝛼𝑑 −𝑛 , where 𝜇0 is the chemical potential of the bulk liquid, 𝑚𝑔ℎ
is the gravitaional potential at height ℎ, and 𝛼𝑑 −𝑛 is the van der Waals potential. In
equilibrium we must have 𝜇film = 𝜇0 , so that:
𝑑=

𝛼 1/𝑛
𝑚𝑔ℎ

(2.24)

where we use the approximation 𝑛 = 3 in the limit of thin films 𝑑 ≤ 0.5 nm [96]. This
dependence has been confirmed by a number of experiments [110, 111]. We plot
the prediction of Eq. (2.24) in Fig. 2.19 using a typical value of (𝛼/𝑚𝑔) 1/3 = 31.5
nm for metal surfaces [109, 141]. Since the transistor inside the packaged device
sits approximately 7 cm from the bottom of the mounting stage, we estimate a film
thickness of at least 16 nm, with the thinnest film occurring immediately prior to the
liquid level dropping below the mounting stage. Assuming a mean atomic spacing
in the liquid of 0.3 nm, this thickness can be interpreted as a superfluid film of 53

50

Fig. 2.19: Thickness of a He II film versus height above the bulk fluid.
atomic layers. We note that superfluidity has been observed in films as thin as 2.4
atomic layers [142]. We used the sharp rise observed in the attenuator and stage
temperature measurements to indicate the complete evaporation of He II from the
stage. The DUT noise was also measured on warming from 1.6 K to 80 K in the
vacuum space left after all liquid was pumped away.
In the second measurement dewar, the noise temperature was again measured in 4.2
K liquid using the same procedure as in the first measurement dewar but without
the subsequent evaporative cooling step. Instead, the liquid bath was allowed to
evaporate under the heating power of the dipstick, enabling measurements to be
taken in a vapor environment at several temperatures after the liquid level dropped
below the mounting stage. The vapor warmed sufficiently slowly (< 1 K/hour) such
that all measurements were effectively in a steady state vapor environment. The
calibrations used for these measurements were the same as for the 4.2 K liquid since
the coaxial cable loss and temperature were observed to change negligibly up to 45
K stage temperature.

51
Chapter 3

MEASUREMENT RESULTS, DEVICE MODELING, AND
INTERPRETATIONS
In this chapter, we present the full suite of noise temperature and gain measurements
taken at various frequencies, biases, physical temperatures, and in different cryogenic
environments, as well as a derivation of the uncertainty analysis of our measurement
scheme. We detail the small-signal model of the device which was developed,
and interpret the measurement results using it. We compare these results with
the predictions of a phonon radiation model of heat transport in the device. We
then examine how our results compare with prior He II interfacial heat transport
studies. We finish by incorporating the bias dependence of the gate temperature
predicted by the radiation model into both the Pospiesalski noise model and a
phenomenological circuit model, in order to demonstrate the benefits that a reduction
in drain temperature through, for instance, improved device engineering would have
on the overall noise temperature.
3.1

Microwave noise temperature versus bias and frequency

Figure 3.1 shows the noise temperature 𝑇e and gain 𝐺 versus bias power 𝑃DC taken
at a fixed frequency of 𝑓 = 4.55 GHz, the frequency at which the device exhibited
its noise minimum as determined by the IMN, and in several different cryogenic
environments. Here, 𝑃DC was varied by changing 𝑉DS for a fixed 𝑉GS . In Fig. 3.1(a),
𝑉DS ranged from 0.21 V to 0.99 V yielding current densities 𝐼DS from 57.9 mA mm−1
to 241.6 mA mm−1 , with a peak gain of 𝐺 = 27.5 at biases above 𝑃DC = 50 mW
mm−1 . In Fig. 3.1(b), 𝑉DS ranged from 0.15 V to 1.25 V yielding current densities
𝐼DS from 29.2 mA mm−1 to 190.5 mA mm−1 , with a peak gain of 𝐺 = 31.5 dB at
biases above 𝑃DC = 75 mW mm−1 . In Fig. 3.1(c), 𝑉DS ranged from 0.19 V to 1.35 V
yielding current densities 𝐼DS from 21.3 mA mm−1 to 169.0 mA mm−1 , with the same
peak gain of 𝐺 = 31.5 dB at biases above 𝑃DC = 75 mW mm−1 . The data taken with
the DUT immersed in vapor environments was taken in the second measurement
dewar, where each sweep was taken up to a smaller maximum power. The gain
varies by less than 0.5 dB across all temperatures, indicating consistent biasing
across each measurement dewar and cryogenic environment. The dependence of
both 𝐺 and 𝑇e on 𝑃DC and 𝑇phys is qualitatively similar for each fixed 𝑉GS . As bias is

52
increased, 𝑇e initially decreases as 𝐺 increases. The noise temperature then bottoms
out as the gain saturates, before 𝑇e begins to increase with bias. Across all cryogenic
environments, the minimum noise temperature occurs between 1.96 K (𝑉GS = −2.8
V and 𝑇phys = 1.6 K in He II, as seen in Fig. 3.1(b)) and 2.81 K (𝑉GS = −3.2 V
and 𝑇phys = 33.8 K in vacuum, as seen in Fig. 3.1(c)), an indication that cryogenic
environment did not play a significant role in mitigating self-heating at these biases.
A divergence in 𝑇e between different physical temperatures is observed beginning at
𝑃DC = 50 mW mm−1 in Fig. 3.1(a) and at 𝑃DC = 75 mW mm−1 in both Figs. 3.1(b)
and 3.1(c). The reason for this divergence is unclear, although we suspect that it is
related to a changing thermal resistance between the channel, gate, and substrate,
and not due to self-heating mitigation by the liquid cryogens.
Figure 3.2 shows 𝑇e and 𝐺 versus drain-source current 𝐼DS taken at a fixed frequency
𝑓 = 4.55 GHz in several different cryogenic environments, where 𝐼DS was varied by
PS output by the biasing power supply. Converting
changing 𝑉GS for a fixed voltage 𝑉DS
𝐼DS to 𝑃DC was not possible for this data due to complications arising from an
asymmetry in the biasing circuits between the two stages, which led to an uncertainty
in the true 𝑉DS across each transistor. Although this uncertainty was also present
in the data shown in Fig. 3.1, in that case since the gate voltage was fixed, the
offset of 𝑉DS caused by the bias asymmetry did not change between data points.
For measurements taken in the first measurement dewar (He II, He I, and vacuum)
the gate-source voltage was swept from 𝑉GS = −4.1 V to 𝑉GS = −1.0 V, whereas
for measurements taken in the second measurement dewar (vapor) this voltage was
swept from 𝑉GS = −4.0 V to 𝑉GS = −2.0 V. In Fig. 3.2(a) the drain-source voltage
output by the power supply was 𝑉DS = 1.0 V. The gain and noise remain relatively
flat with increasing 𝐼DS , with 𝑇e varying by less than 1.5 K and 𝐺 varying by less than
5 dB until a critical bias of DS = 110 mA mm−1 when the gate becomes sufficiently
open that the gain decreases and the noise temperature increases rapidly. The noise
temperature in the flat region varies by less than 1.8 K for the same bias across
different physical temperatures. In Fig. 3.2(b) the drain-source voltage output by
the power supply was 𝑉DS = 1.4 V. In this case, there was a larger spread in 𝑇e
versus 𝑇phys with a maximum difference of 4.2 K between the noise measured in
vacuum and He II at a bias of 99.9 mA mm−1 , and a bump in noise temperature
is also observed between the flat region and the region of rapidly decreasing gain.
The origin of this bump is presently unclear, and is suspected to be associated with
asymmetric biasing.

53

Fig. 3.1: Noise temperature (left axis) and gain (right axis) versus bias measured at
a fixed frequency of 𝑓 = 4.55 GHz in the following cryogenic environments: 1.6 K
He II (blue triangles), 4.2 K He I (cyan triangles), 8.0 K vapor (green diamonds),
19.7 K vapor (green squares), and 33.8 K vacuum (purple circles). The bias was
varied by changing the drain-source voltage for a fixed gate-source voltage of (a)
𝑉GS = −2.2 V, (b) 𝑉GS = −2.8 V, and (c) 𝑉GS = −3.2 V. For each fixed 𝑉GS the
low-noise bias shifts by less than 10 mW mm−1 and the noise temperature at the
low-noise bias changes by less than 1 K across all physical temperatures.
Figure 3.3(a) shows 𝑇e and 𝐺 versus microwave frequency 𝑓 with the device biased
at its low-noise bias 𝑃DC = 24.5 mW mm−1 and immersed in five different bath
conditions ranging from 1.6 K He II to 35.9 K vacuum. The noise varies by less

54

Fig. 3.2: Noise temperature (left axis) and gain (right axis) versus drain-source
current measured at a fixed frequency of 𝑓 = 4.55 GHz in the same cryogenic
environments as in Fig. 3.1. The bias was varied by changing the gate-source
PS=1.0 V
voltage for a fixed drain-source voltage output by the power supply of (a) 𝑉DS
PS=1.4 V.
and (b) 𝑉DS
than 2 K across all measured frequencies and temperatures at this bias. The noise
increases monotonically with increasing physical temperature regardless of bath
condition. The gain varies by approximately 4 dB over the measured frequency
range, peaking at 𝑓 = 4 GHz. The gain variation with physical temperature is less
than 0.5 dB at fixed 𝑓 and 𝑃DC .
Figure 3.3(b) shows 𝑇e and 𝐺 versus 𝑓 with the device immersed in 1.6 K He
II at three different device biases of 𝑃DC = 24.5 mW mm−1 , 𝑃DC = 79.5 mW
mm−1 and 𝑃DC = 120 mW mm−1 . At biases below 𝑃DC = 79.5 mW mm−1
the noise temperature varies by less than 1.5 K for all frequencies, while above
𝑃DC = 79.5 mW mm−1 the noise increases more rapidly with bias. The gain

55
increases monotonically with increasing bias while retaining the same shape versus
frequency, but it appears to asymptotically plateau at approximately the highest gain
shown here, at a bias of 𝑃DC = 120 mW mm−1 .
3.2

Interpretation using small-signal modeling

To interpret these measurements, a microwave model of the full device including the
IMN, monolithic microwave integrated circuit (MMIC) components, and transistor
small-signal model was made using Cadence AWR Microwave Office. Figure 3.4
shows a micrograph image of the 2-stage MMIC, including the external IMN, which
are all housed inside a gold-plated copper chassis. The model used micrograph
measured values of the IMN, foundry schematic values for the MMIC, and independently measured small-signal model values from nominally identical discrete
OMMIC D007IH 4f200 transistors from Ref. [58]. The small-signal model and
IMN parameters were manually tuned, constrained to change by less than 20% from
the starting values to fit both the gain and noise temperature curves. The gain data
from several datasets were used first to tune the IMN microstrip geometry which
determined the shape of the gain versus frequency dependence. The gain data from
each dataset was then used to tune the following small-signal model parameters: the
gate-source capacitance 𝐶gs , the gate-drain capacitance 𝐶gd , the transconductance
𝑔m , and the drain-source conductance 𝑔ds . Finally the noise data was used to tune
and extract 𝑇g and 𝑇d . An image of the transistor model schematic at the low-noise
bias at 1.6 K is shown for reference in Fig. 3.5. The parameters not listed above,
such as the gate, drain, and source inductances 𝐿 g , 𝐿 d , and 𝐿 s , respectively, were not
altered from their starting values since changing them did not measurably impact
the results of the noise modeling.
Representative frequency-dependent results of the model are plotted in Figs. 3.3(a)
and 3.3(b). The modeled and measured gain are in agreement over the full frequency
range, and the model captures the overall trend in noise temperature as a function
of both temperature and bias.
We used this model to extract the gate temperature 𝑇g under various conditions.
Figure 3.6(a) shows the extracted 𝑇g versus 𝑇phys at the device’s low-noise bias of
𝑃DC = 24.5 mW mm−1 . 𝑇g is elevated above 𝑇phys below 20 K even in the presence
of superfluid, and follows 𝑇phys above 20 K, behavior which is in agreement with
prior reports [28, 60, 61]. Figure 3.6(b) shows 𝑇g versus 𝑃DC at 𝑇phys = 1.6 K
physical temperature. Here, 𝑇g changes by less than 2 K for bias powers below 50 –

56

Fig. 3.3: (a) Noise temperature (left axis) and gain (right axis) versus frequency,
measured at the device’s low-noise bias of 𝑃DC = 24.5 mW mm−1 (𝑉DS = 0.56 V,
𝐼DS = 43.9 mA mm−1 , 𝑉GS = −2.7 V) in the following cryogenic environments: 1.6
K He II (blue triangles), 4.2 K He I (cyan triangles), 8.2 K vapor (green diamonds),
20.1 K vapor (green squares), and 35.9 K vacuum (purple circles). Only the gain
under He II conditions is shown for clarity since the gain varies by less than 0.5 dB
across all temperatures. The small-signal model fits for each dataset is also shown
(solid lines). (b) Noise temperature (left axis) and gain (right axis) versus frequency
measured at biases of 𝑃DC = 24.5 mW mm−1 (magenta circles; 𝑉DS = 0.56 V,
𝐼DS = 43.9 mA mm−1 , 𝑉GS = −2.7 V), 𝑃DC = 79.5 mW mm−1 (dark blue circles;
𝑉DS = 1.0 V, 𝐼DS = 79.5 mA mm−1 , 𝑉GS = −2.7 V), and 𝑃DC = 120 mW mm−1
(red circles; 𝑉DS = 1.2 V, 𝐼DS = 100.0 mA mm−1 , 𝑉GS = −2.7 V) with the DUT
submerged in He II at 1.6 K. To vary the bias, the gate-source voltage was held
constant at 𝑉GS = −2.7 V while the drain-source voltage 𝑉DS was varied. The
small-signal model fits (solid lines) are also shown. Where omitted in both (a) and
(b), the vertical error bars are equal to the height of the symbols.
75 mW mm−1 , after which 𝑇g increases more rapidly.
We also used the model to extract the drain temperature 𝑇d under various conditions.

57

Fig. 3.4: High-resolution micrograph image of the packaged device including the
input matching network and MMIC. The inset shows a zoom of the MMIC fabricated
by OMMIC.
Figure 3.7(a) shows the extracted drain temperature 𝑇d versus 𝑇phys under the same
bias conditions as in Fig. 3.6(a). A linear fit is also plotted for reference. The drain
temperature follows this linear trend for all physical temperatures measured here,
although we emphasize the magnitude of the uncertainty caused by the complexity
of modeling a packaged device without access to a full S-pars characterization at
these temperatures. Figure 3.7(b) shows 𝑇d versus 𝑃DC under the same conditions
as in Fig. 3.6(b). The trend in 𝑇d shows reasonable agreement with measurements
of similar HEMTs in the literature (see Fig. 5 of Ref. [22] and Fig. 5 of Ref. [35])

58
as well as with the predictions of recent modeling of drain noise using real-space
transfer (see Fig. 3.(a) of Ref. [57]); however, we emphasize that those studies use
a fixed 𝑉DS with 𝑉GS used to change 𝐼DS .
3.3

Comparison with a phonon radiation model

We compare these results with an equivalent circuit radiation model of the HEMT
structure developed in [62]. The explicit functional form for the gate temperature
derived from this model is:
 1/4
𝑃DC ℛ𝑐𝑠 ℛ𝑔𝑠
𝑇g (𝑇s , 𝑃DC ) = 𝑇s +
(3.1)
𝜎𝑝 (ℛ𝑐𝑠 + ℛ𝑔𝑐 + ℛ𝑔𝑠 )
where 𝑇s is the substrate temperature, 𝜎𝑝 = 850 W m−2 K−4 is the equivalent
Stefan–Boltzmann constant for phonons in GaAs, and ℛ𝑖 𝑗 = 𝐴𝑖 𝐹𝑖 𝑗 is the space
resistance between nodes 𝑖 and 𝑗 with emitting line length 𝐴𝑖 and view factor 𝐹𝑖 𝑗
which quantifies the fraction of power emitted from surface 𝑖 that intercepts surface
𝑗. The subscripts 𝑔, 𝑐, and 𝑠 represent the gate, channel, and substrate, respectively.
Following Ref. [45], we take 𝐴g = 𝐴c = 70 nm and compute 𝐹𝑔𝑐 = 0.3.
The predictions of Eq. (3.1) are also shown in Figs. 3.6(a) and 3.6(b). In Fig. 3.6(a)
the data and model are in quantitative agreement over the full range of physical
temperatures. The model captures the elevated gate temperature behavior at fixed
bias. In Fig. 3.6(b) the data and model agree at biases below 50 mW mm−1 , but the
data deviates sharply from the model above some critical bias between 25–50 mW

Fig. 3.5: Schematic of the transistor model made using Microwave Office.

59

Fig. 3.6: (a) Extracted gate temperature versus physical temperature at the device’s
low-noise bias of 𝑃DC = 24.5 mW mm−1 . Symbols indicate extracted values and
represent the same conditions as in Fig. 3.3(a), along with extracted values in 4.2
K He I (cyan triangles) and 8.1 K vapor (green diamonds). The radiation model
is also shown (dash-dotted black line). (b) Extracted gate temperature versus bias
at 1.6 K physical temperature (blue triangles). The radiation model is also shown
(dash-dotted black line). In both (a) and (b) the error bars were generated by
determining the range of gate temperatures that accounted for the uncertainty in the
frequency-dependent noise temperature data.
mm−1 . The origin of this discrepancy is presently unclear. A possible explanation
is that other noise sources are being attributed to gate thermal noise leading to
artificially high extracted gate temperatures. This additional noise may signal the
onset of impact ionization [143, 144]; however, the usual indications of this process,
such as increased gate-leakage current and decreased gain, were not observed. The

60

Fig. 3.7: (a) Extracted drain temperature versus physical temperature generated
from the same extraction process as in Fig. 3.6(a). A linear fit is also shown (dashed
black line). (b) Extracted drain temperature versus bias generated from the same
extraction process as in Fig. 3.6(b). In both (a) and (b) the error bars were generated
by determining the range of drain temperatures that accounted for the uncertainty in
the frequency-dependent noise temperature data.

61
gain plateaued at 33 dB at the highest measured bias, as shown in Fig. 3.3(b), and
the gate current remained less than 100 𝜇A at all biases. This discrepancy remains
a topic of investigation.

Fig. 3.8: (a) Noise temperature (left axis, blue line) and physical temperature (right
axis, black line) versus time in an evaporating He II bath sampled at 𝑓ENR = 10
Hz and digitally filtered at 1 Hz, taken at a fixed bias 𝑃DC = 80 mW mm−1
(𝑉DS = 1.0 V, 𝐼DS = 80 mA mm−1 , 𝑉GS = −2.8 V) and frequency 𝑓 = 4.55 GHz.
The sharp kink in the physical temperature at time 𝑡 = 0 minutes, interpreted as
the time at which superfluid is no longer present on the attenuator and device, is
not reflected in the noise temperature. (b) Noise temperature (black line) versus
physical temperature obtained from the transient data shown in (a). Symbols show
independently measured noise temperatures representing the same bath conditions
as in Fig. 3.3(a), and the same bias conditions as in (a). The presence of liquid
cryogens does not affect the noise temperature within the measurement uncertainty.

62
3.4

Noise temperature dependence on cryogenic environment

We obtain further insight into how liquid cryogens impact HEMT noise performance
by examining the DUT noise temperature measured continuously in a changing cryogenic environment. Figure 3.8(a) shows the time series of both 𝑇e and 𝑇phys measured
continuously as the He II was pumped out of the measurement dewar, with 𝑡 = 0
minutes chosen as a reference time at which a rise in 𝑇phys was observed, interpreted
as the departure of superfluid from the attenuator and device. A corresponding
feature in the DUT noise temperature is absent. After 𝑡 = 0 minutes the device thermalized with the surrounding 4 He vapor, and 𝑇e was observed to increase smoothly
with increasing physical temperature. After 20 minutes, the remaining He II liquid
below the stage fully evaporated, and the warming rate increased as the mounting stage and DUT passively warmed to room temperature through the mounting
apparatus.
In Fig. 3.8(b) the warming curve of 𝑇e plotted against 𝑇phys is shown from 1.6 K
to 80 K, taken from the time series in Fig. 3.8(a). Again, the noise temperature
measured in vacuum exhibits no sharp features, instead smoothly varying with
physical temperature. Also plotted are noise temperatures measured separately
under various bath conditions, at the same bias and frequency. The discontinuity
in vacuum data reflects the period of time during which continuous measurement
was paused, and frequency and bias dependent data were acquired. We note that the
liquid, vapor, and vacuum data all lie within the reported error bars of the warming
curve. These observations suggest that the liquid and vapor cryogen environments
provide no self-heating mitigation beyond maintaining a fixed ambient temperature.
3.5

Measurement uncertainty analysis

We estimate the contribution of each quantity in Eq. (2.6) to the overall noise temperature measurement uncertainty Δ𝑇𝑒 , as given by the standard error propagation
formula:
𝜕𝑇𝑒
Δ𝑋
(3.2)
Δ𝑇𝑒𝑋 =
𝜕𝑋
where X denotes the measurement error source. All individual error sources are
assumed to be independent unless otherwise stated, and are added in quadrature to
estimate the overall uncertainty. Numerical estimates listed below assume a noise
temperature of 𝑇𝑒 = 2 K. We assume that each VNA loss measurement has an
uncertainty of ±0.01 dB, which is the magnitude of the variation in the measured
loss versus frequency of the calibration cable immediately after calibration. We

63

Fig. 3.9: Gain (top plot) and noise temperature (bottom plot) versus frequency
measured with the device biased at its low-noise bias of 𝑃DC = 24.5 mW (𝑉DS = 0.56
V, 𝐼DS = 43.9 mA mm−1 , 𝑉GS = −2.7 V). The approximately quadratic shape of the
noise temperature curve is determined by the IMN.
also assume that the variance in the measured losses across different calibration and
measurement dewars is 0.03 dB.
In Fig. 3.9 we illustrate the level of precision of our measurements by plotting the
noise temperature and gain measured under the lowest noise conditions achieved in
this experiment, at the low-noise bias of of 𝑃DC = 24.5 mW mm−1 with the device
submerged in He II at 1.6 K. The error bars were generated from the uncertainty
analysis outlined below.

64
Attenuator
The uncertainty in the attenuator loss Δ 𝐿 2 is found by adding the uncertainty
from the VNA measurements in calibration dewars 1 and 2 in quadrature, so that
Δ 𝐿 2 = 0.05𝑑𝐵.
𝑇𝑒 + 𝑇𝐿 2
𝜕𝑇𝑒
𝜕𝐿 2
𝐿2
Δ𝑇𝑒𝐿 2 = 0.036 K
The uncertainty in the attenuator temperature 𝑇𝐿 2 is determined by the temperature
diode calibration. The calibrated Lake Shore DT-670-SD bonded to the attenuator
chip has a manufacturer-reported temperature uncertainty of Δ𝑇𝐿 2 = ±20 mK.
𝐿2 − 1
𝜕𝑇𝑒
𝜕𝑇𝐿 2
𝐿2
𝑇𝐿

Δ𝑇𝑒 2 = 0.020 K
Coaxial cables
The uncertainties in the coaxial cable losses Δ 𝐿 1 and Δ 𝐿 3 are found by adding
the uncertainties from three separate VNA loss measurements: the measurement of
𝐿 coax = 𝐿 1 𝐿 3 in calibration dewar 1, and the measurements of 𝐿 1 and 𝐿 3 at room
temperature to determine the loss asymmetry between the two cables, yielding
Δ 𝐿 1 = 0.06 dB and Δ 𝐿 3 = 0.07 dB.
𝑇𝑒 + 𝑇𝐿 1 𝐿 2−1 + 𝑇𝐿 2 (𝐿 2 − 1)𝐿 2−1
𝜕𝑇𝑒
𝜕𝐿 1
𝐿1
Δ𝑇𝑒𝐿 1 = 0.088 K

𝑇𝐿 3
𝜕𝑇𝑒
𝜕𝐿 3
𝐺 full 𝐿 1 𝐿 2 𝐿 32
Δ𝑇𝑒𝐿 3 = 0.006 K
The uncertainty in the cable temperatures Δ𝑇𝐿 1 and Δ𝑇𝐿 3 are derived from error
analysis of Eq. (2.17). We estimate Δ𝑇coax = ±20 K.
𝜕𝑇𝑒
𝐿1 − 1
𝜕𝑇𝐿 1
𝐿1 𝐿2
𝑇𝐿

Δ𝑇𝑒 1 = 0.100 K

65
𝐿3 − 1
𝜕𝑇𝑒
𝜕𝑇𝐿 3
𝐺 full 𝐿 1 𝐿 2 𝐿 3
𝑇𝐿

Δ𝑇𝑒 3 = 0.032 K
Gain
The uncertainty in the total gain Δ𝐺 full comes directly from the uncertainty of a
single VNA loss measurement so that Δ𝐺 full = ±0.01 dB.
−1

𝑇coax (𝐿 3 − 1)𝐿 3 + 𝑇BE
𝜕𝑇𝑒
𝜕𝐺 full
𝐿 1 𝐿 2 𝐺 2full
Δ𝑇𝑒𝐺 full = 0.002 K
Backend detector
The uncertainty in the backend detector noise temperature Δ𝑇BE is determined by
the temperature and loss uncertainties in the coaxial cable connecting the cooled
load to the backend. We estimate Δ𝑇BE = ±5 K.
𝜕𝑇𝑒
𝜕𝑇BE
𝐺 full 𝐿 1 𝐿 2
Δ𝑇𝑒BE = 0.016 K
Noise source
The uncertainty in the noise source ENR comes from error analysis of Eq. (2.14).
We estimate Δ 𝐸 = ±0.040 dB.
𝑇0
𝜕𝑇𝑒
𝜕𝐸
(𝑌 − 1)𝐿 1 𝐿 2
Δ𝑇𝑒𝐸 = 0.073 K
The uncertainty in the noise source diode temperature Δ𝑇C comes from the uncertainty in the Type T thermocouple temperature measurement of the noise source
chassis. We estimate Δ𝑇C = 1 K.
𝜕𝑇𝑒
𝜕𝑇C
𝐿1 𝐿2
Δ𝑇𝑒𝐸 = 0.005 K
Y-factor power
The Y-factor measurement uncertainty Δ𝑌 accounts for all uncertainty sources
originating after the transduction of microwave power to DC voltage. We report an

66
−4 for a 4 s integration
effective normalized Y-factor error Δ𝑌
𝑌 of better than 3 × 10
time, which used for all steady-state data presented in this paper.

𝑇0 𝐸
𝜕𝑇𝑒
𝜕𝑌
𝐿 1 𝐿 2 (𝑌 − 1) 2
Δ𝑇𝑒Y = 0.003 K
Cable mismatch
There is error introduced from the difference in noise source impedance between the
on and off state, which causes a changing reflection coefficient between the noise
source and the first component in the measurement chain (in our experiment this is
the input coaxial cable). In cases where the impedance match at the output of the
noise source is poor, this error must be considered, and can be corrected for if the
full S-parameters of the noise source in the on and off state and of the cable are
known. This correction procedure is outlined in Appendix A. In our experiment this
error was found to contribute negligibly to the overall uncertainty.
Overall uncertainty
The uncertainty budget is shown in Table 3.1. The uncertainty analysis shown in
this section was used to generate the error bars in the primary noise temperature
datasets.
Error Source
𝐿2
𝑇2
𝐿1
𝐿3
𝑇1
𝑇3
𝐺 full
𝑇BE
𝑇C
RMS Sum

Value
20.00 dB
1.600 K
3.25 dB
3.44 dB
223 K
223 K
1.98 dB
170 K
15.0 dB
301.0 K
6.8

Estimated Error
0.05 dB
0.020 K
0.06 dB
0.07 dB
20 K
20 K
0.01 dB
5.0 K
0.040 dB
1K
0.002

Contribution to 𝑇𝑒
0.036 K
0.020 K
0.088 K
0.006 K
0.100 K
0.032 K
0.002 K
0.016 K
0.073 K
0.009 K
0.003 K
0.162 K

Table 3.1: Table of parameters used to extract 𝑇𝑒 , along with their associated
uncertainties.

67
3.6

Limits on thermal conductance at the helium-gate interface

We consider the finding that He II is unable to mitigate self-heating in the context
of He II transport. In Section 1.3 we made predictions of the order of magnitude
of the heat fluxes between the HEMT gate surface and He II. The persistence of an
elevated gate temperature irrespective of liquid cryogenic environment suggests that
our estimates of the predicted heat fluxes being below what is required to extract a
significant percentage of the heat produced in the channel were correct. Taking 1
mW as an order of magnitude estimate for the cooling power required to measurably
change the noise temperature and making the same assumption for thermal contact
area 𝐴 = 1 𝜇m × 200 𝜇m as in Section 1.3, we estimate the required heat transfer
¤ 𝐴Δ𝑇) −1 to be ℎ = 250 kW m2 K−1 , which is a factor of 10 higher
coefficient ℎ = 𝑄(
than the highest measured Kapitza resistance between He II and a solid (in this case
Hg) [121].
3.7

Implications for noise performance of cryogenic HEMTs

We now examine the implications of our finding that self-heating in HEMTs cannot
be mitigated for their noise performance. The only method to reduce gate noise
below the observed low-temperature plateau at approximately 20 K is to reduce the
dissipated DC power. However, lower power also reduces the gain, which in turn
leads to an increase in the contribution of both drain noise and any noise source
originating after the gain stage of the HEMT, when referred to the input.
We first explore how 𝑇min from the Pospiezalski model [53] varies with bias while
including the explicit bias dependencies of both 𝑇d and 𝑇g . We recall that 𝑇min in
the limit 𝑓
𝑓𝑇 [22] is:
𝑇g𝑇d
(3.3)
𝑇min = 𝑔0
𝑔m
where we have explicitly introduced the bias independent prefactor 𝑔0 = 4𝜋 𝑓 (𝐶gs +
𝐶gd ) (𝑟𝑖 + 𝑅G + 𝑅S )𝑔ds . We assume a 𝑇phys = 4.2 K, a gate-source capacitance
𝐶gs = 150 fF, a drain-source capacitance 𝐶ds = 28 fF, frequency 𝑓 = 5 GHz,
a parasitic gate resistance 𝑅G = 1 Ω, a parasitic source resistance 𝑅S = 1 Ω, an
intrinsic input resistance 𝑟 i = 1 Ω, an intrinsic drain-source conductance 𝑔ds = 15.4
mS, and a transconductance 𝑔m obtained by taking a finite-difference approximation
of the derivative of 𝐼DS − 𝑉DS data for different 𝑉GS separated by 20 mV. All values
are taken from [58]. We also assume a drain noise temperature that varies linearly
from 𝑇d = 20 K at 𝑃DC = 0 mW mm−1 to 𝑇d = 1000 K at 𝑃DC = 68.5 mW mm−1 ,
an approximation of the bias dependence measured in [58], while taking 𝑇d = 20 K

68

Fig. 3.10: (a) Modeled 𝑇min versus bias, shown for a fixed gate temperature 𝑇g = 20
K (dashed green line) and for a gate temperature with bias dependence determined by
a radiation model (solid blue line). The radiation model predicts a gate temperature
below 20 K for powers below 40 mW mm−1 , which is reflected in 𝑇min . (b) Modeled
𝑇50 noise temperature versus bias, shown for 𝑇d = 500 K (dashed red line), 𝑇d = 200
K (dash-dotted gold line), and 𝑇d = 20 K (solid pink line). Both the minimum 𝑇50
and the power required to achieve this minimum decrease with decreasing 𝑇d .
as the zero-bias limit.
Figure 3.10(a) shows 𝑇min versus dissipated power both with and without the 𝑇g ∝
1/4
𝑃DC
dependence predicted by the radiation model. For the case of fixed gate
temperature we assume 𝑇g = 20 K. The radiation model predicts a lower 𝑇min than
the fixed 𝑇g model up to a power of 𝑃DC = 15 mW mm−1 , above which self-heating
raises the gate temperature above 20 K. The minimum 𝑇min predicted by the radiation
model is lower than that predicted by the fixed model by 0.16 K, and the bias at which
the minimum 𝑇min occurs is also lower by 2.7 mW mm−1 . Below this optimal bias,
both models predict an increase in 𝑇min with decreasing power, indicating where the
gain is insufficient to overcome drain noise.

69
A reduction in drain noise at low biases is evidently beneficial. We demonstrate the
effect such a reduction has on the overall noise temperature by using a phenomenological model that both accounts for self-heating and separates the input and output
noise contributions additively, a feature of noisy amplifiers which is not captured
in the expression of 𝑇min . At low enough frequencies such that 𝑓
𝑓𝑇 , the noise
temperature of a HEMT with a 50 Ω source impedance is:
−2
𝑇g (𝑟 i + 𝑅G + 𝑅S ) + 𝑇d 𝑔ds 𝑔m
𝑇50 =
50 Ω

(3.4)

as derived in Ref. [38] in the limit of open 𝐶gs and 𝐶gd . For illustrative purposes,
we assume a constant 𝑇d , a physical temperature of 4.2 K, and all other parameter
values identical to those in the 𝑇min model.
Figure 3.10(b) shows the modeled 𝑇50 noise temperature versus dissipated power at
different drain temperatures. Drain temperatures of 500 K and 200 K were chosen to
approximate state-of-the-art low-power performance in GaAs devices [58] and InP
devices [49], respectively. A knee is observed in each curve, the location of which
indicates where the gate noise and input-referred drain noise become comparable
in magnitude. As 𝑇d is decreased, both the 𝑇50 value at the knee and the bias at
which the knee is observed decrease. This feature is explained as follows. As 𝑇d
decreases, less gain is required to achieve the same contribution of 𝑇d to the overall
noise, which implies that less power is required to bias the device, ultimately leading
to less self-heating and therefore a lower 𝑇g . In this way we see that reducing 𝑇d
leads to a simultaneous improvement in noise temperature and reduction in optimal
low-noise bias.

70
Chapter 4

SUMMARY AND OUTLOOK
In this work, we evaluated the utility of using liquid cryogens to mitigate the selfheating effect, a limiting noise source in low-noise cryogenic microwave HEMT
amplifiers. The primary object of study was a packaged two-stage cryogenic
HEMT amplifier. This device, whose noise was measured in various cryogenic
environments including liquid superfluid helium-4, did not exhibit improved noise
performance beyond what was expected from conventional solid-state conductive
cooling strategies. The work presented here demonstrates a contribution to further
establishing and understanding the noise limits of cryogenic HEMT amplifiers.
In Chapter 2 we demonstrated the design and construction considerations of our
experiment, and showed all calibration procedures which were required. We discussed the vacuum and cryogenic engineering involved in evaporatively cooling the
liquid bath of a helium dewar. We provided the details of our microwave noise
measurement apparatus, which was based on the cold-attenuator Y-factor method,
demonstrating in particular the ability to digitally sample Y-factors at speeds in excess of 600 kHz, limited by the digital sampling rate of our analog-digital converter.
The room temperature characterization of our DUT was presented, as were all calibration techniques required to perform the experiment, both at room and cryogenic
temperatures. We also outlined the measurement procedure which was followed to
characterize the noise performance of our device.
In Chapter 3 we examined the dependence of the device’s noise temperature on
bias, frequency, physical temperature, and cryogenic environment. We developed a
small-signal model of the device which was used to extract its gate temperature and
drain temperature. We compared the extracted gate temperatures to the predictions
of an equivalent circuit phonon radiation model, and found good agreement over a
wide range of cryogenic temperatures and biases. The gate temperature was found
to exhibit the same plateau at temperatures below 20 K as has been observed in
the overall noise temperature measured both in this work and in other published
studies. We also measured the LNA noise temperature as the He II bath in which
it was immersed evaporated from the HEMT surface, and found no change in noise
temperature within the measurement uncertainty. These observations led us to the

71
conclusion that self-heating of cryogenic HEMTs cannot be mitigated through direct
liquid cooling strategies. This observation was found to be in agreement with our
estimates of the He II cooling power accessible to the HEMT gate, based on our
approximation of the HEMT surface area and values of the solid-helium boundary
thermal conductance available in the literature. We explored the consequences of
our null result on the limits of HEMT noise by incorporating the effect of selfheating into noise modelling of FET amplifiers. We found that both the Pospiesalski
noise model’s prediction of the device’s minimum noise temperature, and a phenomenological model of noisy linear amplifiers both predict that a reduction in drain
temperature enables simultaneous improvements in noise performance by reducing
the gain required to overcome the drain noise, and hence reducing the dissipated
power, thereby diminishing the magnitude of self-heating.
4.1

Future work

We outline here the potential direction for future research that extends naturally from
the work presented in this thesis, both by suggesting extensions and improvements to
the characterization measurements performed here, and by discussing other possible
methods of improving HEMT noise performance given the limits imposed by selfheating. Firstly, as discussed in Chapter 2, the cryogenic vacuum engineering can be
improved by including an inner vacuum jacket in the pumping line, so that the room
temperature vacuum seals do not experience convective cooling from the pumped
helium. An alternative is to implement a different design which uses subcooled He
II instead of saturated He II. As discussed briefly in Chapter 1, subcooled helium
presents several engineering advantages, and at high pressures has been shown to
increase the film-boiling heat transfer coefficient ℎfb in certain surface geometries
(see Table 7.5 of Ref. [63]). A yet more sophisticated apparatus might involve the
use of forced-flow convective cooling which has also be shown to increase ℎfb [69,
101]. Finally, different surface passivisation treatments designed to increase the
effective thermal interface area should be explored.
The cold-attenuator Y-factor scheme itself can be improved by moving the noise
diode directly to the input of the cold attenuator, eliminating noise contributions
from the lossy input coaxial cabling. A coaxially coupled, internally calibrated,
packaged noise source capable of continuously variable noise generation from a
chip-mounted thermal source which can be operated at cryogenic chassis temperatures is currently in development [145]. A similar device which employs additional
noise amplification via a travelling wave parametric amplifier has been successfully

72
demonstrated in a dilution refrigeration environment [146]. A larger scale improvement would be to design a semiconductor probe-station, commonly employed for
full DC and microwave discrete transistor characterization [147], capable of containing a He II bath inside its vacuum assembly. This would enable measurements
of discrete HEMTs immersed in He II, enabling a more systematic evaluation of
the possible effects of, for example, the use of subcooled He II, increased thermal
interface area, etc.
The experimental technique used here to measure LNA noise temperatures in liquid
cryogenic environements can also be applied to other LNA technologies. A particular example would be studying the effect of liquid cryogenic cooling on SiGe
heterojunction bipolar transistor (HBT) amplifiers, which possess an inherently different structure and whose "self-heating" is expected to occur over a larger physical
region than in HEMTs [148].
As already alluded to, our findings bolster the conventional wisdom that low-noise
HEMT amplifier design should involve minimizing the power dissipation required
to achieve sufficient gain to overcome drain noise. A more rigorous understanding
of the physical origins of drain noise should continue [57] so that future device
designs can achieve the lowest possible drain temperatures and take full advantage
of the conclusions of Chapter 3, that a reduction in drain noise leads to a cascading
improvement in noise temperature. Future efforts must incorporate developments
from all facets of low-noise amplifier research, including the theory of noise in
HEMTs, device fabrication techniques, and precision measurement schemes.

73

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85
Appendix A

CORRECTION ALGORITHM FOR A CHANGING NOISE
SOURCE IMPEDANCE
We present an algorithm to correct for the error introduced into the measured
noise power by a changing impedance mismatch at the input of a 2-port network
consisting of the following components: (1) a microwave noise source at the input
H and 𝑆 C at its output plane in the hot and cold states
with reflection coefficients 𝑆22
22
BE at its
respectively; (2) a power detector at the output with reflection coefficient 𝑆11
input plane; (3) any series of elements connecting the noise source to the detector
which we can treat as a single element with S-matrix Snet . This scenario is shown
schematically in Fig. A.1(a). The technique relies on mathematically cascading
casc of the entire network which
these quantities to find an effective cascaded gain 𝑆21
is used to modify Eq. (2.6).
(H,C) BE
First each quantity 𝑆22
, 𝑆11 , and 𝑆𝑖net
𝑗 is measured. Next a set of effective S-pars for
H,C
H,C
H,C
BE = 𝑆 BE = 1,
the noise source and detector are defined as 𝑆11
= 0, 𝑆21
= 𝑆12
= 𝑆21
12
BE
and 𝑆22 = 0, so that each component has a fully defined S-matrix. This is equivalent

Fig. A.1: Equivalent noise measurement scheme representations involving a noise
source, an intermediate 2-port network, and a detector where (a) reflections are
attributed to the noise source output plane and detector input plane, (b) the noise
source and detector are treated as ideal and virtual components carry the associated
S-pars, and (c) the noise source and detector are treated as ideal and all S-pars are
cascaded into a single element.

86
to treating the noise source and detector as ideal, perfectly matched elements and
introducing virtual components at the input and output of the network with Smatrices S(H,C) and SBE , respectively, using the S-pars defined above. This network
is shown schematically in Fig. A.1(b), and is equivalent to the network shown in
Fig. A.1(a).
In order to cascade each component, we transform from the scattering matrix (Smatrix) representation to the transfer matrix (T-matrix) representation. Since there
are several T-matrix formalisms (see, for example, the ABCD matrix defined in
Chapter 4 of Ref. [131]), we adopt the definition from Appendix A of Ref. [149]
which does not rely on knowing the reference impedance 𝑍0 . In terms of 𝑆𝑖 𝑗 , the
T-matrix elements 𝑇𝑖 𝑗 are given by:
𝑆11
𝑆12 𝑆21 − 𝑆22 𝑆11
, 𝑇12 =
𝑆21
𝑆21
𝑆22
𝑇21 = −
𝑇22 =
𝑆21
𝑆21
𝑇11 =

(A.1)

The T-matrix TM of 𝑚 cascaded 2-ports is then given by matrix multiplication:
(A.2)

TM = T1 T2 . . . Tm

We use Eq. (A.1) to transform each of the three components in our network, cascade
them using Eq. (A.2), and then transform back to the S-matrix representation using:
𝑇11𝑇22 − 𝑇12𝑇21
𝑇12
, 𝑆12 =
𝑇22
𝑇22
𝑇21
𝑆21 =
, 𝑆22 = −
𝑇22
𝑇22

𝑆11 =

(A.3)

The final cascaded network is shown schematically in Fig. A.1(c) and is equivalent
to the networks shown in Figs. A.1(a) and A.1(b).
This procedure allows us to define effective gains of the cascaded network 𝐺 H
casc =
|𝑆21 | with the noise source on and 𝐺 casc = |𝑆21 | with the noise source off. We use
these to modify Eqs. (2.4) and (2.5) as follows:
𝐺H
𝐺H
(𝐿 1 − 1)
𝑃H
casc
= 𝑇H
+ 𝑇𝐿 1 casc
𝐵𝑘 B
𝐿1 𝐿2 𝐿3
𝐿1 𝐿2 𝐿3
𝐺H
𝐺H
(𝐿 3 − 1)
casc (𝐿 2 − 1)
+ 𝑇𝐿 2
+ 𝑇𝑒 casc + 𝑇𝐿 3
+ 𝑇BE
𝐿2 𝐿3
𝐿3
𝐿3

(A.4)

87
𝐺C
𝐺 C (𝐿 1 − 1)
𝑃C
= 𝑇C casc + 𝑇𝐿 1 casc
𝐵𝑘 B
𝐿1 𝐿2 𝐿3
𝐿1 𝐿2 𝐿3
𝐺 C (𝐿 2 − 1)
𝐺C
(𝐿 3 − 1)
+ 𝑇𝐿 2 casc
+ 𝑇𝑒 casc + 𝑇𝐿 3
+ 𝑇BE
𝐿2 𝐿3
𝐿3
𝐿3

(A.5)

We can again solve for 𝑇e by plugging Eqs. (A.4) and (A.5) into Eq. (2.1). and arrive
at:

𝑌 −1
𝑇0 𝐸
𝐿3 − 1
𝑇e =
− 𝑇C − 𝑇coax (𝐿 1 − 1) − 𝑇𝐿 2 (𝐿 2 − 1)𝐿 1 − H 𝑇3
+ 𝑇BE
𝐿1 𝐿2 𝑌 0 − 1
𝐿3
𝐺 casc
(A.6)
−1
where we have defined 𝑌 = 𝑌 𝐺 casc (𝐺 casc ) .
Fig. A.2 shows the extracted noise temperature 𝑇e from measurements of an identically designed device to the primary LNA studied in this work, taken at room
temperature using the noise source and backend detector described in Section 2.2.
The extraction using Eqs. (2.6) and (A.6) are both shown, along with a separate
measurement taken using a commercial NFA whose noise source uses attenuation
pads to dampen the effect of the impedance change between hot and cold states (a
technique which is not applicable when unattenuated noise power is required). The
DUT has a relatively high reflection coefficient |𝑆11 | ≈ −2 dB over its bandwidth,
and was connected directly to the output of each noise source. Without correction,
oscillations of approximately 20 K magnitude with a roughly 1 GHz period can be
seen seen. The NFA measurement also exhibits noise temperature oscillations with
frequency, but the attenuator pads suppress their magnitude to less than roughly 5
K. With the correction algorithm, the agreement between the NFA measurement
and the extracted 𝑇e improves significantly.

88

Fig. A.2: Noise temperature of packaged HEMT amplifier versus frequency measured at room temperature with a commerical NFA (blue circles) and with the noise
measurement setup described in Section 2.2 using the uncorrected Eq. (2.6) (red
circles) and the corrected Eq. (A.6) (green circles). The agreement between the
two measurements improves upon correction. All measurements were performed at
the same bias. Error bars reflect the overall measurement uncertainty, including the
impedance mismatch in the uncorrected case.