Radio Frequency Studies of Surface Resis

Radio Frequency Studies of Surface Resistance and Critical Magnetic Field of Type I and Type II Superconductors - CaltechTHESIS
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Radio Frequency Studies of Surface Resistance and Critical Magnetic Field of Type I and Type II Superconductors
Citation
Yogi, Tadashi
(1977)
Radio Frequency Studies of Surface Resistance and Critical Magnetic Field of Type I and Type II Superconductors.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/p14g-5751.
Abstract
The surface resistance and the critical magnetic field of lead electroplated on copper were studied at 205 MHz in a half-wave coaxial resonator. The observed surface resistance at a low field level below 4.2°K could be well described by the BCS surface resistance with the addition of a temperature independent residual resistance. The available experimental data suggest that the major fraction of the residual resistance in the present experiment was due to the presence of an oxide layer on the surface. At higher magnetic field levels the surface resistance was found to be enhanced due to surface imperfections.
The attainable rf critical magnetic field between 2.2°K and T_c of lead was found to be limited not by the thermodynamic critical field but rather by the superheating field predicted by the one-dimensional Ginzburg-Landau theory. The observed rf critical field was very close
to the expected superheating field, particularly in the higher reduced temperature range, but showed somewhat stronger temperature dependence than the expected superheating field in the lower reduced temperature
range.
The rf critical magnetic field was also studied at 90 MHz for pure tin and indium, and for a series of SnIn and InBi alloys spanning both type I and type II superconductivity. The samples were spherical with typical diameters of 1-2 mm and a helical resonator was used to generate the rf magnetic field in the measurement. The results of pure samples of tin and indium showed that a vortex-like nucleation of the normal phase was responsible for the superconducting-to-normal phase transition in the rf field at temperatures up to about 0.98-0.99 T_c' where the ideal superheating limit was being reached. The results of the alloy samples showed that the attainable rf critical fields near T_c were well described by the superheating field predicted by the one-dimensional GL theory in both the type I and type II regimes. The measurement was also made at 300 MHz resulting in no significant change in the rf critical field. Thus it was inferred that the nucleation time of the normal phase, once the critical field was reached, was small compared with the rf period in this frequency range.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
(Applied Physics) ; Radio frequency, surface resistance, magnetic field
Degree Grantor:
California Institute of Technology
Division:
Physics, Mathematics and Astronomy
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Mercereau, James E.
Thesis Committee:
Unknown, Unknown
Defense Date:
2 August 1976
Record Number:
CaltechTHESIS:07312014-142921170
Persistent URL:
DOI:
10.7907/p14g-5751
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ID Code:
8624
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CaltechTHESIS
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31 Jul 2014 22:08
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RADIO FREQUENCY STUDIES OF
SURFACE RESISTANCE AND CRITICAL MAGNETIC FIELD
OF TYPE I AND TYPE II SUPERCONDUCTORS
Thesis by
Tadashi Yogi

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

California Institute of Technology
Pasadena, California
1977

(Submitted August 2, 1976)

-ii-

-iii-

To Aiko and Our Parents

-ivACKNOWLEDGMENT
I would like to express my sincere appreciation to the following
persons who have made this work possible directly and/or indirectly.
Professor J. E. Mercereau for his patient advice and interest
throughout the course of this research.
Drs. G. J. Dick and K. W. Shepard for their guidance and suggestions of the problem areas that were essential.
Dr. H. Notarys who was always there, ready to help in the laboratory and who even took the awesome task of correcting my uncorrectable
English.

I cannot thank him enough.

All the graduate student colleagues in the laboratory who created
the 11 10W temperature 11 atmosphere; particularly, J. R. Delayen and H. C.
Yen whose cooperation in various phases of this study is greatly appreciated.
Mr. S. Santantonio for his willing help in the construction of
the necessary equipment.
Mr. E. P. Boud for his technical assistance in the laboratory.
Ms. G. Kusudo for her general assistance in the office.
I am also thankful to Prof. L. B. Slichter (U.C.L.A.) for his
patience and understanding while I was writing this dissertation.
I am grateful to the California Institute of Technology for
financial assistance throughout my graduate study.
The fine typing of this dissertation is due to Mrs. R. Stratton.
Last, but not least, I would like to thank my wife for her
patience and encouragement throughout the course of my graduate study.

-vABSTRACT
The surface resistance and the critical magnetic field of lead
electroplated on copper were studied at 205 MHz in a half-wave coaxial
resonator.

The observed surface resistance at a low field level below

4.2°K could be well described by the BCS surface resistance with the
addition of a temperature independent residual resistance.

The avail-

able experimental data suggest that the major fraction of the residual
resistance in the present experiment was due to the presence of an
oxide layer on the surface.

At higher magnetic field levels the sur-

face resistance was found to be enhanced due to surface imperfections.
The attainable rf critical magnetic field between 2.2°K and Tc of
lead was found to be limited not by the thermodynamic critical field
but rather by the sup~rheating field predicted by the one-dimensional
Ginzburg-Landau theory.

The observed rf critical field was very close

to the expected superheating field, particularly in the higher reduced
temperature range, but showed somewhat stronger temperature dependence
than the expected superheating field in the lower reduced temperature
range.
The rf critical magnetic field was also studied at 90 MHz for
pure tin and indium, and for a series of Sn!n and InBi alloys spanning
both type I and type II superconductivity.

The samples were spherical

with typical diameters of 1-2 mm and a helical resonator was used to
generate the rf magnetic field in the measurement.

The results of pure

samples of tin and indium showed that a vortex-like nucleation of the
normal phase was responsible for the superconducting-to-normal phase

-vi-

transition in the rf field at temperatures up to about 0.98-0.99 Tc'
where the ideal superheating limit was being reached.

The results

of the alloy samples showed that the attainable rf critical fields
near Tc were well described by the superheating field predicted by
the one-dimensional GL theory in both the type I and type II regimes.
The measurement was also made at 300 MHz resulting in no significant
change in the rf critical field.

Thus it was inferred that the nuc-

leation time of the normal phase, once the critical field was reached,
was small compared with the rf period in this frequency range.

-viiTABLE OF CONTENTS
I.

INTRODUCTION

II.

A REVIEW OF THE PAST EXPERIMENTAL RESULTS AND THE
RELEVANT THEORIES

2.1

The Superconducting Surface Resistance

2.1.1

Surface Resistance of Normal Metals at Low
Temperature
2.1.2 Surface Resistance of Superconductors
2.1.3 Temperature Dependence of Superconducting
Surface Resistance
2.1.4 Residual Surface Resistance

Critical Magnetic Field of Superconductors at Radio
Frequencies

2.2.1
2.2.2
2.2.3
2.2.4

12
16
22
23

2.2

III.

General Remarks on the Superconducting State
Ginzburg-Landau Theory
One-Dimensional GL Superheating Theory
Previous RF Critical Field Measurements

2.3 Summary

SURFACE RESISTANCE AND RF CRITICAL MAGNETIC FIELD OF
LEAD AT 205 MHz

3.1

Introduction

3.2 Experimental Method

3.2.1 Half-Wave Coaxial Resonator at 205 MHz
3.2.2 Preparation of the Surface
3.2.3 Cryogenic Apparatus
3.2.4 RF Instrumentation
3.2.5 General Theory of Measurement
3.2.6 Measurement, Calibration and Error Estimation

35
36
38

41
45

-viii3.3

Experimental Results and Discussions

Surface Resistance of Superconducting Lead
at Low Field Level
3.3.2 Surface Resistance at High RF Magnetic Field
3.3.3 RF Critical Magnetic Field of Lead at 205 MHz

3.3.1

IV.

56
60

3. 4 Conclusions

RF CRITICAL FIELD OF SN-IN AND IN-BI ALLOYS AT 90
AND 300 MHz

4.1

Introduction and Description of Overall Scheme

4.2

Experimental Method

4.2. 1 Sample Preparation
4.2.2 Determination of the GL Parameter by Resistivity
4.2.3 DC Critical Fields of Tin, Indium and Their
Alloys
4.2.4 RF Measurement

Experimental Results and Discussion

102

RF Critical Field of Pure Tin and Indium
RF Critical Fields of Snin and InBi Alloys at
90 MHz
4.3.3 Frequency Dependence of RF Critical Field

102
115

4.3

4.3.1
4.3.2

4.4 Conclusions

126
128

Appendix A.

THE RESIDUAL LOSS DUE TO TRAPP~D FLUX

Appendix B.

SURFACE RESISTANCE OF LEAD AT HIGH RF MAGNETIC FIELD 138
LEVEL

BIBLIOGRAPHY

149

REFERENCES

150

130

-1-

Chapter I
INTRODUCTION
The superconducting state is a macroscopic quantum state characterized in a superconductor by the absence of de electrical resistance
(the perfect conductivity) and the expulsion of magnetic flux (the
Meissner effect).

Since its discovery in 1911 by Onnes (02), the des-

cription of superconductivity remained largely phenomenological until
a successful microscopic theory was formulated by Bardeen, Cooper and
Shrieffer (henceforth 8CS) in 1957 (81).

Coupled with advances in

general experimental technique and materials processing, the last two
decades have seen a rapid growth in both the basic understanding of the
phenomenon and the applications ranging from small scale devices such
as Josephson junctions to large scale generation of intense magnetic
fields.
The field of high power rf superconductivity was initiated in
the early 1960s with the proposal to use superconducting rf resonators
for the acceleration of charged particles (83,S3).

Since then exten-

sive investigations have been carried out in various laboratories in
order to generate intense rf fields at frequencies ranging from 100
MHz up to 10 GHz.

(Throughout this dissertation the radio frequency

is loosely defined as this frequency range.)
As a result of these efforts, notably at Stanford and Karls1·uhe
(Germany), it has become possible to attain rf magnetic fields up to
1600 gauss (S4) and rf electric fields up to 70 MV/m (T2) in microwave
resonant cavities.

Also surface resistances which are 3 to 5 orders

-2-

of magnitude smaller than that of copper at 4°K are commonly achieved.
This means, for example, that surface rf magnetic fields of 1,000
gauss can be sustained by a superconductor with a power dissipation
per unit area of 0.03 watts/cm2 as compared to 30 watts/cm2 for a
copper at the same temperature

<~ 1 GHz).

The materials that have been considered and investigated in
detail so far are two elemental superconductors, niobium and lead.
They have been chosen because of their high transition temperatures
(9.46°K for Nb and 7.l8°K for Pb) and high critical magnetic fields
(1944 gauss for Nb and 803 gauss for Pb at 0°K).

More recently, studies

have been started of high field alloys such as Nb 3Sn {PlO).

The mate-

rial parameters that play an important role in the high power rf
application are:
Transition temperature (Tc)
Surface impedance (Z = R + iX)
RF critical magnetic field (H~f)
Work function (¢, field emission property)
Thermal conductivity.
This dissertation deals with two of the above, the surface resistance
and the rf critical magnetic field, with considerably more emphasis on
the latter, in the frequency range between 90 and 300 MHz.

Most of

this work, particularly Chapter 3, is a direct product of the supercon-,
ducting heavy ion accelerator project which began in 1969 at Caltech.
The content of each chapter is as follows:

-3-

Chapter 2 reviews and summarizes the previous experimental results and the relevant theories of surface resistance and rf critical
magnetic field.
Chapter 3 deals with the measurement of the surface resistance
and the rf critical magnettc field of superconducting lead electroplated on copper at 205 MHz.
Chapter 4 reports the measurement of the rf critical fields of
small samples of Sn, In, Snin and InBi alloys.
The system of units used in this dissertation is the rationalized
MKS units except where specified.

-4-

Chapter II
A REVIEW OF THE PAST EXPERIMENTAL RESULTS AND THE RELEVANT THEORIES
In the rf application of superconductivity, two of the most important parameters are the surface resistance, since it determines the
power dissipation, and the critical magnetic field, since it limits
the attainable electromagnetic field level.

Because of their practical

importance they have been studied quite extensively in terms of both
experiment and theory.

In the following sections the results of pre-

vious experiments and the pertinent theories are reviewed separately for
the surface resistance and the critical magnetic field.
2.1

The Superconducting Surface Resistance
A brief historical survey is presented starting from the surface

resistance of an ordinary metal, and leading up to the temperature dependence of the superconducting surface resistance based on the microscopic theory of superconductivity.

The residual (non-superconducting)

processes which, at low temperatures, give rise to an effective surface
resistance are also discussed.
2.1. 1 Surface Resistance of Normal Metals at Low Temperature
The response of a metal to an applied electromagnetic field is
described by a surface impedance Z defined by

z = R + i X = EII /H II

Isurface

(2.1)

where Ell and HI! are respectively the component of electric and magnetic
field parallel to the surface of the conductor.

Then the time-averaged

-5-

dissipation per unit area of the conductor is given by
pd =

2 R HII

(2.2)

In a normal metal at room temperature, the Ohm's law

= crE

(2.3)

holds and the classical surface resistance at radio frequency is given
by

(2.4)

where f is the frequency, lJ the permeabi 1i ty of the metal, a its
electrical conductivity.

We also have the expression for classical

skin depth,

( 1/TiflJO)

l/2

(2.5)

When sufficiently pure metals are cooled to low temperatures,
the number of electron-phonon collisions decreases and the mean free
path of electrons, R. , increases.

As R. becomes comparable and larger

than the rf skin depth, the point relationship between the current and
the electric field no longer holds and a non-local fonn of Eq. (2.3) must
be used to take into account the field variation over the mean free path.
This is called the anomalous skin effect and its theory was developed
by Reuter and Sondheimer (R2).
R. 7

oo,

In the extreme anomalous limit. i.e.,

they obtained the surface resistance for a diffuse reflection of

electrons at the surface,

-6-

2 2

R = (13 Tif J.1 ) l/3 (.&) l/3
00

(2.6)

In this limit the surface resistance is independent of mean free path
since the ratio ~Ia is a constant for a given metal.

Measurements made

by Chambers (Cl) at 1.2 GHz showed that common metals with reasonable
purity are in fact in the extreme anomalous limit at temperatures rang-

2. 1.2 Surface Resistance of Superconductors
Although the electrical resistance of a superconductor against
de transport current vanishes below Tc' the surface resistance in high
frequency fields stays finite at a finite temperature.

The first theory

of superconducting surface resistance was given by H. London (L2) in
1934 based on the two-fluid model of a superconductor.

In this picture,

the conduction electrons are assumed to be composed of two components,
the superfluid and the normal component.

In de fields the superfluid

electrons carry all the current, shielding the applied electric field
from the normal electrons, so that the electrical resistance vanishes.
However, in high frequency fields the shielding of the electric field
by supercurrent is incomplete due to the inertia of electrons, and as a
result the normal electrons 11 feel 11 the electric field leading to dissipation.

Since this dissipation is proportional to the density of normal

electrons which is unity at Tc and smoo~hly goes to zero at 0°K, the
surface resistance also varies. continuously from normal state value at
Tc to zero at 0°K.

London's theory agreed qualitatively with the ex-

perimental results available at that time (Ll).

-7-

With the realization that the anomalous skin effect was important
at low temperatures, Pippard applied the idea to normal electrons and
gave an approximate solution to the London equations for surface resistance (P5).

In order to obtain better agreement with observed

temperature dependence he later made a modification and gave what is
now called the semi-empirical Pippard function f(t}, which is the temperature dependent part of the surface resistance, as (P4)
f(t)

where t = T/T

= t4(1-t2)
(l-t 4 )

for

is the reduced temperature.

(2. 7)

t < 0.9

In later years, more precise

theory of anomalous skin effect was used by Maxwell et al. (Ml} to
treat the normal component of two-fluid model.

Nevertheless, the over-

all situation still remained phenomenological until a microscopic theory
of superconductivity was established in 1957.
2. 1.3 Temperature Dependence of Superconducting Surface Resistance
In 1957 BCS published their theory of superconductivity in which
a wide variety of equilibrium superconducting properties was explained.
Based on this theory, the surface impedance of superconductors subjected to a weak alternating magnetic field was formulated independently by Mattis and Bardeen (MB) and Abrikosov et al. (A2).

In these

formulations, the surface impedance was evaluated as a function of
temperature, frequency and a set of material parameters.

The general

expression, however, is quite complex and reduces to explicit forms
only in very restricted limits of these parameters.

-8-

The first detailed comparison between experimental and theoretical surface impedance at microwave frequencies was made by
Turneaure (T3).

Based on the formulation by Mattis and Bardeen, he

performed a numerical computation of surface impedance.

The material

parameters needed for the computation were
Tc:

Transition temperature

AL:

London penetration depth

vF:

Fermi velocity

Mean free path of electrons in normal state

The values of these parameters were either measured or inferred from
the surface resistance measurement in normal state.

The results were

compared with the surface impedance measurement made on lead and tin
at 2.85 and 11.2 GHz.

He obtained very good agreement for temperature

dependence of the surface resistance.

Also, the superconducting gap

parameter at 0°K, ~(0), which was deduced from the experimental data,
agreed well with the values derived from other measurements.

The

absolute value of the surface resistance, however, did not agree with
the theoretical value by a factor of up to ~2.
Later, Halbritter (H5) reexamined this problem using more accurate values of the material parameters which had become available in

recent years.

Using these revised material parameters, he showed that

the agreement between theory and experiment, particularly for the
absolute magnitude of the surface resistance, was improved for both
niobium and lead to within experimental error.

It was also established

that the following simplified form describes the temperature dependence

-9-

of BCS surface resistance for frequencies much smaller than the gap
frequency (= ~/h, 2.6 x 10 11 Hz for Pb at 0°K)

BCS

where

~(T)

(T) = A' e -~(T)/kT

for T ~ Tc /2

(2.8)

is the temperature dependent gap parameter, k the Boltz-

mann constant.

The factor A is independent of temperature but depends

on the frequency and the material parameters in a non-trivial way so
that a numerical computation is necessary to obtain an explicit value.
This form was later shown (Hl) to be proportional to the number of
thennally excited quasi-particles which approximately corresponds to
the nonnal fluid in London's two-fluid model~
In Chapter 3, this form will be used to compare the experimental
values of surface resistance of lead at 205 MHz with the result of
numerical computation for lead given in the literature.
2. 1.4 Residual Surface Resistance
Since the superconducting surface resistance vanishes at 0°K,
the quality factor, Q , of a superconducting resonator may be expected
to increase indefinitely.

In practice, however, this is not the case,

and Q tends to a finite value as the temperature is reduced.

This is

due to other loss mechanisms only some of which are understood.

This

section describes briefly the residual loss mechanisms that have been
studied up to the present.
Dielectric loss:

This is due to the presence of an·oxide layer on a

metal surface and the dissipation is proportional to a tan o where

is th_e fraction of the electric field in the dielectric and

tan o

-10is its 1oss tangent.
Loss due to trapped flux:

When a superconductor is cooled through its

transition temperature in the presence of an applied magnetic field,
it tends to trap magnetic flux.

Since the cores of the trapped flux

are normal conducting, they will dissipate energy when rf current is
applied to the surface.

This effect has been studied in the past for

its temperature dependence, frequency dependence, and dependence on
applied magnetic fields (H4,P7,P9,Vl).
Loss due to acoustic phonon coupling:

When an electromagnetic wave is

incident on the surface of a superconductor, there is a small but nonnegligible tangential component of electric field within the penetration
depth of the surface.

This electric field acts on the lattice ions thus

giving rise to the coupling between the electromagnetic field and the
acoustic wave.

This coupling is another loss mechanism which puts a

lower limit on the surface resistance of a superconductor.

Residual

losses in some type of cavities were explained by this mechanism (H6).
Passow (P8) compared the residual resistance attained experimentally
with his theory based on this mechanism and found a good correlation
at frequencies between 100 MHz and 10 GHz.
In the present studY of low field surface resistance of a leadplated copper resonator,

the residual surface resistance was observed

to be present. although every effort was made to avoid surface contamination and oxidations.

Estimates for residual resistances due to the

above mentioned mechanisms will be given in the discussion of temperature
dependence of low field surface resistance (Chapter 3).

-11-

2.2 Critical Magnetic Field of Superconductors at Radio Frequencies
In the foregoing sections, we have discussed the superconducting surface resistance which is extremely small compared with the
normal state surface resistance.

Since the resistance vanishes in the

limit of zero frequency we see that the smallness of the superconducting surface resistance at radio frequency is a consequence of the
perfect de conductivity, which is one aspect of the superconducting
phenomenon.

Another aspect of superconductivity is its ability to

exclude an applied magnetic field from a superconducting body up to a
certain field* called the thermodynamic critical magnetic field, He.
This is called the Meissner effect and every superconductor has a
well-defined thermodynamic critical field when measured in a static
magnetic field.
However, when the magnetic field is changing as a function of
time as in the present case at radio frequency, i.e.,
H(t) = Hrfsin(wt)
it is not immediately obvious what will happen when the amplitude of
the rf magnetic field becomes comparable in magnitude to the thermodynamic critical field He.

To put it differently, we ask what is the

critical magnetic field at radio frequencies, H~f, at which the transition to nonmal state takes place.

Our interest in this quantity is

*Strictly speaking, this statement is valid for type I superconductors.
For type II superconductors He has to be replaced by the lower critical field Hcl· These two types are discussed in the following sections.

-12-

both practical and basic.

That is, in the high power application of

superconductivity H~f will put the ultimate limit on the magnetic
field level that can be achieved.

Also, it is physically an interest-

ing phenomenon and the study of rf critical fields may provide new
information on the details of superconducting-to-normal phase transition.
With these motivations, the rf critical field was studied for
two different systems:
(1) Lead electroplated on copper substrate which formed the
walls of a resonant cavity (Chapter 3).
(2) Small samples of tin, indium and their alloys, where the
necessary material parameters were well defined and measurable (Chapter 4).
In the following sections we review the basic theories which are i-mportant in explaining the present experimental results.

Also previous

rf measurements for different superconductors are reviewed and summarized toward the end of this chapter for later comparison.
2.2.1

General Remarks on Superconducting State
In 1933 Meissner and Ochsenfeld (M4) found that a magnetic field

was expelled from an originally normal sample as it was cooled through
its transition temperature.

This property of perfect diamagnetism,

now known as the Meissner effect, is another basic property of superconductor independent of perfect conductivity.

A class of superconduc-

tors called type I (to be discussed soon) expels an applied magnetic
field up to a critical field Hc which is thermodynamically related to
the difference in free energy density g between normal and superconducting states in zero field,

-13-

which is the so-called condensation energy.

Here the subscripts no and

so refer to the normal and superconducting states in zero applied
field.

Empirically, it has been found that Hc(T) is quite well ap-

proximated by a parabolic temperature dependence,
(2.10)

where H0 is the critical field at 0°K and Tc is the transition temperature.

The above form agrees with actual critical fields within 3%

for most superconductors (L3, p.95).
When a superconductor excludes an applied magnetic field Ha
from its body, its free energy density is raised by ~ 0 H~/2, i.e.,
(2.11)

where it is assumed that the metal in normal state is non-magnetic.
Once the free energy difference is given, the entropy difference between normal and superconducting state may be determined as

(2. 12)

and the heat capacity difference as
(2.13)

In the absence of an external magnetic field the superconducting transition at Tc is of the second order, since the entropy is continuous

-14-

whereas there is a discontinuity in the heat capacity.

In the presence

of external magnetic field the transition takes place at a temperature
below Tc and it is of the first order since the entropy is discontinuous, and there is a latent heat in the transition given by
(2.14)

These results so far have been obtained thermodynamically without any
reference to microscopic behavior of superconductors.
In the formal description of superconductivity, it has been found
that two characteristic lengths play a basic role.

They are the pene-

tration depth and the coherence l. ength.
Penetration depth:

In an attempt to describe superconductivity, F. and

H. London (L4) proposed two phenomenological equations governing the
microscopic electric and magnetic fields.

One of the equations which was

designed to describe the Meissner effect was

h = -:.\

2 ajs

L at

(2.15)

where h is the microscopic magnetic field, ns is the number density of
superconducting electrons which contribute the supercurrent density j 5 ,
and m and e are the electron mass and charge respectively.

When combined

with the Maxwell equation

VXh =j

we obtain

(2.16)

-15(2.17}

which implies that a magnetic field is exponentially screened from the
interior of a sample within the length of AL.

This is known as London

penetration depth and it gives a characteristic length over which the
magnetic field and current can vary within a superconducting body.
Coherence length:

In the condensed state, it is found that there are

strong correlations among superconducting electrons.

The concept of

coherence length, ~ 0 , was first proposed by Pippard (P3) in order to
account for the inadequacy of the London equations for certain superconductors. * Later Faber and Pippard (Fl) estimated ~ 0 from the uncertainty principle qS follows:

Only electrons within

kTc of the Fermi

energy can play a major role in the phenomenon and these electrons have
a momentum range of oP ~ kTc/vF where vF is the Fermi velocity.

Then

so that they define

where a is a numerical constant of order unity.

This form was later

confirmed by the microscopic theory of BCS (Bl) with a numerical constant a= 0.18.
In the next section it will be seen that this coherence length ~ 0
is closely related to the characteristic distance over which a density

*Extreme type I superconductors such as aluminum, tin and indium.

-16-

of superconducting electrons can vary within a superconductor.
2.2.2 Ginzburg-landau Theory
In 1950 Ginzburg and landau proposed a phenomenological
11

theory

of superconductivity (henceforth Gl theory) in which a pseudo-wavefunction w(r) was introduced as a complex order parameter. In this theory
lwl 2 was to represent the local density of superconducting electrons,
ns(r).

Upon minimizing the free energy density which was expanded in
powers of lwl 2 and 1Vwl 2 , they obtained a pair of coupled differential

equations for w(r) and the vector potential A(r).

The phenomenologi11

cal .. nature of the theory was later lifted in 1959 when Gor'kov (G2}
showed that the Gl theory was derivable as a limiting case of the
The Gl theory has been successful in treating the

microscopic theory.

macroscopic behavior of superconductors where the overall free energy
is important.

For example, it can successfully predict various critical

fields and the spatial variation of w(r) in nonuniform cases.
Ginzburg and landau postulated that, if w is small and varies
slowly in space, the free energy density fs can be expanded in a series
of the form *

= f

+ a.l•'•l2 + ~ 1·"14 + _1_. /<~ v- e* A)ljJ\2 + h2

2 "'

2m*

(2.19)

8TI

where a. and B are real coefficients in the expansion and e* and m* are
the effective charge and mass of superconducting electrons.

By minimiz-

ing the volume integral of fs with respect to the variations in
A, they obtained two differential equations (Gl equations):
cgs Gaussian units are used up to Eq. (2.27).

1jJ

and

-17-

(2.20)

(2.21)
together with the boundary condition

(~ V 1

e* +A)~
--

normal

(2.22)

which ensures that no,current passes through the surface (i.e., superconductor-insulator interface).
Two special and obvious solutions of the above equations are

= 0 describing the nonmal state;

(1)

(2)

~ = ~oo = {-a/e) 112 describing the superconducting state in a bulk
medium in the absence qf field and gradient.
Also, two characteristic lengths of a superconductor and their

ratio can now be introduced as follows:
Coherence length (~):

We consider a special case where there is no ex-

ternal field, i.e., A= 0.

Letting f ~ ~~~oo in Eq. (2.20) we obtain
(2.23)

where it is natural to define GL coherence length by

- 1-fr

~(T) = J~CfT

(2.24)

as a characteristic length for the variation of order parameter. ~(T)
thus defined turns out to be approximately the same as the Pippard

-18-

coherence length, ~ 0 , for pure metal well away from Tc (T4, p.ll2).*
Penetration depth (A.):

Taking the curl of j in Eq. (2.21) and assum-

ing that the order parameter in weak field is not changed from zero
field value, i.e.,~=~ to first order, we obtain
00

(2.25)
By combining with the Maxwell equation, (2.16), we have

= _12 +h

(2.26)

which is exactly the same result as obtained by using the London equation
(Eq. 2.17) •

Thus the penetration depth A. is a characteristic length

for the variation of magnetic field and current density.
Ginzburg-Landau parameter (K):

The ratio of the penetration depth,

A.(T), to the coherence length, ~(T), is defined as the GL parameter
_ __ilil

-un

(2. 27)

The comment on the same page is essential in distinguishing the Gl
coherence length from the electrodynamic coherence length in the more
general case. See also, L3, p.47.

-19which is expected to remain constant of temperature* si.nce ·ooth A:(T). and
~(T) diverge as (l - (T/Tc))-l/ 2 near Tc.
Type I and Type II superconductors:

In 1957 Abrikosov examined the

magnetic properties of superconductors as a function of the GL parameter
K •

He found that superconductors with K > 1/~ would admit magnetic

flux in the form of quantized vortices without becoming entirely normal
at a lower critical field Hcl and remain in this mixed state up to an
upper critical field Hc2 where it would become entirely normal.

This

behavior was quite contrasting to the superconductors with K < l/~
which would become normal at a well defined critical field He.

Since

then, those superconductors with K > 1/~ are known as type II and those
with K < 1/~ as type I.
The physical distinction between type I and type II behaviors is
due to an interfacial energy (surface tension) which is contained in the
wall separating a normal phase from a superconducting phase.

To clarify

this point we consider a simplified situation where superconducting. and
normal phases are separated by a plane interphase boundary.

Also, we

suppose that there is an applied magnetic field H parallel to the boundary.

This magnetic field is constant within the normal phase and at-

tenuated exponentially within the interphase boundary to reach zero
within the superconducting phase.

On the other hand, the order parameter

(or density of superconducting electrons) 11 heals 11 from its zero value
Although K is presumed to be temperature independent, some dependence on
temperature arises depending on the details of temperature dependence of
A:(T) and ~(T) away from Tc, which is discussed in Chapter 4.

-20-

within the normal phase to its undisturbed value within the superconducting phase.

This healing length is just given by the coherence

length ~(T).

Then the energy of the interphase boundary (as compared

to the situation where the two phases are sharply divided) is lowered
by admitting the applied magnetic field within the penetration depth A

(i.e.,-~ H2A) but is raised by having to suppress the density of
superconducting electrons over the coherence length (i.e.,+~ H~~ ,
where

~ H~

conductor).

is the condensation energy per unit volume of the superTherefore the boundary energy per unit area is *

~ ~ (~ H~ - AH 2)

(2.28)

When the applied field is H the above surface energy is

positive for

> A • i.e.,

negative for

i.e •

(type I)

K > 1

(type I I)

K <

Thus, when there is a boundary separating two phases, type I superconductors try to minimize the interphase boundary area, whereas type II sJperconductors try to maximize the boundary area, resulting in a maximum
possible boundary area when the normal phase exists only in the cores of
quantized vortices.

As mentioned previously, the exact separation be-

tween the two types is at

= 1I l'l .

So far, our discussions involved only thermodynamically stable
states

in static fields.

In the actual process of magnetic phase

A more thorough argument to arrive at this equation is given in reference G3 {p.382).

-21transition from initially superconducting to finally normal state, a
boundary between the two phases must be created so that the boundary
energy as obtained in the preceding paragraph can be expected to play
an important role.
In particular, the positive surface energy for type I superconductors suggests a possible persistence of Meissner state above the
thermodynamic critical field, since the surface energy vanishes only
at an applied field Hsh such that
llo

a =2

(f;Hc - /..Hsh)

= 0

H h = -H
IK c

where

= A/s

(2.29)

We can immediately see that this field Hsh is higher than He for small

(type I) and it has been generally called a (magnetic) superheating

field (to be discussed in the next section).

Noting that the volume

contribution to the free energy above the thermodynamic critical field Eq.
(2.11) favors the transition to the normal state, we see that the above
field is only an upper limit to the (metastable) superheated state.
For type II superconductors it is also possible to have a
similar situation in which the Meissner state persists above the lower
critical field Hcl·

This is generally discussed in tenns of a .. surface-

barrier field 11 opposing the entrance of mgnetic flux at the surface and
this barrier field was found to coincide with He in the limit of large

(01).

For both types of superconductor the existence of superheated
state was first theoretically predicted by Ginzburg (G4) and later ob-

-22served experimentally for tin by Garfunkel and Serin (G5).
2.2.3 One-Dimensional GL Superheating Theory
Ginzburg (G4) first analyzed the superheating field by solving
the GL equations for the one-dimensional case where a half of the spdce
was occupied by a superconductor.

He performed a numerical integration

to obtain the superheating field as a function of the GL parameter
He also examined two limiting behaviors for small and large

K •

and

found the superheating field Hsh in terms of bulk critical field, He
as follows:
( 1)

For K << 1 ,

(extreme type I)

(2.30)

(2)

For K >> 1 ,

(extreme type II)

( 2. 31 )

The same problem was later treated numerically for better accuracy by
Matricon and St. James {M5).
The prediction of one-dimensional theory is considered to put an
upper limit on the superheating field, since possible variations of the
order parameter in other directions such as parallel to the surface may
lower the actual superheating field.

This problem has been studied in

terms of the stability of the superheated Meissner state against spatial
fluctuations of order parameter by various authors (F2,K2,K3).

In par-

ticular, Kramer {K3) found that, for K ~ 0.5, another instability
Later the numerical constant in this limit was shown to be 2- 114 =0.841
rather than 0.89 by the Orsay group on superconductivity (01).

-23occurred at a field, Hsl, lower than Hsh"

For large K limit he ob-

tained

His estimate of the field H51 is shown in the phase diagram of Fig. 2.1
together with the prediction of one-dimensional GL theory as evaluated
by Matricon and St. James {M5).

In this phase diagram all the critical

fields are reduced by the thermodynamic critical field He and shown as
a function of GL parameter K = A/~ .

Thus we have type I superconduc-

tors forK< l//:2 where two thermodynamically stable phases (Meissner
and normal states) are separated by the line H/Hc = 1.0 .

For

K > l//:2 we have type II superconductors which go from Meissner to
mixed state at hcl = Hc 1/Hc and into normal state above hc 2 = Hc2/Hc.
This phase diagram will be found convenient in discussing the present
experimental results in Chapters 3 and 4.
2.2.4 Previous RF Critical Field Measurements
In going from superconducting to normal state, the surface resistance of a metal changes by several orders of magnitude.

Thus if a

small fraction of surface area becomes normal while a high rf magnetic
field is being sustained, it can lead to a thermal runaway situation
where the power dissipated into the normal region heats up the surrounding area above the transition temperature which, in turn, drives more
area normal.

This runaway process is generally called .. magnetic-thermal

breakdown .. and the field at which this process is initiated is called
the breakdown field or rf critical field.

-24-

In
II

2.5

Pb (I)

Pb (2)

750 MHz

9.7 GHz

2.0

INORMAL STATE I
1.5
IMIXEO STATE!

.....

He
1.0

. . . . . . . . ___

-----

[h----"""*---..(

p:r1_- Hsl

__ .... __ _
He

Pb(l)

ONb

!MEISSNER STATE

0.5

TYPE

0,4

--~---TYPE

0.8

GL Parameter

II - - - - - - - - - - - 1

1.6

1.2

2.0

[:= ~]

Fig. 2.1 Phase Diagram of Superconductors: All fields are normalized to thermodynamic critical field and shown as a function of
Ginzburg-Landau parameter K • RF critical fields previously
measured for Sn, In. Pb and Nb are also shown" H h was evalus
ated in M5, Hsl in K3 and Hcl in Hll.

-25Most of the previous measurements of critical magnetic field
at radio frequencies were performed on the walls of resonant cavities.
Among the elemental superconductors studied, tin, indium and lead belong
to type I superconductors and niobium (colombium) to type II superconductors.

Turneaure (T3) measured the critical field of electroplated

tin in 2.85 GHz resonator and found that the rf critical field, H~f ,
was close to the de critical field.

Similar results were later found

for electroplated indium and lead at 2.85 GHz {84).
also reported by

For lead it was

Flecher et al. (F3) that the rf magnetic field higher

than the de critical field by 25% at 4.2°K was observed at 750 MHz.
To the author•s knowledge, this is the only reported observation of
superheating at radio frequencies for type I material.

For niobium

Schnitzke et al. {S4) recently attained rf critical field higher than
the first critical field, Hcl' as a result of improved material processing technique.

Also, more recent work on type II alloys such as

Nb 3Sn (PlO,H9) and Nb 0 . 4Ti 0 . 6 (G7) show that H~f is not limited by Hcl
in type II materials.

These previous rf critical field results are

summarized in the phase diagram of Fig. 2.1.
Concurrently with the above efforts in measurements and improvements in materials processing, there have been phenomenological attempts
to determine what limits the rf magnetic field which can be attained in
superconducting resonators.

Factors that have been considered to be

important in determining the rf critical field include the thermal
stability of cavity surfaces (55), the power dissipation in a single
isolated fluxoid on the resonator surface (M6) and the magnetic boundary

-26-

energy between superconducting and normal phases (HlO).
Theoretical treatment of phase transition from superconducting
to normal state in radio frequency field has been extremely scarce.
Only a very limited qualitative argument has been used in "predicting"
the rf critical field.

Halbritter (HlO) used a surface energy

argument similar to the one given in Section 2.2.2 to arrive at the
same result as the superheating field obtained by Ginzburg (G4). Then
it may be expected that the superheating field will put an intrinsic
limit on the rf magnetic field that can be reached.

Prior to the

present study, there has been only one instance of the observation of
rf magnetic field higher than the thermodynamic critical field for
type I superconductors (F3).

For type II superconductors the rf

critical field has been shown to be somewhat higher than Hcl"
In the present study it was found that indeed the superheating
field puts an upper limit on the rf critical field in the frequency
range between 100 and 300 MHz.

This was initially found for lead at

205 MHz as shown in Chapter 3.

Also, by resorting to material whose

GL parameters could be varied, a similar conclusion was found for

up to about 1.8, which is well into the type II region.
2. 3 Summary
The rf surface resistance of superconductors was reviewed for
its physical origin and for its historical background.

There is now

good agreement between the past rf experiments and the theoretical
surface resistance based on microscopic theory, provided that the rf

-27magnetic field is small compared with the thermodynamic critical
field.

At low temperatures, there are other loss mechanisms that

give rise to an effective surface resistance, only some of which are
understood.
The magnetic behavior of superconductors in a static magnetic
field is well understood in terms of various critical fields.

How-

ever, the situation involving rf magnetic fields is rather uncertain
in that there has been no definite proof, either experimentally or
theoretically, as to what limits the attainable rf magnetic field. It
was once thought (up to about 1971-1972) that the rf critical magnetic field was given by the thermodynamic critical field for type I
superconductors and by the lower critical field, Hcl, for type II
superconductors.

More recent experiments suggest that the above

statement does not hold for type II superconductors.

A large portion

of the present investigation is directed to answering experimentally
the question of the rf critical magnetic field for both type I and
type II superconductors.

-28Chapter II I
SURFACE RESISTANCE AND RF CRITICAL MAGNETIC FIELD OF LEAD AT 205 MHz
3.1

Introduction
Lead is one of the superconductors whose superconducting proper-

ties have been studied quite extensively in the past.

It is an

attractive material for potential rf applications, not only because of
its high critical field (803 gauss at 0°K) and relatively high transition temperature (7.2°K), but also because of its wide availability
and ease in handling.

In particular, low frequency rf resonant cavi-

ties, which are often geometrically complex, can easily be coated with
lead by such means as electroplating on a copper substrate.

The rf

superconducting properties of such lead have been studied previously
in the frequency range of 1-10 GHz.

However, no systematic study has

been undertaken in the lower frequency range (~ 100 MHz) where there
are possible practical rf applications such as the acceleration of heavy
ions using recently developed rf resonant structures.

Thus the present

study, whose results are presented in this chapter, was designed to investigate the current carrying properties of lead electroplated on
copper in terms of its surface resistance and rf critical magnetic field
at this lower end of radio frequencies.
3.2

Experimental Method
The use of rf resonant cavities is a convenient means commonly em-

ployed in the measurement of very low surface resistance as well as in
the generation of high rf magnetic fields.

For the present investigation

-29-

a coaxial resonator was designed so that the surface resistance of lead
could be determined and at the same time a hiqh rf magnetic field comparable to the thermodynamic critical field of lead could be generated
in a well-defined area.

This resonator is described in the next section

where it is also shown how surface resistance is related to the characteristic quantities of the resonator.

In subsequent sections general

experimental procedures are given, followed by a description of how the
characteristic quantities of the resonator are determined from the measured quantities and then related to the quantities of present interest:
the surface resistance and the magnetic field level within the resonator.
3.2.1

Half-Wave Coaxial Resonator at 205 MHz
In order to study the surface resistance and rf critical magnetic

field of superconducting lead electroplated on copper, a coaxial resonator was constructed from OFHC (Oxygen Free High Conductivity) copper as
shown in Fig. 3.1.

The resonator consisted of four parts: the outer con-

ductor, the center conductor and the two end plates.
the center conductor was constricted as shown.
twofold.

The midsection of

The reason for this was

One was to lower the resonant frequency to the desired range.

The other, more important, was to enhance the surface magnetic field as
well as to localize the high magnetic field region to a well-defined
area.

The center conductor was supported from both ends by the sapphire

rods with low loss tangent (Yl).
The electromagnetic field distribution in the resonator was evaluated analytically using transmission line and lumped parameter approximations.

In this approximation, the resonant frequency, the electric

-30-

tTob:J~ing

Port
Dewar Insert
Grounding Clips
Top Plate

Indium

"o" Rings

Drive Probe

Uncalibrated
Pickup Probe
Outer
Conductor
Scale

lcm

,r-_ _ Sapphire

Support Rods

Bottom Plate
Indium

"o" Rings

Calibrated
Pickup Probe

Fig. 3.1 The half-wave 205 MHz coaxial resonator: The center conductor
is supported by two sapphire rods so that it makes no direct electrical connection to other parts. The resonator is coupled to the
outside rf system by three coupling probes as shown. Note the difference in the vertical and horizontal scales.

-31and magnetic field distributions, the total energy content and the
geometrical factor of the resonator were evaluated.

The resonant

frequency thus determined agreed with measurement within 1%, indicating that this approximation serves well for the present purpose. The
relative radial electric field, EP, and the azimuthal magnetic field,
H¢, on the surface of the center conductor are shown in Fig. 3.2.
The total energy content, W , of a resonator at resonance is
given by (P407,B2)

where the time averaging has been performed.

E and H are the spatial

part of the electric and the magnetic field, respectively.

and ~

are, respectively, the permittivity and the permeability within the
volume V of the resonator.

Since the time-average electric and mag-

netic energies are equal at resonance (Ibid), W can be expressed in
terms of either the electric or the magnetic field distribution. Using
the field distribution as obtained above, it was found for this resonator
W= 7.32 X 10- 7 Hmax
- g

-1

Hmax

(joule)
(3.2)

where Hmax is the (amplitude of the)* magnetic field at the midsection
Throughout this dissertation all rf fields (electric and magnetic) are
represented by their amplitude so that, for example, the full expression for the magnetic field at the midsection of the center conductor
is Hmaxsin(wt) where w is the rf angular frequency.

-32-

Em ax

EP----.,

Field Level
(Arb. Scale)
rI

-- -- ---

Fig. 3.2 The radial electric field, Ep , and the azimuthal magnetic
fiel~H
, at the surface of the center conductor are shown in arbitrary¢scale. The vertical scale is the same as in the resonator
shown on the left side where the center conductor is 30.5 em in
length from one end to the other.

-33of the center conductor in gauss, and

2 /W.
g is defined as Hmax

can also be expressed as

w= 3.71

{joule)
(3.3)

where Vmax is the rf voltage at the end of the center conductor in
volts, and

d is defined as Vmax/W.

Both of the above expressions

will be useful in the discussion of calibration (Sec. 3.2.6).
Another quantity of practical importance is the ratio of maximum
magnetic field to maximum electric field within the resonator.

The

locations of their occurrence are shown clearly in Fig. 3.2.

This ratio

was found to be
Hmax/Emax = 190 gauss/(MV/m)

(3.4)

The geometrical factor of a resonator is a quantity which is very
useful in dealing with resonant cavities in general.

This is related

to the quality factor of a resonator which is a measurable quantity and
defined by
(3.5)

where

W is the total energy of the cavity whose resonant frequency

and

Pdiss is the average power dissipated within the resonator.

W is obtained by integrating the energy density over the volume of the
cavity as in Eq. (3.1).

Assuming there is only resistive dissipation

-34-

due to the surface current,Pdiss is given by
Pdiss -- l2 I Rs H2 dS

( 3. 6)

where

R5 is the surface resistance of the wall, S , enclosing the

cavity volume, V . Then

(£E + ].1H ) dV
rrf _,..v______
H dS

( 3. 7)

Since the right hand side is dependent only on the geometry of the
resonator, it is a constant for a given normal mode and defined as
the geometrical factor

r .

For the present resonator, using the known field distribution,
the geometrical factor was evaluated to be

= 7.oo n
This was verified within experimental error (~ 3%) by measuring the Q
of copper (unplated) resonator at room, liquid nitrogen, and liquid
helium temperatures, where Rs was determined independently from the de

-35resistivity of the same material (Chamber's interpolation formula (Cl)
was used to take into account the anomalous skin effect below room
temperature) .
3.2.2

Preparation of the Surface
Once the parts of the resonator were machined, all the surfaces

were polished with emery papers of decreasing grain size (320, 400, 600)
and finally with levigated alumina polishing compound (grain size =
1-3~).

After the mechanical polishing all the surfaces were electro-

plated with lead to a typical thickness of 1.5- 3~ .

Since the general

electroplating procedure and problems are discussed elsewhere (G6, Pll,
T3) only a brief description of the procedure is given below.
There are three main steps in obtaining the lead layer on copper
by this method:
1) Cleaning.
face.

The first step was to remove the oxide on the copper sur-

This was done by electrically polishing the copper surface in

an electropolishing solution* followed by a thorough rinsing with
deionized water.
2) Electroplating.

As soon as the electropolishing solution was rinsed

away, the copper piece was placed in an electroplating solution**
The electropolishing solution was a mixture of 1 part Electro-Glo "200"
(supplied by Electro-Glo Co., Chicago) and 3 parts 85% phosphoric acid
by volume.
**At the time of this study, it was found that the addition of "Shinol
LF-3" (supplied by Harstan Chern. Corp., Brooklyn, N.Y.) in place of
the usual animal glue to lead fluoborate solution gave an improved
surface finish.

-36with appropriately shaped electrodes which were made from 99.9% purity
Initially a few pulses of very short duration <~ 1 sec) were
applied at a current density of 80 mA/cm 2 in order to obtain a uniform
lead foil.

thin lead layer as a "starter". Then the plating continued at a current density of 8 mA/cm2 until the desired thickness was obtained.
3) Rinsing and drying.

After plating, the surface was rinsed in running

deionized water to remove the plating solution.

This was followed by

rinsing with absolute ethanol and then blown dry in a stream of dry
nitrogen.
3.2.3 Cryogenic Apparatus
After all the parts were plated, the resonator was assembled and
attached to a dewar insert.

Indium "0 11 rings were used where parts were

joined in order to obtain vacuum tight seals as well as current carrying
joints* (see Fig. 3.1).

The dewar insert, shown schematically in Fig.

3.3, consisted of four long stainless steel tubes leading up to the top
of the dewar at room temperature.

One of the tubes was used as a pumping

line to pump on the resonator, the two others to guide coaxial lines into
the cavity resonator for rf coupling and the last one to feed helium gas
into the resonator when necessary.
Immediately after the assembly, the resonator was pumped down to
about 10 ~Hg through the dewar insert so as to avoid any possible oxidation of the freshly plated surfaces due to exposure to the atmosphere.
The resonator was then filled with gaseous helium** which had been filtered
No observable loss was introduced by these joints since they carried only
a negligible fraction of the total oscillating current.
**Purity 99.995% (Gardner Cryogenics, Glendale, California).

-37-

Pump

-----.

~-------------

Pressure and Vacuum
Gauge

BNC Connector
Valve

Coaxial Line

4-in
Metal Helium Dewar

Pressure
Regulator
Pressure Gauge
Heat Shield
Liquid Helium Bath

1J -Metal
Magnetic Shield
Liquid Nitrogen Bath

CAVITY

Fig. 3.3 The dewar insert (in bold lines) and cryogenic equipment are
shown schematically. The pressure within the resonator could be
varied by either feeding helium gas or pumping by a small mechanical pump. The temperature of the helium bath in the dewar was
varied by pumping with a large mechanical pump through the pressure regula tor.

-38-

through a liquid nitrogen cold trap.

Once it was full, the resonator

was pumped again and this process of "flushing" with gaseous helium was
repeated several times in order to get rid of impurities that might
condense on the cavity surfaces at liquid helium temperatures.
For the low temperature experiment, a 4-inch metal dewar was
used in conjunction with standard cryogenic equipment shown in Fig.
3.3.

The earth•s magnetic field was reduced by a mu-meta1 shield to

below 15 mi11igauss over the entire experimental space.
working temperature range was between 4.2 and 2°K.

The typical

The temperature of

the helium bath was varied by pumping with a mechanical pump and a
pressure regulator.

The vapor pressure of the helium bath was read on

a Wallace and Tiernan pressure gauge and related to temperature using
the 1958 He 4 scale of temperatures (03).
In order to make a thermal contact to the cavity center conductor, the resonator was filled with helium exchange gas at a pressure
of about 100~ Hg while low power measurements were made.

For high

power measurements, the resonator was filled with liquid helium then
pressurized to 5-15 psi above the atmospheric pressure with gaseous
helium.

The pressurization was necessary in order to avoid spark dis-

charging that took place within the resonator at lower vapor pressures
of helium.
3.2.4

RF Instrumentation

Because a superconducting resonator has an extremely high Q (on
the order of 10 8 at 205 MHz in the present case) the excitation of such
a resonator by a conventional rf oscillator poses a special problem.

-39-

This is because the resonator bandwidth is much smaller than the frequency stability of the oscillator.

The following two methods are

conventionally used to overcome this problem:
1)

Phase-lock operation.

In this mode of operation the resonator is

driven by an external voltage controlled oscillator (VCO) whose
frequency and phase are adjusted to those of the resonator by means
of a feed-back.
2)

Self-resonating loop operation.

If an amplification loop is closed

through the resonator of interest, then a spontaneous oscillation
takes place when the gain and the phase of the loop are optimized.
Throughout this investigation the phase-lock operation was used
more extensively because of its ease in making measurements.

The block

diagram of the rf instrumentation used for the phase-lock operation is
shown in Fig. 3.4.

For a stable oscillation of the resonator, the

driving signal out of VCO had to be at proper phase and frequency. This
condition was achieved in the following manner.

The field within the

resonator was sampled through one of the pickup probes.

The phase dif-

ference between this sampled signal and the VCO signal was detected by
a mixer (here used as a phase detector), then amplified by a de ampli-.
fier and fed into the frequency control input of the VCO.

As a result,

the external oscillator (VCO) was 11 locked 11 to the resonator once the
phase shifter in the loop was adjusted properly at the beginning of
operation.
In order to avoid impedance matching problems, all the electronics
used, including coaxial cables and transmission lines to the resonator,
had a characteristic impedance of 50 ohms.

Pine
p f
re

VOLTMETER~ { Ppu }

OR
POWER METER

DC
AMPLIFIER

(Voltage
...
Controlled
Oscillator)
TUncalibrated TFrequency
RF Output 6Control Input

vco

.. Pref

DIRECTIONAL
COUPLER

Pine

;-Q-

1.,.

.. ---

RESONATOR

I rl

" I

Ppu (Calibrated)

RF
AMPLIFIER

Fig. 3.4 The block diagram of the rf instrumentation used in the phase-lock operat~on of the
superconducting resonator. The external voltage controlled oscillator is "locked" to the
resonator by means of a feedback. Approximate rf envelopes of the incident, reflected,
and pickup voltages are also shown.

OSCILLOSCOPE

FREQUENCY
COUNTER

PULSE
GENERATOR Pulse
Modulation
Input

.rL..

-41-

3.2.5 General Theory of Measurement
In this section it is shown how a set of experimentally measured quantities are related to the characteristic quantities of a
resonant cavity.

In particular, we are interested in the following

two characteristic quantities of a resonator:
1) the total energy content of the resonator, W (to be
related to Hmax
2) the intrinsic quality factor, Q •

The system of a resonant cavity with two coupling ports may be
looked at as a four-terminal passive device.

The coupling ports at

frequencies on the order of 100 MHz are normally coaxial lines whose
ends are coupled either capacitively or inductively to the electromagnetic fields of the cavity.

Measurements are made by sending an rf

signal into one port and looking at a signal picked up through the
other port (see Fig. 3.5a).

The measured quantities are the incident

power, Pine• the reflected power, Pref' the power picked up, Ppu' and
their time variations.

In the following we assume that every quantity

has been averaged over an rf period.* Balancing the power flow into
the system, we have
(3.8)

where

Pdiss is the power dissipated into the cavity itself and dW/dt

is the time rate of change of the total energy content of the cavity,
The time dependence we are dealing with here is on the scale of characteristic decay time of the resonator (~ 100 msec as compared to the
rf period of 5 xlo-9 sec).

-42-

(a)

Pine (t)

RESONATOR
with
Energy Content

1----11~-~ p p u

(t )

w(f)

Pref (t)
(Prod (t))

(b)

r------....,,',~'------.___ Pine (0-)

,_--Pine (0+)

Pref {Vref)
(Prod)- - - - - 1

STEADY
STATE

TRANSIENT
STATE

------~--------~,~--------~~-------------INCIDENT
TIME
t =0
PULSE ON

(PULSE OFF)

Fig. 3.5 (a) The power flow into and out of the resonator is shown.
(b) Typical rf envelopes of the rf voltages when in the pulsed mode
are shown for the case where the incident port is critically coupled
to the resonator. P is proportional to the energy content of the
pu
resonator.

-43W.

By definition, the intrinsic (or unloaded) quality factor is given

by
(3.9)

Qo = wW/Pdiss
where w is the angular resonant frequency.

For a fixed pickup coupl-

ing, Ppu is proportional to the energy content, i.e.,

Ppu = a -1 W
where a

( 3. 10)

is the coupling constant to be determined.

Then the recipro-

cal of Q0 can be expressed as
- p
ln p
ref _ 1 _ a
at-u]
Q-1 _ 1 [ inc
- ~

(3.11)

Ppu

In order to determine a , we consider a situation where the rf incident power has been on for a long time and is turned off at
(see Fig. 3.5b).

Then at

t = 0

t=O- (just before Pine is turned off) the

system is in steady state so that dPpu/dt = 0 • Thus,

( 3. 12)

When the incident power is turned off at

t = 0 , the cavity 1 oses its

stored energy with the characteristic time of the loaded cavity, TL ,
which can be directly measured.
P.

1nc

For t ~ 0

then we have

-44-

where Pref(t) is now the power radiated into the incident port.

There-

fore,

-P rad(O+)
( o+) - 1 ] + pu
WTL

Q~ ( t=O ) = wa [ p

(3.14)

Since the intrinsic cavity Q is the same whether the system is in steady
state or transient state, we can equate the above equation to Eq. (3.12).
Using Ppu(O-) = Ppu(O) = Ppu(O+) we obtain

( 3. 15)

Since all the quantities on the right hand side are measured, this determines the coupling constant, a, which can then be used in Eq. (3.10)
to determine the energy content of the cavity, or in Eq. (3. 11) for
general Q measurement.
The special case of experimental interest is shown in Fig. 3.5b
where the incident port is critically coupled to the resonator so as
In this case the coupling constant is given as
(3.16)
where

So far we have dealt with resonant cavities in general.

Now in

order to relate the field level within the resonator to the measured

-45quantities we need the field distribution within the particular resonator.

The relationship between the energy content and the field within

the present resonator was obtained in Sec. 3.2.1.

Then using Eq. (3.2)

and (3.10) we obtain
Hmax = (gW) l/2

(a.g Ppu)l/2

( 3. 17)

So, the measurement of Ppu directly gives the surface magnetic field at
the midsection of the center conductor once the coupling constant a. is
determined.

Also, the intrinsic quality factor at any field level is

given by Eq. (3.11), which holds for a general time dependent case.

the resonator is in a steady state, then
Q~l

1 [ 1ncp
= __

wa.

- p

ref_ l]

(3.18)

3.2.6 Measurement, Calibration and Error Estimation
The resonator was coupled capacitively to the outside rf system
by way of three probes attached to coaxial transmission lines.

Two

probes at the top of the cavity (see Fig. 3. 1) were coupled to the resonator field and their coupling strength was variable.

One of them was

used to drive the resonator at any desired coupling strength.

The other

one was always coupled weakly and was used only to sample the field of
the resonator for the phase-lock operation.

The third or pickup probe

at the bottom of the resonator was fixed to the end plate and used
solely for the purpose of field level determination upon calibration.

-46The calibration of the resonator field level was of basic importance in this study.

For the present coaxial resonator two

independent calibration procedures were employed.
1) The first method was to use the procedure as outlined in
the general theory of measurement (Sec. 3.2.5).

In this procedure,

the magnetic field level of interest, Hmax' was related to the measured
quantity Ppu by Eq. ( 3. 17).

= (ag p

) l/2

Thus it was only necessary to determine the coupling constant a since
g was evaluated with the known field distribution as given in Sec.
3. 1. 1.

The coupling constant was normally determined by critically

coupling the drive probe so that from Eq. (3.16)
with

Therefore a was determined by measuring four quantities: TL on the
scope,

Pinc(O-}, Ppu(O) and Prad(O+) by the rf voltmeter.

(Note that

only the ratios of these powers are needed here.)
2) In the second method the coupling constant a was determined
by a direct application of an rf voltage to the cavity in the following
manner (Details of this method are discussed in 02).

An rf voltage

was applied between the center and outer conductor at one end of the
resonator by lowering the drive probe until its tip touched the end
face of the center conductor (see Fig. 3.1).

This applied voltage, Va'

-47-

is related to the voltages measured at the incident and the reflected
port of the directional coupler (Fig. 3.4), Vine and Vref
by ( 02 )

respectively,

Va -- ( Vincvref )1/2 under an assumption that the resonator is

an open circuit at resonance in this configuration.
Then by measuring the resultant voltage, Vpu' at the pickup
probe (the bottom probe in Fig. 3.1) we can express Vain terms of the
measured ratio r = (V inc Vref ) 112;v pu as
va = r vpu
Now the magnetic field at the midsection of the center conductor, Hmax'
can be related to the voltage at the end of the center conductor by
combining Eqs. (3.2) and (3.3).

Thus we obtain

Hmax = (g/d) 112 Vmax *
or, since Vmax is just the applied voltage at the end of the center conductor,
.H

max

= (g/d) l/ 2 r Vpu

Finally, expressing Vpu in terms of the power at the pickup probe, Ppu
(i.e., Ppu = V~u/2R where R =50 ohms), we have

*Previously this relationship was verified experimentally (Yl) within
about 5% in a similar resonator by applying an rf voltage in the same
manner as described here and measuring the magnetic field at the midsection of the resonator.

-48-

Hmax = ((2Rr2/d)g Ppu ) 112

(3.19)

Therefore, the coupling constant a in this scheme is given by

which can be determined by the measurements of Vine' Vref and Vpu
since d is a numerical constant given previously.
The fields determined by the above two methods typically agreed
within 6%.

Once their agreement was established, the first method was

routinely used to determine a in different low temperature experiments
on the same resonator, since it was operationally simpler.
The sources of error in the determination of Hmax and Q0 were:
1)

In the determination of TL.

An oscilloscope trace of the decaying

pick-up voltage was photographed and its readings as a function of time
were plotted on a semi-log scale to determine the slope TL.

The error

in TL thus determined was estjmated to be less than 3%.
2)

In the measurements of Pine' Pref and Ppu

In the CW mode their

values were read on a power meter and the reading error was less than
1%.

In the pulsed mode, their corresponding voltages were displayed on

an oscilloscope and read within 1.5%, so that the errors in power determination in the pulsed mode were estimated to be 3% or less.
Combining these errors, we estimate the total errors, ~
the quantities of present interest to be as follows:

, in

-49-

Absolute
~ 6%

.s. 10%

Relative
"' 3%
"'1.5% for low field CW
measurement
"' 4% for high field
pulsed measurement

The errors in temperature determination {which were relatively
unimportant for the present measurement and could be neglected for all
practical purposes) were typically 1 m°K at 4.2°K and increased monotonically to about 10 m°K at 2.0°K.
3.3 Experimental Results and Discussions
In low temperature experiments for a given lead surface the following quantities of interest were typically measured.
{1) The resonator Q0 as a function of temperature at low field
level.
(2) The resonator Q0 as a function of the magnetic field, Hmax'
at two or more different temperatures (typically at 4.2 and
2.2°K},
(3} The rf critical field, Hrf,
as a function' of temperature.
The results are discussed in t~e following three sections.
3.3.1

Surface Resistance of Superconducting Lead at Low Field Level
The quality factor, Q0 , of the resonator was measured as a func-

tion of the helium bath temperature, T , below 4.2°K.

A typical rf

magnetic field level in the resonator during this measurement was kept

-50below 5 gauss* which is less than 1% of the thermodynamic critical
field of lead at these temperatures.

The surface resistance, Rexp(T),

is related to Q0 by Eq. (3.7) as

where r

(3.21)

f/Q (T)

is the geometrical factor defined previously and is 7.00 for

this resonator.

The error in the absolute determination of Rexp was

estimated to be less than 10% and the relative error at different ternperatures was 1ess than 1. 5%.
The following expression was used to analyze the temperature
dependence of the surface resistance;
R(T) = ~ e-~(T)/kT + R
res

(T/T c < 0.5}

(3.22)
where the first term is the BCS surface resistance (Eq. (2.8}} discussed in Chapter 2.

Here

~(T)

is the temperature dependent gap

parameter and Rres is a constant term to describe the residual surface
resistance due to nonsuperconducting loss mechanisms.

The temperature

dependent gap parameter can be approximated by (S2)

~(T}/~(0} = [cos(nt 2/2}] 112

= 0 (t)

(3.23)

*At this field level nonlinear behavior in Q to be described in the
next section was never observed.

51
where t is the reduced temperature T/Tc.

Using the above expression

for ~(T) we have
R(T)

- ~ilil

= t e

kTc

+ R

res

(3.24)

Then a set of experimental data (Rexp(T),T) was fitted to the above
equation by a least-squares fit with three adjustable parameters: A,
~(0)/kTc

and Rres

A typical data set, together with the result of

the fit, is shown in Fig. 3.6.

The three parameters thus obtained are

compared with the existing calculations and previous measurements as
follows.
BCS Surface Resistance at 4.2°K (A and ~(0)/kTcl
Based on the BCS theory the surface resistance of lead was
numerically evaluated by Halbritter (HS) as a function of frequency
for two different mean free paths of the electrons.

His results show

a weak dependence on the mean free path, ~ , and the values he obtained
at 205 MHz were
for
and
for
The mean free path of the present surfaces was estimated to be typically
(1.02-1.29) x10 4 ~ on the basis of their normal state surface resistance
measurements (between 7.2 and l0°K).

Thus, interpolating* the calculated

*The approximate functional dependence of the surface resistance on the
mean free path is logarithmic when ~ is such that 0 0.01 ~ ~o/~ ~ 1.0
where ~ is the coherence length. Since ~ 0 = 1130A for lead, this
condition is satisfied (HS).

-52lo- 7 ----~----~-----r----~--__,

Surface:
Thickness: 2-3 }J(polished)

OOoo
Ooo

Rexp (T)
ooo
0 0

R(T)-Rres= Rscs(T)

i:!

Slope:

(f)

10-8

0::

(f)

~t~) = 1.95± .05

(f)

lo- 9 ~--~----~-----~--~---L~
1.5

2.0

2.5

3.0

3.5

4.0

Tc/T (Tc = 7.18°K)
Fig. 3.6 The surface resistance data, Rexp(T) (the upper circles),
are shown as a function of the inverse reduced temperature. After
subtracting a constant residual term, Rres• the result of a leastsquares fit to RexP.(T) is shown by the solid line (Recs)
together with Rexp(T) - Rres(the lower circles).

-53-

surface resistance for the estimated mean free path, we obtain

which compares very well with the present value of
R(T) - R
= A exp -[MQ}_ ~I
res
kTc
t=O.SS

(3.47 ± 0.17)

10- 8st

where A and 6(0)/kTc were determined by the fit.

Considering the un-

certainties in material parameters that go into the numerical calculation,this good agreement may be somewhat fortuitous.
Energy Gap at 0°K (6(0)/kTcl
In the BCS theory of superconductivity, the ratio of the energy
gap at 0°K to the transition temperature, 6(0)/kT e , is a universal constant given by (Bl)
6(0)/kTc = 1.75
This value depends on their assumption of a weak electron-phonon interaction.

However, a number of superconducting elements such as lead and

mercury were found to have a stronger electron-phonon interaction,
giving rise to a higher value of 6(0)/kTc than the above.
thus called the strong coupling superconductors.
In this study the ratio for lead was found to be
6(0)/kT c = 1.95 ± .05

They are

-54as shown in Fig. 3.6.

This value is well within the range of values

determined previously by different experimental techniques, some of
which are summarized below (taken from H7)
Method

1'1(0)/kTc

Microwave absorption

2. 03- 2.13

Infrared transmission

2.0

Infrared absorption

2.05-2.19

Tunneling

2. 09- 2.19

Ultrasonic absorption

1.8

Thermodynamic measurement

1.98

A typical value of 1'1(0)/kTc

deduced from the surface resistance meas-

urement made in the GHz region is (H5)
t.(O)/kT c = 2.05 ± .05
which is about 5% higher than the present result.

This discrepancy

is on the borderline of the experimental error and it could have been
caused by a relatively large value of Rres which becomes increasingly
difficult to avoid at lower frequencies such as used in the present
investigation.
Residual Surface Resistance (Rresl
The observed residual surface resistance, Rres' varied from
surface to surface and typically it was in the range

-55which is comparable in magnitude to the BCS surface resistance at
4.2°K.

Although we do not yet understand all the residual loss mech-

anisms that may be involved, estimates can be made for particular
types of loss mechanisms which are discussed in Sec. 2.1.4 (the dielectric loss, the trapped flux loss, and the acoustic phonon generation
loss).
An estimate for the dielectric loss was made from the result
of an independent measurement made by J. R. Delayen in the Caltech
laboratory on a similarly prepared lead surface in a highly re-entrant
resonator (Dl).

In such a cavity the quality factor was found to be

limited by a thin dielectric layer on the surface (Tl).

If the dissi-

pation due to the presence of the dielectric layer is properly scaled
to give an equivalent surface resistance, R , in the coaxial resonaE

tor used in this study, we obtain

which is in the range of the observed residual surface resistance.
The residual resistance may also be caused by trapped flux if
a cavity is cooled through its transition temperature in the presence
of a static magnetic field.

Since the shielding of the earth's mag-

netic field in an experimental space is not always complete, the
surface resistance due to trapped flux, RH , is of some practical
importance and it was studied in some detail in the course of this
investigation (see Appendix A).

The results show that the typical

RH for the surfaces under investigation is given by

-56-

for Hd c < 2 gauss

where Hdc is the static magnetic field present during cooling and is
expressed in gauss.

The remnant magnetic field in the present experi-

mental space was measured to be less than 15 milligauss so that

which is much too small compared with the observed residual resistance.
Another possible source of residual resistance is due to the
acoustic phonon generation.

A previous theoretical estimate (P8) gives

for lead at 205 MHz

which may explain a part of the observed residual resistance.
In summary, the above estimates suggest that the major fraction
of Rres comes from the dielectric loss caused by a layer of dielectric
on the surface of the electroplated lead.

Also the phonon generation

may account for a part of the residual surface resistance.
3.3.2 Surface Resistance at High RF Magnetic Field
In the application of_superconducting resonators to the production
of high electromagnetic field levels, the surface resistance at high rf
magnetic field is of practical importance, since it is one of the main
parameters that determine the power dissipation in a low temperature
bath.

Although a considerable amount of previous work has gone into

improving the performance of superconducting resonators at high power

-57levels, our basic understanding of the surface resistance at high
magnetic field is still very limited both theoretically and experimentally.

The main reason for our lack of theoretical understanding

is that the perturbation approach, such as used in the Mattis and
Bardeen formulation of surface resistance, is not adequate when the
rf magnetic field becomes comparable to the thermodynamic critical
field.

Experimentally, the difficulty lies in the fact that even a

very small number of surface defects, when normal, can greatly enhance the averag~ surface resistance, since the surface resistance
typically changes by more than four orders of magnitude in going from
the superconducting to the normal state.
In the present investigation the surface resistance of lead was
studied also for the situation when the rf magnetic field amplitude
was comparable to the thermodynamic critical field.

The resonator Q0

was measured as a function of the rf magnetic field level.

A typical

example of the reciprocal of Q , which is proportional to the surface

resistance, as a function of the rf magnetic field level within the
resonator is shown in Fig. 3.7.

The main features that can be noted

in this figure are as follows.
(1)

The surface resistance rises slowly below a certain field level

which, for convenience, we call Htr ("tr" for transition from one
behavior to another).
(2)

Above Htr the surface resistance rises very sharply, particularly

near the thermodynamic critical field, He (shown by an arrow in the
figure.)

-58-

10------------~~----~----~~----~----~

...........

,... 0

Surface:

Thickness:

6fL

Temperature:

4.2°K

~4
..........
>C
CG

::z::

..........0

.,...

(( Hl.96

max

200

300

400

500

600

Hmax(Gauss)
Fig. 3.7 A typical example of the reciprocal of Q0 , normalized to
its low field value, is shown as a function of the magnetic
field, Hmax' at the constricted section of the center conductor.
The solid line shows a fit of the form aHn
to the experimental
data below the transition field, Htr"
max

-59In order to find the possible causes of the above behavior,
the surface resistance as a function of the field level was studied
for lead surface thicknesses ranging from 0.3 to 14 microns.
tails of this study are given in Appendix B.

The de-

All the observations

made in this study point to the following conclusions:
1)

There is a field at which the increase in surface resistance

changes its character.

We call this field Htr and it was observed for

all the surfaces examined.
2)

Below Htr the increase in surface resistance is approximately pro-

portional to the square of the rf field level.

This portion of the

increase in surface resistance seems to be caused by heating at local; zed defects.
3)

Above Htr surface structures of the electroplated lead surface

(protrusions and spikes) are most likely to be responsible for the
sharp rise in the dissipation.

Also this rise was found to be approx-

imately exponential in the magnetic field level.
These conclusions are necessarily qualitative because of the
practical complications involved in the experimen~ such as surface
structures of the lead surface and a heating effect.

In view of the

experimental difficulties, as well as the absence of any reasonable
theory for comparison, this topic is still largely an open question.
More work will be necessary to determine the intrinsic dependence
of surface resistance upon rf magnetic field.

-60-

3.3.3

RF Critical Magnetic Field of Lead at 205 MHz
As the resonator is driven to a higher field level, eventually

a point is reached where a small fraction of surface area is driven
normal.

Owing to the increased joule dissipation within such an area,

once it is created the normal area quickly spreads by heating the surrounding area above the transition temperature.

This process shows

itself as a sudden reduction in the resonator Q0 on a time scale much
shorter than the characteristic decay time of the superconducting
resonator.

This entire process is generally called the 11 magnetic-

thermal breakdown 11 and the field at which this process is initiated is
called the breakdown field or the rf critical magnetic field.
For the present half-wave resonator this breakdown process was
observed to be initiated at a well defined field level where the resonator Q decreased typically by a factor of 100 or more within about

10 microseconds.

The resultant Q0 after the breakdown was found to be

such that the entire constricted section of the center conductor (see
Fig. 3.1) was normal conducting.
All the measurements at high field levels were obtained by pulsing
the incident power so as to avoid an excessive heating of the cavity
surface.

Consequently the field level within the resonator responds in

time as shown in Fig. 3.8a, where the breakdown process can be seen as
a break in the slope of the field level within the resonator. followed
by a steep_negative slope.

The surface resistance just prior to the

breakdown was found to be equivalent to approximately 1-3% of the area
of the constricted section of the center conductor being normal.

r--

-61Incident Pulse On - - - - - - -

(a)

(b)

Onset
t =O

t= l O

5 .U s ec/ division

.u s e c

Vn ·tical Scale : 65: Gauss Fu ll Scale for bot h (a) and ( h )
Data : 0 . 3 micron Pb Surface at 4 . 2 ° K

Fig . 3.8 The envelope of t he rf f ield level within the resonator is
shown as seen on an oscilloscope . (a) When the incident power
i s kept constant the f i e l d l evel within the resonator reaches the
breakdown f iel d at whi ch all the stored energy is dissipated
quickly and the entire process repeats itself . (b) The cavity
field level irllnedia t el y fo llowing the onset of breakdown on a
much finer time s cale tha n in (a ).

-62-

Once the transition is initiated, the normal region spreads by
thermally driving the surrounding area normal at the expense of the
energy stored in the cavity.

Thus by measuring the cavity Q0 as a

function of time, it was possible to infer how much area was being
driven normal at any instant of time following the initiation of the
breakdown.

To do so we look at the time dependent Q0 of the cavity

(Eqs. (3.8) and (3.9))*
W(t)
Qo(t) - w P (t) - dW(t)/dt
net

(3.25)

where w is the resonant frequency, W(t) is the total energy content
of the cavity, and Pnet(t) is the net power flowing into the cavity.
Here all the quantities are assumed to have been averaged over an ,rf
period as before.
In the photographs of Fig. 3.8 the envelope of the rf voltage
at the pickup probe is shown, and this voltage is proportional to
(W(t)) 112 (see Eq. (3.10)). So the cavity Q (t) in the above equation

can be determined by measuring the pickup voltage as a function of
time on such photographs as shown in Fig. 3.8b.

Pne t(t) can also be

measured in the same manner.
When the field level reaches the critical field, we have dW/dt=O
(this is just an operational definition of the rf critical field in the
present measurement).

Once the breakdown is initiated

dW/dt

in the

denominator of Eq. (3.25) takes over Pnet rapidly, and in a short time
Assuming that the pickup probe is weakly coupled. For all measurements of this type the power picked up was much less than 1% of Pnet
or dW/dt.

-63<~ lO~sec)

the cavity Q0 attain~ the value corresponding to the situa-

tion where the constricted section of the center conductor is normal
conducting.

The asymptotic approach to this Q0 value is shown as a

function of time in Fig. 3.9.

Since the high field region is well

localized within the constricted section which is thermally "anchored"
to the massive tapered section, the normal region does not spread beyond
the edge of the constricted section.
The rf critical field, H~f(T), was measured for electroplated
lead films of different thicknesses (0~3, 1.45, 6.0 and 14 microns) at
different temperatures.

The results are shown in Fig. 3.10 for two tem-

peratures where H~f is shown as a function of the incident power, Pine'
used to drive the resonator.

By increasing the incident power, the time

needed to reach a given field level could be shortened, thus minimizing
the possibility of heating the entire wall of the resonator.
As can be seen in the figure, H~f was found to depend on both the
incident power and on the thickness of the plated layer.

The dependence

on the incident power is relatively small compared with the dependence
on the thickness and could have been caused by a localized heating of
the lead surface where the initial normal area was nucleated.

From the

observed change in Hrf as a function of P.
and the temperature depenc
1nc
dence of H~f,we can estimate the temperature rise, due to such local
heating at the nucleation site, to be 0.35°K for 6 and 14 m1cron surfaces and O.l0°K for 0.3 micron film, in the range of Pine used in the
measurement.

Thus it is relatively unimportant for thinner films.

-64-

q ..__Onset of Breakdown
JOG

= 612 Gauss

Oo
10 5

~',~

-o-

Q 0 of Partly Normal Cavity

10 4

Time (I0- 6 sec)

Fig. 3.9 The resonator Q is shown as a function of time immediately
following the onset of breakdown. The resonator Q0 decreases until
the entire constricted section becomes normal. No~e
the change
in time scale at 10 ~sec.

-65-Absolute

950
900

-Relative
Error Bar
at
900 Gauss

T = 2.2 °K

o--o

850
Hrf

800

[Gauss]

He { 0 o K)

750

-O-m

Hc{2.2 °K)

600

650

pine (Watt)

800
750

Chemically
Polished
Surface

T=4.2 OK

0 {2"'3fL)

700
Surfaces as
Plated

Hrf

650

[Gauss]

-----

__.--x

0.3,.,.

1.45fL
6.o,.,.
14.0fL

x- ~-~
-------500

He (4.2 °K)

Pine (Watt)

Fig. 3.10 The critical rf magnetic field, Hrf, is shown as a function
of the incident power used to drive thechalf-wave resonator. The
measurements were made for various thicknesses of lead layer. The
highest fields were attained for a surface prepared by chemical
polish method (see text).

-66-

On the other hand, the dependence of H~f on the thickness of
the lead film was more pronounced, increasing as the thickness decreased.

This dependence was attributed to the surface structures

on the lead ~urface (protrusions and sharp corners of the crystal
grains of lead) which tended to develop as the thickness of the electroplated layer was increased.* Since the surface magnetic field could
be locally enhanced at such surface structures, the field at which the
initial normal area was created would be lower for the surfaces with
more surface structures, i.e., for thicker films.
The above interpretation was later verified when a new surface
preparation technique involving a chemical polishing of the lead surface to remove the surface structures was developed in this laboratory.**
The lead surface prepared in this manner was found to have superior
field emission properties compared with ordinary electroplated surfaces
(Dl)***, which indicated that there were many fewer sharp corners and
spikes.

The corresponding rf critical fields were improved by a signi-

ficant amount as can be seen in Fig. 3.10, where the highest H~f at a
given temperature was obtained for a chemically polished surface.

*This point is also discussed in Appendix B.
**This preparation technique was developed by G. J. Dick in the low temperature physics group at Caltech.
***The surface electric field at which field emission electron loading
began was improved from 15 MV/m to above 25 MV/m by this preparation (Dl).

-67As discussed in Sec. 2.2.4, there is a possibility of magnetic superheating in the transition from the superconducting-tonormal state in rf magnetic fields, i.e., the superconducting surface
can remain in a metastable Meissner state above the thermodynamic
critical field.

Throughout the present study for lead, various

amounts of superheating were observed.

In Fig. 3.10 the thermodynamic

critical fields at the respective temperatures are shown by the dashed
lines.

For example, at 4.2°K the rf magnetic field could be as high

as 40% above the thermodynamic critical field.
In order to study further this rf superheating behavior in lead,
the rf critical fields were measured between 2.2°K and Tc.

The re-

sults were then compared with the expected superheating field deduced
in the following manner.

As discussed in Sec. 2.2.3, the superheating

field was previously evaluated (M5) as a function of the GL parameter
(see Fig. 2.1).

Smith et al. (S6) experimentally determined the GL

parameter, K , of lead by the observation of supercooling.* They found
that the K for lead varied from 0.515 at 0°K to 0.240 at Tc' and can
be expressed by the empirical relation
K(t) = 0.514 - 0.274 t 2
where

for Pb

is the reduced temperature.

Then, combining the above GL parameter for lead and the results
of the one-dimensional GL theory, we obtain the expected superheating

*The (magnetic) supercooling is a persistence of the normal state below the thermodynamic critical field in a decreasing magnetic field.
Experimentally it has been found to be readily observable, in contrast to the case of superheating. Supercooling was also analyzed
theoretically by Ginzburg (G4).

-68field, hsh' as a function of temperature.

This field and the measured

rf critical field are shown in Fig. 3.11 where both fields are normalized by the thermodynamic critical field, i.e., hsh (=Hsh/Hc) and
hrf = (Hrf/H ).*
c c
In this figure, the rf critical field can be seen to have a very
similar but somewhat stronger temperature dependence than the superheating field.

The rf critical field tends to approach the superheat-

ing field at higher temperatures (near Tc, however, the errors in h~f
due to uncertainty in the temperature determination make the absolute
comparison between herf and hsh difficult). The experimental observation of the expected superheating in de field for lead has been found
to be generally difficult, presumably due to surface defects** which
would ease the nucleation of the normal phase.

The present behavior

of h~f may have been caused by the flaws on the surface of the electroplated lead su~h as grain boundaries and surface structures, which
lowers the field at which the normal phase is nucleated.

Also, there

is a possibility that the nucleation of the normal phase in different
dimensional forms (to be discussed fully in Chapter 4) gives rise to
a stronger temperature dependence of h~f compared with hsh"

(In the

language of Chapter 4, the present rf results for lead fall between
the plane nucleation field and the line nucleation field.)
All the high field measurements were performed by pulsing the

* In the normalization of the rf critical field, the form H (T) =
H0 (1-(T/Tc)2), was used, together with the accepted valuescof
H0 = 803 gauss and Tc = 7.l8°K.
**H. Parr (private communication ). Even when the superheating fields
in lead were observed (S6,S7), they tended to have somewhat erratic
temperature dependence at lower temperatures.

-69-

he vs t

2.0

Surface: Pb
Thickness: 2-31-'
1.8

(Polished)

zhsh=

1.6

Hsh

1.4

Hrt

Hi~

He
1.2

~?£H2

1.0

o.a----~--~--~--~--~--~--~--~--~--~

0.2

0.4

0.6

0.8

1.0

T/Tc
Fig. 3.11 The measured rf reduced critical fields of lead are shown
as a function of reduced temperature. The solid curve shows the
expected superheating field deduced from the temperature dependence
of the Ginzburg-Landau parameter of lead measured by Smith et al.
(Sl).

-70incident power flowing into the cavity.

Below 4.2°K the temperature

was fixed by regulating the helium bath temperature.

Above 4.2°K no

such regulation was available and the resonator was freely warmed up
at the rate of approximately 0.35°K/hr.

Since typical critical field

data could be obtained in less than 5 min, the temperature drift
during the measurement was kept below 30 m°K.

The temperature of the

resonator above 4.2°K was determined by two carbon-resistor thermometers * located at the top and bottom of the outer conductor.

The two

thermometers generally indicated the temperature within 50 m°K so
their average value was used as the temperature of the resonator.
Overall error in temperature above 4.2°K was then estimated to be less
than O.l°K, which did not give rise to an excessive error except very
close to the transition temperature as may be seen in Fig. 3.11.
The error in temperature may also be caused by the heat dissipation from the power density involved in driving the resonator at high
magnetic field.

The rise in temperature of the cavity surface due to

heat dissipation was estimated under a simplifying assumption that the
high field region of the resonator (constricted section of the center
conductor) was thermally anchored to the massive adjoining section
(tapered section).

Then using the thermal conductivity of OFHC copper**

and the measured power dissipation, the temperature rise was estimated
to be less than 0.5°K at 2.2°K and less than 0.2°K at 4.2°K and above.

*Allen-Bradley carbon composite resistors, lOOD, l/10 Watt.
** Thermal conductivity was estimated from the measured electrical resistivity of the OFHC copper by using the Wiedemann-Franz law. The
resistivity ratio of copper was 63.

-71-

The errors in reduced rf critical fields due to these temperature
rises were relatively insignificant in the representation of Fig.
3.11.

3.4 Conclusions
The surface resistance and the rf critical magnetic field of
lead electroplated on copper were studied at 205 MHz in a half-wave
coaxial resonator.

The observed surface resistance at low field

level could be well described by the BCS surface resistance with the
addition of a temperature independent residual resistance, Rres· Good
agreement was found between the present result and the result of previous numerical calculation, once Rres was subtracted.

The sources of

Rres are not all known but the available experimental data and other
estimates suggest that the major fraction of Rres in the present experiment was due to the presence of an oxide layer on the surface.
At higher rf magnetic field levels the surface resistance was
found to depend heavily on the surface condition of the electroplated
layer, which made it difficult to determine the intrinsic dependence
of the surface resistance on the rf magnetic field.
The attainable rf critical magnetic field of lead was found to be
limited not by the thermodynamic critical field, but rather by the
superheating field predicted by the one-dimensional Ginzburg-Landau
theory.

The observed rf. critical field was very close to the super-

heating field but showed somewhat stronger temperature dependence than
the superheating field.

-72-

Chapter IV
RF CRITICAL FIELD OF SN-IN AND IN-BI ALLOYS AT 90 and 300 MHz

4.1

Introduction and Description of Overall Scheme
In Chapter 3 it was found that the rf critical field of lead at

205 MHz approached the superheating field but was always somewhat smaller,
particularly at lower temperatures.

The disadvantages of working with

lead were:
(1) It was not possible to make accurate measurements near Tc (7.18°K)
where the GL theory was presumably more valid and comparison between
experiment and theory more meaningful.
(2) The de critical field of the same cavity surface could not be directly determined because of its geometry.
(3) Due to the high field level involved, it was necessary to use relatively high power which might have raised the temperature of the cavity
surface making the exact temperature somewhat uncertain.
Thus alternative schemes were sought by which we could look at
the rf critical field of superconductors and its relation to other critical fields, particularly the superheating field, while avoiding the abovementioned difficulties for lead.

Since the superheating is caused

mainly by the boundary energy which is a function of the Ginzburgh-Landau
parameter, K , and which changes quite drastically in the low

range

(see Fig. 2.1), it is desirable to measure the rf critical field over a
range of

K .

-73It is well established that the

of a superconductor can be

changed by alloying since the reduction in the mean free path of the
electrons causes changes in the penetration depth and coherence length
in such a way as to increase K (:::: A/F;.).

Thus by starting with a pure

metal which is an extreme type I superconductor, we can obtain a series
of alloys which span a range of

After looking into several pos-

sible metals and their alloy combinations, it was eventually decided
that two alloy systems, tin-indium and indium-bismuth, be used.

Some

of the considerations that dictated the above choices were:
(1) The transition temperature, Tc• should be lower than 4.2°K (normal

boiling point of helium) so that it would be possible to work near Tc.
(2) The magnetic properties in the de field should be reasonably well

known so that results obtained by relatively simple de measurements
could be checked against previous measurements.
(3) Forms of the samples should be such that both de and rf measure-

ments could be made on the same set of samples.
(4) A reasonable range of

should be available.

These requirements were satisfied by the above alloys as listed
below.
A11 oy Sys tern

K Range

Tc Range

Snin

. 09 - 1. 0

3.7- 3.6°K

InBi

.06 - l. 8

3.4 - 4.4°K

It should be noted that the systems span a
type II regimes.

range in both type I and

A range of K was duplicated in order to be able to

discern possible anomalies, if any, of a particular system.

-74-

Both the de and rf critical fields were measured for direct comparison on spherical samples prepared from these alloys, whereas the
GL parameter was determined independently by measuring the resistivity
of cylindrical samples of the same set of alloys.

The results show that

the measured rf critical field near T agrees very closely with the

s~perheating

4.2

field as predicted by the one-dimensional GL theory.

Experimental Method
For both alloys, concentration ranges were chosen so that the re-

sultant alloys were homogeneous solid solutions of one metal in another.
This range was

0-6 at.% of In in Sn (B5) and 0-4 at.% of Bi in In (Hl2).

About ten alloys were made for each system within these concentration
ranges.

The entire experiment consisted of three major stages: sample

preparation, measurements in de field, and measurements in rf field.
They are discussed in the following four sections.
4.2. 1 Sample Preparation
Starting metals* were first chemically etched to remove the surface oxide.

For each concentration, appropriate quantities of each metal

were weighed out on a microbalance.

They were then combined in a hot

bath of glycerin contained in a glass test tube.
was thoroughly mixed using .a glass rod.

Once molten, the metal

Then samples were prepared in

two shapes: spheres and cylinders.

* Indium (5N purity, l/8-in. dia. wire, supplied by Indium Corporation of
America, New York.)
Tin (5N purity, shots, Electronic Space Products, Inc., Los Angeles)
Bismuth (6N purity, ingot, Electronic Space Products, Inc., Los Angeles)

-75-

Spherical Samples.

The molten metal was withdrawn by suction into

one end of a long and thick-walled glass capillary tube.

Once the de-

sired amount of metal was within the capillary, the glass tube was taken
out of the glycerin bath.

When the molten metal was ejected slowly out

of the capillary tube and allowed to fall about 3 em into a bath of cold

glycerin (at room temperature), the metal formed a sphere under its own
surface tension and retained the spherical shape as it froze in falling
through the cold glycerin.

The sample formed this way was spherical

within about 5% of its mean radius.

The surface was generally smooth

except at the top end where the last stage of solidification took place,
showing either a concave ~urface or an irregular surface structure.
These defects were usually confined within a small area at the pole and
their effects were minimized in the measurement by aligning the sample
with respect to the applied field direction so that they were in the
region of minimum field.

The spherical samples were the primary samples

on which both de and rf critical fields were measured.

The typical

dimension of the spheres was 1-2 mm in diameter.
Cylindrical Samples.

For the determination of the GL parameter (to be

described in the next section) it was necessary to measure the resistivity of each alloy sample.

This was done on a cylindrically shaped

sample which was made by a similar method as above.

The molten metal

was withdrawn by suction into a thin-walled and slightly-tapered glass
capillary tube.

The metal was allowed to freeze within the tube, which

was then broken at the edge to remove the solidified cylindrical sample.
The typical sample dimensions were 1 mm in diameter and 2.5 em in length.

-76-

4.2.2

Determination of the GL Parameter by Resistivity

Gor•kov-Goodman Equation.

As discussed previously in Sec. 2.2.2, the

Ginzburg-Landau parameter, K(= A/E,}, plays an essential role in determining the magnetic properties of a superconductor.

Hhen a supercon-

ductor is alloyed with an impurity, the mean free path of the electrons
is reduced, which in turn reduces the coherence length and increases
the penetration depth at the same time.

The net effect of alloying,

then, is that K increases with increasing impurity concentration.

the present case the starting metals, tin and indium, are in the extreme type I region (K = 0.06- 0.1).

By alloying with indium and bis-

muth, respectively, they cross the boundary between the type I and type
II regions (K = l/12) and finally K up to about 1.8 which is well into
the type II regime.
The expression for the Ginzburg-Landau parameter of alloys was
first derived by Gor•kov (G8) for extreme type II alloys and later generalized by Goodman (G9).

It is now known as the Gor•kov-Goodman

equation and given simply as
K = K0 + K·1
( 4.1 )

where the GL parameter, K , of an alloy at Tc is expressed as a sum of
the GL parameter of the pure metal, K0
due to the alloying, Ki .
tivity in the normal state,

and the change in the parameter

Ki is given in terms of the residual resisp (ohm-em),

and the Sommerfeld constant for
the electronic specific heat, y (erg/(cm3deg 2 )).

-77The GL parameters of alloys determined by the above equation were
found to agree within a few percent with those obtained from various
features of the magnetization curves near Tc {K4) * for InBi alloys of
the same composition range as used in the present study.
Resistivity Measurement.

The residual resistivity of cylindrical

samples of alloys was measured between 4.2°K and the transition temperature by a conventional 4-point probe measurement with a lock-in
amplifier as a detector.

Initially, all the alloy samples, except the

most dilute ones, were found to have a broadened transition temperature,
indicating inhomogeneity within the sample.

Thus, they were subse-

quently annealed at 5-l0°C below their melting temperature (solidus
temperature) for 20-100 hrs, depending on their concentration. ** After
the annealing treatment the alloy samples showed, generally, a welldefined transition temperature above which the resistance was constant
up to 4.2°K (the width of Tc will be discussed in the next section in
connection with the Tc of the spherical samples).

For those samples

with Tc above 4.2°K (two samples at the high concentration end of InBi
alloys) the resistivity was measured while the sample was driven normal
by an applied magnetic field.

* Kinsel et al. (K4) assumed K = 0.112 for indium from the supercooling
result of Faber {F5). However, the supercooling field was later shown
to be given by Hc 3 rather than Hc 2 (58). So their value of K should
be modified to K = 0.066. This does not change their results by more
than 6%.
** Spherical samples of the same concentration were annealed at the same
time as the cylindrical samples.

-78The GL parameters of pure samples of tin and indium could not
be determined by resistivity measurements.

However, for both metals,

previous K 0 values were measured to be in the extreme type I region.
The typical values of K0 determined by supercooling observations (S7)
are

= 0.087

for tin

= 0.060

for indium

The GL parameters of the present alloys were then evaluated using the
above values of K0 and the measured values of resistivity.
tronic specific heat constant, y

The elec-

of pure tin and indium (86) * was

used in the expression instead of the y of the a11 oys. **
The K values thus determined for the present samples are tabulated
in Table 4.1 and compared with previously measured values of
4.1.

in Fig.

Very good agreement was found for Snin alloys between the present

results and the results of Smith et al. (57) as shown in Fig. 4. la. For
InBi alloys, the values obtained by Kinsel et al. (K4) show a general
agreement in Fig. 4. lb.

The present results for both alloys show a

good linear relationship between

and the alloy concentration (a: p)

as expected for dilute alloys.
4.2.3

DC Critical Fields of Tin, Indium and Their Alloys
It was desired that both de and rf critical fields be measured

on the same set of samples for a direct comparison.

For this purpose,

*Y = 1.80 ± 0.02 mJ/(mole del) for Sn and 1.60 ± .02 mJ/(mole del) for In.
** Kinsel et al. (K4) found that using the K values of the alloys did not
change the K values by more than 5% for their InBi alloys.

-79-

Table 4.1
The residual resistivities, p(4.2°K), of cylindrical samples
of Snln and InBi alloys are shown with the resulting values of the GL
parameter, K , determined from the Gor'kov-Goodman equation.
values of K0 obtained by Smith et al. (S7) are assumed.
path of electrons, t

is also listed.

The

The mean free

-80Snin Allo~s
At.% In

p(4.2°K)
]£2-cm

(a)
Ki

K(b)

Type

o. 10
0.20
0.27
0.62
1.05
1. 74
2.95
4.21

<6.82x10- 3
4. 34 X 10- 2
9.07xl0- 2
0.151
0.347
0.642
1. 31
1. 84
2.29

<.002
0.0108
0.0226
0.038
0.086
0.160
0.326
0.458
0.570

0.087
0.098
o. 110
0.125
0.173
0.247
0.413
0.545
0.657

4.96
6.26

3.00
3.65

0.747
0.9()9

0.834
0.996

~(c)

l. 89 X 10
> 12 X 10

9.040
5.430
2.360
1.277
626
446
358
273
225

InBi Allo~s
0.25
0.52
0.99
1. 43
1.49

3.25 X 10- 3

7.8x10- 4

0.423
0.907
1. 710
2.38
2.58

o. 101
0.217
0.409
0.569
0.617

0.060
0.161
0.277
0.469
0.629
0.677

1.63
1. 99
2. 80
3.42
3.94

2.78
3.36
4.60
6.06
7.08

0.664
0.803
1.099
1.448
1. 692

0.724
0.863
1.159
1. 508
1. 752

II
II
II

II
II

(a)K; = 7.5 xl0 3y 112p where y values were taken from (86).
(b)K = K + K.
where K values in (S7) were used.
( )
11
c p~ = 0.82 x l0- ~cm 2 for Sn (S7).
(d)P~ = 0.89 x lo- 11 Qcm2 for In (84).

27 X 10 4
2 '100
981
524
374
345
320
299
193
147
126

-81-

(a)

1.2
1.0

Sn In Alloys

~ 0.8
.._

2Q)

e o.6

0..
_]

<..9 0.4

o Present Results

+ Smith et. al. [s7]
o Rosenblum et. a I.

[ R 1]

At.% In
(b)
In Bi Alloys

Present Results
Kinsel et.al. [K4]
Rosenblum et. a I.

[R 1]
At.% Bi

Fig. 4.1 The Ginzburg-Landau parameter (K) of alloys is shown as a function of the solute concentration. The present values were determined
from the Gor'kov-Goodman equation using the measured resistivity of
the samples in the normal state. They are compared with previously
determined values in the limit of Tc for (a) Snln alloys by the ob-servation of supercooling (S7), and (b) InBi alloys by resistivity
as well as various features of magnetization (K4). The results obtained by surface superconductivity observations in Rl are also shown.

-82-

the spherical samples of tin, indium, and their alloys were made as
described in Sec. 4.2. 1 and annealed in the same manner as the cylindrical samples.

The de critical fields were determined by measuring

the low frequency ac susceptibility of the spherical samples in a
slowly varying magnetic field.
In this measurement the de (or slowly varying) magnetic field
was applied by a solenoid * located in the liquid nitrogen bath outside
the helium dewar as shown in Fig. 4.2 where the experimental arrangement is shown schematically.

For the measurement of susceptibility, a

small (less than 50 mgauss in amplitude), low frequency (100-1000 Hz)
magnetic field was superimposed on the applied magnetic field by means
of a drive coil.

This ac component was detected and amplified by a

phase-sensitive detector (PAR lock-in amplifier).

The amplified signal

was then displayed as a function of the applied magnetic field on the
X- Y recorder.
Typical examples of the susceptibility thus determined are shown
in Fig. 4.3 for a pure type I sample (a) and a type II alloy sample (b).
Particular features of the transition that have been found to be important in the determination of the bulk critical field are discussed
separately for both types.

Other susceptibility features involving the

surface superconductivity (Hc 3), upper critical field (Hc 2 ), ideal
supercooling field (Hsc) and ideal superheating field (Hsh) are not

*The solenoid was wound on a lathe from heavy copper wire. The field
was calculated from the known geometry and the field of an approximately infinite solenoid. An overall error in the absolute determination of magnetic field was estimated to be less than 3%.

··83-

Y-Axis
Input

RAMP
GENERATOR

X-Y RECORDER
X-Axis
Signal
Input
Output
.._--e,.....__....,..,___-4 POWER SUPPLY
------4 (Programmable)
LOCK-IN AMPLIFIER
(Internal Mode)
100 ...... 1000 Hz
Signal
Input

Reference
Output

Current Sampling
Resistor
CRYOSTAT
Liquid
Helium Bath
Liquid
Nitrogen Bath
SOLENOID

Fig. 4.2 The experimental arrangement for the measurement of low frequency ac susceptibility is shown schematically.

··84-

Indium Sphere
(Type I)

= 0.06

X"
Arb. )
( Scale

Hdc(=_g_H)
c \

Intermediate
State

In+ 0.28 at. 0/o Bi
Alloy Sphere
(Type II)
K = 1.16

X"
Arb. )
( Scale

Fig. 4.3 The imaginary component of the lo\'J frequency ac suscepti···
bility of a sphere is shown for (a) an indium sphere, and (b) an
alloy sphere (K = 1. 16) as a function of the applied magnetic
fie 1d.

-85-

discussed here.

Discussions on such characteristics may be found in

57.

Type I Case:
An ideal type I superconducting sphere is expected to remain in
the Meissner state up to an applied field, Ha, of (2/3)Hc at which it
goes into the intermediate state and does not become completely normal
until the applied field is equal to H . *

In Fig. 4.3a the imaginary

component of susceptibility (non-inductive component) for an indium
sphere is shown.

For pure samples both superheating and supercooling

were generally observed to be present to some extent.

The extent of

these metastabilities depended mostly on the exact condition of the
spherical samples.

For this particular sample they are shown by arrows

as 2/3 Hsh (superheating field) in an increasing magnetic field and as
Hsc (supercooling field) in a decreasing field.

However, there was no

difficulty in identifying the field at which the intermediate state
began, H~c(= 2/3 He for a perfect sphere), and the field at which the
sample became entirely normal, H , both of which are shown in the figure.

For other type I alloy samples. this field (H~c) was also well defined
since the superheating and supercooling tended to disappear as soon as
the pure metals were alloyed.

*This is generally known as the demagnetization effect and for simple
geometries such as an ellipsoid of revolution, the problem is effectively dealt with by the use of the demagnetization coefficient D . For
a sphere D = l/3 and the field at the equator (polar axis being along
the applied field) is given asH eq = Ha/(1-D) = 1.5 Ha (L3, p.23).

-~-

Type II Case:
In an increasing field a type II spherical sample is expected
to remain in the Meissner state up to 2/3 Hcl at which it goes into the
mixed state.

For type II alloy samples this field was identified as

the first field, H1 , at which there was a distinct change in the imaginary component of the susceptibility in an increasing field.
shown by an arrow in Fig. 4.3b.

This is

The ratio of Hcl to He was evaluated

previously by Harden and Arp (Hll) as a function of the GL parameter
K.

Thus the bulk critical field of the present spherical samples, H~c.

can be obtained by using the values of K determined by the resistivity
measurement (preceding section).
For the spherical samples of both types, the susceptibility data
were obtained in the reduced temperature range 0.9 ~ t ~ 1.0. * The
resultant critical field:,

H~c, as a function of temperature was fitted

to the parabolic expression,

to determine H~c and Tc to be used later for the comparison with rf measurements. ** The results of the fit showed that the deviation of the
Extending the measurement down to t=0.80 did not alter significantly
the determination of H~c and Tc. In the rf measurements, the temperature
range of concern was 0.8 ~ t ~ 1.0.
**The demagnetization effect of the sphere was not included in H~c, i.e.,
the actual critical field would be He= 1.5 H~c if the sample was a perfect sphere. Since the rf critical field will be defined in the same
way, the demagnetization factor drops out when H~f is normalized by H~c.

-87experimental points from the fit was typically within ~3% and no more
than 5% in the worst case.
tabulated in Table 4.2.

The values of H~c and Tc thus obtained are

They were found to be in satisfactory agree-

ment with the existing data when available: He data for InBi (K4,K6),
Tc data for InBi (K4,K6), and Tc data for Snin (S7,M7).
The transition temperature and its width for the spherical
samples were also determined by measuring the inductive component of
the susceptibility as the temperature of the sample was reduced.

The

temperatures at which the transition from the normal to superconducting
state were initiated and completed, defined as Tc 2 and Tel, respectively, were determined.

~T (= Tc

2 -Tel) was
found to be reasonably narrow, i.e., typically 2-30 m°K with only a few

exceptions.

The transition width

They are also listed in Table 4.2.

The transition temper-

atures determined by the extrapolation of Hdc data as described in the

preceding paragraph were in good agreement with these direct measure~n~.

4.2.4 RF Measurement
Once the de properties of the alloy samples were determined and
verified against the existing data, the rf measurements were carried
out on the spherical samples by using a helical resonator constructed
for this purpose as shown in Fig. 4.4.

The helical resonator was chosen

primarily for two reasons:
(1) It was possible to generate rf magnetic fields which were uniform
over the size of the spherical samples.

-88-

Table 4.2
The critical field of spherical sampless H~c(T), was expressed
as H~c (1 - (T/Tc) 2 ) where H~c and Tc were determined by a leastsquares fit.

The transition widths, 6T = Tc 2 - Tc 1,are also listed

where Tc 2 is the onset and Tel the completion of the normal to superconducting transition.

-89Sn!n AllO,lS
At.% In
0.10
0.20
0.27
0.62
1.05
1. 74
2.95
4. 21
4.96
6.26

0.087
0.098
0. 110
o. 125
o. 173
0.247
0.413
0.55
0.66
0.83
1.00

Hdc(o)**
gauss

T*
OK

Tc
OK,

Tc
OK

t.T c

188±1.4
195.5±1.4
194.6 ± 1. 2
188 ± 1.1
186 ± 1. 3
188 ± 1. 6
182 ± 1. 2
202 ± 2
200 ± 2
233 ± 2
261 ± 3

3. 715
3.695
3.685
3.676
3.653
3.643
3.623
3.611
3.613
3.614
3.626

3.715
3.695
3.685
3.676
3.656
3.641
3.625
3.608
3.614
3.614
3.610

3. 717
3. 705
3.693
3.699
3.693
3.650
3.629
3.641
3.648
3.683
3. 703

.002
.010
.008
.023
.037
.009
.005
.033
.034
.069
.093

3.404
3.408
3.463
3.586
3.703
3.720
3. 780
3.835
4.022
4.236

3.406
3.416
3.470
3.599
3. 722
3. 728
3.799
3.866
4.053
4.269
N.A.

.002
.008
.007
.013
.019
.008
.019
. 031
.033
.033

InBi AllO,lS
0.25
0.52
0.99
1.43
1.49
1.63
1.99
2.80
3.42
3.94

0.06
0.16
0.28
0.47
0.63
0.68
o. 72
0.86
1.16
1.51
1. 75

172 ± 2
181 ± 3
188± 2
204± 2
209 ± 2
206 ± 4
209 ± 2
224± 2
249 ± 2
322± 2
310 ± 3

3.405
3.410
3.466
3.593
3. 710
3.726
3. 762
3.856
4.041
4.249
4.411

N.A~**

Absolute determination ofT is estimated to be within + 4 m°K.
** de
He (0) = 3 Hc(O) for a perfect sphere.

*** Not available.

N.A.

-90-

To Dewar Top

Grounding Clips
TOP PLATE

NIOBIUM
HELIX---...-,
vr--.a...

OUTER CAN

Helix Support Rods
(Sapphire, 1/16" Dia)'~""'*--.1

Indium "o" Rings
SCALE

Fig. 4.4 General construction of the helical resonator is shown. The
resonator consists of the niobium helix, outer can and two end
plates. The outer can (copper) and the two end plates (brass) were
electroplated with lead. The schemes of supporting the sample and
the helix are shown in the expanded views.

-91-

(2) There were multiple modes in the resonator so that the measurement
could be made at different frequencies.
The resonator consisted of a helical wire supported by two sapphire rods, an outer can and two end plates.

The helix was wound from

a mechanically polished niobium wire of 2 mm diameter.

It was then
annealed in a quartz tube under pressure of less than 2 x l0- 5mm Hg for
three hours at temperature 1200-1350°C by means of induction heating in
order to remove the internal stress.

After annealing treatment,

the

helix was chemically etched in a mixture of HF and HN0 3 acids to expose
a fresh surface.

Then the helix was secured to the sapphire rods

(1/16-inch diameter) and Teflon spacers with a fine nylon line (fishing
line) as shown in the expanded view within the figure.

The outer can

and two end plates were made of copper and brass, respectively, and
all the inside surfaces were electroplated with lead (Sec. 3.1.2). The
current carrying vacuum seals were made with indium non rings at the
top and bottom of the can as described previously (Sec. 3.1.3).
For the low temperature measurement the spherical sample was attached to one end of the sapphire support rod (1/8-inch diameter) with
a small amount of varnish (see expanded view in the figure).

This was

done by spreading a thin layer of GE varnish (No. 7031) on the end face
of the sapphire rod and.then placing the spherical sample on it.

The

sample rod was then attached to a long stainless steel rod leading up
to the top of the dewar system so that the sample could be moved into
and out of the resonator at will.

-92Exactly the same rf instrumentation and cryogenic apparatus
were used as in Sees. 3.1.3 through 3.1.5.

However, a new calibration

scheme was devised in order to determine the rf magnetic field at the
site of the sample.

This was necessary since the exact calculation of

the field distribution in such a helical resonator* would be extremely
difficult if not impossible.
Calibration
Presently we are interested in the determination of the rf magnetic field level at the site of the spherical sample placed within the
resonator.

This can be related to the measurable quantities of the

resonator and sample as follows.
When a metallic sphere (in the normal state) is placed in an rf
magnetic field, the field is shielded from the interior of the sphere by
the skin effect.

The current and magnetic field distribution on the

surface of the sphere can be calculated with the assumption that the
sphere is diamagnetic as far as the rf field is concerned.

This gives

the surface field as
Hsurface = l2 Hs sin e sin(wt)

(4.2)

where Hs is the applied magnetic field at the site of the sample, and

*For a calculation of field distributions in a helical resonator based
on a simplified model, see, for example, S9.

**This is exactly the same problem as a superconducting sphere in a
static magnetic field, except the rf skin depth is replaced by the
superconducting penetration depth (see, for example, L5, p.34).

-93-

is the polar angle (the polar axis being along the direction of the
applied field).

Then the time-average power dissipation over the en-

tire sample, Ps, can be evaluated as

{4.3)
where

is the radius of the spherical sample and Rn is its normal

state surface resistance.

In the above derivation it is assumed that

the rf skin depth is much smaller than the radius of the sphere* .
Recalling the definition of the quality factor of the resonator
( Eq. ( 3. 9))

{4.4)
where Pdiss is the power dissipated in the resonator itself, W is its
energy content and w0 is the resonant frequency.

Generalizing it for

a loaded resonator (in this case with the spherical sample) we have

Pd.
+ ps
= --=-l~S::...;:S:..,...,...._.::..

= Q-1
+~
wW

(4.5)

where QL is the quality factor of the loaded resonator, w is its
resonant frequency ** , and Ps is the power dissipated due to the spherical

*The rf skin depth for the present samples was less than 14~, whereas
the radius of the samples was 0.5-1.0 mm.

**The shift in resonant frequency, 6w = w - w , due to loading by the

sample is assumed to be small. In the experiment 6w was ~0.3 MHz at
90 MHz, most of which was due to the sample support rod.

-94sample.
Combining Eq. (4.3) and Eq. (4.5) we have

(4.6)
where g is defined in an exact analogy with Eq. (3.2).
All the quantities in g can be determined experimentally:

and Q0 from the measurement of the cavity decay time (Q = wT) and Rn
from the de resistivity, p , of the same alloy samples by using the
expression for the classical surface resistance, Eq. (2.4) *
R = (rrf ~ p) 112

( 4. 7)

According to Eq. (4.6), the measurement of the resonator Q when
a spherical sample is placed anywhere in the resonator gives the local
magnetic field at the site of the sample.

Then, by moving the sample

along the axis of the helix, the axial magnetic field profile can be
determined.

The magnetic field distribution thus obtained is shown in

Fig. 4.5 where the field can be seen to be sufficiently uniform over
the dimension of the sample diameter (also shown) for the two modes
(first and fourth modes) used in the measurement.

Also the measured

field distribution enabled us to identify the positions of the peaks in
magnetic field where the g-factor in Eq. (4.6) was to be determined.

* For the majority of the alloy samples used, the rf skin depth, o , was
much larger than the mean free path of electrons,~ , (typically
100 ~ o/~ ~ 1 ,100). Therefore the use of the classical expression was
well justified.

-95-

1.0
I ST MODE
(90.6 MHz)

-¥J.

i+.~

~~~------+- MAXIMUM -+------t

't·~+"+

Smax

4TH MODE
'+,(300MHz) f
~,

\ 1

-0.5
Top

End
of

I+

SAMPLE
DIAMETERt
(~ 2mm)
~r--

° Middle
of
Helix
(Approx.)

.o _ _ _.........__ __.____ _
Plate

lt \

0.5

Heli;l

....__~

SAMPLE POSITION (Inch)

Fig. 4.5 The magnetic field distribution along the axis of the helix
was determined by moving the spherical sample along the axis and
measuring the resonator Q at each position. The magnetic field along
the axis is normalized by the maximum field at which the g-values were
determined (shown by arrows). Here the first and fourth modes \'ihich
were used in the rf critical field measurements are shown.

-96-

Combining Eqs. (4.6) and (4.7), g is expressed as

(4.8)
where

and

are respectively the loaded and intrinsic decay

times of the resonator.

The measurement of TL when the sample was lo-

cated at the peak of the magnetic field distribution determined g
values for the different modes.

For the first mode (90.6 MHz), g-values

were determined for 20 alloy samples, all of which agreed within ± 6%.
Thus their average was used throughout the measurement as g1

Similarly, for the fourth mode (300 MHz) with four samples we obtained

Once g-factors are determined as above (noting that Eq. (4.6) is
equivalent to Eq. (3.2) for the half wave resonator) the rest of the
measurement proceeds in exactly the same manner as in Sec. 3.2.4.

Then

the magnetic field at the site of the sample, Hs' is related to the
measured power at the pickup probe, Ppu' as
H = (ag p )1/2
pu

(4.9)

where the coupling constant a is determined in the same way as before.
The resulting errors, ~ , in the determination of the magnetic
field at the sample site were estimated to be

-97<

10%

for the absolute error in Hs at a given frequency, and

for the relative error between different samples at the same frequency.
Also the relative error in the determination of magnetic fields at two
different frequencies for a single sample was estimated to be

Sources of error are the same as those discussed in Sec. 3.2.6.
RF Critical Field Measurement
The determination of the rf critical field was carried out in
the same manner as in Sec. 3.3.3.

The quality factor of the resonator-

sample system when the sample was normal was dominated by the dissipation due to the sample, whereas below Tc it was determined by the
intrinsic dissipation inside the resonator itself* (the intrinsic Q of
the resonator was typically 2-4 x 10 7 ). For the measurement of the rf
critical field, the incident power was pulsed as before in order to
avoid excessive heating of the sample.

Upon reaching the critical field

of the sample, the resonator Q was reduced by an amount proportional to
the dissipation due to the normal sample.

This change in Q at the tran-

sition depended on the sample and varied between a factor of about 10

*The loss due to the sample essentially vanished within a narrow temperature range below Tc, reaching less than a few percent of the normal
state value at 0.99 Tc.

-98-

and 100 {the alloy samples being more dissipative than pure samples of
the same size).
The observed pulse shapes are shown in Fig. 4.6 where the envelopes
of the rf voltage at the pickup probe {proportional to the field level
within the resonator) are shown.

When the rf magnetic field at the site

of the sample is small, the sample stays in superconducting state
throughout the pulse duration.

In this case the resonator is driven by

the incident power, while the incident power is on, and after the power
is off it dissipates the stored energy with the intrinsic characteristic
time of the resonator itself as shown in the upper trace of Fig. 4.6a.
As the incident power is increased, eventually a point is reached where
the rf critical field is reached at the equator of the sample and the
entire sample is driven normal by thermal propagation as can be seen in
the lower trace of Fig. 4.6a.
Further increase in the incident power results in a shorter driving time to reach the critical field as shown in Fig. 4.6b.

Looking at

the breakdown field on a much finer time scale (Fig. 4.7a and b) the
entire process of the "magnetic-thermal,. breakdown can be seen to take
place on a time scale much shorter than the characteristic decay time of
the resonator.

Thus this field at which there was a break in the slope

of the energy content of the resonator was operationally defined as the
rf critical field H~f.
As mentioned previously, H~f was defined as the magnitude of the
rf critical magnetic field at the site of the sample (i.e., the demagnetization factor was not included so that the actual magnetic field at

-99-

I ncider.L Power Or.

(\ 'tJ )

Time Sc~le = 20 msec / div

Fig. 4.6 The envelopes of the rf voltage proportional to the field
level with in the resonator are shown. (a) The upper trace shows
no breakdown. As the incident power is increased, the rf
critical field is reached where the sample becomes normal as
shown in the lower trace . (b) Further inc rease in the incident
power results in a shorter driving time to reach the rf critical
field.

-·100I n cident Pmrer On

:' d. )

2 mser / div

ms c·c / eliv

Fig . 4.7 The envelope of the rf f i eld le vel contained in the
res ona tor is shown. {a) The break in the initi al slope of
the reson ator field level determines the rf criti cal field at
which the sample i s dri ven normal. {b) The en tire process of
brea kdow n can be seen to take place on a time scale much
s ho,~ter than the decay time of the resonator wi th the sample
in t he normal state {the trailing edge in (a)).

-101-

the equator of the sample at the time of breakdown would be 1.5 H~f
if the sample were a perfect sphere).

Since H~f was normalized by H~c,

which was similarly defined, there was no need to determine the demagnetization factor of each sample,* thus enabling the direct comparison
between Hrf and Hdc to be made.

*The orientations of samples were kept the same in both measurements.

-1024.3

Experimental Results and Discussion
For every sample of tin, indium and their alloys the rf critical

fields were typically measured in the reduced temperature range
0.8 ~ t ~ 1.0.

It was found that the following form for the reduced

rf critical field displayed the temperature dependence very clearly:

(4.10)

H~c and Tc values were determined for each sample* in the preceding
sections.
4.3. 1 RF Critical Field of Pure Tin and Indium
Before proceeding to the details of experimental results, a simple
extension of the boundary energy argument (Sec. 2.2.2) is proposed for
different

dimensions

of nucleation of the normal phase.

In the one-dimensional formulation of the GL theory, the superheating field defines the field at which the order parameter (the density
of superconducting electrons) is driven to zero at the surface of a
superconductor filling a half-space.

This can be pictured as a nuclea-

tion of the normal phase in the form of a plane at the surface. In this

* For the spherical samples of pure indium, because of the difficulty of
handling to be described in Sec. 4.3. 1, H~c was determined for two out
of four samples used in the experiment. They agreed within 2% and
their typical value was used for all four samples of indium.

-103-

case we previously obtained (in Sec. 2.2.2) the superheating field
for type I superconductors by balancing the two contributions to the
free energy, i.e., the diamagnetic energy, A(t ~ 0 H 2 ), and the loss

in condensation energy, ~( 2
~oHc), so that the boundary energy van-

ished at H (:: Hsh) such that
AH 2 = ~ H2c
or,
Hsh

~ plane

- IA =

(4.11)

It(

This energy balance can be extended to other dimensional forms
of nucleation such as a line and a point.

The line nucleation can be

pictured as a vortex-like nucleation where the normal phase is
created in the form of a line lying at the surface.

In this case the

magnetic field distribution and the order parameter variation are centered about the nucleus and are semi-cylindrical with their characteristic lengths.

Then the diamagnetic energy is proportional to
00

J [H(r)i 2rrp dp
00

= J H2 e- 2p/A 2rrp dp

where the field distribution was assumed to be exponential in the

-104radial coordinate

p •

Similarly, the loss in condensation energy

is proportional to

Then balancing the above two contributions we obtain

~lin~

a:§_=l

(4.12)

A similar argument applies to the case of the nucleation at a

point where the magnetic field and the order parameter distributions
are semi-spherical and centered about the point of nucleation so that

~· jpoint

l/K3/2

(4.13)

It will be argued later that the proportionality constants in the above
equations may be determined near T , where the effect of localized
nucleation becomes weak due to the divergence of the order parameter
near Tc·
In the limit of low

K (extreme

type I) the result of the one-

dimensional GL theory (01) for the superheating field (the plane
nucleation field in the present scheme) is (Eq. (2.30))
Hsh

0.841

(4. 14)

This derivation depended on the assumption of the local relation between

-105the current and the vector potential {Eq. {2.21)) so that it is valid
only near Tc where the penetration depth, AL{T),is much larger than
the electrodynamic coherence length, ~ 0 • *

For tin and indium,

empirical values of ~ 0 are 23ooR and 44ooR, whereas the penetration
depth at 0°K, AL{O) are 510R and 640R, respectively {L3, p.73).

Thus

in most of the range of the present experiment (0.8 ~ t ~ 1.0) both
metals are in the anomalous limit, i.e., ~ 0 >> AL(t) = AL(0)/(2(1-t)) 112
except very near Tc {~ 0 ~ AL(t)

t ~ 0.97-0.99).

In this situa-

tion where the non-local electrodynamics is important, the range of
validity of Eq. (4.14) may be extended phenomenologically by using the
penetration depth derived microscopically in the extreme anomalous
limit as obtained by BCS (Bl) for diffuse scattering at the surface

(4.15)
where AL(T) is the London penetration depth and J{O,T) is the function
introduced by BCS varying smoothly from 1.0 at 0°K to 1.33 at Tc.
For the present temperature range AL(T) can be approximated by
(T4, p. 113)
(4.16)
so that

*This condition (SlO, p.27) restricts the applicability of Eq. (4.14)
for tin and indium in the reduced temperature range of
.01-.03.

(1 - t) <<

-106-

A ( T)

(0) 1/3
= _ __;;.:______,~ ( ~o(1 AL- t)
(4rr J(O,T)) 1

31/6

(4.17)

We also have the expression for the GL coherence length for a pure
meta 1 as

~(T) = 0. 74 ~i(l - t) l/ 2

(4.18)

Then using the above two equations in the definition of K , we obtain
A00 (T)

2 3
-= ~(T) = 0 . 635(A L (0)/~o ) 1 (1- t) l/

(4.19)

so that the superheating field in Eq. (4.14) is now given by

Hshl

~plane

~ l.0 4 K-l/3(l- t)-1/12

where the expression for

(4.20)

in a pure metal at Tc (G2), Kp =

0 was used. The same expression was obtained previously

0.96AL(0)/~ ,

(57) in the framework of the one-dimensional GL theory, where the
numerical constant turned out to be 1.36 instead of 1.04.

The above

temperature dependence of the superheating field was previously observed experimentally (57) for a collection of small spheres (1-10
microns in diameter) of tin and indium.

In the same experiment thl:

value of the numerical constant in Eq. (4.20) was found to be~ 0.9.
For the case of line nucleation, using the expression for Koo in
Eq. (4.19), we obtain the temperature dependence of the superheating

-107-

field as

a: _1 a:(l-t)-1/6

He hine

(4.21)

and similarly for the point nucleation

.. a:

j pol nt

1n a: (1-t)-1/4

(4.22)

The rf critical fields were measur~d for several samples of pure
tin and indium spheres 1.5-2 mm in diameter.

It was found that their

rf critical fields were much higher than their bulk critical fields by
a factor of 2-3.5, depending on the temperature, and increasing toward
the transition temperature.

In the vicinity of Tc the observed criti-

cal fields were very close to the superheating fields predicted by the
one-dimensional Ginzburg-Landau theory.

From the observed temperature

dependence of the rf critical field, a line nucleation is inferred for
both tin and indium.
Tin

The rf critical fields were measured at 90 MHz for two spherical
samples of tin.

The reduced rf critical field, h~f = H~f(T)/H~c(T}, is

shown as a function of the reduced temperature, T/Tc, in Fig. 4.9.

The

temperature dependence of h~f was found to be described very closely by
that of the superheating field for the line nucleation (Eq. (4.21)).
Thus it was inferred that the nucleation of the normal phase in the form
of a line was responsible for the transition in the rf field. This type
of nucleation may well be caused by inhomogeneities at the surface, such

-108-

r-------.------------r------------.-----------~4.0

h~f vs t at 90 MHz
3.5
Tin Spherical Sample

Tc = 3.715 °K
hsh (S7)

Plane Nucleation

----------

..._-hi~ {S7)

(1-t)l/12~ ................

---------

h~~ {F5)
h~h (57)
hsh (F6)

h~h(F5)

Line Nucleation

2.0

0.85

0.90

0.95

T/Tc
Fig. 4.9 The reduced rf critical field of tin is shown as a function
of the reduced temperature. The solid line is a least-squares fit
for the line nucleation model. The upper curve represents the de
superheating field (plane nucleation field) observed by Smith et
al. Previously observed de superheating fields and the expected
superheating fields (see text) are shown by arrows on the right hand
side.

-109as grain boundaries which would ease the nucleation of the normal
phase. * For the present spherical samples only moderate amounts of
superheating in de field were observed (up to 1.2 He for tin and up
to 1.6 He for the best indium sample), which suggested that nucleation centers were actually present in some form in these samples.
The data of h~f as a function of the reduced temperature were
fitted to the following expression for the line nucleation
hrf = C/{1 _ t) l/6
with one adjustable parameter C.

(4.23)

The result shows a good fit as can

be seen in Fig. 4.9.

The upper curve in the same figure represents the
de superheating field (plane nucleation field tt (l-t)-l/l 2 ) observed by

Smith et al. (S7) ** for a collection of tin spheres.

The lower curve
shows the temperature dependenGe for a point nucleation, (1-t)-l/ 4 ,

where its coefficient is adjusted, for the purpose of illustration, so
that it agrees with the other two experimental curves at t =0.99.
The present rf results, h~f, can be directly compared with the
previous experimental values of de superheating fields, hsh (= Hsh/Hc),
that have been observed.

By doing so, it should be possible to deter-

mine, at least empirically~ if the plane nucleation limit is reached
near Tc in the present rf experiment.

For a collection of tin spheres

In fact, previous observations suggest that the ideal (the plane nucleation) superheating could be observed only for single crystal
samples (G5,F5) and for spherical samples of small diameter where
surface flaws are avoided (S7,F6).
** This line is the same as the fit to their data within a few percent.

-110of 1-10 microns in diameter, Smith et al. (S7) observed a superheating
up to
at

0.99

The temperature dependence they observed was that due to the plane
nucleation. Feder and Mclachlan (F6) also measured the superheating in
single spheres, typically 10 microns in diameter, and found
at

t = 1 by extrapolation

For the purpose of comparison, the expected superheating field
(for plane nucleation) may be deduced from Eq. (4.14)*

Hsn

-1/4 K-1/2

by using the GL parameter, Ksc' obtained by supercooling observations.
Previous supercooling observations give Ksc = 0.087 (S7) for a collection of small spheres and Ksc = 0.097 (FS) for cylindrical samples of
single crystal so that the expected superheating fields are

and

*At the time of writing a preprint was made available (Pl3) in which
an analytical expression for the superheating field correct to second
order in K was evaluated as
(4.26)
H h/H = 2-l/ 4 K-l/ 2 (1 + 15 /2 K/32)
The expected superheating fields using Ksc in this equation are shown as
h~~(Ref.) in the figure.

-111-

All the above values are shown by arrows in Fig. 4.9 where the
rf critical fields near t = 0.98-0.99 agree very closely with the previously observed superheating fields and the expected superheating
fields deduced from Ksc

Therefore it may be concluded that the present

rf critical field reaches the plane nucleation limit at t=0.98-0.99.*
A possible explanation for this behavior (h~f approaching the ideal
hsh) is that the line nucleation is made ineffective as the GL coherence length, ~(t), becomes larger than a typical size of nucleation
centers (~(t) ~ 2 microns at t = 0.99 for tin).
Under the above circumstance, we may impose a condition that
the line nucleation field approaches the plane nucleation field so that
equating Eqs. (4.23) and (4.20) we obtain
(4.?4)
where ti is the temperature at which the line nucleation becomes ineffective and is estimated to be 0.98-0.99 for the present case.

Then

using the value of KP obtained by a previous supercooling observation
(Kp = 0.0926, F6), we can evaluate Cas

c = 1.57- 1.66
which compares favorably with the value of C = 1.46 determined from
the fit to the present experimental data.

*Above the temperature of 0.99 T , because of the normalization by
Hdc(T) = Hdc(l-(T/T )2 ) used incobtaining hrf, the error in hcrf rapidc
ly increases due to the uncertainty in temperature measurement (~1 moK)
and the uncertainty in the absolute T determination between different
low temperature experiments (6Tc ~ 4 m°K due to the hydrostatic temperature variations).

-112Indium
Four spherical samples of indium, typically 2 mm in diameter,
were used in the rf critical field measurements.

The results showeG

essentially the same temperature dependence as for tin so that it was
inferred again that the line nucleation was responsible for the phase
transition in the rf field.

Since indium is very similar to tin in

terms of superconducting parameters such as the electrodynamic coherence length and the penetration depth, all the preceding arguments used
for tin apply to indium with no significant modifications.
The results on all four samples are shown in Fig. 4.10.

The tem-

perature dependence was found to be the same for all samples. However,
the absolute magnitude was somewhat dependent on the exact condition
of the surface of the sample, which was not unreasonable since the normal
phase would be nucleated at the 11 Weakest 11 spot on the surface.

Ini-

tially it was found that the surface of indium samples could be easily
scratched in the process of preparation for a low temperature experiment,
even though a soft camel-hair (or nylon) brush was used in their handling.

For such a sample with obvious scratches on the surface, the rf

critical field (the lowest line in the figure) was found to be slightly
lower than that of a sample which was handled as little as possible
before the low temperature experiment (the second line from the bottom).
Eventually a chemical etching solution for indium* was found to
remove the scratches on the surface, leaving a shiny, very smooth surface.

This process improved the critical field by about 15% on the

*A mixture of one part of H 0 (30%), one part of HCl and three parts
of H o by volume.

2 2

-113-

4.0
h~f

vs t at 90 MHz

Indium Spherical Samples

h~~(F5,PI4)

(typdia::::::2mm)
Tc = 3.405 °K

h~h (S7)
he 2 (F5)

* { ~} Chern ically Etched

sh
hsh (S7)
hsh (F6)

h~h (F5)

Line Nucleation

H~f

1.78
( 1- t) 1/6

Hgc
2.5

2.0

Sample As Made

0.85

0.90

0.95

T/Tc
Fig. 4.10 The rf critical fields of four spherical samples of indium
at 90 MHz. Upper three curves are for chemically etched samples.
The solid lines are the fit to the experimental data for the line
nucleation model.

-114-

same sample as shown in the same figure.

Once the indium samples

were prepared in this manner, their results showed that the critical
fields were within ±5% of each other.as can be seen for three chemically etched samples (upper three curves).
The present results are again compared with the past measurements in exactly the same manner as for tin.

Previously observed

superheating fields in static magnetic field are
at

= 0.99

for a collection of small spherical samples of 1-10 microns in diameter
and
at

t = 1.0 by extrapolation

for single spheres of typically 10 microns in diameter.

Also, the

expected superheating fields deduced from the supercooling data are

for a collection of small spheres (Ksc = 0.060) and

for a cylindrical sample of single crystal (K sc = 0.062).
All of the above values are shown by arrows in the figure, which
leads to the same conclusion as for tin, i.e., the line nucleation
model explains the observed temperature dependence up to t = 0.98-0.99
where the ideal (plane) nucleation field is reached.

-1154.3.2

RF Critical Fields of Snln and InBi Alloys at 90 MHz
The rf critical fields were measured at 90 MHz for 10 Snin and

10 InBi alloys.

These alloys covered both the type I and type II

regimes and their GL parameter was as high as 1.75, which was well
into the type II region.

The results are shown* for Snin alloys in

Fig. 4.11 and for InBi alloys in Fig. 4.12, where the reduced critical fields

hrf (= Hrf/Hdc) are shown as a function of the reduced
c '
temperature, t(= T/T c ), as before. As can be seen in the figures,
varying amounts of superheating were observed for both types, i.e.,

H~f higher than He for the type I and higher than Hcl for the type II
superconductors.

At a given temperature, the amount of superheating

consistently decreased as the GL parameter increased (or equivalently
as the alloy concentration increased).

Also, a systematic tendency

of the h~f to increase toward the transition temperature was noted.
However, it was found that the exact manner of this temperature dependence was somewhat different between the samples of the two alloy
systems for similar

K •

In particular, Snin samples were generally

found to show less temperature dependence than InBi samples of a similar

K •

Before going further into the possible causes of the above
behavior, the expected superheating field and its temperature dependence in the local limit is discussed.

In the preceding section it

was noted that, for pure tin and indium, the ordinary GL formulation
(local electrodynamics) was valid only in the vicinity of Tc.

*A few samples were omitted because they crowd the figure.

The

-116-

( 0)

- 2.5
h~f vs t

( +)

Sn In Samples
at 90 MHz

- 2.0

+ + +

+ ++ ++

0 DO O OX

(o)

- 1.5

(x)
(A)

( 0)

A.
AAAA

~ ( v)

1---

-o-- -<>-- <> -o- _<>_<>_ <>¢ ¢ o <>-o-<>~Q..

Legend At.% In
0.27
0.62
1.74
2.95
4.21
4.96
6.26

<>
'il

v v v v v

Type
0.125
0.173
0.413 I
0.545 I
0.657 I
0.834 II
0.996 IT

v vvv

- 0.5

0.85

0.90

0.95

T/Tc
Fig. 4.11 The rf critical fields of Snln alloys at 90 MHz are shown
as a function of the reduced temperature. The superheating field
of each sample expected from the one-dimensional GL theory is
shown by an arrow on the right-hand side.

-117-

.----.-----------.----------,2.5

,..-,--------------.------------,2.5

(o)

h~ vs t

ln. Bi Samples
at 90 MHz
2.0
oO

(x)

(o)

++ + + ++

+ +

(o)
( +)
(o)

(x)

------~~-------~0

X X

1.0

X XX

Legend At.% Bi

0.90

0.25
0.99
1.49
3.94

Type
0.161
0.469
0.677 I
II
1.75
0.95

T/Tc

0.5

Legend At.% Bi

1.0°

0.90

0.52
1.99
2.80
3.42

Type
0.277
0.863
II
1.16
IT
1.51
II

0.95

0.5

1.0°

T/Tc

Fig. 4.12 The rf critical fields of InBi alloys at 90 MHz are
shown as a function of the reduced temperature. The superheating field of each sample expected from the one-dimensional
GL theory is shown by an arrow on the right-hand side.

-118reason for the restriction is that the local electrodynamics is
valid only if the electrodynamic coherence length, ~ 0
smaller than the penetration depth, AL(t).

is much

However, as soon as the

pure metals are alloyed, the mean free path of the electrons, ~ , is
reduced, which in turn reduces the effective electrodynamic coherence
length, ~eff' and increases the effective penetration depth, Aeff' at
the same time. *
This means that the local electrodynamics becomes important
where

becomes comparable or less than the ~ 0 of pure metals, which

. the present case. **
happens with very little impurity concentration 1n
Then for most of the present alloy samples except the very dilute ones,
the local GL equations are expected to be valid in the temperature
range of the present measurement.
An approximate temperature dependence of the superheating
field, Hsh' in the ordinary Ginzburg-Landau framework (local electrodynamics) can be obtained by expressing

in terms of Hc(T) and

\ (T) as (T4)
( 4. 27)

*Approximate forms of these two lengths are ~~}f = ~~l + {a~f where
a~ 0.8 (P3) and Aeff(t) = \(t)/x 112 where X is the function intraduced by Gor'kov (G8) and can be approximated by X= (1+~/~)-1
(L3, p. 48).

** For example, for Snln alloys ~ = 2,360A at 0.62 at.% In, whereas
2,300A, and for InBi ~ = 2,100A at 0.25 at.% Bi, whereas
~ (In) = 4,400A.
~ is listed in Table 4.1 for the present samples.
~ (Sn) =

-119where

is the flux quantum equal to hc/2e. With the empirical
approximations He~ (1 - t 2 ) and A~ ~ (1 - t 4 )-l, we have

(4.28)
Using the above

in Eq. (4.14)*, the approximate temperature depen-

dence of the superheating field for the plane nucleation is

(4.29)

Similarly for the line and point nucleations we obtain, using Eq
(4.12) and (4.13)
Hsh
He 1ine

(1 + t 2 )

{4.30)

and
Hsh
~ (1 + t2)3/2
He point

(4.31)

The above estimates give the variation in the superheating field
between Tc and 0.9 Tc of 5%, 10% and 16% for the plane, the line, and
the point nucleation, respectively.

These variations are generally

too small to account for the observed temperature dependence for most
of the samples except for several Snin samples (Sn + 0.62, 1.05, 1.74

*Since Eq. (4.14) is a limiting expression for small K we are still
restricted to type I superconductors. For higher K the above expression will give an upper limit on the temperature variation of hsh
since the actual superheating field has a weaker dependence on K than
in Eq. ( 4. 14) .

-120and 4.96 at.% In). *
The experimental observation of superheating in a static magnetic field for alloys has been found in general to be difficult.
However, very recently Parr (Pl4) has successfully observed the superheating field for a series of dilute InBi alloys (up to 0.6 at.% Bi)

in the form of spheres typically 18 microns in diameter.

According to

his observation, the switch-over from the non-local type of temperature dependence (hsh ~ (1 - t)- 1112 ) to the local type (hsh ~ canst.
to first order in t) is complete before the concentration of Bi
reaches 0.4 at.%.

This means

for the present InBi alloy samples

(0.25-3.94 at.% Bi) that the superheating field is expected to have
only a weak temperature dependence in the temperature range of the
present measurements, whereas the rf critical fields for InBi samples
change by as much as 50% between Tc and 0.9 Tc .
For the present alloy samples, considering the above observation as well as the discrepancy in the temperature dependence of h~f
between the two alloy systems of similar K

it is quite probable

that defective areas on the surface of the samples (or "weak" spots)
are responsible for the nucleation which causes the large temperature
dependence in the rf critical field.

The visual observation of the

samples under a microscope showed that a somewhat coarse surface involving dendritic structures tended to develop as the concentration of
alloy was increased.

This surface structure was particularly notable

*However, among them only one sample (Sn + 1. 74 at.% In) shows the expected superheating field toward Tc (see Fig. 4.11).

-121-

on the hemisphere where the solidification took place last when the
samples were made.
It was also noted when making the. spherical samples, that the
Sn-based spherical samples were much easier to make than the In-based
samples in terms of their sphericity and the smoothness of the surface,
probably due to the higher surface tension of tin in the molten state
compared to indium.

This resulted in better success rates for Snin

alloys than for InBi alloys, which might explain the smaller temperature dependence of h~f for Snin samples compared to InBi samples.
Also as noted in the previous sections on de measurements, the
original samples (as made) showed a broadened transition (O.l-0.2°K)
indicating the presence of inhomogeneous solute concentration within
the samples and possibly of different metallurgical phases.

After the

annealing treatment already described, the samples showed a reasonably
narrow width in the transition temperature (see Table 4.2) as well as
well-defined de critical fields (entry field) in a static magnetic
field, which indicated that the bulk of the samples was a homogeneous
solid solution.

However, there remained a possibility of localized

inhomogeneities being present on the surface of the samples which would
not have been detected by the susceptibility measurement, since it
generally measured the average magnetic property of an entire spherical
sample.
Another observation that is worthwhile noting at this point is
that the surface resistance of the spherical samples in the normal
state was consistent within ±6% with the classical surface resistance

-122deduced from the bulk resistivity of the cylindrical alloy samples
(see calibration in Sec. 4.2.4).

Since the skin depth in that mea-

surement sampled a surface layer of typically between 0.5 and 15
microns, the agreement with the classical expression meant that the
major portion of the surface layer within such depths was on the
average of the same composition as the bulk of the sample.

This again

suggests that the defects, if present, are not global but localized on
the surface.
Even with the presence of localized defects, it may be expected
that their effects would tend to be diminished as the transition ternperature is approached, since the GL coherence length diverges as
(1- tf 112 thus making the defects invisible to the superconducting
11

electrons.

Thus the following approach was used in presenting the sum-

mary result on alloys.

Since the highest temperature where a reliable

rf critical field measurement was made was about 0.99 Tc' the observed

h~f around 0.99 Tc would give a lower bound on the attainable field.
Thus these points, h~f (t=0.99), are taken as the basic experimental
points* and shown as a function of the GL parameter in Fig. 4.13.
Furthermore, whenever there was a systematic trend in h~f below ~0.99 Tc
it was extrapolated to Tc and shown at the tip of an arrow originating
from the t = 0.99 experimental point.
The absolute errors in hr (shown in the figure) were estimated to be
less than 20% where the ~rror in the absolute determination of H~f (10%)
and the estimated errors in H~c and Tc were combined.

n- Sn In

.06 .07 .08 .09 0.1

Absolute
0.2

0.3

~iL<.

0.4

TYPE II

0.5 0.6 0.7 0.8 0.9 1.0

[MEISSNER STATE!

GL Parameter K

Err~r Bor

..- h~t(t!::::!0.99)

h~f vs Kat 90 MHz

InBi

LEGEND:

:::t ' ' ' '

~~~

L------------

~uP{RHEATED-STATElJ

I!!

.,.,,.

!NORMAL-STATE!

TYPE I

. ,

2.0

2.5

-~:.-

[MIXED STATE!

He2
He

Fig. 4.13 The rf critical fields that were actually measured (near t=0.99) for all Snln and InBi
alloy samples are shown as a function of the GL parameter K • Also, the values extrapolated
to Tc, when possible, are shown by arrows originating from the data points.

2.0

3.0

4.0~--~~~~---------.------.----.--~--.-,-~~-------.----.--.

-124Figure 4.13 shows the phase diagram of critical fields versus
the Ginzburg-Landau parameter K plotted on a log-log scale (this is
the same phase diagram as Fig. 2.1).

Also shown in the same figure

are the superheating field predicted by the one-dimensional GL theory,
hsh(= Hsh/Hc) * and the fluctuation-limited superheating field estimated
by Kramer (K3, see Sec. 2.2.4), Hs 1/Hc.

As can be seen, the present

results on both alloys show a fair agreement with the superheating field
in both type I and type II regimes where the following conclusions are
obtai ned.
Type I
In this region the rf critical field is not limited by the thermodynamic critical field but rather by the superheating field determined
from the one-dimensional GL theory.

The results from lead discussed in

Chapter 3 give more support for this conclusion.

This is the first

time**, to the author•s knowledge, that rf critical fields approaching
the superheating field have been systematically observed in experiments
involving type I superconductors.
Transition Region from Type I to Type II
The rf critical fields that can be attained show no signific,1nt
discontinuity at K = 1/1:2 = 0.707 where the transition from type I to
type II superconductivity takes place.

This is unlike the situation

*The numerical values of hsh were obtained by reading the graph given
in M5 for K ::: 0. 25 and by the analytic expression for small K given
in P13 forK~ 0.25.
** Except for the single instance where an rf superheating in lead (by
25%) was reported at 4.2°K (F3).

-125in the static field where the two critical fields Hcl and Hc 2 separate
the three possible phases, and it is again consistent with the superheating field which is continuous across the boundary.
Type II
In this regime a total of 7 alloy samples (2 Snln and 5 InBi
samples) were used in the rf measurements.

All of the samples showed

rf critical fields higher than the first critical field Hcl and all
of them, except one, showed h~f equal to or higher than the thermodynamic critical field.

Thus it is concluded that the lower critical field

does not limit the attainable rf fields.

This is consistent with the

previous measurements made on niobium (54) and the alloys such as Nb 3Sn
(PlO) and Nb 0 . 4Ti 0 . 6 (G7).
The cause for this behavior (h~f above hcl) was previously attributed (HlO) to the time it takes to nucleate a flux line at Hcl in
type II superconductors.

The nucleation time of a flux line was
measured (F7) to be on the order of l0- 6sec which is long compared with
the present rf period (~ l0- 8sec for this study).
The uppermost limit of the rf critical field for the present
type II superconductors is placed by the superheating field rather than
the thermodynamic critical field, although the results are somewhat on
the borderline of the absolute experimental error.

This is again the

first observation of hrf equal to or higher than h for type II superc
conductors. The difference between the one-dimensional superheating
field, hsh' and the fluctuation-limited superheating field (hsl shown
by a dashed line in the figure) is too small to be distinguished, based
on the present experimental results.

Also, the time scale in which hsl

-126-

becomes effective is not established so that the comparison of hsl
to h~f is somewhat ambiguous.

Therefore this point is an open ques-

tion at this stage.
4.3.3

Frequency Dependence of RF Critical Field
The rf critical fields were also measured at 300 MHz by exciting

the fourth mode of the helical resonator where the rf magnetic field
was reasonably uniform over the dimension of the sample.

The results

obtained at 300 MHz are compared with those obtained at 90 MHz in Fig.
4.14 for four samples.

It can be seen that the rf critical field is

generally insensitive to the frequency in this range, varying by up to
10% over the frequency change of a factor of 3. *
This relative insensitivity of h~f to the frequency of the magnetic field suggests that the nucleation time of the normal phase,
once the critical field is reached, is much shorter than the typical
rf period used in the experiment.

There has been a very limited amount

of work with regard to the nucleation time of the normal phase, particularly in the superheated state as found in the present experiment.
According to one recent experiment (V2) on small spheres of mercury in
the superheated state, the nucleation time was estimated to be
4 xlo- 11 -lo- 12 sec which is much smaller than the present rf period of
10- 8 - 3 x 10- 9sec.

*rhe lower h~f at 300 MHz may be due to a localized heating at the site
of nucleation, as the superconducting surface resistance varies as
~f 2 . However, the magnitude of the effect (0-10%) and the relative uncertainties in the determination of the rf magnetic fields at two different frequencies (~ 12%) do not warrant any definite conclusion in
this regard.

-127,-------------------,3.0
rl

MHz

300 MHz

coo

- 2.0

Pure In
K = 0.060

Sn + 0.62 At.% In
K = 0.173

H''c

= 0.677

In+ 3.42 At.% Bi

1. 508

~---L-·-------~·-------·~------~0
0.85

0.90

0.95

1.0

Yl-c
Fig. 4.14 The rf critical fields measured at two frequencies,
90 and 300 MHz are shown for four samples.

-128-

Theoretically the situation is even more uncertain.

Neverthe-

less an order of magnitude estimate can be obtained from the characteristic time for the variation of the order parameter, TGL' which
appears in the time-dependent Ginzburg-Landau theory (A4, and Y2, p.22)
as
(4.34)
This gives for the present temperature range of 0.8 ~ t < 0.99
5 X 10

-12

~ T GL < 10-

which again is much smaller than the present rf period of measurement.
These estimates, incidentally, justify the use of the ordinary
(time-independent) GL theory in the frequency range used in the present
experiment and at the same time may account for a generally good agreement between the present rf results and the prediction of the GL superheating theory.
4.4 Conclusions
The rf critical magnetic fields were measured at 90 MHz for pure
tin and indium, and for a series of Snln and InBi alloys spanning both
type I and type II superconductivity.

The results of pure samples of

tin and indium showed that a vortex-like nucleation of the normal phase
was responsible for the superconducting-to-normal phase transition in
the rf field at temperatures up to about 0.98-0.99 Tc , where the ideal
superheating limit was being reached.

The results of the alloy samples

-129-

showed that the attainable rf critical fields near Tc were well described by the superheating field predicted by the one-dimensional
Ginzburg-Landau theory in both the type I and type II regimes.

The

measurement was also made at 300 MHz resulting in no significant
change in the rf critical field.

Thus it was inferred that the

nucleation time of the normal phase, once the critical field was
reached, was small compared with the rf period in this frequency
range.

-130Appendix A
THE RESIDUAL LOSS DUE TO TRAPPED FLUX
If a resonator is cooled through its transition temperature in
the presence of a static magnetic field, the walls of the resonator
tend to trap magnetic flux.

Since the cores of the trapped flux are

normal-conducting, they will dissipate energy when an rf current flows
on the surface of the resonator.

This loss mechanism can be charac-

terized by an effective surface resistance, RH , which is a function
of the applied magnetic field, Hdc' and temperature, T , when measured
at low rf field level.

Since the shielding of the earth•s magnetic

field is not always complete in a low temperature dewar apparatus, it
is of practical interest to determine the magnitude of RH and its dependence on the applied field and temperature.
The effect of a static magnetic field on the residual surface
resistance was investigated by slowly cooling the cavity through the
transition temperature of lead while a constant magnetic field was
present.

The de magnetic field was applied horizontally (perpendicular

to the axis of the coaxial resonator) by a pair of Helmholtz coils
while the vertical component of the earth•s magnetic field was cancelled
down to about 15 milligauss by another pair of Helmholtz coils.

The

uniformity of the applied field above 0.2 gauss was within 10% over the
entire region of the cavity.
The temperature dependence of cavity Q was measured for the applied field ranging from 0 up to 8 Gauss.

Assuming that all the applied

magnetic flux which intercepts the cavity surface is trapped, the

-131partial geometrical factor,* rH , may be calculated using the known
current distribution as discussed in Sec. 3.2.1.
Q0 RH = rH = 13 ohms.

It was found to be

Then the effective surface resistance is given

by
(A. 1 )

where

Q~ 1 (0,T) is the result in a zero applied field.
The dependence of RH on temperature was found to be

(A.2)

where

t = T/Tc with Tc = 7.l8°K for lead.

This temperature dependence

holds for the entire Hdc range studied as shown in Fig. A.l.

This is

distinguishably different from the functional form observed for the
same loss mechanism by Victor and Hartwig (Vl) which is

in the same frequency range for a rolled foil resonant circuit.
The presently observed temperature dependence may be explained
by a simple model where the dissipation is proportional to the effective volume of the normal region through which a current can flow. The
conservation of flux requires that the fraction of the normal surface

*The partial geometrical factor, rH, is defined in an analogous manner
as the geometrical factor r in Eq. (3.7) except the surface integral
is now over the effective surface area instead of the entire area.

-132-

2.0

2.5

3.0

3.5

4.0

Hdc = 3 Gauss

-.g2

10- 8 ~~~~~~

2XI6 0~------------~--------------~------------~
10 LOG((H~)3/2)
Fig. A. 1 The quantity, Q~ 1 (Hdc)- Q~ 1 (o), which is proportional to
the effective surface resistance induced by the presence of trapped
flux is shown as a function of the characteristic temperature quantity of the present trapped flux mode 1.

-133area is proportional to

Hdc/Hc(T) at all temperatures.

If we assume

that the depth to which the current can flow is limited by the typical dimension of each flux tube, then the volume, Veff' in which the
dissipation is effective is proportional to
Veff

where

normal area x depth

is the radius of a typical flux tube.

Since nr2 oc Hdc/Hc(T)

we have

(H /H (0)) 312 (1 - t 2)- 312
de c

(A.4)

where the usual parabolic law for the critical field is used.

Below

Hdc of 1 gauss, RH was seen to depend on H~~ 2 as shown in Fig. A.2.
Also the above expression gives the observed temperature dependence.
However, in order to obtain the proportionality constant in
the above equation we need a detailed knowledge of the intermediate
state (e.g., the size of a typical flux tube) which cannot be uniquely
calculated with the present experimental configuration.

We can never-

theless determine the proportionality constant empirically in the
above model from the experimental data.

For the 10~ surface studied

it was found to be

R = 1.13 x 10-7 ( de )3/2~
1 - t2
where Hdc is expressed in gauss.

(A.5)

Thus, for example, in the earth's

-134-

1.5

Surface: Pb
Thickness: I 0 fL
Temperature: 4.2°K

,..........,
CX>

...._,

"'0

...._, 0.5
10

o~----._~----~--~--~----._----~

0.5
312

Hde

1.0

(Gauss 312 )

Fig. A.2 Q~ (Hdc- Q~ 1 (o) {proportional to the effective surface re-sistance due to trapped flux) is shown as a function of H~~ 2 where
Hdc is the static field present when the resonator was cooled
through its transition temperature.

-135-

magnetic field of 0.5 gauss, we have

(A.6)
which is approximately the same as the BCS surface resistance at 4.2°K.
In a typical dewar apparatus with a mu-metal magnetic shield the remnant magnetic field is below 10 milligauss so that
(A. 7)

which is totally negligible compared with RBCS (4.2°K).
Aside from the remnant magnetic field there is a possibility of
a thermoelectrically generated magnetic field (Pll) which may be
trapped during the cooling process.

In the worst case of this process

Pierce suggested a field up to 100 milligauss.

Even in this case

and this is about an order of magnitude smaller than the observed
residual resistance in this experiment (Rres = 2-5 x 10- 8n).
The dependence of RH on Hdc above 2 gauss was found to be approximately linear as shown in Fig. A.3.

However, the temperature depen-

dence remains essentially the same as for the lower fields.
crepancy cannot be explained in the present model.

This dis-

The exact cause of

this behavior is difficult to determine due to several possible complications such as (1) the demagnetizing effect of the complicated surface geometry of the present resonator, (2) the size effect due to
surface roughness as previously advanced by Pierce (Pll), and (3) the

-136-

,.......---,
CX)

L--..1

..........

'-"

1 0 10

T = 4.2°K

! 1011- surface in

I 0 J.L surface
reverse field
2.5fL surface

Hdc (Gauss)

Fig. A.3 The reciprocal of Q measured at 4.2°K is shown for two
surfaces examined as a fugction of the static magnetic field,
Hdc' which was present during the cooling process.

-137-

intermediate state structure which is not known~ priori.

Thus in

order to obtain a more definitive answer, it is necessary to design
an experiment for this purpose only, where the intermediate state can
be observed directly.
Finally, it should be pointed out that the loss due to trapped
flux is not well understood in low frequency resonators (order of
100 MHz) as reported by Piosczyketal. for niobium helical resonators
(P9).

They report a complicated dependence of RH on temperature as

well as on the rf magnetic field, which is in contrast to the loss due
to the same mechanism in GHz region where the loss is well described
by the dissipation in the .. quasi-normal cores .. of fluxoids.

-138-

Appendix B
SURFACE RESISTANCE OF LEAD AT HIGH RF MAGNETIC FIELD LEVEL
The surface resistance of superconducting lead of various thicknesses was investigated at rf magnetic field levels comparable to the
thermodynamic critical field.

The thicknesses of 0.3, 1.45, 6 and 14

microns were obtained by varying the deposition time at constant current
density in the electroplating.

For each thickness the resonator quality

factor, Q0 , was measured as a function of the rf magnetic field at two
temperatures (4.2 and 2.2°K).

The surface resistance was observed to

increase with rf field level for all the surfaces examined.

A typical

manner of this increase is shown in Fig. B.l, where the reciprocal of

Q0 (proportional to the surface resistance) is shown as a function of
the rf magnetic field, Hmax' at the midsection of the center conductor.
The general features are characterized by an initial slow increase followed by a rapid increase near He.

In analyzing this behavior (Q~ (Hmax'T) versus Hmax), it was
found that the expression

Hmax n

Q~ (Hmax'T) = Q~ {O,T)(l + a(H (T)) )

with

n ~ 2.0

(B.l)

below Htr

described the behavior below a certain field Htr well.

Here Hc(T) is the thermodynamic critical field of lead, and
are the parameters to be determined for every surface.

a and n

The justifica-

tion of this form will be given later, and the meaning of Htr will soon
become clear.

The parameters

a and

n were found by plotting

-139-

ICY 6
He(4.2°K)

He (2.2°K)

He(4.2°K)

..!

10-7

....--...

)(

1.0
400

600

Hmax (Gauss)

:c
..........

104

Surface: Pb
Thickness: 14,u.
Temperature
+ 4.2°K
2.2°K

CY-:

......

....

H tr(4.2°K) Htr(2.2°K)

IX1~ ~--~-------~--~-------~--~----~----~--~

200

400

Hmax (Gauss)

600

800

Fig. B. 1 The recipnocal of Q0 (proportional to the surface resistance is shown as a function of the magnetic field, Hmax' at the
mi dsect·i on of the center conductor. The solid 1 ines are the power1 aw fit to the experimental data bel ow Htr· The inset shows the
additional dissipation above Htr·

-140-

versus

Hmax

on a log-log plot and drawing a straight line in the portion where the
slope was constant.

Examples of the term thus determined are shown as

solid curves, together with experimental points in Fig. B.l, where
Htr can be seen as the field at which the experimental points start
deviating from this simple power law term (i.e., transition from one
Above Htr the increase in Q~ 1 is much more rapid

behavior to another).
than the H~ax term.

Taking the difference between the experimental

points and the extrapolated values of H~ax term above Htr' we define
f(Hmax) by
above Htr

(B.2)

which is a measure of the additional loss above Htr"
With the above procedure, a, n, Htr and f(Hmax) were found for
every surface examined.

The values of a, n, and Htr are listed in Table

B. 1.

Before discussing the significance of these parameters we need to
look at the real surfaces of lead for different thicknesses.

Direct

observation under a microscope shows that there are at least two kinds
of surface imperfections:
(1) The first kind is the imperfect plating at the grain boundaries of
the copper substrate.

This is particularly pronounced for large-grained

substrate as shown in Fig. B.2a, which shows a discontinuity in surface

-141-

Table B.l

The parameters, a, n, and Htr' determined by the present
analysis are shown for all surfaces studied at two ternperatures.

Temp.

Hc(T)

Thickness
or
Pb Layer

gauss

10-8

4.20

528

0.3

2.04

3. 83

1.45

2.08

6.0

2.20

728

Q-1
(O,T )

gauss

Hrf
gauss

1. 84 ± 0. 21

580 ± 20

627

2.07

2. 61 ± 0. 46

510 ± 20

600

1.36

1.77

1.96 ± 0.26

400 ± 20

580

1.77

o. 76

1.77±0.15

375 ± 25

566

0.3

1.02

4.42

1. 88 ± 0. 21

740 ± 20

791

1.45

1.18

4.46

2.25±0.28

690 ± 20

734

6.0

0.80

3.83

2.05 ± 0.32

575 ± 25

717

1. 36

1. 38

1. 66 ± 0. 39

490 ± 40

681

. tltr

-142-

(a)

0.23~

(b) 4.5~

(d)

4 . 5~

Fig. B.2 The surface of electroplated l ead is shown for a largegrained copper substrate (a,b) and for a fine- grained copper sub strate (c ,d) . For both types of substrates, the thin lead film
replicates the substrate, whereas the thicker lead film tends to
develop its awn surface structures.

-143-

texture.

Also distinct edges develop as the thickness of the plated

layer increases as shown in Fig. B.2b.
(2) The second kind is the rough surface features that tend to develop
as the thickness increases.

At low coverage the surface of lead tends

to replicate the substrate (see Fig. B.2a and c).

As the thickness of

the plated layer increases, the lead surface develops its own structure
such as protrusions and sharp corners as can be seen in Fig. B.2b and d.
Once these observations are noted we can proceed to discuss the increase
in surface resistance at high field levels.

This can best be done in

two separate field ranges.
Below Htr
We have noted that the observed increase in surface resistance is
given by
(8.3)

For all the surfaces examined it was found that

n was very close to

2.0 (1.66 ~ n ~ 2.61, see Table B.l).
It is proposed that this portion of the additional loss is associated with localized defective areas or weakly superconducting regions
that are located at or near the grain boundaries of the substrate. The
proportionality to H2 can be seen in the following simplified argument.
The power dissipated in a localized defect is given by

(B.4)

-144-

where

and Ad are, respectively, the effective surface resistance

and the area of the defect.
ture

Now this dissipation raises the tempera-

T of the surrounding area above the bath temperature Tb by an

amount

where

K is the effective thermal conductivity between the lead film

and the substrate.

Then the surface resistance of the surrounding

area is given by
R(T) = R(Tb) + ~~~Tb

1 dRI

RdAd H2

= R(Tb) + 2 dT Tb -K-

1 RdAd dRI
R(T)- R(Tb) = (2K
dT Tb)H 2

Since Q~ 1 is proportional to the surface resistance

Q0-1( H,T ) - Q0-1 {O,T )
-1

(8.5)

R we have

ex:

(8.6)

Qo (O,T)

where it is assumed that the area at the elevated temperature remains
constant.
The coefficient, a , in front of H~ax in this picture will be
proportional to the density of defect sites which is expected to decrease as the thickness increases, since smaller defects will be covered as the lead layer builds up.

The coefficient, a , was indeed ob-

served to decrease with thickness as shown in Table 8.1 .

-145-

Another support for the above interpretation in terms of localized
heating comes from the fact that heating of the entire surface cannot
account for the observed increase in the dissipation.

The global heat-

ing is another possible mechanism (H8) in which the total power dissipated within the resonator raises the overall temperature of the
resonator with a resultant increase in surface resistance, which is
also proportional to H2 . The temperature rise of the cavity surface
in this scheme was calculated using the measured power dissipation in
the cavity and the known thermal conductivity of the copper substrate.
The expected increase in Q0-1 thus calculated was smaller than the
observed increase by typically more than two orders of magnitude which
clearly indicated that the global heating was not the cause of the increase in surface resistance.
Above Htr
At the transition field Htr an additional loss mechanism enters
that enhances the surface resistance or, equivalently, increases Q~ 1 .
This additional mechanism can be characterized by Eq. (8.2) for f(Hmax)
which was found to be almost exponential in Hmax as shown in the inset
of Fig. B. 1.

In this figure f(H max ) is shown for two temperatures,
4.2

and 2.2°K and the arrows show the thermodynamic critical fields of lead
at the respective temperatures.

Their approximate correlations suggest

that this additional loss mechanism is related to the critical magnetic
field.

A plausible phenomenological explanation for this mechanism is

as follows.

Since the surface magnetic field is enhanced by surface

structures such as protrusions and sharp corners (see photographs in

-146-

Fig. 8.2), it is possible to reach the rf critical magnetic field of
lead locally at these structures.

Once this field is reached locally

it will give rise to quasi-normal
11

regions (HlO) which contribute to

the increase in surface dissipation.
This explanation, although extremely qualitative in nature, is
supported by the observation that Htr (Table B. 1) is lower for thicker
films, i.e., this additional loss mechanism comes into effect at lower
field for thicker films, since there are more surface structures in
these films that enhance the surface magnetic field locally. Exactly
the same argument applies for the upward trend in f(Hmax) as the film
thickness increases as shown in Fig. 8.3.
All the above features are summarized in· the diagram shown in
Fig. B.4 where the transition field, Htr' and the rf critical field,

H~f, are plotted as a function of the thickness of the lead film. The
area below Htr is characterized by the fact that the additional surface resistance is proportional to H2. All the observations made suggest that this is due to a localized heating at defect sites such as
at grain boundaries of the substrate.

In the region between Htr and

H~f the additional surface resistance is dominated by the surface
structures of the lead surface.

As a reference the thermodynamic

critical field of lead is shown by a dashed line.

It clearly shows

that the superconducting state can persist above the thermodynamic
critical field at this frequency.

-147-

.........,._

)(

:::c

....

..............

1.45~

1.0

0.8

400

lo:3fL

500
Hmax (Gauss)

600

Fig. B.3 The quantity f (Eq. (B.2)) is shown as a function of Hmax for
different surfaces at 4.2°K. This quantity is a measure of the
additional loss mechanism above Htr·

-148-

Temperature

= 4.2°K

Normal State

600

+- Hrf

...<;>,.c- ,,

-+----s·1,1}

9(J.Q

~0/''

.s,o

-400
(/)
(/)

::J

<.9

............

::c 300
R5 «
200

n ~ 2.0

Meissner State with Localized Heating

100

Thickness (micron)
Fig. 8.4 The transition field, Htr' and the rf critical field, H~f ,
are shown as a function of the film thickness.

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