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THESIS

1"”

This is to certify that the

dissertation entitled

Non-Equilibrium Proximity Effect
In a SNS Junction

presented by

JIAN HUANG

has been accepted towards fulfillment
of the requirements for

Ph. D degree in Physics

flow 0 a?

Major .professor U
N . Blrge

Date December 10, 2001

MS U is an Affirmative Action/liq ual Opportunity Institution 0-12771

 

 

LIBRARY
Michigan State
University

 

 

 

PLACE IN RETURN Box to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE ' DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 c:/ClRC/DateDue.p65—p.15

NON-EQUILIBRIUM PROXIMITY EFFECT IN A SNS Junction
By
JIAN HUANG

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Physics
2001

ABSTRACT
NON-EQUILIBRIUM PROXIMITY IN JOSEPHSON DEVICE
By
JIAN HUANG

Our goal is to study the quasiparticle-induced nonequilibrium proximity effect in
a SNS device. We first. performed a dangling arm experiment which demonstrates
the counter-flowing quasiparticle current and supercurrent. Then, a second exper-
iment. was performed to demonstrate the appearance of a 7r junction. Baselmans
et a1. [1] have recently shown that the direction of the supercurrent in a super-
conductor/ normal / superconductor Josephson junction can be reversed by applying,
perpendicularly to the supercurrent, a sufficiently large control current between two
normal reservoirs. The novel behavior of their 4-termina1 device (called a controllable
n-junction) arises from the nonequilibrium electron energy distribution established in
the normal wire between the two superconductors. We have observed a similar su-
percurrent reversal in a 3-terminal device, where the control current passes from a
single normal reservoir into the two superconductors. We show theoretically that
this behavior, although intuitively less obvious, arises from the same nonequilibrium
physics present in the 4-terminal device. The difference lies in the scattering of the
Cooper pair states by quasiparticles with zero momentum (Baselmans’) and with a
finite momentum (our device). Moreover, we argue that the amplitude of the 7r-state
critical current should be at least as large in the 3-termina1 device as in a comparable

4—terminal device.

To God Who is loving and merciful
and

To my dearest wife and daughter

iii

ACKNOWLEDGEMENTS

I would like to thank my advisor, Professor Norman Birge, for his constant support
and care for the past four years. For this work would have been impossible without
him. I really appreciate this opportunity of working with him and it has been a
tremendous experience. I enjoy many things I have learned from him and I have
always found encouragement through various things we worked on together.

My thanks go to Professor Mahanti, Professor Phil Duxbery, Professor Mark Dyk-
man, and Professor Zelevinsky for many enlightening discussions with them. Their
knowledge on many subjects will always encourage and spur me for better work. I
would like to thank Professor V. Ambegaokar, J. Saul and Professor Abrikosov for
their teaching on superconductivity and Green’s functions which brings better under-
standing. My thanks go to Professor C. Lobb and D. Ralph for their comments on
the metal-metal contact issues. Thanks to Frank Wilhelm for many explanations of
the quasiclassical theory. I would like to thank Dr. Baokang Bi for his generosity and
patience in teaching me the fabrication techniques. My thanks go to H. Pothier and
D. Esteve at Saclay group for their support and insights. I want to thank Fred Pierre
for his help with the experiment and many discussions. My thanks go to Reza Loloee
for his help with the SQUID measurement. My deep appreciation also goes to Prefes-
sor I. O. Kulik who shared key ideas for our experiment. I also extend my thanks to
Professor Volkov for a helpful comment regarding the theoretical calculations.

This work is supported by the National Science Foundation (grants DMR-9801841
and DMR-0104178) and the Keck Microfabrication Facility.

iv

Table of Contents

LIST OF TABLES ix
LIST OF FIGURES x
1 Introduction 1
1.1 Background ................................ 1
1.2 Motivation ................................. 2

2 Review 6
2.1 From Simple Metal to Superconductor ................. 6
2.1.1 Dielectric Function ........................ 6

2.1.2 BCS Theory ............................ 10

2.2 NS System ................................. 11

2.2.1 Modified States at the NS interface and Bogolubov- de Gennes

Equation .............................. 11

2.2.2 Andreev Reflection ........................ 14
2.2.3 Proximity Effect ......................... 16

2.3 SNS Josephson Effect and Andreev Bound States ........... 18
2.4 Spectral Supercurrent Density ...................... 19
2.4.1 Ballistic case ........................... 20
2.4.2 Diffusive case ........................... 21

2.5 Distribution Functions .......................... 22

2.6 7r Junction ................................. 25

2.7 Summary ................................. 27
3 Fabrication Methods 29
3.0.1 Surface Preparation ........................ 29

3.1 Photolithograhy .............................. 30
3.1.1 Photo Resist Coating ....................... 30
3.1.2 Photo Resist Baking ....................... 32
3.1.3 Exposure ............................. 32

3.1.4 Development ........................... 33

3.1.5 Evaporation and Liftoff ...................... 33
3.1.6 Photo-alignment ......................... 34

3.2 Electron Beam Lithography (EBL) ................... 34
3.2.1 Electron-Solid Interaction and Proximity Effect ........ 36
3.2.2 Electron Beam Writing ...................... 39
3.2.3 Development ........................... 40
3.2.4 Electron Beam Alignment .................... 40

3.3 Actual Sample Making .......................... 43
3.3.1 Review of Methods ........................ 43
3.3.2 Method 1: E—beam First, Photolithography Second ...... 45
3.3.3 Method 2: Photolithography First, E-beam Second ...... 49

vi

4 Electronics for Measurement 52

4.1 Sample Mounting ............................. 52
4.2 Electronics ................................. 52
4.2.1 DC Measurement ......................... 55

4.2.2 AC Measurement ......................... 55

5 First Experiment: Dangling Arm 57
5.1 DC and AC Measurement between the N and S Terminals ...... 58
5.2 DC and AC Measurement between the S and S Terminals ....... 60
5.2.1 Analysis .............................. 60

5.3 Temperature Dependence ........................ 63
5.4 Summary ................................. 64

6 Tr-Junction 65
6.1 Observation of the 7r-Junction ...................... 65
6.1.1 Experiment ............................ 65

6.1.2 Discussion on the appearance of 7r junction ........... 69

6.2 7r-Junction affected by the rounding of f0 ................ 70
6.2.1 Temperature ............................ 70

6.2.2 6’6" interaction .......................... 71

6.3 Comparison with Baselmans’ Experiment ................ 75
6.4 Sub-harmonic Gap Structure ....................... 78

vii

7 Quasiclassical Theory
7.1 Review of the quasiclassical formalism .................

7.2 Analytical Solution with High-barrier Approximation .........

Appendices

A Tunnel Junctions

B High Barrier (Rb >> RN) Approximation

C Diffusive region

LIST OF REFERENCES

viii

80

80

87

94

95

97

99

101

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

5.1

List of Tables

Metal deposition current and rate for Edward Auto306 thermal evap-

orator ................................... 34
3—level alignment ............................. 41
Spin and baking conditions for 7mm x 7mm Si substrate ....... 46
E—beam writing conditions for single path ................ 47
E—beam resist Development ....................... 47
Thermal evaporation of the nanowire .................. 47
photo printing condition ......................... 48
Thermal evaporation of Silver reservoir ................. 48
Ion-milling ................................. 48
Experimental results vs. prediction ................... 58

ix

1.1

1.2

1.3

1.4

1.5

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

List of Figures

Schematic of Sample and measurement. circuit .............
Two current flow regimes .........................
Expected V vs. I for dangling arm experiment .............
Schematic picture of the measurement with varied normal injection

Schematic picture of the expected V-I measurement. corresponding to
a 7r junction at large values of injected current Imj ...........

Schematic picture of a metal .......................
Schematic picture of Andreev reflection .................
Schematic picture of the proximity effect .................

Schematic picture of the Andreev bound states formed in a SN S long
junction with L < L¢, ...........................

Energy levels of the excitations in the normal region at a fixed value
of electron velocity [2] ..........................

Spectral supercurrent density vs. excitation energy for different phase

differences for a 3-terminal superconductor-semiconductor-superconductor

device [3] ..................................

Calculated spectral supercurrent density by quasiclassical approach.
The units for energy is meV .......................

Distribution functions for different sample lengths for N-n—N structures.
Upper right: L > Le_ph; lower left: Le-.. < L < Le_ph; lower right:
L < Le_e, Le—ph [4] ............................

A S-n-N system with N elevated by voltage V .............

14

16

18

21

22

23

2.10 A schematic picture of how a quasiparticle distribution function varies

2.11

2.12

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

in space. The solid dot stand for an particle and the circle stands for an
Andreev reflected hole. The size of the circles stand for the amplitude
of distribution function of the particle/ hole ...............

A schematic picture of the distribution function with correction due to
both temperature and proximity effects .................

Antisymmetric distribution function f0 (upper graph) and the spectral
supercurrent density ...........................

A schematic picture of photo printing ..................

E—beam Process: (a) Spin-coat the substrate with e-beam resist. Ex—
pose the resist with electron beam. (b) Develop the resist. (c) Metal
deposition. ((1) After lift-off .......................

Block diagram showing the major components of a typical electron
beam lithography system .........................

Simulated profile of the energy absorbed from an electron beam expo-
sure [5] ...................................

SEM pictures of a series of single paths (Ag lines) with following doses
in nC/cm: 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, and 2.8 ..........

Width of the single path versus dose ..................

Schematics of an angle evaporation method. The two sets of patterns
are from evaporations at different angles ................

Three Steps of making the the sample ..................

SEM pictures of the sample. The upper picture was taken at 10kX and
the lower picture was taken at 18kX ...................

3.10 The dimensions of the actual sample ..................

3.11

4.1

SEM picture (taken at 20kX) of the sample by method 2 .......

Schematics of the sample holder .....................

xi

24

25

27

32

35

35

37

39

40

44

46

50

51

53

4.2

4.3

4.4

5.1

5.2

5.3

5.4

5.5

5.6

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

Schematics of the sample mounting ................... 53

electronic setup for the DC measurement ................. 54
Electronic setup for ac measurement .................. 56
SEM picture of the sample ........................ 57
DC measurement of V-I ......................... 58
AC measurement of (dV/dI) vs. I .................... 59
DC measurement between the superconducting terminals ....... 60
AC measurement between the superconducting terminals ....... 61
Temperature dependence of the critical currents [C(S N ) and IC(S N S ) 63

SEM picture of the sample and the measurement circuit. The picture
is take at 10kX. The distance between 8 terminals is 1.1um, and the
distance between the cross point and N terminal(bottom of the picture)
is 4.5,um .................................. 65

dc measurement of the V-I characteristics for different values of I N . 66

Selected curves from the previous graph for following values of I N:

0.53, 0.70, 1.01, 1.23, 1.89, 2.18, 3.15 MA ................ 67
[C(SNS)vs.I,-nj at 40 mK ........................ 67
1C vs. ij ................................ 68

Absence of the 7r junction resulted from the rounding of the distribution

function due to the increase of temperature ............... 71
V-I characteristics for 96mK, 200mK, and 304mK ........... 72
[C vs. [W and 10 vs. ij for 96mK, 200mK, and 304mK ....... 73

xii

6.9

6.10

6.11

6.12

6.13

6.14

7.1

7.2

7.3

7.4

7.5

7.6

B1

Schematic picture of Baselmann’s four terminal device and measure-
ment setup [1] ...............................

Observed 7r junction and 10 vs. Vconml in Baselmans et al’s experiment

[1] .....................................

Difference of changes of R. f f(S N ) at zero phase state and 7t state . .

Comparison of the total distribution functions between baselmans’ de-
vice (top) and a four-terminal device with current flow equivalent to
that in our device (bottom) .......................

Schematic picture of multi-Andreev processes .............

Observation of the Sub—harmonic Gap Structure ............

The diagram equation for Green’s functions ..............

Comparison of the experiment and theory for IC(S N S ) vs. T for the
dangling arm case .............................

Experimental data for [C(SN) vs. T for the dangling arm case . . . .
Theoretical calculations of 1+, I _, and 10 at 40 mK ..........

Comparison of the I m[j (E )] vs. ij from the numerical calculation
and the high-barrier approximation for the 7r junction case ......

Comparison of [C vs. ij between the theory and experiment for the
7r junction case ..............................

SN S system ................................

xiii

75

77

78

79

82

9O

91

92

92

97

Chapter 1: Introduction

1 . 1 Background

The study of normal conductor-superconductor systems first started in the 60’s.
The phenomenon of superconductivity is a remarkable example of quantum effect
operating on a truly macroscopic scale, while the macroscopic properties of normal
metal are rather classical. Only when the normal metal is made mesoscopic,
quantum phenomena are found on a scale of the phase-coherencelength L¢. The
proximity effect was first studied within the framework of Ginzburg-Landau [6] in
which the superconducting order is expressed by a complex order parameter \Il(r)
which is related to the pair correlations. Later, a more rigorous treatment was done
by using Bogoliubov-de Gennes equations (BdG) [7], where the eigenstates describe
the electron-like and hole-like excitations. It was understood then that proximity
effect describes the phenomenon where the superconducting order can extend far
into the normal region even in absence of the attractive potential between the
electrons. Since the 1990’s, there has been a revival of interest in studying the
proximity effect, which started with bringing together superconductivity and
mesoscopic physics (Altshuler et a1 1991 [8]). Meanwhile, modern fabrication
techniques enable us to make devices with sub-micrometer dimensions. The first
experiment that measured the phase coherent transport properties in submicron
superconductor devices were performed by Petrashov and Antonov [9] (1991) and
Kastalskii et al [10] (1991). Since then, there have been active studies in
understanding the physics as well as applications. The significant progress made
during the past ten years or so signals that the seed planted 30 years ago is growing
into a rich field of study. As a result of much work, the equilibrium proximity effect

is well understood now. But, nonequilibrium situations are still under investigation.

1.2 Motivation

 

 

 

 

 

 

 

 

 

Figure 1.1: Schematic of Sample and measurement circuit

We are particularly interested in studying the non—equilibrium transport
phenomena in a long superconductor-normal metal-superconductor(SNS) junction
(in the diffusive regime) where the nonequilibrium is introduced by injecting
quasiparticle current from a normal reservoir. We considere the system shown in
figure 1.1. It is a SNS long junction with an extra normal lead connected to the
middle of the normal wire to inject quasiparticle current. If a normal current is
driven from the normal reservoir to one of the superconducting(SC) reservoirs, the
peaceful equilibrium is destroyed. Despite the fact that there are complicated
microscopic changes, the system will respond in one of the following two ways
illustrated in figure 1.2.

If the current goes directly into the left SC reservoir, a potential difference is

 

Figure 1.2: Two current flow regimes

 

_y

/

Figure 1.3: Expected V vs. I for dangling arm experiment

 

created between the SC reservoirs. There arises an ac Josephson current due to this
finite voltage difference. The effective resistance between the N reservoir and the
left SC reservoir is simply Raff = R0 + R1. We believe this is what happens when
the driving current exceeds a certain critical current. At an injection current smaller
than the critical current, the dc Josepshon effect may take place. The normal
current can split at the cross point going into both SC reservoirs so that the voltage
drops from the cross point to the two SC reservoirs are the same. To satisfy the
Kirchhoff ’s law, a dc Josephson current arises going from right to left so that the
total current in the right arm is zero. The SC reservoirs are as if tied together. The
effective resistance between the N reservoir and the left SC reservoir is then

Reff = R0 + R1//R2. So, we expect a V-I relationship like the one shown in figure
1.3. If we indeed see this type of V-I curve, we will know that when I < IC, there
are counter-flowing quasiparticle current and supercurrent in the dangling arm.

We also performed a second measurement (shown in Figure 1.4) where we drive
the system farther away from equilibrium to see if a 7r junction appears by
controlling the normal current injection. We also aimed to understand if the same
physics appears studied by Baselmanns et a1 [1]. Unlike the dangling arm situation,

this experiment does not have the constraint that 215 2’ I N. This means the driving

 

 

 

 

 

 

 

 

Figure 1.4: Schematic picture of the measurement with varied normal injection

current can be much higher than the critical current [C before the dc Josephson
effect goes away. The condition for a 1r junction is that the distribution function of
the quasiparticles inside the normal region has a staircase shape. We fabricated our
sample so that the length of the normal region is smaller than the phase breaking
length L4,, so that there is little energy exchange among the electrons. We also
fabricated thick normal reservoir (220nm thick) to ensure efficient electron-phonon
interaction to avoid heating. Thus, the staircase distribution function is achieved.
We measured the V-I characteristics between the two SC reservoirs with different
values of normal injection. The V-I characteristics should be like those shown in
figure 1.5, where the critical current shrinks first as the injection increases, goes to
zero at a certain point, then, opens up again, then goes to zero eventually. If we see
this type of data, we know the 1r junction has occurred.

By comparing our data with the quasiclassical theory, we want to evaluate, as
the sample goes further away from equilibrium, if the electron-electron interaction
becomes an important factor.

There are some lengths involved in characterizing the transport properties.
They are defined as following:

Fermi wave length /\p

 

 

Figure 1.5: Schematic picture of the expected V-I measurement corresponding to a 7r
junction at large values of injected current 1",]-

 

Phase breaking length L4, is a characteristic length for the interference of the
electron wave functions. It is a length on which a wave packet can travel without
loosing its phase coherence.

Thermal Length: LT is also called normal conductor phase coherence length. It
is a length on which quasiparticles lose phase coherence.

Correlation Length for proximity effect L E = W is an energy-dependent

length scale on which the Cooper-pair-like states lose phase coherence.

Chapter 2: Review

This chapter starts with introducing the nature of the attractive potential
between the electrons through the dielectric function with both electron-electron
repulsion and electron-phonon interaction. Then, BCS theory, based on the
attractive potential between the partners inside a Cooper pair, is reviewed. Then, in
discussing modified states of NS systems, the four-fold degenerate states in the
normal region, a result of solving the Bogoliubov-de Gennes equation, is presented.
This lays the ground work for the next discussion of Andreev reflection and
proximity effect. Then, the proximity effect in long SNS junctions is discussed with
the language of Andreev bound states. This is followed by more in-depth discussion
of spectral supercurrent density (energy spectrum of the system) and the
distribution function (occupation of the energy states). Then, 7r junction is
explained in terms of spectral supercurrent density and antisymmetric distribution

function. The chapter ends with a brief summary of current theoretical approaches.

2.1 From Simple Metal to Superconductor

2 . 1 . 1 Dielectric Function

[Note: the derivations below are inspired by Ambegaokar’s notes on review of
superconductivity[11]]
In a simple metal shown in figure 2.1, the basic ingredients of the interactions

are:
0 e' in periodic potential + Coulomb repulsion
o Ions + Ion-Ion interaction —> phonons

o e‘-phonon interaction

 

 

 

a; t
t L
- . \v ..
._1 fl. 5‘
. ..
lon .‘ - v’. ‘I‘ -
f ~( . ‘ ' -
O J "
‘.‘. . , . . . . . - < - ‘ .
j , . . .

 

Figure 2.1: Schematic picture of a metal
0 Impurities

Despite the strong interaction among the electrons, the system is reasonably
described by the independent-particle model. This is based on the fact, traced by
Landau [12], that single particle excitations obey Fermi-Dirac statistics and the
number of the excitations is small at normal temperature. These independent
particles are called quasipariticles which are the basic bricks of Fermi-liquid theory.

When in the superconducting phase, the electron-phonon interaction, as
Frohlich [13] pointed out, mediates the electron-electron interaction. Now, let’s look
at the dielectric function in the presence of both e-e repulsion and e-phonon

interaction. For an electronic system, at sufficiently long wavelengths or at

frequencies u) << top; = t/47rne2 / m, we have
-,o -' -' an -v —o
J = —eDVn — qui z —eDat-Vp — ant) (2.1)

where D is the diffusion constant, e is the charge, a is the conductivity, and (25 is the
electric potential. In equilibrium, a + 8gb = constant, and j = 0, which leads to the

Einstein relation

a = 2—D (2.2)

If we plug 2.1 into the charge conservation 6871/ at + V - j = 0, we have

% = DV2n + 2N(0)DeV2q§ (2.3)

where 872/811 = 2N (0) (N (0) is the density of states per spin). Now, V% = 47rpT
where pr = pm + 716 is the total charge. The Fourier transformation of the equation

above gives

(iw + Dq2)en(q,w) = —87rN(0)e2DpT E —k§DpT (2.4)

where .1332 = 87rN(0)e2. The dielectric function is defined by epT 2 pm which can

also be written as
1

€(q, w)

 

en<st) = i — llpext(qvw) (2'5)

Thus, we have the dielectric constant with the correction of the screening of the

electrons
k2
—z'w / D + q2

 

e(q,w) 2’ 1+ (2.6)

for q << kp and w << wpl. Now, we also need to include phonons. For the unscreened

ions,

ajion _ nion(Ze)2
at — M W (2'7)

 

where -Ze is the ion charge, M is ion mass, and mm, is ion density. Again, following
the similar derivation with charge conservation as that for electrons shown above,
one obtains e = 1 — Sigh/a)2 where 9%,, = 47rn,0n(Ze)2/M. Then, the total dielectric

function is roughly

.2 ‘2
k5 Qph

€(q7w)§1+_iw/D+q2 — “)2

 

(2.8)

Coulomb forces thus screen longitudinal phonons and electron—electron interactions.
If we ignore the —2'w/ D correction (low frequencies), the screened interaction is

roughly

47re2 _ 47re2 [1+ Din/“’2 ]
(New) ‘ «12+ k3 (12+ k3 — nit/w?

 

(2.9)

Screened Coulomb repulsion plus exchange of screened longitudinal phonons
becomes attractive at low frequencies. Though the repulsion and the attraction are
both short-ranged, they work at different band width: The attraction has a
frequency range of cap which is the phonon frequency. The repulsion affects the
electrons with energies on the scale of the Fermi energy. Averaging over the Fermi

surface, one obtains
47re2 02 1 + a2

=—-1
q2+k§> 2 n (12

 

 

< s aN(O) (2.10)

where 02 E Egg/41$. To put this on the same band width with the attraction one

introduces a pseudo-potential

. _ u
— 1+ N(0)#ln(€F/WD)

 

u (2.11)

With this approximation one reaches the BCS-Gorkov local effective interaction
——2- (1 Two i/Jorwa'il’a (2.12)

where 21),, E“ E’e‘k'xcka. The prime means the sum over a shell of width wD.

1.

A = A — a“ is the net effect of phonon induced attraction and coulomb repulsion

(Notezx is the same V appearing in the review on the BdG equation).

2.1.2 BCS Theory

The key to understanding superconductivity is the attractive interaction
between the electrons. The section above, rough but physically sound, should be
treated more rigorously. In 1957, Bardeen, Cooper, and Schriffer (BCS)[14]
published a microscopic theory based upon Cooper pair electrons bound by phonon
interactions. The Cooper pair states, which are doubly occupied or empty
quasiparticle states, show a lower energy with respect to the Fermi energy. So, in a
superconductor, the Fermi sea is replaced with the condensate of Cooper pair states.
The pairs greatly overlap with each other in space, and it is the strong pair-pair
correlations in addition to correlations between the mates of a pair which are
ultimately responsible for the rigidity of the superconducting wave function. The
energy gap A, the energy required to create an excitation (or, it takes 2A break a
pair) also comes from these correlations. At zero temperature, the condensate is
complete and all the electrons take part in forming the superfluid. At finite
temperatures, a small fraction of electrons evaporated from the condensate form a
weakly—interacting gas of excitations which also extends through the whole system.
[151 [16] [171

The BCS Hamiltonian is

H = Z EkCZaCka + Z Vk.k’C;TC:k[C—k’lck’1 (2.13)
k,U kik’

where C's are creation/anihilation operators, and V”, is the effective attractive

potential. The ground state,

[0) = H(uk + vkewczTcileO) (2.14)
k

is a condensate which is a phase coherent superposition of Cooper pair states. [0) is

10

the vacuum state. The phase 43 turns out to be the phase of the macroscopic wave
function of the condensate. This macroscopic phase of a single superconductor is
completely arbitrary when talking about a single piece of superconductor. (Note
that the CL equation does not determine the absolute phase also). This phase
becomes important when determining the coupling between weakly connected

superconductors (ie. Josephson effect).

2.2 NS System

 

 

 

What happens when a superconductor is in contact with a normal metal?

2.2.1 Modified States at the NS interface and Bogolubov-

de Gennes Equation

The states close to the interface are modified as a consequence of the
competition of the N and S orders. Before going into the transport phenomena, let’s
first look at some solutions from the famous Bogoliubov—de Gennes(BdG) equation

for a ballistic situation. The BCS-Gorkov Hamiltonian is

He” = / dr2w3<r>Howa<r> + / «Ml/é") Zwaamrwwewaei (2.15)

 

Where V(r) is the pseudo-potential A that appeared in the last section, H0 is the

single particle Hamiltonian, 0 is the spin, and r is the position. The mean field

11

approximation for the spin singlet is

H.,,=E0+/d:zm How] camera)»: (Wan/210)} (2.16)

where EO— — f V( f)(|2f f(r‘ ) =<< ¢1(T)¢’T(T) >> is the pairing amplitude and
<<>> stands for an average over all states. Then, H =2 C + E0 + Heff (C stands
for C number which arises from the interchange of the two field operators in the

kinetic energy), where

H.” = / dr ( w: at) )Hm W) (2.17)

with

H('r)= HOW NT) (2.18)
A*(r) —H0(r)

This is the Bogoliubov—de Gennes Hamiltonian. The eigenvectors

un('r)
\Iln(r) = (2.19)
vn(r)
satisfy the Bogohubov de Gennes equation
7‘ u" r
H (r) M ) = E... ( ) (2.20)
vn("‘) vn(7‘)

with eigen-energies E3 = 63, + A(r)2. Heff can be diagonalized by the

transformation

A

2.21)

12

and it yields

Hen = Ea + Z Enid)... (2.22)

n>0g

where the ground state energy is

E(; = C + E0 + 2 En = —2 2 En /‘CII.T'I’UT,(T')I2 (2.23)

n>0 n>0

The pairing amplitude is

f(T)=<<w1()wI(7‘)>>= Zum PM )<< 7...,(7’ MAT) >>
n>0 n>0
(2.24)
For a ballistic SNS system (the size of the normal region is on the scale of the

elastic mean free path I), the solution for the normal region includes four-fold

degenerate solutions which can be written as [2] [3] [18]

1
as: = exp(:tiq+r) (2.25)
0
with hq+ = \/2m\/Ep + E. And,
0
21);: = exp(iiq’r) (2.26)
1

with hq‘ = v2m\/Ep -— E. Here, the state is the total state and E is the total

energy. In the superconductor, the solutions are electron-like and hole-like bogolons

i uei“5
(be = exp(:l:ik+r) (2.27)
v

13

i we”
11),, = exp(:l:ik_r) (2.28)
u

where hk+ = \/2m Ep + (E2 — A2)”2 and My = ,/2m EF _ (E2 _ A2)1/2, and

u = %[1+(1— g—ZWQW’Z (2.29)
1 A2 1212
v=E[1—(1—E)/]/ (2.30)

2.2.2 Andreev Reflection

 

 

Figure 2.2: Schematic picture of Andreev reflection

The supercurrent inside the bulk superconductor is carried by Cooper pair
states. When a normal current runs into a superconductor through a NS interface,
part of the current has to transform into supercurrent, the rest will go through
charge relaxation and then become supercurrent. If the energy of the electron in the
normal metal is higher than the superconducting energy gap A, it will go into the
superconductor and go through charge relaxation which can occur on the scale

much longer than the superconducting phase coherence length. But, if the electron

14

energy is less than the energy gap, it can not go in because there are no available
single particle states. Andreev [19] came up with a novel double-particle reflection
process that solved the problem (shown in Figure 2.2) Andreev discovered that the

solutions for the normal region of the Heisenberg equation of motion are of the form

u l ,k 0 .k
= A e1 1’ + B e1 2” (2.31)
v 0 1
where k1 = fiw/v and k2 = —f1w/v are the wave-vectors for an electron and a hole

respectively. 73. is the unit vector of momentum and v = pp / m (pp is the Fermi
momentum). The first term represents an incident particle(hole) and the second
term represents a reflected hole(particle). Notice the [Cl and k2 have completely
opposite sign. For energy less than A, u and v decrease exponentially inside the
superconductor. The solution certainly reminds us of the results from the BdG
equations. The four-fold degenerate solutions describe the NS interface as a
conjugate ”mirror” where we find the ”object” states (i.e. particles) and their
”image” states (time reversed holes). As shown in figure 2.2, an incident electron
with energy 5 above the Fermi-level is reflected as a time-reversed hole and charge
of 2e is transferred into the superconductor, in another words, adding a Cooper pair
into the superconductor. Likewise, a hole can be reflected as an electron and 26 (or
a C00per pair) is removed from the superconductor. This actually should be noticed
from the off-diagonal terms of the Hamiltonian in the BdG equation with A
corresponding to creating a pair, and A“ corresponding to annihilating a pair. On
the superconductor side

\/1+vnk3/w a”

e ‘ (2.32)

c
2 . -
v —2\/1—vnk3/w
where nk, = v'lx/ 1.22 — A2. The interesting aspect of the Andreev reflection is the

15

 

 

 

 

l
.
I
‘2
7:

gap ——‘\

 

 

 

pair
amplitude \\

 

Figure 2.3: Schematic picture of the proximity effect

coherent interference of these modified states because particles and holes do obtain
the superconducting phase through Andreev reflection. As will be discussed further
in the next section, the phase coherence near the interface is critical for the
proximity effect.

[For more exposure on both experimental details and theories regarding

Andreev reflection can be found in this excellent paper by Blonder et a1. [20]]

2.2.3 Proximity Effect

Proximity effect describes the presence of superconducting order inside the
normal metal in a NS system, even though there is no attractive potential between
the electrons. [7] [21] Obviously, this is just the continuation of the last section
because proximity effect is originated from Andreev reflection. The pairing
amplitude, as shown in figure 2.3, is nonzero in the normal side. The
superconducting order parameter decays exponentially inside the normal metal over
a characteristic length (thermal length) LT = W.

For a NS system, the phase coherence length, an energy-dependent quantity, is

16

different for the ballistic case and the diffusive case. [22]

hl/‘F

LE=UFt=—2—E'

(2.33)

where t = h/ 2E is the time that the electron and hole wave—functions remain

phase-coherent, and ’Up is the Fermi velocity. For the diffusive case,

120
= t: — .
LE VD 2E (2 34)

In the diffusive limit, from both the Andreev’s solutions and the BdG solutions, the
conductance depends on the interference of the states in the normal region close to

the interface and is proportional to

0 ~ my). + «m? = anal? + |u>h|2 + 2Re(t/Jel/)Z)l (2.35)

which is an energy-dependent quantity. So, the Ginzberg-Landau order parameter
112(x) overlooks this energy-dependence on a scale less than L E in which the
interference of the states is energy—dependent. When E < A, Reamer/1,2] ~ cosweh.
For E = O,

RCWJPZ) N COSSDeh = 0 (236)

For E> 0,

 

 

k. — k, = (be; + 2mE/h2 — \/k,2.. — 2mE/h2 = 2E/h2vp (2.37)

4,96,, builds up as we go away from the interface. The interference is lost on a scale of
L E = 5222p / 2E for ballistic systems or LE = VhD/ZE for diffusive systems. Note
that the energy E is the total particle energy which includes the thermal energy.

The superconducting order can extend through out the normal region if the sample

17

length L is less than the phase coherence length L4,. When L > L», the phase
coherent inference is cut off by L¢. Superconductivity can be thought of as a sort of
rigidity. And, this rigidity can be maintained even in the normal region by phase

coherent Cooper pair states.

2.3 SNS Josephson Effect and Andreev Bound
States

The Josepshon effect [23] [24] describes how a superconductor with long-range
order couples with another superconductor through a weak link. The coupling can
be expressed in the form of

Is = 103m.) (2.38)

where 10 is the critical current (maximum supercurrent can pass through the
device), (15 is the phase difference between the two superconductors, and I 5 is the

supercurrent.

 

Figure 2.4: Schematic picture of the Andreev bound states formed in a SNS long
junction with L < L¢

The microscopic picture of the Josephson effect is the Andreev bound states
formed between the two NS interfaces. [2] As shown in figure 2.4, an electron
reflected as a hole at the left interface gets to the other NS interface without losing

phase, and gets reflected again, thus, forms an Andreev bound state. This is an

18

energy-dependent process because L E = W~ [25] For the states with large E,
phase coherence is hard to be maintained. While, for the low energy states, the
phase coherence is maintained because LE is long enough. The Thouless energy
ET), = hD/L2, which is the energy broadening as a particle goes across the sample,
is a consequence of the system coupled with the environment. It turns out that, as
A governs the energy scale for ballistic SNS transport, ET), [26] sets the energy scale
for long diffusive junctions.

The supercurrent. inside the normal region can be derived from the

quasiclassical theory(reviewed in chapter 7) and has the form

IS = (UN/2) /d51m[j(5)]f0(5) (2.39)

in which the integrand is a product of Spectral supercurrent density I m[](E)]

and an asymmetric distribution function f0.

2.4 Spectral Supercurrent Density

Spectral supercurrent density I m[j(E )] is also called the energy spectrum of
supercurrent density. For a long SNS junction, it depends on D(diffusion constant),
sample geometry, inelastic scattering, and NS interface property. For a SNS device,
I m[](E )] is conserved with respect to the position. The expression of I m[j (E )] for
the diffusive situation is given in chapter 7 and the AppendixC. Let us first look at

the ballistic case.

19

 

Figure 2.5: Energy levels of the excitations in the normal region at a fixed value of
electron velocity [2]

2.4. 1 Ballistic case

Exact solutions [2] can be obtained by solving the BdG equation for a Ballistic

SN S junction. The eigenenergies are

 

E: = 2:”; [2(mr + M» 3F (>1 (2.40)

where 90(E) = arccos(E / A) is a slow varying function of energy. At low energies

E << A, we have (p(E) z 7r/2. So we have

h’Up
Ei =
" 2L*

 

[27r(n + 1/2) a: a] (2.41)

which depends only on (,b (the phase difference between the superconductors). The
effective sample length (with proximity correction) is L“ = L + hvp/Ao. E:
corresponds to the energy states going to right and left. When phase difference is
zero, energy states going both directions coincide and the total supercurrent is zero.
When the phase difference is not zero, the energy spectra for the counter-flowing
supercurrents are separated so that net current density appears. As is shown in
figure 2.5, due to the phase difference (15, the energy separation between two adjacent
levels going into the same direction is d2 2 7rvp/L", and the separation between the

levels going opposite directions is d1 = d2 X (qb/ 7r) which depends on (15. A graph of

20

 

v Y v V v v

 

 

 

 

 

 

g ”8 II
V
Q mu
g - _
Q
E 13/16“
5 1
5 15116 n f
3 .
U} . <-
» ll (
0
f l
0—~ AAAAA T 1.3

0.5 1 A
Excitation Energy (A)

Figure 2.6: Spectral supercurrent density vs. excitation energy for different phase
differences for a 3-terminal superconductor-semiconductor-superconductor device [3]
I m[j(E )] for a three-terminal superconductor-semiconductor-superconductor device

is plotted versus energy in Figure 2.6, in which each curve corresponds to a fixed

phase difference [3].

2.4.2 Diffusive case

For long SNS junctions, things are complicated by the diffusion process and the
broadening of the energy band. One has to solve for the Green’s functions of the
quasiclassical equations to obtain I m[](E )] A result. from my calculations by the
quasiclassical approach is shown in Figure 2.7. Compared with the ballistic case,
there are two dramatic differences. (1) The 6 function for the peaks are widened by
the factor of ET). which is about 5pV. (2) The amplitude of I m[j (E)] is reduced at
higher energies because of the energy dependence of the correlation length

LE = VhD/ 2E. For higher energies, fewer Cooper pairs can survive the trip from

21

 

 

 

 

Figure 2.7: Calculated spectral supercurrent density by quasiclassical approach. The
units for energy is meV

one NS interface to the other.

2.5 Distribution Functions

The distribution function describes how the occupation of the states is affected
by temperature and applied potential. It turns out the distribution function
depends on the sample length compared with electron-phonon length Le_ph and
electron-electron length Le_e. In figure 2.8, distribution functions are shown for the
N-n—N device at zero temperature. When L > Le_ph, the electrons completely
thermalize because the electron phonon interaction carries away all the heat
generated inside the normal wire. When Le_e < L < Le_ph, electrons exchange
energies with each other so that the effective temperature can reach much higher
than the lattice temperature. Local heating is produced because there is not
sufficient electron-phonon interaction to carry out the heat. This is called the hot

electron regime. When L < Le_e, Le_ph, the single electron energy is conserved. The

22

aevi‘ 1 f(x. E)

 

 

 

 

 

 

 

 

 

 

 

 

0 >
E
f(x, E)
1
o .E

 

 

 

 

 

 

 

Figure 2.8: Distribution functions for different sample lengths for N—n—N structures.
Upper right: L > Le_ph; lower left: Le_e < L < Le_ph; lower right: L < Le_e, Lew),

[4]

23

 

Figure 2.10: A schematic picture of how a quasiparticle distribution function varies
in space. The solid dot stand for an particle and the circle stands for an Andreev
reflected hole. The size of the circles stand for the amplitude of distribution function
of the particle/ hole

Boltzmann equation with zero inelastic collision (a; f = 0) gives a distribution of
staircase [4]. This is the regime in which our experiment is performed.

The distribution function for the SN system shown in Figure 2.9 is quite
different than the N-n-N. A schematic picture showing how the distribution function
changes inside the normal wire is shown in figure 2.10. First of all, the Andreev
reflection is a two-particle process. So, the excitation ranges from -eV to eV. The
second is that the distribution function still varies linearly, but over the distance of
2L instead of L. For a particle at the normal reservoir going to the left, its
distribution function reaches 1 / 2 at the right NS interface. This can be obtained
through the relationships f (E) = f (—E)(two particle process) and
1 - f h(‘E) = f (E)(boundary condition). Figure 2.11 shows schematically a real

distribution (including the correction of proximity effect) at a point very close to the

24

 

 

 

E
-eV 0 eV
Figure 2.11: A schematic picture of the distribution function with correction due to
both temperature and proximity effects
NS interface. The small wiggles are due to the proximity effect.

For the nonequilibrium proximity effect, it is convenient to resolve the

distribution function into a symmetrical part and a symmetrical part

2f(E)=1-(fo + f1) (2-42)

where f0 is the antisymmetric part which corresponds to the effective
temperature change in the system, and the fl is the symmetric part which
corresponds to the effective chemical potential shift or charge imbalance. This is
why f0 and f1 have different x-dependence inside the wire. For a SNS system, f0
and f1, which are the driven forces for heat current and the charge current, are

coupled through supercurrent density I m[j (E )]

2.6 7r Junction

7r junction refers to a 7r phase shift to the phase difference (25 in the Josephson
relation so that it changes from I S = [0 sin qb to I s = [C sin(7r + ¢). As discussed

above, inside a. long Josephson junction (1 << L < L¢), there are supercurrent

25

carrying states flowing into both directions and the net current

IS = (UN/2)/delm[j(5)]f0(e) (2.43)

is the result sum of all supercurrent-carrying states. When the phase difference (b is
zero, the supercurrent. flowing in both directions cancels each other [3] and

Im[j(5)] = 0. When a finite qb # 0 is produced (by either applying magnetic flux or
driving current), the currents carried by states going into opposite directions no
longer cancel each other. The direction of the net supercurrent depends on which
direction most current carrying states are going. In addition to the phase difference
cf) which determines each supercurrent-carrying state, the antisymmetric distribution
function f0(occupation of the states) selects which states to exist. As a normal
injection increases, the energies of the quasiparticles change also, so, the occupation
of the states changes. By shifting the occupations of the quasiparticle states along
the energy spectrum, the direction of the net supercurrent is alternated. So, when
we increase the injected quasiparticle current from zero, the supercurrent decreases
until the net supercurrent changes direction when it reaches a certain energy. This
is when 7r junction occurs. It appears in the Josephson relation as a 7r phase shift to
the phase difference d.

The appearance of the 7r junction can take place by controlling the width of the
staircase distribution function f0. For our experiment in which we inject
quasiparticle current, the width of f0 is 26V (from —eV to 6V). (figure 2.12) The
spectral supercurrrent density is an odd function of energy and is not affected by
either normal injection or the temperature. Then, the sign of the integrand of the
net supercurrent 2.43 depends on the value of eV. For eVN < E0, the junction is at
zero phase state(the Sign of the product of f0 and I m[j (E)] is positive); for

eVN ~ E0, 1C goes to zero; and for eVN > E0, 7r phase state appears (the sign of the

26

 

 

 

 

 

Figure 2.12: Antisymmetric distribution function f0 (upper graph) and the spectral
supercurrent density

product of f0 and I m[ ](E)] is negative).

2.7 Summary

As discussed in earlier sections, the BdG formalism can strictly solve the NS
problem for the ballistic case, but is difficult to solve for the diffusive case in which
the length greatly exceeds the elastic mean free path I. The Quasi-classical theory, a
application of the Gorkov equations to the dirty limit, has become a powerful tool
for many practical diffusive calculations. I will discuss quasiclassical theory in

chapter 7. There is also the matrix approach which is based on Landauer’s formula.

27

More details on discussing the differences is provided in the paper by Ralph and
Ambegaokar. [27]

28

Chapter 3: Fabrication Methods

Introduction: Micro-fabrication has become the key technology that enables the
dynamic growth of the IC industry and fundamental research of small systems. The

fabrication techniques include

o Photolithography

Electron beam lithography

X-ray lithography

soft-lithography

deep UV resist technology

polymer and organic methods

It seems like organic methods will be the future of nano—fabrication since they pull
together physics, chemistry, electrical engineering, chemical engineering, biology,
and even medical sciences. The fabrication methods involved in this thesis are

surface preparation, photolithography, electron beam lithography, and etching.

3.0.1 Surface Preparation

Contaminants cause serious problems with lithography processes so they must

be removed properly before resist coating. Typical contaminants are:
0 Dust from dicing the wafer
0 Atmospheric dust
o Resist and developer residue from previous lithography

o Bacteria

29

o Solvent residue from previous chemical processes
0 H20 residue
0 Oil

A nitrogen gun can remove dust that falls upon the surface of the substrate,
laser scribing can remove the dust from dicing the wafer, Micro (a detergent) can
remove the oil particles, plasma etching can remove resist residue, and deionized
(DI) water can remove most of the bacteria.

After removing these contaminants, start the fine cleaning process by first
putting the substrate into acetone and put in the ultrasonic bath for 5 to 10
minutes. Often it helps to preheat the acetone. But, the temperature should not
exceed 680C which is the flash point of acetone. Then, rinse the substrate with
Isopropyl alchohol(IPA). Last, rinse again with DI water. Finally, blow dry it with

nitrogen gun.

3. 1 Photolithograhy

3.1.1 Photo Resist Coating

Thin film spin coating is often the first step of the whole fabrication. It is a
process to coat the substrate with layer(s) of resist (such as photo resist or e-beam
resist) by using a spinner with controllable spin speed. It normally includes the

following steps:

a Mount substrate on the stage
The substrate should be placed in the center of the stage so that the film will

be uniform. Then, apply vacuum to hold the substrate down.

0 Use Nitrogen gun to clean the surface of the substrate.

30

This is to blow away any dirt that may fall onto the surface while transferring.
Use gentle blow if there are already some features on the substrate to avoid

damage.

Use pipette to apply liquid resist onto the center of the substrate
The resist. will form a ball at the center. Squeeze the pipette uniformly to

avoid bubbles because they can cause singularities on the film.

Immediately start spinning by pressing the starting button.
Centrifugal force will cause the drop of resist to move in waves away from the
center of the substrate until a film of the resist is formed. The normal spin

time is 1 minute.

Wait
Before removing the substrate, wait for 15 seconds to allow good adhesion

between the resist and the substrate surface.

Remove the substrate and put it into the baking oven.

One ought to practise in order to control the thickness and uniformity of the

film. The thickness is normally determined by the concentration of the resist and

spin speed. The thickness vs. spin speed curves for various resists with different

concentrations are provided in the clean room. These numbers are generally good

for wafer substrates. But, when the size of the substrate is very small, the film

becomes thicker because the centrifugal force is reduced compared with the surface

tension at the edge. For example, when spinning 4% PMMA onto a 7mm x 7mm

Silicon substrate and onto a 5mm x 5mm substrate, the thicknesses of the films are

240nm and 400nm, respectively.

Though the range of the speed can be chosen from 1000 rpm to 6000 rpm, 2500

rpm is generally the minimum speed to ensure the uniformity of the film. The

31

thickness and the concentration of the resist film should be chosen to optimize the
lithography processes. For example, if one wants to fabricate very fine features on
the substrate, the aspect ratio can not be too large. This means the resist can not be
too thick. Meanwhile, the resist can not be too thin because it can cause problems

with the lift—off. Normally, the resist should be 3 times the height of the features.

3.1.2 Photo Resist Baking

The baking time is minimum 45 minutes. Extending the baking time helps to
improve the uniformity of the resist. This is only necessary when very fine features
are desired. The baking temperature is usually 95°C. This temperature is not fixed
under certain circumstances. For example, if there are already some features (i.e.
nano wires) underneath the resist, baking at 95°C may cause nucleation and

damage the wires. One has to lower the temperature to avoid damage.

3. 1 .3 Exposure

 

(x.y.0) alignment of uniform UV exposure illumination
masmsubmt" uuuuuuuuu
_ a he chrome on glass photomask
I I I l I l l

photoresist (PR)

  
   

i *1 substrate wafer

latent image created In I 11355;».53335‘133193‘ :;::::: 0m,

photoresist afler exposure

/ wet chemical development \

I_-
.

    

. u” -
,. . I . .
32"” ‘2‘; 3231‘Y4‘3f‘: 33:33:33.“ o'n'tlx‘T‘lif‘i‘iV‘I‘M'tmwa k""i'l’"-"!\".'1‘n‘nC‘PJTnbmm'aW'fi-K‘vfl‘fim‘m9.

NEGATIV PHOTORESIST POSITIVE PHOTORESIST
Photoresist is photopolymerized where Exposure decomposes a development
exposed and rendered insoluble to the inhibitor and developer solution only
developer solution. dissolves photoresist in the exposed areas.

Figure 3.1: A schematic picture of photo printing

Exposing the photo resist(contact printing shown in figure 3.1) is done by using

UV light (wavelength 345 nm) shining through the mask which is in contact with the

32

resist. Because the exposure is through the mask, the volume of the resist exposed
depends on the interference and diffraction of the UV light, as well as the thickness
and density of the resist. The proximity effect, which describe the widening effect of
the features due to the interference of the UV light, becomes critical, especially
when the feature size is very small. The proximity effect. can be reduced by
decreasing the thickness of the photo resist. But, the resist can not be too thin
because it. might be difficult to produce a clear undercut and cause trouble for the

liftoff. Generally, it is harder to have good undercut when the resist is too thin.

3. 1.4 Development

Right after the exposure is the resist development. Holding a corner of the
substrate with your tweezer, submerge into photo-developer 452. Slowly shake it in
the liquid back and forth for 45 seconds. Then, take it out quickly and submerge
into DI water for 30 seconds. Then, blow dry with the nitrogen gun. Be careful that
you do not point too close to the substrate if you already have some features on it.

The pressure of nitrogen may damage them.

3.1.5 Evaporation and Liftoff

The metal evaporation is done in a Edward Aut0306 thermal evaporator which
has a base pressure 2 x 10‘7 torr. The current and evaporation rate for some
common metals are listed in table 3.1. The liftoff is a process to remove the
excessive metal and resist on the substrate. This is normally done in hot acetone
(heated to 680C). Observe closely and take the substrate out as soon as the liftoff is

complete. Rinse with Acetone, IPA, and DI water.

33

 

 

 

metal current (Amp) evaporation rate/l/s
Ag 2.2 3

A1 2.6 2.8

An 2.4 3.2

 

 

 

Table 3.1: Metal deposition current and rate for Edward Auto306 thermal evaporator

3. 1.6 Photo-alignment

Photo—alignment is to optically align the substrate with the mask so that the
exposure takes place at the desired areas on the substrate. This is necessary when
putting together different metal patterns. The goal of alignment is to overlay some
alignment marks on the substrate with the alignment marks on the mask.
Obviously, the marks put down on the substrate have to be exactly the same as
those on the mask. This has to be emphasized because some choose to put down
alignment marks by e-beam writing. The magnification can be off a little bit so that
the marks may be slightly different than those on the mask. This will reduce the
alignment accuracy. The best method is to have two masks with identical alignment
patterns on them. This way, one can put down the alignment marks on the
substrate by optical lithography. Another factor that affects the alignment accuracy
is the magnification of the lens. 200x magnification or better is needed when 0.1
micron accuracy is desired. The third factor that affects the alignment accuracy is
the thickness of the resist. The resist is in contact with the mask. So, the distance
between the marks on the mask and the marks on the substrate is at least the
thickness of the resist. When it is too thick, it is impossible to have both sets of

marks in focus simultaneously.

3.2 Electron Beam Lithography (EBL)

Electron beam lithography (EBL) is a specialized technique for creating the

very fine patterns through scanning a beam of electrons across a substrate covered

34

with a e-beam resist film which is sensitive to the electrons. Similarly to the photo

development, the exposed resist is removed through e—beam development. Figure 3.2
is a schematic picture of a typical e-beam process. The process of forming the beam
of electrons and scanning it across a surface is very similar to what happens inside a

television, but EBL typically has three orders of magnitude better resolution.

ElectronlBeam lmetal

resrs ‘ "
[ Jsubstrate . ..-.7...‘4._resl§,
ubstrate

(a) (c)

». A - .1.-..,.'.... 1
up

J

W‘lll‘-r‘..z".lst, '

   
 

 

   
 
  

 

 

   

‘
substrate I

.
.

-a

'r.

lIn\ \ 72.3

I I U substrate

(b) (d)

 

 

Figure 3.2: E-beam Process: (a) Spin-coat the substrate with e-beam resist. Expose
the resist with electron beam. (b) Develop the resist. (c) Metal deposition. ((1) After

lift-off

 

[1 Pulse]
Scanning D” ° PG

corls \68 8 RE)“. D A C

 

Beam
/ blanker

 

 

 

 

 

 

 

 

 

 

 

v

 

 

 

 

 

 

 

 

 

 

lnt.
Va'< ADC
L d :3 HP Vectra
°a & NPGS

 

 

 

 

 

 

Lock

 

 

 

CRT] Scan Blanking
SEM Control

 

 

 

 

 

 

 

 

 

 

Figure 3.3: Block diagram showing the major components of a typical electron beam
lithography system

The main attributes of the EBL are

35

0 very high resolution, almost to the atomic level

0 flexible technique that can work with a variety of materials and an almost

infinite number of patterns
0 slow, being one or more orders of magnitude slower than optical lithography

o expensive and complicated - electron beam lithography tools can cost many

millions of dollars and require frequent service to stay properly maintained

The scanning electron microscope (SEM) JEOL 840 (see figure 3.3 is used for
imaging samples. It is also equipped with a computer-controlled pattern generating
system enabling the writing of patterns for devices with feature sizes as small as 50

nm. Some of the features on the SEM/EBL system include:

o Acceleration voltage: up to 40 kV

Filament type: Tungsten

0 Beam size: 8 nm at 35 kV

Pattern design: DesignCad

Pattern formation: Vector scan

Pattern generating: Nabity Lithography system

Beam blanker: 8 MHz electromagnetic

3.2.1 Electron-Solid Interaction and Proximity Effect

[The following is a brief description of how the electron solid interact when

electrons going into the substrate. For more exposure, refer to the book on

microlithography by Mark A. McCord and Michael J. Rooks [5].]

36

 

 

  

- lOkV

4 l l l I I I I I I I I I I I I I I I

Z pm (a) Z um (b)

 

 

 

 

 

 

Figure 3.4: Simulated profile of the energy absorbed from an electron beam exposure
[5]

As the electrons penetrate the resist, they experience many small angle
scattering events (forward scattering), which tend to broaden the initial beam
diameter. As the electrons penetrate through the resist into the substrate, they
occasionally undergo large angle scattering events (backscattering) which are
responsible for causing the proximity effect. During this process the electrons are
continuously slowing down, producing a cascade of low voltage electrons called
secondary electrons. Figure 3.4 is a simulated picture of the traces of the scattering

of the incident electrons.

Forward Scattering

As the electrons penetrate the resist, some fraction of them will undergo small
angle scattering events, which can result in a significantly broader beam profile at
the bottom of the resist (see figure 3.4). The increase in effective beam diameter in
nanometers due to forward scattering is given empirically by the formula
d f = 0.9(Rt/Vb)1'5 [5], where R, is the resist thickness in nanometers and Vb is the
beam voltage in kilovolts. Forward scattering is minimized by using the thinnest

possible resist and the highest available accelerating voltage.

37

Backscattering

For those electrons experiencing large angle scattering, they can return back
through the resist at a significant distance from the incident beam, causing
additional resist exposure. This is called the electron beam proximity effect. The
range of the electrons depends on both the energy of the primary electrons and the
type of substrate. The fraction of electrons that are backscattered, is roughly
independent of beam energy, although it does depend on the substrate material.

Normally, backscattering decreases with lower atomic number material.

Secondary Electrons

As the primary electrons slow down, much of their energy is dissipated in the
form of secondary electrons with energies from 2 to 50 eV. They are responsible for
the bulk resist exposure. Since their range in resist is only a few nanometers, they
contribute little to the proximity effect. Instead, the net result can be considered to
be an effective widening of the beam diameter by roughly 10 nm. A small fraction
of secondary electrons may have significant energies, on the order of 1 keV. These
so-called fast secondaries can contribute to the proximity effect in the range of a few

tenths of a micron.

Proximity Effect Avoidance

To optimize the dose is always the first thing to do. Also, using bilayer resist
with different e—beam sensitivity can reduce the forward scattering effect. Higher
beam voltages, from 50 kV to 100 kV or more, also minimize forward scattering,
although in some cases this can increase the backscattering. This method is
extremely good if the substrate is very thin so that most of the electrons go right
through. Meanwhile, by going to very low beam energies, where the electron range

is smaller than the minimum feature size, the proximity effect can be eliminated.

38

The penalty is that the thickness of a single layer resist must also be less than the
minimum feature size so that the electrons can expose the entire film thickness. It

also makes is hard to focus. So, it is a limited method.

Proximity Effect Versus Dose of Exposure

I have done a series of tests to show how the dose affects the proximity effect
(see SEM pictures in figure 3.5), and the relationship, shown in figure 3.6, is

obtained.

 

Figure 3.5: SEM pictures of a series of single paths (Ag lines) with following doses in
nC/cm: 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, and 2.8.

3.2.2 Electron Beam Writing

The pattern design is done by using DC6 designCAD and stored as designCAD
file. A run file ”filename.mef” is produced by the ”mrf” command. This is the file
that has all the writing conditions for each element of the pattern. Before writing,

get the microscope in focus by focusing on a subject on the substrate. Try focusing

39

, line width vs. dose

width(nm)
f

45-]

 

 

T j l ' I ' I V I ' l

2.0 2.2 2.4 2.6 2.8 3.0
line dose (nC/cm)

Figure 3.6: Width of the single path versus dose

in well at 200KX or better. Then, turn on the external mode, start the writing by a

command ”pg filename”. The details are provided in the Nabity Menu book [28].

3.2.3 Development

Right after the writing is the resist development. Submerge the substrate into
e-beam-developer MIBK(1:3). Slowly shake it back and forth for 55 seconds. Then,
take it out quickly and submerge into IPA for 40 seconds. Then, rinse it in DI water
and blow dry with a nitrogen gun. Be careful that you do not point too close to the
substrate if you already have some features on it. The pressure of nitrogen may

damage them. Evaporation and liftoff are identical to the photolithography.

3.2.4 Electron Beam Alignment

When multi-level Lithography process is needed, e-beam alignment becomes

critical because the small features demand very high alignment accuracy. The first

40

step is to create an alignment file which includes the alignment marks inside some
scanning windows. The alignment run file will control the beam to scan the
windows and find the marks. Then, overlay the marks with the marks of the file.
An offset will be computed when the overlay is done. This offset will be the offset
when writing the pattern.

Take 0.1nm alignment accuracy for example, the smallest alignment window is

around 4nmr4nm. So, a 3-level alignment process is adopted. (table 3.2

 

 

 

size of the marks size of the windows magnification
Levell 1mm 1mmx1mm 20X
Level2 30umx30pm 40am x 40pm 100X
Level3 3nmx3um 4am x 4pm 1000X

 

 

 

 

Table 3.2: 3-1evel alignment.

The successful alignment has to be done by using both mechanical alignment
(by moving the stage) and computational alignment (computer calculated matrix to
shift the beam). The first is used at low magnification and the adjustment is rough.
The purpose for the mechanical alignment is to set the sample as close to the center
of the screen as possible. If you do not, the 3rd level alignment will be out of the
range of the screen. The computational alignment is used for 2nd and 3rd level
alignments (lst level too if you choose). Each time you do alignment the program
deals with both translation and rotation by computing the matrix that adjusts the

beam.

0 Alignment at 20X:
Overlay the alignment pattern from your alignment file with alignment marks
as well as possible by moving the stage. On the JEOL monitor you can
roughly see where you are. But, for more accurate position you have to trust
only the computer monitor screen. You can check how close you are by going
into the 2nd alignment level. If you see, at the first scan, that the marks are

pretty much inside the windows (say 75%) you are ok. But, if you only see a

41

little bit of each mark appearing in the scanning windows, you will probably
be in trouble when going into the 3rd level because the center of the sample
may well be out of the range of the screen. Then, you have to go back to 20X,
adjust the sample mechanically while scanning until the both sets marks

overlay. At this moment it is ready to go into the 2nd level.

100X level:
'Ify to overlay the pattern and the marks by moving each individual mark
onto the real mark, then hit ”r” (return) key. Repeat this process until you

get. exact overlay. Now, you are ready to go into 3rd level

1000X level:

Note that there is still another factor that needs to be taken into
consideration: shift of the center of beam with changing magnification from
100X to 1000X. This factor becomes important when going into high
magnification. You have to measure the offsets and put them into the offset
settings in your alignment file. The typical offset from 100X to 1000X is
(4.6nm and 8.1pm). The alignment process is the same as that at 100X. Right

after it is done, write pattern immediately with the command
Pg filename a

”a.” calls for the computed matrix offset. It is important to remember to input

the offset to your writing file also.

42

3.3 Actual Sample Making

3.3.1 Review of Methods
Fabrication Methods

I have tried several different fabrication methods to fabricate the sample. One
of the reasons is that we did not initially have ion—milling in the evaporation

chamber.

0 The first method
A gold pattern was first put down to serve as the alignment marks. Then an
insulating layer of SiO (70 nm) was deposited on top of gold. Then, the
normal wire and normal reservoir were put down by e-beam lithography with
e-beam alignment. The last step was to put down the superconductor (Al)
using the same alignment marks. The problem was the visibility of the Au
pattern. SiO is an insulator, it is very hard to see through it. That made the
alignment a pain. We could not go any thinner because the SiO has a

minimum thickness about 70nm to maintain its insulating behavoir.

0 Angle Evaporation
The second idea was based on the angle evaporation method. The whole
pattern is one e—beam pattern (figure 3.7). On the pattern, the reservoirs and
the wire are separated. The idea is to bring the wire and the reservoirs in
contact by evaporating at different angles. It requires an angle tilting during
the evaporation. Though we designed the pattern and developed the skills for
e-beam lithography, we did not pursue this idea because the tilting sample

holder was not ready at the time.

I then tried two other methods both of which worked. They are described in

detail in the later sections.

43

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.7: Schematics of an angle evaporation method. The two sets of patterns are
from evaporations at different angles

Material Combinations

Meanwhile, I also tried different metal combinations for the normal conductor

and superconductor. Below is a list of combinations I have tried.

0 Pb«In-Au(90%,8%, and 2%) and silver
Choosing Pb—In—Au was due to the fact that the surface oxidation of Pb—In—Au
is very little. The problem is that when the film of Pb-In-Au is less than

100nm thick, pin-holes were formed.

0 silver(Ag) and aluminum (Al)
This works fine except that silver is not as stable as we desired. Ion-milling

has to be applied to the first metal before the second metal is deposited.

o gold(Au) and aluminum (Al)
I spent a lot of time on this because one of the first tests on a 0.5,um wide and
2am long Au wire in contact with Al reservoirs had a resistance of 8 ohms.
But, when I fabricated narrow wires the interface became insulating. This

turns out to be the well known purple plague. There are alloys formed

44

between gold and aluminum and these alloys are insulators.

0 Nb and Au
I had problems with heating the substrate while sputtering Nb due to the
small size of the substrate. The Nb film does not stick to the substrate very

well. It becomes problematic during liftoff.

So, after all these tests, we chose Ag and Al.

3.3.2 Method 1: E-beam First, Photolithography Second

Unlike the traditional way of making samples (lay down the big features first by
photo processes, then e—beam process to put down the nano wires), I first write the
nano features and alignment marks with e-beam lithography. Then, with
photo-alignment, I put down the normal reservoir and superconducting reservoirs as
illustrated in figure 3.8. This method shortened the whole process almost by half.
And, it avoids ion-milling the Al which can be rather difficult due to the tough

oxide.

Substrate preparation

The commercial wafer is very clean already. I first spin-coat the surface with
photo resist 81813. Then, bake it at 95°C for 40 minutes. This is to protect the
surface when it is diced into 7mm x 7mm square pieces. The surface is protected by
the resist until it is washed away with acetone and IPA before spin coating the

e—beam resist.

E-beam Process

0 Resist Preparation

The e-beam resist is a bilayer resist in order to obtain good undercut. The

45

 

Figure 3.8: Three Steps of making the the sample

 

 

Bilayer Resist Spin Thickness Bake
P(MMA/MAA)6% 5750 rpm (1 min.) 260nm 160°C, 1 hour
PMMA 4% 5950 rmp (1 min) 140 nm 160%, 2 hours

 

 

 

 

Table 3.3: Spin and baking conditions for 7mm X 7mm Si substrate

magnification measured beam current dose step size
1000x 7 pC 2.8 pC/cm 13.7 .i

 

 

 

 

 

 

Table 3.4: E—beam writing conditions for single path

develop rinse
MIBK 1:3 for 55 seconds DI water for 30 seconds

 

 

 

Table 3.5: E—beam resist Development

bottom layer is co—polymer (P(MMA/MAA)6%) The top is PMMA. The total
thickness of the bilayer resist is around 400nm. Two small drops of silver paint

are put on top of the resist symmetrically for e-beam focusing.

E—beam writing

Instead of using mask as in photolithography, pattern is designed with CAD
and saved as a CAD file. A run file can be made by using software DC6 so
that all the writing conditions for each part of the pattern are defined. These
conditions include layer, color, dose, offset, beam current, step size, line
spacing, magnifications, pause, etc. Then, the ”pg filename” command starts
the writing. For a single path feature, the writing and development conditions

are shown in Tables 3.4 and 3.5.
Thermal evaporation of Ag

Liftoff
The liftoff is done in heated acetone (58°C) for 1 minute. Often it takes much

longer when the undercut is not good or the feature is very small.

Metal current deposition Rate thickness
Silver (99.999%) 2.3 Amp 3.2 21/8 65 nm

 

 

 

 

 

Table 3.6: Thermal evaporation of the nanowire

47

resist spin thickness bake expose develop
81805 4500 rpm 5000A 70°C 3.4 seconds 45 seconds

 

 

 

 

 

 

 

Table 3.7: photo printing condition

Metal current deposition Rate thickness
Silver (99.999%) 2.3 Amp 3.2 2178 220 nm

 

 

 

 

 

Table 3.8: Thermal evaporation of Silver reservoir

First Photolithography

Notice the baking temperature is very low (Table 3.7). This is because there is
already the silver wire on the substrate. It will be definitely damaged if the baking
temperature is 95°C. I have tested baking at different temperatures ranging from
40°C to 90°C and found that photo exposure can be successfully done even when
the resist is baked at 55°C for three hours. I used 70°C for baking the real sample.
Before exposing the resist, alignment has to be done so that the normal reservoir
will sit on the nanowire. By moving the stage where the substrate sits while
watching through the microscope, overlay the alignment marks on the substrate
with the marks on the mask. The magnification of the lens is 200x. Before
exposing, make sure the resist and the bottom of the mask are in good contact.

liftoff time is 30 seconds

Second Photolithography

Conditions are the same as in the first photolithography procedure except that
a gentle ion-etching is applied before deposition of the superconducting reservoirs
(Aluminum) to ensure good contact between Ag and Al. We used high energy Ar

gas to ”bombard” the surface of the substrate to remove the dirt before putting

voltage current chamber pressure Ar flow
250 V 10 mA 3 x 10“4 torr 3 sccm

 

 

 

 

 

Table 3.9: Ion-milling

48

metal upon it. The conditions for the ion-etching are in Table 3.9.

The SEM pictures of the sample are shown in figure3.9.

The actual dimensions are given in figure 3.10. The T-shaped Ag wire, 70nm
wide and 60mm thick, is connected to two S reservoirs (100nm of Al) and one N
reservoir (220nm of Ag). The distance between S electrodes is 1.1um, while the

distance from the top of the “T” to the N reservoir is 4.5nm.

3.3.3 Method 2: Photolithography First, E—beam Second

Another method starting with photolithography was also used. This is to put
down first the SC reservoirs by photolithography. Then, put down the normal wire
and the normal reservoir by e-beam alignment and e—beam writing. Most of
processes are very similar to the method above except for the e—beam alignment and
the ion-etching of the Al surface.

This method requires an extra e—beam process to write down the alignment
marks of Au. Note that the Al marks will not be seen because its atomic number is
too close to that of Si. The alignment details are given in section 3.2.2.

The ion—etching was difficult because the A1203 is very hard to remove. This
problem was solved by putting an addition Al layer between the copolymer and
PMMA so that A1203 will serve as a mask for etching. This is a rather messy
process. It results in a widening of the pattern. Figure 3.11 is a SEM picture of the

sample.

49

I I
7.0kV 23.2mm x18 0L SEiU) 8/17/2001 10:31

 

Figure 3.9: SEM pictures of the sample. The upper picture was taken at 10kX and
the lower picture was taken at 18kX

50

l.l um

     

Figure 3.10: The dimensions of the actual sample

 

Figure 3.11: SEM picture (taken at 20kX) of the sample by method 2

51

Chapter 4: Electronics for

Measurement

4. 1 Sample Mounting

The sample is measured in a mixture of H e4-H e3 in a dilution refrigerator
which has a base temperature 28 mK. The sample, which is on a 7mm x 7mm
square shaped substrate, is mounted on a sample holder which has a diameter of 12
cm (see Figure 4.1). There are six isolated cells on the holder each of which is
covered with Cu sheet. Copper wires are soldered onto each cell, then connected to
the sample through Au wires which are fixed by silver paint on the holder and the
substrate. There are three current (I) leads (twisted together) for driving currents
and there are three voltage (V) leads (also twisted together) for the voltage
measurement. Figure 4.2 shows the sample actually sitting on the sample holder.
On the substrate, V and I pads alternate. So, the advantage for the design of the
holder is that one V lead and one I lead switch sides so that there are no

possibilities of crossing the Au wires when connecting them onto the pads.

4.2 Electronics

All electrical leads to the sample are filtered twice - at the top of the cryostat
with commercial LC “7r” filters with a 3dB roll-off at about 100 MHz, and in the

mixing chamber with discrete RC filters with a 3dB roll-off at 160 kHz.

52

silver paint solder joint

   

Cu wire

Figure 4.1: Schematics of the sample holder

 

Figure 4.2: Schematics of the sample mounting

53

 

 

mSSmon

our

005::
So

 

ooFum

 

 

 

 

 

 

54

 

 

 

 

 

 

 

 

Figure 4.3: electronic setup for the DC measurement

4.2.1 DC Measurement

The setup for the dc measurement is shown in Figure 4.3. The source is a HP
function generator. The typical signal used for our measurement has a frequency of
0.01 Hz for the slow dc measurement and 23Hz to 46 Hz for the fast dc averaging
measurement. The signal measured from the sample goes through two pre—amps

before going into the DMM.

4.2.2 AC Measurement
Summing Box

The driving signal has a fast small ac signal riding upon a slow dc sweeping
signal. This is done by using a summing box (see figure 4.4) which has a gain of 0.1

for the dc signal and a gain of 0.01 for the ac signal.

Data Analysis

As shown in figure 4.4, a lock-in amplifier is used to measure the potential
change in response to the ac signal. The output is then converted into differential

resistance through

dV __ VLOCk_,n * lock — insensitivity * RB
d1 _ Vac 10V gpre—amp

 

 

(4.1)

where Vac).-." is the dc voltage at the output of the lock—in amplifier, and Va,C is the
ac drive signal at the output of the summing box, and Rb is the ballast resistor

shown in figure 4.4.

55

 

 

0.2 vnnm

 

grammes.

 

 

 

 

 

 

 

 

, omw mm
OO—hm EOE—3w
ma 3 553.5
53.8..
xon 02.88%
+ |_ 9.2
- 9.8?
Cx _.

 

 

So

ill Ooowum

 

 

 

 

 

09:40”

ml

 

 

 

, 8» mm -
, .2255
o< ‘

 

 

Figure 4.4: Electronic setup for ac measurement

56

Chapter 5: First Experiment:

Dangling Arm

 

Figure 5.1: SEM picture of the sample

The purpose of this experiment is to test our hypothesis of the counter-flowing
quasiparticle current and supercurrent which was discussed in the introduction. To
make the discussion easier, let us assume that R1 = R2 for the moment. If the dc
Josephson effect does occur when I N is less than a critical value [C(S N ), the normal
current splits into two going into both superconductors so that the two SC reservoirs
are at the same potential. Here, IC(S N ) is defined as the maximum injected current
at which the switching from dc Josephson state to the ac Josephson state takes
place. So, when IN < IC(SN), IN E’ 215 = 2IC sin Q5, where 10 is the critical
supercurrent of the SNS Josepshon junction. The effective resistance between the N
reservoir and the left SC reservoir is then Ref; = R0(q§) + R1(¢) / / R2(¢). When
IN > IC(SN), the effective resistance will become Reff = Ro(qb) + R1(¢). Notice
that the effective resistance depends on the proximity effect which is reflected in the

phase difference (b. This is why the resistance is written as a function of q).

57

2,- T=28mK
20, SIope=RO+R1

     
  
 

15-1

 

 

 

Slepe= ,
R0+R1IIR2

 

V(uV)

 

 

 

Figure 5.2: DC measurement of V—I

5.1 DC and AC Measurement between the N and
S Terminals

The resistances of each arm (R0, R1, and R2) were obtained beforehand by
four-terminal measurement at large values of injection current(IN > [C(SN)). The
measurement were done between all three pairs of terminals so that we obtained
values for R0 + R1, R0 + R2, and R1 + R2. Then, the values for each arm were

determined (first three columns in table 5.1).

Be R1 R2 Ro+R1//R2 Ro+R1
experiment 7.39 4.00 4.80 9.4 9 11.352
prediction base on 120,121,122 - - - 9.50 -

 

 

 

 

 

 

 

 

Table 5.1: Experimental results vs. prediction

The experimental results shown in figure 5.2 are from the dc measurement
between the normal terminal and the left superconductor terminal. The data
indicates that the effective resistance, which is the slope of the V-I curve, changes

when crossing some critical current IC(S N ) Their values agree very well with the

58

T=28mK

171

16-
15-1

14-

 
   

R=R0+R1

.,...,\

R=R0+R1I/R2

_______________________ /

V V . I I V l V r Y 1' fi
-15 .10 .05 0.0 fix 1.0 1.5

l(uA) |C(SN)

13-

 

dV/dl (ohms)

 

 

Figure 5.3: AC measurement. of (dV/dI) vs. I

prediction base on the values for R0, R1, and R2.(Table 5.1) It convinces us that the
effective resistance indeed changes from R0015) + R1(qb)//R2((,/>) to ROM) + R1(¢)

when passing the critical current [0.

The ac measurement is aimed to reproduce the dc results and to look into more
details that the dc measurement might have missed. The data of dV/dI versus I are
shown in figure 5.3. The measurement shows a result 94$) for Ref; for
[N < [C(SN) and 11.69 for IN > [C(SN). Again, they agree very well with the dc
measurement. Meanwhile, the ac measurement does reveal some details which the
dc measurement did not detect. The switching region from dc Josephson to ac
Josephson shows some small oscillations. This may be because a finite potential
difference appear introduced by the dramatic change in the resistance so that a
small voltage appears when the switching takes place. [29] Also, this also has to do

with the fact that the two SN interfaces are not quite the same.

59

”-51 DC-SNS
10.0 - T=28mK

 

 

 

 

 

. . , .
-1 .0 -0.5 0.0 0.5 1 .0

KM)
Figure 5.4: DC measurement between the superconducting terminals

5.2 DC and AC Measurement between the S and
S Terminals

DC V-I Measurement is also performed between the superconducting terminals
(results are shown in figure 5.4). The critical current is then labeled IC(S N S ) The
effective resistance is zero when I N < [C(S N S ) due to the fact that the two S
terminals are at the same potential. When I N > IC(S N S ), it switches to the ac
Josephson state. The effective resistance is then Ref; = R1 + R2. We also performed
ac measured between 8-8 terminals and the results are shown in Figure 5.5.

Agreeing with the other measurement, it shows that the resistance is zero when

I < IC(SNS), R1+ R2 when I > [C(SNS).

5.2. 1 Analysis

Both dc and ac results confirmed the fact that the quasiparticle current splits
at the cross point. Theoretical analysis of the dangling arm experiment was carried

out by Shaikhaidarov et al [30], who performed an experiment similar to ours. The

60

 

SNS
T=81 mK
— T=193 mK

 

 

 

R (n)

 

 

[ill-A)
Figure 5.5: AC measurement between the superconducting terminals

resistance, a quantity which is affected by the proximity effect, is phase dependent

R(¢) [30]. Take the effective resistance between the NS terminals
R, f f(N S ) = Ro(¢) + R1(¢) / / 122(45) for example, the resistance can be expressed as

a normal resistance R2” = R0 + R1//R2 plus some correction 6Rcos¢ due the

proximity effect. So,

where the second term is the proximity contribution. Then, the total normal

current in the horizontal arm is

VN VN 1
1,, = = 5.2
R2” — 6Rcosq§ RS” 1 — gig—lewd ( )

Expand it to the first order, we have

VN VN 6R cosgb = 2(I+ + 1-60890)

1": 0 + 0
ECU Refngff

61

where the factor of 2 arises from the definition of I. and 1+. Because

IN = 215 = 2IC(SNS) sin (1), then,

[C(NS) = (210(SNSl sin ¢)]V=VC (5‘4)

ritical

Vmumz is the maximum applied voltage at which [C(NS) occurs. In the dangling

arm, the total current is
Id = ICsinqb — (1,. + Leosgb) = 0. (5.5)

thml is the maximum value that satisfies the equation above. Putting the Sinai

and coscfb terms together, we have

—1+ + mg + IEsz'n(¢ + 6) = 0 (5.6)

where 6056 = IC/ #13; + 13. The above equation has a maximum 1+ when
sin(¢ + 9) = 1. Because 1+ increases as VN increases, when 1,. reaches its

maximum, VN reaches Van-“cal. So, we have

 

[+(Vcritical) : \[IgU/critical) + 1.2—(Verified) (57)

Once we have expressions for 1., IC, and I- as functions of VN, we can solve for

Venn-ca). So, we obtain

 

Sin¢(vcritical) = IC(Vcritical)/\/I(2;'(Vcritical) + 13(Vcritical) (58)

which leads to

 

[C(NS) = 21C(Vcritical)2/\/I(2}(Vcritical) + 1.2.(Vcritz‘caz) (59)

62

Notice that sin¢(Vc,,-t,-ca¢) can reach 1 only when I-(Vc,.,-t,-ca1) is negligible, which

occurs in the limit of very weak proximity effect.

5.3 Temperature Dependence

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.50. DC measurement
0'45: I I SNS
0.4m A ‘ A NS , :
a I A 81 ; ___‘
0.35-1 ' ‘ '
. A $2
2
if» ‘ _ N
I
A
I
A
I
I
go ' 260 ' zéo i ario ‘
T(mK)

Figure 5.6: Temperature dependence of the critical currents [C(S N ) and IC(S N S )

Figure 5.6 contains two sets of data: [C(S N S )vs.T (square points) and
[C(S N )vs.T (triangular points). The relationship between them shows to be
temperature-dependent. At high temperatures, IC(S N ) g 2IC(SN S). While at
lower temperatures, both [C(SN) and [C(3 N S) increase. Yet, IC(S'N) shows
considerable suppression much more than that of [C(S N S ) [C(SN ) even goes
below [C(S N S ) at very low temperatures. The reason is that, at low temperatures,
the injected normal current can be very high since [C(S' N S ) is much larger. The
suppression due to the normal current is very high. This is reflected in the rising I -
term in equation 5.9 relative to [C(SN S). When T is high, the injected normal

current is limited to a very small value because IC(S N S) is small. The suppression

63

due to the quasiparticle injection is buried because the temperature becomes the
main factor. The distribution function is indeed affected by both temperature and

applied voltage. More discussion will be provided in chapter 6.

5.4 Summary

0 The amount of normal injection I N is limited for the dangling arm case. When
the dc Josephson effect takes place, the system responds to the normal
injection with a supercurrent 213 g I N. The dc Josephson effect goes away if

I N goes above 210.

o The supercurrent is a result of the microscopic changes in the distribution
function caused by normal current injection. The existence of the net
supercurrent indicates that there are more pairs added into the
superconductor than being removed at one NS interface while, at the other NS
interface, more pairs are removed from the superconductor than being added.
This is because that Andreev reflection depends on the available states inside
the normal region. The occupation of the states (distribution function) are
determined by temperature and injected quasiparticle energy. This will be

further discussed in chapter 6 and 7.

64

Chapter 6: 7r-Junction

The theoretical discussion of a 7r junction was provided in chapter 2. In this
chapter, our experimental results are presented which followed by more discussions
on microscopic interpretation of how 7r junction appears. Then, rounding factors of
the distribution function due to temperature and electron-electron (e-e) interaction
is also discussed. Then, a comparison is made with Baselmans’ experiment. It ends

with the subharmonic gap structures observed in the experiment.

6.1 Observation of the 7r-Junction

6. 1. 1 Experiment

VSNS I SNS

IQA

    

Figure 6.1: SEM picture of the sample and the measurement circuit. The picture is
take at 10kX. The distance between S terminals is 1.1,um, and the distance between
the cross point and N terminal(bottom of the picture) is 4.5nm

There is a minimum energy that is required for the 7r junction to take place.
This may require the injected current to be much higher than even the critical

current [0. This is impossible for the dangling arm experiment because the dc

65

V(uV)
s

\.

My”

”4'
,x

3 ,/
g //%

/_,,v

 

 

 

130 ff. — -' '
12° " f
110 J ,-

 

 

l(uA)
Figure 6.2: dc measurement of the V-I characteristics for different values of I N

Josephson effect goes away when the I N exceeds 210. In order to drive the system
farther away from equilibrium, a more general setup shown in figure 6.1 is adopted.
The Ag wire has a phase coherence length L,» of several micrometers at sub-Kelvin
temperatures, hence we expect to observe a substantial Josephson effect between
the two S terminals.

We perform the V-I measurement between the S terminals with the presence of
an injected normal current I N from the normal reservoir. In another words, we
sweep the current between the S terminals to determine [C with each given I N. We
have chosen for I N the values from 0,uA to 4pA with steps 0.21124. The results are
shown in figure 6.2 in which each curve corresponds to a fixed value of I N. The
measurement at zero I N shows a standard Josepshon junction behavior with a
critical current 0.76uA. As I N increases, the critical current [C(S N S) decreases.
[The horizontal shift of the curves is due to the asymmetric injection]. A selected

subset of the curves from figure 6.2 are magnified in figure 6.3

66

 

VSNSWV)

 

 

 

 

 

_5 r 1 1 M a 1 .
-O.4 -0.2 0.0 0.2 0.4
| (NA)

Figure 6.3: Selected curves from the previous graph for following values of I N: 0.53,
0.70, 1.01, 1.23, 1.89, 2.18, 3.15 pA

 

 

 

08‘ DCSNSG40mK
O
0.6 - '
O
a 0.4 —
z
a) o
v
2 0.2 -(
I
I
0.0 - . . . . . . .
l
‘0-2 Y i T n ' I ' I ' l T l r r r I
0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
l(lnlection) (uA)

Figure 6.4: [C(SNS)vs.I,-nj at 40 mK

67

 

 

 

 

0.3 « IC vs. V(injection)
1)
. DC40mK
0.6-1 '
O
0.4«
A
<
3 o
v
_0
0.2~
I
O
0.0 v
' o o 0 ’ ° .
’02 Trwn'r'I'I'InI'IYIYIYIVI‘rTIYfi
010 20 30 40 50 60 70 80 90100110120130140150

V( V)

Figure 6.5: [C vs. ij

By measuring the 10 in figure 6.2, we obtained 10 vs. I N which is shown in

figure 6.4. Multiplying I N by the effective resistance, 10(5' N S) vs. ij is also

obtained (figure 6.5). The critical current decreases to zero when I N reaches 1.0pA.

The corresponding energy is ~ 48uV. As I N continues to increase, the critical

current reaches a local maximum at I N ~ 1.5nA which corresponds an energy

~ 65/1V. Finally, Io goes to zero at large injection. This behavior of IC is, as will be

discussed later, a signature of the 7r junction. Note [C is the maximum current that

the Josephson junction allows to pass. It should always be positive because it is an

absolute value. In order to show the sign reversal of the supercurrent, we purposely

plotted [C as negative after it reaches zero for the first time. The 7r junction

appears when the dominant occupation of the energy states changes from one

direction to the other.

68

6.1.2 Discussion on the appearance of 7r junction

As discussed in chapter 2, the appearance of the 7r junction is controlled by
controlling the width of the staircase distribution function f0 which is 2eV (from
—eV to eV). (see figure 2.12) This determines which energy states exist in the
supercurrent energy spectrum. Then, the sign of the integrand of the net

supercurrent

Is =(UN/2)/d€1mlj(€)lfo(€) (6.1)

depends on the value of eV. For eVN < E0, the junction is at zero phase
state(the sign of the product of f0 and I m[J(E)] is positive); for eVN ~ E0, 10 goes
to zero; and for eVN > E0, 7r phase state appears (the sign of the product of f0 and
Im[J(E)] is negative). As discussed in chapter 2, the antisymmetric distribution
function f0 and symmetric distribution function f1 are coupled through I m[J(E)]
The exact relations are shown in two coupled equations which are derived from the

Keldysh part of the Usadel equation [31] (refer to chapter 7).

vf‘w) = V(D1Vf1) + 2Im[j(E)]Vfo = 2% f1 (6.2)

and

VflE) = V(Dono) + 21mlj<E>lVfT = 0 (3.3)

J C(E ) is the charge current density and jQ(E) is the heat current density.

I m[j(E)] is the spectral supercurrent density, Do and D1 are the diffusion
constants for heat and charge diffusion. R, the contribution due to the gap, is the
zero in the normal metal. jC(E), a constant through out the normal wire, includes

both normal current part and supercurrent part.

69

is = ”7” dE1m[j(E)lfo (6.4)

and

jN = 9% 611‘:lefo (6.5)

When d is zero, I m[J(E )] = 0. The two equations are decoupled. When qb ¢ 0,
they are coupled in a more complicated way. A simplification occurs in the vertical
arm of the sample. First, the heat current jQ(E) can not exist for energies less than
A because Andreev reflection does not allow heat transfer through the NS interface.
Because the vertical arm has a constant (15, the supercurrent density is zero in that
arm. So, equation 6.3 leads to a constant f0 in the vertical arm. Thus, Equation 6.3
then leads to a constant f0. So, f0 at the cross point takes the value of f0 in the
normal reservoir:

E + eV E — eV

2kBT +tanh 2k3T ] (6.6)

 

 

f8 = %[tanh

6.2 7r-Junction affected by the rounding of f0

There are two processes that lead to the rounding of f0 which suppresses IC

[32]: temperature and local heating caused by electron-electron interaction.

6.2. 1 Temperature

The dc measurements were performed at fridge temperatures 40 mK, 96 mK,
154 mK, 200 mK, 244 mK, and 304 mK. (data for 96mK, 200mK, and 304mK are
shown in figure 6.7. It is extrapolated that the 7r junction disappears around 350
mK. This corresponds to 55 pV which is around the quasiparticle energy at which 7r
junction appears. The 10 vs. Imj and 10 vs. ij are also plotted for the three

temperatures in figure 6.8. At very low temperature close to T = 0, the step shape

70

 

 

 

 

 

 

Figure 6.6: Absence of the 1r junction resulted from the rounding of the distribution
function due to the increase of temperature

of the distribution function is maintained because the energy of each quasiparticle is
conserved. When T increases, the distribution function becomes more and more
rounded until the integration of the product of I m[J(E )] and f0 can no longer be
negative(see figure 6.6). Then, the 7r phase shift can no longer be observed. Another
factor that contributes to the temperature of the system is the reservoir heating
[33]. Due to the very high heat impedance at the NS interface, the heat carried by
the injected quasiparticles has to be removed by the normal reservoir to avoid
heating in the wire. So, one has to increase the electron-phonon interaction inside
the normal reservoir so that the heat is carried away efficiently. The effective
cooling by electron-phonon interaction is proportional to the volume of the normal
metal. This is why we fabricated a 220nm thick normal reservoir in which the

estimated heating is about 1K/mV.

6.2.2 e‘e‘ interaction

The energy of the injected quasiparticles not only determines the width of the

staircase of f0, but also brings rounding effect to f0. This is because, as discussed in

71

T890 mK

150
140

130

 

T=200mK
120

110

V(uV)

 

W
80
70
60
50

 

 

 

0% . do ' o.'2 ' oh . ole
l(uA)

Figure 6.7: V-I characteristics for 96mK, 200mK, and 304mK

72

0.8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

" . 40mK
0.6 - I . 96mK -(
p A 200mK
. I
0.4 ~ ' -
‘3? ~ '
_o I
0.2 3 ‘ . -
. ‘ ‘ I
A
0.0 1% .gi I? f ' I
_02 1 1 1 m 1 L 1 m 1
0.0 05 10 1.5 20 25 3
|.,,(11A)
0.8 I ' l T I ' I ' I ' I ' I
I
’ - 40mK
0.6 - II . 96mK -
A ZOOmK
. I
g 0.4 - ° -
.0 _ o .
I
0.2 -‘ ‘ ‘ . -
A ‘ ‘
A
0.0 W I: U I I’"

 

 

 

l l J
O 20 40 60 80 100 120
vfliuA)

Figure 6.8: [C vs. [W and [C vs. Vin]- for 96mK, 200mK, and 304mK

73

chapter 2, ee inelastic collisions brings energy exchanges among the electrons and
results in local heating of the electrons which can be much higher than the lattice
temperature. This heating becomes another factor that causes the rounding of f0.
We can calculate the shape of the distribution function by solving the Boltzmann’s
equation in the diffusive limit [34]

i 6%, E)

m 312 + 1W, E) = o (6.7)

where the inelastic collision contribution is expressed as

[0011(23, E) = i" (x, E) — [W‘(:1:, E) (6.8)

col I call

And,

2321(35» E) = [dEdE’flSCa E + €)(1 — f(IL‘a E)) X ”(ElflE'Xl — f(iEa E’ + 5) (6-9)
and

135(22, E) = / dedE/f(a:,E)(1 — f(x,E — 5)) x n(5)f(E’)(1 — f(z,E/+ 5) (6.10)

5(5) is called the interaction kernel which depends on the exchange energy 5. The
calculation of 19(5) in a diffusive wire was done by Altshuler in 1980 [35]. He used
Fermi’s Golden rule with corrected potential due to polaron and diffusion process
and obtained

32(8) = K3/25_3/2 (6.11)

where n3/2 = (\/ D/27rh3/2va)‘1. A is the cross section of the wire. Whether or
not this scattering plays a big effect depends on how the scattering time T3 (inverse

of the scattering rate PE) is compared to the diffusion time T0.

74

TD = 4(L5/2 + LN)2/D, where D = 15001712/3 is the diffusion constant, L5(half of
the length between S terminals) is 0.5pm, and LN (length of the vertical arm) is
4.51mi. This gives 6ns for the T0. The scattering time can be approximated as 1 /1"E
where FE = (1/2)k3/2\/l§——€l/_. 1:3/2 z 0.5ns‘1meV‘1/2 is an approximation of n(5)
for low energies. For 8V 2 0.05meV, one has 3ns for T5. So, because of the long
vertical arm, electron-electron interaction should be taken into consideration. Using
a program written by Frederic Pierre during his Ph.D thesis, we have solve equation
6.7 numerically for our sample to determine the rounding of the distribution

function f E.

6.3 Comparison with Baselmans’ Experiment

 

 

 

 

Figure 6.9: Schematic picture of Baselmann’s four terminal device and measurement
setup [1]

Baselmans et a1 [1] [36] used a four-terminal device shown in figure 6.9 with two
S and two N electrodes. The normal current is injected from one N reservoir to the
other by putting 2V voltage difference between them. They have measured the V-I
characteristic between the two S terminals of which the results are shown in figure
6.10. The effective resistance Re I f(S N ) between the S and N terminals is also
measured at both zero phase state and the 7r phase state. The change of the

resistance AR... f f(S N ) depends on how strong the proximity effect is. At zero phase

75

 

 

 

 

 

'V'U'T1
f//
123‘ o

v

   

 

 

on u u 15 u
U (HIV)

 

Figure 6.10: Observed 1r junction and [C vs. Vamtro; in Baselmans et al’s experiment

[1]

.€L___/_\

AR

v

 

 

-30 -15 o 1E3 3'6

Figure 6.11: Difference of changes of R, f f(S N ) at zero phase state and 7r state

state, the AReff(S N) reaches its minimum for ISNS = 0 because the proximity is
the strongest. With the same situation in the 7r state, ARE; [(8 N ) reaches its
maximum because the proximity effect is the weakest. The differences in
ARC f f(S N ) is the direct indication that the 7r phase state indeed occurs. This lays
the foundation for our statement earlier about the signature of 7r junction.

In the horizontal arm of Baselmann’s four terminal device, there is no normal
current flowing even though there are normal quasiparticles present. The

quasiparticles are not current carrying states, so the normal current

76

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f(E)“
1 E
f(E) {—1 eV
E
. l -
f(E) 11 -eV eV ,
_| E
-eV F
f(E)“
| E
f(E) ,( eV
___, E
f(E) 11 -e1V -eV ‘7
. 1 E
eV '

Figure 6.12: Comparison of the total distribution functions between baselmans’ device
(top) and a four-terminal device with current flow equivalent to that in our device
(bottom)

77

quasiparticles are not current carrying states, so the normal current

jN = ”7” dED1(Vf1) = 0 (6.12)

A comparison of the total distribution functions is shown in figure 6.12 In our three
terminal case, normal current does appear inside the horizontal arm so that f0 and
f1 are more complicated coupled than the four-terminal case. The net momentum of
the quasiparticles is due to the finite potential at the cross point. While, in
Baselmann’s setup, the potential is zero at the cross point. So, the scattering of the
Cooper pair-like states due to the normal injection is different for the two setups

because of the difference in quasiparticle states. [37]

6.4 Sub-harmonic Gap Structure

 

Figure 6.13: Schematic picture of multi-Andreev processes

Sub-harmonic Gap Structure(SGS) is an amplified normal conductance related
to the multi-Andreev processes(flgure 6.13). It only happens when there is a finite
potential difference between the SC reservoirs(IN < IC). After a particle is reflected
as a hole at the NS interface, it travels to the other NS and gets reflected as an
electron again. During this propagation, the hole gains eV. (V is the potential
difference). And, this electron travels back and gets reflected again. The energy of

the particle keeps increasing as it travels back and forth. Until, after nth reflection,

78

32—

304 382pV=2A/e

28..
.1

264

24-

20...

dV/dl(ohms)
f

18-

16 ~ \
. 191 pV=2AI2e

14-

 

 

12 Y I T I V 1 Y j’ I I r T T I
0 200 400 EX) 8m 1000 1200 1400

V(uV)

Figure 6.14: Observation of the Sub-harmonic Gap Structure

the particle has enough energy overcoming the gap of the superconductor, it goes
into the superconductor and becomes a superconducting excitation after a charge
relaxation time. Actually, this single particle tunnelling happens at the fractions of
the gap voltage (2A/ 8) / n, where n = 1, 2, 3.... [38] These correspond to the peaks of
the measured differential resistance shown in figure 6.14.

The gap obtained from the measurement is around 0.191meV. The typical
value for the gap is about 0.2meV. The slight difference is ascribed to the presence

of the normal metal surrounding the SC reservoirs.

79

Chapter 7: Quasiclassical Theory

7 .1 Review of the quasiclassical formalism

There are several excellent review articles on this topic. The review by Hammer
and Smith [39] (1986) provides great details. This thesis follows closely the review
by Lambert and Raimondi [40]( 1998) which follows mainly Larkin and
Ovchinnikov’s (1986) approach [41].

The quasi-classical approach to the theory of superconductivity was first
initiated by Eilenberger(1968) [42] who applied Gorkov’s equation to the dirty limit.
This method was further developed by Larkin and Ovchinnkov (1969), Usadel [43]
(1970), Eliashberg (1971), Larkin and Ovchinnikov (1973, 1975), and has been
largely used to analyze transport phenomena in dirty hybrid systems. In contrast
with the BdG equations, this approach, due to its diffusion-like nature, can be used
on scales significantly exceeding the Fermi wavelength AI: = 27r/kp, and even the
elastic main free path I. The following is a brief review of the quasi-classical Green’s
functions approach.

It is convenient to express the Green functions in matrix form with the matrix
elements being the ordinary and anomalous Green functions

iG>(1,2)= <wT(1>w+(2>> Mame» (7.1)

WW (1 1(1)) —<¢f(1 W?»
and

W’HQWTU» wzmu»
-— (wf(2)¢:(1)> —<wf(2)wi(1)>

Where 1 E (T1, t1), and 2 5 (r2, t2). Following from the Keldysh formalism, in the

iC<(1, 2) = (7.2)

Nambu space [44] where the spinor operators

80

\I'(T) = WT) (7.3)

W0")

W) = ( 727(7) 7m ) (7.4)

are introduced to treat the pairing, the advanced, retarded, and Keldysh matrices of

Green’s functions [45] [46] are defined

(23(1, 2) = 6(t1 — t2)[C>(1, 2) — G<(1,2)] (7.5)
(2"(1, 2) = —6(t2 — t1)[C>(1, 2) — CH1, 2)] (7.6)
éK(1,2) = c‘;>(1,2) + é<(1,2) (7.7)

These can also be written as GR/A = Gil/Adz + FR/A = GREIZ + H6113?“ + dyFyR/A)
and GK = GR}f — fGA, where G and F are the Gorkov normal and condensate

amplitudes and f is the filling matrix. Now, the system can be written as

- GR GK
G = . (7.8)
0 GA

and the i3 is the matrix of self-energies with retarded, advanced, and Keldysh parts

- £33 it"
)3 = . (7.9)

0 2"

They satisfy the nonequilibrium Dyson equation
((331 — at; = i (7.10)

81

I \
: + l l +

Figure 7.1: The diagram equation for Green’s functions

 

and its conjugate equation

v V

C(Ggl — 2) :1 (7.11)

To cancel the large terms, one takes the difference between the two equations
[651 — 2, G] = 0 (7.12)

The equations for G averaged over realizations of the elastic scattering
potential is given by the diagram shown in figure 7.1 [47]. For the diffusion case in
which the spatial scale is much larger than the mean free path, it can be integrated
over 5 = p2 / 2m — 11 because only G now has the sharp dependence on p2. This
equation is simplified by going to the center of mass and relative coordinates (R, T)

and (7“, t) defined as
T1.2=R:l:T/2 t1’2=Tit/2

and Fourier transforming with respect to r and t, and introducing the quasi-classical

Green’s function defined by

' +00
9(R,T;15,e) = f; / d5 G<R.T;p,e) (7.13)

—00

where f = p2 / 2m — 11 is the energy measured from the Fermi level and 13 is the unit
vector in the direction of the momentum p. With the assumption that the

self-energy depends weakly on the energy 5, one sets g = O in :3, which yields

5T{fz,9}+’UFI5'aR§-i€ [29] =0 (7-14)

82

This is basically the equation for the quasi-classical Green’s function derived by
Eilenberger. 7": is a block-diagonal matrix with diagonal block entries (71. The

normalization condition is

(715)

Q(
(Q(
ll
p-A

In the dirty limit, the effect of non—magnetic impurities can be described by the

self-energy
2'

‘2 = 7; (g) (7.16)

and the equation was considerably simplified by Usadel who expanded g in spherical

harmonics keeping the only the s-wave and p-wave terms:

9(COS(9)) = go + cos(0)§1 (7.17)

with go and 9, not depending on 008(6) and cos(6)g1 << go. Now insert the above

equation into equation 7.14, and with 571 expressed in terms of go,

91 = 49031290 (7.18)

where l = ’UFT is the mean free path. For 510 one obtains a diffusion-like equation:

D 31257031290 + i5 [52, £70] — 3T {52, 90} = 0 (7-19)

with D = UFlT / 3 the diffusion coefficient. The first term is the variation of go, the
second term is the energy, and the third term is due to the inelastic scattering which
can also be written as (7/ 2) [62 go 62, go] where 7 is the inelastic scattering rate.

Then, writing 90 as (7, one has

D V(QVQ) + [6.91 — (mt/2) [6. g 6., g) = 0 (7.20)

83

Practically, the equation has to be solved in the presence of boundary conditions

contributed by the work of Zaitsev [48]

GT

‘2? [925, 915] (7.21)

0 0' 0 0
I = E9253392s =

where ”5” refer to the symmetric part of the Green’s funtion matrices.

For tunnel junctions (see AppendixA), with the boundary conditions shown

above, the tunnel current is

' GT +00
] = — d€(IJ + [131) (7.22)
88 _00
where
1, =1sin<¢w§<Ff — 1?;fo + F?) + 79w? + Ff)(F1R — F0] (723)
and

1m = [(Gf - 019(052- 02") + COS(¢)(F2R + F?)(F1R + FIA)l(fIZ - fzz)l (7-24)

I J is the Josephson current, while I p] is sometimes referred to as the quasi-particle

and interference current. The detailed derivation is provided in the appendix A.
Things get more complicated when solving the quasiclassical equation in a

diffusive region (see also AppendixC). The components of the diffusion-like equation

have the diagonal part

3R(§7R(A)3R§RW) = ie[&z,gR(A)] + (WV/2) [52 Q 62, 9] (7-25)

84

which describes the energy spectrum of the states and the Keldysh part
33(9RBRQ + 9339A) = 0 (7-26)

which describes the occupation of these states. The first equation gives the energy
spectrum. Using 9 = (")3 f — f§A(result from the normalization condition) and the

equation for QR“), the equation for the Keldysh part becomes
magi — mam — (9R339R)(3Rf) — (OHM/WM) = 0 (7.27)

f = foéo + f1 {72 is the filling matrix or matrix for the distribution functions. f0 is
the asymmetric part which reflects the effective temperature, and fl is the
symmetric part which reflects the chemical potential change. For NS systems, the
equation becomes easier to solve because QR = 0362 + iFRéy which does not have

superconducting phase variance. The Keldysh part of the equations becomes
aR[(1 — GRG" — FRFA)8Rf1] = 0 (7.28)

But, for the SN S systems, both the amplitude and the phase of the Green’s
functions inside the normal region are complex. The Keldysh part gives the

supercurrent and normal current components
IS = 383 Traz[FRaFR — FAaFA1deE (7.29)
and
IN = 7% TT&Z[1 — GRG" + FRFA]6Z(8f1)dE (7.30)

where IS is the supercurrent and I N is the normal current. Another equivalent way

85

to express the currents follows from converting the Keldysh equation into a pair of

equations [31]

V(Dlvm + 21mlj(EllVfo = 2%. (7.31)
and
V(Dovfol + 21m[j(E)]VfT = 0 (732)
where
2 Imlj<E>1 = (1/4)TT&z[§R3§R — wag/‘1 (7.33)

is the spectral supercurrent density, and

Do = (1/4)Tr(9’*§2") (7.34)
01 = (1/4)Tr(1 — QRcngéz) (7.35)
R = (1 /4)TrA(gR + 57") (7.36)

D0 and D1, affected by the proximity effect, are the energy-dependent diffusion

coefficients for heat diffusion and charge diffusion. The current densities are

is = ”7” dB Imij<E>170 (7.37)
and
m = ”7” dED1(Vf1) (7.38)

The R in equation 7.31 is zero for the normal region it becomes the conservation of
the total charge current. Similarly, equation 7.32 represents the conservation of the
thermal current.

[Note: The Usadel equation can be written (often more convenient) in terms of

86

the amplitude and phase of the Green’s functions 6 and 05 both of which are

complex in the normal region of a SN device. [49]]

7 .2 Analytical Solution with High-barrier
Approximation

The Usadel equation is highly nonlinear so it is very difficult to solve
analytically. So, we adopt the way of linearizing the Usadel equation with
high-barrier approximation developed by Volkov. [50] [51] [52] [53] [54] When the
barrier resistance of the NS interface is much higher than the resistance of the

normal metal, the solutions for the Green’s functions have the form
FR”) = tug-1155",“) + 5215;“) ) /(5 +17) (7.39)

The details of the calculations are provided in Appendix B.

When the resistance of the interface RS/N is comparable to the normal wire
resistance R N, the solutions above are not valid. But, the Usadel equation can be
linearized by using F R = sinh 713 z 113. Then, the diagonal part of the Usadel
equation becomes

aufiRW — (kR<A>)2FR<A> = 0 (7.40)

(kRU‘U2 = 7 $ 21E. For our sample, the equation has to be solved in all three arms

with correct boundary conditions at the two NS interfaces

8F p
5&2:th = EFSIxziLS (7-41)

where p is the resistivity of the normal metal, Rb is the resistance times the area of

the SN interface, F 3 is the condensate amplitude in the bulk superconductor. The

87

boundary condition at the n—N interface is
FlrzLN = 0 (7.42)

In addition to the continuity of F at the cross point, the conservation of the spectral

supercurrent density is conserved which is equivalent to

26mm = 0 (7.43)

From these equations, we obtained the solutions for the Green’s functions

FR = 7(dex + chiy) in the horizontal arm.

_ (p/Rb)FSL5 SlIthCIII) , (b
F, — 93 cosh 63 sm 2 (7.44)

 

(p/Rb)F3L5 (2 sinh 9N — cosh 0N) cosh[k(a: + L5)]

F1! 2 08 { 23inh BS sinh 0N + COSh 65 COSl’l 6N

 

+ e_k”}e_05 cos g (7.45)

 

where 65).; = \/('7 - 22'E)L§,N/hD (7 is the inelastic scattering rate), and phases of
the two SC reservoirs are set to 03/ 2 and —¢/ 2. Next, we found the values of F,t and
E, on the interface and plug them into the current (in terms of Green’s function

matrices) across the interface

u}

1J:_16Ra

 

77-3. / (1405313? — fiAfig‘wo + fos) + (6R6; — FAF§)(fo — 63)]

(7.46)

w A A A A A . . ~ .
= —16R Traz/d€l(FR-FA)(F§+F§)fo+ (FR+FA)(F§—F§)fos (7-47)

 

88

and

U.)

I r = —
A 16170

 

Tr / deKGR — CW6? —G‘§> + (FR+F")(F§+F§‘)](f1 — fig) (748)
In terms of FJr and Fy, the currents are written as

15 = gTr/d€{1m(F§)[Re(F§) cos; — Re(F,5) sin glfo (7.49)

+ Re(F§)[Im(F{}) sin; — Im(F§) cos §]f03} (7.50)

IN = gTr/d€[(Re[GR]Re[G§] — (Im[F£]sz'ng + Im[F,5]cos-(2é)1m[FSR])f1 (7.51)

f 15 = 0 when the potential in that superconductor is zero. The current consists of

three components discussed in chapter 5,

 

 

 

 

I = 1+ +I_ cos¢+ICsin¢ (7.52)
where
1 °° F W
1, = — / dE{Im[Fsl Iml S il}f1 (753)
CR 0 BS
1 °° F W. F W.
[C = — / dE{Im[FS] Re[ 3 ]f0 + Re[FS] Im[ S ]f03} (754)
BR 0 63 65
and
sinh 6N
W- _ 65(sinh 0N sinh 63 + 0.5 cosh 6N cosh 03) (7°55)
W+ _ smh(203)cosh(01v) + 23mh(0~)cosh(05) (7.56)

 

— 65 (sinh 0N sinh 03 + 0.5 cosh 6N cosh 63)

89

0.9 -

 

 

 

0'8 “ —-—- SNS-theory

 

\ I NS-expedment

0.7 q \
0.5 -
0.5 d

0.4 d

|C(HA)

0.3 '-
02 -i

0.1 -

 

0.0 I T I Y I T ' l I I l I I I l
0 50 100 150 200 250 300 350
T(mK)

 

Figure 7.2: Comparison of the experiment and theory for IC(SNS) vs. T for the
dangling arm case

Here, R = 1/(G3RN) where 0;, is the conductance of the barrier, RN is the
resistance of the normal wire.

For the dangling arm case, we first calculated [C(S N S ) vs. T. The parameters
used for this calulation are D = 120cm2/s, Lg = 0.5um, LN = 4.5pm,
A = 0.191meV, ’7 = 0.113, RN : R4, = 1 : 1, and R = 89. We have found good
agreement between the theory and the experiment (shown in figure 7.2.

Then, we computed IC(S N S ) vs. T. We first choose the temperature points
from the measurement and find the 1+, 1-, and 10. Figure 7.4 shows the calculated

results for 40 mK. Vmuwl is determined by solving

 

1+ (Veritical) = \/I(2}(Vcritical) + 13(Vcritical) (757)

Then, plug the thm1 into [C(thical), we obtained [C(S N ) vs. T. We did not find
good agreement with the experiment (shown in figure 7 .3). There are two reasons

for that. The first is that the scattering part of the retarded Green’s function of the

90

0.55 -
0.50 4

0.45 d

0.35 -

0.30 -

usmw

025 ~

0.20 -

0.15-

 

0.10.,...,.r..
050100150200250300

T(mK)

 

Figure 7.3: Experimental data for [C(S N ) vs. T for the dangling arm case

Usadel equation becomes energy dependent when the energy of the injected
quasiparticles is high. So, the solution from the equation is not correct. The other
reason is that both expressions for 1+ and I _ use the equilibrium value in the
normal reservoir for the antisymmetric distribution function f1. But, f1 can be quite
different from the equilibrium value due to the presence of the long vertical arm.

For the 7r junction case, we first compared our numerical result and high-barrier
approximation for the spectral supercurrent density I m ](E) They are not very
different from each other (figure 7.5). We then computed the 10 vs. VN and
compared it with the experimental result (figure 7.6). The discrepancy remains as it
is in the I C(S N ) vs. T calculation. We also tried computing the distribution
function with Altshuler’s approach plus the NS boundary conditions. It still did not
improve much.

The quasiclassical approximation uses instantaneous short-range two particle
interaction to treat electron-electron (ee) interaction. This may become a problem

when the system is far away from equilibrium. For example, when the injected

91

current (uA)

 

 

 

1
E lc H
0.3. \x
. /
0.6:
0.4E
o_zi l'
: / Vn (mV)
’ ——.v—.'—. ....... x .........
: 0.01 0.02 0. 0.04 0.05 0.0
-0.2E

Figure 7.4: Theoretical calculations of 1+, I _, and [C at 40 mK

 

 

 

 

 

 

 

 

 

 

 

 

1.0 -
* Numerical
0.8 - —— High-barrier Approx.
0.6 -
a 4 .
E 0.
E
0.2 —
00-»~
0.2 I I T T 1
o 100 200 300 40°
E(u~0

Figure 7.5: Comparison of the I m[j(E)] vs. ij from the numerical calculation and
the high-barrier approximation for the 7r junction case

92

 

0-8 r I Experiment
Theory(high barrier approx.)

 

‘ .

 

 

 

0.6 d

0.4 -‘

A
3 ‘ I
00.2-

 

'H

 

 

 

V I V I V I U I V I i I ‘
0 20 40 60 80 100 120 140 160

v.,,(uA)

Figure 7.6: Comparison of [C vs. ij between the theory and experiment for the 7r
junction case

quasiparticle energy is high, the phase space opens up for ee interaction so that the
electrons can no longer be treated instantaneously. So, the quasiclassical averaging
is not sufficient to deal this kind of situations. The key is the correct form of the
scattering part in the Usadel equation. [55] The exchange of the Green’s functions
must provide the right form for ee interactions. Once this is solved, we will be able
to solve the transport properties by solving the Keldysh part of the Usadel equation

with correct boundary conditions.

93

Appendices

94

Appendix A: Tunnel Junctions

The physical current is obtained from the Keldysh component of the Usadel

equation.

[92:91lx=9291+9291 — 9192— 919? (A.1)
where we have droppeds ” ’ .The normalization condition, 95'] = 1, allows us to
choose

91.2 = 9f2f12 - 13,2914; (A2)

Where the f can be taken to be diagonal (Larkin and Ovchinnikov 1975):
f1,2 = 11.102570 + 1112,2572 (A3)

As a result, multiplying 62 and taking the trace yields

GT

J=iE

__ 00 d6 T 7'[c‘7.z(f{’ia + fSib + ff!) + fizidfl (M)
where
i. = [92 (91 — 91) — (91‘ — 91595]
= —91["é(R—ga‘)(é‘—Qa‘)éi‘]
=[91122(9i{92 — (A729?) _ (9f52— 0291A)92]
id = —[9f‘<gé‘&z —- 67295)— of oz — 0292 )gi‘]

RM)

Due to the normalization g g3“) = 1, we have

QR‘A)= 9W 052ng (A6)

95

where 93”) = (z'FRM) sin(q’>), z'z'FRM) cos(q§), Gm“) and 05 is the phase of the

superconducting order parameter. Hence,

. GT +00

J — 8e dEUJ + 1P1) (A5)

where
IJ = isinwi — ¢2)[f§(F2" — FJ‘XFI“ + Ff‘) + fi‘(F2" + Fa‘fo‘ — Fm] (A?)
and

IF; = [(G{‘ — Gi‘)(G§ — G?) + cos(¢1— 09%" + Fé‘)(FF + Ff‘)](flz — ff )] (A8)

I J is the Josephson current, while [p] is sometimes referred to as the quasi-particle

and interference current.

96

Appendix B: High Barrier

(Rb >> R N) Approximation

Normally, one does numerical calculations because that the exact analytical
result for such a problem is very difficult to obtain. Another method is to simplify

the calculations with low barrier transmittance approximation.

 

 

 

 

 

 

Figure 8.1: SNS system

I define r = Rb/RN as the ratio of the barrier resistance and the resistance of

the normal wire. The quasiclassical equation for the Green’s function matrices is

D V(gvg) + 1502,57] — (w/Z) [6. a 6., 9] = 0 (Bi)

Where D is the diffusion constant in the N region, and 7N = (1 / 275) + ng is the
deparing rate. T3 is the spin-flip scattering time and p3 = H /\ / (1)0 is the condensate
momentum due to an applied field H. The Green’s functions in the superconductors

and the normal metal are related through a boundary condition

Thugvgflinterface = [9N 95] (B2)

Where l is the elastic mean free path. Tb is a dimensionless constant characterizing
the transparency of the interface, and it is related to its resistance per unit area Rbg
by rbl = 2Rbg / p, where p is the resistivity of the N region. Define a characteristic

energy EN = Dp/2RbgLN for describing the transmittance of the barrier. We can

97

rewrite equation 8.1 for the advanced and retarded parts by averaging over L N and

using the boundary conditions, we obtain
Six/[9305]“) + 2'6 [fiaélmm - (“m/2) [52 9 62, film” = 0 (B3)
For a SN S device, it becomes
6109511“ + 520. 0521’“) + 2's [6... 01"“) — mm [6. 9 6., 91"“) = 0 (13.4)

To solve this equation, we write
gRM) = GR(A)&Z + FRM) = GR(A)&z + (z'Ff(A)&x + iFf(A)&y), Plug into the equation

above, assuming 51, 52 << ’7, I obtained
FR“) = mfg") + 521%") /(5 + 2'7) (13.5)

where FEM = idyA/ERM), and F53“) = 2(6I sincb + (3,, cos ¢)A/€R(A).
QR“) = [(5 + if“)2 — A2]1/2, and (b is the phase difference between the two
superconductors one of which has zero phase. To make it easier to calculate, I
assume 51 = 52 = 8N. Then,

is

- iEN . A A N .A . .- A
FR(A) _—_ j;(€ + 2'7)(F;(A)+F:2( )) = i0: + z,Fy)(zc7,,,smg15+wy(1+cos(15))E—m—A-i (B.6)

 

 

The current is obtained by plugging the solved Green’s functions in the expression

of the currents at the interface (written in terms of Green’s function matrices)

 

 

w A A A A A A A A A
11 = _16R Tmz/deFRFsB — FAFslon +f05) + (FRFé‘ — FAquXfo - fosll
(3.7)
w A A A A A A A A A
= _16R TTUz/dEKFR — FA)(F§2+F§1)f0+(FR+FA)(F§‘st‘lfos (13-8)

98

Appendix C: Diffusive region

The diagonal part of the Usadel equation is
aRgRlA>aRgR<A> = “[02, WM] (0.1)

and

83038;;9 + @8352") = 0 (0.2)

Using Q = QR f — fQA and the equation for QR”), the equation for the Keldysh part

becomes

aaaaf — Waflf‘m‘] — (9363950313) —- (emu/‘65") = 0 (0.3)

f (the filling matrix) can be solve once 0” and 9A are determined by the diagonal

part of the Usadel equation.

A

f = f05'0 + f15z (C.4)

In a NS system, one chooses QR = GRdz + iFRéy for the normal region. After

multiplying by €72 and taking the trace, we have
aR[(1 — GRG” — FRFA)BRf1] = 0 ((3.5)

These equations are no longer true if the inelastic scattering is considered. The
second term is still zero after taking the trace of the product with Pauli matrix.
Note, we have to multiply Pauli matrix and take trace for the Keldysh equation is
because the physical current requires so. But, for the advanced and retarted
equations we don’t need to do this. For those we just find the Green’s functions

inside the normal region in terms of the Green’s functions in the superconductors.

99

For a SN S system, the Green’s function is more complicated because the phase

X varies in the normal region.
QR = GREIZ + iFR(chsinx + a‘xsinx) (C.6)

By working out equation G3, the total current has both supercurrent and normal

current components

13 = 3g. / Tr&z(FRBFR — FA6FA)f0dE ((3.7)

1,, = 0% Tr(1— CIRGA + FRFA)8f1dE (0.8)

100

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