ELECTRONIC AND OPTICAL PROPERTIESOF THIN-FILM SUPERCONDUCTORS AND

SUPERCONDUCTOR – SEMICONDUCTOR INTERFACES.

BY

IGOR VLADIMIROVICH ROSHCHIN

Magister, Moscow Institute of Physics and Technology, 1993M.S., University of Illinois at Urbana-Champaign, 1999

THESIS

Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Physics

in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2000

Urbana, Illinois

Copyright by Igor Vladimirovich Roshchin, 2000

iii

ELECTRONIC AND OPTICAL PROPERTIES OF THIN-FILMSUPERCONDUCTORS AND SUPERCONDUCTOR-SEMICONDUCTOR

INTERFACES.

Igor Vladimirovich Roshchin, Ph.D.Department of Physics

University of Illinois at Urbana-Champaign, 2000Laura H. Greene, Advisor

Recently, the interest in the properties of superconductor-normal metal and

especially superconductor-semiconductor interfaces increased in the basic research area.

The superconducting proximity effect, known for about four decades, still poses many

unanswered questions. Use of a semiconductor instead of a metal broadens the

possibilities: superconductor-semiconductor interfaces are tunable, and their parameters

can be varied in a wide range. This type of interfaces may also be used in future

superconducting electronic device applications.

Our studies of the interfaces between high-quality thin film Nb and NbN

superconductors, and (100) n+-InAs (1019cm-3) are done by transport and optical

measurements. For the first time, the proximity effect in superconductor-semiconductor

interfaces is observed using Raman scattering. The InAs longitudinal optical phonon LO

mode (237cm-1) and the plasmon-phonon coupled modes, L– (221cm-1) and L+ (position

is carrier concentration dependent), are studied. The intensity ratio of the LO mode

(associated with the near-surface charge accumulation region, CAR in InAs) to that of the

L– mode (associated with bulk InAs) is observed to increase by up to 40%, when the

temperature is decreased below the superconducting transition temperature Tc. Further

experimental and theoretical studies are required to find the underlying mechanism.

Effects of surface damage and surface passivation on the electronic properties of

InAs are studied. Damage, produced by Ar-ion etching, reduces the surface band

iv

bending. Passivation of the InAs surface by application of alkanethiols is found to reduce

the surface band bending and to preserve the surface of InAs from oxidation.

In close collaboration with A. Shnirman, a theoretical model for the Andreev

reflection contributions to the current at the SN interfaces is developed using the

tunneling Hamiltonian method. Both, the interference due to multiple Andreev reflection,

and the electron-electron interaction in the normal metal, contribute to the zero-bias

conductance peak (ZBCP). The shape of the ZBCP is calculated as a function of

temperature, strength of the interaction and other parameters of the normal metal.

Application of an external ac field results in steps in the current-voltage

characteristics at Vn = nhω/2e. Experimental parameters of a highly-doped n+-InAs are

used to obtain estimates for future experiments to be conducted to test the predictions of

the model.

v

To my mom and dad,

Aida and Vladimir Roshchin.

vi

ACKNOWLEDGMENTS

All time that I have been doing research at the University of Illinois at Urbana-

Champaign, I enjoyed the great atmosphere established at the Department of Physics. I

also enjoyed many discussions and collaborations with people from many different

institutions.

First, I would like to thank my advisor L. H. Greene for the unique opportunity to

work with her and in her group. Working on very interesting projects suggested by her, I

enjoyed the freedom of choosing my own ways of solving problems, and also choosing

other projects and collaborations. Her guidance and valuable advice which were not

limited to science, and also her moral and financial (through the research assistantship)

support are greatly appreciated. I am grateful to her for sharing her experience in many

aspects of scientific life, including communication and presentation skills. Our work

together would not be possible without Laura’s sense of humor.

Secondly, I would like to thank W. L. Feldmann whom I consider my technical

advisor. The role of his talented technical assistance, his invaluable experience in various

every-day aspects of the experimental research, his handyman approach in designing,

building and improving experimental systems can not be underestimated.

I am happy to acknowledge the close and enjoyable collaboration with the group

from Chemistry Department: J. F. Dorsten, T. A. Tanzer, and P. W. Bohn, with whom all

Raman scattering experiments are done.

vii

I would like to thank G. C. Spalding and W. L. Murphy (who also was a summer

student in our group) from Illinois Wesleyan University who did a lot of work for

establishing the nano-fabrication process.

I also want to thank P.-W. Han and P. F. Miceli from University of Columbia-

Missouri for very interesting collaboration, including x-ray reflectivity measurements,

analysis, and discussions, and also I. K. Robinson (UIUC) who performed the first x-ray

reflectivity measurements on our films.

I would like to acknowledge very productive collaboration on optical studies in

millimeter and submillimeter range with the groups from Augsburg University and

General Institute of Physics of RAS (Moscow), and in particular, A. Pronin, and M.

Dressel.

I want to thank J. F. Klem from Sandia National Laboratories not only for the great

InAs materials that he prepared for our studies, but also for very helpful discussions and

ideas on the structure and properties of InAs.

I had many productive and joyful discussions with people from both UIUC and

other institutions: G. Blumberg, E. Burnstein, M. Cardona, P. M. Goldbart, D. J. Van

Harlingen, Yu. Lyanda-Geller, D. Maslov, A. Pinczuk, B. Plourde, R.R. Ramazashvili, J.

Sauls, M. Turlakov, and many others. I would like to express my special thanks to A.

Shnirman for theoretical support and many fruitful discussions. His clarity of thinking

helped me to understand the very details of the superconducting proximity effects. I am

very grateful to R. R. Ramazashvili who proofread this thesis and made many valuable

suggestions.

viii

I am glad to acknowledge assistance from many people at the Center for

Microanalysis of Materials and Microfabrication Center at Frederick Seitz Materials

Research Laboratory (MRL). They have been very helpful not only providing various

modern experimental equipment, but also sharing their expertise in different areas of

materials research. Special thanks to T. R. Banks, S. A. Burdin, N. Fennigan, R T.

Haasch, V. Petrova, M. R. Sardela, Jr.

Readiness to help, and provision of normal condition for the research by the

technical personnel, secretaries, and business departments of both MRL and Physics

Department is greatly acknowledged. My sincere thanks to the librarians, and especially

M. K. Newman and G. Youngen for their assistance in complicated bibliographical

searches without which the research would not be as effective.

All this time, from the very first day that we both joined this group, I enjoyed

working shoulder-to-shoulder with A. C. Abeyta. I appreciate his help in various

situations of every-day work. Whenever it was necessary, I could rely on him and his

knowledge of the experimental equipment and experimental procedures. I also appreciate

his patience in frequent discussions on both scientific and non-scientific topics. I am very

thankful for his help in improving my skills in spoken English, and introducing me to

various aspects of American culture.

I also enjoyed working with an exchange student from Regensburg University G.

Kuhler, who played significant role in establishing procedure for preparation of NbN

films. I would like to acknowledge M. Matney for doing resistivity measurements for the

studies of how Nb films degrade with the time.

ix

I also want to thank all other people, past and current students and postdocs in our

group, for the nice atmosphere established in the group, interesting conversations on both

scientific and general topics: M. W. Covington, N. Haas, M. Aprili, H. Aubin,

E. Paraoanu, D. E. Pugel, P. J. Hentges, M. Dittrich.

With great pleasure I would like to thank D. M. Ginsberg for his informal advice,

and great sense of humor.

This research was funded by Department of Energy through Materials Research

Laboratory (Grant no. DEFG02-91ER45439). I am also grateful for receiving John D.

and Catherine T. McArthur Fellowship in 1994. Assistantship from the Department of

Physics in 1995 is acknowledged.

Finally, I would like to thank my close friends and family members for their support

and encouragement.

x

TABLE OF CONTENTS

1. INTRODUCTION ................................................................................... 1

1.1. Superconducting proximity effect – background information..........................1

1.2. Motivation...............................................................................................................4

2. EXPERIMENTAL TECHNIQUES .............................................................. 7

2.1. Sample preparation...............................................................................................7

2.1.1. UHV-compatible system for magnetron sputter deposition and Ar-ion etch ...............7

2.1.2. DC magnetron sputter deposition.........................................................................10

2.1.3. Surface preparation: Ar-ion etch, As-capping.......................................................12

2.2. Measuring techniques..........................................................................................13

2.2.1. Surface, thickness, and chemical composition characterization ..............................13

2.2.2. Transport measurements .....................................................................................15

2.2.3. Millimeter and submillimeter conductivity measurements ......................................16

2.2.4. Raman spectroscopy ...........................................................................................18

3. MATERIALS AND THEIR CHARACTERIZATION ..................................... 21

3.1. Niobium.................................................................................................................21

3.1.1. Film structure (SEM and X-ray results)................................................................21

3.1.2. Mean free path and RRR .....................................................................................24

3.1.3. Tc and its dependence on thickness .......................................................................27

3.1.4. Superconducting gap...........................................................................................34

3.2. Niobium nitride ....................................................................................................36

3.2.1. Tc dependence on growth parameters ...................................................................36

xi

3.2.2. Tc dependence on the film thickness......................................................................38

3.3. InAs .......................................................................................................................39

3.3.1. InAs materials ....................................................................................................39

3.3.2. General information about InAs and its surface ....................................................40

3.3.3. Doping level of InAs and Raman modes. L+ measurements as a tool for doping level

determination...............................................................................................................42

3.3.4. Charge Accumulation Region. I(LO)/I(L-) as a measuring tool..............................45

3.3.5. Difference between bulk InAs and MBE-grown InAs .............................................48

4. SEMICONDUCTORS: SURFACE DAMAGE AND PASSIVATION STUDIES ..... 49

5. OBSERVATION OF THE SUPERCONDUCTING PROXIMITY EFFECT BY

RAMAN SCATTERING .............................................................................. 51

5.1. Raman measurements of SN structure ..............................................................51

5.2. L+ mode for Nb/InAs samples............................................................................55

5.3. Discussion .............................................................................................................56

5.4. Conclusions ...........................................................................................................59

6. PROXIMITY EFFECT: ZERO-BIAS FEATURES AND “SHAPIRO-LIKE” STEPS

IN THE IVC OF AN SN STRUCTURE.......................................................... 60

6.1. Background ..........................................................................................................60

6.2. Zero-bias features of the IV characteristics in the Andreev scattering

regime............................................................................................................................63

6.3. Experimental predictions and discussion ..........................................................66

6.4. Conclusions ...........................................................................................................69

xii

7. FUTURE WORK .................................................................................. 71

7.1. Temperature evolution of the I(LO)/I(L-) change ............................................71

7.2. Nano-line structures ............................................................................................71

7.3. Magnetic and electric field dependence .............................................................75

APPENDIX .............................................................................................. 76

A.1. Andreev current and tunneling Hamiltonian method......................................76

A.2. Tien Gordon.........................................................................................................79

A.3. Purely diffusive regime, no electron-electron interaction................................80

A.4. Effects of the electron-electron interaction, paraconductivity ........................83

A.5. Simple example ....................................................................................................84

A.6. A case of a thick normal metal layer..................................................................85

REFERENCES.......................................................................................... 87

VITA ...................................................................................................... 93

1

1. INTRODUCTION

1.1. Superconducting proximity effect – background information

The standard description of the proximity effect, the influence of the

superconductor on the normal metal, is based on the coherent coupling of electrons and

holes in the metal, produced by the Andreev reflection (AR) process [1]. In this process,

an electron in the normal metal is reflected from the interface with a superconductor as a

hole along the time-reversed path. Along this path, the hole has the same phase of the

wave function as the electron, and therefore they are coherent. Inside the superconductor,

this process breaks and recreates a Cooper pair, a pair of coherent electrons, that carries

the current into the superconductor. Andreev reflection, as described by Bogoliubov-de-

Gennes (BdG) equations [2], mixes electron and hole states, while the quasiparticle

current is converted into the supercurrent. It is important that the current conversion

occurs inside the superconductor in the region limited by the superconducting coherence

length ξS [2]. This length determines the distance at which the superconducting

condensate wave function can vary, and characterizes the roll-off of the order parameter

on the superconductor side. It is typically of order of 100-1000Å for conventional

superconductors. Another equivalent description of the same process is when the

superconductor induces the condensate wave function in the normal metal as provided by

the continuity conditions at the superconductor-normal metal (SN) interface [3].

The strength of the proximity effect, and therefore the transport characteristics,

such as the current-voltage characteristic (IVC), are affected by the quality of the

2

interface, and by the phase-breaking processes, i.e. processes which change the phase of

the electron wave function, such as inelastic scattering. The BTK theory [4] proposed in

1982 models the transport across such an interface using the barrier strength as a

parameter. Tuning the barrier strength allows for investigation of the crossover from the

quasiparticle tunneling, where single electrons and hole tunnel across the interface, to the

Andreev reflection regime. When the effective barrier is strong, i.e. in the low-

transmittance regime which occurs either in the case of a high potential barrier or when

the Fermi-surface parameters of the two materials are disparate, tunneling dominates [4].

Although this elegant theory is still one of the most quoted theories on the transport

across SN interfaces, there are several experimental results which can not be fit by this

theory. It has been shown that the two transmission channels alone, quasiparticle

tunneling for low-transmittance junctions and AR for high-transmittance junctions, do

not account for all the transport phenomena observed. An example of this was found in

1992, when an observation of a zero bias conductance peak (ZBCP) was attributed to a

new transmission channel in high-transmittance junctions [5], namely the Cooper pair

tunneling into N. A weak magnetic field, well below 100 mT, suppresses this excess

conductance. This is qualitatively similar to a supercurrent suppression by an applied

field in a Josephson junction, that is in a normal metal or an insulator between two

superconductors.

These same experimental results were later accounted for by a phase-coherent

Andreev reflection processes or “reflectionless tunneling” [6, 7]. In this case, scattering

of the time-reversal electron-hole pair from impurities leads to multiple Andreev

reflections (MAR) at the SN interface, and to the resulting interference in the closed loop.

3

These processes increase low-bias conductance [6]. The role of the MAR is to increase

the effective probability of the electrons to undergo the Andreev process. The increase in

conductivity due to Andreev processes is limited by the factor of two. This limiting case

is achieved when the probability of the Andreev reflection is 100%, such as with an

ideally transparent interface.

When the electron-electron interaction in the normal metal is attractive, such as in

case of a paraconductor (a superconductor above its Tc), the superconducting order

parameter, corresponding to the off-diagonal element of the electron-hole interaction

matrix, has a finite value inside the normal metal. In this case, additional Andreev

processes occur inside the normal metal at a distance defined by the order parameter

induced inside the normal metal, and phase-breaking processes. It is important to note

that this length can be different from the coherence length of the non-interacting normal

metal that determines the distance of the penetration of the Cooper pair wave function

F=<ψ↑ψ↓> inside the normal metal in the absence of the electron-electron interaction.

The latter length can vary from only a few angstroms to more than a micron, depending

on the temperature and the scattering rate in the normal material, and is defined as

ξN=hvFN/2πkBT for the clean metal case (lN>ξN), and ξN=(hD/2πkBT)1/2 for the dirty

metal (lN<ξN) [3], where D is a diffusion constant of the normal metal, and lN is the mean

free path. As a result of these Andreev processes, some portion of the current can

propagate as a supercurrent inside the paraconductor, thus increasing the conductivity

value at low bias.

It is important to distinguish the order parameter from the energy gap in the density

of states. The presence of a finite order parameter usually causes a gap in the density of

4

states. However, a similar gap can occur as a result of some other processes where the

order parameter can be zero. Such a gap is very often called a pseudo-gap. A good

example is a theoretical model suggested by Volkov [8], where such a gap opens in the

density of states in a wire with impurities or a thin film of a normal metal, that is brought

in a contact with a superconductor. The normal metal is considered to have a zero

electron-electron interaction constant in the Cooper channel, and therefore no

superconducting order parameter can be induced in the normal metal.

In the bulk of a superconductor, where the order parameter is uniform, the order

parameter and the energy gap in the density of states are directly related.

The superconducting order parameter on the N side was first considered by

Ferrell [9] and Scalapino [10], which they called “fluctuation”. It was experimentally

observed in tin-tin oxide–lead tunnel junctions by Anderson and Goldman [11] as an

excess tunneling current.

1.2. Motivation

More recently, studies of superconductor-semiconductor (S-Sm) interfaces have

been done by many groups around the world (see for example review by Kleinsasser et

al. [12]). One of the main reasons for interest in such interfaces is that they may be used

in the electronic devices, such as the Josephson field effect transistors (JOFETs). These

devices operate with much lower voltages and much lower power loss than in the existing

semiconductor devices.

5

The replacement of the normal metal by a semiconductor opens several new

opportunities for investigation of the SN interfaces. First, the electronic properties of

semiconductors are rather disparate from those of the conventional superconductors:

typically, carrier concentration is 102-105 times lower, and the typical Fermi momentum

and the effective mass are 10-20 times lower in a semiconductor than in a

superconductor. Most semiconductors tend to form a depletion layer (Schottky barrier) at

the interface. A few narrow gap semiconductors, such as InAs, can form a charge

accumulation region (CAR). The width of both depletion and accumulation layers is

determined primarily by the doping level (see section 3.3). The parameters of the space-

charge layer also depend on the choice of a semiconductor. They can also be modified by

external electric field that can change the effective carrier concentration and band

bending in the near-surface region. As it is shown and discussed in Chapter 4, the

parameters of the space-charge region can also be significantly modified by surface

treatment, such as surface etch, metal deposition or surface passivation which affect the

surface states. The band bending, and therefore, ultimately, the carrier concentration in

the space-charge region is also affected(Chapters 3 and 4).

Another advantage of using a semiconductor as a normal metal is that optical

methods can be employed. Even in the highly doped semiconductors, the carrier

concentration is about two orders of magnitude lower than it is in metals. The visible

light penetration depth, determined by the skin depth, is a few hundreds of angstroms. So,

optical methods, such as Raman scattering, can be used.

Raman spectroscopy has been proven to be a powerful probe of the near-surface

electronic structure of semiconductors, as well as of material processing on interfaces

6

[13-18]. Light scattering can yeild significant information about modification of phonons

due to coupling to electronic excitations and vice-versa. It can also reveal the information

on electron layers at heterostructure interface and space charge layers created by surface

pinning of the Fermi energy. For historical survey on Raman spectroscopy applications,

see e.g. Ref. [19].

The Raman-active vibrational modes of accumulated or depleted surfaces of

semiconductors are different from those modes of the bulk. This difference is due to the

difference in the electronic character of the region of origin

Distinct differences in the Raman-active vibrational modes of accumulated or

depleted surfaces of semiconductors from those modes of the bulk are due to the

difference in the electronic character of the region of origin. Raman scattering is therefore

an effective tool for comparison of the surface and the bulk electronic properties. It also

provides information on the carrier concentration and the band bending associated with

the interface. Using all these advantages, we have, for the first time, employed Raman

scattering to study the influence of the normal-state and superconducting Nb films on the

near-surface properties of n+-InAs.

7

2. EXPERIMENTAL TECHNIQUES

2.1. Sample preparation

2.1.1. UHV-compatible system for magnetron sputter deposition and Ar-ion etch

A UHV-compatible system is used for Nb deposition and Ar-ion etching (Figure 1).

The system consists of two major parts: vacuum chamber and an Edwards EO6K

diffusion pump – VZCCT150 vertical liquid nitrogen cold trap combination from

Vacuum Generators. These two parts are separated by a manual 10” gate valve that

allows throttling. The automatic filling system controls the level of the liquid nitrogen in

the cold trap. Roughing of the sample chamber and pumping on the back stage of the

diffusion pump is done with E2M12 Edwards 12-liter mechanical pump. This setup

allows reaching a pressure of 0.6–2.0⋅10-8 Torr in the sample chamber. Just before the

sputtering process, the Meissner trap, a cold trap that surrounds the processing space

inside the vacuum chamber, is filled with liquid nitrogen in order to absorb the water

vapor that constitutes the major part of the gas remaining in the system at low pressures

(below10-7 Torr). This provides a significant decrease in pressure (down to low-10-9-mid-

10-10 Torr). Overall cleanness of the system is improved by 15-30 minutes of

presputtering performed just prior to film deposition, due to the gettering of the fresh

metal.

The power for the 2-inch sputtering guns (180W for the samples described in this

work) is supplied using PPS7901 plasma power supply (Basic Sputtering, Inc.).

8

4

5

7668

9

12

11

10

3

1LN2 LN2

to LN autofill

2

11

2

Figure 1. UHV-compatible system for dc-sputter deposition and Ar-ion etch:1. liquid nitrogen trap inlets2. VZCCT150 liquid nitrogen cold trap3. main gate valve4. Edwards E06K diffusion pump5. ballast6. valves7. E2M12 mechanical pump

8. sputtering guns9. 3cm ion source for surface cleaning10. vacuum chamber11. liquid nitrogen inlets for the Meissnertrap12. insert with rotary-linear feedthrough.

9

An insert with a rotary-linear feedthrough (Huntington, Inc.) drives the sample

holder and provides the possibility to place the sample above one of the three ion guns

that have sputtering targets of different materials. A rotating manual shutter shields the

sample holder from the sputtering guns. There is also a horizontally installed standard

3cm ion source (Commonwealth Scientific Corp.), used with a single mesh nickel grid

used. The linear motion of the feedthrough allows to tilt the sample holder and to expose

the samples to the ion beam. This allows an in-situ ion etch of the substrate surface prior

to metal deposition, without breaking vacuum.

The temperature of the substrates is monitored with a thermocouple digital

thermometer HH-99-A-K from Omega Engineering, Inc., using a thermocouple mounted

inside the stainless steel sample holder, directly below the substrates, just above the

tungsten wire heater, located on the opposite side of the ~ 2mm stainless steel sample

holder plate.

For the reactive sputter deposition of NbN films, the grade “zero” (99.95%) N2 is

introduced in the vacuum system through ports with a stainless steel tube. For both the

sputtering and Ar-etching procedures, ultra-high purity (99.999%) Ar is supplied through

a tube in the ion source. MKS Instruments 100 sccm mass flow controllers are used for

both Ar and N2 gases for the reactive sputter deposition of NbN. For sputter deposition of

Nb, and also for Ar-etching, a bypass in the Ar line with a manual leak-valve is used. In

order to facilitate the optimal throttling rate, just before the Ar is leaked in the vacuum

chamber, the main gate valve, separating the diffusion pump and nitrogen cold trap from

the vacuum chamber, is usually closed all the way and then opened one turn for the

deposition process, and two turns for the Ar-ion etching.

10

The pressure at different points inside the growth chamber is monitored with

Granville-Phillips vacuum gauge controller 307, and Perkin-Elmer Vacuum Products

digital gauge controller III, using a convectron gauge, thermistor gauge, and ion gauge

tubes. For early experiments, the ion gauge was used to monitor the pressure during the

sputtering process. The range of the pressures measured by this ion gauge is limited by

low 10-2 Torr. For the later experiments, and especially for the Nb deposition at higher Ar

pressure (3.5·10-2Torr), the convectron is used. Since the convectron is located closer to

the diffusion pump, and the ion gauge is closer to the port where the Ar is introduced in

the system, the pressure readings from the two sensors are different. However, a good

consistency between the two values is observed, and an accurate mapping between the

two sensor readings is done. Throughout this thesis, the quoted Ar pressure during

sputtering processes of Nb is either measured by the convectron, or calculated from the

ion-gauge measurement, what it would be if it were measured by the convectron. For the

NbNx deposition process, the pressure measured by the ion gauge is reported.

2.1.2. DC magnetron sputter deposition

Niobium films are grown in the UHV-compatible chamber described above by

planar DC magnetron sputter deposition using three slightly different procedures:

The first procedure is used for the highest quality Nb films. 10×10 mm2 and

3.5×11mm2 Nb films are grown on a plane-parallel 0.45 mm thick sapphire substrate

oriented in the (1102 ) plane. To clean the sapphire surface of water and hydrocarbons,

an in-situ bake at 490-500 °C is performed simultaneously with approximately a 50-hour

11

pumpdown. When a base pressure lower than 4·10-8Torr is achieved, the Meissner trap is

filled with liquid nitrogen, and the base pressure typically falls below 5·10-9Torr. High

purity Ar is leaked into the system to a pressure of 7.0·10-3Torr, and immediately

following the 15-30 minutes of presputtering, the shutter is opened, and the Nb film is

deposited at an approximate rate of 8 Å/sec on the substrates kept at 450-470°C

The second procedure is developed to grow Nb films on InAs where long heating at

a temperature above 200°C causes loss of As from the surface. Rectangular 3.5×11 mm2

and parallelogram shaped with the sides of 3.2mmx11mm at 45°, niobium films are

grown on a plane-parallel 0.45 mm thick sapphire substrate oriented in the (1102 ) plane,

and on (100) surface of highly-doped n+-InAs. To clean the substrate surface of water and

hydrocarbons, an in-situ bake at 60-100°C is performed simultaneously with

approximately a 30-hour pumpdown. When a base pressure lower than 3·10-8Torr is

achieved, the Meissner trap is filled with liquid nitrogen and the base pressure is typically

reduced below 2·10-9Torr. High purity Ar is leaked into the system to a pressure of

7.0·10-3 Torr, and immediately following 15-30 minutes of presputtering, the shutter is

opened, and the Nb film is deposited at an approximate rate of 8 Å/sec on the substrate

kept at 100°C.

Some of the Nb films are sputtered at a higher Ar pressure (3.5·10-2Torr) in order to

reduce the damage to the surface of InAs (see Chapter 4).

NbNx samples are grown by reactive DC magnetron sputter deposition using the

procedure similar to that for Nb films, except that a mixture of high-purity argon and

nitrogen is used. The ratios of the partial gas pressures and flow rates are optimized for

the highest Tc. Details are given in Chapter 3.

12

2.1.3. Surface preparation: Ar-ion etch, As-capping

Two different types of InAs are used: bulk and MBE grown. Bulk InAs is

purchased from OMK, Slovakia. The rest of the InAs is grown by John F. Klem at Sandia

National Laboratory by solid-source molecular beam epitaxy at a growth rate of

approximately 2.1Å per second with a substrate temperature of 460-480°C. The incident

As4 flux is approximately 1.1 monolayers per second as measured by As-limited

reflection high-energy electron diffraction (RHEED) oscillation measurements with

GaAs. Silicon is used as the n-type dopant. Following the growth, some samples are

capped with arsenic by exposing the surface of the sample to an incident As4 flux of

approximately 0.5 monolayers per second for a period of 5.5 hours. During this time, the

sample is cooled from the growth temperature to less than 0°C. Arsenic adsorption can

only occur below 0°C.

For the InAs substrates not capped with As, a gentle Ar-ion etch (beam current:

1mA, beam voltage: 75V, 30 seconds unless specified otherwise) is performed in situ on

the InAs, just prior to Nb deposition, in order to remove the surface oxide layer estimated

to be 20-30A thick.

For the experiments when only the Ar-ion etch on InAs is performed, and no Nb is

deposited, the pumpdown time is shorter, usually 6-12 hours, and the base pressure

achieved is 10-7–10-8Torr.

For the As-capped InAs, arsenic is removed in-situ just before depositing Nb. For

this purpose, the InAs is slowly heated to ~ 420°C, at a rate of 3-7 degrees per minute.

The As cap is then evaporated from the surface, as determined by a pressure spike,

at ~ 390°C.

13

2.2. Measuring techniques

2.2.1. Surface, thickness, and chemical composition characterization

SEM

In order to characterize the surface of the Nb and NbN films, a Hitachi S-800 field

emission scanning electron microscope is used at the beam voltage of 20kV, providing

the highest available magnification of ×130k. Also, a JEOL-6400 scanning electron

microscope is used at 40kV for imaging, and for e-beam writing, using Joe Nabity

software to control and to raster the electron beam.

X-ray diffraction measurements

A powder diffraction Rigaku D-Max X-ray diffractometer with a copper source,

line focus, 1° divergency slit, and graphite monochromator, operated at 40kV and 20mA

emission current, is used to determine the crystal orientation of the primary phase (Nb

and NbNx), and existence of secondary phases.

In order to characterize the flatness and the surface oxide formation on the

Nb/sapphire and Nb/InAs samples, and the Nb-InAs interface quality (roughness and

materials interdiffusion), X-ray reflectivity measurements are performed by I. K.

Robinson on beamline X16C at the National Synchrotron Light Source (NSLS) at

Brookhaven. Similar X-ray measurements and modeling are also done by S.-W. Han and

P. F. Miceli at the University of Missouri-Columbia using a rotating anode line beam

X-ray generator, with Mo anode (wave length: 0.70926Å) and 0.15mm-wide beam at the

sample.

14

Thickness measurements

DC-magnetron sputter deposition rate is constant during the deposition process.

Thus, the thickness of the film is determined by the actual sputtering time. The sputtering

rate is determined by measuring the thickness of a 2000-4000Å film using a Dektak3 ST

surface profiler. The rate is then determined as a ratio of the thickness to the sputtering

time.

AFM

Surface characterization of the samples, patterned with e-beam lithography, are

done using the Digital Instruments Dimension-3100 atomic force microscope with oxide-

sharpened silicon nitride probes in contact mode. This instrument allows to do scans of

up to 100 micron scan length, with a vertical sensitivity of the order of a few angstroms.

For features smaller than a few hundred angstroms, i.e. comparable to the probe size, the

measured height range can be limited by the geometric aspect ratio of the probe.

XPS

Physical Electronics PHI-5400 X-ray Photoelectron Spectrometer (XPS) with Mg

Kα line (1253.6 eV) is utilized to characterize the chemical composition of the NbNx

films and the InAs surface composition. The spectra are obtained at pass energies (energy

of electrons at the analyzer) of 178.95eV for surveys and 35.75 eV for the high-resolution

(“multiplex”) scans.

15

2.2.2. Transport measurements

The dc resistivity measurements are done in the standard four-probe geometry.

Gold wires are used as the probe leads and are attached using In-Ag alloy solder.

The voltage drop across the sample and the signal from the thermometers are

measured using digital voltmeters HP34401A and HP3456A. The sample is current

biased with current sources Keithley 220 and Keithley 224. Sample current is chosen

from the range of 1 to 100µA to keep Ohmic conductance, and to avoid sample heating

and exceeding the value of the critical current density Jc in the entire range of

temperatures.

The temperature dependence of the resistivity is measured in the range of 1.5-

300K. Temperature is monitored using a Lake Shore silicon diode DT470-LR for the 35-

300K temperature range, and germanium resistor GR-200B thermometers for the 1.5-35K

range, installed inside the copper sample holder. A mercury manometer is also used for

the range of 1.5-4K. Lake Shore temperature controller DRC 81 is used for current

biasing the thermometers, and for the temperature reading from the silicon diode using a

built-in calibration. A digital voltmeter HP34401A is used to measure the signal from the

germanium resistor. The voltage is converted into temperature in real time by software,

using a calibration table and a linear interpolation. The temperature calibration point

separation is 1K between 20K and 35K, 0.5K between 6K and 20K, 0,2K between 4K

and 6K, and 0.1K below 4K.

All devices are controlled via GP-IB bus using a PC-386 computer and a custom

written program.

16

The superconducting critical temperature is determined from the temperature

dependence of the resistivity using the 50% criterion of the normal-state resistance just

above TC. Transition width is defined as the distance between the temperatures where

10% and 90% of the normal state resistance is achieved.

2.2.3. Millimeter and submillimeter conductivity measurements

The measurements in the millimeter and submillimeter frequency range are

performed at Augsburg University [20] using a coherent source spectrometer [21]

sketched in Figure 2. In order to cover the range from 5 cm-1 to 30 cm-1, four backward

wave oscillators operating in partially overlapping frequency ranges are utilized as

monochromatic and continuously tunable sources. Both the intensity and the phase shift

of the transmitted far-infrared radiation can be measured simultaneously using the Mach-

Zehnder interferometer arrangement. These transmission measurements are considered to

be a powerful method of analyzing the electrodynamic properties of a wide variety of

high-conductivity thin-film materials [22, 23]. Most of the experiments in this energy

range measure only the surface resistance or the transmission. Measuring simultaneously

both the amplitude and the phase of the transmission through superconducting films

allows the direct determination of both, the real and the imaginary, components of the

complex conductivity.

In addition, a Bruker IFS 113v Fourier transform interferometer is used to perform

temperature dependent reflectivity measurements in the FIR (20 cm-1 to 500 cm-1). The

resonant technique developed in Ref. [24] is used to increase the sensitivity of reflection

17

measurements. The substrate is orientated with the film facing the back in order to use the

sapphire substrate as a Fabry-Perot resonator. Both real and imaginary components of the

complex conductivity can be determined without performing a Kramers-Kronig analysis

from reflection amplitude and frequency of the interference fringes in the substrate [22].

For the measurements with both spectrometers, the temperature of the film is varied from

4.5 to 300K using a He cryostat, with the sample in He exchange gas.

Source

Mirror 1

Analyzer

Wire-gridmirror

Sample Mirror 2

Detector

Figure 2. Setup of the coherent source spectrometer used for the opticalconductivity measurements in the frequency range from 2 to 50 cm-1. (After Ref. [20]).

The energy gap of most of the conventional superconductors is in the millimeter

and submillimeter wavelength range. Therefore, the discussed method can be used as a

18

direct probe of the superconducting energy gap, and for studies of the excitation spectrum

of the quasiparticles.

2.2.4. Raman spectroscopy

Optical setup for Raman measurements

The Raman measurements are done in a near backscattering geometry with an

incidence angle of approximately 15°. The x(y,z) x configuration is used primarily,

where x, y, and z represent the (100), (010), and (001) directions respectively. The first,

x, and the last, x , symbols correspond to the direction of propagation, or k-vector, of the

incident and scattered light respectively. The symbols inside the parentheses, (y, z),

correspond to the polarization of the incident and scattered light respectively.

An Argon ion laser (Coherent 90-5) is used in all Raman measurements at four

wavelengths: 457.9 nm, 488 nm, 496.5 nm, and 514.5 nm. The size of the laser aperture

is 2 mm with a divergence of 5 milliradians.

In order to guide the beam to the correct height and to disperse the laser wavelength

from the plasma lines, thus avoiding any interference with the Raman spectra, a high

index equilateral triangular prism is used. After going through a polarizer that defines the

incident polarization, the light is focused by either a spherical or plano-convex lens, both

with a focal length of 30 cm. The former lens produces a 100µm2 circular spot, while the

latter one focuses light in a rectangle of approximately 100µm × 1cm. The plano-convex

lens, focusing the laser beam in a larger area, allows for higher laser power to be used

19

while keeping the power density the same or lower, and therefore for a stronger spatially

integrated signal for a given power density. This is extremely important for the low

temperature measurements to avoid heating of the sample and driving a superconductor

into normal state.

1 2 3 4 5

6

7

8

9

1214

13

14 10

11

11

1. Laser 2. Equilateral prism 3. Aperture 4. Polarizer 5. Focusing lens 6. Sample holder 7. Camera lens with attached directing mirrors 8. Polarizer 9. Depolarizer10. Holographic filter11. Focusing lens12. Single monochromator13. Triple monochromator14. CCD camera

In order to use the triple monochromator a directing mirror is placed as indicated by the double line.

Figure 3. Optical system for Raman measurements. (After Ref. [25]).

The collimated Raman-scattered light is passed through a polarizer to select the

appropriate polarization. It is then passed through a depolarizer (Newport Optics) to

eliminate polarization effects in the monochromator. Finally, a holographic filter (Kaiser

Optical) is used to remove the very strong laser line, corresponding to the light scattered

at the original frequency. Finally, the light is guided into the monochromator with a

spherical lens.

20

A Spex Triplemate 1877 AG triple spectrometer and a Spex 0.5m single

monochromator are used. The former allows for a higher resolution and lower signal-to-

noise ratio, while the latter has a higher throughput, thus allowing the use of shorter

collection times and lower laser power, important for the low temperature experiments.

A liquid nitrogen cooled Photometrics 200 CCD is used for collecting the spectra,

with a resolution of 1.5–3.0cm-1, depending on the excitation wavelength used. An Ar

pen-lamp that provides a calibrated spectrum is used for calibration of the CCD, in order

to map the pixels to the corresponding frequency.

Low temperature Raman measurements

Low temperature measurements are performed in a Janis Model 8DT

Supervaritemp optical cryostat with a cold finger that has four optical windows. Liquid

He is used for temperatures down to 4K; temperatures as low as 1.7K are achieved by

pumping on the helium with a mechanical pump.

The temperature is monitored with a Si diode. The temperature is controlled with

the valve that adjusts the pumping rate, and a 25mW resistive heater mounted on the

sample holder and controlled by a LakeShore 330 autotuning temperature controller.

Independent temperature monitoring is done by measuring the resistance of the sample in

the standard four-probe geometry, and comparing it to the temperature dependence of the

samples determined before the Raman experiments as described earlier.

21

3. MATERIALS AND THEIR CHARACTERIZATION

3.1. Niobium

3.1.1. Film structure (SEM and X-ray results)

Nb has a body centered cubic structure. The structural quality of the Nb films is

noticeably high. An X-ray diffraction Θ-2Θ scan for a typical Nb film is presented in

Figure 4.

30 35 40 45 500

200

400

600

800

(110) Nb

Substrate peaks

Inte

nsity

(cps

)

2Θ (degrees)

Figure 4. X-ray diffraction: Θ-2Θ scan for a 5700Å-thick Nb film on sapphire.

22

The only peak that is not related to the sapphire substrate is at 38.46° and

corresponds to (110)-Nb. The x-ray diffreaction peaks at 25.58°, 47.15°, 50.20°, and

52.55° are due to the sapphire substrate. The 25.58° peak arises from the the Cu(Kα) line

on (012)-Al2O3 diffraction. 47.15°, 50.20°, and 52.55° peaks are all due to the (024)-

Al2O3 diffraction from the X-ray source of Cu(Kβ), W(Kα), and Cu(Kα), respectively.

Observation of a single peak indicates a highly-oriented film with no secondary

phases or orientations observable.

The smoothness of the Nb films is demonstrated by an SEM picture of a 100Å-

thick Nb film which is featureless to the resolution of 30Å (Figure 5).

Figure 5. SEM image of a 100Å-thick Nb film surface grown on sapphire.

23

X-ray reflectivity measurement results for the Nb/InAs sample AR050595-B- are

shown in Figure 6. The reflectivity is plotted as a function of the wave-vector transfer

q=2ksinθ, where θ is the incident angle of the X-ray beam. For a single uniform layer of

material, the value of 1/∆q, where ∆q is the period of the oscillations, is proportional to

the film thickness, and the flatness of the film is determined from the maximum wave-

vector transfer at which oscillations are observed. In order to deduce thickness and

roughness of all layers of the sample, an accurate fit to a reflectivity model, using

Parratt’s method [26], is performed by S.-W. Han and plotted as a line in Figure 6. The

best fit provides us with the following data: substrate (InAs) roughness: 1.4 ± 1.0Å,

thickness of Nb film: 97.2 ± 0.8Å, roughness 2.1 ± 1.6Å. The surface oxide layer is 22 ±

1Å with roughness 7 ± 5Å

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0101

102

103

104

105

106

107

108

Nb/InAsAR050595B

experimental data best fit

Ref

lect

ed in

tens

ity (a

.u.)

Scattering wavevector, q (nm-1)

Figure 6. X-ray reflectivity vs. vector transfer for a thin Nb film on InAs.

24

For the Nb/Al2O3, the 144.1 ± 0.3Å Nb film is rougher: 16.6 ± 2.5Å. This may be

due to apparently rougher surface of the substrate (sapphire) which is 11.9 ± 1.7Å. The

surface oxide layer is found to be 15.8 ± 0.2Å with roughness of 2.3 ± 0.2Å

3.1.2. Mean free path and RRR

In general, the measured metallic resistivity, ρ, is a function of temperature, but

near absolute zero it approaches a constant value, known as the residual resistivity ρ0.

The quantity ρ 0 is determined by the presence of defects, impurities, and strains in the

metal lattice. In a pure, clean metal with little lattice distortion, it constitutes only a small

fraction of the total resistivity at room temperature. Thus, the values of ρ 0 and the total

room-temperature resistivity ρ 300 can be used to characterize the material’s state of purity

and presence or absence of a strain in the crystal lattice.

Since Nb becomes a superconductor below 9.2K, the resistivity at 10K (or just

above the Tc) is used to characterize the quality of the Nb films. The residual resistivity

ratio RRR=ρ 300/ρ 10 is used as a parameter. While for most of the Nb used in industry and

research the RRR is of the order of 10, for the best Nb films grown in our sputtering

system, this ratio exceeds the value of 200. These, usually ~4000Å-thick films, grown

using the first procedure described in Chapter 2, have the ρ 10 in the range of 0.09-

0.1µΩcm, and ρ 300 ~ 14-17µΩcm. A typical plot of resistivity vs. temperature for a high-

quality ~4500Å Nb film grown at 450-470°C is presented in Figure 7. The room

25

temperature resistivity of this film is about 14µΩcm, and the resistivity just above Tc is

about 0.1µΩcm, yields a RRR of 140.

0 50 100 150 200 250 3000

2

4

6

8

10

12

14

1614 µΩcm

0.10 µΩcm

IR081098-C

R

esis

tivity

(µΩ

cm)

T(K)

Figure 7. Typical resistivity vs. temperature plot for a high-quality ~4500Å Nb film

grown at ~470°C.

Such a high value of the RRR indicates a single-crystal like quality of our Nb films.

The latter is also confirmed by observing the peaks in the X-ray diffraction spectrum,

corresponding to a single crystal orientation.

Yet another evidence of a good crystal structure is the observation of a weak

channeling effect with Rutherford backscattering spectroscopy (RBS), when the alpha-

particles can propagate through the film at some particular incidence angle(s) with a

reduced scattering rate.

26

Let us reproduce here how the resistivity depends on the mean free path and on the

geometry of the sample. In the absence of an applied magnetic field, the resistivity can be

separated into temperature independent ρ 0 and temperature dependent ρ T contributions:

ρ = ρ 0 + ρ T (Matthiessen’s rule). The latter is caused by the interaction of the conduction

electrons with the thermal vibrations of the ions of the lattice.

Considering the Drude form for conductivity σ = ne2τ /m, and substituting the τ

with l/vF, where l is a mean free path, and vF is a Fermi velocity, we obtain the relation

for the temperature independent part of bulk resistivity: ρ 0 :

ρ 0 = mvF /ne2l0. ( 3.1 )

When the thickness of the film is reduced to a value comparable to the mean free

path, the electrons are scattered from the boundaries of the film, and the effective mean

free path is reduced from its value for the bulk specimen. If we consider a thin wire of a

diameter d, where d is comparable to or less than l, then the resistivity of such wire will

be greater than that of the bulk material. The additional resistivity can be attributed to the

surface scattering, and therefore, as a rough approximation, d can be considered as a

mean free path for the surface scattering. Following Matthiessen’s rule, we write:

ρ 0 = mvF/ne2 (1/l + 1/d), and ρ 0/ρ 0b = 1 + l/d. (after Ref. [27]), where ρ 0b refers to the

bulk value of the resistivity and is defined by Eq.( 3.1 ).

More rigorous consideration for the case when d << l gives the result: ρ 0/ρ 0b → l/d

for a thin wire [27, 28], and ρ 0/ρ 0b = 4/3 l/d 1/ln(l/d) for a thin film of a thickness d [27,

29].

27

Using the Nb parameters: carrier concentration n=5.56⋅1022cm-3 from Ref. [30], and

mean field value for the Fermi velocity vF=0.3⋅108cm/s from e.g. Ref. [31-33], equation (

3.1 ) yields ρ 0l 0 = 1.9⋅10-6µΩcm2. The values for l0, that we obtain using this product for

films of measured thickness 50-100Å, are about 20-30Å less than the measured thickness

of the films. This is attributed to the formation of an oxide layer, and also to some

reduction in the crystalline quality of the films at lower thicknesses.

The RRR is significantly reduced in the presence of any sort of impurities or

irregularities that might limit the mean free path. Thus, the RRR is an extremely useful

parameter for characterization of the structural quality of metallic films. RRR also

decreases with a decrease in the film thickness.

3.1.3. Tc and its dependence on thickness

We observe the Tc of high-quality films that are 4000-5000Å to be slightly higher

(~9.28K) than the Tc value for bulk Nb of 9.2K [31]. A typical resistivity vs. temperature

dependence just around Tc for a high-quality Nb films grown at ~450-470°C is plotted in

Figure 8. The width of the superconducting transition of ~0.02K is typical for these films.

28

9.26 9.28 9.30 9.32 9.340.00

0.02

0.04

0.06

0.08

0.10

0.12

IR081098-C

Res

istiv

ity (µ

Ωcm

)

T(K)

Figure 8. Typical resistivity vs. temperature plot near Tc for a high-quality ~4500Å

Nb film grown at 470°C.

Nb films, thicker than 2000Å, grown on both sapphire and InAs, show the Tc of

bulk Nb, 9.2K. Figure 9 shows that as the film thickness is decreased, a gradual reduction

in Tc is observed with a significant fall off for the films thinner than 100Å, i.e. in the

range that is important for the proximity effect studies by optical measurements (see

Chapter 5).

There have been a lot of experimental and theoretical studies of the decrease in Tc

when the thickness is reduced. However, so far no model has been able to account

quantitatively for all the available experimental data. Many papers (e.g. [35]) discuss this

issue from two perspectives: the proximity effect [36] and the electron localization effect

in two-dimensional systems [37]. A brief description of these two models and a

comparison of our data with them follows.

29

100 10000

2

4

6

8

10

12

14

16

NbN Nb

T c (K

)

Film thickness (Å)

Figure 9. Critical temperature of Nb and NbN films as a function of film thickness.(After Ref. [34]).

In the case of the proximity effect scenario, the Tc should decrease as lnTc(d) ~ -1/d

as suggested by Cooper [36] and McMillan [38]:

[ ] )/(/

00

2da

cB

Dcc

SN

TkTT

νν

πωγ

−

=

h.

Here Tc0 is the transition temperature of bulk specimen, γ = 1.781, ωD is the Debye

frequency, νS and νN are the densities of states for superconducting and normal layers,

and d and a are the thicknesses of the superconducting and normal layers, respectively.

30

0.000 0.005 0.010 0.015 0.0201

10

NbN Nb

T c (K

)

1/d (Å-1)

Figure 10. Tc vs inverse film thickness d -1.

The data presented in Figure 9 is replotted in Figure 10 with Tc on a logarithmic

scale as a function of inverse film thickness d -1. It can be clearly seen that while the data

for NbN can be fit with a straight line on this plot, the data for Nb has significant

deviation from the linear behavior for the thinnest films. The estimate based on the best

fit for NbN yields the thickness of the normal layer to be a ≈ νN/νS ⋅9Å. Assuming

νN/νS ≈ 1, we get that the normal layer should be about 1-3 monolayers thick. This might

be reasonable for NbN which is known to form many conductive oxides near the surface.

In the Maekawa-Fukuyama model [37] based on the localization effect, Tc

depression of the two-dimensional superconductor is described as follows:

31

3

0022

22

0022

2

0

5.5ln23

15.5ln

221

ln

−

−=

c

cS

c

cS

c

c

TT

lR

eTT

lR

eTT ξ

πλνξ

πλν

hh,

where λ is the electron-electron coupling constant, νS is the density of states, ξ0 is the

coherence length, l is the mean free path, and R2 is the residual sheet resistance defined as

R2L/a, where L and a are the the length and the width of the film.

0 5 10 15 20 256

8

10

12

14

Nb

T c (K

)

Sheet resistance R2 of Nb films (Ω)

0 200 400 600 800

Sheet resistance R2 of NbN films (Ω)

NbN

Figure 11. Tc vs. sheet resistance R2 for Nb and NbN films.

The plot of Tc versus sheet resistance is presented in Figure 11. Attempts to fit the

data for Nb requires the electron-electron coupling constant, which is usually about

0.1-0.3 for the conventional superconductors, to be greater than 50. Thus, we conclude

that for our films the proximity-effect model is not suitable to describe the reduction in Tc

due to decreasing film thickness.

32

The quality of the films changes with time. Both Tc and resistivity show

degradation of the film quality during first few days after the deposition of the film, as

presented in Figure 12. Temperature dependence of the sheet resistance of a high-quality

150Å Nb films grown on sapphire (sample -C-) is plotted as a function of time after

deposition. The Tc and the RRR decrease, and the resistivity at all temperatures increases.

This is attributed to the oxidation of the surface layer and formation of one or several

niobium oxides which effectively reduces the thickness of the film. This agrees with

studies by Halbritter and coauthors [39, 40] of the formation and oxidation of Nb films.

The sheet resistance of an identical sample -B-, which has been preserved in liquid

nitrogen for 14 days, is also plotted in Figure 12. The Tc and resistance values changed

much less than those of the sample -C- which was exposed to the air all this time. This

allows us to conclude that the nitrogen preserves the films from oxidation and therefore

from degradation.

Decrease in the Tc of a few tenths of a degree is also observed after the optical

measurements, described in Chapter 5. This change is possibly due to some reactions at

the interface with InAs, such as oxidation or interdiffusion, catalyzed by the laser

irradiation. However, no systematic study of this effect has been done.

33

8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.00

1

2

3

4

5

-B-Day 14

-C-Day 14

-C-Day 1

She

et re

sist

ance

(Ω)

T (K)

0 50 100 150 200 250 3000

5

10

15

20

25

Day 1

IR080399 -B- and -C-

Sample -B- after 14 days in Liquid N2

Sample -C- : Day 1 Day 11 Day 2 Day 13 Day 4 Day 14 Day 9

She

et r

esis

tanc

e (Ω

)

T (K)

Figure 12. Sheet resistance R2 vs. temperature T as a function of time afterdeposition for a 150Å Nb film (sample -C-). An identical sample -B- was preserved inliquid nitrogen for 14 days and then measured.

34

3.1.4. Superconducting gap

Reflection, and the amplitude and the phase of the transmission through a high-

quality superconducting Nb film are measured in the millimeter and submillimeter

spectral range. The 150Å-thick 10×10mm2 Nb film grown at ~470°C as described in

section 2.2, has a transition temperature at Tc=8.31K and a transition width of 0.02K, as

determined by 10% and 90% values of the resistivity drop. The frequency dependent real

and imaginary parts of conductivity derived from the transmission data are plotted in

Figure 13.

The value of the energy gap obtained from a fit to the BCS theory yields

2∆(0)=24±1.5cm-1. The value of 2∆(0)/kBTc = 4.1±0.3 deviates from the universal value

of 3.53, corresponding to the weak-coupling BCS limit. This indicates a strong or

intermediate type of electron-phonon coupling.

The summary of measured and evaluated values are presented in Table 1.

Transitiontemperature

Tc,K

Energygap

2∆(0),cm-1

2∆(0)/kBTc

Coherencelength

ξ0=hvF/π∆(0),Å

Meanfree pathl(T=9K),

Å

πξ0/lObserved

penetration depthλ(0),

Å

Londonpenetration depth

λL(0),Å

Plasmafrequency

hωp,eV

8.31 24 4.1 390 90 13 900 350 5.8

Table 1. Measured and evaluated electrodynamical properties of niobium. (AfterRef. [20]).

35

0 10 20 30

0

5

b)

σ 2 (10

5 Ω-1cm

-1)

Frequency (cm-1)

0 2 4 6 80

10

20

BCS

Tc = 8.31 K

Nb

2∆ (c

m-1)

T (K)

0

1

2

a)

9 K 7.5 K 7 K 6 K 4.5 K

σ 1 (10

5 Ω-1cm

-1)

Figure 13. (a) Real and (b) imaginary parts of conductivity of Nb vs. frequency forseveral temperatures above and below Tc=8.31K. The solid lines are guides to the eye.

The dashed line represents 1/ν behavior predicted for low temperatures. The inset shows

the temperature dependence of the superconducting energy gap 2∆(T). The line

corresponds to the BCS prediction with Tc=8.31K and 2∆(0)/kBTc=4.1. (After Ref. [20]).

36

3.2. Niobium nitride

NbN usually has a B1 (fcc, NaCl-type) crystal structure. There are several

advantages of using NbN instead of Nb for the Raman experiments discussed in

Chapter 5. First, it has higher Tc: 13-17K, depending on the preparation. The coherence

length is shorter than that of Nb, so, the film thickness should have smaller effect on the

Tc. Also, the carrier concentration in NbN is lower than in Nb. So, the light attenuation

by the superconducting film, discussed in Chapter 5, is smaller for NbN films.

3.2.1. Tc dependence on growth parameters

NbNx has many non-stoichiometric compounds with different “x” values, and most

of them can superconduct, but have different values for Tc [41]. So, the initial goal was to

optimize the growth procedure for the highest Tc achievable using magnetron sputter

deposition in our system. The highest Tc reported for NbNx is 17.3K [42], and ~15.7K for

the stoichiometric NbN films with B1 structure. The values of Tc usually achieved using

reactive dc-magnetron sputter deposition are in the range of 13-16.5K [43-45].

The techniques of optimizing the Tc of the deposited films depend on the NbN

preparation technique used (see e.g. discussion in Ref. [46]). Control of the flow rates (or

partial pressures) of Ar and N2 is proven [43-46] to be an effective method for controlling

the chemical composition and therefore such properties as Tc of dc-magnetron sputtered

NbN. In this technique a sputter discharge is operated at constant power. All pressure

values reported here are measured using the ion gauge. While the flow of the Ar is kept at

a constant rate of 10.8 sccm, the pressure is tuned to 7 10-3 Torr by changing the throttling

37

rate, using the main gate valve between the vacuum system and the diffusion pump.

Thus, the run-to-run consistency of the throttling rate is established. Then nitrogen is

allowed into the system at a constant rate in the range of 1.5-3.0 sccm. Then the flow

ratio is fixed, and the overall pressure is brought down to 8.4 10-2 Torr by adjusting the

flow rates.

Figure 14 shows the Tc dependence on the ratio of the N2 flow rate to that of the Ar.

The ratio of 108:15 is chosen for the further NbN film growth procedures.

10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.00

100

200

300

400

500

600

Ar:N2 flow ratio:

108:15

108:20

108:25

108:30

Res

istiv

ity (µ

Ωcm

)

Temperature (K)

Figure 14. Resistivity vs. temperature around Tc of NbN thick films for differentN2:Ar flow rates.

Analysis of the film chemical composition using XPS showed that for the NbNx

films with the maximum Tc, the value of x is between 0.9 and 1. This correlates with the

previous studies [41] in which higher Tc’s are observed for stoichiometric or slightly

38

nitrogen-deficient NbNx compounds. After the optimum Ar:N 2 flow rate ratio is

established, it is then used in all of the subsequent growth processes.

3.2.2. Tc dependence on the film thickness

Much like Nb films, NbN films show a strong dependence of the Tc on the

thickness of the film (Figure 9). However, since the Tc of the bulk NbN is higher than

that of Nb, it remains higher at any given thickness.

In the normal state, our NbN films have a small negative slope of the R(T) curve in

the normal state. The RRR for these films ranges between 0.85 and 0.95, with the highest

values corresponding to thicker films. The room temperature resistivity ranges between

90 and 150 µΩcm for thicker films, and becomes 250-300 µΩcm for the 50-60Å-thick

films.

The film quality changes with time. Both the Tc and the resistivity increase during

the first two weeks after the deposition of the film as presented in Figure 15. This is

attributed to the oxidation of the surface layer and formation of one or several insulating

oxides, which effectively reduces the thickness of the film. This agrees with the results

obtained by Halbritter and coauthors (e.g. see [40] ) on the formation and the oxidation of

NbN.

39

0 2 4 6 8 10 12 148.0

8.2

8.4

8.6

8.8

9.0

Tc

T c (K

)

Time (days)

220

240

260

280

300

320

340

360

380

ρ 300K

(µΩ

cm)

ρ300K

Figure 15. Room temperature resistivity and Tc as a function of time afterdeposition.

3.3. InAs

3.3.1. InAs materials

We use two types of InAs. The first type is bulk-grown single crystals at too doping

levels, purchased from OMK. The highly Sn+S-doped n+-InAs has a carrier concentration

of n=1.2×1019cm–3, a mobility of 5.1×103 cm2/s, and a resistivity of 1×10-4 Ωcm as

measured at 77K. The nominally undoped InAs has a concentration of n=1.8×1016cm-3, a

mobility of 5.0×104cm2/s, and a resistivity of 6.9×10-3 Ωcm as measured at 77K. The

second type of InAs was MBE grown InAs by John F. Klem, as described in

40

section 2.1.3. It consists of a 100nm-thick highly Si-doped n+-InAs layer (1-2×1019cm3)

grown on top of a 120nm of a 1-2×1018cm3 buffer layer of InAs which is grown on an

undoped InAs.

3.3.2. General information about InAs and its surface

InAs is a direct III-V semiconductor with a zincblend structure. The minimum of

the conduction band (Γ6) is situated in the center of the Brillouin zone. The value of the

direct gap is 356meV at room temperature, and 418meV at 4.2K [47]. Since, near the

minimum, E(k) is isotropic but non-parabolic, the effective electron mass is a scalar and

depends strongly on the electron concentration. The value of the effective electron mass

ranges from 0.02 to 0.12.

The free surface of III-V semiconductors (GaAs, InAs, InGaAs, GaSb, InSb, InP,

etc.) tends to form a space charge region near the surface (interface). The band bending

occurs due to the presence of surface states, which act as an uncompensated charge. In

order to preserve electroneutrality, electrons are either pushed away from or attracted to

the near surface region. Depending on crystal termination (i.e. surface orientation, surface

treatment, etc.), surface coverage, surface reconstruction, presence of the surface states,

etc., either a charge depletion region (Schottky barrier) or a charge accumulation region

(inversion region) can be formed at the surface/interface of the semiconductor [13.a].

The width of such a space-charge region is determined primarily by the density of

the electrons, and in a rather crude approximation can be found from the Poisson

equation as a Debye length. More accurate evaluation of the size of the space charge

41

region requires rigorous consideration of the semiconductor parameters, such as doping

level, degeneracy, and actual band structure [48].

While for most semiconductors the bands are bending upwards, thus forming a

Schottky barrier, a charge accumulation region (CAR) is formed at the (100) surface of n-

doped InAs, where bands are bent down, and the Fermi level at the surface is pinned in

the conduction band (e.g. see Ref. [49]).

Metal deposited on top of a semiconductor can also influence the band bending.

There were several different theoretical models for the electronic band bending at

semiconductor/metal interfaces, starting with the first ones proposed by Schottky [50]

and Mott [51]. Systematic experimental studies of the influence of the metal coverage on

the band bending of n-InSb and n- and p-InAs were done by Corden [52]. Evaporating

Au, Al, Sn and In on the surface of the semiconductors, he found that they change the

band bending as measured by the intensity of the LO-mode (see section 3.3.4) is different

in each case and can not be directly correlated with the work function values. An

extensive review of studies of band bending modification for different surface coverage is

due to Guerts [13.b]. However, there is still no single theoretical model which would

account for all of the observed electronic near-surface properties, and which, in

particular, would predict the band configuration when a given arbitrary metal and an

arbitrary semiconductor are joined.

42

3.3.3. Doping level of InAs and Raman modes. L+ measurements as a tool for

doping level determination.

There are two types of optical phonon modes in InAs: transversal and longitudinal.

The transversal optical (TO) mode does not depend on the presence of the free carriers, as

predicted by group theory for a zinc-blend insulator [53], and the optical dielectric

constant ε for the transversal mode does not change with carrier concentration. The

longitudinal optical mode (LO) is affected by the presence of the free carriers at

concentrations higher than ~ 1017 cm-3 [54]. The frequencies of these two modes are

related to each other via Lyddane-Sachs-Teller relation ωεε

ωLO TO=∞

( )( )

/0

1 2

, where ωLO

and ωTO are the frequencies of LO and TO modes, and ε(0) and ε(∞) are the dielectric

constants in the limit of very small and very large frequencies. At higher levels of

concentration, the longitudinal phonon mode couples to the plasmon mode, producing L-

and L+ coupled phonon-plasmon modes whose properties differ from those found for a

pure longitudinal optic (LO) mode.

The coupling between the lattice vibrations and the plasma oscillations becomes

much weaker at high carrier concentrations due to the increasing difference in the eigen

frequencies of these two oscillations. As a result, the center frequency of the lower

branch, L-, becomes independent of the carrier concentration, while the high-frequency

L+ mode has a very strong dependence on n. The frequency of these two modes, Ω– and

Ω+, respectively, are governed by the following formula [13.c]:

( ) ( ) ( ) ( )

−+−±+=Ω ± 222222222

421

TOLOpLOpLOp ωωωωωωω , ( 3.2 )

43

where ω p=*

2

)(4

mne

∞επ

.

Note that due to dependence of the effective mass on the carrier concentration, a

deviation from the square root dependence of ω p on n is expected.

0 1 2 3 4 50

250

500

750

1000

1250

1500

1750

L- L+, max L+, avg L+, min L+, measured

Fre

quen

cy (c

m-1)

n1/2 (109cm-3/2)

Figure 16. Phonon mode frequency as a function of the square root of the carrierconcentration. A range of values for the effective mass is taken into account, providingthree curves for the L+ mode.

The frequencies of the L+ and L- modes are plotted in Figure 16 as a function of

the square root of the concentration, as calculated in Eq. ( 3.2 ). The effective electron

mass values are, experimentally obtained by different methods, are taken from the

reference tables [47, 55] and range from 0.02, for the undoped InAs, to 0.07-0.10, for

n =1019cm-3. The three curves for the high-frequency plasmon-phonon coupled mode are

shown to illustrate the effect of the effective mass, and are calculated using the

44

maximum, minimum, and average values for the effective mass, obtained experimentally

by different methods [47, 55].

600 800 1000 1200 1400 1600 1800

epi ~1, 2, (2?) 1019cm-3

#4

#5

#3

EB0269

EB0272

Bulk "omk" 1.2 1019cm-3

EB0308

Inte

nsity

(a.u

.)

Raman Shift (cm-1)

Figure 17. L+ Plasmon-phonon coupled mode for different InAs wafers.

Since the position of the L+ mode shows a strong and well-defined dependence of

its central frequency on the carrier concentration at sufficiently high doping levels, it can

serve as a tool to determine the carrier concentration and/or its possible change. The plot

of Raman spectra for the L+ mode for different InAs wafers with different concentrations

is presented in Figure 17. The following wafers are tested: three different bulk-grown

wafers (#3, #4, #5) with the concentration of 1.2 1019cm-3, and MBE–grown InAs wafers

at n=1 1019cm-3 and 2 1019cm-3 (EB0308, EB0272, EB0269). We also plot these

experimental points in Figure 16, and they show reasonably good agreement with the

theory.

45

This type of plot is especially important for the MBE-grown semiconductors which

have several different layers with different doping levels, and therefore their doping level

cannot be simply measured using the Hall effect, as it is usually done for bulk

semiconductors.

3.3.4. Charge Accumulation Region. I(LO)/I(L-) as a measuring tool

The measurements of the LO and L- modes allow to measure the width of the

CAR. [13.d].

TO

ω = ω p

nn* √

ω

L +

ωT

ωL

ω πp= 4 ne /m)(

2 1/2

NbNbNbNb

Nb

CAR

CAR

n -InAs+

n -InAs+

L +

L+

d = 35 ÅCAR

,

Figure 18. Phonon and plasmon-phonon coupled modes, observed in InAs.

46

The LO mode is not observable in highly-doped bulk InAs when n is of the order of

1018–1019cm-3. However, it is detected in the charge accumulation region, despite the fact

that the carrier concentration is even higher in the CAR.

In the CAR, the uncertainty of the longitudinal phonon’s wave number is governed

by the uncertainty principle ∆k∆x ≥ 1. Since the CAR width is small (typically about 30-

50 Å), the uncertainty in k becomes larger than the Thomas-Fermi screening wave-

vector. As a result, the longitudinal phonon mode is no longer effectively screened by

electrons in the CAR, and can be observed at the original LO frequency.

In x(y,z) x geometry of the Raman measurements (see section 2.2.4), we observe

the L- and L+ modes from the bulk, and the LO mode from the near surface CAR.

190 200 210 220 230 240 250 260 2700

50

100

150

200

L- LO

Inte

nsity

(a.

u.)

Raman shift (cm-1)

Figure 19. Typical Raman spectrum for n+-InAs (1.2 1019cm-3).

Intensities of the L- and LO modes can serve as a tool to study the strength of the

band bending. This is possible because with increasing band bending, the intensity of the

47

L- mode decreases, while that of the LO mode increases. The increase of the CAR width

means an increase of the volume where the LO mode exists.

The intensity ratio of the LO and L- mode in a highly doped InAs depends on the

CAR width dCAR: )1)2(exp( −=−−

dRR

II

L

LO

L

LO α [13.e], where RLO and RL- are the Raman

scattering tensors for the LO and L-, respectively. The probing depth 1/2α (α is the

extinction coefficient), defined as half of the light penetration depth (skin depth), depends

on the light frequency, ω, and the dielectric constant, ε.

The LO intensity ILO from the CAR can be related to the LO intensity of an

undoped sample IL- [13.d]: ILO=IL-[1-exp(-2αdCAR)].

An example of the LO/L- test done on OMK single crystals of highly doped n+InAs

(n=1.2×1019cm –3), and undoped InAs (n=1.8×1016cm –3) is presented in Table 2. Raman

measurements are performed in the x(y,z) x geometry, using all four available

wavelengths at room temperature and at 4K. The E1 gap at the Brillouin zone boundary in

InAs is 2.62eV at room temperature and 2.50eV at 4K. Therefore all laser frequencies

appear to be close to the E1-gap resonance. The Raman signal is significantly larger for

the lines that are nearly at the InAs E1 optical resonance: 457.9nm (at 2-10K) and

496.5nm (at 300K), than for those lines that are further away from the resonance:

488.0nm and 514.5nm respectively. The intensity ratios as well as the probing depth

estimated from the extinction coefficient for all four laser wavelengths used are presented

in Table 2.

The CAR width, dCAR, for the same highly-doped InAs (n=1.2·1019cm-3) is found to

be 35±3Å.

48

Probe Wavelength(nm)

Sampling Depth(Å)

Intensity ratioI LO / I L-

resonant 457.9 75 1.14non-resonant 488.0 92 0.71

resonant 496.5 96 0.80non-resonant 514.5 133 0.58

Table 2. Intensity ratios of the LO phonon mode to the coupled phonon-plasmonmode based on sampling depth for several wavelengths. Resonance is with the E1 gap ofInAs. (After Refs. [34, 56]).

3.3.5. Difference between bulk InAs and MBE-grown InAs

A similar spectrum of the phonon modes is observed on the MBE grown InAs at

approximately the same doping level. The only difference in the spectra of the MBE InAs

from the spectra of the bulk-grown single-crystal InAs is that the intensity of the LO

mode in the MBE-grown InAs is smaller. The LO-mode in the MBE-grown InAs is also

more sensitive to the surface treatment: the reduction in its intensity due to Ar-ion etching

and Nb deposition is somewhat stronger than that in the bulk-grown InAs (see

Chapter 4).

The initial treatment of the tested InAs samples is different: bulk-grown single-

crystal InAs was cut and polished, while the surface of the epitaxially grown InAs was

either just capped with As or not treated at all (see section 2.1.3). This difference in the

surface treatment may be responsible for the observed spectral difference for different

InAs.

49

4. SEMICONDUCTORS: SURFACE DAMAGE AND

PASSIVATION STUDIES

In order to understand the influence of damage and surface passivation on the

optical and electronic properties of the InAs surface, we undertook a series of

experiments. The ramification of these results also impacts device applications. Ability to

control and modify the interfacial properties is crucial in design and production of

semiconductor electronic devices. Passivation of the surface prevents surface oxidation,

improving yield and stability of the devices.

Damage on the (100) InAs surface is produced by Ar-ion etching. The effects of

damage and surface passivation are studied using Raman scattering and XPS [57, 58].

Strength of the band bending is proportional to the LO mode intensity as measured by

Raman scattering. The carrier concentration in InAs is determined using Raman

measurements of the L+ mode frequency. The chemical composition of the near-surface

region is determined using XPS.

In order to control surface band bending, which is primarily governed by surface

oxidation, alkanethiols, R-SH, where R is CH3(CH2)n, both neat (i.e. pure) and in

ethanolic solutions, are used. Removal of the native oxide with a Br:CH3OH

chemomechannical etch reduces the surface band bending, but without subsequent

passivation, the band bending is restored to the original value due to exposure to

atmospheric oxygen. Passivation with ethanolic alkanethiol solutions is found to reduce

the band bending, and to be stable for more than one week in atmospheric oxygen.

50

Passivation with neat alkanethiol is less effective. Passivation layer formation is inhibited

by the exposure of the InAs to the atmosphere between etching and passivation.

The Ar-ion etch, even under the mildest conditions, is found to produce lattice

damage and consequently reduce the LO mode intensity. Etching using a 75V beam

voltage creates an In-rich surface. Etching at higher voltages creates states which increase

the carrier concentration at a depth comparable to the Raman probing depth (~100Å).

Annealing at 500°C in UHV restores the material to the initial, unprocessed state: It

reduces carrier concentration, as determined by L+ peak center frequency; removes etch-

induced lattice damage in the near-surface region, as determined by the intensity of LO

peak; and restores the stoichiometric surface composition, as determined by XPS

analysis.

51

5. OBSERVATION OF THE SUPERCONDUCTING PROXIMITY

EFFECT BY RAMAN SCATTERING

5.1. Raman measurements of SN structure

The studies of the superconducting proximity effect by Raman scattering are done

using a 60-100Å-thick superconducting film (Nb or NbN) sputter deposited on an OMK

single crystal of highly doped (1.2⋅1019cm-3) n+-InAs. Raman measurements are

performed in the x(y,z) x geometry using a probe wavelength of 457.9nm, near resonant

to the E1 gap at helium temperature. The schematic of the experiment is presented in

Figure 20.

NbNbNbLASER

DETECTOR

60-100Å

35Å - CAR(Charge Accumulation

Region)

100Å

Figure 20. Schematic of the Raman measurements used for detection of thesuperconducting proximity effect.

52

Two modes, LO and L-, are measured at two temperatures, 10K and 2K, which are

above and below Tc, respectively. The LO phonon mode, as described in section 3.3.4, is

associated with the near surface charge accumulation region (CAR), while the L-

plasmon-phonon coupled mode arises from the bulk. The amount of light scattering is

different at this two temperatures, because at 2K the sample is immersed in superfluid

He. For this reason, the absolute mode intensity can not be used for temperature

dependent comparisons. Therefore, the ratio of the intensities of the LO and L- modes is

used. It is observed that this ratio, I(LO)/I(L-), measured at 2K and 10K, is enhanced

below Tc (2.5–5K) by up to 40% (Figure 21).

190 200 210 220 230 240 250 260 2700

50

100

150

200

L- LONb / InAs

above Tc

below Tc

Inte

nsity

(a.u

.)

Shift (cm-1)

Figure 21. LO and L- phonon modes at 10K and 2K, above and below Tc= 4.5K.

This observed change in the phonon spectrum is reversible upon cycling through

the superconducting transition temperature. At cryogenic temperatures, the laser power

can be high enough to heat the sample by several degrees. This allows temperature

53

cycling to be done by two different methods: by changing the He-bath temperature and

by changing the laser power. In both cases, the change of the sample resistance at the

superconducting transition is monitored, as described in section 2.2.2.

Raman measurements are also done on the bare InAs, off the Nb film. No change in

the Raman peak intensities with temperature between 2K and 10K is observed in this

case.

Raman measurements are also performed on an identical sample, except that no

surface etch was done prior to Nb deposition, i.e. when the native oxide layer, estimated

to be about 30Å thick, is present at the interface between the superconductor and the

semiconductor. No spectroscopic change is observed in this material over the same

temperature range. This confirms that the intensity change only occurs when the Nb is in

a good electrical contact with the InAs.

It is important to note that Nb is not optically active at these frequencies, so it does

not participate in the Raman scattering process itself. Thus, we conclude that the intensity

change occurs in the InAs in good electrical contact with the Nb, and is therefore due to a

superconducting proximity effect.

Results, similar to those for the Nb are also observed for the NbN (Figure 22).

54

190 200 210 220 230 240 250 260 270 2800

200

400

600

800

L-

LO

NbN/InAs

below Tc above Tc

Inte

nsity

(a.

u.)

Shift (cm-1)

Figure 22. NbN on InAs: LO and L- phonon modes above and below Tc.

The observation of this proximity effect is very sensitive to the interface quality,

and to the InAs material itself. It is mentioned that the surface oxide is removed by an in-

situ gentle Ar-Ion etch. Our studies (see Chapter 4) show that an increase in either the

etch time or the ion energy causes significant damage to the InAs surface, thus reducing

the intensity of the LO-mode. When the LO peak is very weak compared to the L- (e.g.

when it is just a small shoulder on L-, and its height is less than 10% of that of the L-) it

is difficult to measure any change in the intensity, and therefore, to perform studies using

the LO-mode.

As it is discussed in section 3.3.5, the surface of the MBE-grown InAs has different

properties from those of the single-crystal bulk-grown InAs. We are unable to observe the

proximity effect using MBE-grown InAs.

55

5.2. L+ mode for Nb/InAs samples

It is suggested that the charge density in the near surface region of InAs can change

upon the superconducting transition. Since the center frequency of the L+ mode is very

sensitive to the carrier concentration, measurements of the L+ mode at the temperatures

above and below the Tc provide a measure of any change in the charge density. No

changes are observed between 2K and 10K, either in the position or in the strength of the

L+ peak within the available experimental resolution. We estimate that our measurements

allow at least 5⋅1017cm-3 resolution in carrier concentration. However it is worth repeating

that L+ mode peak is more than 100cm-1 wide and its intensity is more than 20 times

smaller than that of the L- and LO modes. Also, because of the large uncertainty in the k-

vector due to a small size of the CAR, the L+ mode in the CAR may be damped by the

single-electron excitations. In this case the signal from the L+ in the CAR may not be

detected. Therefore, our result cannot be considered conclusive.

There is a report of an experiment [59] where intersubband absorption resonance in

a Nb/InAs/AlSb quantum well is measured. This resonance is sensitive to the electron

density in the InAs. No change in absorption with the temperature decreased below Tc,

was found in those tests, indicating that no significant charge transfer occurs at the

superconducting transition.

56

5.3. Discussion

A change in the intensity of the LO phonon and L-, plasmon-phonon coupled mode

of the InAs is observed upon the change in temperature from above to below the

superconducting critical temperature Tc. This change is only observed when the InAs is

in good electrical contact with the superconductor (Nb or NbN). These changes in the

Raman spectrum cannot be due to the superconductivity in the Nb alone. First, since a

superconducting transition in a zero applied magnetic field is a second-order transition,

and, therefore, is not accompanied by a volume change, it does not cause changes in the

crystal lattice to produce strain on InAs. Second, the changes in the electronic structure of

the Nb upon crossing Tc, i.e. the superconducting gap opening, are negligible on the

energy scale of the Raman scattering experiment. The Nb superconducting gap in our

films is 1.5meV or smaller, while all of the other energy scales are much larger: The

phonon energy is ~29meV, the Fermi energy counted from the bottom of the conduction

band, estimated for this high level of doping is ~280meV, the direct optical gap of InAs is

~418meV, and the probing light frequency corresponds to ~2.5eV.

Thus, the observed effect is due to the change in the near surface electronic

properties of the InAs upon the superconducting transition. This is in accord with the fact

that the Raman scattering process involves electrons, and the intensity of the Raman

modes is sensitive to the near surface electronic configuration (band bending). We

conclude that we observe a superconducting proximity effect.

The observed change may be attributed to one or several of the following processes

occurring in the surface charge accumulation region: 1) increase in the scattering volume

of the LO-phonon due to an increase in the CAR width; 2) decrease in the probing length

57

due to an increase in the extinction coefficient; 3) increase in the carrier concentration; 4)

phase correlation of carriers; 5) change in the conductivity of the InAs; and 6) influence

on the electron-phonon interaction in the InAs. Each is briefly discussed below.

The scattering cross-section of the LO-phonon with respect to that of the L- mode

would increase if the scattering volume of the LO-phonon increases due to an increase in

the width of the CAR. Such a change in the CAR width can be caused by a change in the

SN interface states that determine the band bending.

An increase of the extinction coefficient would lead to a decrease in the light

penetration depth, thus reducing the probing depth. The width of the CAR, the region of

the LO mode existence, is smaller than the probing depth. Therefore the intensity of the

LO mode should remain the same. At the same time, the sampled volume of the bulk,

where L- exists, is reduced. Since the volume of the bulk probed is reduced compared to

the volume of the CAR probed, the relative intensity of the LO mode to that of the L-

mode is increased.

Additional charges leaking into InAs from Nb would increase the carrier

concentration in the InAs. The Raman cross-section would then be certainly increased in

InAs, and the LO mode would be enhanced. There is no obvious mechanism for this

process, and no change in the concentration was detected within our experimental

resolution.

Below the superconducting transition temperature, electrons in the InAs CAR

participate in the Andreev reflection, therefore developing some degree of phase

correlation. The fraction of electrons undergoing Andreev reflection may be estimated by

taking into account the differences in the effective masses and Fermi momenta of Nb and

58

InAs. Assuming a clean interface, we follow the method suggested by Blonder [60].

Considering an electron as a plane wave, and matching the boundary conditions, we

estimate the probability for Andreev reflection for electrons near the Fermi surface to be

A = (1 + (1-r)2/2r)-2, where r is the Fermi velocity ratio for the Nb and InAs. The mean

field value for the Fermi velocity of Nb is vF = 0.3⋅108cm/s from e.g. Ref. [31-33]. For

InAs (n=1.2⋅1019cm-3, meff=0.08me), we find vF=h(3π2n)1/2/meff=1.03⋅108cm/s. In this case

we can estimate the ideal probability for Andreev reflection to be 29%. Other estimates

that use experimental values for the Fermi energy of similar InAs, provide much higher

values for the probability of Andreev reflection. This may be attributed to the difference

in InAs concentration. As it was explained earlier, the electrons and the holes

participating in the Andreev reflection are coherent. We stress that the effect of a

correlated electron plasma on the Raman phonon modes has not been studied

experimentally or theoretically, but an argument as to why this electron correlation can

influence the Raman spectrum is given below.

Proximity-induced electron coherence certainly increases the low-frequency

conductivity of the near-interface region of the Nb. The effect on the conductivity at

frequencies much higher than the superconducting energy gap is very small. However, at

frequencies very close to an optical gap, such as the E1 gap in our case, the dielectric

constant ε changes very rapidly with frequency. The derivative dε/dω, and therefore

dσ/dω, has a very large value. As a consequence, a tiny shift in the resonance frequency

can result in a significant change in optical properties of InAs, including the light

penetration depth. For this reason we do not completely rule out a possibility that a small

effect of the superconductor induced electron correlation on the high-frequency

59

conductivity can be significantly amplified due to resonance effects in the vicinity of the

E1`optical resonance in InAs.

It may be possible that the interaction between the longitudinal phonons and the

electrons change due to the presence of the superconductor, changing the Raman tensor.

It is worth noting that the scale of the LO phonon-electron interaction can be estimated

from the ωLO-ωL- difference, and is about 18cm-1, that is ~ 2.25 meV. This value is

comparable to the energy scale of the electron-phonon interaction in the superconductor

defined as twice the superconducting energy gap, 2∆. For bulk Nb, 2∆ ≈ 3meV (see

Ref. [20] and references within), and for NbN, 2∆ ranges from 3.4 to 6.6 meV (see e.g.

Refs. [45, 61, 62]), depending on the material quality.

5.4. Conclusions

For the first time, the superconducting proximity effect is observed by optical

measurements. A change in the electronic properties of the near-surface charge

accumulation region of InAs with the superconducting transition is reported. A few

possibilities for the mechanism of this effect are suggested. Further studies, both

experimental and theoretical, are required to find the exact mechanism responsible for the

observed effect.

60

6. PROXIMITY EFFECT: ZERO-BIAS FEATURES AND

“SHAPIRO-LIKE” STEPS IN THE IVC OF AN SN

STRUCTURE

6.1. Background

In 1982, Kadin and Goldman [63] predicted the observation of a “second-order

Josephson effect” (later called pair-field susceptibility) in S-S’ (SN) structures, where S

is a superconductor, and S’ is another superconductor above its superconducting

transition temperature, playing the role of N. They predicted hω/2e steps in the I-V

characteristics of such a junction, where ω is the frequency of the applied ac-voltage.

Point-contact tunneling experiments were performed by Han et al. [64-67], using

Nb or Ta superconducting tips. Measurements on superconductors UBe13, CeCu2Si2, Ta,

In and Mo, at temperatures above their Tc,, showed Shapiro-like steps in the I-V curves.

Han et al. also observed a periodic behavior of the critical current in magnetic field,

where the critical current is defined as a current at which the dI/dV significantly changes

from its near-zero-bias value. This periodic field dependence is much like one expected

for Jc(H) dependence of a Josephson (e.g. SNS) junction. Later, they performed

systematic point-contact measurements on several different materials, again using Nb or

Ta superconducting tips, and tunneling into normal state Ta, In, Mo, UBe13.

These experimental observations were explained first by Han et al. [64, 65] as a

“Proximity induced Josephson effect” (PIJE), which would be a first-order Josephson

61

effect between the superconductor and a region of the normal metal where

superconductivity is proximity induced.

Geshkenbein and Sokol [68], and then later Han et al. [67] suggested that the kinks

in the current-voltage relation are not Shapiro steps but rather steps caused by “photon

assisted tunneling” in the junction. This is a process in which absorption or emission of a

photon by the electrons increases the tunneling probability.

Although it was not clearly stated by Geshkenbein and Sokol, and just briefly

mentioned by Han et al., the essence of this phenomenon had been known for long time

as a Tien-Gordon mechanism [69] where the low-bias nonlinearity of the IVC could be

cloned by an applied ac field. We will discuss the details of the Tien-Gordon mechanism

further in this chapter.

Geshkenbein and Sokol obtained the nonlinearity of the IVC at low bias, solving

one-dimensional time-dependent Ginzburg-Landau (TDGL) equation, assuming a very

limiting set of parameters of the structure: a dirty gapless superconductor and a

temperature just above the Tc of the normal metal.

Kadin [70] suggested that the observed effect arises from phase slip centers in the

superconducting tip. His consideration is based on the assumption that the phase slip

occurs in the superconductor within a couple of superconducting coherence lengths from

the interface with the normal metal. However, no solution is provided for the case when

ac microwave irradiation is applied.

More recently, Thuneberg [71] developed a microscopic theory. Using the

quasiclassical approach, he presented a unified derivation of the results earlier obtained

62

by Geshkenbein and Sokol, and Han et al., who had used time-dependent Ginzburg-

Landau(TDGL) theory. Thuneberg approximates the induced order parameter as a step

function, determining self-consistently both the amplitude and the phase of the induced

order parameter, and the width of the step. His calculations are done without any

restrictions on the temperature or on the transmission coefficient of the junction. These

calculations provide a time-dependent solution of the order parameter at all currents or

voltages. Two cases are considered. The first one is the TDGL limit (gapless limit),

considered earlier by Geshkenbein and Sokol, when the inelastic scattering time τE, due

to by phonon scattering, and the order parameter ∆ satisfy the condition τE∆ « 1. This

case is very restrictive. With the reference to the earlier work of Watts-Tobin,

Krähenbühl and Kramer (WKK) [72], this calculation can be extended beyond the

gapless limit. Thuneberg makes a qualitative suggestion that there might be a regime

where the Andreev reflection at the SN interface makes an important contribution to the

nonlinearity of the conductivity. He also makes a reference to the work of Volkov [8]. In

this work Volkov uses the idea suggested by van Wees et al. [6], that multiple Andreev

reflection (MAR) contributes to the zero bias conductance peak (ZBCP). Based on this

idea, Volkov calculates the width of the ZBCP for different geometries of samples with

an SN interface.

Tuneberg’s theoretical consideration seems to be the only attempt to link together

two phenomena: the ZBCP, occurring as a result of MAR, and the Tien-Gordon

mechanism of cloning the differential conductivity peaks by photon-assisted tunneling

process when an external ac field is applied.

6.2. Zero-bias features of the IV characteristics in the An-

dreev scattering regime

A theoretical comparison of the two mechanisms producing a zero-bias conduc-

tance peak (ZBCP) is presented here. The two mechanisms are multiple Andreev

re ection and penetration of the superconducting order parameter inside the normal

metal. Evolution of the peak as a function of temperature, normal metal electron-

electron interaction constant, geometry and dephasing is discussed. We also combine

these two mechanisms of the ZBCP with the photon-assisted tunneling processes

(Tien-Gordon), and show that the ZBCP of either origin can be cloned.

These calculations are done in close collaboration with A. Shnirman. The details

of the calculations are presented in the Appendix. Here, we reproduce a brief summary

of the main results and discussion.

We consider the Andreev re ection at an SN interface. We are interested in the

corrections to the BTK [4] picture due to the interference and the electron-electron

interaction in the normal metal.

Using the tunneling Hamiltonian method and the quasi-classical approximation

pursued by Hekking, Nazarov and Averin [73-75], we obtain (A.1) the expression for

the Andreev current in terms of the Cooperon (Eq. A.7.):

I(t) =1

e32NIm

ZBd2r1

ZBd2r2g(r1)g(r2)

Z

0d2 e

2i(~(1)~(2)) CN(r1; r2; 1; 2)1!it; ~(1)!(t)

; (6.1)

where CN(r1; r2; 1; 2) CN(r1; r1; r2; r2; 1; 1; 2; 2) is the imaginary time

63

Cooperon, g(r) is the normal tunneling conductivity per unit area, and N is the

density of states per unit volume in the normal metal.

It is important that this expression is valid for any applied voltage. Then we

consider an applied ac-voltage in the following form (Eq. A.8):

V (t) = V0 + V1 cos(!t) : (6.2)

Using Tien-Gordon [69] expansion we deduce the full IVC (Eq. A.12):

I(V0 + V1 cos(!t)) jDC =Xn

J2n(2eV1

h!) I0(V0 +

nh!

2e) ; (6.3)

where I0(V ) is the dc IV characteristic of the system (Andreev contribution to it).

This equation shows that any non-linearity in the I0(V ), and therefore the peak

in conductivity 0(V ) will be cloned by the applied ac voltage. The position of the

conductivity peaks, and the distance between them is dened by

Vn =h!

2en ; (6.4)

where ! is the the frequency of the external ac eld.

This relation is the same as the Josephson relation governing the Shapiro steps

in the IVC observed in the ac-Josephson eect in SNS or other Josephson junctions.

It is worth reiterating here, that cloning of the ZBCP in both standard ac-Josephson

eect (SNS or SIS junctions) and in the case of the SN junction is essentially done by

the same Tien-Gordon mechanism. However, the origin of the zero-bias conductance

peak is dierent in these two cases.

64

In the ac-Josephson eect the peak is due to presence of a non-dissipative su-

percurrent in SNS or SIS structures. In the SN junctions, this peak is due to the

interference eects at the SN interface occurring as a result of Andreev processes, as

shown below.

First we consider a suciently thick superconductor and a normal metal layer

of a thickness d, separated by a tunnel barrier with area SB = LB LB. To analyze

the most general situation we rst assume that the barrier area is a fraction of the

total area of the normal layer SN = LN LN.

Considering diusive regime with the diusion coecient D, we obtain the most

general expression for the current (Eq. A.22):

I0(V )=

=4h

e3NIm

1

X!j>0

1

SNd

Xki

RB d

2rd2r0g(r)g(r0)2i=1 [zki cos(kiri) cos(kir

0

i)]

+Dk2 + 2!j i2eVh

: (6.5)

Here, !j = (2j + 1) kBTh, is the Matsubara frequency, j = 0; 1; :::; i = 1; 2; 3 is the

index of the spatial direction; ki is quantized: k1LN = n1, k2LN = n2, k3d = n3,

n3 0; zki = 1 if ni = 0, zki = 2 if ni > 0; and is a phenomenological dephasing

rate.

In the simplest case, when LN = LB and g(r) = GT=SB, where GT is the full

normal tunneling conductance, Eq. (6.5) reduces to (Eq. A.23)

I0(V ) =4hG2

T

e3NSBdIm

1

X!j>0

Xk3

1

+Dk23 + 2!j i2eVh

; (6.6)

If the electron-electron interaction takes place in the Cooper channel (A.4), and

the electron-electron coupling constant is given by , the expression for the current

65

in the same simple geometry can be rewritten as follows (Eqs. A.28, A.29):

I0(V ) =G2T

e3NSBdIm

CN(k;i2eVh)

1 + hCN(k;i

2eVh); (6.7)

where

CN(k;i2eV

h) =

4hN

X!j>0

Xk3

1

+Dk23 + 2!j i2eVh

: (6.8)

In the limit of a thin normal layer, and small temperature and voltage, such

that hD=d2 kBT; eV , and in the absence of dephasing, = 0, we can obtain (see

A.6) a simple analytical expression for the conductance peak at zero bias (Eq. A.32):

dI0=dV jV=0 =G2Th

4e2NSBdkBT

1 + N log(

2e h!DkBT

)

!2

: (6.9)

It is obvious that the sign and the value of the interaction constant determine

the height of the ZBCP.

In order to determine the IVC in the full range of parameters, the Eqs. A.28

and A.29 are used. The numerical calculations are performed using Mathematica

software, using parameters discussed below.

6.3. Experimental predictions and discussion

We consider the case when the n+-InAs plays the role of the normal metal. As

it is already mentioned, lower carrier concentration and small eective electron mass

in InAs provide rather large values for the diusion constant, and reduced values for

the density of states near the Fermi energy. This makes it simpler to observe the

66

ZBCP in such SN structure. Plugging in the values for the InAs into the Eq. 6.8, we

do the calculations using Mathematica. We use the following material parameters:

TD = 260K, GT = 101, SB = 20 20m2, d = 500A, n = 1019cm3, D = =e2N,

where = 0:1(m)1 is the bulk value of conductivity, N =kFme

22h2, kF = (32n)1=3.

The dephasing is neglected for simplicity.

The area of the contact and the thickness of the InAs are chosen small enough,

that the contribution to the current is measurable. It follows from the Eq. 6.7 that

the current scales with the inverse area of the contact and the inverse thickness of

the normal layer.

The interaction constant in N is chosen so that the product N is -0.05, 0,

and +0.05, which corresponds to a weak attraction, no interaction, and a repulsion

in the Cooper channel, respectively.

The plot of the conductance as a function of voltage, = dI0(V )=dV , as

determined using Eqs. A.28 and A.29, and the chosen parameters, is presented in

Figure 23.

First, we see that for 0 the ZBCP width is of the order of the temperature,

while the peak is signicantly narrower and much higher for negative , i.e. the

paraconductor. These results for the peak width are qualitatively similar to the

estimates obtained by Volkov [8].

A rather interesting result can be observed for a paraconductor case ( < 0).

For kBT , eV , where is the energy gap in S, the Andreev contribution to

the conductance may have a minimum at some nite voltage, just below 0.2mV for

67

considered here parameters (see Figure 23). The strength of the electron-electron

interaction (), and also the vicinity to the Tc of the paraconductor determine how

well pronounced this minimum is.

If all other contributions to the current are comparable to the Andreev current,

such a conductance minimum may be observed as a conductance reentrance eect.

If the dierential conductance of such an SN interface is measured at a constant low

voltage bias, while decreasing the temperature, one may observe a decrease in the

conductance at some low nite temperature. This is due to reduction of the ZBCP

width when the temperature is reduced. A detailed analysis of this possible eect is

required.

Now we x the temperature at T = 0:4K, and apply external ac voltage

V1 = 4V, with the frequency ! = 400GHz. The frequency is chosen so, that the

corresponding voltage V =h!

2eis larger than the temperature in corresponding units.

The strength of the steps in IVC calculated using Eq. 6.3 as well as the overall con-

tribution to the current from the considered mechanisms depend on the value and

sign of . The IVC plots in the external ac eld are presented in Figure 24.

Two important issues are noted: First, the obtained IVC does not represent

the total current, but the contribution to the current due to Andreev processes. So,

it is important to have other contributions reasonably small, so that the steps can

be observed. Second, it is important that the calculated conductance, =dI

dV, be

smaller than the conductance of the tunnel barrier GT. Otherwise the perturbation

theory used is not valid, and a more rigorous calculation is required.

68

While this work was in progress, we learned of a calculation by Feigel'man et al.

[78]. They consider a current across an SN interface in the presence of an electron-

electron interaction in N, using Keldysh functional approach and renormalization

group theory. Although no direct quantitative comparison can be made, qualitatively,

the results obtained for the contribution from Andreev processes to the current are

very similar.

6.4. Conclusions

The solution for the current contribution due to the superconducting proximity

eect in the SN structure is obtained using the tunneling Hamiltonian method. De-

pendence of this current on temperature, interaction constant, and other parameters

of the normal layer is studied. The contribution to the current from the Andreev

processes, including multiple Andreev re ections results in a zero-bias conductance

peak in which the height and width depend on the the normal layer parameters, in-

cluding the strength of the electron-electron interaction. Simple analytical form for

the height of the ZBCP is derived for the case when hD=d2 kBT; eV .

Application of an external ac eld produces steps in the current-voltage char-

acteristics due to the Tien-Gordon mechanism. The steps appear at voltages dened

by the relation Vn =h!

2en.

Experimental parameters of a highly-doped n+-InAs are then used to obtain

reasonable estimates for an experiment to be conducted to test the predicted model.

69

70

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.0

0.1

0.2

0.3

0.4

λνN = -0.05 λνN = 0 λνN = 0.05

Con

duct

ance

(Ω-1)

Voltage (mV)

Figure 23. Zero-bias conductance peak for different values of interaction constant λ.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

λνN = -0.05 λνN = 0 λνN = 0.05

Cur

rent

(µA

)

Voltage (mV)

Figure 24. Current-voltage characteristic with steps due to external ac-voltage for

different values of interaction constant λ.

71

7. FUTURE WORK

7.1. Temperature evolution of the I(LO)/I(L-) change

Depending on the actual mechanism of the observed proximity effect, the change in

the Raman mode intensity would have different dependence on the temperature. If the

temperature evolution of this spectroscopic change is continuous, and the I(LO)/I(L-)

ratio increases as the temperature decreases, it may follow the temperature evolution of

the superconducting gap in Nb, according to the BCS theory, but these measurement have

not yet been done.

Thus, the measurements of the temperature evolution of the ratio I(LO)/I(L-) may

provide very important information for understanding the underlying mechanism.

7.2. Nano-line structures

The geometry of the experiment with Raman measurements described in Chapter 5

places serious limitations on the superconductor thickness. Because of the backscattering

geometry, the film must be thin enough for the light signal to traverse the film twice. At

the same time, since the Tc is significantly reduced as the film thickness is decreased, the

film must be thick enough to have a Tc that is obtainable in our liquid helium optical

cryostat. These two conditions imply a narrow range of thicknesses: 60-100Å, dictating

the highest Tc of Nb film that can be used for these experiments: 4.5-5K. Due to the

instability of liquid helium between the boiling temperature (4.2K) and the λ-point where

72

it becomes superfluid (2.17K), it is difficult to perform optical measurements over this

temperature range.

In order to overcome this problem, another geometry of the sample is suggested:

“nano-lines” (Figure 25). The sample consists of ~2000Å-wide Nb lines separated by

1000-2000Å, covering an area of at least 200×200µm2. The height of each Nb line may

be 1000- 2000Å, providing a near-bulk value of Tc.

InAs

Nb

200-400nm

100-200nm

200-600nm

Figure 25. Nano-line structure: Nb lines on top of n+-InAs.

In this geometry, the Raman signal is collected from the bare InAs which is in

lateral proximity to bulk-like (Tc≈9.2K) Nb. The area of InAs surface participating in the

Raman process is much smaller than in the first experiment due to the Nb lines covering

and shadowing a fraction of InAs surface. However, unlike in the experiment described

in Chapter 5, the signal is not attenuated by traversal through the Nb in this geometry.

The Nb lines are fabricated using e-beam lithography. First, the Nb film is

deposited on InAs as described earlier. Then a ~2000Å-thick layer of 950K PMMA (4%

solution in chlorobenzene) is spun onto the surface at 6000rpm for 1 minute, following an

73

immediate 1-hour bake on a hot plate kept at 180°C. Then the pattern, created in

DesignCad, is written on the PMMA by e-beam. The exposure is optimized such that the

whole thickness of PMMA is exposed, while avoiding an overlap of the features caused

by the scattered electrons, due to exposure proximity effect.

After the pattern is written, the sample is developed in 1:3 MIBK:Isopropanol

solution for 1 minute, following an immediate rinse in isopropanol. The pattern is

transferred from PMMA to Nb using an SF6 reactive ion etch (RIE). To provide

consistency of etching, a descum, cleaning of the RIE chamber from residues, with 120

mTorr of oxygen at 120W is performed for 15 minutes. The etching procedure has two

steps: first, at 75 mTorr and 55W to remove the harder surface oxide layer and then at

45W to etch the Nb to the desired depth. The timing is determined experimentally by

etching films with macroscopic features (100µm-wide lines), and using Dektak3 ST

profiler to measure the etched depth.

It is important that the RIE on Nb/PMMA is performed in one step, because the

products of the RIE tend to form hard, teflon-like, compounds on the surface of Nb.

It is easy to verify the presence of the pattern on both PMMA and Nb, as well as to

distinguish between different line spacings, due to light interference on the nano-line

periodic structure. This is because the period of the nano-line structure is comparable to

the visible light wavelengh. The patterns with different spacing between the lines produce

different colors at the same viewing angle.

Removal of PMMA from the surface is complicated, because soaking in acetone or

other solvant, as well as sonicating it, peels off the Nb film. However, PMMA removal is

74

not required, because the PMMA does not have active Raman lines in the spectral range

of interest.

Figure 26. SEM and AFM images of the Nb nano-lines on sapphire.

Since the required accuracy of the RIE is very high, and the etch rate is rather fast,

a different procedure of either RIE, or even evaporation of the Nb on the PMMA with the

subsequent liftoff instead of RIE may be considered.

75

7.3. Magnetic and electric field dependence

It has been observed [79] that an applied external static electric field changes

parameters of the CAR, changing the bend at the interface. The width of the CAR, the

effective carrier concentration in the CAR, and the strength of the LO mode should

strongly depend on the applied electric field. Thus, studies of the dependence of the

observed proximity effect on the applied electric field may not only provide new

information about the mechanism, but also lead to some interesting device applications.

The observed proximity effect should also be sensitive to an external magnetic

field. It is expected that at some relatively small value of the magnetic field Hb, the effect

is suppressed, very much like superconductivity is suppressed by the termodynamic

critical field Hc. This breakdown field value may also depend on the temperature. It is,

however, important to keep in mind that the superconducting transition in the external

magnetic field is the first order transition, and, therefore, may introduce an additional

strain on the lattice of InAs, thus changing the phonon modes. It may result in an

additional mechanism of LO-scattering that can be observed, thus increasing the intensity

of the LO mode. However, depending on the particular mechanism of the observed

proximity effect, this LO-scattering may be influenced differently by the proximity of the

superconductor or not influenced at all.

APPENDIX

A.1. Andreev current and tunneling Hamiltonian method

We study the Andreev re ection in the superconductor-normal metal (SN) in-

terface. We are interested in the corrections to the BTK [4] picture due to the

interference and the electron-electron interaction in the normal metal.

We approach the problem from the \high barrier" side thus using the tunneling

Hamiltonian method. The Andreev current is given, in this method, by a diagram

corresponding to a second order in GT (the normal tunneling conductance). Formally,

however, this is a third order diagram in the perturbation expansion.

The tunneling Hamiltonian in the interaction representation reads:

HT(t) =X

ZNd3x

ZSd3y

yN;(x)T (x; y)S;(y)e

i(t) + h:c: ; (A.1)

where yi; is the creation operator of an electron on the side i (i = N; S) with spin

, T (x; y) is the tunneling matrix element, and (t) Rt dt0 eV (t0)=h.

The current operator in the interaction representation is:

I(t) =i e

h

X

ZNd3x

ZSd3y

yN;(x)T (x; y)S;(y)e

i(t) + h:c: ; (A.2)

The calculation is performed in the imaginary time with subsequent the ana-

lytical continuation to the real time. To this end we dene the operators HT() and

I() by making the following substitution

t! i; (t)! ~() ; (A.3)

76

and we restrict ~() to satisfy the periodic boundary condition: exp(i~(0)) =

exp(i~()) = 1, where is Matsubara time:1

=

kBT

h. All this allows us to calculate

the current as

I(t) = hT I() exp [Z

0d 0HT(

0)]i!it; ~()!(t)

: (A.4)

The Andreev scattering shows up in one of the third order terms of the pertur-

bation expansion of (A.4). This is

I(t) =4e

h4Imh Z

Nd3x1d

3x2d3x3d

3x4

ZSd3y1d

3y2d3y3d

3y4

Z

0d2d3d4

T (x1; y1)T (x2; y2)T(x3; y3)T

(x4; y4) ei(~(1)+~(2)~(3)~(4))

CN(x1; x2; x3; x4; 1; 2; 3; 4)CS(y4; y3; y2; y1; 4; 3; 2; 1)i 1!it;

~(1)!(t)

; (A.5)

where Cj(:::) hTyj;"(x1; 1)

yj;#(x2; 2)j;#(x3; 3)j;"(x4; 4)i, where j = N; S.

y1 x1

x2

x3

x4

y2

y3

y4

S N

C = + + + ...

Figure 27. Diagram for the current through an SN interface due to the interfer-

ence in the normal metal without electron-electron interaction in the Cooper channel

in the normal metal.

At low voltages the main contribution to the current in the normal metal comes

77

from the single electron interference. This is the interference of an electron fol-

lowing dierent paths while scattering from impurities; the process that leads to

a weak localization eect (see e.g. [80]). Since the disorder average of the prod-

uct of two Green's functions is not equal to the product of the separate averages,

the CN may not be factorized. In the superconductor, CS may be factorized as:

CS F y(y4; y3; 4; 3)F (y2; y1; 2; 1). If the frequencies involved are much smaller

then =h, i.e. if ~() does not change much on the time scale of h= and for temper-

atures kBT , the following approximation holds F (y1; y2; 1; 2) F (y1; y2;!j

0)(1 2).

We may further use the quasi-classical approximation pursued by Hekking,

Nazarov, and Averin [73-75] which gives:

ZSd3y1d

3y2F (y2; y1;! = 0)T (x1; y1)T (x2; y2) =h2

2e2Ng(rjj)

(3)(x1 x2)(1)(r?) ;

(A.6)

where r (x1 + x2)=2, rjj is the projection of r on the plane of the barrier, r? is the

distance from the barrier, g(r) is normal tunneling conductivity per unit area, and

N is the density of states per unit volume in the normal metal. The relation (A.6)

should not be understood too strictly. The delta-functions are of the \width" of order

F and (A.6) holds only if the quasi-classical description is used for the normal metal

as well.

Finally, one gets for the current:

78

I(t) =1

e32NIm

ZBd2r1

ZBd2r2g(r1)g(r2)

Z

0d2 e

2i(~(1)~(2)) CN(r1; r2; 1; 2)1!it; ~(1)!(t)

; (A.7)

where CN(r1; r2; 1; 2) CN(r1; r1; r2; r2; 1; 1; 2; 2) is the imaginary time

Cooperon.

A.2. Tien-Gordon [69]

The formula (A.7) is valid for any applied voltage. We are particularly interested

in a voltage of the following form

V (t) = V0 + V1 cos(!t) ; (A.8)

(t) =eV0t

h+eV1

h!sin(!t) : (A.9)

After the analytical continuation the following time-dependent factor appears

in Eq. (A.7):

exp(2i((t1) (t2)) =

Xn

Jn(2eV1

h!) exp

i(2eV0

h+ n!)t1

!

Xm

Jm(2eV1

h!) exp

i(2eV0

h+m!)t2

!: (A.10)

If we are to calculate the DC current, we need only those terms of (A.10) that depend

on the time dierence t1 t2 only. This is so since CN(:::) depends only on the time

dierence 1 2. Thus we collect only the terms with m = n:

exp(2i((t1) (t2)) =Xn

J2n(2eV1

h!) exp

i(2eV0

h+ n!)(t1 t2)

+ ::: (A.11)

79

From Eq. (A.11) one may deduce the full IVC as

I(V0 + V1 cos(!t)) jDC =Xn

J2n(2eV1

h!) I0(V0 +

nh!

2e) ; (A.12)

where I0(V ) is the DC IV characteristic of the system (Andreev contribution to it).

For I0(V ) one gets from Eq. (A.7)

I0(V ) =1

e32NIm

ZBd2r1

ZBd2r2g(r1)g(r2)CN(r1; r2; )

i!2eV=h

; (A.13)

where is a Bosonic frequency. Eq. (A.13) is the one we will analyze in various

regimes and geometries.

A.3. Purely diusive regime, no electron-electron interaction

We analyze a situation when the tunnel barrier is between a suciently thick

superconductor and a layer of a normal metal. The thickness of the layer is d and the

area of the barrier is SB = LBLB. To analyze the most general situation we assume

that the barrier area is a fraction of the total area of the normal layer SN = LNLN.

Although there exist diagrammatic techniques to calculate the Cooperon, we

present here a more straightforward way due to de Gennes [76]. One gets

CN(r; r0; ) =

1

X!j

Xm;l

1

(i!j m)

1

(i i!j l)m(r)l(r)m(r

0)l(r0) ; (A.14)

where m;l are the real wave functions of the one-electron states in the normal metal

and m;l are their energies (relative to the Fermi energy). The product of four wave

functions may be connected to a correlation function via

80

K(r; r0; ) Xl

hm(r)l(r)m(r0)l(r

0)i(m l + ) ; (A.15)

where the average h::i is taken over all states with a xed energy m. The Fourier

transform of K(r; r0; ) is nothing but a one-electron diusion correlator

K(r; r0; t) Zd exp(it)K(r; r0; ) = h(r(0) r) (r(t) r0)i ; (A.16)

where the average is again over all states with a xed energy m (there is no depen-

dence on m near the Fermi energy). K(r; r0; t) satises the diusion equation, that

is

K(r; r0; t) = ~K(r; r0; t) + ~K(r; r0;t) ;

~K(r; r0; t < 0) = 0 ;

@

@t~K Dr2

r~K = const (r r0)(t) : (A.17)

We solve (A.17) for our geometry (a normal slab of dimension LN LN d)

using an appropriate basis of the diusion modes which satisfy the diusion boundary

condition rrK(r; r0)j? = 0. Placing the origin in one of the corners of the slab we

get

~K(r; r0; ) = constXki

i [zki cos(kiri) cos(kir0i)]

Dk2 + i; (A.18)

where i = 1; 2; 3 is the index of the spatial direction, k1LN = n1, k2LN = n2,

k3d = n3, ni 0, zki = 1 if ni = 0 and zki = 2 if ni > 0.

Using Eqs. (A.14,A.15,A.18) one obtains

CN(r; r0; ) = const

Xki

1

X!j>0

i [zki cos(kiri) cos(kir0i)]

Dk2 + 2!j + ; (A.19)

81

where we assumed 0. In this equation !j = (2j + 1) kBT

h, is the Matsubara

frequency, j = 0; 1; :::.

The constant in (A.19) may be xed by noting directly from (A.14) that

Zdr0CN(r; r

0; 0) =1

X!j

Xm

1

2m+ !2

j

2m(r) =

1

X!j>0

2h

!j

N : (A.20)

Thus

CN(r; r0; )=

1

VN

Xki

i [zki cos(kiri) cos(kir0i)] CN(k; )

=4hN

X!j>0

1

VN

Xki

i [zki cos(kiri) cos(kir0i)]

+Dk2 + 2!j + : (A.21)

Here we have dened the Cooperon in the basis of the diusion modes CN(k; )

and have introduced phenomenologically the dephasing rate . The most general

expression for the current thus reads

I0(V )=

=1

e32NIm

1

VN

Xki

ZBd2rd2r0g(r)g(r0)i=1;2 [zki cos(kiri) cos(kir

0i)] CN(k;i

2eV

h)

=4h

e3NIm

1

X!j>0

1

SNd

Xki

RB d

2rd2r0g(r)g(r0)2i=1 [zki cos(kiri) cos(kir

0i)]

+Dk2 + 2!j i2eVh

: (A.22)

In the simplest case, when LN = LB and g(r) = GT=SB, where GT is the full normal

tunneling conductance, Eq. (A.22) reduces to

I0(V ) =4hG2

T

e3NSBdIm

1

X!j>0

Xk3

1

+Dk23 + 2!j i2eVh

; (A.23)

If the layer is so thin that hD=d2 kBT; eV it is clear that only k3 = 0 will contribute,

which is equivalent to a two-dimensional diusion in the normal layer.

82

A.4. Eects of the electron-electron interaction, paraconduc-

tivity

Here we assume that electron-electron interaction takes place in the Cooper

channel. We denote the electron-electron coupling constant . The corresponding

diagram is presented in Figure 28.

C = + + + ...

y1 x1

x2

x3

x4

y2

y3

y4

S N

Figure 28. Diagram for the current through an SN interface due to the inter-

ference in the normal metal with electron-electron interaction in the Cooper channel

in the normal metal.

We replace the previously calculated Cooperon (A.21) by an interaction ladder:

DN(r; r0; ) = CN(r; r

0; )

h

Zdr00CN(r; r

00; )DN(r00; r0; ) : (A.24)

In the basis of the diusion modes this becomes:

DN(k; ) = CN(k; )

hCN(k; )DN(k; ) ; (A.25)

and, therefore, one gets

83

DN(k; ) =CN(k; )

1 +

hCN(k; )

: (A.26)

Now, to calculate the current one has to substitute CN by DN in Eq. (A.22):

I0(V ) =1

e32NIm

1

V

Xki

ZBd2rd2r0g(r)g(r0)i=1;2 [zki cos(kiri) cos(kir

0i)]

CN(k;i2eVh )

1 +

hCN(k;i2eVh )

: (A.27)

Considering the simple geometry when LN = LB and g(r) = GT=SB, where GT

is the full normal tunneling conductance, we can simplify Eq. A.27 to

I0(V ) =G2T

e3NSBd

Xk3

ImCN(k1 = 0; k2 = 0; k3;i2eVh )

1 +

hCN(k1 = 0; k2 = 0; k3;i2eVh )

; (A.28)

where

CN(k1 = 0; k2 = 0; k3;i2eV

h) =

4hN

X!j>0

1

+Dk23 + 2!j i2eVh

: (A.29)

Due to niteness of the phonon spectrum, the frequency summation has to be

cut o at an frequency of the order of the Debye frequency !D.

A.5. A simple example

Consider the simplest situation: LN = LB, = 0, hD=d2 kBT; eV . Then we

can write down the current as:

I0(V ) =G2T

e32NSBdImDN(k = 0;i2eV

h) : (A.30)

We want to calculate only the zero peak height, i.e. dI=dV for V ! 0. Thus we

assume eV; hDk2; kBT and expand the Cooperon as follows:

84

CN(k; )4hN

X!j>0

1

+Dk2 + 2!j +

hN

"log(

2e h!D

kBT)

h

8

! +Dk2 +

kBT

#: (A.31)

And we immediately nd

dI0=dV jV=0 =G2Th

4e2NSBdkBT

1 + N log(

2e h!D

kBT)

!2: (A.32)

A.6. A case of a thick normal metal layer

We should also consider the case when the normal metal layer can not be con-

sidered thin, i.e. when either of the conditions: hD=d2 kBT; eV breaks. In this

situation we can no longer assume k3 = 0, and, therefore, need to do the summation

over the k3. For a reasonably thick normal layer: hD=d2 kBT; eV , the sum1

d

Xk3>0

in Eq. (A.23) can be replaced with the integral:1

2

Zdk

2. (For the sake of simplicity,

we do this consideration in the absence of the electron-electron coupling, i.e. = 0)

Upon integration we obtain:

I0(V ) =4hG2

T

e3NSBIm

1

X!j>0

1

4pD

1q + 2!j i2eV

h

: (A.33)

Let us further assume = 0. The ZBCP height, i.e. dI=dV for V ! 0 can be

found as:

dI0=dV jV=0 =G2T

4e2NSBpD

kBT

h

!1=2 1Xj=0

1

(2(2j + 1))3=2: (A.34)

The sum in the right hand side of Eq.A.34 is just a number which can be expressed

in terms of the Riemann -function, and is equal to 0.107226.

85

These results show that now the peak hight is proportional to the inverse of

only the square root of temperature, and not to 1/T as in the case of a thin N layer.

It is important to realize limitations of the validity of our solution for this case.

The tunneling Hamiltonian approximation breaks down for a very thick normal layer,

where, as it was shown by Zhou et al. [77], a signicant non-linearity of the potential

distribution in the normal layer develops.

86

87

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93

VITA

Igor Vladimirovich Roshchin was born on March 14, 1970, to Aida and Vladimir

Roshchin in Chelyabinsk, Russia. In 1987, he began studies at the Department of

Problems of Physics and Energy of Moscow Instutute of Physics and Technology,

Moscow, Russia, where he obtained a Diploma of Higher Education (with distinction),

and a M.S. (Magister) degree (Cum laude) in Solid State Physics in 1993. He enrolled in

the Ph.D. program in physics at the Moscow Institute of Physics and Technology, and

transferred to the University of Illinois at Urbana-Champaign in 1994. In 1994 he joined

Laura H. Greene’s Superconducting Group. In 1999 he received a M.S. degree in physics

from the University of Illinois. During the period 1991-1992 he was a recipient of

Moscow Government Fellowship for Best Students. During 1994, he was a recipient of

McArthur Graduate Fellowship from the University of Illinois, and from 1994 to present,

he has been supported by the Department of Physics and by Department of Energy

through Materials Research Laboratory (Grant DEFG02-91ER45439).