transport, magneto-transport and electron tunneling studies on
TRANSCRIPT
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Transport, magneto-transport and electron tunneling
studies on disordered superconductors
A Thesis
Submitted to the
Tata Institute of Fundamental Research, Mumbai
for the degree of Doctor of Philosophy
in Physics
by
Madhavi Chand
Department of Condensed Matter Physics and Materials Science Tata Institute of Fundamental Research
Mumbai August, 2012
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Table of Contents Declaration ................................................................................................................................ 9
Preamble.................................................................................................................................. 11
Statement of joint work .......................................................................................................... 13
Acknowledgements ................................................................................................................. 15
List of Publications ................................................................................................................. 17
Glossary of Symbols and Abbreviations used in this thesis ................................................... 21
Synopsis ................................................................................................................................... 23
Chapter I: Introduction .......................................................................................................... 49
1.1 Fundamentals of superconductivity and the effects of disorder........................................ 50
1.1.1 Phenomenology of superconductivity ................................................................... 50
1.1.2 London equations and electrodynamics of a superconductor ................................. 51
1.1.2.1 London equations and the magnetic penetration depth ................................... 51
1.1.2.2 Pippards coherence length ............................................................................ 52
1.1.3 BCS: the microscopic theory for conventional superconductors ............................ 53
1.1.3.1 Elementary excitations over the ground state: BCS theory at finite temperatures ................................................................................................. 56
1.1.3.2 Strong coupling superconductors ................................................................... 57
1.1.3.3 Application of BCS theory to disordered superconductors: Andersons theorem .................................................................................................................... 58
1.1.4 The Ginzburg Landau theory of superconductivity ................................................ 59
1.1.4.1 GL theory and fluctuations ............................................................................ 61
1.1.4.1.1 Amplitude fluctuations........................................................................... 61
1.1.4.1.2 Thermal and quantum phase fluctuations ............................................... 63
1.2 Effects of disorder on normal state electronic properties ................................................. 65
1.2.1 Localization of wave functions ............................................................................. 65
1.2.1.1 Weak localization .......................................................................................... 65
1.2.1.2 Anderson localization .................................................................................... 66
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1.2.1.3 Scaling theory of localization........................................................................ 67
1.2.1.4 Localization and superconductivity ............................................................... 69
1.2.2 Coulomb repulsion ............................................................................................... 70
1.2.2.1 Coulomb repulsion and the normal state: resistivity, MR and Hall effect ...... 70
1.2.2.2 Coulomb repulsion and superconductivity .................................................... 71
1.3 Recent studies on disordered superconductors ................................................................ 72
1.3.1 Experimental results ............................................................................................. 72
1.3.2 Theoretical models and numerical studies ............................................................ 75
1.4 Our chosen system: 3-dimensional NbN thin films ......................................................... 79
1.4.1 NbN films: properties and fabrication................................................................... 79
1.4.2 Classification of disorder using the Ioffe Regel parameter kFl ............................... 80
References ........................................................................................................................... 82
Chapter 2: Experimental Details ........................................................................................... 91
2.1 Fabrication of NbN thin films by reactive DC magnetron sputtering .............................. 91
2.1.1 Thin films and epitaxial growth ............................................................................ 91
2.1.2 Sputtering: basic concepts .................................................................................... 91
2.1.3 Fabrication of NbN thin films using reactive DC magnetron sputtering ................ 94
2.2 Techniques used for structural characterization .............................................................. 96
2.2.1 X-ray diffraction .................................................................................................. 97
2.2.2 Four-circle goniometry ......................................................................................... 97
2.2.3 Cross sectional transmission electron microscopy ................................................ 98
2.3 Transport Measurements ...............................................................................................101
2.3.1 Sample patterns, physical measurements, wiring configurations and electrical contacts ...............................................................................................................101
2.3.2 Electronics used for transport measurements .......................................................102
2.3.3 Low temperature measurements ..........................................................................104
2.3.3.1 Basic cryogenic concepts .............................................................................104
2.3.3.2 Continuous flow cryostat .............................................................................105
2.3.3.3 4He VTI in 12T cryostat...............................................................................106
2.3.3.4 3He insert in 12T cryostat ............................................................................109
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2.3.3.5 3He cryostat with sample dipped in liquid .................................................... 110
2.4 Electron tunneling ........................................................................................................ 111
2.4.1 Principle of tunneling measurements ................................................................... 111
2.4.2 Devices used for tunneling measurements ........................................................... 115
2.4.2 Optimization of tunnel junction devices .............................................................. 117
References .......................................................................................................................... 119
Chapter 3: Synthesis and characterization of superconducting NbN thin films ................ 123
3.1 Synthesis and optimization of NbN films ...................................................................... 123
3.1.1 Measurement of critical temperature and resistivity ............................................ 123
3.1.2 Structural characterization of the NbN films ....................................................... 125
3.1.2.1 Characterization by X-ray diffraction .......................................................... 125
3.1.2.2 -scans using 4-circle goniometer................................................................ 126
3.1.2.3 Cross sectional transmission electron microscopy........................................ 126
3.1.3 Analysis of transport and X-ray data to understand the roles of different sputtering parameters and establish a method to control disorder ......................................... 127
3.2 Transport characterization of 3D disordered NbN thin films ......................................... 129
3.2.1 Measurement of critical temperature, resistivity and carrier density; and the calculation of kFl ................................................................................................. 129
3.2.2 Correlation between measured physical quantities .............................................. 134
3.2.3 Measurement of Hc2 and GL ............................................................................... 134
3.2.4 A note on errors in the transport characterization ................................................ 138
3.3 Summary ...................................................................................................................... 139
References .......................................................................................................................... 139
Chapter 4: Normal state properties of disordered NbN films ............................................. 141
4.1 Introduction .................................................................................................................. 141
4.2 Theoretical background ................................................................................................ 142
4.3 Results .......................................................................................................................... 144
4.4 Discussion .................................................................................................................... 147
4.4.1 Understanding the temperature dependence of resistivity .................................... 147
4.4.1.1 Role of electron-phonon interactions ........................................................... 147
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4.4.1.2 Are the samples insulating? .........................................................................148
4.4.1.3 Details of (T) in the metallic regime ..........................................................150
4.4.2 Analysis of the temperature dependence of the Hall coefficient ...........................152
4.5 Summary ......................................................................................................................156
References ..........................................................................................................................156
Chapter 5: Superconducting properties and phase diagram of strongly disordered NbN films close to the metal-insulator transition .........................................................................159
5.1 Introduction ..................................................................................................................159
5.2 Evolution of superconducting properties with disorder: emergence of the pseudogap state ....................................................................................................................................160
5.2.1 Planar tunnel junction measurements...................................................................160
5.2.2 Results from scanning tunneling spectroscopy.....................................................165
5.2.3 Magnetoresistance studies on disordered films ....................................................166
5.3 Establishing the phase diagram .....................................................................................170
5.4 Discussion of the phase diagram ...................................................................................170
5.4.1 Region I ..............................................................................................................171
5.4.2 Region II .............................................................................................................172
5.4.3. Region III...........................................................................................................177
5.5 Quantitative analysis of the tunneling spectra ................................................................179
5.6 Summary ......................................................................................................................183
Appendix 5A: Dependence of Tc on the Coulomb pseudopotential and reduced density of states at Fermi level .....................................................................................................184
Appendix 5B: Quantitative analysis of the decay of MR in region I of the phase diagram ...186
References ..........................................................................................................................188
Conclusions, open questions and future directions ..............................................................193
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Declaration
This thesis is a presentation of my original research work. Wherever contributions of
others are involved, every effort is made to indicate this clearly, with due reference to the
literature, and acknowledgement of collaborative research and discussions.
The work was done under the guidance of Professor Pratap Raychaudhuri, at the Tata
Institute of Fundamental Research, Mumbai.
Madhavi Chand
In my capacity as supervisor of the candidates thesis, I certify that the above statements
are true to the best of my knowledge.
Prof. Pratap Raychaudhuri
Date:
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Preamble
The work presented in my doctoral thesis is an experimental investigation of the
electronic and superconducting properties of disordered superconductors through transport,
magneto-transport, Hall effect and electron tunneling measurements on niobium nitride thin
films.
The effect of disorder on superconductivity has been studied on a variety of systems for
many decades. However, the unprecedented technological development of novel experimental
probes at low temperatures over the past decade has made it possible to access physical
quantities in disordered superconductors, which were not easily accessible before. As I will show
in this thesis, investigation of disordered superconductors using these probes enable us to verify
the validity of a number of theoretical predictions made for systems close to the critical disorder
for destruction of the superconducting state. The reliability of such an experimental scheme
depends on two factors: (i) The ability to synthesize high quality samples with controlled
amounts of disorder and (ii) obtain information from complementary measurements on the same
samples. Towards this goal, in Prof. Pratap Raychaudhuris lab, our strategy was to use a
combination of scanning tunneling spectroscopy, Giaever tunneling spectroscopy, penetration
depth measurement and conventional magneto-transport and Hall effect studies on NbN thin
films with controlled amounts of structural disorder in the form of Nb vacancies in the crystalline
lattice. The experiments have been carried out collectively by four graduate students each
concentrating on a different specialized technique: Mintu Mondal (low frequency electrodynamic
measurements), Anand Kamlapure and Garima Saraswat (low temperature scanning tunneling
spectroscopy) and myself (magneto-transport, Hall effect and Giaever tunneling measurements).
While I will concentrate primarily on the last set of measurements in this thesis, I will use the
insight obtained from these complementary measurements to establish physical points as and
when required.
The thesis is organized as follows: In Chapter 1, I provide a theoretical background of
superconductivity and disordered materials, introduce the field of disordered superconductors
and motivate our experiments. I will also introduce our chosen system of study: homogenously
disordered 3D NbN thin films and the Ioffe-Regel parameter which is used to classify the
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samples. Chapter 2 contains the experimental details of the fabrication of samples by magnetron
sputtering and the measurement techniques including the cryogenics and electronics involved.
This chapter also explains the use of electron tunneling as a tool to study the spectral properties
of superconducting materials. Chapter 3 deals with the standardization, characterization and
classification of sputtered NbN films. In this chapter I discuss properties of the system like
transition temperature, normal state resistivity, number density and upper critical field. Chapter 4
contains the results of Hall effect studies and investigates the unusual normal-state properties of
NbN. In Chapter 5, I will use tunneling and magnetoresistance data to establish a
phenomenological phase diagram for NbN and demarcate three different regimes of disorder. I
will then discuss the physical phenomena that govern the superconducting properties in each of
these regimes.
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Statement of joint work
The experiments reported in this thesis have been carried out in the Department of
Condensed Matter Physics and Materials Science at the Tata Institute of Fundamental Research
under the supervision of Prof. Pratap Raychaudhuri. The results of the major portions of the
work presented in this thesis have already been published in refereed journals.
Most of the experiments discussed in this thesis have been carried out by me. For the sake of
clarity and completeness, I have included some of the data and analyses of experiments
performed by others. These experiments are listed in this statement.
The fabrication and characterization of NbN films, and tunnel junction devices as well as
transport, magneto-transport, Hall effect and planar tunneling measurements, which form the
major part of this thesis, were carried out primarily by me. Some of the above synthesis and
characterization were done jointly with John Jesudasan and Vivas Bagwe and some of the
measurements were performed jointly with Dr. S. P. Chockalingam. All the scanning tunneling
spectroscopy data shown in this thesis were performed by Anand Kamlapure and Garima
Saraswat. Penetration depth measurements, which I have occasionally referred to, were carried
out by Mintu Mondal and Dr. Sanjeev Kumar. Measurements on Be films were carried out in the
group of Prof. P. W. Adams by Dr. Y. M. Xiong at Louisiana State University and the analysis
based on their data was performed by me. Theoretical support for all the analysis presented here
was obtained from Prof. Vikram Tripathi in the Department of Theoretical Physics and Dr. Lara
Benfatto at Sapienza University, Rome, Italy.
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Acknowledgements
Pursuing a PhD at TIFR has been a very rewarding experience and at the same time an
extremely challenging one. I have learnt many valuable lessons while working in this beautiful
institution; I have broadened my horizon and deepened my understanding of physics. To reach
here and complete this degree, I have leaned heavily on the help and support of many people and
I would like to take this opportunity to express my gratitude.
I would like to begin by thanking my thesis advisor Prof. Pratap Raychaudhuri from
whom I have learnt a lot about solid state physics and superconductivity, low temperature
physics and experimental techniques. More importantly, I am grateful for the valuable lessons in
hard work, dedication, perseverance and perfectionism that he always taught by example. Most
of all, I would like to thank him for the faith he had in my abilities as a scientist and for showing
tremendous patience and tolerance as a guide.
I am extremely grateful to my lab-mates John Jesudasan, Vivas Bagwe, S. P.
Chockalingam, Mintu Mondal, Anand Kamlapure, Garima Saraswat, Archana Mishra, Sanjeev
Kumar and Charudatta Galande for all their help in the lab, especially for the innumerable
helium transfers and taking care of the experimental systems when I was not around. I would
also like to thank them for their company and moral support and for making the lab a home away
from home.
I am especially thankful to Subhash Pai and Atul Raut for their assistance with the
fabrication of equipment used in the various experiments. I am also grateful to Bhagyashree
Chalke, Rudheer Bapat, Shashank Purandhare, Nilesh Kulkarni, V. M Chopde, Abdul Kadir,
Masihhur Lashkar and Mahesh Kulkarni for their help with the structural characterization of the
samples discussed here i.e, the SEM, TEM, XRD and 4-circle measurements. I would like to
thank the entire staff of the low temperature facility (LTF) for providing our laboratory with
huge quantities of liquid helium, sometimes even at odd times and on weekends.
I would like to express my gratitude to Prof. Vikram Tripathi for the many explanations
and discussions. I am also grateful to Dr. Lara Benfatto, Dr. Sudhansu Mandal, Prof. Nandini
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Trivedi, Prof. Mohit Randeria, Prof. Mikhail Feigelman and Prof. T. V. Ramakrishnan for useful
and enlightening discussions.
I am grateful to my teachers, especially Dr. Bikram Phookun, colleagues and students at
St. Stephens College who have provided inspiration for me to pursue my studies in physics.
I am extremely grateful to my family who have been extremely supportive and
understanding during these past five years. A special thanks is due to the folks in Bombay whose
homes and help were never more than a phone call away.
I owe a lot of what I am to my parents who encouraged me to be myself and not follow
the crowd. My mother taught me do my level best in anything I undertake, and ensured that Ive
always had the space to follow my dreams. My father, who was an academic at heart, inspired
me to study science and be a thinking human being. This thesis is dedicated to him.
In the end I would like to attempt the impossible; to express my gratitude to my husband
and best friend for always being there, for his reassurance and encouragement and for boosting
my self-confidence whenever it was low. Without his undying love and support, this thesis
would never have come into being.
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List of Publications
In refereed journals, directly related to the material presented here:
Superconducting properties and Hall effect of epitaxial NbN thin films: S. P.
Chockalingam, Madhavi Chand, John Jesudasan, Vikram Tripathi and Pratap
Raychaudhuri, Physical Review B 77 214503 (2008)
Tunneling studies in a homogenously disordered s-wave superconductor: NbN: S. P.
Chockalingam, Madhavi Chand, Anand Kamlapure, John Jesudasan, Archana Mishra,
Vikram Tripathi and Pratap Raychaudhuri, Physical Review B 79 094509 (2009)
Temperature dependence of resistivity and Hall-coefficient in a strongly disordered
metal: NbN: Madhavi Chand, Archana Mishra, Y. M. Xiong, Anand Kamlapure, S. P.
Chockalingam, John Jesudasan, Vivas Bagwe, Mintu Mondal, P. W. Adams, Vikram
Tripathi and Pratap Raychaudhuri, Physical Review B 80 134514 (2009)
Phase diagram and upper critical field of homogeneously disordered epitaxial 3-
dimensional NbN films: Mintu Mondal, Madhavi Chand, Anand Kamlapure, John
Jesudasan, Vivas C. Bagwe, Sanjeev Kumar, Garima Saraswat, Vikram Tripathi and
Pratap Raychaudhuri, Journal of Superconductivity and Novel Magnetism 24 341 (2011)
Phase fluctuations in a strongly disordered s-wave NbN superconductor close to the
metal-insulator transition: Mintu Mondal, Anand Kamlapure, Madhavi Chand, Garima
Saraswat, Sanjeev Kumar, John Jesudasan, Lara Benfatto, Vikram Tripathi and Pratap
Raychaudhuri, Physical Review Letters 106 047001 (2011)
Phase diagram of the strongly disordered s-wave superconductor, NbN, close to the
metal-insulator transition: Madhavi Chand, Garima Saraswat, Anand Kamlapure, Mintu
Mondal, Sanjeev Kumar, John Jesudasan, Vivas Bagwe, Lara Benfatto, Vikram Tripathi
and Pratap Raychaudhuri, Physical Review B 85 014508 (2012)
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In refereed journals, not directly related to the material presented here:
Measurement of magnetic penetration depth and superconducting energy gap in very thin
epitaxial NbN films: Anand Kamlapure, Mintu Mondal, Madhavi Chand, Archana
Mishra, John Jesudasan, Vivas Bagwe, Lara Benfatto, Vikram Tripathi and Pratap
Raychaudhuri, Applied Physics Letters 96 072509 (2010)
Role of the vortex-core energy on the Beresinkii-Kosterlitz-Thouless
transition in thin films of NbN: Mintu Mondal, Sanjeev Kumar, Madhavi Chand, Anand
Kamlapure, Garima Saraswat, Goetz Seibold, Lara Benfatto and Pratap Raychaudhuri,
Physical Review Letters 107 217003 (2011)
As conference proceedings:
Evolution of superconducting properties with disorder in epitaxial NbN films: S. P.
Chockalingam, Madhavi Chand, John Jesudasan, Vikram Tripathi, and Pratap
Raychaudhuri, Journal of Physics: Conference Series 150 052035 (2009)
Fabrication and characterization of epitaxial NbN thin films and tunnel junctions, John
Jesudasan, S. P. Chockalingam, Madhavi Chand, Vivas Bagwe, S. R. Barman and Pratap
Raychaudhuri, Proceedings of the 53rd DAE Solid State Physics Symposium (2008)
Effect of point disorder on superconducting properties of ultrathin epitaxial NbN film,
John Jesudasan,Vivas Bagwe, Mintu Mondal, Madhavi Chand, Archana Mishra, Anand
Kamlapure, S. P. Pai, Pratap Raychaudhuri, Proceedings of the 54th DAE Solid State
Physics Symposium (2009)
Effect of Phase Fluctuations on the Superconducting Properties of Strongly Disordered
3D NbN Thin Films: Madhavi Chand, Mintu Mondal, Anand Kamlapure, Garima
Saraswat, Archana Mishra, John Jesudasan, Vivas C. Bagwe, Sanjeev Kumar, Vikram
Tripathi, Lara Benfatto, and Pratap Raychaudhuri, Journal of Physics: Conference Series
273 012071 (2011)
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Study of Pseudogap State in NbN using Scanning Tunneling Spectroscopy; Madhavi
Chand, Anand Kamlapure, Garima Saraswat, Sanjeev Kumar, John Jesudasan, Mintu
Mondal, Vivas C. Bagwe, Vikram Tripathi and Pratap Raychaudhuri, Proceedings of the
55th DAE Solid State Physics Symposium (2010)
Upper critical field and coherence length of homogenously disordered epitaxial 3-
dimensional NbN films; John Jesudasan, Mintu Mondal, Madhavi Chand, Anand
Kamlapure, Vivas C. Bagwe, Sanjeev Kumar, Garima Saraswat, Vikram Tripathi and
Pratap Raychaudhuri, Proceedings of the 55th DAE Solid State Physics Symposium
(2010)
Phase diagram of a strongly disordered s-wave superconductor, NbN: Madhavi Chand;
to appear in Physics Teacher
Evolution of Kosterlitz-Thouless-Berezinskii (BKT) Transition in Ultra-Thin NbN Films;
Mintu Mondal, Sanjeev Kumar, Madhavi Chand, Anand Kamlapure,Garima Saraswat,
Vivas C. Bagwe, John Jesudasan, Lara Benfatto and Pratap Raychaudhuri, to appear in
Journal of Physics: Conference Series
Effect of disorder and phase diagram of s-wave superconductor, NbN; Anand
Kamlapure, Garima Saraswat, Madhavi Chand, Mintu Mondal, Sanjeev Kumar, John
Jesudasan, Vivas Bagwe, Lara Benfatto, Vikram Tripathi and Pratap Raychaudhuri, to
appear in Journal of Physics: Conference Series
Magnetoresistance studies of homogenously disordered 3-dimensional NbN thin films,
Madhavi Chand, Mintu Mondal, Sanjeev Kumar, Anand Kamlapure, Garima Saraswat, S.
P. Chockalingam, John Jesudasan, Vivas Bagwe, Vikram Tripathi, Lara Benfatto and
Pratap Raychaudhuri, to appear in Journal of Physics: Conference Series
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Glossary of Symbols and Abbreviations used in this thesis
SYMBOLS a lattice constant or characteristic length scale of phase fluctuations e electronic charge Ec mobility edge EF Fermi energy G conductance
=h/2 h is Planck's constant Hc2 upper critical field Hpeak position of MR peak J superfluid stiffness kB Boltzmann constant kF Fermi wave-number kFl Ioffe Regel parameter l mean free path me mass of electron n number density/ electronic carrier density nn number of electrons that remain normal ns superfluid density N(0) density of states at Fermi level R resistance RH Hall coefficient Rsq sheet resistance for a 2-dimensional system t sample thickness T temperature Tc superconducting critical temperature vF Fermi velocity VH Hall voltage
Coulomb pseudopotential coherence length 0 Pippard Coherence length BCS BCS Coherence length GL Ginzburg Landau coherence length resistivity m maximum/peak resistivity n normal state resistivity peak peak resistivity
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electron scattering time penetration depth 0 flux quantum conductivity 0 minimum conductivity D Debye temperature
superconducting energy gap ABBREVIATIONS 2D two dimensions 3D three dimensions AA Altshuler and Aronov AL Aslamazov and Larkin BCS Bardeen, Cooper and Schreiffer DOS density of states GL Ginzburg Landau IVC inner vacuum chamber MIT metal-insulator transition MR magnetoresistance MT Maki and Thompson PID proportional-integral-derivative SIT superconductor-insulator transition STM scanning tunneling microscope STS scanning tunneling spectroscopy TEM transmission electron microscopy VTI variable temperature insert XRD X-ray Diffraction
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Synopsis
I. Introduction
The effect of disorder on superconducting properties has been a topic of active research
for many years. The earliest understanding was due to Andersons theorem1 which predicts that
the superconducting transition temperature (Tc) of a superconductor is not affected by disorder
scattering as long as the system remains a metal. This prediction is based on the spherical
symmetry of the superconducting order parameter as suggested by the Bardeen-Cooper-
Schrieffer (BCS) theory of superconductivity2.
However, it was later realized that this theorem does not hold true in strongly disordered
systems. The Tc does decrease with increasing disorder and sufficiently strong disorder can in
fact suppress superconductivity altogether. Since this disorder driven transition from a
superconducting ground state to a non-superconducting ground state occurs at zero temperature it
is a quantum phase transition (QPT). In 3-dimensional (3D) systems, the transition could be to a
metal or an insulator, while in 2-dimensional (2D) systems, it is to an insulating state as there is
no true metallic state in 2D (Ref. 3). Therefore, the transition is usually referred to in the
literature as the superconductor-insulator transition4 (SIT). In actual experiments, a number of
different parameters can be adjusted to tune the effective disorder. These include atomic scale
disorder in the form of defects and vacancies, granularity, and the effective increase in disorder
by reducing film thickness in 2D samples. A similar QPT can also be observed by the application
of magnetic field.
This field has received renewed attention in recent times because of the observation of
novel phenomena in many strongly disordered superconductors, especially in the vicinity of the
critical disorder for destruction of superconductivity. These include the spontaneous formation of
superconducting islands in physically homogenous films of TiN (Ref. 5) and NbN (Ref. 6),
observation of large magnetoresistance (MR) peaks in thin films of InOx (Ref. 7), TiN (Ref. 8),
amorphous Bi (Ref. 9 ) and NbN (Ref. 6), evidence of Cooper pairing via magnetic flux
quantization in a-Bi (Ref. 10) samples with a non-superconducting ground state, the persistence
of superfluid stiffness11 at temperatures above the mean field transition temperature and the
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existence of a pseudogap in the density of states above Tc (Ref. 12,13,14), similar to that seen
in the High Tc cuprates.
Many of these phenomena are also manifest in numerical studies. A homogenously
disordered 2D system without any physical granularity is seen to spontaneously form domains of
superconducting and non-superconducting regions15,16. This is accompanied by a decoupling of
the energy gap and superfluid stiffness suggesting the existence of a pseudogap state and
persistence of superconducting correlations on the insulating side17 of the SIT as well as a peak18
in the MR.
Understanding these phenomena is a challenging problem because they arise due to a
competition of different many-body effects namely Cooper pairing, Coulomb repulsion and
localization of wave functions19. In addition they may also be governed by universal scaling laws
arising from proximity to a quantum critical point20. It requires us to go beyond mean field
approaches like the BCS theory and perturbative corrections to the same; and look towards
radically different ideas like fluctuations in the phase of the order parameter21, localization of
Cooper pairs22,23 and condensation of vortices24 in the presence of strong disorder.
The system that we have used for this study is a set of 3D homogenously disordered NbN
thin films with different levels of disorder. NbN is a conventional s-wave superconductor with a
relatively high Tc of ~17K. Epitaxial thin films of this material can be grown by reactive DC
magnetron sputtering on nearly lattice matched (~5% mismatch) [1 0 0]-oriented MgO
substrates25. Obtaining films with Tc of 16-17K requires very specific deposition parameters.
Deviations from these optimal conditions results in samples with suppressed Tc and varying
disorder. Thus by adjusting the deposition parameters, we were able to tune the disorder to
obtain samples ranging from Tc~16.8K all the way down to samples in which no
superconducting transition is seen even at 300mK. In addition, these samples are chemically
stable, have good mechanical strength26 and can be thermally recycled from cryogenic to room
temperature without degradation of their superconducting properties.
Most studies of disordered superconductors have concentrated on the 2D limit, with the
tuning parameter being thickness, granularity or magnetic field. Our emphasis has been on the
study of 3D superconducting films as a function of disorder. Our films are homogenously
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disordered, containing defects on the atomic scale, and have thicknesses t 50nm, which is much
larger than the dirty limit coherence length (2.5dd7nm)27 thereby ensuring that they are in the
3D regime as far as superconducting correlations are concerned.
We have carried out transport, magneto-transport, Hall effect and electron tunneling
experiments to study the evolution of normal state resistivity, number density, density of states
(DOS) and MR. We find that the system evolves from a conventional BCS superconductor to
one with an unusual normal state, a pseudogap above Tc, a superconducting transition governed
by phase fluctuations and a peak in the MR for the samples without a superconducting transition.
I will present these results and use them to establish a phenomenological phase diagram and
highlight the different physical interactions that govern the normal and superconducting states at
varying degrees of disorder.
II. Experimental Details
1. Fabrication and structural characterization of NbN thin films
The films were grown by reactive DC magnetron sputtering a Nb target in an Ar/N2 gas
mixture onto [1 0 0] oriented single crystal MgO substrates. Deposition parameters like substrate
temperature and sputtering gas pressure were kept constant at 600C and 5 mTorr respectively.
The disorder was controlled by varying either one or both of the sputtering power (between 15W
and 250W) and Ar:N2 ratio in the gas mixture (between 90:10 and 30:70). Both decreasing the
sputtering power and increasing the partial pressure of nitrogen reduce the Nb concentration in
the sputtering plasma, thus creating Nb vacancies in the deposited film. It is these vacancies that
form the atomic scale disorder in the system. The time of deposition was adjusted to ensure that
the thickness of each sample was t50nm, which is much larger than 2.57 nm. Therefore
these films are effectively in the 3D limit as far as superconducting correlations are concerned.
The structural properties of the samples were examined through X-ray diffraction -2 scans
using the Cu-K line, -scans using a 4-circle goniometer and cross sectional transmission
electron microscopy (TEM).
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2. Low temperature transport measurements
Resistance (R), MR and Hall effect measurements were carried out using standard AC
and DC four-probe techniques from 285K down to 300mK using a combination of 4He and 3He
cryogenic techniques. The 4He measurements were done in a continuous flow cryostat with the
ability to go down to 2.2K and a home-built conventional 4He variable temperature insert (VTI)
in a 12T magnet operating down to 1.7K. The latter was specially designed to have a stability
better than 10 mK in the temperature range 1.7K - 150K and better than 100 mK in the
temperature range 150K - 285K. This level of stability was essential to obtain reliable Hall effect
data (described later). The 3He measurements were done on a custom built 3He insert in the 12T
cryostat as well as a separate sample-in-liquid 3He cryostat, with a lowest temperature of 350mK
and magnetic field up to 5T.
The AC measurements were carried out using a two-lock-in circuit which allows
independent measurement of the excitation current and induced voltage. For the Hall and MR
measurements, we used the offset and expand functions in the voltage measurement to detect
small changes in voltage over a large background.
The Hall coefficient (RH) was calculated by sweeping the magnetic field (H) from 12T
to -12T, thus subtracting out the resistive contribution. The upper critical field (Hc2) for several
samples was measured from either R-T scans at different H or R-H scans at different T. In all
these measurements, H was perpendicular to the plane of the film.
3. Electron tunneling
Quantum mechanical tunneling provides a useful tool for studying the spectral properties
of superconductors28 . When an electron tunnels across a thin insulating barrier, its energy
remains unchanged and it can therefore be used to carry spectroscopic information. We have
studied normal metal-insulator-superconductor (NIS) tunneling which is used to measure the
single particle density of states for the following reason: In a typical superconductor, the
superconducting energy gap feature extends over a few milli-electron-volts around the Fermi
energy, whereas the Fermi energy for a typical metal is in electron-volts. Therefore, in the
vicinity of EF, the DOS of the normal metal can be considered constant. At each value of bias
-
27
voltage, the tunneling current will depend on Ns(E) and Nn(E) i.e. the densities of states of the
superconducting and normal metal electrodes respectively as well as the Fermi-Dirac distribution
function f(E). The differential conductance G(V) which is proportional to the superconducting
DOS is obtained from the tunneling equation:
( ) ( ) ( ) ( ) ( ){ }
dEeVEfEfeVENENdV
d
dV
dIVG
ns
V
The tunneling experiments were carried out using NbN/native oxide/Ag planar tunnel
junctions down to 300mK. The geometry of the tunnel junction devices was designed so that
current (I)-voltage (V) and R-T could both be measured on the same sample without having to
take it out of the cryostat29. The I-V curves were then differentiated numerically to obtain the
differential conductance.
III. Characterization of NbN Samples
In order to classify the NbN samples and have a quantitative characterization of disorder,
we use the Ioffe Regel parameter kFl (which is a measure of the mean free path in units of de-
Broglie wavelength). This quantity decreases with increasing disorder. At kFl ~1, the Bloch state
is completely destroyed 30 and the system becomes insulating. The kFl values are extracted
experimentally from RH and resistivity () measurements and calculated from the free electron
formula: ( ) ( )[ ]{ } ( )[ ]3/53/13/22 2852853 eKKRlk HF h= . The usage of free electron formulae here is justified by the very short mean free path even in the samples with relatively high Tc. The
significant disorder scattering in each of the samples is expected to smear out any fine structure
in the electronic density of states leaving an almost free electron like structure.
I will demonstrate in a later chapter that strong el-el interactions are present in disordered
NbN. Since ne
RH1
= is truly valid only in the absence of el-el interactions, we have calculated kFl
at the highest temperature of our measurements i.e. 285K, where the interactions are believed to
be small31. The set of films described here range from the moderately disordered (kFl ~10) to
very strongly disordered, with kFl ~0.4.
-
28
We begin with the structural characterization of the NbN films. The films have a NaCl
structure and grow epitaxially on MgO substrates oriented along the [1 0 0] direction. Figure
S1(a) shows the -2 scans for NbN films of varying disorder, deposited with the same gas ratio
but different sputtering powers. The [2 0 0] MgO peak is ~43 and the [2 0 0] NbN peak is
observed ~ 41 We find that the optimal NbN film, i.e. the one with the highest Tc is at a
structural phase boundary between NbN and Nb2N. Films for which the sputtering plasma has a
greater Nb concentration than the optimal one form a different phase, Nb2N which is non-
superconducting.
An example of this is the sample
deposited at 250W shown in Figure S1(a). This is
evident from the fact that it does not show the
NbN peak, but instead shows a peak ~38.6,
which corresponds to the [1 0 1] peak of Nb2N.
Although many of our samples showed a small
peak corresponding to the Nb2N impurity phase,
on comparing the area under the peaks we
estimate that the volume fraction of the impurity
was ~ 0.5% for these films. Only such films, with
a significant NbN peak and very small (or absent)
Nb2N peak were considered in further experiments. The lattice constants of the different films
Figure S2: -scans of samples with kFl~4 and 9 showing epitaxy of the films.
0 60 120 180 240 300 360
kFl~9
Inte
nsity (
arb
. u
nits)
(degrees)
kFl~4
Figure S1 (a) X-ray diffraction spectra for NbN films deposited with a Ar:N2 ratio of 80:20 and sputtering powers 40W, 150W, 200W and 250W. (b) Lattice constants derived from the X-ray spectra for all samples deposited with the same Ar:N2 ratio and different sputtering power.
40 80 120 160 200
4.39
4.40
4.41
4.42
4.43
38 39 40 41 42 43 44 45
(b)
a (
)
Sputtering Power (W)
(200)
MgO
*
*
NbN-40W NbN-150W
NbN-200W
NbN-250W
Lo
g In
ten
sit
y (
arb
un
its
)
2 (degrees)
(200)
NbN
(a)
-
29
were calculated from the position of the [2 0 0] NbN peak. These were found to vary only by a
small amount [Figure S1(b)].
The epitaxial nature of the films was verified by taking -scans around the [1 1 1]
direction using a 4-circle goniometer. Representative scans for samples with kFl ~4 and 9
showing the peaks separated by 90 are presented in Figure S2. Further verification of the
epitaxial growth is seen from cross sectional TEM images of samples with kFl~3 and kFl~9 in
which the lattice planes are clearly seen to be parallel to the corresponding ones in the MgO
substrate (Figure S3(a) and (b)). The inset in Figure S3(b) shows the high resolution (HR) TEM
image with atomic scale resolution for the ordered NbN film.
Figure S3 (a) and (b) Cross sectional TEM images of NbN films with kFl~3 (a) and 9 (b) showing the lattice planes both in the NbN films and the MgO substrates. A high resolution TEM image is shown for the latter in the inset of 3(b).
From the preceding structural characterization we can see that the NbN thin films are
homogenously disordered and there is no evidence of any physical granularity.
We now proceed to the transport characterization of our films. Figure S4(a) shows the
resistivity vs. temperature for samples with different disorder in the temperature range 300mK to
300K. An enlarged picture of the low-temperature region is shown in the inset. The Tc is
extracted from these curves by recording the temperature at which the resistance reaches 1% of
MgO MgO (b) (b)
NbN kFl~3
kFl~9
NbN
(a)
-
30
its normal state value just above the transition. In Figure S4(b), Tc is shown as a function of
disorder (kFl). We have observed samples with Tc ranging from ~17K down to samples which do
not show a transition even at 300mK. A number of explanations are afoot for this strong decrease
of Tc with disorder, like Coulomb effects driven by loss of screening and phase fluctuations in
the superconducting order parameter. At present I will not elaborate on these and defer our
understanding of this decrease in Tc to the section V. In Figure S4(a) we see that all except the
least disordered sample have a negative temperature coefficient of resistance that becomes more
and more pronounced as disorder in increased. This is also evident from Figure S4(c) where the
resistivity at the peak (for superconducting samples) and at 300mK (for non-superconducting
samples) as well as the resistivity at 285K have been plotted in the log scale as functions of kFl.
This figure clearly shows that the disparity between low temperature and room temperature
resistivities gets magnified with increasing disorder. Hall effect measurements have been carried
out for samples of different disorder and the number density values so obtained are also plotted
in Figure S4(c) on the right hand axis. The number density for the least disordered sample comes
out to be ~2.63X1029 electrons/m3, which is close to the theoretical estimate32 of 2.39X1029
electrons/m3. This estimate has been arrived at by counting the number of electrons per Nb atom
and assuming that the nitrogen electrons do not contribute to electrical conduction. The
calculation assumes perfect stoichiometry, which suggests that our least disordered sample is
very close to stoichiometric and the disordered samples have varying degree of Nb vacancy.
50 100 150 200 250 3000.1
1
10
100
1000
10000
0 2 4 6 8 100
2
4
6
8
10
12
14
16
18
0 2 4 6 8 10
1
10
100
1000
10000
0 2 4 6 8 10 12 14 16 18 200.1
1
10
100
1000
10000
10.1
(
m
)
T (K)
kFl~0.42
(a) (b) (c)
Tc (K
)
kFl
(
m
)
kFl
0
5
10
15
20
25
n (
X 1
028m
-3)
(
m
)
T (K)
Figure S4: (a) vs. T for films with kFl~0.42, 0.49, 0.82, 1.23, 1.58, 1.68, 1.96, 3.27, 3.65, 4.98, 5.5, 8.02, 8.13, 8.82 amd 10.12. The inset shows an expanded plot of the low temperature region. (b) Tc vs. kFl for all the films and (c) Peak (black squares) and room temperature (blue triangles) values of resistivity as well as number density (purple stars) as functions of kFl.
-
31
The upper critical field (Hc2) and coherence length were also measured for the different
samples. For the less disordered samples, for which the upper critical fields were not directly
accessible at 12T, they were measured through -T measurements at different fields and then
calculated from the dirty limit relation 33 cTT
c
ccdT
dHTH
=
= 22 693.0)0( where dHc2/dT was
calculated at the point where the resistance becomes 90% of its normal state value.
10 12 14 16 180.0
0.2
0.4
0.6
0.8
1.0
6 8 10 12 14 160
2
4
6
0 2 4 6 8 100
5
10
15
20
25
1 2 3 4 5 6 7 80
2
4
6
8
10
12
14
16
kFl~8.7
(
m
)
T(K)
B=0
B=5.8T
(a)
(d)(c)
Hc2(T
) (T
)
T (K)
Hc2 (
0)
(T)
kFl
4
5
6
7
8
(
nm
)
(b) B=11T
(
m
)
T(K)
B=0
kFl~2.2
Figure S5: (a) and (b) vs. T at different magnetic fields for samples with kFl~8.7 and 2.2 respectively. (c) Hc2(T) vs. T for films with kFl~ 8.7, 7.09, 5.5, 4.6, 4.38, 3.91, 3.5, 2.85 and 1.5. (d) Hc2(0) and for all the films.
Representative plots of -T measurements at different fields are shown in Figures S5(a)
and S5(b). In S5(b), the Hc2(0) is directly read off the graph as the field for which the
superconducting downturn is no longer present. The dHc2/dT lines close to Tc for all the samples
that are measured using the dirty limit relation are shown in Figure S5(c). The Ginzburg-Landau
coherence length is then extracted from the upper critical field using)0(2 2
0
c
GLH
= . Both Hc2
and GL have been plotted as functions of kFl in Figure S5(d). The initial increase in Hc2 with
-
32
disorder (5.5dkFld10) seems counter-intuitive, but can be understood in the following way: In
the low disorder regime, Andersons theorem applies to some extent and the Tc and (the
superconducting energy gap) are not significantly affected. Therefore, in this regime, the Pippard
coherence length, BCS (=vF/), where vF is the Fermi velocity, does not change either.
However, GL depends on both BCS and the mean free path l ( lBCSGL = ), the latter of which
decreases with disorder even in the low disorder regime, resulting in a net decrease in GL. At
higher disorder, starts to decrease, thereby increasing BCS. The competition between BCS and l
results in a net increase of GL and decrease in Hc2.
Finally, I make some observations on the sources of error that affect the basic quantities
that we measure. The single largest source of error comes from the measurement of thickness of
the samples (using a stylus profilometer), which affects RH and measurements. This error is
~15%. In addition the values also depend on the widths and lengths of the patterned devices.
However, since all the samples are deposited using a shadow mask, there will be no relative error
and we estimate the absolute error to be not more than 1-2%, which is the tolerance of our mask.
In comparison to these mechanical quantities, the electronic quantities (V and I) are measured to
within a few parts per million. Therefore the errors in and RH are estimated to be ~15% and in
kFl ( RH1/3/ thickness-2/3) ~10%. The temperature measurements are accurate to within 0.1%.
In the measurements from 1.8K to 300K, using the 4He VTI, the sample and sensor are both in
contact with the same copper block, and also the experiment is carried out in a He atmosphere,
ensuring that thermal gradients are negligible. In the 3He insert, where the sample is in vacuum
there are two sensors, one near the 3He-pot and the other close to the sample. These two sensors
are a few centimeters apart and can have a gradient of up to 100mK. However, the distance
between the sample and the sensor closest to it is much smaller, therefore we do not expect a
significant error in the estimation of temperature.
IV. Understanding the normal state of disordered NbN
A number of factors control the transport properties of a metal. In most metals, resistance
increases with temperature primarily due to phonon-scattering and follows Mattheissens rule34.
In the weakly disordered regime, the role of disorder is to increase the temperature independent
-
33
part of the resistance. However, strong disorder can localize states at the band edges and can
drive the system into an insulating state30, even for a partially filled band, when the all the states
up to Fermi level get localized. This is known as the Anderson metal-insulator transition (MIT)
and corresponds to kFl~1. The scaling theory of localization3 studies the effect of disorder
scattering right down to the MIT and describes the eigenstates of a system of size (2L)d (where d
is the dimensionality) in terms of the eigenstates of the corresponding systems of size Ld. i.e. it is
used to study how the properties of a system scale with the system size. It predicts that the metal-
insulator transition is continuous in 3D without any minimum conductivity and predicts power
law dependence of conductivity on temperature.
Another phenomenon that affects the transport properties is el-el Coulomb repulsion,
which becomes important in disordered systems because of the effective loss of screening in the
diffusive transport regime. Coulomb repulsions have also been shown to change the sign of
d/dT. In the weak scattering regime in 3D, the Altshuler-Aronov35 theory for el-el correlation
predicts a conductivity of the form ,
DTk
Fe
B
hh)2
33
4(2
3.1
4
1 ~2
2
= .. (1)
where D =
is the diffusivity, and F
~is a Fermi liquid parameter such that
F~
23
34
is of the order of unity.
Since the temperature dependence of resistivity cannot be unambiguously attributed to
either disorder scattering or e-e repulsion, one can study Hall effect as a function of temperature
to distinguish between the two. Shapiro and Abrahams36 have applied the scaling theory of
localization to the Hall effect and they predict that for a 3D disordered system without e-e
interactions, RH will retain its metallic character down to the metal-insulator transition i.e. it will
be temperature independent. On the other hand, according to the A-A theory, xy is unaffected
by temperature 37 and therefore RH=xy/xx2 varies twice as fast with temperature as i.e.
=
2H
H
R
R (Ref 38).
-
34
In this chapter I will present a detailed study of resistivity and Hall effect studies to
understand the effect of these different phenomena on the normal state properties of disordered
NbN films.
3 6 9 12 15 18
0
1
2
3
4
5
6
7
8
0 30 60 90 120 150 1800
1
2
3
4
5
6
7
8
180 210 240 270 300
0.736
0.738
0.740
0.742
0.744
0 1 2 3 40
200
400
600
800
1000
0 50 100 150 200 250 3000.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(d)
1.68
5.50
kFl
(
10
5
-1 m
-1)
T1/2
(K1/2
)
4.98
3.65
3.27
(c)
1.68
(b)
3.27
3.654.98
(
10
5
-1 m
-1)
T (K)
kFl
5.50
(
m
)
T (K)
kFl ~ 8.82
~0.002 m
kFl~0.82
kFl~0.49
kFl~0.42
(
-1)
T (K)
kFl~10.12
(a)
(
m
)
T (K)
kFl~8.82
Figure S6: (a) vs. T for samples with kFl 10.12 and 8.82. The inset shows an expanded view for the sample with kFl ~8.82 at high temperatures, vs. T for samples with wide range of disorder. (c) vs. T for the most disordred samples at low temperature. (d) vs. T1/2 for the same samples as in (b)
We begin by looking at the role of phonon scattering. Figure S6(a) shows -T for samples
with kFl~ 10.12 and 8.82 up to 300K. The positive slope seen for the sample with kFl ~ 10.12
indicates that this sample obeys Matthiessens rule and we can attribute the temperature
dependence largely to phonon scattering. In the next sample (kFl~8.82), only a small region at
higher temperature has a positive slope (see expanded view of -T for the film with kFl~ 8.82
shown in the inset) which can be attributed to the el-ph scattering. This results in a resistivity
increase of 0.002 m in the interval 240 K to 300 K. At lower temperatures, this sample too
has a negative temperature coefficient of resistance. Therefore the contribution of el-ph
scattering to the overall (T) is less than 2% for this sample. For the more disordered samples
-
35
where we do not see any positive slope at all in the -T curves, this contribution would be
negligible compared to impurity scattering.
To explore the reasons for the negative d/dT for all the films with kFl 8.13, we turn
our attention to the precise temperature dependence of for the disordered samples. For the
more disordered samples, varies linearly with T from 40 K to 150 K (Figure S6(b)). The films
with kFl>1 show an upward deviation from the linear T behaviour at temperatures below ~40 K,
but for the samples with kFl
-
36
In Figure S8(a)-(b) we plot ( )
( )
=
KR
KRTR
R
R
H
HH
H
H
285285)(
vs.
=
)285(
)285()(K
KT
for
all NbN samples with 1 kFl 8.13. All the curves are linear within error bars of our
measurements and the slope,
=
dR
Rd
H
H has a universal value 0.680.1. The
variation in is shown as a function of kFl in the inset of Figure S8(a), which emphasizes the fact
that there is no monotonic trend in as a function of kFl. The observed value of =0.68 differs
significantly from the predictions from localization (=0) as well as AA theory of interactions
(=2).
The first argument to support being different from 2 is the following: Since we expect
both localization and interaction effects to be present, could in principle take any intermediate
value between 0 and 2. However, this scenario can be ruled out for two reasons. First, since for
0 2 4 6 8 10 12-16.48
-14.12
-11.77
-9.42
-7.06
-4.71
-2.35
0.00
0 2 4 6 8 10 12
-3.23
-2.69
-2.15
-1.61
-1.08
-0.54
0.00
0 50 100 150 200 250 3000.0
0.4
0.8
1.2
1.6
2.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0 50 100 150 200 250 3000
1
2
3
4
5
6
7
8
0.3
0.6
0.9
1.2
B (T)
(a) kFl ~ 3.27
12K 25K
50K 75K 100K
150K 195K
240K 285K
xy (
10
-4
m
)
B (T)
(b)
20K 40K
70K 105K
145K 180K 232K
285K
xy (
10
-4
m
)
kFl ~ 8.82
(c)
RH (
10
-10m
3C
-1)
T (K)
RH (
10
-10m
3C
-1)
(d)
(
m
)
T (K)
RH (
10
-10m
3C
-1)
Figure S7: (a) and (b) xy vs. applied field at different temperatures. (c) RH vs. T for samples with kFl ~1.68, 3.27, 3.65, 4.98, 5.5, 8.01, 8.13, 8.82 and 10.12. (d) and RH as functions of temperature for the sample with kFl~3.27, superposed to illustrate the linear relationship between them.
-
37
samples with larger kFl, el-el interactions should be
more predominant than localization effects, a
systematic deviation of towards 2 should have
been observed with increasing kFl. Secondly, in 3D
the temperature dependence of due to
localization and el-el interactions are different.
Since RH is affected only by el-el interactions, the
linear relation between (T) and RH is not expected
over a large temperature range.
Another possibility is that our films are not
in the weak scattering regime (kFl>>1) so we
should not consider AA theory at all for the more
disordered samples. However, since the level of
disorder in the films shown in Fig. 7 spans a large
range of kFl~3.27-8.13, one would have expected
to be asymptotic to the theoretical value of 2 with
increasing kFl. Such a systematic change was
actually observed in a 2D electron gas 39 in Si
inversion layers where a gradual increase of
towards the theoretical value of 2 was observed as
the sheet resistance Rsq0. No such systematic
variation is observed in our data.
It can also be argued that a match is not
expected as the AA calculation is strictly valid for
the limit H0. The importance of this limit is also
illustrated in Ref. 39. However, taking this limit would not alter our results since for our samples
xy is linear over the entire magnetic field range (Figures S7(a)-(b)).
To check if the observed behavior is specific to NbN, similar measurements were also
performed at Louisiana State University on a 2nm thick Be film with resistance, Rsq3.61 k40.
Figure S8: (a) and (b) RH/RH vs. //// for NbN films with kFl~1.68, 3.27, 3.65 and 4.98 and 5.5, 8.01 and 8.13 respectively. The inset in (a) shows , the slope of the plots as a function of kFl (c) RH/RH vs. Rsq/Rsq for a 2nm thick Be film.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.0
0.4
0.8
1.2
1.6
2.0
2.4
2 4 6 80.0
0.2
0.4
0.6
0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.20.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.1 0.2 0.3 0.4 0.5 0.60.0
0.1
0.2
0.3
0.4
kFl
1.68
3.27
3.65
4.98
R
H/R
H
/
(a) NbN
kFl
Be(c)
R
H/R
H
Rsq
/Rsq
= 0.69
kFl~2
NbN
kFl
5.50
8.01
8.13
(b)
R
H/R
H
/
-
38
While for this thickness the film is not superconducting the thickness is much smaller that the
coherence length (~20nm) of its superconducting thick counterpart. This film is thus in the 2D
limit with kFl~2. The Rsq(T) and RH(T) measured in this film is summarized in Figure S8(c).
While this film displays a logarithmic temperature dependence of Rsq typical of a disordered 2-D
system, the slope, 69.0
=
sq
sq
H
H
R
Rd
R
Rd , is strikingly similar to the value observed in
disordered NbN. Since at present we are able to compare with only one such Be film, it would be
interesting to investigate whether the value of in the 2D Be films remains robust as a function
of disorder (Rsq), similar to NbN.
The theoretical value of 2 has been observed in 2D electron gas in Si inversion
layers41 in the limit of large sheet resistance and at intermediate magnetic fields (~0.1-0.5 T).
However, decreases from 2 for both very low fields as well as higher field values. The former
is due to the antilocalization effect of magnetic field even at relatively small fields (where the
localization effects get suppressed and increases from zero) and the latter is presumed to be
attributed to breakdown of the low field limit where this calculation is valid. The dominance of
localization effects is also observed in 3D disordered In2O3 films42 where RH measured at very
low magnetic field was reported to be temperature independent. While the very low limit of RH is
below our experimental resolution, we do not observe any non-linearity in xy vs. H at fields
above ~ 0.2 T.
On the other hand, Hall effect measurements on uncompensated Si:As samples43 in the
metallic regime showed that ranges from 0.4 to 0.6 for different samples. This is speculated to
be due to the anomalous Hall effect44 driven by spin-orbit interaction45. For the various reasons
listed above, it is evident that these Hall effect data cannot be understood on the basis of existing
theories and further theoretical considerations may be needed to establish the value of in a
disordered metallic system.
-
39
V. Tunneling studies, pseudogap, phase fluctuations and phase diagram
To understand the superconducting properties of disordered NbN, we begin by studying
the electronic DOS through tunneling measurements. Figures S9(a), (d) show the conductance
spectra observed in tunneling measurements for samples of Tc ~14.9 and 7.7K at temperatures
ranging from 2.2K to Tc. These spectra are fitted to the tunneling equation after subtracting a
linear background to symmetrize the data. The density of states used for fitting is the BCS
density of states( )
=
22Re)(
iE
iEEN s where is the superconducting energy gap and
(=1/) is the inverse of the lifetime of the quasiparticle excitations but phenomenologically
takes into account all non-thermal sources of broadening. The symmetrized data, along with the
BCS fits are shown in Figures S9(b) and (e). The best fit values of and are then plotted as a
function of temperature along with the theoretical BCS curve for (T) in Figures S9(c) and (f) .
The resistances vs. temperature curves of the underlying films are also shown on the
same graphs. In Figure S9(c) we see that the temperature dependence of closely follows the
Figure S9: (a) and (d): Raw dI/dV curves for tunnel junctions with kFl~7.9 and 3.5. (b) and (e): Symmetrized and fitted conductance curves for the same two junctions as in (a) and (d) respectrively. (c) and (f) and as function of temperature, with the R-T measurement on the underlying films shown alongside on the right hand axes.
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0
1
2
3
4
5
2 4 6 8 10 12 14 16 18 200.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
-3 0 30.0
0.2
0.4
0.6
0.8
1.0
1.2
-6 -3 0 3 60.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
(
m
)
T(K)
,
(m
eV
)
BCS
(f)
(c)
, (
me
V)
T(K)
(
m
)
BCS
-10 -5 0 5 100.0
0.2
0.4
0.6
0.8
1.0
1.2
1.9K
2.8K
3.6K
4.4K
5.2K
6K
6.8K
7.5K
8.8K
dI/d
V (
-1)
V (mV)
(d)
kFl ~ 3.5
kFl ~ 3.5
(e)
1.9K
2.8K
3.6K
4.4K
5.2K
6K
7.5K
8.1K
dI/
dV
(
-1)
V (mV)
-15 -10 -5 0 5 10 150.0
0.5
1.0
1.5
kFl ~ 7.9
(a)
1.9K
3K
4.5K
6K
7.5k
9K
10.5K
12K
13.5K
14.4K
15K
dI/
dV
(
-1)
V (mV)
(b)k
Fl ~ 7.9
3K
4.5K
6K
7.5K
9K
10.5K
12K
13.5K
15K
dI/d
V (
-1
)
V (mV)
-
40
BCS curve and goes to zero at Tc. However, the sample with Tc ~7.7K shows a significant
deviation from the BCS curve and does not seem to go towards zero close to Tc. At the same
time, the value of increases with temperature and becomes comparable to close to Tc. If we
extrapolate (T) beyond Tc (dashed line in Figure S9(f)), we can assume that the gap in the DOS
will persist in the normal state. Such a gap above Tc is known as a pseudogap and is usually
seen in the underdoped high Tc cuprates. Unfortunately, it was not possible to directly measure
the tunneling DOS above Tc using these planar tunnel junction devices because of the low
resistance of our tunnel junctions which get overpowered by the resistance of the NbN film in the
normal state. However, this problem was overcome by using a scanning tunneling microscope
(STM) as the tunnel barrier in that case is of the order of hundreds of megaohms. Scanning
tunneling spectroscopy (STS) was carried out on the NbN samples by my lab-mates Anand
Kamlapure and Garima Saraswat.
Figure S10: Temperature evolution of the DOS for samples with Tc: (a) 11.9K; (b) 4.1K; (c) 2K and (d)
-
41
Figure S10 shows the temperature evolution of the tunneling DOS in the form of a colour
plot for samples of Tc ~ 11.9K, 4.11K, 2K and one that shows an incomplete transition even at
300mK. The spectra shown here are obtained by taking the average of 32 spectra measured along
a 150nm line. In each of the plots, the x-axis is temperature and the y-axis is the bias voltage.
The color scale shows the normalized conductance values. These conductance values have been
arrived at after subtracting a V-shaped background from the conductance curves which is due to
AA type corrections to the DOS that persist up to high bias and the highest temperatures at
which STS has been performed here. In the lower part of each panel, the R-T for the same sample
is also shown. In Figure S10(a), for the sample with Tc 11.9K, we see that the superconducting
gap feature vanishes at Tc whereas for the other, more disordered, samples it is seen to persist
well above Tc. This persisting gap is called the pseudogap and the temperature at which it finally
vanishes is known as T*.
Figures S11(a)-(d) show the resistivity vs. perpendicular magnetic field for strongly
disordered samples with kFl~0.42, 0.49, 0.82 and 1.23. The samples with kFlTc. T* is apparently independent of disorder in this regime. In
region III, superconductivity is suppressed altogether and the samples do not show any
superconducting transition at all. In addition, a peak is observed in the MR which vanishes close
to T*.
-
42
We now discuss each of these regimes in some detail and try to understand which
physical interactions are dominant in each of them. The superconducting order parameter is
represented as a complex number ei, where is the magnitude of the binding energy of the
Cooper pair and is the phase of the macroscopic wave function which must be constant
throughout the sample in order for it to be in the zero resistance state. Destruction of the
superconducting state, therefore, can have two main causes: reduction of due to weakening of
the pairing interaction, and phase decoherence between different parts of the sample without
suppression of . Within the BCS theory, and Tc are coupled together by the relation
0 1 2 3 4 5 6 7 8 9 10 11 125000
10000
15000
20000
25000
0 2 4 6 8 10 122000
3000
4000
5000
6000
7000
0 2 4 6 8 10 12
2000
2500
3000
3500
4000
0 2 4 6 8 10 120
10
20
30
40
50
60
70
0 2 4 6 8 10
-150
-100
-50
0
50
0 2 4 6 8 10
-90
-60
-30
0
30
60
300mK
500mK
750mK
1K
1.25K
1.6K
2K
kFl~0.42
(
m)
0H (T)
(b)
300mK
500mK
750mK
1.1K
2K
(
m)
0H (T)
kFl~0.49
(c)300mK
600mK
900mK
1.2K
2K
kFl~0.82
(
m)
0H (T)
(d)
300mK
500mK
1K
2K
3K
(
m
)
0H (T)
kFl~1.23
0H
c2~7.55T
(f)(e)
2K 2.5K 3.5K
4.5K 5K 5.5K
6K 8K
(
)
(0)
(
m)
0H (T)
kFl~0.82
kFl~0.49
2K 2.5K 3K
3.5K 4K 4.5K
5K 5.5K 6K
(
)(
0) (
m)
0H (T)
Figure S11: as a function of magnetic field at different temperatures for four strongly disordered NbN thin films with kFl~(a) ~0.42, (b) ~0.49, (c) ~0.82 and (d) ~1.23. The samples with kFl
-
43
2/kBTc=constant and the transition
temperature is when goes to zero.
Here phase fluctuations have no role to
play in the superconducting transition.
In region I of the phase diagram,
Tc reduces with increasing disorder, but
the superconducting gap vanishes at Tc
in accordance with BCS theory. We
believe that this decrease in Tc can be
attributed to two effects. The first is
electron-electron (el-el) Coulomb
repulsion. Due to disorder scattering,
the electrons are in the diffusive
regime and electron-screening is less
efficient. This effectively weakens the
attractive Cooper pairing, reducing
(Ref. 46 , 47 ). The second effect
arises because with increasing disorder, not only do the band edges get localized, but also the
bandwidth increases, thus reducing the DOS at the Fermi level (EF) (Ref. 48). Both these effects
can be included in a modified BCS relation49:
=
*)0(1
expVN
T Dc where D~Debye
temperature, V is the attractive electron-phonon potential, N(0) is the DOS at EF and * is a
Coulomb pseudopotential. Although the current theoretical development is not sufficient to
calculate * and attempt a quantitative fit, the combination of enhanced * and suppressed N(0)
qualitatively explains our data.
As disorder is increased and we enter regime II, an additional energy scale i.e. the
pseudogap temperature T* becomes relevant. In this region, Tc continues to decrease
monotonically but T* does not vary significantly. These two temperatures can be associated with
two energy scales: The strength of the pairing interaction and the phase stiffness of the
Figure S12: Phase diagram of strongly disordered NbN, showing Tc and T
* as functions of kFl. Tc is obtained from transport measurements while T* is the crossover temperature at which the low bias feature disappears from the observed tunneling conductance. The samples with kFl
-
44
superfluid J. In conventional BCS
superconductors, J>> and therefore
is not relevant for the
superconducting transition. In order
to compare these two energy scales
for our system, we estimate J using
the relation: *)4/()( 2 manJ sh=
where a is the distance over which
the phase fluctuates, ns is the
superfluid density and m* is the
effective mass of the electron. ns is
obtained from low temperature
penetration depth measurements12
and a is taken to be ~ (Ref. 27). The values of J thus obtained are plotted in temperature units
against Tc in Figure S13. A dotted line separates regions I and II in this figure. We can see that
in region I, J>>Tc, and therefore the superconducting transition will be governed by the
temperature at which (T) becomes zero. On crossing over to region II the superfluid density is
reduced due to disorder scattering and J~kBTc. This renders the system susceptible to both
quantum and thermal phase fluctuations. It has also been observed that the ns values for samples
in regime II deviate from the dirty limit BCS predictions both at zero and finite temperatures, but
these deviations can be accounted for by the presence of dissipation and phase fluctuations12.
This certifies the assertion that phase fluctuations are the dominant phenomenon in this regime.
The STM studies also observe that the strongly disordered samples show a spontaneous
inhomogeneity in the DOS6 as predicted in the literature15,18. Therefore in this regime, the
superconducting state is destroyed due to phase fluctuations between the superconducting islands
while phase incoherent Cooper pairs continue to persist up to T*.
In region III, there is no superconducting ground state, but superconducting correlations
are expected to persist as T* is finite at the boundary between regions II and III. We therefore
believe that in this regime, the system comprises of superconducting islands that are unable to
Figure S13: Superfluid stiffness (J/kB) and penetration depth ((T0)) for NbN films with different Tc. The solid line corresponds to J/kB=Tc. Regions I and II corresponding to the phase diagram are segregated by the dashed vertical line.
3 6 9 12 15 181
10
100
II
J/
k B (
K)
Tc (K)
I
0
2000
4000
6000
(0
) (n
m)
-
45
form a zero resistance ground state
due to quantum phase fluctuations.
The MR peak vanishes close to T*
(Figures S11(e) and (f)) and is
therefore attributed to
superconducting correlations.
The peak can be understood
on the basis of the superconducting
islands scenario as follows16: at low
fields, MR is positive as the
superconducting paths shrink due to
magnetic field. Beyond a certain
field, the superconducting paths become small enough that the chosen conduction path now goes
through the normal regions. The gradual decrease in MR at higher fields is because of the slowly
increasing normal regions as the superconducting islands shrink further. The peak field (Hpeak) is
then interpreted as the field at which the superconducting correlations are almost destroyed and
is expected to evolve from Hc2 for the superconducting samples. To verify this, we plot Hc2(0 or
300mK) and Hpeak(300mK) as functions of kFl in Figure S14. For the strongly disordered
samples, Hc2 values are taken as the field at which the resistance reaches 90% of its normal state
value and for the more ordered samples whose Hc2 is not directly accessible up to 12 T, we use
the dirty limit formula as in Chapter 3. We observe that Hc2 decreases with disorder and
smoothly evolves into Hpeak for kFl1) and Hp (for kFl
-
46
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-
49
Chapter I: Introduction
Superconductivity is a phenomenon that is extremely resilient to disorder. Experimentally
it is seen that many superconducting materials continue to show superconductivity with
reasonably high critical temperatures even when they are significantly disordered. Indeed, many
of the common superconducting materials used for making superconducting magnets are alloys
like Nb-Sn and Nb-Ti (Ref. 1). Even the high-Tc superconductors like cuprates and pnictides are
doped, disordered materials. The earliest theoretical understanding was due to Anderson2 who
said that as long as the disorder is caused by non- magnetic impurities*3 it does not significantly
affect the superconducting critical temperature. However, this turned out to be true only at low
disorder and is in fact invalid for many experimental systems 4 , 5 , 6 , 7 , 8 . At strong disorder,
electronic and superconducting properties undergo drastic changes, to the extent that both
metallicity and superconductivity can be destroyed completely by sufficiently strong disorder9,10.
In recent times, a number of novel phenomena11,12,13,14,15,16,17,18,19,20,21 have been observed in
disordered superconductors