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Spin-flip Scattering in Superconducting Granular Aluminum Films

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Spin-flip Scattering in
Superconducting Granular Aluminum
Films
By
Nimrod Bachar
Submitted In Partial Fulfillment Of The
Requirements For The Degree Of
Doctor Of Philosophy
Thesis Advisors:
Prof. Guy Deutscher
Dr. Eli Farber
submitted to
the Senate of Tel-Aviv University
on October 2014
Abstract
Granular Al consists of nano Al grains coupled through thin insulating barrier. It shows an
elevated superconducting temperature Tcof 3.2 K which is higher than that of bulk Al having
Tcof 1.2 K. We show in this experimental work evidence for magnetic properties, an approach to a
metal to insulator transition of the Mott type and signatures for unconventional superconductivity
in granular Al films. The normal state resistivity as a function of temperature shows a negative
d2ρ/dT 2at high temperatures, a resistivity minimum at low temperatures and log(T) dependence
all of which are typical characteristics of Kondo systems, such as dilute magnetic alloys and dense
Kondo compounds. The magnetoresistance of our films is negative and scales as B/T . We show
a (B/T )ndependence at low magnetic fields and high temperatures with an exponent nclose to
2. This behavior is a striking evidence for a spin-flip scattering mechanism between conduction
electrons and localized magnetic moments. We confirm the existence of these magnetic moments by
a muon spin relaxation measurements showing a resistivity independent spin concentration of about
400 ppm, similar to typical values for Kondo systems. We analyze the increasing magnetoresistance
as a function of the resistivity that we observe in our films. It suggests that a decreasing effective
Fermi energy of the system of coupled grains is at the origin of the magnetoresistance increase. We
identify a critical resistivity for a metal to insulator transition of the Mott type where the effective
Fermi energy is down to the electrostatic charging energy of the grain. In the vicinity of such
a transition, the effective mass is predicted to enhance and we do observe a Heavy-Fermion like
behavior in the magnetoresistance of high resistivity films. We probe the superconducting energy
gap by THz spectroscopy and show that the spectra deviate from the conventional BCS theory at
low frequencies and high resistivities. Altogether we show the coexistence of spin flip scattering
mechanism, enhanced superconducting properties and increasing electron-electron interactions in
granular Al films which could help to shed some light on the unconventional pairing mechanism of
this system.
To Yaara
II
Acknowledgments
I would like to express my deepest gratitude to my supervisor, Prof. Guy Deutscher, for his
guidance and support during my PhD. research. I thank him for many fruitful scientific discussions
which motivated me during my work. I specially thank him for granting me his unique approach
of examining experimental results, writing scientific essays and of course for his endless knowledge
of scientific citations. There is no doubt that all of the above and more provided a solid ground for
our success in this work.
My deepest gratitude also goes to my co-supervisor, Dr. Eli Farber, for his outstanding belief in
me. Short while after starting to work under his supervision on my B.Sc. degree, I have joined him
for our work on THz spectroscopy in YBCO films in Prof. Dirk van der Marel’s group in Geneva.
During the years, Dr. Farber was constantly pushing me forward to additional collaborations and
to present our work in scientific conferences. He was the one that introduced me to the group of
Prof. Deutscher while I’m now closing a circle and intended to start my PostDoc term in Prof. van
der Marel’s group. I wish to thank Dr. Farber for showing me the way to the academic life and for
all of his support along the past years.
Many thanks are deserved to Dr. Boaz Almog for our joint work in the past several years. We
have met when I was a B.Sc. student and he was starting his PhD. term. During the years I learned
to admire his unique personality and to acquire from him a lot of knowledge about experiential
work, measurements, deposition of films and way of life of a hard working scientist being also a
devoted parent. His preliminary work on granular Al films has motivated me to take the reins and
to turn it to the center of my PhD. research. Since then many important and exciting things were
discovered but there is no doubt that his work and support along the way were very helpful to our
success.
Special thanks is sent to Dr. Shachar Lerer with whom I enjoyed to work on granular Al along
the years. We spent many hours together in the lab growing films, measuring them, analyzing and
discussing about the data while glaring at the computer screen trying to understand our work. Our
team work was a remarkable milestone in the success of both of our PhD. research.
I would like to thank all my research colleagues and friends in the past years, those who left
and returned, those who left and will return, those who are still working in the lab and those who
left without saying goodbye. Thanks to Dr. Shay Hacohen-Gourgy for many helpful discussions
and for his important contributions to our work. Thanks to Dr. Guy Leibovich and Dr. Roy
Beck Barkay for teaching me the secrets of film deposition. Thanks to Prof. Yoram Dagan for his
remarkable support, useful discussions about granular Al and solid state physics and of course for
the huge amount of PPMS measurement time which its fruits I’m showing in this work. Thanks to
Dr. Itay Diamant, Dr. Moshe Ben Shalom, Dr. Miri Sachs, Michal Petrushevsky, Alon Rabinowicz,
Alon Ron, Eran Maniv, Erez Flekser (I will cherish his memory), Aviad Levy, Eran Greenberg, Dr.
Amir Segal, Dr. Itay Kishon, Prof. Sasha Gerber, Rafi Hevroni, Lior Tzarfati and Prof. Alexander
Palveski for help and discussions along the years. Special place in my heart is kept for Prof. Enrique
Grunbaum for his excellent reviews of our manuscripts and of course for his warm personality and
IV
kindness. I am deeply sorry that he is not with us anymore and hope that he would have like this
dissertation as this subject was close to his heart.
Many thanks To my colleagues all around the globe: Dr. Daniel Sherman (Bar Ilan University)
for our mutual work on THz spectroscopy in the past few years where we inspired each other on
granular and disordered superconductors. I also thank Prof. Aviad Frydman (Bar Ilan University)
for useful discussions about my work and Prof. Efrat Shimshoni (Bar Ilan university) for her
incredible remark on H/T scaling. Prof. Martin Dressel, Dr. Marc Scheffler and in particular
Uwe Pracht (Stuttgart University) for our recent collaboration on THz spectroscopy of granular Al
films. Prof. Antonio Garcia-Garcia and James Mayoh (University of Cambridge) for enlightened
discussions on shell effects in granular Al which resulted in a recent publication. Prof. Elvezio
Morenzoni, Dr. Zaher Salman and Dr. Hassan Saadaui (Paul Scherrer Institute -PSI) for our
collaboration in µSR measurements. I would like to acknowledge travel support within the Access
Programm of FP7 given to us during our stay at PSI. Dr. Uli Zeitler, Dr. Steffen Wiedmann and
Dr. Veerendra Guduru for our work in the High Field Magnetic lab in Nijmegen, Netherlands.
Part of this work has been supported by the EuroMagNET II under the EU contract. To the
colleagues and friends in Ariel university: Raziel Itzhak, Yoni Bechor, Moshe Mizrachi, Elichai
Glassner, Vitaly Khavkin and Eldad Holdenberg. To the workshop staff in Tel Aviv University in
particular to Avner Yechezkel, Natan Bonik and Dov Lupo.
My last thanks are of course granted to my wife, Yaara, which supported me during this hard
work and always reminded me how much family is important even when you have some sample
to measure. Her remarkable advice to leave some experimental problems aside and try to solve it
again in the following day with fresh strength and clear mind has really proved itself useful.
V
Preface
An enhanced superconducting temperature in Granular Al films was discovered in 1966 by
Abeles and Cohen [1]. Since then much work was done in order to understand the anomalous
superconducting properties such as the enhanced Tcand the resistivity dependent dome-like phase
diagram. The morphology of the films showing grain size distribution was observed by the Tel
Aviv group using Dark Electron Microscopy. Theoretical work was done in order to shed more
light on the peculiar existence of superconductivity in a system of grains which by themselves
should not be superconducting at all. Special interest was dedicated to understand the interplay
between electron-electron interactions, weak localization effects and superconducting fluctuations
by the Rutgers group.
Other unconventional superconductors such as Heavy Fermions, Pnictides, organic supercon-
ductors and oxide interfaces were discovered after granular Al. In particular it was the discovery
of high Tcsuperconductors in 1986 that has drawn much attention from the low Tcgranular Al
superconductor. Yet there are many similarities between these materials and granular Al films.
The nowadays well known phase diagram of high Tcsuperconductors was in fact observed first for
granular Al films. However, almost 30 years after the celebrated discovery of the Cuprates family,
there is no clear understanding of the enhanced Tcin these compounds, their pairing mechanism
and their anomalous normal state properties. Nowadays, much work is focused on measurements
of low Tcsystems such as the extreme low doping of Cuprates and Heavy Fermions. Although
superconductivity is reduced in these regimes, most of the work is focused on shedding light on the
interplay between the normal and superconducting state. This was one of the motivations to go
back to granular Al in the search for answers to some of the opened questions in unconventional
superconductors physics.
The current work on granular Al films makes use of several experimental methods and covers
reviews of a broad range of physical phenomena. Therefore, this dissertation is not written in
the typical format of introduction, theoretical background, experimental results, discussion and
conclusions. Instead it devotes each chapter to a particular experimental observation or analysis
in our work. Mostly this dissertation reviews the existence of spin-flip scattering mechanism in
superconducting granular Al films as shown by our experimental work. We use our results to shed
light on the underlying physics of the normal and superconducting state of granular Al films.
A work on granular Al system cannot be presented without proper introduction to the previous
work and interpretation that was done in the past 50 years. Therefore the dissertation is presented
as follows: Chapter 1 introduces the reader to the morphology of granular Al films, the growth
mechanism and first observations of enhanced superconductivity in these films. Chapter 2 gives a
theoretical review of superconductivity in nano particles as conventional superconductivity is not
obvious at such small scale. Chapter 3 gives an experimental review of the resistive transition as
a function of temperature and magnetic field obtained in films grown in this work, compared to
previous reported results and sheds light on the transition between the superconducting state and
the normal state.
VI
Our work is focused to measurements at relatively high temperatures where we carefully mea-
sured the resistivity temperature dependence and the magnetoresistance (MR) of granular Al films.
In chapter 4 we review the anomalous R(T) of our samples which show a Kondo-like behavior. In
chapter 5 we review our new MR results and compare them with previous reported results on granu-
lar Al films at lower temperatures and magnetic fields. We show that the MR at high temperatures
and low magnetic fields can be interpreted by a spin-flip scattering mechanism. Chapter 6 discusses
the origin of localized magnetic moments in Al grain and gives a direct observation of the spins
by µSR measurements. Chapter 7 shows the analysis of our results in terms of an increasing spin
scattering due to a decreasing effective Fermi energy. We extrapolate the decrease of the effective
Fermi energy of the system of coupled grains down to the electrostatic energy of the grain. We
suggest then that our system approaches a metal to insulator transition of the Mott type at high
resistivities. In particular we show similarities between such films and Heavy Fermions system as
the effective mass is predicted to increase toward the transition. The anomalous enhanced super-
conductivity have motivated us to carefully probe the superconducting gap in order to observe
signatures for unconventional superconductivity. In Chapter 8 We review our results obtained
by THz spectroscopy and show several deviations from the BCS theory. Altogether we conclude
the observation of a coexistence between enhanced superconductivity, magnetism and increasing
electron-electron interactions.
During the past few years we had many useful discussions regarding our work. Among these
discussions we had the pleasure of presenting our results with Prof. Huebener. He suggested to
write a review on the extended knowledge that we obtained in the past few years regarding the new
physics of granular Al films. We hope that this dissertation will act as such review and hopefully
will induce future exciting work in granular Al in particular and unconventional superconductors
in general.
VII
Contents
1 Introduction 1
1.1 Properties Of Clean Aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 TheGranularMorphology................................. 3
1.2.1 Thin Films Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Grain Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Growth Procedure And Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Enhanced Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 The Superconducting Phase Diagram . . . . . . . . . . . . . . . . . . . . . . 9
2 Enhanced Superconductivity 13
2.1 SCinSmallParticles.................................... 13
2.1.1 Electronic Properties Of Small Metallic Particles . . . . . . . . . . . . . . . . 13
2.1.2 Anderson Criteria For Superconductivity . . . . . . . . . . . . . . . . . . . . 14
2.1.3 Enhancement Of TcBy Quantum Size Effects . . . . . . . . . . . . . . . . . . 15
2.1.4 Electronic Shell Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 IntergrainCoupling..................................... 17
2.2.1 Level Spacing And Charge Fluctuations . . . . . . . . . . . . . . . . . . . . . 17
2.2.2 ChargingEnergy .................................. 18
2.2.3 JosephsonEnergy ................................. 18
2.2.4 Superconductivity in Granular Al . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Superconducting Properties 21
3.1 TheResistiveTransition.................................. 21
3.1.1 ExperimentalSetup ................................ 21
3.1.2 R(T) Of Granular Al Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.3 PhaseDiagram ................................... 22
3.1.4 Magneto-Resistance ................................ 24
3.1.5 CriticalFields.................................... 25
3.1.6 Effective Mean Free Path in Granular Al . . . . . . . . . . . . . . . . . . . . 29
3.1.7 Abrikosov-Maki-de Gennes Theory . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 SCFluctuations ...................................... 31
VIII
CONTENTS CONTENTS
3.2.1 Fluctuating Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2 Superconducting Fluctuations In Granular Al . . . . . . . . . . . . . . . . . . 33
3.2.3 Nernst Effect In Granular Al . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.4 Superconductivity Beyond The Coherent Limit . . . . . . . . . . . . . . . . . 36
3.3 Summary .......................................... 40
4 Kondo-Like Behavior 41
4.1 The Normal State Resistivity Of Granular Al Films . . . . . . . . . . . . . . . . . . 41
4.2 TheKondoEect...................................... 41
4.2.1 The R(T) Anomaly In Dilute Alloys . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.2 KondoEect .................................... 43
4.2.3 Kondo Lattices And Heavy Fermions . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Kondo-LikeBehavior.................................... 46
4.3.1 R(T) Of Granular Al Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.2 Temperature Scales Of Kondo-Like Features . . . . . . . . . . . . . . . . . . . 50
4.4 Summary .......................................... 53
5 Spin Flip Scattering 55
5.1 SpinFlipinTransport................................... 55
5.1.1 Non Spin-Flip Scattering Mechanisms . . . . . . . . . . . . . . . . . . . . . . 55
5.1.2 Magnetoresistance In The Presence Of Spin Flip Scattering . . . . . . . . . . 56
5.1.3 Negative Magnetoresistance Of Dilute Magnetic Alloys . . . . . . . . . . . . . 57
5.2 NegativeMRMechanisms................................. 59
5.2.1 WeakLocalization ................................. 59
5.3 LowTemperatureMR ................................... 60
5.3.1 PreviousResults .................................. 60
5.3.2 NewResults..................................... 63
5.4 HighTemperatureMR................................... 63
5.4.1 B/TScaling..................................... 66
5.5 Summary .......................................... 70
6 Localized Magnetic Moments 71
6.1 MagnetismInMetals.................................... 71
6.1.1 Magnetism At Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 MagneticMoments..................................... 72
6.2.1 Spin Concentration Estimate from Magnetoresistance Measurements . . . . . 72
6.2.2 Direct Observation Of Free Spins . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.3 Summary .......................................... 78
IX
CONTENTS CONTENTS
7 M/I Transition 79
7.1 Increasing Spin-Flip Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.1.1 Effective Fermi Energy of A System of Coupled Grains . . . . . . . . . . . . . 79
7.2 BandwidthControl..................................... 82
7.2.1 Effective Bandwidth Of Coupled Grains . . . . . . . . . . . . . . . . . . . . . 82
7.2.2 Charging Energy And Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . 83
7.3 M/I Transition in Granular Al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.3.1 Mott-HubbardModel ............................... 84
7.3.2 Bandwidth-Controlled Transition . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.3.3 MottLimit ..................................... 87
7.3.4 Transport Determination of the Metal to Insulator Transition . . . . . . . . . 87
7.3.5 Effective mass of Granular Al . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.4 SCnearM/ITransition .................................. 89
7.4.1 Conventional Superconductivity In The Presence Of Spin Scattering . . . . . 89
7.4.2 Superconductivity of Granular Al in the presence of spins . . . . . . . . . . . 89
7.4.3 Superconductivity and the approach to MIT . . . . . . . . . . . . . . . . . . 90
7.5 Heavy Fermion Like Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.6 Summary .......................................... 91
8 Gap Spectroscopy 93
8.1 ElectrodynamicsofSC................................... 93
8.1.1 TheEnergyGap .................................. 94
8.1.2 TheNormalState ................................. 95
8.1.3 The Superfluid Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
8.1.4 BCS Dynamical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.2 THzSpectroscopy ..................................... 99
8.2.1 The Quasi Optical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.2.2 Complex Spectra Of Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.3 GranularAlGap ......................................104
8.3.1 SampleFabrication.................................104
8.3.2 Complex Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.3.3 Complex Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.3.4 Deviations From The Bcs Theory . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.4 Gapless Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.4.1 CollectiveModes..................................111
8.4.2 SpinFlipScattering ................................111
8.5 Summary ..........................................112
9 Conclusions 113
X
CONTENTS CONTENTS
Bibliography 131
Appendix 133
XI
List of Figures
1.1 BCSIllustration ...................................... 3
1.2 GranularMorphology ................................... 4
1.3 EvaporationSystem .................................... 5
1.4 HallBarGeometry..................................... 6
1.5 GrainSize-TEM ..................................... 7
1.6 Grain Size Distribution - 300 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 Grain Size Distribution - 77 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.8 PhaseDiagram ....................................... 10
2.1 Electroniclevelspacing .................................. 17
3.1 Typical Resistive Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Superconducting Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Typical Magnetoresistance curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Perpendicular Critical Field as a function of Temperature . . . . . . . . . . . . . . . 27
3.5 ρ(B)-Sample65...................................... 28
3.6 Phase Diagram of the Upper Critical Field . . . . . . . . . . . . . . . . . . . . . . . . 30
3.7 Upper Critical Field as a function of Temperature (Maki-de Gennes) . . . . . . . . . 32
3.8 Fluctuating Resistivity in Granular Al . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.9 Scaling of MR and Nernst Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.10 MR and Nernst below Tc................................. 37
3.11 BRmax ingranularAlsample ............................... 38
3.12MagneticFieldScales ................................... 40
4.1 Normalized R(T) - Full range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 DiagramofKondoEect ................................. 44
4.3 Temperature dependence - Kondo effect . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 R(T)ofKondoLattice................................... 46
4.5 NormalizedR(T)...................................... 47
4.6 ρ(T)-Sample237 ..................................... 49
4.7 ρ(T)Sample310 ...................................... 50
XII
LIST OF FIGURES LIST OF FIGURES
4.8 ρ(T)Sample2425...................................... 51
4.9 ρ(T)-Sample65 ...................................... 51
4.10Kondoeecttemperatures................................. 52
4.11 ρ(T)-Sample21550 .................................... 53
4.12KondoFittoR(T) ..................................... 54
5.1 NegativeMR-DiluteAlloys ............................... 58
5.2 Reports on Negative MR in Granular Al . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3 InelasticScatteringtime.................................. 62
5.4 ρ(B)............................................ 64
5.5 HigheldMR........................................ 65
5.6 Negative MR- High temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.7 R(T)inmagneticeld................................... 68
5.8 MR as a function of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.9 B/T Scaling......................................... 69
5.10 Scaling of MR as a function of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.1 ∆ρcomparison ....................................... 73
6.2 µSR-Polarizationspectra................................. 76
6.3 Muon Relaxation Rate - Low resistivity . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.4 Muon Relaxation Rate - Higher resistivities . . . . . . . . . . . . . . . . . . . . . . . 77
7.1 ∆ρPowerlawincrease................................... 80
7.2 EectiveFermiEnergy................................... 82
7.3 HubbardBandDiagram.................................. 85
7.4 DMFT calculations - Metal to Insulator Transition . . . . . . . . . . . . . . . . . . . 86
7.5 Magnetoresistance of CeAl3................................ 92
7.6 Magnetoresistance of Sample 13145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.1 BCSDensityofStates ................................... 95
8.2 ∆(T)-BCSandGranularAl ............................... 96
8.3 Drude Complex Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.4 Complex Conductivity - Mattis Bardeen Theory . . . . . . . . . . . . . . . . . . . . 99
8.5 QuasiOpticalSetup ....................................101
8.6 Transmission Spectra - Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.7 THz Samples - Superconducting Phase Diagram . . . . . . . . . . . . . . . . . . . . 105
8.8 Transmission Spectra - ρ300 µcm .........................106
8.9 Complex Conductivity - ρ300 µcm ........................107
8.10 Complex Conductivity - ρ500 µcm ........................109
8.11 2∆(T) - Sample ρ500 µcm .............................110
8.12 2∆(0)/kBTc.........................................110
XIII
List of Tables
3.1 Transport samples - Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Meanfreepathanalysis .................................. 29
4.1 Kondo behavior - Summary of typical temperatures . . . . . . . . . . . . . . . . . . 48
8.1 THzSamples-Summary .................................108
XIV
List of Abbreviations
1D One Dimensional
2D Two Dimensional
3D Three Dimensional
AG Abrikosov-Gor’kov
AL Aslamazov-Larkin
BCS Bardeen-Cooper-Schrieffer
BWO Backward Wave Oscillator
DMFT Dynamical Mean Field Theory
DOS Density of States
FCC Face Centered Cubic
GCF Ghost Critical Field
GL Ginzburg-Landau
µSR Muon Spin Relaxation/Rotation
MC Magnetoconductance
MIT Metal to Insulator Transition
MR Magnetoresistance
MT Maki-Thompson
ppm Parts per million
QO Quasi Optical
RKKY Ruderman-Kittel-Kasuya-Yosida
SC Superconductor, Superconducting, Superconductivity
SCF Superconducting Fluctuations
SQUID Supercondcuting Quantum Interference Device
STM Scanning Tunneling Microscopy
TEM Transmission Electron Microscopy
WL Weak Localization
XVI
Chapter 1
Introduction
Aluminum is one of the well-known conventional superconductors [2, 3]. As such, it obeys the
Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity in which attractive interaction via
electron-phonon coupling results in the emergence of the Cooper pair, i.e. two electrons of opposite
momentum and spin [4]. The BCS theory succeeded in explaining the superconducting phase in
metals such as Hg, In, Pb, Al and more [5].
Aluminum is also one of the most used materials in the community of superconducting integrated
circuits [6]. The well controlled growth parameters and in particular the apparently easy to control
oxidation layer turned Aluminum into the major player in superconducting quantum computing
devices despite of its low superconducting transition temperature. However, as will be elaborated in
this work, the metal-oxide interface of Al can lead to several interesting and challenging phenomena.
The normal and superconducting bulk properties of Al are well known [7] but they are altered
as Al is confined to nano scale dimensions. For example an increase of the superconducting critical
temperature, Tc, was observed for clean Al thin films with thicknesses of about 10 nm and below [8].
The confinement of Al into small nano grains, i.e. granular Al, also leads to several interesting
features both in the normal state and in the superconducting state.
In this chapter, we will introduce some of the properties of granular Al reported in previous
works. These properties will be compared with the bulk values in order to understand the effect of
nano scaling on the underlying physics of granular Al. As will be seen, the effects are not obvious
as at the nano scale size of such grains, conventional superconductivity is unexpected.
1.1 Properties Of Clean Aluminum
In this chapter we will review the properties of clean Aluminum both in the normal metallic
phase and in the superconducting phase. As will be seen in the following chapters, the values given
here are important in order to understand the properties of granular Al and will be used in our
further analysis and discussion.
The normal properties of Aluminum are usually given by the Drude-Sommerfeld theory of
metals which is sufficient to explain most of the transport properties. Here is a summary of several
1
1.1. PROPERTIES OF CLEAN ALUMINUM CHAPTER 1. INTRODUCTION
important parameters [9]:
Atomic number - 13.
Valence electrons per atom Z= 3.
Unit cell - Face Centered Cubic (FCC).
Lattice constant - a= 0.405 nm.
Carrier concentration n=Ncell ·Z/V = 18.1×1028 m3where Ncell = 4 atoms per unit cell
of FCC and a cubic volume V=a3.
Room temperature resistivity - ρ= 2.75 µcm [10].
The Hall coefficient RHis known to be negative, i.e. determined by the electrical transport
of holes. For pure Al RH=1/ne ≈ −0.34 ×1010m/T .
Fermi wave vector - kF= 1.75 ×1010 m1.
Fermi energy - EF=~2k2
F/2m= 11.7eV .
Fermi velocity - vF=~kF/m = 2.03 ×106m/sec
Density of states (DOS) at the Fermi level - g(EF) = 3n/2EF= 2.3×1028 eV 1m3.
Mean free path l=vFτ- usually given by ρl 1.6×105µcm2[11]. For ρ= 2.75µcm we
get l= 5.81 ×106cm = 58.1nm.
Drude relaxation time - τ=l/vF= 2.8×1014 sec.
Aluminum is a conventional BCS superconductor whose pairing mechanism can be explained
in terms of the electron-phonon coupling. In this picture, an electron with a positive momentum
and spin up excites a phonon mode at low temperatures (Figure 1.1(a)). A second electron with
an opposite momentum and spin is attracted toward this excitation (Figure 1.1(b)). These two
electrons, attracted by interaction through the lattice form a Cooper pair which is the quantum
particle of the superconducting state. The superconducting properties of bulk and pure Al are as
follows [7]:
The critical temperature - Tc= 1.19 K.
Energy gap - 2∆ = 0.36 meV (Tunneling spectroscopy) [7, 12].
Critical field Hc- Al is a type I superconductor with Hc= 10 mT at T= 0.
Coherence length - ξ0=~vF/Π∆ = 1.6µm
London’s Penetration depth λL= 15.7nm
2
1.2. THE GRANULAR MORPHOLOGY CHAPTER 1. INTRODUCTION
Figure 1.1: ”Classical” illustration of the electron-phonon interaction in the BCS theory for super-
conductivity [After Ref. [13]]. An electron with wave vector kand spin up is deforming the lattice
thus inducing a phonon. A second electron with wave vector k(opposite direction) and spin down
is attracted to the lattice deformation in order to lower its energy. As a result the two electrons are
attracted by electron-phonon-electron interaction to form a Cooper pair. The ”classical” picture
cannot account for the macroscopic wave function of these new bosonic particles which seems to
be rather local and temporal in the above illustration [13].
We will make use of the bulk values given above in the following chapters in discussing the
transport properties of granular Al. The most interesting phenomenon in this case is the increase
of Tc.
1.2 The Granular Morphology
Granular Al is formed of nano metallic Al grains usually on the order of 2 to 3 nm in size.
The nano grains are embedded in an insulating amorphous matrix of Al2O3or Ge (Figure 1.1).
The grains are separated by thin insulating barriers resulting in tunneling coupling which is barrier
thickness dependent.
The main parameters in the morphology of granular Al are the grain size dand the barrier
thickness s[14]. The smaller the grain size the higher the maximum Tcobserved in granular Al
films [15]. On the other hand, the barrier thickness controls the inter-grain resistance R. These
two parameters enable us to estimate the resistivity of the film ρ=Rd [16]. The resistivity will
be a convenient parameter in the following chapters for a classification of the samples as strongly
coupled grains (small Rthus low ρtypically .100 µcm ) versus weakly coupled grains (large R
thus high ρtypically &1,000 µcm ).
1.2.1 Thin Films Fabrication
Deposition of granular Al films is obtained by thermal evaporation of clean Al pellets in a partial
pressure of O2gas (Figure 1.3) [18]. Alternately, Al can be co-evaporated with Ge by electron beam
gun [19]. The deposition can be done on a water cooled substrate [18]. Alternately the substrate
can be held at liquid nitrogen temperature [15]. The substrate temperature has a significant effect
on the films’ properties [15].
3
1.2. THE GRANULAR MORPHOLOGY CHAPTER 1. INTRODUCTION
(a) (b)
Figure 1.2: [1.2(a)] Illustration of the granular morphology. The dark areas are the metallic nano
grains while the bright areas are the amorphous insulating material. The grains are coupled through
a thin insulating barrier typically on the scale of one atomic layer. [1.2(b)] Bright field TEM image
of a typical granular Al sample [17]. Selected crystallographic areas are marked in the image (Full
line circle) coupled through thin oxide barrier. As the sample is 3D, there can be seen mixing of
crystallographic order (dashed line circle). Amorphous areas can also be seen in the sample (dotted
line circle).
In the current work we have used a similar deposition method as was described elsewhere [15].
Granular Al films were deposited by using 99.999% pure Aluminum pellets and O2partial pressure
range of 1 ×105to 3.5×105Torr. The evaporation chamber was kept at the desired pressure
for approximately 1 hour in order to achieve equilibrium between the bleeding O2hose located at
top of the chamber and the vacuum pressure gauge located at the bottom of the chamber. The
films’ room temperature resistivities were measured to be between less than 100 µcm to several
10,000 µcm. Resistivities of less than 100 µcm were obtained by evaporation at a base pressure
of 1 ×107Torr. The films were deposited from an Alumina coated Mb boat which was replaced
after each growth in order to avoid contamination.
If not mentioned otherwise, the substrates were cooled to liquid nitrogen temperature and held
at this temperature for several minutes prior to evaporation in order to obtain thermal equilibrium
of the substrate’s temperature. Substrates were typically attached to the sample holder using
double sided Carbon tape. Several substrates were used in this work among them SiO2terminated
h111iSilicon substrate, glass plates, R plane cut Sapphire and MgO. No differences were observed
in the transport properties of the films evaporated on these various substrates.
Silicon substrates were mainly used for electrical transport measurements. Most of the samples
in this work were measured in a Hall bar geometry (Figure 1.4). The photolithography was done
using AZ 1518 photoresist and AZ 726 (MIF) Developer. The bars were prepared in the following
4
1.2. THE GRANULAR MORPHOLOGY CHAPTER 1. INTRODUCTION
(a) (b) (c)
Figure 1.3: Thermal evaporation chamber [1.3(a)] with a Load-Lock sample holder [1.3(b)] allowing
to keep the chamber at base temperature of <1×107Torr with the option to cool the substrate to
liquid Nitrogen temperature by an inner tube thermally coupled to the sample space (not shown).
Al is evaporated from a typical Alumina coated Mb boat [1.3(c)].
process: first a thin Ti/Au layer (Ti thickness 5 nm and Au thickness 15 nm) was deposited by
electron beam evaporation in order to create a shifted contact pattern via a designated mask. Ti
was evaporated first in order to obtain good adhesion of the Au layer. The Au layer was used in
order to achieve high transparency contacts with no oxidation layer. After a chemical development
of the first stage, the second stage included the photolithography of Hall bar geometry by the same
process and with alignment of the bars contacts to the Ti/Au contacts. Last, granular Al films were
evaporated on the substrate and by using standard lift-off technique we obtained well defined Hall
bars with width of 3, 5, 10, 20 and 40 µm. Length between adjacent voltage pads was measured
to be 400 µm and typical thickness of the films was 100 nm.
Granular Al samples were grown on specific substrates for measurements not requiring pho-
tolithography processes. Thin films of 100 nm were deposited on square Si/SiO2substrates for
electrical transport measurements by using the Van Der Pauw method. Glass substrates were used
for Nernst effect measurements due to the low thermal conductivity of the substrate. The special
pattern of the films for Nernst effect measurements is described elsewhere [17]. Thin films of 100 nm
were deposited on narrow, typically 3 mm wide, strips for magnetic properties measurements in a
SQUID apparatus. Glass substrates were used in order to deposit thin films with typical thickness
of 100 nm on large areas, typically 20 mm by 20 mm, for µSR measurements. Thin films of 100 nm
and 40 nm thickness were deposited on Square Sapphire and MgO substrates for THz spectroscopy
measurements.
5
1.2. THE GRANULAR MORPHOLOGY CHAPTER 1. INTRODUCTION
Figure 1.4: Typical Hall bar geometry used in this work. The thickness of the bar is 100 nm. The
width of the bar in this figure is 10 µm. The distance between adjacent voltage pads of the bar
(Yellow arrow) is 400 µm. The Hall bar is deposited on shifted Ti/Au contacts.
1.2.2 Grain Size Distribution
Granular Al films are composed of nano grains embedded in an insulating matrix. Transmission
Electron Microscopy (TEM) of films grown on top of TEM grids allows the microscopy of crystallized
areas on the order of sub-nm length. However, as the microscopic image is 2 dimensional while the
physical morphology is 3 dimensional it is very hard to separate the grains in the TEM image as
the bright field image is formed from layers of layers of different grains (Figure 1.2).
Several techniques were implemented in order to obtain the grain size distribution of these films.
The most successful one is dark field microscopy. As the grains are crystallized, it is possible to
image a preferred reciprocal lattice vector ~
K. By that, only grains diffracting in the chosen ~
Kvector
are observed with the highest contrast in the picture while other areas remain dark, therefore the
method’s name. In this way it is possible to get the grain size statistics according to the separated
crystalline areas.
Deutscher et al. [18] implemented this technique in granular Al thin films (Figure 1.5). The
films in that work were grown on substrates held at room temperature. At low O2partial pressure
the grain size distribution is broad, with a mean value of about 5 nm for a film with a room
temperature resistivity of 26 µcm (Figure 1.6).
At higher resistivities, typically above 100 µcm a narrow distribution is observed (Figure 1.6)
which remains narrow and symmetric around a mean grain size up to high resistivities (Figure 1.8).
For substrates held at room temperature the mean grain size was obtained to be 3 nm (Figure 1.6).
However, as soon as the substrate is cooled to liquid Nitrogen temperatures during growth, the
mean grain size was obtained to be 2 nm (Figure 1.7).
We have mainly used samples grown on liquid Nitrogen cooled substrates in order to obtain a
2 nm grain size as shown in Figure 1.7.
1.2.3 Growth Procedure And Mechanism
The grain size distribution of granular Al films is clearly correlated with the temperature of the
substrate and concentration of the insulating phase during growth. The process of achieving small
6
1.2. THE GRANULAR MORPHOLOGY CHAPTER 1. INTRODUCTION
(a) (b)
Figure 1.5: Typical dark field images obtained by TEM of granular Al films (1.5(a) from Ref. [18],
1.5(b) from Ref. [17]). The bright spots are grains with similar crystal orientation.
(a) (b)
Figure 1.6: Grain size distribution obtained from the analysis of a dark field TEM image [18].
Samples were grown on a water cooled substrate. For a low resistivity sample (ρ300K26 µcm )
the distribution is wide with a mean value of about 4.5 nm [1.6(a)]. For a high resistivity sample
(ρ300K2,300 µcm ) the distribution is narrow with a mean value of about 3 nm (1.6(b)). The
critical transition temperature of the two samples is similar, 2.05 K, and is broader at the high
resistivity sample [18].
7
1.2. THE GRANULAR MORPHOLOGY CHAPTER 1. INTRODUCTION
(a) (b)
Figure 1.7: Grain size distribution of granular Al film deposited on a glass substrate held at 100 K
[1.7(a)] [15]. The grain size distribution is narrow with mean grain size of 1.9 nm. Analysis of the
dark field TEM image of Figure 1.5 gives a similar distribution [1.7(b)] [17].
nano particles in chemical growth is known in the field of nano particles. A very generic process
for achieving small particles in the presence of an insulating shell was described in chemistry as
the LaMer process [20]. In short, a narrow distribution of nano particles can be obtained if the
process of nucleation of the particle, its growth and the increasing capping layer on the surface of
the particle reach an equilibrium state. Several examples of such a process can be found in the
literature especially for metallic nano particles [21].
The growth mechanism of granular Al was described by Shapira and Deutscher [22]. The
formation of crystalline areas is usually given in two stages: nucleation and then growth. Following
the nucleation of small particles, there is continuous expulsion of the dielectric material toward its
periphery. As a result the dielectric molecules pushed to the outer part of the particle are now
coating the particle’s surface. The particle growth continues until it is completely coated with
the insulating interface, i.e. there is a metal-insulator-metal phase separation. At this stage the
dielectric barrier thickness is of the order of one atomic layer. The thickness of the barrier can be
increased by adding more dielectric material to the surface while the metallic grain is not growing in
size. A thicker layer is progressively formed thus separating the grain from its neighboring particles
or in other words the grains are progressively decoupled.
Two main parameters are important for this growth procedure. The first is the temperature
of the substrate which controls the diffusion of atoms and molecules during the initial nucleation
process. The lower the temperature the smaller are the crystalline areas as diffusion is slower. The
second parameter is the metal to dielectric ratio during growth. At low dielectric concentrations
the metallic areas are large. The grain can grow larger until an insulating layer is completely
8
1.3. ENHANCED SUPERCONDUCTIVITY CHAPTER 1. INTRODUCTION
formed on the grain surface. At high concentrations the metallic areas nucleate and grow into
small grains but cannot grow further as there are already separated by the surface insulating layer.
The grain distribution as described above can be interpreted by the interplay of the metal-dielectric
concentration ratio and substrate temperature as given in the mechanism proposed by Shapira and
Deutscher [22].
1.3 Enhanced Superconductivity In Granular Aluminum
Enhancement of superconductivity in granular metallic films was observed by Abeles et al. in
1966 [1]. The highest increase in Tcwas obtained for Al with a factor of 2.6 from bulk Al giving
Tc= 3 Kfor films grown by thermal evaporation. In a later work it was shown that as soon as the
films develop small nano-metric crystallized areas the Tcis increasing by a factor of 3 from that
of the bulk [23]. These small metallic regions define what is now known as the metallic grains in
granular Al.
1.3.1 The Superconducting Phase Diagram
Several important observations can be obtained from the work of Cohen and Abeles [23]. The
transition temperature was seen to be increasing with the decreasing temperature of the substrate.
This was the first observation of superconductivity enhancement which later on was correlated with
the grain size distribution by Deutscher et al. [15]. It was shown that for films grown at room
temperature the maximum Tcis 2.2 K while for films grown at liquid Nitrogen temperatures the
maximum Tcis 3.2 K. In addition, the Tcwas dependent in the room temperature resistivity of
the films (Figure 1.8).
The resistivity of the samples is determined by the barrier thickness between the grains as was
noted in the previous section. Thus by increasing the concentration of the insulating material one
can control the Tcof the sample. This is correlated also with the grain size distribution, starting as
a wide distribution including large grains and converging into a narrow distribution with a constant
mean grain size. Tcfirst increases with the increasing resistivity and the formation of grains and
barriers in the film. It reaches a maximum at several 100 µcm when the grain size distribution is
already narrow and symmetric around its mean grain size value (Figure 1.8). At higher resistivities,
the barrier thickness is increasing and thus grains are decoupled, resistivity increases and as was
previously reported, Tcstarts to decrease too.
The Phase diagram of Tcas a function of the resistivity in granular Al shows a dome-like shape.
In high temperature superconductors a dome phase diagram shows a doping dependent Tcwith a
maximum Tcat an optimum doping while decreasing in the regimes of over-doping or under-doping
of the doping parameter, typically the carrier concentration. For granular Al, the resistivity of the
film is the experimental parameter. The increasing resistivity points out to the decoupling of the
grains. As will be shown in following chapters the carrier concentration decreases in this regime.
A dome-like phase diagram of the superconducting transition temperature in granular Al was
9
1.3. ENHANCED SUPERCONDUCTIVITY CHAPTER 1. INTRODUCTION
(a)
(b)
Figure 1.8: The phase diagram of granular Al films grown on water cooled substrates show a
maximum Tcof 2.2 K [1.8(a), dashed line] [18]. The transition broadens as resistivity increases
due to strong superconducting fluctuations. Above the resistivity of 100 µcm , the grain size
distribution is narrow with a resistivity independent mean value of 3 nm [1.8(a), full line]. The
increasing resistivity is attributed to the increasing barrier thickness. The phase diagram of films
deposited on substrates cooled to 100 K shows a maximum Tcof 3.0 K [1.8(b)] [15]. The increase in
Tcis allegedly attributed to the decreasing grain size from 3 nm (Figure 1.6) to 2 nm (Figure 1.7)
10
1.3. ENHANCED SUPERCONDUCTIVITY CHAPTER 1. INTRODUCTION
originally observed in the 1970’s [18]. This behavior of Tcis quite common these days and ob-
served for several unconventional superconductors among them are the famous Cuprates which
were discovered only 15 years later [24, 25].
As will be elaborated in the following chapters, the anomalous increase of superconductivity
together with the observation of a dome-like phase diagram is only one of the main and important
features that granular Al has in common with other unconventional superconductors. We revealed
additional properties which bring granular Al closer to the family of unconventional superconductors
and in particular to Heavy Fermions. These experimental observations have enabled us to shed
more light on the superconducting mechanism of granular Al.
11
Chapter 2
Enhanced Superconductivity:
Theoretical Background
In the previous chapter we introduced the granular morphology and the experimental obser-
vations of nano size grains. In this chapter we will review some important aspects regarding
superconductivity at this length scale which cannot be strictly explained by conventional theories.
We will review some of the suggested theories and try to shed some light on the enhancement of
Tcat the nanometer scale.
2.1 Superconductivity at the Nano Scale
Granular Al films show enhancement in the superconducting critical temperature. However,
this experimental observation is not in agreement with conventional theoretical predictions for the
existence of superconductivity at nano size particles. In fact superconductivity in a 2 nm Al grain
should be quenched in terms of BCS superconductivity while it is enhanced in granular Al films. We
will try to shed some light on this anomalous behavior which might suggest that superconductivity
in granular Al is not of the conventional type.
2.1.1 Electronic Properties Of Small Metallic Particles
The confinement of electrons in small cavities with dimensions of the order of the electron wave-
length, λF(E), leads to the discretization of the electronic energy eigenvalues. The equivalent of
this quantum size effects can be described in terms of waveguides for electromagnetic field. It is
known that when an electromagnetic wave is injected into a waveguide it will propagate in specific
modes which are determined by the dimensions of the waveguide and the radiation wavelength [26].
By tuning the dimensions of the waveguide one can choose a specific propagation mode. As a result
only a specific frequency and its harmonics at higher orders will propagate through the waveguide
while other modes will be quenched [27]. In this view, the electron wave packet propagates through
the sample which acts as the waveguide. By changing the sample dimensions, for example by low-
13
2.1. SC IN SMALL PARTICLES CHAPTER 2. ENHANCED SUPERCONDUCTIVITY
ering the thickness, one can control the specific modes of propagation of the electron wave packet.
The electron wave packet will propagate only when the sample dimensions will correspond to an
integer number of its wavelength which will lead to a discretization of the electron energy levels.
The change in the electronic levels due to the confinement of the electrons leads to a change in
the density of states. In this case it cannot any more be a continuous broad band structure [9] but
rather should be presented by discrete energy levels as that of a quantum well [28]. Therefore, the
goal of this chapter is to understand the quantum physics of small metallic particles at the nano
meter scale as in granular Al films.
Kubo has predicted that finite size particles will develop quantized electronic states with energy
level average spacing of
δV =1
g(EF)(2.1)
where g(EF) is the density of states per unit volume and δis the level spacing [29].
Taking into account that for aluminum g(EF)=2.3×1022 eV 1cm3and the volume of 2 nm
spherical grain is V= 4πr3/3 = 4.189 ×1021 cm3we get
δ=1
g(EF)·V10 meV (2.2)
assuming that the density of states is indeed that of the bulk. Another important fact is that in
granular Al films the metallic particles are coupled through a thin oxide barrier. Therefore one
should consider that the level splitting of an isolated particle is not exactly that of coupled particles.
Quantum size effects due to the confinement of the electrons lead to a change in the electronic
heat capacity and spin paramagnetism of the metallic particle from that of the bulk material [29].
For the sake of this work, we will emphasize the latter effect. We consider particles with even number
of electrons and particles with odd number of electrons. For the even particle, the highest occupied
level is completely filled, i.e. two electrons of spin up and down, and the magnetic susceptibility
is given by χeven = 3.177µ2g(EF) not far from that of the Pauli paramagnetism susceptibility
χP auli = 2µ2g(EF) [29]. In the case of an odd particle, the highest occupied level is filled with
only one electron having a magnetic moment of 1 Bohr magneton. The magnetic susceptibility of
such particle is given by χodd =µ2/kBT. As the temperature is decreased the contribution to the
magnetic susceptibility by odd particles is increasing similar to a Curie law.
The energy level spacing should be compared with the thermal energy kBTand the Zeeman
energy µBH. A convenient unit conversion between energy, temperature and magnetic field is given
by 1meV
=10K
=10T. Therefore, a level spacing of δ10 meV corresponds to approximately
T= 100 Kand B= 100 T[29]. In order to observe the electronic properties of finite size particles,
one should stay at sufficient low temperatures and low magnetic field.
2.1.2 Anderson Criteria For Superconductivity
We now consider that the particle is made out of a superconducting material. A basic property
14
2.1. SC IN SMALL PARTICLES CHAPTER 2. ENHANCED SUPERCONDUCTIVITY
of the superconducting state is the energy gap ∆. We noted that when a metal is confined to small
particles, discrete electronic levels start to emerge with a mean level spacing δ. If this metal is also
a superconductor then, according to Anderson [30], superconductivity is unaffected by the grain
size as long as δ < ∆. In that case superconducting pairs can still be formed in the particle. The
critical temperature of this particle will be that of the bulk. However, if δ > ∆ BCS coupling is no
longer applicable and Cooper pairs cannot be formed. This is known as the Anderson’s criteria for
superconductivity.
For a 2 nm Al grain the level spacing δis approximately 10 meV. On the other hand the
superconducting energy gap of granular Al is ∆ 0.5 meV [23]. According to Anderson’s criterion
we are well in the limit of δ∆ therefore superconductivity should be quenched in a 2 nm Al
grain. Indeed quenching of superconductivity was observed experimentally in Al particles smaller
than 5 nm in size by Black et al. [31].
We noted in the previous chapter that granular Al is a superconductor with an enhanced
Tccompared to that of the bulk. The observation of coherent superconductivity in granular Al is
in disagreement with the Anderson criteria which deals with isolated grain. It could be that the
inter-grain coupling is at the origin of this disagreement and indeed an array of grains could become
superconducting and even show Tcwhich is higher than that of the parent bulk material. Before
discussing the effects of inter-grain coupling on superconductivity we first discuss some theories
regarding the enhancement of Tcin small isolated particles.
2.1.3 Enhancement Of TcBy Quantum Size Effects
Discrete energy levels could contribute to the enhancement of Tc. The BCS equation predicts
that [4]:
kBTc= 1.13~ωDe1
N(0)V(2.3)
where kBis the Boltzmann’s constant, ~is the Planck’s constant, ωDis the Debye frequency, V
is the interaction energy and N(0) is the Density of States (DOS) at the Fermi level. A peak in
N(0) will result in the enhancement of Tc[4]. Blatt and Thompson [32] have discussed this issue
in the case of thin films with a thickness tcomparable to the electron’s Fermi wavelength λF. As
was noted before, the film acts as a cavity to the single electron’s wave function resulting in an
oscillatory density of states as a function of the decreasing film thickness. By setting the Fermi
energy at one of the peaks in the oscillatory DOS function, an enhancement of Tcis predicted [32].
A similar mechanism of enhancement was proposed by Parmenter [33] for granular systems, in
particular granular Al. Here the grain’s size dacts as the dimensional parameter for the quantum
size effects and should be compared with a characteristic length
L= (λ2
Fξ0)1/3(2.4)
rather than to the sample thickness as in Blatt and Thompson [32]. Here ξ0is the Pippard co-
herence length at T= 0. We introduced the discrete spectrum of the single electron energy levels
15
2.1. SC IN SMALL PARTICLES CHAPTER 2. ENHANCED SUPERCONDUCTIVITY
separated by the level splitting δin a grain which replaces the continuous spectrum of the bulk.
An enhancement of Tcis predicted when the DOS peaks fit the discrete energy spectrum. For
example, in aluminum with λF'0.36 nm and ξ0'1.6µm,Lis approximately 6.2 nm. In the case
of granular aluminum, an enhancement of the Tcby a factor of 4 from that of the bulk is possible.
2.1.4 Electronic Shell Effects
Tuning the Fermi energy of a small particle such that it will fall exactly on a peak in the density
of states seems to be experimentally ambitious. As there is some finite grain size distribution there
is also a finite level spacing distribution for each grain. Such distribution of the level spacing will
result in a distribution in the energy gap of the system due to different grain sizes.
A peak in the DOS due to quantum size effect is indeed hard to implement in a system of small
particles. However it can be implemented more easily in an energy level structure showing large
degeneracy near the Fermi level. A high degree of degeneracy can be obtained in clusters of atoms
which show an electronic shell structure as in atoms [34].
An electronic shell structure in clusters consisting of a small number of atoms was observed in
1984 by Knight et al. [35]. Typically clusters are obtained by aggregation of the order of 100 atoms.
As in atoms, closed shell clusters, called ”magic” clusters, with specific numbers of electrons, e.g.
Nm= 8,20,40,58,92,132,138..., are known to be stable. A shell structure is not observed for
every type of metal. For example, Al clusters do exhibit a shell structure [36] while Nb clusters do
not [37]. It was also observed in large Al clusters similar in size to a 2 nm Al grain [38].
A cluster shell structure is classified by its orbital momentum Land radial number n[34].
Magic clusters shells are completely full and have a spherical symmetry. However, there are also
shells which are incomplete and their shape deviates from the spherical symmetry to a more elliptic
structure [34]. In this case the electronic levels and cluster classification is done by the projection
of the angular momentum m.
High Tcsuperconductivity is predicted in these metallic clusters [39]. The origin of the en-
hancement in Tcis very similar to the explanation given above regarding the peak in the DOS of
a small particle with discrete electronic levels. In the case of clusters the discrete electronic levels
are described by the shell structure. As in atoms this structure is not equally spaced in energy and
is more dense at high n(Figure 2.1). In addition, there is also a degeneracy of 2 (2L+ 1) which
depends on the value of the orbital momentum. In this scenario it is theoretically easier to achieve
a sharp peak in the DOS at the Fermi energy due to the shell structure. The level spacing between
the degenerate states is also much lower than that of the equally spaced levels and is of the order of
several meV in nearly magic clusters [39]. Therefore pairing can be obtained and an enhancement
of Tcis then expected. It is worth noting that although the splitting of the levels in nearly magic
clusters leads to smearing of the density of states, high Tcvalues are still expected [34] in this case.
Recent experimental and theoretical work has shed light on the shell structure effect in nano
particles. An enhancement of the gap is observed by Scanning Tunneling Microscopy (STM) mea-
surements on Sn particles which was attributed to the shell structure of Sn clusters [40].
16
2.2. INTERGRAIN COUPLING CHAPTER 2. ENHANCED SUPERCONDUCTIVITY
Figure 2.1: Illustration of the electronic level spacing in small particles. Equally spaced levels
(Left) have larger ∆Egap (Red arrow) between the Highest Occupied Level/Shell (HOL) to the
Lowest Unoccupied Level/Shell (LUS) than that of the electronic shell effect (Right) [as shown in
Ref. [34]].
2.2 Intergrain Coupling
We showed that superconducting pairing can be obtained in small metallic particle. However,
this entire picture is true for an isolated particle, while in the granular system, one should consider
the above effects in an array of particles. Inter-particle coupling could quench the discrete electronic
levels thus removing any enhancement to Tcby quantum size effects. In the following section we
will discuss the effects of intergrain coupling on superconductivity in an array of particles.
2.2.1 Level Spacing And Charge Fluctuations
Discretization of the electronic energy levels is expected inside a small isolated metallic particle.
However, if the particle is coupled to other particles, as in granular Al system, electrons can tunnel
into the particle and cause charge fluctuations. As a result, the electronic levels will be smeared
and the level spacing will be irrelevant. However, if certain conditions are met, the discretization
of the electronic energy levels is still valid even in the case of coupled particles.
Kawabata showed that in the case where the electrostatic energy Ecin the grain is strong,
electrons are localized in each grain therefore the energy levels remains quantized [41]. The con-
ditions for the above situation are defined by two critical electrostatic energies Uc1zt2and
Uc210zt2. Here zis the number of adjacent particles, tis the tunneling matrix and δis the
electronic level splitting. If the electrostatic energy is larger than Uc2, the levels are not smeared.
In this case several important properties of finite size particles will be observed in particular the
magnetic properties of a particle with an odd number of electrons.
In order to estimate the electrostatic energy in the terms of the tunneling matrix it is necessary
to compare it to the transmission coefficient of the barrier and is given by:
zt2/U δ e2