One-Dimensional Superconductivity in Nanowires E-Book

Fabio Altomare

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 570

Veröffentlichungsjahr: 2013

Beschreibung

Devoted to the topic of superconductivity in very narrow metallic wires, the goal of this book is to produce a relatively self-contained introduction to the theoretical, experimental and phenomenological aspects of the 1-dimensional superconducting nanowire system.

Leseprobe

Contents

Preface

Abbreviations and Symbols

Part One: Theoretical Aspects of Superconductivity in 1D Nanowires

1 Superconductivity: Basics and Formulation

1.1. Introduction

1.2. BCS Theory

1.3. Bogoliubov–de Gennes Equations – Quasiparticle Excitations

1.4. Ginzburg–Landau Theory

1.5. Gorkov Green’s Functions, Eilenberger–Larkin–Ovchinnikov Equations, and the Usadel Equation

1.6. Path Integral Formulation

References

2 1D Superconductivity: Basic Notions

2.1. Introduction

2.2. Shape Resonances – Oscillations in Superconductivity Properties

2.3. Superconductivity in Carbon Nanotubes – Single-Walled Bundles and Individual Multiwalled Nanotubes

2.4. Phase Slips

References

3 Quantum Phase Slips and Quantum Phase Transitions

3.1. Introduction

3.2. Zaikin–Golubev Theory

3.3. Short-Wire Superconductor–Insulator Transition: Büchler, Geshkenbein and Blatter Theory

3.4. Refael, Demler, Oreg, Fisher Theory – 1D Josephson Junction Chains and Nanowires

3.5. Khlebnikov–Pryadko Theory – Momentum Conservation

3.6. Quantum Criticality and Pair-Breaking – Universal Conductance and Thermal Transport in Short Wires

References

4 Duality

4.1. Introduction

4.2. Mooij–Nazarov Theory of Duality – QPS Junctions

4.3. Khlebnikov Theory of Interacting Phase Slips in Short Wires: Quark Confinement Physics

References

5 Proximity Related Phenomena

5.1. Introduction

5.2. Transport Properties of Normal-Superconducting Nanowire-Normal (N-SCNW-N) Junctions

5.3. Superconductor–Semiconductor Nanowire–Superconductor Junctions

5.4. Majorana Fermion in S-SmNW-S Systems with Strong Spin–Orbit Interaction in the Semiconductor

References

Part Two Review of Experiments on 1D Superconductivity

6 Experimental Technique for Nanowire Fabrication

6.1. Experimental Technique for the Fabrication of Ultra Narrow Nanowires

6.2. Introduction to the Techniques

6.3. Step-Edge Lithographic Technique

6.4. Molecular Templating

6.5. Semiconducting Stencils

6.6. Natelson and Willet

6.7. SNAP Technique

6.8. Chang and Altomare

6.9. Template Synthesis

6.10. Other Methods

6.11. Future Developments

References

7 Experimental Review of Experiments on 1D Superconducting Nanowires

7.1. Introduction

7.2. Filtering

7.3. Phase Slips

7.4. Overview of the Experimental Results

7.5. Other Effects in1D Superconducting Nanowires

7.6. Antiproximity Effect

References

8 Coherent Quantum Phase Slips

8.1. Introduction

8.2. A Single-Charge Transistor Based on the Charge-Phase Duality of a Superconducting Nanowire Circuit

8.3. Quantum Phase-Slip Phenomenon in Ultranarrow Superconducting Nanorings

8.4. Coherent Quantum Phase Slip

8.5. Conclusion

References

9 1D Superconductivity in a Related System

9.1. Introduction

9.2. Carbon Nanotubes

9.3. Majorana Experiments

9.4. Superconducting Nanowires as Single-Photon Detectors

References

10 Concluding Remarks

Index

Related Titles

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The Authors

Dr. Fabio AltomareD-Wave Systems Inc. 100 - 4401 Still Creek Drive Burnaby, BC, V5C 6G9 Canada

Prof.Albert M.ChangDuke University Department of Physics Physics Building Science Drive Durham, NC 27708-0305 USA

Cover FigureArtistic rendering of an 8 nm wide, 20 µm long nanowire fabricated on an InP semiconducting stencil. Electrical contacts to the nanowire (left and right pads) are realized during the nanowire deposition. At the center, a QPS junction is current biased by a current source: The number of windings of the phase of the order parameter decreases because of a phase slippage event. The diagram does not indicate the actual nanowire connection in a circuit. Rendering of the windings courtesy of A. Del Maestro.

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To Maria Elena, Mattia, Gabriele, Giulia and to my parents: Thank you for your love and support.

Fabio

To my family: Ying, Emily and Austin, my sister Margaret, my brother Winston, my father Bertrand Tsu-Shen, and my late mother, Virginia Pang-Ying.

Albert

Preface

This book is devoted to the topic of superconductivity in very narrow metallic wires. Interest in such wires is driven by the continuing drive for miniaturization in the electronics industry, where the reduction of heat dissipation by the use of interconnects which are superconducting may be necessary. This has led to the invention of new methods of producing very narrow wires with good quality and uniformity in dimensions, and opened up the possibility of novel device paradigms.

The superconducting state is a state of coherent pairs of electrons, held together by the mechanism of the Cooper instability. When the lateral dimensions are small, a new pathway opens up for the superconductor to produce dissipation, associated with the enhanced rate at which fluctuations can occur in the complex order parameter – a quantity which describes the magnitude and phase of the Cooper pairs. In many regimes, the fluctuations of the phase of the Cooper pairs is often the dominant process, in analogy with what occurs in dissipative Josephson junctions. This new pathway is related with the motion of vortices in type-II superconductors, which gives rise to dissipation in 2D (two-dimensional) and bulk superconductors, in which a rapid change of phase occurs as a vortex passes by. Here, because of the physical dimension of the system (comparable to 10 nm), the vortex quickly passes across the entire narrow wire, producing dissipation in the process.

Although nanowires are small in their transverse dimensions, they are still much larger than the Fermi wavelength (λF) of the electronic system, which is of the order of a few angstrom in all metals. In conventional superconductors such as Al, Pb, Sn, Nb, MoGe, and so on, in thin film or nanowire form, the coherence length is ξ ≈ 5–100 nm, typically 10–1000 times the Fermi wavelength. The system we will consider is therefore in a sort of mixed-dimensional regime. From the condensate perspective, the system is 1D (one-dimensional), in the sense that the transverse dimensions are smaller than the Cooper pair size: for this reason, the wave function, or alternatively, the order parameter, describing the Cooper pairs, is uniform, and thus position-independent, in the transverse directions. From the fermionic quasi-particle excitation perspective, the system is effectively 3D (three-dimensional; λF is much smaller than the lateral dimension) and there is a large number of transverse channels (from ≈ 100 in a multiwalled carbon nanotube, to ≈ 3000 in an ≈ 8 nm diameter aluminum nanowire), analogous to transverse modes in a wave guide.

In this limit, the dominant collective excitations are no longer pair-breaking excitations across the superconducting energy gap, but rather “phase-slips,” which are topological defects in the ground state configuration. Phase slips are related to the motion of vortices in type-II superconductors which give rise to dissipation in 2D and bulk superconductors: in a 1D system, they produce a sudden change of phase by 2π across a core region of reduced superconducting correlation which gives rise to a voltage pulse.

Another intriguing aspect arises from the fact that even in wires which are not ballistic along the wire length, the typical level spacing in the transverse direction can significantly exceed the superconducting gap energy scale. Thus, the possibility of singularity in the density of the electronic state in the normal system associated with each transverse channel can cause oscillatory behaviors in the superconducting properties.

From the above discussion, it is clear that the regime of interest is delineated by the condition that the size of the Cooper pairs, or the superconducting coherence length, be larger than the transverse directions perpendicular to the length of the wire. In this limit the order behavior of the Cooper pair is largely uniform across the wire length, and only variations along the wire length need to be considered. At temperatures well below the superconducting transition temperature, the scale for the observation of dimensionality effects is in the 10–100 nm range (5–50 nm in radius): in this regime, new physics have been predicted, including universal scaling laws in the conductance of the wire.

To access the regime where quantum processes become dominant, however, a more stringent requirement is necessary, that of a sufficiently large probability of fluctuations to occur: this more stringent criterion places the scale requirement in the 10 nm (5 nm in radius) range. Nanowires in this regime may either be a single monolithic wire, such as nanowires made of MoGe, Al, Nb, In, PbIn, Sn, and so on, or coupled wires such as in carbon nanotube bundles.

It should be noted that, in the strict sense, only systems in 2D or 3D have a true, sharp, thermodynamic phase transition into the superconducting state at a finite temperature Tc. In the 2D case, it is a Berenzinskii–Thouless–Kosterlitz type of second-order phase transition, while in 3D, it is a second-order continuous phase transition in the Ginzburg–Landau sense, at least for type-II superconductors supporting the existence of vortices within the bulk. In contrast, in a 1D nanowire, the fluctuations cause the transition to become smeared, so that the resistance remains finite, albeit small, below the transition. Early theoretical analyses were motivated by experimental observations of such behavior, and attempted to quantify the amount of residual resistance. Thus, naturally, questions arise as to whether the resistance vanishes at zero T, and whether novel excitations are able to limit the supercurrent below the depairing limit.

From a broader point of view, 1D superconducting nanowires are interesting from a variety of perspectives, including many body physics, quantum phase transition (QPT), macroscopic quantum tunneling (MQT) processes, and device applications. The field, by nature, involves rather technically sophisticated methods. This is true from either the experimental or theoretical side.

On the experimental side, there are many technical challenges, and thus it is not for the faint of heart. Such challenges include nanowire fabrication, the delicate nature of nanowires with respect to damage–they act as excellent fuses, and their sensitivity to environmental interference from external noise sources, and so on. At the level of 10 nm in transverse dimensions, corresponding roughly to 40 atoms across, even width fluctuations of a few atoms can have significant influence on the energetics and properties. Thus, to obtain intrinsic behaviors of relevance to a uniform wire, rather than behavior limited to weak-links or a very thin region in a nonuniform wire, fabrication is exceedingly demanding. Arguably, only in recent years has the emergence of fabrication techniques come into existence with sufficient precision for producing unusually uniform nanowires. Thus, substantial progress is occurring on the experimental front.

On the theoretical side, analyses invariably involve sophisticated quantum field theory (QFT), quantum phase transition (QPT), Bogoliubov–de Gennes BdG, Ginzburg–Landau (GL), Gorkov–Eilenberger–Usadel self-consistent (including nonlinearities) techniques, all of which require a rather advanced level of understanding of the theoretical machinery. This is often compounded by the fact that the concept of the quantum-mechanical tunneling in the phase of the superconducting order-parameter is difficult to motivate from a classical perspective, since, unlike coordinates or momenta, the phase is a concept born out of wave mechanics. Instead, the tunneling in the phase degree of freedom is usually introduced via the Feynman path integral type of formulation, as one finds a more natural description, for this phenomenon, in terms of instantons in field theory language.

The goal of this book is to produce a relatively self-contained introduction to the experimental and theoretical aspects of the 1D superconducting nanowire system. The aim is to convey what the important issues are, from experimental, phenomenological, and theoretical aspects. Emphasis is placed on the basic concepts relevant and unique to 1D, on identifying novel behaviors and concepts in this unique system, as well as on the prospects for potentially new device applications based on such new concepts and behaviors. The latest experimental techniques and results in the field are summarized. On the theoretical side, much of the field theoretic methods for analyzing the various quantum phase transitions, such as superconductor–metal transitions, superconductor–insulation transitions, and so on, brought about by disorder, are highly technical and the details are beyond the scope of this book. Nevertheless, an attempt is made to summarize the relevant issues and predictions, to pave the way for understanding the formalisms and issues addressed in available journal literature. It is the hope of the authors that this book will serve as a starting point for those interested in joining this exciting field, as well as serving as a useful reference for active researchers.

To this end, our philosophy is to present the field as an active, exciting, and ongoing discourse, rather than one that is fully established. Thus, many of the concepts and experiments are still fraught with a certain degree of healthy controversy. Thus, an attempt is made to convey a sense of openness to the discourse in the field.

The book is organized as follows: Chapter 1 contains a brief history of the field, and a succinct summary of the various theoretical methodologies for understanding conventional superconductivity. These methods are widely used in analyzing 1D superconducting nanowire systems. Chapter 2 is devoted to the basic concepts of 1D superconductivity, including size quantization and its influence on superconducting properties, leading to the phenomenon of shape resonances, the phase-slip phenomenon, which originated from an attempt to explain the broadened temperature transition and the finite voltage along the wire below but near the transition, as well as the conditions and relevant energy scales in molecular systems such as carbon nanotubes. In Chapter 3, the quantum theory based on path integral formulation is summarized. The various types of quantum phase transitions and competing physical scenarios are described. Chapter 4 explores new concepts and potentially new devices based on the idea of a duality between Cooper pairs and the phase slip. Novel QPS junctions are described. These are believed to offer new venues for a current version of the Shapiro steps, as well as a platform for qubits. Nonlinear and nonequilibrium effects based on the Usadel equations are described in Chapter 5.

On the experimental side, the all-important description of the state-of-the-art fabrication methodologies is presented in Chapter 6. Experimental techniques, such as filtering to remove external environmental noise are summarized in Chapter 7. Finally, in Chapter 8, we discuss the current state of experimental progress, and the many open questions, as well as future prospects. To conclude, in Chapter 9, we describe recent experimental results in superconducting nanowire single-photon detector that are now approaching the 1D superconducting limit and devices that are related to 1D superconductivity via the proximity effect: in this class, we find nanotubes and semiconducting nanowires, which have recently indicated of the presence of Majorana fermions.

The authors are indebted to Sergei Khlebnikov (Purdue University), in particular for sharing his insight and for providing helpful comments on the entire theory section. The authors acknowledge the help of (in alphabetical order) G. Berdiyorov (University of Antwerp), E. Demler (Harvard University), A. Del Maestro (University of Vermont), D. Golubev (Karlsruhe Institute of Technology), F. Peeters (University of Antwerp), G. Refael (Caltech), S. Sachdev (Harvard University), A. Zaikin (Lebedev Inst. of Physics and Karlsruhe Institute of Technology) for critically reading the manuscript. The authors would also like to thank M.R. Melloch (Purdue University), C.W. Tu (University of California at San Diego), P. Li, P.M. Wu, G. Finkelstein, Y. Bomze, I. Borzenets (Duke University), and Li Lu (Institute of Physics, CAS, Beijing) for their help and fruitful discussions.

September, 2012

F. Altomare and A.M. Chang

Abbreviations and Symbols

Acronyms

APE

Anti-proximity Effect

BdG

Bogoliubov–de–Gennes

BCS

Bardeen–Cooper–Schreiffer

CPR

Current–Phase Relation

cQPS

Coherent Quantum Phase-Slip

DCR

Dark Current Rate

DQM

Dissipative Quantum Mechanics

e-beam

Electron beam

GIO

Giordano expression for the resistance due to quantum phase slips, or to QPS and TAPS

GL

Ginzburg–Landau

HQS

Silsesquioxane, a type of negative electron beam resist

IRFP

Infinite-Randomness Fixed Point

JJ

Josephson Junction

KQPS

Khlebnikov Quantum Phase-Slip

KTB

Kosterlitz–Thouless–Berezinskii

LA

Langer–Ambegaokar

LAMH

Langer–Ambegaokar–McCumber–Halperin

MBE

Molecular-Beam-Epitaxy

MH

McCumber–Halperin

PMMA

Polymethylmethacrylate: probably the most common electron beam resist, mainly used as positive tone resist

PSC

Phase-Slip Center

QPS

Quantum Phase-Slip

RTFIM

Random-Transverse-Field Ising Model

SC

Superconducting

SG

sine-Gordon

SEM

Scanning Electron Microscope

SIT

Superconductor-Insulator Transition

SmNW

Semiconducting Nanowire

SMT

Superconductor-Metal Transition

SNAP

Superlattice Nanowire Pattern Transfer

SNAP

Superconducting Nanowire Avalanche Photo Detector

SNSPD

Superconducting Nanowire Single-Photon Detector

SQUID

Superconducting Interference Device

SSPD

Superconducting Single-Photon Detector

SWNT

Single-Walled Nanotube

SWCNT

Single-Walled Carbon Nanotube

TAP or TAPS

Thermally-Activated Phase Slip

Symbols

Part One

Theoretical Aspects of Superconductivity in 1D Nanowires

1

Superconductivity: Basics and Formulation

1.1Introduction

The Bardeen–Cooper–Schrieffer (BCS) theory captures the essential physics that gives rise to the condensation of the Cooper pairs into a coherent superconducting state. It specifically deals with a homogeneous and clean system in three dimensions and is essentially a self-consistent, mean field treatment. Following its success, subsequent theories are based on a mean field treatment on the BCS or related models. In order to account for inhomogeneous systems, such as those occurring in multilayer systems of superconductor- insulators, or normal metals, de Gennes, and independently Bogoliubov, derived a system of coupled equations between electrons and holes that yield the solutions for the fermionic quasiparticles above the superconducting condensate, which are separated from the condensate by the superconducting energy gap, Δ. Such a formalism is useful for computing the properties of ultrathin SC films and clean SC nanowires, as will be discussed in Chapter 2.

Prior to the BCS microscopic theory, Ginzburg and Landau sought to generalize the Landau theory of second order phase transitions to the superconductivity problem. The result is the celebrated Ginzburg–Landau (GL) theory of superconductivity. The free energy is written in terms of the superconducting order parameter, which in this case is a complex quantity. This theory is expected to be valid near Tc, and was derived from a microscopic BCS model by Gorkov.

Gorkov invented a powerful methodology by deducing the equations of motion for the Green’s functions. An anomalous Green’s function, F, accounting for pair-correlations was introduced in addition to the normal electron Green’s function. The two Green’s functions form a closed set of equations, the solutions of which yield all the results of the BCS theory, and moreover, can be readily extended to incorporate dirty systems with impurities, as well as deal with nonlinearities, dynamics, and so on. Thus, type-II superconductors can readily be described. From the perspective of this text, the central importance of the Gorkov equations is the ultimate deduction of the Usadel equation for related Green’s functions (to the Gorkov GF’s), in the limit of dirty systems. The derivation of the Usadel diffusion equation was based on the works of Eilenberger and of Larkin and Ovchinnikov, who independently applied the quasiclassical approximation to the Gorkov equations, and identified an energy-integrated version of the Gorkov Green’s functions. These approaches led to a simplification of the Gorkov equations into Boltzmann transport-like equations for these modified Green’s functions.

The Usadel diffusion equation is much more tractable and amenable to numerical implementations, enabling realistic experimental geometries and situations to be analyzed. In particular, issues of quasiparticle injection at the normal-superconductor interface, nonequilibrium quasiparticle distribution, and so on, are readily computed. These methodologies based on the Usadel equation are naturally suited to analyzing systems with 1D SC nanowires. In fact, Dr. Pauli Virtanen provides downloadable C++ programs for such computations on his website: http://ltl.tkk.fi/~theory/usadel1/!

Beyond these standard methodologies, a powerful technique for analyzing complex superconducting phenomena has been developed over the past 20 years, which is particularly well-suited for understanding quantum phase transitions, such as those in dissipative Josephson junction arrays of 2D and 1D varieties, dissipative single Josephson junctions, vortex states and associated transitions, as well as transitions in the 1D superconducting nanowire system. This technique is based on the path integral formulation. Thus, introducing such a formulation is important in bridging the gap to enable the study of quantum-tunneling processes, such as the macroscopic quantum tunneling of the phase slip, which is central to the understanding of the unique behaviors in 1D superconducting nanowires.

1.2BCS Theory

Conventional superconductivity has, as its key components, the binding of the Cooper pairs, and the Bose condensation of such Cooper pairs. The binding arises from an attractive interaction between electrons mediated by phonons or other excitations, for example, plasmons. The essential behaviors and outstanding characteristics are captured by a self-consistent mean field approximation. The celebrated BCS theory, proposed by Bardeen, Cooper, and Schrieffer in 1957 [1], provides both a microscopic model, which provides the underpinning of our current understanding of conventional superconductivity and a basic formalism, which is most conveniently cast in terms of the second-quantized formalism, that is, in terms of the creation and destruction operators of the single particle electronic states.

The Cooper instability [2] indicates that the Fermi sea is unstable to the formation of bound-pairs (Cooper pairs) of electrons, in the presence of an effective, attractive interaction. In the original formulation, and in most conventional superconductors to date, the attractive interaction is a time-retarded interaction mediated by lattice vibration phonons. A passing electron, whose velocity, given by the Fermi velocity, is much larger than the phonon propagation speed, polarizes the local ions: before the lattice can relax, a second electron arrives and feels the attraction from the still-polarized positive ions. This attraction produces a pairing of electrons which gives rise to an entity which is bosonic in character. The two electrons thus form a Cooper pair whose size-scale is denoted by the superconducting coherence length ξ.

The Bose condensation of such Cooper pairs into a macroscopic, coherent quantum ground state is the source of the unusual physical properties associated with superconductivity. In a weak-coupling superconductor, where the electron–phonon coupling is much smaller than the Fermi energy, the size of the Cooper pair, or ξ, is typically much larger than the Fermi wavelength λF. In a homogeneous system, the attraction is maximal when the relative total momentum of the two electrons is zero, that is, when the center of mass is stationary. This gives the requirement of pairing between +k and –k states. At the same time, the s-wave channel has the largest attractive potential, leading to an s-wave superconductor, for which the energy gap created by the condensation is isotropic in k-space. As the spatial wave-function is symmetric, in order to satisfy Pauli’s exclusion principle, the spin must be in an antisymmetric, singlet state.

The second quantization formulation is then equivalent to a formulation based on the occupation of single-particle states – in this case of a homogeneous, disorder-free system, the k-states. The Pauli exclusion in the occupation dictating that a given state to be either singly occupied or unoccupied, with multi-occupation forbidden, is encoded in the anticommutation relation of the creation and annihilation operators. In the k-representation, the model Hamiltonian is given by

(1.1)

(1.2)

between electrons at positions ri, rj, where V is the interacting strength, V is the system volume, and g the volume-independent Gorkov coupling.

To find the ground state energy of the system, a variational wavefunction in the form is assumed

(1.3)

(1.4)

for the ground state energy

(1.5)

Changing to the customary variables of Ek and Δk, where

(1.6)

and

(1.7)

with

(1.8)

the minimization condition yields the self-consistency expression

(1.9)

This is the BCS gap equation for the superconducting gap of a quasi-particle of momentum k. The restriction in k′ leads to a k-independent gap

(1.10)

and

(1.11)

Converting to an integral using the normal electron density of states, and approximating the density of states with its value at the Fermi energy N(0) gives

(1.12)

The solution yields the celebrated BCS expression for the energy gap Δ

(1.13)

with g ≡ VV the Gorkov coupling constant. The nonanalytic dependence on g indicates that this result cannot be obtained by a perturbation calculation in the small parameter V/EF!

(1.14)

At Tc, the gap vanishes, giving the condition

(1.15)

and the relation

(1.16)

The BCS gap, Δ(T), occupies a central role in the phenomenon of conventional superconductivity. From the existence of the gap, many important properties can be derived, such as the quasiparticle excitation spectrum

(1.17)

and the density of states

(1.18)

the London equations [3] accounting for the Meissner effect of flux expulsion, flux quantization, specific heat, critical current, and so on. Our purpose here is to introduce the essential aspect of BCS superconductivity, and to wet the appetite of those readers unfamiliar with this subject matter. We refer to the excellent standard textbooks, for example, de Gennes [4], Tinkham [5], Ashcroft and Mermin [6], Fetter and Walecka [7], and so on, for a comprehensive treatment.

1.3Bogoliubov–de Gennes Equations – Quasiparticle Excitations

The BCS theory in the previous section is for a 3D, homogeneous system without disorder. An essential feature of the theory is the pairing between time-reversed states. The role of time-reversed states was highlighted by Anderson’s well-known theoretical analysis, demonstrating that nonmagnetic impurities do not suppress the superconducting gap [8]. To go beyond a translational invariant system, and account for boundaries, and for the possibility of tunnel junctions or SNS bridges, the theories of Bogoliubov [9, 10] and de Gennes [11] are required.

In the method of Bogoliubov and de Gennes, a solution is sought for the fermionic electron-like and hole-like quasiparticle excitations above the gap, which separates the condensed Cooper pairs from these unbound, quasiparticles. The theory was developed in part to address situations where there are boundaries or coupling to normal metals [11]. Because the presence of boundaries is properly accounted for, this formalism lends itself readily to deduce the behaviors of clean, 1D superconducting nanowires in which the boundaries are in the two lateral directions, perpendicular to the length of the nanowire, as was done in the work of Shanenko, Croitoru, Peeters, and collaborators [12–14] (please see Chapter 2).

The fermionic electron- and hole-like quasiparticles are each linear combinations of the electron and hole wavefunctions in the normal state. However, because of the spatial dependences introduced by the boundaries, the linear combinations are in general not as simple as in the homogeneous case. The point contact attractive interaction between electrons may be spatially dependent: – V(r)Vδ(r) ≡ –g(r)δ(r) (see (1.2)).

The starting point is the equation of motion for the field operator , which creates an electron of spin σ at position r. In the special case of a homogeneous, clean system, it is related to the creation operator for the kσ state by the (inverse-)Fourier transform

(1.19)

The usual commutation with the BCS Hamiltonian in real space gives the equation of motion:

(1.20)

Here, the one-electron potential Uo(r) includes the boundary and static impurity potentials. Using a Hartree–Fock mean field approximation, the term cubic in the field operators is replaced in the anomalous channel by

(1.21)

and, in the normal channel by

(1.22)

Here, the averaging denotes a thermal average. For the anomalous channel, in anticipation of a coupling similar to s-wave coupling in the homogeneous case, only the dominant terms, which come from the down- and up-spin correlators (and up- and down-), are kept, that is,

(1.23)

which defines the partial potential Δ†(r). For the normal channel, the self-consistent potential contributes a term

(1.24)

(1.25)

where the energies En are positive. The eigenfunctions un(r) and vn(r) satisfy the following coupled equations, namely,

(1.26)

The pair potential satisfies the self-consistency condition

(1.27)

In addition,

(1.28)

In the presence of an external magnetic field, these equations are supplemented, of course, by the London equations. As expected, in the homogeneous case in three dimensions, they reproduce the BCS results.

1.4Ginzburg–Landau Theory

The Ginzburg–Landau (GL) [15] approach to the description of conventional superconductivity is based on the notion of an order parameter, and follows Landau’s phenomenological theory of second order phase transitions. The free energy of the superconducting state is envisioned to differ from the normal state by a contribution from the condensate when the order parameter becomes nonzero below a transition temperature Tc. The thermodynamic state is given by the solution which minimizes the free energy with respect to variations in the order parameter. In the case of a charged superconductor, which couples to the electromagnetic scalar and vector field, minimization with respect to the potentials leads to the London equations. The GL equations therefore represent a mean field treatment and are valid below, but near Tc. Very close to Tc, additional fluctuations may arise. Gorkov showed that the GL equations (see below) can be derived from a microscopic theory starting from a BCS model Hamiltonian [16].

For a superconductor, the order parameter Ψ is a complex quantity, having a magnitude, |Ψ|, and a phase, φ. The physical interpretation of this order parameter is that its modulus squared 2|Ψ|2 yields the density fraction of superfluid component. Thus, Ψ may be thought of as the "wavefunction" of the Cooper pairs; these pairs are coherent below Tc and are described by a single “wavefunction.” The GL theory was used by Little [17], and Langer and Ambegaokar to analyze the phase-slip process, to account for the generation of a finite voltage below Tc in thin superconducting whiskers, at current levels below the expected critical current [18]. Thus, in some sense, the GL approach has been essential in providing a physical picture of a phase-slip defect. A time-dependent version was used by McCumber and Halperin to more accurately estimate the prefactor in the rate for the thermal generation of phase slips for an activated process just below Tc [19]. In this approach, relaxation to equilibrium from an external perturbation is characterized by a relaxation time, and modeled by the addition of a relaxation term which is proportional to the variation of the free energy density with respect to the order parameter.

The phenomenological GL free energy, in cgs units, has the following form:

(1.29)

Here, M refers to the mass of the Cooper pair, and its charge is –2 e. The inclusion of the gradient term with the covariant derivative (ħ/i∇ + 2e/cA) accounts for the kinetic energy term when the superfluid is in motion, such as when a current is flowing.

The coefficients are dependent on temperature. Near Tc,

(1.30)

(1.31)

where a and b are positive constants, and

(1.32)

Note that α changes sign and is negative below Tc. This leads to a minimum energy state which has a nonzero value for the order parameter.

Minimization with respect to the order parameter Ψ* yields the GL equations

(1.33)

Near Tc, the solution for a uniform state is given by

(1.34)

Thus, the superfluid density, ns, goes as

(1.35)

Variation with respect to the vector potential gives

(1.36)

with

(1.37)

These are the phenomenological London equations, which were put forth by London [3] to account for the complete expulsion of magnetic flux from the interior of a bulk SC sample, beyond the London penetration depth λ.

In the Ginzburg–Landau theory, the coherence length ξ characterizing the size of the Cooper pair is given by

(1.38)

while the penetration depth

(1.39)

1.4.1Time-Dependent Ginzburg–Landau Theory

In equilibrium, the minimization of the free energy yields the lowest energy state, for which the free energy does not change, to first order, with any variation of the order parameter or the fields. If, however, the system is not in an equilibrium state, the system should relax toward the equilibrium state. To account for the time-dependence of this relaxation process, the time-dependent Ginzburg–Landau equation is often used to describe the temporal behavior. The relaxation rate is assumed proportional to the variation of the free energy density with the order parameter ψ*. The TDGL equation is written as

(1.40)

This form of the TDGL is somewhat controversial and is believed to be accurate when close to Tc [20, 21]. It was used by McCumber and Halperin to estimate the attempt frequency to surmount the free energy barrier due to a phase slip in the vicinity of Tc. Review papers on applying the TDGL approach to the phase-slip phenomena in narrow superconducting devices are readily available in the literature [22, 23]. In addition, the generalized TDGL approach of Kramer and coworkers [24, 25] has been widely used to describe the property of both 1D [26, 27] and 2D [28, 29] superconducting systems, for example.

1.5Gorkov Green’s Functions, Eilenberger–Larkin–Ovchinnikov Equations, and the Usadel Equation

Gorkov developed a powerful method for understanding superconductivity by introducing a set of coupled equations for the dynamics (time evolution) of the Green’s functions [30]. The equations couple the normal Green’s functions, G, and the anomalous Green’s, F, functions relevant to Cooper pairing. The equations of motion follow from the time evolution of the fermion field operators (see (1.20)), in which the interaction terms are again treated in a mean field manner. From such equations, essentially all interesting physical quantities can be obtained.

However, these coupled equations are difficult to solve. Starting from these equations, Eilenberger [31], and separately Larkin and Ovchinnikov [32], developed transport-like equations for a set of Green’s functions closely related to Gorkov’s Green’s functions; these Green’s functions are related to Gorkov’s via an integration over the energy variable. By exploiting the fact that most quantities of interest are derivable from such Green’s functions integrated over energy, and exploiting the quasiclassical approximation (which amounts to the neglect of the second order spatial derivatives relative to terms involving kF times the first order derivatives), after averaging over the position of impurities, the number of Green’s functions was reduced from four to two, and a transport-like equation for these Green’s functions integrated over energy emerged.

Going one step further, Usadel noted that in the dirty limit, the Eilenberger–Larkin–Ovchinnikov Green’s functions are nearly isotropic in space [33]. By exploiting this condition, the Eilenberger equations are further simplified to the transport-like Usadel equations describing diffusive motion of the Cooper pairs and the normal electrons. These equations now form the corner stone of many analyses of the dynamics of superconducting nanowires and SNS (superconductor–normal metal–superconductor) bridges. The purpose of this section is to provide a brief summary and some intuitive understanding of the transport-like Usadel equations, starting from the Gorkov formulation. These equations will be essential to understanding 1D nanowires, when either parts of the nanowires are driven into the normal state in a nonequilibrium situation, or else are connected to normal leads at one or multiple points along the nanowire.

The BCS Hamiltonian, which contains a contact interaction for the attraction between electrons, is written in the form

(1.41)

(1.42)

and

(1.43)

satisfy the following equations of motion:

(1.44)

where σ′ ≠ σ and the terms –μ have been added to measure the energy relative to the chemical potential μ. Defining the normal Green’s function, G, and anomalous Green’s function, F, which measures pair correlations

(1.45)

For these Green’s functions, their dynamical equations of motion form a closed set of coupled equations. The coupled equations are derived from the equations of motion for ψ and ψ†, making use of the same mean field approximation employed in the BCS solution as well as the BdG formulations to account for pair correlations. Defining a Green’s function matrix Ĝ,

(1.46)

and going over to imaginary time with t → –iτ, the dynamical equations of motion for the Green’s functions in matrix notation are found to be [30–32]

(1.47)

Here,

(1.48)

is the gap function.

The potential u(r) may include the random impurity potentials, as well as that from an applied electric potential ϕ. From these equations, many properties, including nonequilibrium properties, can be obtained. However, the equations are difficult to solve in realistic situations of relevance to experiment, except for special cases, such as thin films at weak perturbation of external drive.

(1.49)

Here, the choice of singling out the relative coordinate is predicated on the weak dependence of the Green’s functions on the center-of-mass coordinate (R ≡ (r + r′)/2), once averaged over impurity positions. (Note that Eilenberger introduced extra gauge potential phase factors in his definition of the Green’s functions, and thus these matrix equations are in a slightly different but equivalent form [34].) Following an average over the impurity positions, one arrives at a set of equations which are now first order in the covariant derivatives, rather than second order. In the absence of paramagnetic impurities that flip spins, the Gorkov equations become (from here on, we denote R by r)

(1.50)

where

(1.51)

and

(1.52)

(1.53)

A key observation significantly simplifies the situation. As it turns out, physical quantities of interest, such as the self-consistent gap equation

(1.54)

and the supercurrent density, and so on, can be written in terms of the energy integrated Green’s functions

(1.55)

Here, the wave vector k is to be taken to indicate the direction on the Fermi surface, or in essence kF, which points in the same direction as the vector k, and the energy integration must be taken as the principal value for large energies ξ. The energy-variable integrated Green’s functions satisfy the normalization condition

(1.56)

(1.57)

where the normalization condition now reads

(1.58)

the resultant equations of motion of these Green’s functions that are integrated over the energy variable are given by

(1.59)

where SF denotes the Fermi surface.

Defining the Fourier transform

(1.60)

and the inverse transform

(1.61)

the Eilenberger–Larkin–Ovchinnikov equations read

(1.62)

These equations are reminiscent of Boltzmann transport equations, albeit for quantities which are complex, and thus can account for quantum-mechanical interference. The second term on the right-hand side has the appearance of a collision integral. An additional equation for g(kF, ω; r) is redundant.

These equations are supplemented by the self-consistent equations for the gap Δ (r) and for the relation between the magnetic field and the supercurrent, which now become

(1.63)

Although these equations are substantially more convenient than the Gorkov equations, and are more amenable to numerical implementation for computing the physical quantities under realistic experimental conditions, one further development rendered this entire theoretical machinery far more tractable. Usadel, following the earlier work by Lüders [38–41], recognized that a crucial further simplification can be obtained in the case of a very dirty superconductor. Starting with the Eilenberger–Larkin–Ovchinnikov equations in the form of (1.62), Usadel noted that the large amount of scattering in a dirty system renders the Green’s functions nearly isotropic in space, and they can thus be written as the sum of a dominant isotropic part, plus a smaller part dependent on the direction on the Fermi surface. Based on this idea, and keeping the leading terms, Usadel transformed the equations into a diffusion equation valid in this dirty limit.

Separating to the dominant isotropic term and the Fermi velocity-dependent term

(1.64)

the quantities for g are expressible in terms of those for f via the normalization condition, which now, to leading order, reads

(1.65)

yielding, in addition,

(1.66)

Under the conditions

(1.67)

and of course , the resultant diffusion equation is given by

(1.68)

with the diffusion constant , where vF is the Fermi velocity. The transport time is given by the average of the scattering rate over the Fermi surface SF

(1.69)

1.6Path Integral Formulation

The techniques described thus far are powerful and have been extremely successful in describing most of the properties of conventional superconductors and the phenomena associated with them as well. However, in order to formulate a fully quantum theory that captures the physics of 1D superconducting nanowires in the dirty limit, in which the mean free path Lmfpξ0, where ξ0 is the clean limit coherence length, a path integral formulation has proven to be a convenient starting point. This formulation enables to go beyond their description close to Tc, and is also naturally suited to describe issues pertaining to quantum phase transitions, such as those occurring in Josephson junctions coupled to a dissipative environment [42–44], such as those occurring in arrays of Josephson junctions in 2D and 1D, or in vortex matter [45].

In the path integral formulation, the action plays a central role. Theoretical developments in the 1970s [42] enabled one to make a connection from the computation of quantum evolution in time, to the partition function, with the introduction of the imaginary time. The nonequilibrium case, such as under current or voltage bias, is also readily incorporated through the generalization of the Keldysh formulation, with the ordered imaginary-time integrals in the expansion of the path integral [42, 46–49]. More to the point, the computation of the low temperature behaviors of a 1D superconducting nanowire has required the use of path integral techniques based on instantons [50–52]. Instantons are saddle point solutions of the action and describe the quantum tunneling processes. Thus, such calculations yield the quantum tunneling rates for the phase slip.

Here, we summarize a formulation of conventional superconductors in a path integral formulation put forth by Otterlo, Golubev, Zaikin, and Blatter [49], which has developed out of earlier works [42, 46–48]. Such an approach not only reproduces the static results of the BCS theory, but enables the treatment of dynamical responses, including relaxation, collective modes, particularly the Carlson-Goldman [53] and Mooij–Schön modes [54], the latter being of direct relevance to 1D superconducting nanowires. Topological defects, such as the motion and the tunneling of vortices, as well as the quantum tunneling of phase slips in one dimension, are also included within this formalism. The Mooij–Schön mode will emerge to play a central role in the dynamics of phase slips and will be discussed in Chapter 3. The standard starting point to compute the partition function in the BCS model is written in terms of the path integral in the imaginary-time formulation [42, 49]

(1.70)

(1.71)

(1.72)

where

(1.73)

and

(1.74)

Ĝ denotes the Green’s function matrix in Nambu space:

(1.75)

while the inverse matrix

(1.76)

The trace Tr is taken over the matrix in Nambu space and also over internal coordinates of frequency and momenta, and the curly brackets {…,…} denote an anticommutator. In addition, the gauge invariant linear combinations of the electromagnetic fields and the order parameter phase φ are introduced:

(1.77)

The action may be expanded in terms of these gauge invariant quantities about a saddle point delineated below.

The Green’s functions in space and real time are defined as

(1.78)

The variation of (1.72) with respect to V, A, and Δ, respectively, yields the equations describing the Thomas–Fermi screening, London screening, and the BCS-gap equations.

(1.79)

for the saddle point solution, such that

(1.80)

and

(1.81)

The trace of the natural log of the inverse Green’s function can thus be expanded as

(1.82)

From here on, various treatments diverge based on the assumption of Galilean invariance [46–48], gauge invariance, and so on [49]. In the formulation due to Otterlo et al., by using Ward identities [49, 55] due to gauge invariance and charge and particles number conservation, they recast the effective action into four contributions:

(1.83)

To express these action terms explicitly, we make use of the gauge invariant quantities

(1.84)

(1.85)

where the approximation holds for small fluctuations, that is, |Δ1| Δ0. The second order expansion leads to to terms of the form tr[GOG′O]. Here, G and G′ symbolize any of the Green functions G, , F, or , and O is an operator. To evaluate the traces over the internal coordinates, tr[GOG′O′], we define

(1.86)

(1.87)

where B denotes a function of frequency-momentum p and q. For example,

(1.88)

(1.89)

The four contributions to the effective action can now be written as

(1.90)

(1.91)

(1.92)

and

(1.93)