Superconductivity in Nanowires E-Book

Alexey Bezryadin

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

Seitenzahl: 585

Veröffentlichungsjahr: 2013

Beschreibung

The importance and actuality of nanotechnology is unabated and will be for years to come. A main challenge is to understand the various properties of certain nanostructures, and how to generate structures with specific properties for use in actual applications in Electrical Engineering and Medicine.

One of the most important structures are nanowires, in particular superconducting ones. They are highly promising for future electronics, transporting current without resistance and at scales of a few nanometers. To fabricate wires to certain defined standards however, is a major challenge, and so is the investigation and understanding of these properties in the first place. A promising approach is to use carbon nanotubes as well as DNA structures as templates.

Many fundamental theoretical questions are still unanswered, e.g. related to the role of quantum fluctuations. This work is tackling them and provides a detailed analysis of the transport properties of such ultrathin wires. It presents an account of theoretical models, charge transport experiments, and also conveys the latest experimental findings regarding fabrication, measurements, and theoretical analysis. In particular, it is the only available resource for the approach of using DNA and carbon nanotubes for nanowire fabrication.

It is intended for graduate students and young researchers interested in nanoscale superconductivity. The readers are assumed to have knowledge of the basics of quantum mechanics and superconductivity.

Leseprobe

Contents

Preface – Superconductivity and Little’s Phase Slips in Nanowires

Abbreviations – Short List

Notations – Short List

1 Introduction

2 Selected Theoretical Topics Relevant to Superconducting Nanowires

2.1 Free or Usable Energy of Superconducting Condensates

2.2 Helmholtz and Gibbs Free Energies

2.3 Fluctuation Probabilities

2.4 Work Performed by a Current Source on the Condensate in a Thin Wire

2.5 Helmholtz Energy of Superconducting Wires

2.6 Gibbs Energy of Superconducting Wires

2.7 Relationship between Gibbs and Helmholtz Energy Densities

2.8 Relationship between Thermal Fluctuations and Usable Energy

2.9 Calculus of Variations

2.10 Ginzburg–Landau Equations

2.11 Little–Parks Effect

2.12 Kinetic Inductance and the CPR of a Thin Wire

2.13 Drude Formula and the Density of States

3 Stewart–McCumber Model

3.1 Kinetic Inductance and the Amplitude of Small Oscillations

3.2 Mechanical Analogy for the Stewart–McCumber Model

3.3 Macroscopic Quantum Phenomena in the Stewart–McCumber Model

3.4 Schmid–Bulgadaev Quantum Phase Transition in Shunted Junctions

3.5 Stewart–McCumber Model with Normalized Variables

4 Fabrication of Nanowires Using Molecular Templates

4.1 Choice of Templating Molecules

4.2 DNA Molecules as Templates

4.3 Significance of the So-Called “White Spots”

5 Experimental Methods

5.1 Sample Installation

5.2 Electronic Transport Measurements

6 Resistance of Nanowires Made of Superconducting Materials

6.1 Basic Properties

6.2 Little’s Phase Slips

6.3 Little’s Fit

6.4 LAMH Model of Phase Slippage at Low Bias Currents

6.5 Comparing LAMH and Little’s Fit

7 Golubev and Zaikin Theory of Thermally Activated Phase Slips

8 Stochastic Premature Switching and Kurkijärvi Theory

8.1 Stochastic Switching Revealed by V–I Characteristics

8.2 “Geiger Counter” for Little’s Phase Slips

8.3 Measuring Switching Current Distributions

8.4 Kurkijärvi–Fulton–Dunkleberger (KFD) Transformation

8.5 Examples of Applying KFD Transformations

8.6 Inverse KFD Transformation

8.7 Universal 3/2 Power Law for Phase Slip Barrier

8.8 Rate of Thermally Activated Phase Slips at High Currents

8.9 Kurkijärvi Dispersion Power Laws of 2/3 and 1/3

9 Macroscopic Quantum Tunneling in Thin Wires

9.1 Giordano Model of Quantum Phase Slips (QPS) in Thin Wires

9.2 Experimental Tests of the Giordano Model

9.3 Golubev and Zaikin QPS Theory

9.4 Khlebnikov Theory

9.5 Spheres of Influence of QPS and TAPS Regimes

9.6 Kurkijärvi–Garg Model

9.7 Theorem: Inverse Relationship between Dispersion and the Slope of the Switching Rate Curve

10 Superconductor–Insulator Transition (SIT) in Thin and Short Wires

10.1 Simple Model of SIT in Thin Wires

11 Bardeen Formula for the Temperature Dependence of the Critical Current

Appendix A: Superconductivity in MoGe Alloys

Appendix B: Variance and the Variance Estimator

Appendix C: Problems and Solutions

References

Index

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The Author

Prof. Alexey Bezryadin

Department of Physics

University of Illinois

1110 West Green Street

Champaign IL 61801

USA

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Preface – Superconductivity and Little’s Phase Slips in Nanowires

This book presents an account of fabrication techniques, charge transport experiments, and theoretical models related to superconducting nanowires, that is, thin wires made of superconducting materials and having dimensions at the nanometer scale. A wire is classified as one-dimensional (1D) if its diameter d is smaller than the superconducting coherence length ξ. In such wires, the superconducting condensate wavefunction only depends on the position along the wire, while it is independent of the position within the wire cross section. A nanowire is classified as quasi-one-dimensional (quasi-1D) if . Such condition ensures that vortices are not energetically stable in the wire. Therefore, the order parameter is approximately constant within the wire cross section. Since ξs diverges at the critical temperature Tc, it is not too hard to make a wire which is quasi-1D near Tc. It is more challenging to fabricate nanowires which are 1D or quasi-1D even in the limit of zero temperature. Recent perfection of nanotechnology allows us to make wire as thin as 5 nm, which can be regarded as quasi-1D even at zero temperature. This book provides a detailed analysis of the transport properties of such ultrathin wires. Among other things, the small size of the wires causes them to behave as quantum objects. Due to the Heisenberg uncertainty principle, such wires might not be true superconductors. Their state becomes uncertain because the wires exist in a quantum superposition of normal and superconducting states. As the admixture of the normal phase increases, nominally superconducting nanowires can lose their ability to carry supercurrent and can become either normal or, what it more typical, slightly insulating.

Even if quantum fluctuations are not sufficiently strong to destroy superconductivity, the ohmic resistance R of a thin wire is always greater than zero, provided that the temperature, T, is above absolute zero. Due to thermal fluctuations, the phase of the superconducting condensate wavefunction jumps irreversibly, leading to a decay of the supercurrent. Such jumps are called thermally activated phase slips (TAPS) or Little’s phase slips (LPS). A detailed discussion of LPS is one of the main topics of this book.

The book is intended for undergraduate and beginning graduate students interested in nanoscale superconductivity. The readers are assumed to have some familiarity with the basics of quantum mechanics and superconductivity. Recommended introductory books on general aspects of superconductivity are Introduction to Superconductivity by M. Tinkham [1] and Superconductivity of Metals and Alloys by P.G. de Gennes [2]. The present monograph is focused on selected topics related to thin superconducting nanowires.

It is my pleasure to acknowledge my indebtedness to the graduate students and postdocs I have supervised since 2000. Of special importance is the advice and the helpful discussions with Liliya Simpson, Anthony T. Bollinger, Ulas C. Coskun, Celia Elliott, David S. Hopkins, Robert C. Dinsmore III, Mitrabhanu Sahu, Sergei Khlebnikov, Thomas Aref, Matthew W. Brenner, Paul Goldbart, Andrey Rogachev, Andrey Belkin, Andrey Zaikin, Konstantin Arutyunov, and Victor Vakaryuk. The work was supported by the DOE grant DE-FG02-07ER46453 and by the NSF grant DMR 10-05645.

Campaign, August 2012

Alexey Bezryadin

Abbreviations – Short List

BCS

Bardeen, Copper, and Schrieffer theory of superconductivity.

BCS condensate

a coherent quantum states of a macroscopic number of electrons appearing in the BCS theory.

CP

Cooper pair.

CPR

Current–phase relationship.

GL

Ginzburg and Landau phenomenological theory.

GLAG

Ginzburg, Landau, Abrikosov, and Gor’kov theory, which is a microscopic theory derived from BCS, but similar in form to the GL theory.

GZ

Golubev and Zaikin theory (either of TAPS or of QPS).

I-type wire

an insulating nanowire, i.e., a nanowire made of a superconducting metal, but, due to various reasons, having its zerotemperature resistance larger than its normal-state resistance.

JJ

Josepshon junction.

KG

Kurkijärvi–Garg model.

LAMH

Langer, Ambegaokar, McCumber, and Halperin combined model.

LPS

Little’s phase slip.

MT

molecular templating.

MQT

Macroscopic quantum tunneling.

N-type wire

A normal nanowire, that is, a nanowire made of a superconducting metal, but, due to various reasons, having its zero-temperature resistance larger than zero but lower or equal compared to its normal-state resistance, which is measured at

T

>

T

c.

QPS

quantum phase slips, usually in quasi-1D superconducting wires. QPS is a particular case of MQT.

SB

Schmid–Bulgadaev quantum phase transition.

SIS

Superconductor–insulator–superconductor junction.

SNS

Superconductor–normal metal–superconductor junction.

SIT

Superconductor to insulator quantum phase transition. The meaning of it is that on the S-side the wire has zero resistance at zero temperature, while on the I-side of the transition the wire acts as an insulator or a bad conductor.

SNT

superconductor–normal quantum phase transition.

SRT

“superconducting” to “resistive” quantum phase transition. The meaning of the phase transition is that in the S-phase the wire has zero resistance at zero temperature, while in the R-phase the wire is resistive. SNT and SIT are examples or particular cases of the more general SRT.

S-type wire

A nanowire characterized by zero resistance at zero temperature.

SM

Stewart and McCumber model, predicting the washboard potential.

TAPS

Thermally activated phase slips, that is, Little’s phase slips energized by thermal fluctuations.

Notations – Short List

1

Introduction

In 1913, the Nobel Prize in Physics was awarded to Heike Kamerlingh Onnes “for his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium.” His crowning achievement was the liquefaction of helium in 1908, which pioneered a new era in low-temperature physics and enabled him to discover superconductivity [3] in 1911.

The theory of superconductivity was developed by Bardeen, Cooper, and Schrieffer (BCS) in 1957 [4]. “In the competitive world of theoretical physics, the BCS theory was the triumphant solution of a long-standing riddle. Between 1911 and 1957, all the best theorists in the world . . . had tried and failed to explain superconductivity.” [5]. The path to the development of the theory of superconductivity was cleared by the pioneering work of Bardeen and Pines [6], who examined the superconducting isotope effect, took into account the electron–phonon interactions, and determined that electrons could overcome the Coulomb repulsion and attract each other. This weak attraction between the electrons inside the superconducting material is the key to explaining the condensation of electrons and their superconductivity.

In traditional superconductors, the electron–electron attraction, which translates into a small but noticeable reduction of the total energy, occurs between two electrons having opposite wavevectors and opposite spins. At low temperatures, because of the electron–electron attraction, the electrons form a condensate, which is a collective bound state having zero entropy and a reduced total energy, compared with the zero-temperature Fermi distribution energy. This reduction of energy is achieved by allowing, even at zero temperature, for some fraction of the electrons in the superconducting material to have momenta larger than the Fermi momentum, thus maximizing their interaction potential.

Starting from the BCS theory, it was rigorously derived [7] that the condensate can be described by a collective wavefunction, also called superconducting order parameter. The order parameter provides the most complete description of the ensemble of superconducting electrons and depends on three spatial coordinates only, as

(1.1)

The normalization of the wavefunction is chosen such that the square of the absolute value of the wavefunction equals half the local density of the condensed electrons ncond(r, t).1) Like the single-particle wavefunction, the collective condensate wavefunction is a complex function of the radius-vector r and time t. The fact that the effective wavefunction of many electrons only depends on one radius-vector r reflects how all electrons behave “coherently,” that is, as a single particle.

If the gradient of the phase of the order parameter is not equal to zero (i.e., if φ(r, t) ≠ 0), then the condensate “flows,” that is, it carries a nonzero electrical current, called “supercurrent.” (Remember that the gradient operator is a vector having three components given by the spatial partial derivative operators ∂/∂x, ∂/∂y, ∂/∂z, where x, y, and z are the Cartesian coordinates.) In some sense, the BCS condensate acts as a huge “macromolecule” of electrons. As with actual molecules, one needs to perform a positive work on the condensate to free an electron from such a huge electronic macromolecule. This work is called the superconducting energy gap Δ. The energy of the condensate is reduced due to the attractive interactions between the electrons. The most important property of the BCS condensate is that it can flow through the lattice of positively charged ions without friction. This happens because slowing the entire bound state of all electrons is much more difficult than slowing down single unbound electrons, which exist in normal (i.e., nonsuperconducting) metals.