Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

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Andreev Billiards 133 Fig. 2. Quantum dot (central square of dimensions 500 nm×500 nm) fabricated in a high-mobility InAs/AlSb heterostructure and contacted by four superconducting Nb electrodes. Device made by A. T. Filip, Groningen University (unpublished figure) below the gap ∆ in the bulk superconductor (hence the name “minigap”). It may also take the form of a level density that vanishes smoothly (typically linearly) upon approaching the Fermi level, without an actual gap. The presence or absence of a gap is a quantum signature of chaos. That is a fundamental difference between normal billiards and Andreev billiards, since in a normal billiard the level density can not distinguish chaotic from integrable classical dynamics. (It depends only on the area of the billiard, not on its shape.) A powerful technique to determine the spectrum of a chaotic system is random-matrix theory (RMT) [3, 9, 10]. Although the appearance of an excitation gap is a quantum mechanical effect, the corresponding time scale �/Egap as it follows from RMT is a classical (meaning �-independent) quantity: It is the mean time τdwell that an electron or hole excitation dwells in the billiard between two subsequent Andreev reflections. A major development of the last few years has been the discovery of a competing quantum mechanical time scale τE ∝|ln �|. (The subscript E stands for Ehrenfest.) RMT breaks down if τE > ∼ τdwell and a new theory is needed to determine the excitation gap in this regime. Two different phenomenological approaches have now reached a consistent description of the τE-dependence of the gap, although some disagreement remains. The plan of this review is as follows. The next four sections contain background material on Andreev reflection (Sect. 2), on the minigap in NS junctions (Sect. 3), on the scattering theory of Andreev billiards (Sect. 4), and on a stroboscopic model used in computer simulations (Sect. 5). The regime of RMT (when τE ≪ τdwell) is described in Sect. 6 and the quasiclassical regime (when τE ≫ τdwell) is described in Sect. 7. The crossover from Egap � �/τdwell to Egap � �/τE is the topic of Sect. 8. We conclude in Sect. 9.
134 C.W.J. Beenakker 2 Andreev Reflection The quantum mechanical description of Andreev reflection starts from a pair of Schrödinger equations for electron and hole wave functions u(r) andv(r), coupled by the pair potential ∆(r). These socalled Bogoliubov-De Gennes (BdG) equations [11] take the form � � � � u u HBG = E , (1) v v � H ∆(r) HBG = ∆∗ (r) −H∗ � . (2) The Hamiltonian H =(p + eA) 2 /2m + V − EF is the single-electron Hamiltonian in the field of a vector potential A(r) and electrostatic potential V (r). The excitation energy E is measured relative to the Fermi energy EF .If(u, v) is an eigenfunction with eigenvalue E, then (−v∗ ,u∗ ) is also an eigenfunction, with eigenvalue −E. The complete set of eigenvalues thus lies symmetrically around zero. The quasiparticle excitation spectrum consists of all positive E. In a uniform system with ∆(r) ≡ ∆, A(r) ≡ 0, V (r) ≡ 0, the solution of the BdG equations is E = � (� 2 k 2 /2m − EF ) 2 + ∆ 2� 1/2 , (3) u(r) =(2E) −1/2 � E + � 2 k 2 /2m − EF � 1/2 e ik·r , (4) � 1/2 e ik·r . (5) v(r) =(2E) −1/2 � E − � 2 k 2 /2m + EF The excitation spectrum is continuous, with excitation gap ∆. The eigenfunctions (u, v) are plane waves characterized by a wavevector k. The coefficients of the plane waves are the two coherence factors of the BCS (Bardeen-Cooper- Schrieffer) theory. At an interface between a normal metal and a superconductor the pairing interaction drops to zero over atomic distances at the normal side. (We assume non-interacting electrons in the normal region.) Therefore, ∆(r) ≡ 0inthe normal region. At the superconducting side of the NS interface, ∆(r) recovers its bulk value ∆ only at some distance from the interface. This suppression of ∆(r) is neglected in the step-function model � ∆ if r ∈ S, ∆(r) = (6) 0 if r ∈ N. The step-function pair potential is also referred to in the literature as a “rigid boundary condition” [12]. It greatly simplifies the analysis of the problem without changing the results in any qualitative way. Since we will only be considering a single superconductor, the phase of the superconducting order parameter is irrelevant and we may take ∆ real. We refer to [13] for a tutorial introduction to mesoscopic Josephson junctions, such as a quantum dot connected to two superconductors, and to [14] fora comprehensive review of the current-phase relation in Josephson junctions.
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Lecture Notes in Physics Editorial
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W. Dieter Heiss (Ed.) Quantum Dots:
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Preface Nanoscale physics, nowadays
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Contents A Guide for the Reader ...
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A Guide for the Reader Quantum dots
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The Renormalization Group Approach
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RG for Interacting Fermions 5 where
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where � is a shorthand: � ≡ 3
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RG for Interacting Fermions 9 These
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RG for Interacting Fermions 11 is i
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RG for Interacting Fermions 13 the
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RG for Interacting Fermions 15 this
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RG for Interacting Fermions 17 equa
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� p � dθp = g 2π RG for Inter
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RG for Interacting Fermions 21 the
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References RG for Interacting Fermi
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Semiconductor Few-Electron Quantum
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1.2 Implementations Semiconductor F
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GaAs n-AlGaAs AlGaAs GaAs Semicondu
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ε1 ε0 e Semiconductor Few-Electro
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250 Ω twisted pair 20 MΩ powder
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V R (V) V R (V) Semiconductor Few-E
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a RESERVOIR 200 nm Semiconductor Fe
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a reservoir dot Semiconductor Few-E
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