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

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Andreev Billiards 139 For energies 0
140 C.W.J. Beenakker A stroboscopic model of an Andreev billiard was constructed by Jacquod et al. [27], building on an existing model for open normal billiards called the open kicked rotator [28]. The Andreev kicked rotator possesses the same phenomenology as the Andreev billiard, but is much more tractable numerically. 3 In this subsection we discuss how it is formulated. Some results obtained by this numerical method will be compared in subsequent sections with results obtained by analytical means. A compact unitary map is represented in quantum mechanics by the Floquet operator F , which gives the stroboscopic time evolution u(pτ0) =F pu(0) of an initial wave function u(0). (We set the stroboscopic period τ0 =1in most equations.) The unitary M × M matrix F has eigenvalues exp(−iεm), with the quasi-energies εm ∈ (−π, π) (measured in units of �/τ0). This describes the electron excitations above the Fermi level. Hole excitations below the Fermi level have Floquet operator F ∗ and wave function v(p) =(F∗ ) pv(0). The mean level spacing of electrons and holes separately is δ =2π/M. An electron is converted into a hole by Andreev reflection at the NS interface, with phase shift −i for ε ≪ τ0∆/� [cf. (20)]. In the stroboscopic description one assumes that Andreev reflection occurs only at times which are multiples of τ0. TheN × M matrix P projects onto the NS interface. Its elements are Pnm =1ifm = n ∈{n1,n2,...nN } and Pnm = 0 otherwise. The dwell time of a quasiparticle excitation in the normal metal is τdwell = M/N, equal to the mean time between Andreev reflections. Putting all this together one constructs the quantum Andreev map from the matrix product � F 0 F = P 0 F ∗ � � T T 1 − P P −iP P , P = −iP TP 1 − P T � . (23) P (The superscript “T” indicates the transpose of a matrix.) The particle-hole wave function Ψ =(u, v) evolves in time as Ψ(p) =F pΨ(0). The Floquet operator can be symmetrized (without changing its eigenvalues) by the unitary transformation F→P−1/2FP1/2 , with � √ � 1 T 1 1 − (1 − 2)P P −i P 1/2 = −i 1 2 √ T 2 2 2P P √ √ T 1 T 2P P 1 − (1 − 2 2)P P . (24) The quantization condition det(F −e−iε ) = 0 can be written equivalently as [27] Det [1 + S(ε)S(−ε) ∗ ]=0, (25) in terms of the N × N scattering matrix [28, 29] S(ε) =P [e −iε − F (1 − P T P )] −1 FP T . (26) 3 The largest simulation to date of a two-dimensional Andreev billiard has N =30, while for the Andreev kicked rotator N =10 5 is within reach, cf. Fig. 24.
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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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10 5 0 -5 Semiconductor Few-Electro
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G ( e / h) 2 2 1 0 Semiconductor Fe
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V R (V) V R (V) Semiconductor Few-E
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V L (V) -1.084 -1.082 -1.080 -1.078
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B // Semiconductor Few-Electron Qua
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a RESERVOIR 200 nm Semiconductor Fe
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4.5 QPC Versus SET Semiconductor Fe
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a reservoir dot Semiconductor Few-E
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a b c V P (mV) 10 5 0 ∆I QPC E F
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Spin-down counts 200 100 a Semicond
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Page 155 and 156: 146 C.W.J. Beenakker Fig. 6. Ensemb
Page 157 and 158: 148 C.W.J. Beenakker 1.4 1.2 1.0 0.
Page 159 and 160: 150 C.W.J. Beenakker 〈ρ(E)〉 =
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Page 165 and 166: 156 C.W.J. Beenakker tunnel/ Egap 4
Page 167 and 168: 158 C.W.J. Beenakker its momentum).
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Page 171 and 172: 162 C.W.J. Beenakker 〈ρ(E)〉 δ
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