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

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Andreev Billiards 167 The scattering time of the stochastic model plays the role of the Ehrenfest time in the deterministic chaotic dynamics. The advantage of a stochastic description is that one can average over different realizations of the disorder potential. This provides for an established set of analytical techniques. The disadvantage is that one does not know how well stochastic scattering mimics quantum diffraction. Vavilov and Larkin [40] have used the stochastic model to study the crossover from the Thouless regime to the Ehrenfest regime in an Andreev billiard. They discovered that the rapid turn-on of quantum diffraction at τE > ∼ τdwell not only causes an excitation gap to open at �/τE, but that it also causes oscillations with period �/τE in the ensemble-averaged density of states 〈ρ(E)〉 at high energies E > ∼ ET . In normal billiards oscillations with this periodicity appear in the level-level correlation function [77], but not in the level density itself. The predicted oscillatory high-energy tail of 〈ρ(E)〉 is plotted in Fig. 23,for the case τE/τdwell = 3, together with the smooth results of RMT (τE/τdwell → 0) and Bohr-Sommerfeld (BS) theory (τE/τdwell →∞). 1.02 1.01 1.00 10 20 E/ET 30 E BS RMT dwell Fig. 23. Oscillatory density of states at finite Ehrenfest time (solid curve), compared with the smooth limits of zero (RMT) and infinite (BS) Ehrenfest times. The solid curve is the result of the stochastic model of Vavilov and Larkin, for τE =3τdwell = 3�/2ET . (The definition (84) of the Ehrenfest time used here differs by a factor of two from that used by those authors.) Adapted from [40] Independent analytical support for the existence of oscillations in the density of states with period �/τE comes from the singular perturbation theory of [78]. Support from numerical simulations is still lacking. Jacquod et al. [27] did find pronounced oscillations for E > ∼ ET in the level density of the Andreev kicked rotator. However, since these could be described by the Bohr- Sommerfeld theory they can not be the result of quantum diffraction, but must be due to nonergodic trajectories [79]. / = 3
168 C.W.J. Beenakker The τE-dependence of the gap obtained by Vavilov and Larkin is plotted in Fig. 21 (dashed curve). It is close to the result of the effective RMT (solid curve). The two theories predict the same limit Egap → π�/2τE for τE/τdwell →∞. The asymptotes given in [40] are Egap = γ5/2 � � � τE 1 − 0.23 , τE≪τdwell , (92) τdwell 2τdwell Egap = π� � 1 − 2τE 2τdwell � , τE≫τdwell . (93) τE Both are different from the results (88) and (89) of the effective RMT. 8 8.4 Numerical Simulations Because the Ehrenfest time grows only logarithmically with the size of the system, it is exceedingly difficult to do numerical simulations deep in the Ehrenfest regime. Two simulations [27, 80] have been able to probe the initial decay of the excitation gap, when τE < ∼ τdwell. We show the results of both simulations in Fig. 24 (closed and open circles), together with the full decay as predicted by the effective RMT of Sect. 8.2 (solid curve) and by the stochastic model of Sect. 8.3 (dashed curve). The closed circles were obtained by Jacquod et al. [27] using the stroboscopic model of Sect. 5 (the Andreev kicked rotator). The number of modes N in the contact to the superconductor was increased from 10 2 to 10 5 at fixed dwell time τdwell = M/N = 5 and kicking strength K = 14 (corresponding to a Lyapunov exponent α ≈ ln(K/2) = 1.95). In this way all classical properties of the billiard remain the same while the effective Planck constant heff =1/M =1/N τdwell is reduced by three orders of magnitude. To plot the data as a function of τE/τdwell, (84) was used for the Ehrenfest time. The unspecified terms of order unity in that equation were treated as a single fit parameter. (This amounts to a horizontal shift by −0.286 of the data points in Fig. 24.) The open circles were obtained by Kormányos et al. [80] for the chaotic Sinai billiard shown in the inset. The number of modes N was varied from 18 to 30 by varying the width of the contact to the superconductor. The Lyapunov exponent α ≈ 1.7 was fixed, but τdwell was not kept constant in this simulation. The Ehrenfest time was computed by means of the same formula (84), with M =2LckF /π and Lc the average length of a trajectory between two consecutive bounces at the curved boundary segment. The data points from both simulations have substantial error bars (up to 10%). Because of that and because of their limited range, we can not conclude that the simulations clearly favor one theory over the other. 8 Since 2γ − 1=0.236, the small-τE asymptote of Vavilov and Larkin differs by a factor of two from that of the effective RMT.
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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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6.7 Unresolved Issues Semiconductor
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