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

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Andreev Billiards 165 form (30) – with effective parameters Neff and δeff. The determinant (22) for the excitation spectrum then takes the form � Det 1 − α(E) 2 e 2iEτE/� S0(E)S0(−E) ∗� =0. (87) We can safely replace α(E) ≡ exp[−i arccos(E/∆)] →−i (since E ≪ ∆), but the energy dependence of the phase factor e2iEτE/� can not be omitted. In App. A we calculate the smallest positive E that solves (87), which is the excitation gap Egap of the effective RMT. The result is plotted in Fig. 21 (solid curve), as a function of τE/τdwell. The two asymptotes (dotted lines) are Egap = γ5/2 � � 1 − (2γ − 1) τdwell τE � , τE≪τdwell , (88) τdwell Egap = π� � 1 − (3 + 2τE √ 8) τdwell � , τE≫τdwell , (89) τE with γ = 1 2 (√5 − 1) the golden number. Fig. 21. Excitation gap of the Andreev billiard in the crossover from Thouless to Ehrenfest regimes. The solid curve is the solution of the effective RMT of [61], derived in App. A. Thedotted lines are the two asymptotes (88) and(89). The dashed curve is the result of the stochastic model of [40], discussed in Sect. 8.3 The τE time delay characteristic of the effective RMT was introduced in [61], but its effect on the excitation gap was not evaluated properly. 7 As a consequence the formula for the gap given in that paper, 7 I am indebted to P. W. Brouwer for spotting the error.
166 C.W.J. Beenakker 0.30 � Egap = = 〈T 〉∗ 0.30 � , (90) τE + τdwell provides only a qualitative description of the actual crossover. The inverse correlation (90) between the gap and the dwell time of long trajectories was observed in a computer simulation of the Andreev kicked rotator [75]. The data points in Fig. 22 track the excitation gap as the location in phase space of the NS interface is varied. The solid curve is a plot of 1 = , (91) � ∞ T 〈T 〉∗ ∗ P (T )dT � ∞ T ∗ TP(T )dT with P (T ) the classical dwell time distribution and T ∗ = 7. We see that the sample-to-sample fluctuations in the gap correlate very well with the fluctuations in the mean dwell time of long trajectories. The correlation is not sensitive to the choice of T ∗ ,aslongasitisgreaterthanτE =4.4. 0.54 E gap E T 0.52 0.5 0.084 〈T〉 * 0.082 0 0.5 lead position 0.08 1 Fig. 22. The data points (left axis) are the quantum mechanical gap values Egap of the Andreev kicked rotator as a function of the location of the NS interface, for parameter values M = 131072, τdwell = M/N =5,K = 14. The solid curve (right axis) is the reciprocal of the mean dwell time (91) of classical trajectories longer than T ∗ = 7. Adapted from [75] 8.3 Stochastic Model Small-angle scattering by a smooth disorder potential provides a stochastic model for the quantum diffraction of a wave packet in a chaotic billiard [76]. 1
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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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Page 155 and 156: 146 C.W.J. Beenakker Fig. 6. Ensemb
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Page 171 and 172: 162 C.W.J. Beenakker 〈ρ(E)〉 δ
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Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

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