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

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Andreev Billiards 161 Fig. 17. Histograms: smoothed density of states of a billiard coupled by a ballistic N-mode lead to a superconductor, determined by (22) and averaged over a range of Fermi energies at fixed N. The scattering matrix is computed numerically by matching wave functions in the billiard to transverse modes in the lead. A chaotic Sinai billiard (top inset, solid histogram, N = 20) is contrasted with an integrable circular billiard (bottom inset, dashed histogram, N =30).Thesolid curve is the prediction (49) from RMT for a chaotic system and the dashed curve is the Bohr- Sommerfeld result (79), with dwell time distribution P (T ) calculated from classical trajectories in the circular billiard. Adapted from [45] 7.3 Chaotic Dynamics A chaotic billiard has an exponential dwell time distribution, P (T ) ∝ e −T/τdwell , instead of a power law [68]. (The mean dwell time is τdwell = 2π�/N δ ≡ �/2ET .) Substitution into the Bohr-Sommerfeld rule (79) gives the density of states [71] 〈ρ(E)〉 = 2 δ (πET /E) 2 cosh(πET /E) sinh 2 , (83) (πET /E) which vanishes ∝ e −πET /E as E → 0. This is a much more rapid decay than for integrable systems, but not quite the hard gap predicted by RMT [22]. The two densities of states are compared in Fig. 18. When the qualitative difference between the random-matrix and Bohr- Sommerfeld theories was discovered [22], it was believed to be a short-coming of the quasiclassical approximation underlying the latter theory. Lodder and Nazarov [23] realized that the two theoretical predictions are actually both correct, in different limits. As the ratio τE/τdwell of Ehrenfest time and dwell time is increased, the density of states crosses over from the RMT form (49) to
162 C.W.J. Beenakker 〈ρ(E)〉 δ/2 1 0.5 RMT BS 0 0 0.5 1 1.5 E/ET Fig. 18. Comparison of the smoothed density of states in a chaotic Andreev billiard as it follows from RMT ((49), with a hard gap) and as it follows from the Bohr-Sommerfeld (BS) rule ((83), without a hard gap). These are the two limiting distributions when the Ehrenfest time τE is, respectively, much smaller or much larger than the mean dwell time τdwell the Bohr-Sommerfeld form (83). We investigate this crossover in the following section. 8 Quantum-To-Classical Crossover 8.1 Thouless Versus Ehrenfest According to Ehrenfest’s theorem, the propagation of a quantum mechanical wave packet is described for short times by classical equations of motion. The time scale at which this correspondence between quantum and classical dynamics breaks down in a chaotic system is called the Ehrenfest time τE [72]. 5 As explained in Fig. 19, it depends logarithmically on Planck’s constant: τE = α −1 ln(Scl/h), with Scl the characteristic classical action of the dynamical system and α the Lyapunov exponent. This logarithmic h-dependence distinguishes the Ehrenfest time from other characteristic time scales of a chaotic system, which are either h-independent (dwell time, ergodic time) or algebraically dependent on h (Heisenberg time ∝ 1/δ). That the quasiclassical theory of superconductivity breaks down on time scales greater than τE was noticed already in 1968 by Larkin and Ovchinnikov [74]. The choice of Scl depends on the physical quantity which one is studying. For the density of states of the Andreev billiard (area A, opening of width W ≪ A 1/2 , range of transverse momenta ∆p � pF inside the constriction) 5 The name “Ehrenfest time” was coined in [73].
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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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