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

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Andreev Billiards 163 Fig. 19. Two trajectories entering a chaotic billiard at a small separation δx(0) diverge exponentially in time, δx(t) =δx(0)e αt . The rate of divergence α is the Lyapunov exponent. An initial microscopic separation λF becomes macroscopic at the Ehrenfest time τE = α −1 ln(L ∗ /λF ). The macroscopic length L ∗ is determined by the size and shape of the billiard. The Ehrenfest time depends logarithmically on Planck’s constant: τE = α −1 ln(Scl/h), with Scl = mvF L ∗ the characteristic classical action. The evolution of a quantum mechanical wave packet is well described by a classical trajectory only for times less than τE the characteristic classical action is 6 Scl = mvF W 2 /A 1/2 [40]. The Ehrenfest time then takes the form τE = α −1 [ln(N 2 /M )+O(1)] . (84) Here M = kF A 1/2 /π and N = kF W/π are, respectively, the number of modes in a cross-section of the billiard and in the point contact. Equation (84) holds for N > √ √ ∼ M.ForN < ∼ M the Ehrenfest time may be set to zero, because the wave packet then spreads over the entire billiard within the ergodic time [62]. Chaotic dynamics requires α −1 ≪ τdwell. The relative magnitude of τE and τdwell thus depends on whether the ratio N 2 /M is large or small compared to the exponentially large number e ατdwell . The result of RMT [22], cf. Sect. 6.2, is that the excitation gap in an Andreev billiard is of the order of the Thouless energy ET � �/τdwell. It was realized by Lodder and Nazarov [23] that this result requires τE ≪ τdwell. More generally, the excitation gap Egap � min (ET , �/τE) is determined by the smallest of the Thouless and Ehrenfest energy scales. The Bohr-Sommerfeld theory [22], cf. Sect. 7.3, holds in the limit τE →∞and therefore produces a gapless density of states. 8.2 Effective RMT A phenomenological description of the crossover from the Thouless to the Ehrenfest regime is provided by the “effective RMT” of Silvestrov et al. [61]. 6 The simpler expression Scl = mvF W of [61] applies to the symmetric case W/A 1/2 � ∆p/pF ≪ 1.
164 C.W.J. Beenakker As described in Sect. 7.1, the quasiclassical adiabatic quantization allows to quantize only the trajectories with periods T ≤ T0 ≡ τE. The excitation gap of the Andreev billiard is determined by the part of phase space with periods longer than τE. Effective RMT is based on the hypothesis that this part of phase space can be quantized by a scattering matrix Seff in the circular ensemble of RMT, with a reduced dimensionality � ∞ Neff = N P (T ) dT = Ne −τE/τdwell . (85) τE The energy dependence of Seff(E) is that of a chaotic cavity with mean level spacing δeff, coupled to the superconductor by a long lead with Neff propagating modes. (See Fig. 20.) The lead introduces a mode-independent delay time τE between Andreev reflections, to ensure that P (T ) is cut off for TτE between Andreev reflections is represented by a chaotic cavity (mean level spacing δeff), connected to the superconductor by a long lead (Neff propagating modes, one-way delay time τE/2 for each mode). Between two Andreev reflections an electron or hole spends, on average, a time τE in the lead and a time τdwell in the cavity. The scattering matrix of lead plus cavity is exp(iEτE/�)S0(E), with S0(E) distributed according to the circular ensemble of RMT (with effective parameters Neff, δeff). The complete excitation spectrum of the Andreev billiard consists of the levels of the effective RMT (periods >τE) plus the levels obtained by adiabatic quantization (periods
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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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98 M. Pustilnik and L.I. Glazman Co
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Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

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