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

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Andreev Billiards 169 Fig. 24. Ehrenfest-time dependence of the excitation gap in an Andreev billiard, according to the effective RMT (solid curve, calculated in App. A) and according to the stochastic model (dashed curve, calculated in [40]). The data points result from the simulation of the Andreev kicked rotator [27] (closed circles, in the range N =10 2 −10 5 ) and of the Sinai billiard shown in the inset [80] (open circles, inthe range N =18−30) 9 Conclusion Looking back at what we have learned from the study of Andreev billiards, we would single out the breakdown of random-matrix theory as the most unexpected discovery and the one with the most far-reaching implications for the field of quantum chaos. In an isolated chaotic billiard RMT provides an accurate description of the spectral statistics on energy scales below �/τerg (the inverse ergodic time). The weak coupling to a superconductor causes RMT to fail at a much smaller energy scale of �/τdwell (the inverse of the mean time between Andreev reflections), once the Ehrenfest time τE becomes greater than τdwell. In the limit τE →∞, the quasiclassical Bohr-Sommerfeld theory takes over from RMT. While in isolated billiards such an approach can only be used for integrable dynamics, the Bohr-Sommerfeld theory of Andreev billiards applies regardless of whether the classical motion is integrable or chaotic. This is a demonstration of how the time-reversing property of Andreev reflection unravels chaotic dynamics. What is lacking is a conclusive theory for finite τE > ∼ τdwell. The two phenomenological approaches of Sects. 8.2 and 8.3 agree on the asymptotic be- havior lim �→0 Egap = π�α , (94) 2| ln �| + constant in the classical � → 0 limit (understood as N →∞at fixed τdwell). There is still some disagreement on how this limit is approached. We would hope
170 C.W.J. Beenakker that a fully microscopic approach, for example based on the ballistic σ-model [81, 82], could provide a conclusive answer. At present technical difficulties still stand in the way of a solution along those lines [83]. A new direction of research is to investigate the effects of a nonisotropic superconducting order parameter on the Andreev billiard. The case of d-wave symmetry is most interesting because of its relevance for high-temperature superconductors. The key ingredients needed for a theoretical description exist, notably RMT [84], quasiclassics [85], and a numerically efficient Andreev map [86]. Acknowledgments While writing this review, I benefitted from correspondence and discussions with W. Belzig, P. W. Brouwer, J. Cserti, P. M. Ostrovsky, P. G. Silvestrov, and M. G. Vavilov. The work was supported by the Dutch Science Foundation NWO/FOM. A Excitation Gap in Effective RMT and Relationship with Delay Times We seek the edge of the excitation spectrum as it follows from the determinant (87), which in zero magnetic field and for E ≪ ∆ takes the form � Det 1+e 2iEτE/� S0(E)S0(−E) †� =0. (95) The unitary symmetric matrix S0 has the RMT distribution of a chaotic cavity with effective parameters Neff and δeff given by (85) and (86). The mean dwell time associated with S0 is τdwell. The calculation for Neff ≫ 1 follows the method described in Sects. 6.1 and 6.2, modified as in [37] to account for the energy dependent phase factor in the determinant. Since S0 is of the RMT form (30), we can write (95) in the Hamiltonian form (32). The extra phase factor exp(2iEτE/�) introduces an energy dependence of the coupling matrix, W0(E) = π cos u � W0W T 0 sin u W0WT 0 W0W T 0 W0W T � , (96) 0 sin u where we have abbreviated u = EτE/�. The subscript 0 reminds us that the coupling matrix refers to the reduced set of Neff channels in the effective RMT. Since there is no tunnel barrier in this case, the matrix W0 is determined by (31) with Γn ≡ 1. The Hamiltonian � � H0 0 H0 = (97) 0 −H0
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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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