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

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Andreev Billiards 149 Fig. 8. Magnetic field dependence of the density of states for the case of a ballistic point contact (Γn ≡ 1), computed from (43a), (44), and (56). The microscopic gap of order δ which persists when Φ>Φc is not resolved in this calculation. Adapted from [45] 6.5 Broken Time-Reversal Symmetry A microscopic suppression of the density of states around E = 0, on an energy scale of the order of the level spacing, persists even if time-reversal symmetry is fully broken. The suppression is a consequence of the level repulsion between the lowest excitation energy E1 and its mirror image −E1, which itself follows from the CT -antisymmetry (37) of the Hamiltonian. Because of this mirror symmetry, the effective Hamiltonian Heff of the Andreev billiard can be factorized as Heff = U � E 0 0 −E � U † , (57) with U a2M × 2M unitary matrix and E = diag(E1,E2,...EM) a diagonal matrix containing the positive excitation energies. Altland and Zirnbauer [39] have surmised that an ensemble of Andreev billiards in a strong magnetic field would have a distribution of Hamiltonians of the Wigner-Dyson form (28), constrained by (57). This constraint changes the Jacobian from the space of matrix elements to the space of eigenvalues, so that the eigenvalue probability distribution is changed from the form (29) (with β =2)into P ({En}) ∝ � � E 2 i − E 2 �2 � j E 2 ke −V (Ek)−V (−Ek) . (58) i
150 C.W.J. Beenakker 〈ρ(E)〉 = 2 δ � 1 − sin(4πE/δ) � . (59) 4πE/δ All of this is qualitatively different from the “folded GUE” that one would obtain by simply combining two independent GUE’s of electrons and holes [48]. A derivation of Altland and Zirnbauer’s surmise has been given by Frahm et al. [38], who showed that the LUE for the effective Hamiltonian Heff of the Andreev billiard follows from the GUE for the Hamiltonian H of the isolated billiard, provided that the coupling to the superconductor is sufficiently strong. To compute the spectral statistics on the scale of the level spacing, a non-perturbative technique is needed. This is provided by the supersymmetric method [49]. (See [50] for an alternative approach using quantum graphs.) The resulting average density of states is [38] 〈ρ(E)〉 = 2 � ∞ sin(2πE/δ) − ds e δ πE 0 −s � � 2πE × cos 1+ δ 4s � , (60) gA N� 2Γ gA = 2 n . (61) (2 − Γn) 2 n=1 The parameter gA is the Andreev conductance of the point contact that couples the billiard to the superconductor [51]. The Andreev conductance can be much smaller than the normal-state conductance g = � N n=1 Γn. (Both conductances are in units of 2e 2 /h.) In the tunneling limit Γn ≡ Γ ≪ 1 one has g = NΓ while gA = 1 2 NΓ2 . Equation (9) describes the crossover from the GUE result ρ(E) =2/δ for gA ≪ 1 to the LUE result (59) forgA ≫ 1. The opening of the gap as the coupling to the superconductor is increased is plotted in Fig. 9. The CT -antisymmetry becomes effective at an energy E for gA > ∼ E/δ. For small energies E ≪ δ min( √ gA, 1) the density of states vanishes quadratically, regardless of how weak the coupling is. 6.6 Mesoscopic Fluctuations of the Gap The smallest excitation energy E1 in the Andreev billiard fluctuates from one member of the ensemble to the other. Vavilov et al. [36] have surmised that the distribution of these fluctuations is identical upon rescaling to the known distribution [52] of the lowest eigenvalue in the Gaussian ensembles of RMT. This surmise was proven using the supersymmetry technique by Ostrovsky, Skvortsov, and Feigelman [53] and by Lamacraft and Simons [54]. Rescaling amounts to a change of variables from E1 to x =(E1−Egap)/∆gap, where Egap and ∆gap parameterize the square-root dependence (46) of the mean density
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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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