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

More documents

Recommendations

Info

� Andreev Billiards 159 pdq =2π�(m + ν/4), m =0, 1, 2,... (77) enclosed by each of the two topologically independent contours on the torus. Equation (77) ensures that the wave functions are single valued. The integer ν counts the number of caustics (Maslov index) and in this case should also include the number of Andreev reflections. The first contour follows the quasiperiodic orbit of (75), leading to ET =(m + 1 2 )π�, m =0, 1, 2,... (78) The quantization condition (78) is sufficient to determine the smoothed (or ensemble averaged) density of states � ∞ ∞� 〈ρ(E)〉 = N dT P (T ) 0 m=0 δ � E − (m + 1 2 )π�/T � , (79) using the classical probability distribution P (T ) for the time between Andreev reflections. (The distribution P (T ) is defined with a uniform measure in the surface of section (x, px) at the interface with the superconductor.) Equation (79) is the “Bohr-Sommerfeld rule” of Melsen et al. [22]. It generalizes the familiar Bohr-Sommerfeld quantization rule for translationally invariant geometries [cf. (7)] to arbitrary geometries. The quantization rule refers to classical periodic motion with period 2T and phase increment per period of 2ET/� − π, consisting of a part 2ET/� because of the energy difference 2E between electron and hole, plus a phase shift of −π from two Andreev reflections. If E is not ≪ ∆, this latter phase shift should be replaced by −2 arccos(E/∆) [64, 65, 66], cf. (20). In the presence of a magnetic field an extra phase increment proportional to the enclosed flux should be included [67]. Equation (79) can also be derived from the Eilenberger equation for the quasiclassical Green function [23]. To find the location of individual energy levels a second quantization condition is needed [61]. It is provided by the area � T pxdx enclosed by the isochronous contours, � pxdx =2π�(n + ν/4), n =0, 1, 2,... (80) T Equation (80) amounts to a quantization of the period T , which together with (78) leads to a quantization of E. ForeachTn there is a ladder of Andreev levels Enm =(m + 1 2 )π�/Tn. While the classical T can become arbitrarily large, the quantized Tn has a cutoff. The cutoff follows from the maximal area (76) enclosed by an isochronous contour. Since (80) requires Amax > ∼ h, the longest quantized period is T0 = α −1 [ln N + O(1)]. The lowest Andreev level associated with an adiabatically invariant torus is therefore
160 C.W.J. Beenakker E00 = π� = 2T0 π�α . (81) 2lnN The time scale T0 ∝|ln �| is the Ehrenfest time τE of the Andreev billiard, to which we will return in Sect. 8. The range of validity of adiabatic quantization is determined by the requirement that the drift δx, δpx upon one iteration of the Poincaré map should be small compared to the characteristic values W, pF . An estimate is [61] δx W � δpx pF � Enm �αN eαTn � (m + 1 2 )e−α(T0−Tn) . (82) αTn For low-lying levels (m ∼ 1) the dimensionless drift is ≪ 1forTn
Page 1 and 2:
Lecture Notes in Physics Editorial
Page 3 and 4:
W. Dieter Heiss (Ed.) Quantum Dots:
Page 5:
Preface Nanoscale physics, nowadays
Page 9 and 10:
Contents A Guide for the Reader ...
Page 11 and 12:
A Guide for the Reader Quantum dots
Page 13 and 14:
The Renormalization Group Approach
Page 15 and 16:
RG for Interacting Fermions 5 where
Page 17 and 18:
where � is a shorthand: � ≡ 3
Page 19 and 20:
RG for Interacting Fermions 9 These
Page 21 and 22:
RG for Interacting Fermions 11 is i
Page 23 and 24:
RG for Interacting Fermions 13 the
Page 25 and 26:
RG for Interacting Fermions 15 this
Page 27 and 28:
RG for Interacting Fermions 17 equa
Page 29 and 30:
� p � dθp = g 2π RG for Inter
Page 31 and 32:
RG for Interacting Fermions 21 the
Page 33 and 34:
References RG for Interacting Fermi
Page 35 and 36:
Semiconductor Few-Electron Quantum
Page 37 and 38:
Page 39 and 40:
1.2 Implementations Semiconductor F
Page 41 and 42:
Page 43 and 44:
GaAs n-AlGaAs AlGaAs GaAs Semicondu
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
ε1 ε0 e Semiconductor Few-Electro
Page 51 and 52:
Page 53 and 54:
250 Ω twisted pair 20 MΩ powder
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
10 5 0 -5 Semiconductor Few-Electro
Page 63 and 64:
G ( e / h) 2 2 1 0 Semiconductor Fe
Page 65 and 66:
Page 67 and 68:
V R (V) V R (V) Semiconductor Few-E
Page 69 and 70:
V L (V) -1.084 -1.082 -1.080 -1.078
Page 71 and 72:
B // Semiconductor Few-Electron Qua
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
a RESERVOIR 200 nm Semiconductor Fe
Page 79 and 80:
Page 81 and 82:
4.5 QPC Versus SET Semiconductor Fe
Page 83 and 84:
a reservoir dot Semiconductor Few-E
Page 85 and 86:
a b c V P (mV) 10 5 0 ∆I QPC E F
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Spin-down counts 200 100 a Semicond
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
6.7 Unresolved Issues Semiconductor
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
98 M. Pustilnik and L.I. Glazman Co
Page 109 and 110:
100 M. Pustilnik and L.I. Glazman F
Page 111 and 112:
102 M. Pustilnik and L.I. Glazman (
Page 113 and 114:
104 M. Pustilnik and L.I. Glazman A
Page 115 and 116:
106 M. Pustilnik and L.I. Glazman t
Page 117 and 118: 108 M. Pustilnik and L.I. Glazman A
Page 119 and 120: 110 M. Pustilnik and L.I. Glazman
Page 121 and 122: 112 M. Pustilnik and L.I. Glazman f
Page 123 and 124: 114 M. Pustilnik and L.I. Glazman 5
Page 125 and 126: 116 M. Pustilnik and L.I. Glazman U
Page 129 and 130: 120 M. Pustilnik and L.I. Glazman A
Page 131 and 132: 122 M. Pustilnik and L.I. Glazman d
Page 133 and 134: 124 M. Pustilnik and L.I. Glazman G
Page 135 and 136: 126 M. Pustilnik and L.I. Glazman d
Page 141 and 142: 132 C.W.J. Beenakker e e N I N S Fi
Page 143 and 144: 134 C.W.J. Beenakker 2 Andreev Refl
Page 145 and 146: 136 C.W.J. Beenakker F E x 8d/hv ga
Page 147 and 148: 138 C.W.J. Beenakker and we have de
Page 149 and 150: 140 C.W.J. Beenakker A stroboscopic
Page 151 and 152: 142 C.W.J. Beenakker largely indepe
Page 153 and 154: 144 C.W.J. Beenakker The effective
Page 155 and 156: 146 C.W.J. Beenakker Fig. 6. Ensemb
Page 157 and 158: 148 C.W.J. Beenakker 1.4 1.2 1.0 0.
Page 159 and 160: 150 C.W.J. Beenakker 〈ρ(E)〉 =
Page 161 and 162: 152 C.W.J. Beenakker P 0.5 0.4 0.3
Page 163 and 164: 154 C.W.J. Beenakker P(x) 0.4 0.3 0
Page 165 and 166: 156 C.W.J. Beenakker tunnel/ Egap 4
Page 167: 158 C.W.J. Beenakker its momentum).
Page 171 and 172: 162 C.W.J. Beenakker 〈ρ(E)〉 δ
Page 173 and 174: 164 C.W.J. Beenakker As described i
Page 175 and 176: 166 C.W.J. Beenakker 0.30 � Egap
Page 177 and 178: 168 C.W.J. Beenakker The τE-depend
Page 179 and 180: 170 C.W.J. Beenakker that a fully m
Page 181 and 182: 172 C.W.J. Beenakker (τ−,τ+), w
Page 183: 174 C.W.J. Beenakker 61. P. G. Silv
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?