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

More documents

Recommendations

Info

Andreev Billiards 155 The suppression of the excitation gap with increasing EC is plotted in Fig. 13, for the case Γ ≪ 1, ∆ ≪ �/τerg [55]. The initial decay is a square root, 1 − ∆eff/Egap = 1 � ECδ 2 E2 �1/2 ≪ 1 , gap ln(2∆/Egap) (73) and the final decay is exponential, ∆eff/∆ = 2 exp � − 2ECδ/E 2 � gap ≪ 1 . (74) Here ∆eff refers to the gap in the presence of Coulomb interactions and Egap = NΓδ/4π is the noninteracting value (51). eff / Egap 1.0 0.8 0.6 0.4 0.2 0.0 0 1 2 3 4 E C 2 E gap Fig. 13. Suppression due to Coulomb interactions of the gap ∆eff in the density of states of an Andreev billiard coupled by a tunnel junction to a superconductor, relative to the noninteracting gap Egap = NΓδ/4π (with Γ ≪ 1 ≪ NΓ). The plot is for the case ∆ = e 5 Egap ≪ �/τerg. Thedashed lines are the asymptotes (73) and (74). Adapted from [55] The gap ∆eff governs the thermodynamic properties of the Andreev billiard, most importantly the critical current. It is not, however, the relevant energy scale for transport properties. Injection of charge into the billiard via a separate tunnel contact measures the tunneling density of states ρtunnel, which differs in the presence of Coulomb interactions from the thermodynamic density of states ρ considered so far. The gap ∆tunnel in ρtunnel crosses over from the proximity gap Egap when EC ≪ EJ to the Coulomb gap EC when EC ≫ EJ, see Fig. 14. The single peak in ρtunnel at ∆tunnel splits into two peaks when EC and EJ are of comparable magnitude [55]. This peak splitting happens because two states of charge +e and −e having the same charging energy are mixed by Andreev reflection into symmetric and antisymmetric linear combinations with a slightly different energy.
156 C.W.J. Beenakker tunnel/ Egap 40 30 20 10 ρ tunnel 1 0 0 20 40 E/Egap 60 0 0 1 2 3 4 EC 2 E gap Fig. 14. Main plot: gap ∆tunnel in the tunneling density of states as a function of the charging energy (for ∆ = e 5 Egap and NΓ =40π). The initial decay (barely visible on the scale of the plot) follows (73) and crosses over to an increase (∆tunnel → EC). Inset: tunneling density of states at ECδ/E 2 gap =2.8 (corresponding to the dot in the main plot). Adapted from [55] 7 Quasiclassical Theory It was noticed by Kosztin, Maslov, and Goldbart [5] that the classical dynamics at the Fermi energy in an Andreev billiard is integrable –evenif the dynamics in the isolated billiard is chaotic. Andreev reflection suppresses chaotic dynamics because it introduces a periodicity into the orbits: The trajectory of an electron is retraced by the Andreev reflected hole. At the Fermi energy the hole is precisely the time reverse of the electron, so that the motion is strictly periodic. For finite excitation energy or in a non-zero magnetic field the electron and the hole follow slightly different trajectories, so the orbit does not quite close and drifts around in phase space [5, 57, 58, 59, 60]. The near-periodicity of the orbits implies the existence of an adiabatic invariant. Quantization of this invariant leads to the quasiclassical theory of Silvestrov et al. [61]. 7.1 Adiabatic Quantization Figures 15 and 16 illustrate the nearly periodic motion in a particular Andreev billiard. Figure 15 shows a trajectory in real space while Fig. 16 is a section of phase space at the interface with the superconductor (y = 0). The tangential component px of the electron momentum is plotted as a function of the coordinate x along the interface. Each point in this Poincaré map corresponds to one collision of an electron with the interface. (The collisions of holes are not plotted.) The electron is retroreflected as a hole with the same px. At the Fermi level (E = 0) the component py is also the same, and so the hole retraces the path of the electron (the hole velocity being opposite to
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 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 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: 154 C.W.J. Beenakker P(x) 0.4 0.3 0
Page 167 and 168: 158 C.W.J. Beenakker its momentum).
Page 169 and 170: 160 C.W.J. Beenakker E00 = π� =
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

Create successful ePaper yourself

Delete template?

Save as template?