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

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Andreev Billiards 147 〈ρ(E)〉 = 2E δ (E2 − E 2 T ) −1/2 ,E>Egap = ET . (51) In this limit the density of states of the Andreev billiard has the same form as in the BCS theory for a bulk superconductor [44], with a reduced value of the gap (“minigap”). The inverse square-root singularity at the gap is cut off for any finite Γ , cf. Fig. 6. 6.3 Effect of Impurity Scattering Impurity scattering in a chaotic Andreev billiard reduces the magnitude of the excitation gap by increasing the mean time τdwell between Andreev reflections. This effect was calculated by Vavilov and Larkin [40] using the method of impurity-averaged Green functions [21]. The minigap in a disordered quantum dot is qualitatively similar to that in a disordered NS junction, cf. Sect. 3. The main parameter is the ratio of the mean free path l and the width of the contact W . (We assume that there is no barrier in the point contact, otherwise the tunnel probability Γ would enter as well.) For l ≫ W the mean dwell time saturates at the ballistic value τdwell = 2π� Nδ = πA vF W , if l ≫ W. (52) In the opposite limite l ≪ W the mean dwell time is determined by the twodimensional diffusion equation. Up to a geometry-dependent coefficient c of order unity, one has τdwell = c A vF l ln(A/W 2 ) , if l ≪ W. (53) The density of states in the two limits is shown in Fig. 7. There is little difference, once the energy is scaled by τdwell. Forl ≫ W the excitation gap is given by the RMT result Egap =0.300 �/τdwell, cf.(49). For l ≪ W Vavilov and Larkin find Egap =0.331 �/τdwell. 6.4 Magnetic Field Dependence A magnetic field B, perpendicular to the billiard, breaks time-reversal symmetry, thereby suppressing the excitation gap. A perturbative treatment remains possible as long as Egap(B) remains large compared to δ [45]. The appropriate RMT ensemble for the isolated billiard is described by the Pandey-Mehta distribution [9, 46] � P (H) ∝ exp × M� i,j=1 − π2 (1 + b 2 ) 4Mδ 2 � (Re Hij) 2 + b −2 (Im Hij) 2� � . (54)
148 C.W.J. Beenakker 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 0.5 1 1.5 2 Fig. 7. The dashed curve is the Γ = 1 result of Fig. 6, corresponding to a quantum dot with weak impurity scattering (mean free path l much larger than the width W of the point contact). The solid curve is the corresponding result for strong impurity scattering (l ≪ W ). The line shape is almost the same, but the energy scale is different (given by (52) and(53), respectively). Adapted from [40] The parameter b ∈ [0, 1] measures the strength of the time-reversal symmetry breaking. The invariance of P (H) under unitary transformations is broken if b �= 0, 1. The relation between b and the magnetic flux Φ through the billiard is [3] Mb 2 2 �vF = c(Φe/h) δ √ , (55) A with c a numerical coefficient that depends only on the shape of the billiard. Time-reversal symmetry is effectively broken when Mb2 � g, which occurs for Φ � (h/e) � τerg/τdwell ≪ h/e. The effect of such weak magnetic fields on the bulk superconductor can be ignored. The selfconsistency equation for the Green function is the same as (41), with one difference: On the right-hand-side the terms G12 and G21 are multiplied by the factor (1 − b2 )/(1 + b2 ). In the limit M →∞, b → 0, Mb2 finite, the first (43a) still holds, but the second (43b) is replaced by dwell (2πE/δ − 4Mb 2 G11)G12 = G11 × N� (−G12 +1− 2/Γn) −1 . (56) n=1 The resulting magnetic field dependence of the average density of states is plotted in Fig. 8, for the case Γn ≡ 1 of a ballistic point contact. The gap closes when Mb 2 = NΓ/8. The corresponding critical flux Φc follows from (55).
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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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