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

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14 R. Shankar HU = � α,s εαc † α,scα,s + U0 2 N 2 − J 2 S2 � � + λ c α † α,↑c† � α,↓ ⎛ ⎝ � β cβ,↓cβ,↑ ⎞ ⎠ (26) where s is single-particle spin and S is the total spin. The Cooper coupling λ does not play a major role, but the inclusion of the exchange coupling J brings the theoretical predictions [9, 14, 15] into better accord with experiments, especially if one-body “scrambling” [19, 20, 21, 22] and finite temperature effects are taken into account. However, some discrepancies still remain in relation to numerical [16] and experimental results [18] atrs ≥ 2. We now see that the following dot-related questions naturally arise. Given that adding more refined interactions (culminating in the universal hamiltonian) led to better descriptions of the dot, should one not seek a more systematic way to to decide what interactions should be included from the outset? Does our past experience with clean systems and bulk systems tell us how to proceed? Once we have written down a comprehensive hamiltonian, is there a way to go beyond perturbation theory to unearth nonperturbative physics in the dot, including possible phases and transitions between them? What will be the experimental signatures of these novel phases and the transitions between them if indeed they do exist? These questions will now be addressed. 4.1 Interactions and Disorder: Exact Results on the Dot The first crucial step towards this goal was taken by Murthy and Mathur [23]. Their ideas was as follows. • Step 1: Use the clean system RG described earlier [2] (eliminating momentum states on either side of the Fermi surface) to eliminate all states far from the Fermi surface till one comes down to the Thouless band, that is, within ET of EF . We have seen that this process inevitably leads to Landau’s Fermi liquid interaction (spin has been suppressed): V = ∞� m=0 um∆ 2 � k,k ′ cos[m(θ − θ ′ )]c † k † ckc k ′ck ′ (27) where θ, θ ′ are the angles of k, k ′ on the Fermi circle, and um is defined by F (θ) = � ume imθ . (28) m A few words before we proceed. First, some experts will point out that the interaction one gets from the RG allows for small momentum transfer, i.e., there should be an additional sum over a small values q in (27) allowing k → k + q and k ′ → k ′ − q. It can be shown that in the large g limit
RG for Interacting Fermions 15 this sum has just one term, at q = 0. Unlike in a clean system, there is no singular behavior associated with q → 0 and this assumption is a good one. Others have asked how one can introduce the Landau interaction that respects momentum conservation in a dot that does not conserve momentum or anything else except energy. To them I say this. Just think of a pair of molecules colliding in a room. As long as the collisions take place in a time scale smaller than the time between collisions with the walls, the interaction will be momentum conserving. That this is true for a collision in the dot for particles moving at vF , subject an interaction of range equal to the Thomas Fermi screening length (the typical range) is readily demonstrated. Like it or not, momentum is a special variable even in a chaotic but ballistic dot since it is tied to translation invariance, and that that is operative for realistic collisions within the dot. • Step 2: Switch to the exact basis states of the chaotic dot, writing the kinetic and interaction terms in this basis. Run the RG by eliminating exact energy eigenstates within ET . While this looks like a reasonable plan, it is not clear how it is going to be executed since knowledge of the exact eigenfunctions is needed to even write down the Landau interaction in the disordered basis: � u(θ − θ ′ � ) φ∗ α(k)φ∗ β (k′ ) − φ∗ α(k ′ )φ∗ β (k) � Vαβγδ = ∆ 4 kk ′ × [φγ(k ′ )φδ(k) − φγ(k)φδ(k ′ )] (29) where k and k ′ take g possible values. These are chosen as follows. Consider the momentum states of energy within ET of EF . In a dot momentum is defined with an uncertainty ∆k � 1/L in either direction. Thus one must form packets in k space obeying this condition. It can be easily shown that g of them will fit into this band. One way to pick such packets is to simply take plane waves of precise k and chop them off at the edges of the dot and normalize the remains. The g values of k can be chosen with an angular spacing 2π/g. It can be readily verified that such states are very nearly orthogonal. The wavefunction φδ(k) is the projection of exact dot eigenstate δ on the state k as defined above. We will see that one can go a long way without detailed knowledge of the wavefunctions φδ(k). First, one can take the view of the Universal Hamiltonian (UH) adherents and consider the ensemble average (enclosed in 〈〉) of the interactions. RMT tells us that to leading order in 1/g, � ∗ φα(k1)φ ∗ β(k2)φγ(k3)φδ(k4) � δαδδβγδk1k4 δk2k3 = g2 + δαγδβδδk1k3 δk2k4 g2 + δαβδγδδk1,−k2 δk3,−k4 g2 (30) It is seen that only matrix elements in (29) for which the indices αβγδ are pairwise equal survive disorder-averaging, and also that the average has no
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: RG for Interacting Fermions 13 the
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 39 and 40: 1.2 Implementations Semiconductor F
Page 43 and 44: GaAs n-AlGaAs AlGaAs GaAs Semicondu
Page 49 and 50: ε1 ε0 e Semiconductor Few-Electro
Page 53 and 54: 250 Ω twisted pair 20 MΩ powder
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 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
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Semiconductor Few-Electron Quantum
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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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132 C.W.J. Beenakker e e N I N S Fi
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134 C.W.J. Beenakker 2 Andreev Refl
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136 C.W.J. Beenakker F E x 8d/hv ga
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138 C.W.J. Beenakker and we have de
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140 C.W.J. Beenakker A stroboscopic
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142 C.W.J. Beenakker largely indepe
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144 C.W.J. Beenakker The effective
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146 C.W.J. Beenakker Fig. 6. Ensemb
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148 C.W.J. Beenakker 1.4 1.2 1.0 0.
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150 C.W.J. Beenakker 〈ρ(E)〉 =
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152 C.W.J. Beenakker P 0.5 0.4 0.3
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154 C.W.J. Beenakker P(x) 0.4 0.3 0
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156 C.W.J. Beenakker tunnel/ Egap 4
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158 C.W.J. Beenakker its momentum).
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160 C.W.J. Beenakker E00 = π� =
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162 C.W.J. Beenakker 〈ρ(E)〉 δ
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164 C.W.J. Beenakker As described i
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166 C.W.J. Beenakker 0.30 � Egap
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168 C.W.J. Beenakker The τE-depend
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170 C.W.J. Beenakker that a fully m
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172 C.W.J. Beenakker (τ−,τ+), w
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174 C.W.J. Beenakker 61. P. G. Silv
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