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

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34 J.M. Elzerman et al. a Ohmic contact gate depleted region AlGaAs 2DEG GaAs b 400 nm Fig. 6. Lateral quantum dot device defined by metal surface electrodes. (a) Schematic view of a device. Negative voltages applied to metal gate electrodes (dark gray) lead to depleted regions (white) inthe2DEG(light gray). Ohmic contacts (light gray columns) enable bonding wires (not shown) to make electrical contact to the 2DEG reservoirs. (b) Scanning electron microscope image of an actual device, showing the gate electrodes (light gray) on top of the surface (dark gray). The two white dots indicate two quantum dots, connected via tunable tunnel barriers to a source (S) and drain (D) reservoir, indicated in white. Thetwoupper gates can be used to create two quantum point contacts, in order to detect changes in the number of electrons on the dot The electron beam can accurately write very small patterns with a resolution of about 20 nm, allowing us to make very complicated gate structures (Fig. 6). By applying negative voltages to the gates, the 2DEG is locally depleted, creating one or more small islands that are isolated from the large 2DEG reservoirs. These islands are the quantum dots. In order to probe them, we need to make electrical contact to the reservoirs. For this, we use rapid thermal annealing to diffuse AuGeNi from the surface to the 2DEG below. This forms ohmic contacts that connect the 2DEG source and drain reservoirs electrically to metal bonding pads on the surface. Metal wires bonded to these pads run toward the current or voltage probes, enabling us to perform transport measurements. 1.5 Transport Though Quantum Dots We use two different ways to probe the behavior of electrons on a quantum dot. In this work, we mostly rely on a nearby quantum point contact (QPC) to detect changes in the number of electrons on the dot. In addition, we can perform conventional transport experiments. These experiments are conveniently understood using the constant interaction (CI) model [27]. This model makes two important assumptions. First, the Coulomb interactions among electrons in the dot are captured by a single constant capacitance, C. This is the total capacitance to the outside world, i.e. C = CS + CD + Cg, where CS is the capacitance to the source, CD that to the drain, and Cg to the gate. Second, S D
Semiconductor Few-Electron Quantum Dots as Spin Qubits 35 the discrete energy spectrum is independent of the number of electrons on the dot. Under these assumptions the total energy of a N-electron dot with the source-drain voltage, VSD, applied to the source (and the drain grounded), is given by U(N) = [−|e|(N − N0)+CSVSD + CgVg] 2 2C + N� En(B) (1) where −|e| is the electron charge and N0 the number of electrons in the dot at zero gate voltage, which compensates the positive background charge originating from the donors in the heterostructure. The terms CSVSD and CgVg can change continuously and represent the charge on the dot that is induced by the bias voltage (through the capacitance CS) and by the gate voltage Vg (through the capacitance Cg), respectively. The last term of (1) is a sum over the occupied single-particle energy levels En(B), which are separated by an energy ∆En = En −En−1. These energy levels depend on the characteristics of the confinement potential. Note that, within the CI model, only these single-particle states depend on magnetic field, B. To describe transport experiments, it is often more convenient to use the electrochemical potential. This is defined as the energy required to add an electron to the quantum dot: µ(N) ≡ U(N) − U(N − 1) = � N − N0 − 1 2 � n=1 EC − EC |e| (CSVSD + CgVg)+EN where EC = e 2 /C is the charging energy. The electrochemical potential for different electron numbers N is shown in Fig. 7a. The discrete levels are spaced by the so-called addition energy: Eadd(N) =µ(N +1)− µ(N) =EC + ∆E . (2) The addition energy consists of a purely electrostatic part, the charging energy EC, plus the energy spacing between two discrete quantum levels, ∆E. Note that ∆E can be zero, when two consecutive electrons are added to the same spin-degenerate level. Of course, for transport to occur, energy conservation needs to be satisfied. This is the case when an electrochemical potential level falls within the “bias window” between the electrochemical potential (Fermi energy) of the source (µS) and the drain (µD), i.e. µS ≥ µ ≥ µD with −|e|VSD = µS − µD. Only then can an electron tunnel from the source onto the dot, and then tunnel off to the drain without losing or gaining energy. The important point to realize is that since the dot is very small, it has a very small capacitance and therefore a large charging energy – for typical dots EC ≈ afewmeV. If the electrochemical potential levels are as shown in Fig. 7a, this energy is not available (at low temperatures and small bias voltage). So, the number of
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 39 and 40: 1.2 Implementations Semiconductor F
Page 43: 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
Page 77 and 78: a RESERVOIR 200 nm Semiconductor Fe
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 91 and 92: Spin-down counts 200 100 a Semicond
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Semiconductor Few-Electron Quantum
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6.7 Unresolved Issues Semiconductor
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98 M. Pustilnik and L.I. Glazman Co
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126 M. Pustilnik and L.I. Glazman d
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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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