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

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60 J.M. Elzerman et al. voltage [27]. This requires that the quantum dot be connected to two leads with a tunnel coupling large enough to obtain a measurable current [43]. Coupling to the leads unavoidably introduces decoherence of the qubit: even if the number of electrons on the dot is fixed due to Coulomb blockade, an electron can tunnel out of the dot and be replaced by another electron through a second-order tunnelling process, causing the quantum information to be irretrievably lost. Therefore, to optimally store qubits in quantum dots, higher-order tunnelling has to be suppressed, i.e. the coupling to the leads must be made as small as possible. Furthermore, real-time observation of electron tunnelling, important for single-shot read-out of spin qubits via spinto-charge conversion, also requires a small coupling of the dot to the leads. In this regime, current through the dot would be very hard or even impossible to measure. Therefore an alternative spectroscopic technique is needed, which does not rely on electron transport through the quantum dot. Here we present spectroscopy measurements using charge detection. Our method resembles experiments on superconducting Cooper-pair boxes and semiconductor disks which have only one tunnel junction so that no net current can flow. Information on the energy spectrum can then be obtained by measuring the energy for adding an electron or Cooper-pair to the box, using a single-electron transistor (SET) operated as a charge detector [51, 52, 53]. We are interested in the excitation spectrum for a given number of electrons on the box, rather than the addition spectra. We use a quantum point contact (QPC) as an electrometer [44] and excitation pulses with repetition rates comparable to the tunnel rates to the lead, to measure the discrete energy spectrum of a nearly isolated one- and two-electron quantum dot. 3.2 Tuning the Tunnel Barriers The quantum dot and QPC are defined in the two-dimensional electron gas (2DEG) in a GaAs/Al0.27Ga0.73As heterostructure by dc voltages on gates T,M,R and Q (Fig. 21a). The dot’s plunger gate, P , is connected to a coaxial cable, to which we can apply voltage pulses (rise time 1.5 ns). The QPC charge detector is operated at a conductance of about e 2 /h with source-drain voltage VSD =0.2 mV. All data are taken with a magnetic field B // = 10 T applied in the plane of the 2DEG, at an effective electron temperature of about 300 mK. We first describe the procedure for setting the gate voltages such that tunnelling in and out of the dot take place through one barrier only (i.e. the other is completely closed), and the remaining tunnel rate be well controlled. For gate voltages far away from a charge transition in the quantum dot, a pulse applied to gate P (Fig. 21b) modulates the QPC current via the crosscapacitance only (solid trace in Fig. 21c). Near a charge transition, the dot can become occupied with an extra electron during the high stage of the pulse (Fig. 21d). The extra electron on the dot reduces the current through the QPC. The QPC response to the pulse is thus smaller when tunnelling takes place
B // Semiconductor Few-Electron Quantum Dots as Spin Qubits 61 a b T DRAIN RESERVOIR 200 nm M P R IQPC Q SOURCE c d 0 -V P ∆I QPC E F Γ τ τ Γ time time Fig. 21. QPC response to a pulse train applied to the plunger gate. (a) Scanning electron micrograph of a quantum dot and quantum point contact, showing only the gates used in the present experiment (the complete device is described in [55]) and Sect. 2. (b) Pulse train applied to gate P .(c) Schematic response in QPC current, ∆IQP C, when the charge on the dot is unchanged by the pulse (solid line) or increased by one electron charge during the “high” stage of the pulse (dashed). (d) Schematic electrochemical potential diagrams during the high (left) andlow (right) pulse stage, when the ground state is pulsed across the Fermi level in the reservoir, EF (dotted trace in Fig. 21c). We denote the amplitude of the difference between solid and dotted traces as the “electron response”. Now, even when tunnelling is allowed energetically, the electron response is only non-zero when an electron has sufficient time to actually tunnel into the dot during the pulse time, τ. By measuring the electron response as a function of τ, we can extract the tunnel rate, Γ , as demonstrated in Fig. 22a. We apply a pulse train to gate P with equal up and down times, so the repetition rate is f =1/(2τ) (Fig. 21b). The QPC response is measured using lock-in detection at frequency f [45], and is plotted versus the dc voltage on gate M. For long pulses (lowest curves) the traces show a dip, which is due to the electron response when crossing the zero-to-one electron transition. Here, f ≪ Γ and tunnelling occurs quickly on the scale of the pulse duration. For shorter pulses the dip gradually disappears. We find analytically 1 that the dip height is proportional to 1 − π 2 /(Γ 2 τ 2 + π 2 ), so the dip height should equal half its maximum value when Γτ = π. From the data (inset to Fig. 22a), we find that this happens for τ ≈ 120 µs, giving Γ ≈ (40 µs) −1 . Using this value 1 This expression is obtained by multiplying the probability that the dot is empty, P (t), with a sine-wave of frequency f (as is done in the lock-in amplifier), and averaging the resulting signal over one period. P (t) is given by exp(−Γt)(1 − exp(−Γτ))/(1−exp(−2Γτ)) during the high stage of the pulse, and by 1−P (t−τ) during the low stage.
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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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Page 23 and 24: RG for Interacting Fermions 13 the
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Page 27 and 28: RG for Interacting Fermions 17 equa
Page 29 and 30: � p � dθp = g 2π RG for Inter
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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
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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: V L (V) -1.084 -1.082 -1.080 -1.078
Page 77 and 78: a RESERVOIR 200 nm Semiconductor Fe
Page 81 and 82: 4.5 QPC Versus SET Semiconductor Fe
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Page 91 and 92: Spin-down counts 200 100 a Semicond
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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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