4 Coulomb blockade

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70 4 <strong>Coulomb</strong> <strong>blockade</strong> Assume now, that the basis states |α〉 are the states with definite spin quantum number σα. It means, that only one spin component of the wave function, namely ψα(σα) is nonzero, and ψα(¯σα) = 0. In this case the only nonzero matrix elements are those with σα = σγ and σβ = σδ, theyare Vαβ,γδ = dr dr ′ ψ ∗ α(r)ψ ∗ β(r ′ )V (|r − r ′ |)ψγ(r)ψδ(r ′ ). (4.12) In the case of delocalized basis states ψα(r), the main matrix elements are those with α = γ and β = δ, because the wave functions of two different states with the same spin are orthogonal in real space and their contribution is small. It is also true for the systems with localized wave functions ψα(r), when the overlap between two different states is weak. In these cases it is enough to replace the interacting part by the Anderson-Hubbard Hamiltonian, describing only density-density interaction ˆHAH = 1 Uαβ ˆnαˆnβ. (4.13) 2 α=β with the Hubbard interaction defined as Uαβ = dr dr ′ |ψα(r)| 2 |ψβ(r ′ )| 2 V (|r − r ′ |). (4.14) In the limit of a single-level quantum dot (which is, however, a two-level system because of spin degeneration) we get the Anderson impurity model (AIM) ˆHAIM = σ=↑↓ ɛσd † σdσ + U ˆn↑ˆn↓. (4.15) The other important limit is the constant interaction model (CIM), which is valid when many levels interact with similar energies, so that approximately, assuming Uαβ = U for any states α and β ˆHAH = 1 Uαβ ˆnαˆnβ ≈ 2 α=β U 2 ˆnα 2 α − U ˆn 2 α 2 α = U ˆ N( ˆ N − 1) . 2 (4.16) where we used ˆn 2 =ˆn, forlargeNit is equivalent to (4.2). Thus, the CIM reproduces the charging energy considered above, and the Hamiltonian of an isolated system is ˆHCIM = αβ ˜ɛαβd † αdβ + E(N). (4.17) Note, that the equilibrium compensating charge density can be easily introduced into the AH Hamiltonian ˆHAH = 1 Uαβ (ˆnα − ¯nα)(ˆnβ − ¯nβ) , (4.18) 2 α=β and at N ≫ 1 we obtain the electrostatic energy (4.2) with C = U/e 2 .
4.2 Single-electron box 4.2 Single-electron box 71 The simplest nanosystem, where the effects of single charge tunneling are important, is the single-electron box (SEB), a metallic island with dense electron spectrum coupled to the lead by a weak tunneling contact with the capacitance CL and the resistance RL (Fig. 4.1). Weak coupling means, that the tunneling resistance RT of the junction is rather high RT ≫ RK = h 25.8 kΩ, (4.19) e2 the RK is a resistance quantum (it is defined only by the fundamental constants). Note, that this resistance is well known from the Landauer-Büttiker theory, where it plays the role of the minimum resistance of a single quantum channel. Thus, we understand now better the limits of weak (RT ≫ RK) and strong (RT ≤ RK) tunneling. We allow also to apply some voltage VGL to the system through the gate capacitance CG. The circuit is broken in this place, and the current can not flow though a system. The gate capacitance CG is ”true” capacitance, while the tunnel junction includes the classical capacitance CL and quantum ”possibility to tunnel” (shown by the cross). It is not equivalent to a classical resistance, in particular, the finite voltage at the capacitance CL does not imply necessarily the current through the tunnel junction. Now let us consider the following question: what is the charge state of the SEB at arbitrary voltage VGL? The important point is, that if the condition (4.19) is satisfied, the probability of tunneling is small, and electrons can be considered as localized inside the system almost all time. Therefore, the electronic charge of the SEB is quantized and is always en with integer n. However, the voltage VGL is arbitrary and tries to charge the capacitance by the charge CGVGL, which is not possible in general. The solution, found by L CLRL VGL System Gate CG CLRL VGL System Fig. 4.1. A single-electron box and its electrical scheme. CG
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Page 25: Additional reading 4.7 Problems 91

4 Coulomb blockade

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