Effect of Quantum Confinement on Electron Tunneling through a Quantum Dot

Employing the Anderson impurity model, we study tunneling properties through an ideal quantum dot near the conductance minima. Considering the Coulomb blockade and the quantum confinement on an equal footing, we have obtained current contributions from various types of tunneling processes; inelastic cotunneling, elastic cotunneling, and resonant tunneling of thermally activated electrons. We have found that the inelastic cotunneling is suppressed in the quantum confinement limit, and thus the conductance near its minima is determined by the elastic cotunneling at low temperature (kBT ≪ Γ, Γ: dot-reservoir coupling constant), or by the resonant tunneling of single electrons at high temperature (kBT ≫ Γ). PACS numbers: 73.20.Dx, 73.40.Gk Typeset using REVTEX 1 During last decade, there has been a rapid advance in the field of single electronics, and accordingly much scientific attention has been given to transport properties through ultra-small tunnel junctions such as GaAs quantum dot [1–3]. In a quantum dot with small capacitance, “Coulomb blockade” of tunneling occurs for small bias voltage V when the charging energy (Coulomb energy U) in the dot is sufficiently large as compared to a thermal energy kBT . It occurs because even a single tunneling event increases the electrostatic Coulomb energy of the system considerably. However, even in this regime, a finite current can flow via virtual intermediate states arising from the quantum fluctuation of macroscopic electric charge in the central electrode of the system. This process in a quantum dot, so called cotunneling or macroscopic-quantum-tunneling, was first pointed out by Averin and coworkers [4], and is considered as setting a limit to the performance accuracy of the single electron transistor. They have shown that the transport near conductance minima is dominated by the inelastic cotunneling process involving the creation of an electron-hole excitation in the central electrode, and predicted an algebraic variation of the leakage current with applied voltage (∼ V ) and temperature (∼ T ). The theory of cotunneling has been derived within the lowest order perturbation when the energy discreteness in a quantum dot is not important, i.e. the continuous energy spectrum is assumed in the central electrode. The inelastic cotunneling has been observed both in metal-insulator-metal tunnel junctions [5,6] and in a 2D electron system of GaAs/Ga(Al)As heterostructure [7,8]. When dealing with an ultra-small quantum dot, the effect of level discreteness (energy quantization: ∆) becomes very important. The effect will be more prominent in semiconductor systems than in metallic systems, due to much lower electron concentration (ρ) and lower effective electron mass (m) in semiconductor systems (recall that a free electron approximation yields ∆ = 1 g(εF )V = 2h̄ π V m∗(3π2ρ)1/3 for 3D systems). There have been quite a few experimental evidences exhibiting coexistence of the charge and energy quantization in the tunneling properties [3]. Furthermore, it is shown that the level spacing ∆ can be even comparable to the Coulomb energy U for a Si-based quantum dot transistor [9]. For such systems, the “quantum confinement” will become significant as much as the Coulomb 2 blockade for the tunneling in the single electron transistor. This kind of electronic confinement could be realized for an ultra-small dot at relatively high temperature. Illustrating typical parameters, a Si dot with diameter of 20nm would have ∆ ∼ 25 meV, and then the confinement of electron could be realized even at T < O(100)K for U ∼ 15 meV [9]. We address in this paper whether the inelastic cotunneling phenomenon is really a limiting factor in operating single electron quantum dot devices. For this purpose, we have examined tunneling properties of an ideal quantum dot coupled to two reservoirs in terms of the Anderson impurity model, where the quantum confinement is important as much as the Coulomb blockade. Special attention is focused on the temperature dependence of the inelastic cotunneling by treating the Coulomb blockade and the quantum confinement on an equal footing. We have found that the characteristic of the tunneling in the quantum confined system far from conductance maxima is qualitatively different from the case where the level discreteness can be neglected. In this case, the “inelastic” cotunneling is substantially suppressed at low temperature, while the “elastic” cotunneling or the resonant single electron tunneling of thermally activated electrons dominates in the system. Therefore it is expected that, in small enough quantum dots, there would be no substantial limitation on the performance accuracy in practical devices by a macroscopic quantum tunneling of charge. We start with the simplest Anderson model Hamiltonian [10] to describe an ideal quantum dot (labeled by D) weakly coupled to two electron reservoirs (labeled by L and R): H = HL +HR +HD +HT HL(R) = ∑ k,α∈L(R) εkc † kαckα (1) HD = ∑