Nonequilibrium transport and population inversion in double quantum dot systems

We present a microscopic theory for both equilibrium and nonequilibrium transport properties of coupled double quantum dots (DQD). A general formula for current tunneling through the DQD is derived by the nonequilibrium Green’s function method. Using a Hartree-Fock approach, effects of multilevel coupling and nonequilibrium electron distributions in resonant tunneling are considered. We find that the peak in the resonant tunneling current through two symmetric dots will split only when the inter-dot coupling is stronger than dot-lead coupling. We predict that population inversion can be achieved in one dot in the nonequilibrium regime. 73.20.Dx, 73.40.Gk, 73.50.Fq Typeset using REVTEX 1 Coupled quantum dot systems have received much attention recently [1–9]. Resonant tunneling through zero dimensional (0D) states of coupled quantum dots has been studied experimentally by several groups in both equilibrium (|μL−μR| ≪ kBT ) and nonequilibrium (|μL−μR| ≫ kBT ) regimes [1–3] (where μL and μR are chemical potentials of leads externally attached to the quantum dots from left and right, respectively). The Coulomb blockade theory (CBT) was used to study the equilibrium properties of electronic tunneling in a double quantum dot (DQD) system [4], and more recently [8,9], has been extended to explain the tunneling current peak splitting observed in the experiment of Waugh et al [3]. Most previous theoretical studies on DQD systems [4–6,8,9] concentrated on the equilibrium properties. Interesting properties in the nonequilibrium regimes [2], where each dot can have different thermal nonequilibrium states driven by the two leads, have received much less attention. In this paper, we use a microscopic tunneling model to study the electronic transport through DQD systems based on the nonequilibrium Green’s function (NGF) method. Using the NGF method, the resonant tunneling current can then be derived exactly in both equilibrium and nonequilibrium regimes for tunneling through an interacting quantum dot [10] and non-interacting multi-quantum dots [11]. Here we give a general expression for the current for tunneling through interacting multi-level double quantum dot systems. The general current we derived can be used to obtain the well-known current equations in the equilibrium regime. In the nonequilibrium regime, it is usually difficult to satisfy detailed current balance between different dots and leads in various approximations [12], except in the case that the contribution to the imaginary self-energy is only from the lead-dot coupling. In this case, the current equation is much simplified and can be written in a compact form. In this paper, we study resonant tunnelings through double quantum dots, each of them having more than 50 electrons. In these systems, only the quasi-particle states are relevant to the resonant tunneling. The spectrum of quasi-particle states near the Fermi surface will not change significantly with changing the electron number from N to N + 1 except for a slight energy level shift mainly due to the “charging effect”. This has been confirmed for 2 quantum dot systems with N > 30 by microscopic calculations [20]. In general, various scattering contributions to the imaginary self-energy (scattering-in scattering-out processes) can be discussed phenomenologically using our current equation below, which is useful for interpretation of experiments. We will use the general current equation derived here to discuss experimental results performed in the nonequilibrium regime [2]; some of our explanations are different from previous ones [2]. In particular, we will see that the study of nonequilibrium electron distributions in the quantum dots are crucial to understand nonequilibrium experiments [2]. If the couplings of lead-dot and dot-dot are Γ and |t|, respectively, we predict that in symmetric equilibrium experiments [3] a peak of the resonant tunneling current will split only when |t| > Γ/2 with a splitting magnitude 2(|t| − Γ/4). Particularly, we predict that a population inversion can be achieved in one of the two quantum dots when the tunneling between the two dots is in resonance. This population inversion should be easily realized under conditions such as the low temperature experiments of Ref. [2,3]. At high temperatures, the population inversion critically depends on electronphonon (e-p) or electron-electron (e-e) scattering induced relaxation times. This population inversion is similar to that in quantum cascade lasers (QCL) [15] except that the former co-exists with resonant tunneling and the tunneling is not photon-assisted. Thus population inversion can be achieved in the double quantum wells under similar conditions as for QCL. The structure of this paper is as follows: in section I we discuss the model and derive the general current equations; in section II we discuss various approximations and the simplified current equations; in section III we discuss the non-equilibrium distribution of electrons in a double quantum dot system and the possibility of population inversion; in section IV, we discuss two recent experiments; and we conclude in section V. I. MODEL AND CURRENT EQUATIONS We model the coupled DQD system with two attached leads by the Hamiltonian [13] H = H0 +Ht +HI , with