Landauer-Büttiker formula for time-dependent transport through resonant-tunneling structures: A nonequilibrium Green’s function approach

Abstract

A nonequilibrium Green's-function formalism is employed to study the time-dependent transport through resonant-tunneling structures. With this formalism, we derive a time-dependent Landauer-Büttiker formula that guarantees current conservation and gauge invariance. Furthermore, we apply the formula to calculate the response behaviors of the resonant-tunneling structures in the presence of rectangular-pulse and harmonicmodulation fields. The results show that the displacement current plays the role of retarding the tunneling current.The study of semiconductor nanostructures is one of the most active areas in condensed-matter physics. Many phenomena of electrical conduction in the dc situation have been successfully explained for mesoscopic systems 1 since the pioneering works of Landauer 2 and Büttiker et al. 3 Recently, time-dependent transport through mesoscopic systems has aroused much interest and the Keldysh nonequilibrium Green's functions are widely applied to analyze such phenomenon. 4 However, in many studies, only the tunneling current is calculated, while the displacement current induced by a time-dependent field is ignored. In recent years, this incompleteness has begun to be recognized. [5][6][7] As a matter of fact, the importance of the displacement current was first noted by Büttiker and co-workers. [8][9][10] To analyze the electron transport in the presence of ac fields, they developed a linear, low-frequency theory 8,9 based on the scattering matrix formulation, in which current conservation and gauge invariance are guaranteed. Very recently, Wang et al. 7 further studied the ac problem by means of nonequilibrium Green's functions and also derived an expression for the linear dynamic conductance that conserves electric current and satisfies the gauge invariance. In the present paper, we investigate the time-dependent transport through resonant-tunneling structures. We also adopt the nonequilibrium Green's-function formalism such that far-fromequilibrium situations can be analyzed conveniently. In contrast to previous works by Büttiker et al. 8,9 and Wang et al., 7 we conduct our calculations directly in the time domain and a time-dependent Landauer-Büttiker formula that guarantees current conservation and gauge invariance is derived. In our approach, the derived time-dependent current formula is applicable to both small and large bias-voltage situations. Finally, the Landauer-Büttiker formula is applied to the rectangular-pulse and harmonic-modulation cases to reveal the response behaviors of the resonant-tunneling structures.We first consider the system described by the HamiltonianThe first term describes the leads. The operator c k † creates an electron with momentum k and spin in either the left ͑L͒ or right ͑R͒ lead. The second term models the scattering region and c d † creates an electron with a time dependent energy level ⑀ d (t)ϭ⑀ 0 ϩ⌬(t). The third term characterizes the tunneling transfer of electrons between the leads and the scattering region. This Hamiltonian can be used to model e...

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