Nonlinear resonant tunneling in systems coupled to quantum reservoirs

Abstract

An adiabatic approximation in terms of instantaneous resonances is developed to study the steady-state and time-dependent transport of interacting electrons in biased resonant tunneling heterostructures. The resulting model consists of quantum reservoirs coupled to regions where the system is described by nonlinear ordinary differential equations and has a general conceptual interest. 73.40.Gk, 05.45.+b The mathematical method recently proposed in [1] provides a significant advance in the solution of timedependent scattering problems for Schrödinger equations with nonlinearities concentrated near the resonances of the corresponding potential. The method consists in the separation of the original system in two coupled subsystems through the splitting of the wavefunction in two components. In this way, we can separately study the simple problem of a reservoir-like subsystem having only extended states and couple its solution to the other subsystem having resonance states. The solution of this second problem is then simplified by an adiabatic approximation in terms of instantaneous resonances.The situation investigated in [1] depicts ballistic transport in a double barrier heterostructure, i.e., the scattering of a wave-packet on a double barrier potential. The nonlinearity, concentrated in the well between the barriers, is due to the interaction of the electrons represented by the wave-packet (mean-field). Here, we generalize the approach [1] to the case of biased heterostructures where a band of scattering states are to be considered. For these systems, hysteresis in the current-voltage characteristics has been observed [2] and recognized as a consequence of the mutual interaction of the electrons trapped in the resonance [2,3]. We show that the approach [1] allows us to quantitatively reproduce experimental results like [4] and predict new time-dependent properties. Although illustrated in the case of heterostructures, these results are very general and have applications in fields like the theory of electric systems [5] and nonlinear optics [6,7].Let us consider a heterostructure whose conduction band edge profile consists of two barriers of height V 0 located in [a, b] and [c, d] with a < b < c < d along the growth direction x. Translational invariance is assumed in the plane parallel to the junctions. Suppose that a bias energy ∆V is applied between the emitter (x < a) and collector (x > d) regions uniformly doped with net donor concentration n D . At thermal equilibrium with temperature T , transport is due to a band of scattering states with Fermi energy E F = (3π 2 n D ) 2/3 (we use everywhere effective atomic unitsh = 2m * = 1 and e 2 /ε = 2a −1 B , where m * is the electron effective mass, ε the dielectric constant and a B =h 2 ε/(m * e 2 ) the effective Bohr radius). Due to the translational invariance in the plane parallel to the junctions, the single-electron scattering state at energy E along the x direction is described by the one-dimensional Schrödinger equationwhere V cb (x) is the step-like c...

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A linearized model of quantum transport in the asymptotic regime of quantum wells

“…This clarifies the idea of our approach: the modification of the physical model (9) by hdependent artificial interface conditions, although introducing a small error on the solution A h 0 (t) (controlled by a power of h), allows us to work in the complex deformed setting (59) where, under the condition (18), an adiabatic approximation holds for the deformed dynamicsũ h (k, ·, t).…”

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