Microscopic model for sequential tunneling in semiconductor multiple quantum wells

We propose a self-consistent microscopic model of vertical sequential tunneling through multiple quantum wells. The model includes a detailed description of the contacts, uses the transfer Hamiltonian for expressions of the current and it treats the Coulomb interaction within a mean-field approximation. We analyze the current density through a double well and a superlattice and study the formation of electric-field domains and multistability coming from the Coulomb interaction. Phase diagrams of parameter regions ~bias and doping in the heterostructure and in the contacts, etc . , where the different solutions exist, are given. @S0163-1829~97! 50128-7# Coulomb interaction in heterostructures with large area wells is a small effect compared with the energy difference between noninteracting eigenstates of the structure. Therefore, a mean-field model gives, for many purposes, a good description of the system. Among features of the transport properties having their origin in Coulomb interaction, intrinsic bistability has great importance. This physical phenomenon arises from the nonlinear effect of the electric charge on the induced electrostatic potential, and it has been predicted and observed in double barrier structures ~DB! 1‐4 Furthermore, in the presence of a laser polarized in the sample growth direction, additional bistability regions have been theoretically predicted. 5 In this paper, we deal with the statics and dynamics of vertical transport through biased heterostructures whose main mechanism is sequential tunneling. This is a topic that has attracted a great deal of attention in recent times. In weakly coupled superlattices, multistability due to domain formation, has been much studied both theoretically and experimentally. 6‐9 When the charge in the superlattice is small due to lower doping in the wells, selfsustained current oscillations and chaos due to domain dynamics are possible. 10‐12 So far, the most successful modeling of these phenomena use discrete rate equations for the electron density and electric field in each well, plus constitutive laws for the current, bias, boundary, and initial conditions. 7,8,13 The laws may be phenomenological 8 or obtained from microscopic considerations. 7,14,15 In all cases cited, the boundary conditions were selected in a more or less ad hoc manner by using the available information from experiments. This is particularly annoying because the boundary conditions select the relevant dynamics of electricfield domains in the oscillatory regime. 16 In this paper, we present a microscopic model that includes in a natural way boundary conditions due to the emitter and collector regions of a multiwell structure ~MW! We then solve it for the cases of a double quantum well ~DQW! and a superlattice ~SL! The presence of intrinsic bistability is demonstrated through phase diagrams and I-V characteristics obtained by numerical simulation and by means of numerical continuation of stationary solution branches. The main ingredients of our model are as follows: we assume that the characteristic time of intersubband relaxation due to scattering ~about 0.1 ps for optical phonon scattering 17 ! is much smaller than the tunneling time ~less than 0.5 ns! , which is in turn much smaller than the dielectric relaxation times responsible for reaching a steady state ~about 10 ns for the 9 nm/4 nm GaAs/AlAs superlattices of Ref. 10! This separation of time scales, as well as the configuration of a typical sample allows us to consider that only the ground state of each well is populated and that the tunneling processes are stationary. Our assumptions then justify using rate equations for the electron densities at each well with relations for the currents calculated by means of the transfer Hamiltonian ~TH! 18 The rate equations for the electron densities imply that the interwell currents and the currents from the emitter and to the collector are all equal to the total current in the stationary case ~a form of