Numerical method for hydrodynamic modulation equations describing Bloch oscillations in semiconductor superlattices

Abstract

We present a finite difference method to solve a new type of nonlocal hydrodynamic equations that arise in the theory of spatially inhomogeneous Bloch oscillations in semiconductor superlattices. The hydrodynamic equations describe the evolution of the electron density, electric field and the complex amplitude of the Bloch oscillations for the electron current density and the mean energy density. These equations contain averages over the Bloch phase which are integrals of the unknown electric field and are derived by singular perturbation methods. Among the solutions of the hydrodynamic equations, at a 70 K lattice temperature, there are spatially inhomogeneous Bloch oscillations coexisting with moving electric field domains and Gunn-type oscillations of the current. At higher temperature (300 K) only Bloch oscillations remain. These novel solutions are found for restitution coefficients in a narrow interval below their critical values and disappear for larger values. We use an efficient numerical method based on an implicit second-order finite difference scheme for both the electric field equation (of drift-diffusion type) and the parabolic equation for the complex amplitude. Double integrals appearing in the nonlocal hydrodynamic equations are calculated by means of expansions in modified Bessel functions. We use numerical simulations to ascertain the convergence of the method. If the complex amplitude equation is * Corresponding author.Email addresses: mariano.alvaro@uc3m.es (M.Álvaro), manuel.carretero@uc3m.es (M. Carretero), bonilla@ing.uc3m.es (L. L. Bonilla) URL: http://scala.uc3m.es (L. L. Bonilla)Preprint submitted to Elsevier January 1, 2018solved using a first order scheme for restitution coefficients near their critical values, a spurious convection arises that annihilates the complex amplitude in the part of the superlattice that is closer to the cathode. This numerical artifact disappears if the space step is appropriately reduced or we use the second-order numerical scheme.

Charge transport in a superlattice: a numerical study using moment methods

“…Except for a narrow parameter range, Bloch oscillations are not stable states in doped SLs [BAC11,ACB12]. However there are other stable self-sustained oscillations (SSCO) of the current that are observed in a DC voltage biased SL.…”

“…These SLs had finitely many spatial periods and were subject to an appropriate DC voltage bias. In SLs made out of doped semiconductors, scattering usually destroys the Bloch oscillations but, in theory, they can persist even in the hydrodynamic regime for a SL with long scattering times [BAC11,ACB12].…”

Finite difference numerical method for the superlattice Boltzmann transport equation and case comparison of CPU(C) and GPU(CUDA) implementations

“…Fully functional solver for PMOSFET devices, which among other things can utilize FDM for solving BTE, was presented in [8]. Numerical method for spatially non-homogeneous 1D BTE in application to semiconductor superlattices was recently considered in [9].…”