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

TL;DR

This paper introduces a finite difference numerical method to solve nonlocal hydrodynamic equations modeling Bloch oscillations in semiconductor superlattices, revealing complex oscillatory behaviors at different temperatures.

Contribution

The paper develops an efficient second-order implicit finite difference scheme for nonlocal hydrodynamic equations describing Bloch oscillations, including novel solutions and analysis of numerical artifacts.

Findings

Spatially inhomogeneous Bloch oscillations coexist with electric field domains at 70 K.

At 300 K, only Bloch oscillations are observed.

Numerical artifacts are mitigated by using second-order schemes or smaller space steps.

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 solved 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.