Abstract
Charge transport in nanosized electronic systems is described by semiclassical or quantum kinetic equations that are often costly to solve numerically and difficult to reduce systematically to macroscopic balance equations for densities, currents, temperatures and other moments of macroscopic variables. The maximum entropy principle can be used to close the system of equations for the moments but its accuracy or range of validity are not always clear. In this paper, we compare numerical solutions of balance equations for nonlinear electron transport in semiconductor superlattices. The equations have been obtained from Boltzmann–Poisson kinetic equations very far from equilibrium for strong fields, either by the maximum entropy principle or by a systematic Chapman–Enskog perturbation procedure. Both approaches produce the same current-voltage characteristic curve for uniform fields. When the superlattices are DC voltage biased in a region where there are stable time periodic solutions corresponding to recycling and motion of electric field pulses, the differences between the numerical solutions produced by numerically solving both types of balance equations are smaller than the expansion parameter used in the perturbation procedure. These results and possible new research venues are discussed.
Highlights
The maximum entropy principle is often used to derive macroscopic equations from a microscopic theory [1]
In previous works [21], we have shown that the balance equations obtained by using the Chapman–Enskog method provide an excellent approximation to the kinetic Boltzmann–Poisson system of equations for realistic parameter values appearing in experiments [20]
We have shown how to derive macroscopic balance equations for the electric field and the electron density for nonlinear charge transport in SLs by combining the maximum entropy closure of moment equations and a simple perturbation expansion that holds in the limit as the Bloch frequency, eFM l, and the collision frequency νen τe are of the same order and dominate all other terms in the kinetic equation
Summary
The maximum entropy principle is often used to derive macroscopic equations from a microscopic theory [1]. A typical situation is that macroscopic quantities are written in terms of averages over some density or distribution function and that exact equations for them can be found that involve those macroscopic quantities and unknown averages of magnitudes or fluxes of magnitudes [2]. These averages are unknown because the microscopic density is not known. The maximum entropy closure produces macroscopic balance equations in a straightforward manner once the macroscopic variables, that are moments of the distribution function, have been chosen. This closure is different from a systematic perturbation procedure and the validity of the resulting macroscopic equations has to be decided by some other means: by comparison of their solutions with numerical
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