An accurate
description
of non-local effects is of utmost importance for modern semiconductor
devices. In particular, the distribution function has to be modeled
properly, as it is inherently linked with hot-carrier effects. In the
framework of hydrodynamic and energy-transport models only the average
energy is known. This does not provide enough information about the
shape of the distribution function.
To overcome the limitations of the available energy-transport models, a
consistent transport model based on six moments of Boltzmann's equation
has been developed. In addition to the concentration and the carrier
temperature, as provided by the energy-transport models, we obtain the
average of the square of the energy, which we map to a new solution
variable,
the kurtosis of the distribution function.
Compared with the simple drift-diffusion model, higher order methods
give stronger coupling between the device equations, as the physical
models for mobility are functions of the local carrier temperature. The
success and practical applicability of numerical simulation depends
critically on the convergence rate as well as on the control of
numerical
instabilities arising from discretization errors.
The aim of our research is to increase the robustness of the solution
algorithm for the six moments model. Higher order moment systems are
complex nonlinear systems. A robust solution requires a lot of
simulation
experience and fine tuning of the nonlinear solver.
As a first step in the investigation, a proposed algorithm using a
generalized Scharfetter Gummel scheme was implemented in one dimension.
Correctness of the implementation has been checked by comparison with
the analytical result known from the bulk case.
To find out the best highest order moment closure for the six moments
model, a variety of approximations for the sixth moment were tested.
Among them are a generalized Maxwellian closure with a free parameter
c, a cumulant closure, a closure derived from the diffusion
approximation and finally a closure from higher order statistics. Our
results suggest that the generalized Maxwellian closure works best on
practical examples. We obtain c by requiring consistency with bulk
Monte Carlo simulations.
To investigate the accuracy of the six moments model and its
corresponding energy transport model, we consider a series of
one-dimensional n+ - n - n+ test structures with
varying channel length.
The figure shows the results of numerical solutions of six moments
models
and compares them to self-consistent Monte Carlo data (SCMC) and to
results from an energy transport model. The energy transport models
show the well-known overestimation of the device currents. The results
of the six moments model, on the other hand, stay close to the SCMC
results, which makes the six moments model a good choice for TCAD
applications.
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