Abstract
We discuss the numerical aspects of the Boltzmann transport equation (BE) for electrons in semiconductor devices, which is stabilized by Godunov’s scheme. The k-space is discretized with a grid based on the total energy to suppress spurious diffusion in the stationary case. Band structures of arbitrary shape can be handled. In the stationary case, the discrete BE yields always nonnegative distribution functions and the corresponding system matrix has only eigenvalues with positive real parts (diagonally dominant matrix) resulting in an excellent numerical stability. In the transient case, this property yields an upper limit for the time step ensuring the stability of the CPU-efficient forward Euler scheme and a positive distribution function. Similar to the Monte-Carlo (MC) method, the discrete BE can be solved in time together with the Poisson equation (PE), where the time steps for the PE are split into shorter time steps for the BE, which can be performed at minor additional computational cost. Thus, similar to the MC method, the transient approach is matrix-free and the solution of memory and CPU intensive large systems of linear equations is avoided. The numerical properties of the approach are demonstrated for a silicon nanowire NMOSFET, for which the stationary I–V characteristics, small-signal admittance parameters and the switching behavior are simulated with and without strong scattering. The spurious damping introduced by Godunov’s (upwind) scheme is found to be negligible in the technically relevant frequency range. The inherent asymmetry of the upwind scheme results in an error for very strong scattering that can be alleviated by a finer grid in transport direction.
Highlights
The semiconductor industry is moving toward nanowire gate-all-around MOSFETs to improve the gate control [1,2,3]
The Boltzmann transport equation (BE) is a partial differential equation with the possibility of discontinuous solutions, which can lead to numerical problems when it is mapped onto a grid in phase space
Since the motion of particles is similar to waves in fluids, we use methods developed for computational fluid dynamics, which can handle, for example, discontinuous distribution functions [21]
Summary
The semiconductor industry is moving toward nanowire gate-all-around MOSFETs to improve the gate control [1,2,3]. By choosing a different set of functions that are nonnegative themselves for projection, this problem can be avoided [12, 13, 16,17,18,19,20] Such functions are directly defined in the phase space and are piecewise constant with a value of either zero or one. For the tessellated phase space, a discrete equivalent of the BE has to be found that conserves the nonnegativeness of the distribution function and that is numerically stable. Since the motion of particles is similar to waves in fluids, we use methods developed for computational fluid dynamics, which can handle, for example, discontinuous distribution functions [21]. Such an approach was presented in Ref. We discuss the problems of the spurious damping and asymmetry introduced by Godunov’s scheme
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