## An efficient multigrid Poisson solver for device simulations

- Marco Saraniti, Achim Rein, Günther Zandler, Peter Vogl, Paolo Lugli
- IEEE Trans. on CAD of Integrated Circuits and…
- 1996

- Published 2004

We present a discussion of various concepts of cellular automata for semiconductor transport in the context of device simulation. A newly developed transformation for the kinetic terms of the Boltzmann equation into deterministic transition rules are found to be superior to probabilistic rules, allowing a complete suppression of statistical errors without any loss in numerical performance. To take advantage of the high speed of the resulting Cellular Automaton, a fast and flexible multigrid-solver for the Poisson equation has been developed. This enables us to study also fluctuations of transport quantities, which determine the high frequency noise behavior of MOSFETs, within the Cellular Automata approach. The reliability of the new CA approach for nanostructured devices is demonstrated by a study of gate length influence onto the drain current characteristics of a novel vertically grown MOSFET. Cellular Automata for transport simulations in semiconductor devices: Concepts and recent developments The physical effects involved in scaled down sub-pm devices require accurate physical simulation tools (1-3). Such simulation tools must be well beyond the level of the drift-diffusion approach to allow for a correct predictive description of high field transport. The Monte Carlo (MC) technique (43 is at present the most valuable approach to account for hot carrier effects and non local transport phenomena typical for such devices. Unfortunately, it is also one of the numerically most costly methods and remained therefore limited to university and laboratory research. Recently, a cellular automaton (CA) approach (6) has been developed as an efficient and discrete variant of the MC. In general, a cellular automaton consists of a lattice with a finite number of states attached to each lattice site. The realization of these states can be interpreted as a population with pseudoparticles. Their fictitious dynamics evolves on the microworld of the given lattice and can be updated simultaneously according to deterministic or nondeterministic rules in discrete time steps. Importantly, the dynamics of CA are governed by local rules, i.e. the updating of site variables involves only a small number of neighbors in each time step. For this reason, CA constitute one of the very few algorithms for physical processes which can optimally utilize massively parallel computer technology. In addition, the representation on a discrete lattice together with the locality of the dynamical rules allows an efficient and flexible treatment of complex geometries. The continuous physical quantities are obtained in practice by taking averages over many lattice sites. In our present implementation for the solution of the Boltzmann equation, the microscale of the CA consists of a hexagonal two dimensional lattice in position space, each site of which has a finite number of momentum cells. These momentum cells are defined on the nodes of a periodic hexagonal close-packed lattice in three dimensional momentum space. The transition rules between these states, associated to collision events and semiclassical motion , are determined from the quantum mechanical scattering rates (in the same way as in MC) and from the classical equations of motion. Due to the locality in position space of quantum mechanical scattering events, which is a basic assumption underlying the Boltzmann equation , there is no principle problem to convert these transitions in momentum space into spatially local CA-rules (6). In a typical simulation, the number of momentum states are of the order of some lo4. The scattering rules are tabulated in coarse grained hierarchical tables to avoid excessive usage of memory. This results in moderate memory requirements for the pretabulated scattering rates, typically on the order of tens of megabytes for a standard subpm-MOSFET simulation. The hierarchical tables mentioned above allow the efficient determination of the transition to the final state.

@inproceedings{Zandler200431,
title={3 - 1 ( Invited ) Cellular Automata for Device Simulation - Concepts and Applications},
author={G{\"{u}nther Zandler and Marco Saraniti and Alexander Rein and Peter Vogl},
year={2004}
}