Abstract

Physical Modeling and Monte-Carlo-Simulation
of Electron Mobility in Silicon

The development of integrated circuits requires physical understanding of charge transport and carrier mobility, respectively. The aim of this work is to investigate the validity of state-of-the-art electron mobility models and to evaluate critically their underlying assumptions and approximations. Furthermore, new approaches will be discussed to improve the agreement between experiment and theory.

In doped silicon the electron mobility is predominantly decreased by ionized impurity scattering. Relying on a single site picture the frequently used Brooks-Herring impurity scattering model has been improved in several aspects.

The Thomas-Fermi atomic model is used to describe the spatial extent of the impurity's electron distribution. Assuming a screened Coulomb charge density for the electron distribution we calculate the total energy of the impurity atom. Minimizing the energy functional one obtains a size parameter for each dopant species, which is a measure of the spatial extent of the electron charge distribution.

Assuming linear superposition a general expression for the scattering potential of N spatially extented impurities can be derived. Having no further information about the impurity distribution we restrict the study to pair scattering, which is shown to have a strong impact on the electron mobility over a large concentration range.

Finally, we incorporate dispersive screening in our electron mobility model by a rational approximation of the Lindhard dielectric function. Its influence on the majority electron mobility becomes significant above an impurity concentration of 10¹⁸ cm^-3.

We use the Born approximation up to second order to calculate the scattering cross section. An exact expression for the second scattering amplitude for a screened Coulomb charge density is derived.

Assuming a spatially extented impurity the electron mobility dependence on the chemical nature of the dopant atom is explained in a natural way without introducing any parameter. The results confirm not only the lower majority electron mobility in As- and Sb-doped silicon compared to P-doped samples, but also the higher minority electron mobility and its different temperature behavior in B-doped Si. Recent simulation results of n- and p-doped GaAs and InP confirm the validity of our theoretical model as well for compound semiconductors.

Calculating the size parameter we are confronted with uncertainties about the correct value for the dielectric constant. In fact, no final decision about the influence of the spatially extented density can be made, as the size parameter depends on the dielectric constant. However, if further studies confirm the validity of the vacuum dielectric constant on an atomic scale, we again would have to rethink the problem of the dependence of the electron mobility on the chemical nature of the dopant atom.

Monte-Carlo simulations of the low field electron mobility are presented for silicon using an analytic band structure and Fermi-Dirac statistics. To improve the efficiency of the Monte-Carlo simulator we used a piecewise linear total scattering rate decreasing the self-scattering events far below ten percent. Additionally we proposed a new method reducing small-angle scattering significantly. Hence, far less scattering events are necessary to obtain significant simulation results. The impurity scattering rate need not to be calculated but the scattering process is selected by a rejection/acceptance scheme.

From these Monte-Carlo data analytical expressions for the majority- and minority electron mobility as a function of impurity concentration, temperature, and dopant species have been derived. To extend the application area of the analytical mobility model we have fitted the expressions to experimental data rather to simulations when the validity of the underlying physical models breaks down. This way the mobility formulas are valid in the temperature range [77-500] K and the impurity concentration range [10¹⁴, 10²²] cm^-3. This is to date the first analytical mobility model for device simulation capable of considering all common dopants in silicon.