4.3.1 Low Field Carrier Mobility and Monte Carlo Techniques

The low field carrier mobility of is an important kinetic property of a bulk semiconductor. It is used to analyze carrier transport in semiconductor devices at low applied voltages and enters expressions for high field mobility models as an additional parameter. Thus the knowledge of the low field carrier mobility and its correct dependence on the material properties such as the doping concentration are necessary to adequately simulate carrier transport in semiconductor devices.

The standard Monte Carlo approach for obtaining the low field carrier mobility is a single particle Monte Carlo method. In order to calculate the low field mobility along the direction of the electric field one has to carefully choose the magnitude of the applied electric field. On the one hand, the magnitude of the electric field must be as low as possible. In principle it is desirable to have zero electric field. However, there exist limitations related to the increase of the variance of standard Monte Carlo methods. On the other hand, the field must not be too high to avoid a mobility reduction due to carrier heating.

In addition to these disadvantages, the standard approaches only give one component of the carrier mobility, namely the component in the direction of the electric field. For isotropic conditions it does not make any difference since the mobility tensor is diagonal and all diagonal values are equal. However, when anisotropy is present, for example in strained semiconductors, the mobility tensor elements may be different and several Monte Carlo simulations are required to obtain all the components of the tensor.

To overcome these problems associated with the standard Monte Carlo methods a new Monte Carlo algorithm has been suggested recently [73], which solves the Boltzmann equation for zero electric field and represents a limiting case of the small signal algorithm obtained in [74]. One of the most remarkable properties of the algorithm is the absence of self-scattering that allows to significantly reduce calculation time and achieve very good accuracy of the results. This method is restricted to the simulation of lowly doped semiconductors. The quantum mechanical Pauli exclusion principle is not included in the scattering term of the Boltzmann equation used for the derivation of the algorithm. As a result there are limitations on the doping level of the materials analyzed by this technique. It allows to obtain excellent results at low and intermediate doping levels while results obtained for higher doping levels, where the effects of degenerate statistics are more pronounced, are incorrect. As the standard Monte Carlo methods exhibit a very high variance especially in the degenerate case, it is thus desirable to have a powerful technique to analyze the carrier mobility at high doping levels.

In this chapter a zero field algorithm [75] used to account for degenerate statistics in strained SiGe layers is described. The Pauli exclusion principle is taken into consideration in the scattering term of the Boltzmann equation. As a result the Boltzmann equation becomes nonlinear. Using this nonlinear equation a generalized zero field algorithm applicable for the analysis of highly doped materials is derived.

S. Smirnov: