3.5.3 A Hierarchy of Transport Models

In the following sections, the method of moment is applied to Boltzmann's equation in order to systematically derive a set of equations for the carriers within a semiconductor. Depending on the number of equations involved, different levels of physical complexity can be described with the according models. Furthermore, additional driving forces caused by graded material alloys and hetero-structures can be considered or neglected, dependent on the device under analysis. All models have Poisson's equation in common, describing electrostatics within the device.

The simplest model is the isothermal drift-diffusion model, which is also accessible by a phenomenological approach. It consists of carrier balance equations and current equations for electrons and holes, respectively accounting for carrier drift induced by external driving forces as well as diffusion. The unknowns for the equation system are the electrostatic potential and carrier densities as well as current densities for both electrons and holes.

For the treatment of non-isothermal conditions the lattice temperature has to be included as an additional solution variable. The temperature is addressed by the heat-flow equation, which has to be solved self-consistently with the carrier balance and flux equations. For the non-isothermal drift-diffusion model, the equation series is truncated after the first moment and the equation system is closed by the condition of local thermal equilibrium $\ensuremath{\langle \ensuremath{\mathcal{E}} \rangle}_{\ensuremath{\mathrm{\en... ...\ensuremath{\langle \ensuremath{\mathcal{E}} \rangle}_{\ensuremath{\mathrm{L}}}$ . An elaborate discussion of the energy exchange between carriers and lattice is given in Section 3.5.10 and Section 3.5.11 based on a systematic and a phenomenological thermodynamic approach, respectively.

The set of models incorporating equations up to the first moment -- the particle current equations -- is able to accurately describe the behavior of comparably large structures. Their only parameters are the carrier mobilities, which are easily accessible by measurement. Besides their dependencies on temperature and dopant concentration, the mobilities depend on the shape of the distribution function. However, due to the lack of additional information incorporated in the equation system, they are modeled as field dependent. This approximation incorporates the neglection of non-local effects such as the velocity-overshoot [85], which can be addressed by proper modeling of an energy-dependent mobility. In small devices, non-local effects play an important role and the assumption of local thermal equilibrium breaks down, since carriers gain energy from strong electric fields and carrier temperatures much above the lattice temperatures are possible.

In order to include appropriate description of carriers driven far from equilibrium, the carrier energies as well as the according energy fluxes have to be included as additional solution variables. In order to maintain the additional unknowns, the energy balance equation as well as the energy flux equation for each of the carrier subsystems is included. In terms of the method of moments, the highest-order moment is the third one. This class of transport models is referred to as hydrodynamic transport models [86].

The systematic approach to the formulation of transport models can be continued by the inclusion of further equations obtained by higher moments of the Boltzmann transport equation. With each moment, more information on the carrier distribution function is included. A truncation after the fifth moment results in the six moments model including the kurtosis as well as the according flux as additional unknowns [87,78,76]. However, the consideration of these high moments is only of importance in devices driven far from equilibrium, such as modern CMOS devices of the deca-nanometer regime used in today's microprocessors and DRAM memories. In this work, transport models including equations up to the third moment are derived in order to accurately reproduce the energy relations within large thermoelectric devices.

M. Wagner: Simulation of Thermoelectric Devices