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Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

2.4 The Boltzmann Transport Equation

In classical transport theory, the Boltzmann Transport Equation (BTE) describes the kinetics of gas. In semi-classical carrier transport, this equation is used to describe the kinetics of particles with a quantum-mechanical extension [51, 95]. The BTE combines motion in \( \vec {r} \) and \( \vec {k} \) space as well as scattering processes [51]. The distribution function \( f(\vec {r},\vec {k},t) \) represents the probability of finding a carrier with crystal momentum \( \vec {p} \), at location \( \vec {r} \), at time \( t \). The BTE can be interpreted as a “bookkeeping” equation for the distribution function [65]. Thus it can be seen as a continuity equation for carriers in the six-dimensional phase-space [60], see Fig. 2.10. The following equation represents the BTE for multiple bands [49, 51, 65, 66]:

(2.26) \{begin}{multline} \left \{\frac {\partial }{\partial t} + \vec {F}_n(\vec {r},t) \cdot \nabla _{\vec {k}} + \vec {v}_n(\vec {k}) \cdot \nabla _{\vec {r}} \right \} f_n(\vec {r},\vec {k},t) = \\ \sum \limits _{n’}\int \limits _\mathrm {BZ} S_{n’,n}(\vec {k}’,\vec {k})f_{n’}(\vec {r},\vec {k’},t) -S_{n,n’}(\vec {k},\vec {k}’)f_{n}(\vec {r},\vec {k},t) \,\mathrm {d}^3 k’ , \label {eq:BTE} \{end}{multline}

where \( n \) represents the band index. The left hand side represents the total time derivative of the distribution function \( f(\vec {r},\vec {k},t) \) [60]. The right hand side of the BTE describes all scattering processes into the state \( (n,\vec {r},\vec {k}) \) and also out of the same state to any arbitrary state [37, 51].

(-tikz- diagram)

Figure 2.10: Illustration of the continuity aspect in \( \vec {k} \) and \( \vec {r} \) space of the BTE

The solutions to the BTE are the distribution functions for all bands, \( f_n(\vec {r},\vec {k},t) \). With the knowledge of \( f_n \), all quantities of interest could be calculated. However, the numerical solution of the BTE is difficult. Consequently, different approaches have been developed over the past decades to achieve a satisfying solution. The following chapter shall introduce the most commonly used approaches.

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