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Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

A Integration over the Brillouin Zone

A.1 The partition function

This section is based on a previous work [P3]. The partition function [74] defined by (3.58) is evaluated by numerical integration. The contribution of band \( n \) is given by

(A.1) \{begin}{align} \label {eq:partfunc-appendix} Z_n= \int \limits _\mathrm {BZ} \e ^{-\beta \Epsilon _n(\vec {k})}\dk \{end}{align}

In VMC, only the irreducible wedge of the BZ is decomposed into \( M \) tetrahedra. One octant of the BZ can be represented by six mirroring operations of the irreducible wedge. In this case the partition function is calculated as

(A.2) \{begin}{align} Z_n=& 48 \sum \limits _{m=1}^M \hspace {2mm}\int \limits _{\text {Tet }m} \e ^{-\beta \Epsilon _n(\vec {k})}\dk , \{end}{align}

The integration over the irreducible wedge is now split into integrals over tetrahedra which can be performed using barycentric coordinates [P3]. The values of the attributes are given at the vertices of the tetrahedron. A linear interpolation of the values inside the tetrahedron is assumed. \( \{\Epsilon _0,\Epsilon _1,\Epsilon _2,\Epsilon _3\} \) are the energy values in the tetrahedron vertices and \( V_m \) is the volume of the \( m^\mathrm {th} \) tetrahedron. From the following integral

(A.3) \{begin}{align} =V_m \int \limits _0^1 \dint \xi _0\, \frac {1}{\beta ^2\left (\Epsilon _2-\Epsilon _3\right )\left (\Epsilon _1-\Epsilon _2\right )}\left (\e ^{-\beta \left (\xi _0\left (\Epsilon _0-\Epsilon _1\right )+\Epsilon _1\right )}-\e ^{-\beta \left (\xi _0\left (\Epsilon _0-\Epsilon _2\right )+\Epsilon _2\right )}\right ) \nonumber \\ -\frac {1}{\beta ^2\left (\Epsilon _2-\Epsilon _3\right )\left (\Epsilon _1-\Epsilon _3\right )}\left (\e ^{-\beta \left (\xi _0\left (\Epsilon _0-\Epsilon _1\right )+\Epsilon _1\right )}-\e ^{-\beta \left (\xi _0\left (\Epsilon _0-\Epsilon _3\right )+\Epsilon _3\right )}\right ) \{end}{align}

follows that

(A.4) \{begin}{align} Z=& \sum \limits _n Z_n = 48 \sum \limits _n \sum \limits _m V_m \sum \limits _{i=0}^3 \frac {\e ^{-\beta \Epsilon _i}}{\beta ^3 \prod \limits _{\substack {j=0 \\ j\neq i}}^3 \left ( \Epsilon _j - \Epsilon _i \right )} \{end}{align}

The spacial case where \( \Epsilon _j = \Epsilon _i \) is discussed in the next section.

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