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Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

A.2 Average Energy and Injection Velocity

The statistical average (3.59) can be calculated with the band-structure given in the irreducible wedge and decomposed in \( M \) thetrahedra, as follows:

(A.5) \{begin}{align} \braket {A}=&\frac {48}{Z}\sum \limits _n \sum \limits _{m=1}^M \hspace {2mm}\int \limits _{\text {Tet }m} A_n(\vec {k})\e ^{-\beta \Epsilon _n(\vec {k})}\dk . \label {eq:est_wedge} \{end}{align}

\( \{A_0,A_1,A_2,A_3\} \) are the attribute values in the tetrahedron vertices. The following derivation assumes a linear variation of the attribute and the energy in the tetrahedron.

(A.6) \{begin}{align} = V_m \int \limits _0^1 \dint \xi _0 \int \limits _0^{1-\xi _0} \dint \xi _1 \int \limits _0^{1-\xi _0 - \xi _1} \dint \xi _2\, \left ( \xi _0\left (A_0-A_3\right ) + \xi _1\left (A_1-A_3\right ) \right . \nonumber \\ +\left .\xi _2\left (A_2-A_3\right ) +A_3 \right ) \, \e ^{-\beta \left (\xi _0 \left (\Epsilon _0-\Epsilon _3\right ) + \xi _1 \left (\Epsilon _1-\Epsilon _3\right ) + \xi _2 \left (\Epsilon _2-\Epsilon _3\right ) + \Epsilon _3 \right )} \{end}{align}

The statistical averages (A.5) becomes

(A.7) \{begin}{align} \braket {A}=&\frac {48 }{Z}\sum \limits _n \sum \limits _m V_m \sum \limits _{i=0}^3 \frac {\e ^{-\beta \Epsilon _i} \bigg ( A_i+\sum \limits _{\substack {j=0 \\ j\neq i}}^3\left ( \frac {A_j - A_i}{\beta \left ( \Epsilon _j - \Epsilon _i \right )} \right ) \bigg )}{\beta ^3 \prod \limits _{\substack {j=0 \\ j\neq i}}^3 \left ( \Epsilon _j - \Epsilon _i \right )}. \{end}{align}

If the attribute of interest is the energy, \( A(k)=\Epsilon (k) \), the average gets reduced to

(A.8) \{begin}{align} \label {eq:energyest} \braket {\Epsilon }=&\frac {48 }{Z}\sum \limits _n \sum \limits _m V_m \sum \limits _{i=0}^3 \frac {\e ^{-\beta \Epsilon _i} \left ( \frac {3}{\beta } + \Epsilon _i \right )}{\beta ^3 \prod \limits _{\substack {j=0 \\ j\neq i}}^3 \left ( \Epsilon _j - \Epsilon _i \right )}. \{end}{align}

In the case where the desired attribute is the injection velocity, \( A(k)=|v_x(k)| \) which is constant in each tetrahedron, the average (4.30) is reduced to

(A.9) \{begin}{align} \label {eq:vxest} \braket {|v_{x}|}=&\frac {48 }{Z}\sum \limits _n \sum \limits _m V_m\,|v_{x,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}

where \( v_{x,m} \) is the constant velocity in \( x \)-direction in the tetrahedron with the index \( m \). Because of symmetry, the absolute value has to be used, otherwise the result would be zero.

In the special case where \( \Epsilon _i=\Epsilon _k \) the expression for the average energy can be written as:

(A.10) \{begin}{align} \braket {\Epsilon }=&\frac {48 }{Z}\sum \limits _n \sum \limits _m V_m \sum \limits _{i=0}^3 \frac {\e ^{-\beta \Epsilon _i} \left ( \frac {3}{\beta } + \Epsilon _i \right )}{\beta ^2 \prod \limits _{\substack {j=0 \\ j\neq i\\j\neq k}}^3 \left ( \Epsilon _j - \Epsilon _i \right )}, \{end}{align}

because of the limit

(A.11) \{begin}{align} \lim \limits _{\Delta \rightarrow 0} \frac {1-\e ^{-\beta \Delta }}{\Delta }=\beta . \{end}{align}

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