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Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

4.3 Multi-Band Semiconductors

The formalism in the sections above is assuming carrier transport in one band only. However, the extension of the formalism to a many band structure is straightforward. The definitions of the normalization factors (3.58) and (4.30) have to be extended by a summation over the band index \( n \) [P4]:

(4.67–4.68) \{begin}{align} Z(T) &= \sum \limits _n\int \limits _\mathrm {BZ} \e ^{-\beta (T)\, \Epsilon _n(\vec {k})}\dk \\ V(T) &= \sum \limits _n \int \limits _\mathrm {BZ}|v_x^{(n)}(\kv _0)|\,\e ^{-\beta (T)\,\Epsilon _n(\kv _0)} \dk \{end}{align}

The band energy \( \Epsilon _n \) denotes the energy of an electron in band \( n \) with respect to the band edge energy \( E_C \). Equation (4.32) defining the electron concentration at equilibrium has to account for a summation over the band index as well.

(4.69) \{begin}{align} n(\rv )=&\frac {\mathcal {C}(\rv )}{4\pi ^3}\sum \limits _n \int \limits _\mathrm {BZ} \, \e ^{-\beta (T)\, \Epsilon _n(\vec {k})}\dk \{end}{align}

The definition (4.33) of normalization factor \( \mathcal {C} \) remains unchanged. Sampling an equilibrium trajectory in a multi-band simulation yields random injection states of the form \( (n_0, \kv _0) \), where \( n_0 \) is the initial band index.

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