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Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

5.2 Two-particle Transition Rate

The transition rate from the initial state \( \ket {\kv _1, \kv _2} \) to the final state \( \ket {\kv _1’, \kv _2’} \) is expressed by Fermi’s golden rule:

(5.12) \{begin}{align} P(\kv _1, \kv _2,\kv _1’, \kv _2’) = \frac {2\pi }{\hbar }\abs {\braket {\kv _1’, \kv _2’|U_s|\kv _1, \kv _2}}^2\,\delta \left ( E(\kv _1’, \kv _2’) - E(\kv _1, \kv _2) \right )\,. \label {eq:fermi-ees-2p} \{end}{align}

The total energy of state \( \ket {\kv _1,\kv _2} \) is equal to \( E(\kv _1,\kv _2) = \Epsilon (\kv _1) + \Epsilon (\kv _2) \). Inserting the matrix element (5.11) in (5.12), the transition rate becomes:

(5.13) \{begin}{multline} P_2(\kv _1,\kv _2,\kv _1’,\kv _2’) = \\ \frac {2\pi }{\hbar } \left (\frac {e^2}{\epsilon _s\Omega }\right )^2 \frac {\delta _{\kv _1 + \kv _2, \kv _1’ + \kv _2’}}{ \left (\abs {\kv _1 -\kv _1’}^2 + \beta _s^2\right )^2}\; \delta \bigl [\Epsilon (\kv _1’) + \Epsilon (\kv _2’) - \Epsilon (\kv _1) - \Epsilon (\kv _2) \bigl ]. \label {eq:ee-trans} \{end}{multline}

This derivation is based on the assumptions of a finite normalization volume \( \Omega \) and the resulting discreteness of the \( \kv \)-vectors. Thus, a Kronecker-delta is obtained, which is idempotent.

(5.14) \{begin}{align} \delta ^2_{\kv _1 + \kv _2, \kv _1’ + \kv _2’} = \delta _{\kv _1 + \kv _2, \kv _1’ + \kv _2’} \{end}{align}

5.2.1 Principle of Detailed Balance

In Section 4.1.2 we introduced a novel backward Monte Carlo method, which utilizes the principle of detailed balance. This section shows that the expression for electron-electron scattering also obeys the principle of detailed balance.

The transition rate (5.13) is conserving the total energy of the two particles involved. Thus, it describes an elastic scattering process. The principle of detailed balance states that for elastic processes the transition rate is symmetric [49, 62].

(5.15) \{begin}{align} P_2(\kv _1,\kv _2,\kv _1’,\kv _2’) = P_2(\kv _1’,\kv _2’,\kv _1,\kv _2) \label {eq:symm} \{end}{align}

The symmetry property can be seen in the original definition of the transition rate:

(5.16) \{begin}{align} P_2(\kv _1,\kv _2,\kv _1’,\kv _2’) = \frac {2\pi }{\hbar } \left | \bra {\kv _1’,\kv _2’} U_s \ket {\kv _1,\kv _2} \right |^2\; \delta \bigl [\Epsilon (\kv _1’) + \Epsilon (\kv _2’) - \Epsilon (\kv _1) - \Epsilon (\kv _2) \bigl ]\qquad \{end}{align}

Since both, the absolute value of the matrix element and the \( \delta \)-function, are invariant under interchange of initial and final state, so is the transition rate.

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