« PreviousUpNext »Contents
Previous: A.3 Fourier Transform of the Screened Coulomb Potential    Top: A Integration over the Brillouin Zone    Next: A.5 The Normalized Fermi-Dirac Distribution

Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

A.4 The Normalized Maxwell-Boltzmann Distribution

The normalization factor for Maxwell-Boltzmann statistics and parabolic bands is defined as follows:

(A.12) \{begin}{align} C_\mathrm {MB}&=\int f_0(\kv ) \dint ^3\,k = \int \e ^{(E_F-\Epsilon (\kv ))/k_BT}\dint ^3\,k \{end}{align}

Introducing scaled variables (5.50) and (5.51) the integral becomes:

(A.13) \{begin}{align} C_\mathrm {MB}&=\e ^\eta \,\int \limits _0^\infty \e ^{-k^2/\tau ^2}\,4\pi \,k^2\dint k = 2\pi \,\tau ^3\,\e ^\eta \,\int \limits _0^\infty \e ^{-z}\,z^{1/2}\dint z \nonumber \\ &= 2\pi \,\tau ^3\,\e ^\eta \,\underbrace {\Gamma (3/2)}_{\sqrt {\pi }/2} =\pi ^{3/2}\,\tau ^3\,\e ^\eta \{end}{align}

The last integral on the right-hand side defines the Gamma function:

(A.14) \{begin}{align} \Gamma (3/2)= \frac {\sqrt {\pi }}{2}\,. \{end}{align}

A.4.1 Evaluating the Transition Rate Integral

With the normalized Boltzmann distribution \( p_0 \), the integral in the transition rate (5.48) can be evaluated as follows:

(A.15) \{begin}{align} \int \limits _{\abs {\kappa }}^\infty \mean {w}_0(\Epsilon _2,\Delta _1)\,k \dint k &= \int \limits _{\abs {\kappa }}^\infty p_0(\Epsilon )\,k \dint k = \frac {1}{C_\mathrm {MB}} \int \limits _{\abs {\kappa }}^\infty \e ^{\eta -k^2/\tau ^2}\,k \dint k \nonumber \\ &=\frac {\tau ^2\,\e ^\eta }{C_\mathrm {MB}} \int \limits _{\abs {\kappa }/\tau }^\infty \e ^{-u^2} u \dint \,u = \frac {\tau ^2\,\e ^\eta }{2\,C_\mathrm {MB}} \int \limits _{\abs {\kappa }^2/\tau ^2}^\infty \e ^{-z} \dint \,z \nonumber \\ &= \frac {\e {-\kappa ^2/\tau ^2}}{2\pi ^{3/2}\,\tau }= \frac {\hbar \,\e ^{-\kappa ^2/\tau ^2}}{(2\pi )^{3/2}\,\sqrt {m\,k_BT}} \label {eq:analytic-MB-limit} \{end}{align}

« PreviousUpNext »Contents
Previous: A.3 Fourier Transform of the Screened Coulomb Potential    Top: A Integration over the Brillouin Zone    Next: A.5 The Normalized Fermi-Dirac Distribution