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Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

A.5 The Normalized Fermi-Dirac Distribution

The normalization factor for Fermi-Dirac statistics is defined as:

(A.16) \{begin}{align} C_\mathrm {FD}&=\int f_0(\kv )\dint ^3 k = \int \limits _0^\infty \frac {4\pi \,k^2}{\e ^{k^2/\tau ^2-\eta }+1} \dint k=4\pi \,\tau ^3\int \limits _0^\infty \frac {u^2}{e^{u^2-\eta }+1}\dint u\nonumber \\ &=2\pi \,\tau ^3\int \limits _0^\infty \frac {z^{1/2}}{e^{z-\eta }+1}\dint z= 2\pi \,\tau ^3\,\underbrace {\Gamma (3/2)}_{\sqrt {\pi }/2}\,\mathcal {F}_{1/2}(\eta ) =\pi ^{3/2}\,\tau ^3\,\mathcal {F}_{1/2}(\eta ) \{end}{align}

Here, \( \mathcal {F}_{1/2} \) denotes the Fermi integral of order \( 1/2 \). The Fermi integral of order \( j \) is defined as:

(A.17) \{begin}{align} \mathcal {F}_{j}(x) = \frac {1}{\Gamma (j+1)}\int \limits _0^\infty \frac {z^j}{\e ^{z-x}+1}\dint z \{end}{align}

A.5.1 Evaluating the Transition Rate Integral

Substituting \( \mean {w}_0=w_0/C_\mathrm {FD} \), the integral in the transition rate (5.48) can be rearranged as follows:

(A.18) \{begin}{align} \int \limits _{\abs {\kappa }}^\infty \mean {w}_0(\Epsilon _2,\Delta _1)\,k \dint k &= \left (\frac {\tau ^2\e ^\eta }{2C_\mathrm {FD}} \right ) \int \limits _{\abs {\kappa }}^\infty \e ^{-\eta } w_0(\Epsilon _2,\Delta _1)\,\frac {2k}{\tau ^2} \dint k \label {eq:norm-FD-rearange} \{end}{align}

Using the relation (5.30), the function w can be reformulated as

(A.19) \{begin}{align} w_0(\Epsilon _2,\Delta _1)&=f_\mathrm {FD}(\Epsilon _2)\left [ 1- f_\mathrm {FD}(\Epsilon _2 -\Delta _1 \right ]\nonumber \\ &=f_\mathrm {FD}(\Epsilon _2)f_\mathrm {FD}(\Epsilon _2-\Delta _1)\e ^{\beta (\Epsilon _2-\Delta _1-E_F)} \{end}{align}

With the variable substitutions

(1.20–A.19) \{begin}{align} d_1 = \beta \Delta _1, \qquad u=\frac {k^2}{\tau ^2},\qquad \dint u=\frac {2k}{\tau ^2}\dint k, \qquad u_1=\frac {\kappa ^2}{\tau ^2}\„\nonumber \{end}{align}

the integral in (A.18) can be evaluated in the manner of:

(A.20) \{begin}{align} I &=\int \limits _{\abs {\kappa }}^\infty \e ^{-\eta } w_0(\Epsilon _2,\Delta _1)\,\frac {2k}{\tau ^2} \,d^3k = \int \limits _{\abs {\kappa }}^\infty \frac {\e ^{k^2/\tau ^2-\beta \Delta _1-2\eta }}{(\e ^{k^2/\tau ^2-\eta }+1)(\e ^{k^2/\tau ^2-\beta \Delta _1-\eta }+1)} \frac {2k}{\tau ^2} \dint k\nonumber \\ &=\int \limits _{u_1}^\infty \frac {\e ^{u-d_1-2\eta }}{(\e ^{u-\eta }+1)(\e ^{u-d_1-\eta }+1)}\dint u = \e ^{-\eta }\int \limits _{u_1}^\infty \frac {\e ^{\eta -u}}{(1+\e ^{\eta -u})(1+\e ^{\eta +d_1-u})} \dint u \{end}{align}

With the further substitutions

(A.21) \{begin}{align} &v=\e ^{\eta -u},\qquad &&v_1=v(u_1)=\e ^{\eta -\kappa ^2/\tau ^2},\qquad v_2=v(\infty )=0, \\ &\dint v= -\e ^{\eta -u}\dint u, \qquad &&a=\e ^{d_1},\nonumber \{end}{align}

the integral gets transformed to:

(A.22) \{begin}{align} I&= \e ^{-\eta }\int \limits _{0}^{v_1} \frac {1}{(1+v)(1+a\,v)}\dint v = \frac {\e ^{-\eta }}{1-a} \left [ \ln (1+v_1) -\ln (1+a\,v_1)\right ]\nonumber \\ &= \frac {\e ^{-\eta }}{1-\e ^{d_1}} \left [ \ln (1+\e ^{\eta -\kappa ^2/\tau ^2}) -\ln (1+\e ^{\eta +d_1-\kappa ^2/\tau ^2})\right ] =\frac {\e ^{-\eta }}{1-\e ^{d_1}} \ln \left ( \frac {1+\e ^{\eta -\kappa ^2/\tau ^2}}{1+\e ^{\eta +d_1-\kappa ^2/\tau ^2}} \right ) \{end}{align}

A.5.2 Asymptotic behaviour of EES
\( F(p,\gamma ) \) at low Energies

With the relation

(1.23–A.22) \{begin}{align*} \frac {\e ^{x}-\e ^{-x}}{2}=\sinh \,x \{end}{align*}

the formula (5.85) can be rewritten to

(A.23) \{begin}{align} F(p,\gamma ) = \frac {\e ^{-p^2}}{p}\int \limits _0^\infty \frac {s^2}{s^2+\gamma ^2}\,\sinh {(2ps)}\, \e ^{-s^2} \dint s \„ \{end}{align}

where the limit for \( p \rightarrow 0 \) can be obtained. With the relation

(A.24) \{begin}{align} \lim \limits _{p\rightarrow 0} \frac {\sinh {2ps}}{2ps}=1 \{end}{align}

the limit of \( F(p\rightarrow 0,\gamma ) \) becomes

(A.25) \{begin}{align} F(p\rightarrow 0,\gamma ) = 2 \int \limits _0^\infty \frac {s^3}{s^2+\gamma ^2}\e ^{-s^2}\dint s = \int \limits _0^\infty \frac {u}{u+\gamma ^2}\e ^{-u}\dint u \label {eq:asym-p0} \{end}{align}

With the substitution \( t=u+\gamma ^2 \) and \( x=\gamma ^2 \) the integral can be reformulated

(A.26) \{begin}{align} F(p\rightarrow 0,x) =\e ^x \left ( \int \limits _x^\infty \frac {\e ^{-t}}{t}\dint t \,-x \int \limits _x^\infty \frac {\e ^{-t}}{t}\dint t\right ) = 1+ x\,\e ^x\,\mathrm {Ei}(-x)\,. \{end}{align}

Here, Ei denotes the exponential integral function. This result shows that also an electron with zero kinetic energy (\( p=0 \)) is affected by e-e scattering.

\( F(p,\gamma ) \) at low Concentrations

From the formula (5.85) the limit for \( \gamma \rightarrow 0 \) can be obtained. In this case, the concentration \( n \) is zero and therefore there is no screening.

(A.27) \{begin}{align} F(p,\gamma \rightarrow 0) &= \frac {1}{2p} \int \limits _0^\infty \left ( \e ^{-(s-p)^2} - \e ^{-(s+p)^2} \right ) \dint s = \frac {1}{2p} \left ( \int \limits _{-p}^\infty \e ^{-t^2} \dint t -\int \limits _p^\infty \e ^{-t^2} \dint t \right )\nonumber \\ &=\frac {1}{2p}\int \limits _{-p}^p \e ^{-t^2}\dint t = \frac {\sqrt {\pi }}{2p}\mathrm {erf}(p) \{end}{align}

The function \( \mathrm {erf}(x) \) denotes the error function

(A.28) \{begin}{align} \mathrm {erf}(x)=\frac {2}{\sqrt {\pi }}\int \limits ^x_0 \e ^{-t^2}\dint t \,. \{end}{align}

Setting \( \gamma =0 \) in (A.25), the asymptotic maximum of the function \( F(p,\gamma ) \) can be derived

(A.29) \{begin}{align} F(0,0) = \int \limits _0^\infty \e ^{-u}\dint u = 1\,. \{end}{align}

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