Predictive and Efficient Modeling of Hot Carrier Degradation with Drift-Diffusion Based Carrier Transport Models

Chapter B Calculating the Moments of the Maxwell Distribution

The $i^{th}$ moment of the maxwellian distribution can be calculated from:

$\left \langle \phi _{i}\right \rangle =\int k^{i}A\exp \left [\frac {-\varepsilon }{k_{\mathrm {B}}T_{\mathrm {n}}}\right ]d^{3}k$

by transforming the $k$ -space into spherical polar coordinates as: $\int d^{3}k=\int _{0}^{\infty }4\pi k^{2}dk$ .

$(B.1) \begin{equation} \left \langle \phi _{i}\right \rangle =\int _{0}^{\infty }k^{i}A\exp \left [\frac {-\varepsilon }{k_{\mathrm {B}}T_{\mathrm {n}}}\right ]4\pi k^{2}dk\label {eq:0th_moment} \end{equation}$

Assuming a parabolic band dispersion relation $\varepsilon =\frac {\hbar ^{2}k^{2}}{2m}$ , Equation B.1 becomes

$(B.2) \begin{equation} \left \langle \phi _{0}\right \rangle =2\pi A\left (\frac {2m}{\hbar ^{2}}\right )^{\frac {3}{2}}\int _{0}^{\infty }\varepsilon ^{i+\frac {1}{2}}\exp \left [\frac {-\varepsilon }{k_{\mathrm {B}}T_{\mathrm {n}}}\right ]d\varepsilon \end{equation}$

Using the value of the gamma function ( $\Gamma (x)=\int _{0}^{\infty }e^{-u}u^{x-1}du$ ) at $1/2$ , $\Gamma (1/2)=\sqrt {\pi }$ , and the property $\Gamma \left (x+1\right )=x\Gamma \left (x\right )$ the $i^{th}$ moment is reduced to:

$(B.3) \begin{equation} \left \langle \phi _{i}\right \rangle =2\pi A\left (\frac {2mk_{\mathrm {B}}T_{\mathrm {n}}}{\hbar ^{2}}\right )^{\frac {3}{2}}\left (k_{\mathrm {B}}T_{\mathrm {n}}\right )^{i}\left (i+\frac {1}{2}\right )\Gamma \left (i+\frac {1}{2}\right )\label {eq:ith moment} \end{equation}$

The parameter A in Equation B.3 can be evaluated by normalizing the distribution function.