Part 2 : Nanostuctures Module 3 : Electron Transport
Lecture 2 : BTE (Contd...)
 
using the definition of $ \Gamma$-function as
$\displaystyle \Gamma(j+1)=j!=\int_0^{\infty}\,x^j\, e^{-x}\,dx,$
and $ \Gamma(3/2)=\sqrt{\pi}/2$. Although this result has been obtained using classical ideas, for the quantum case of electrons one can use the Fermi-Dirac distribution for $ f_{eq}(\epsilon)$, $ \it i.e.$,
$\displaystyle f_{eq}(\epsilon)= \frac{1}{e^{\beta(\epsilon-\epsilon_F)}+1},$
where $ \epsilon_F$ is the Fermi energy for the electrons, so that, putting $ x=\beta\epsilon$ and $ \eta=\beta\epsilon_F$, one gets (integrate the numerator by parts assuming $ r+3/2>0$),
$\displaystyle <\tau_m> = \frac{2}{3}\,(r+3/2)\tau_o\, \frac{\int_0^{\infty} \, ... ...=\frac{4}{3\sqrt{\pi}}\,(r+3/2)!\tau_o\, \frac{F_{r+1/2}(\eta)}{F_{1/2}(\eta)},$
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