Problem $5\mathrm{.}8$:

Thermal velocity and mean free path in GaAs

Given that the electron effective mass me$$m$\_$e$^*$me$$ for GaAs is $0\mathrm{.}067$me$0\mathrm{.}067$m$\_$e$0\mathrm{.}067$me$,$ calculate the thermal velocity of the conduction band $($CB$)$ electrons. The electron drift mobility e$\backslash$mu$\_$ee$$ depends on the mean free time e$\backslash$tau$\_$ee$$ between electron scattering events $($between electrons and lattice vibrations$).$ Given:

e$=$ee$/$me$$ande$=8500$cm$2$V$1$s$1$for$$GaAs$,\backslash$mu$\_$e $=$ e $\backslash$tau$\_$e $/$ m$\_$e$^*\backslash$quad $\backslash$text$\{$and$\}\backslash$quad $\backslash$mu$\_$e $=8500\backslash ,\backslash$text$\{$cm$\}^2\backslash ,\backslash$text$\{$V$\}^\{-1\}\backslash ,\backslash$text$\{$s$\}^\{-1\}\backslash ,\backslash$text$\{$for GaAs$\},$e$=$ee$/$me$$ande$=8500$cm$2$V$1$s$1$for$$GaAs$,$

calculate e$\backslash$tau$\_$ee$,$ and hence the mean free path $\backslash$ell$$ of CB electrons.

How many unit cells is $\backslash$ell$$ if the lattice constant aaa of GaAs is $0\mathrm{.}565$nm$0\mathrm{.}565\backslash ,\backslash$text$\{$nm$\}0\mathrm{.}565$nm$?$

Calculate the drift velocity vd$=$eEv$\_$d $=\backslash$mu$\_$e $\backslash$mathcal$\{$E$\}$vd$=$e$$E of the CB electrons in an applied field E$=104$V$/$m$\backslash$mathcal$\{$E$\}=10^4\backslash ,\backslash$text$\{$V$/$m$\}$E$=104$V$/$m$.$

What is your conclusion?