Solution details

asymptotic solution details for Boltzmann equation for a hole has a uniform electric field

$(\frac {\partial^2{}}{\partial{x^2}} +\frac {\partial^2{}}{\partial{x^2}})$n + $D_L \frac {\partial^2{}}{\partial{z^2}}$ - $W \frac {\partial{}}{\partial{z}}$ n = 0

Steps to solve Boltzmann equation

for the previous equation let consider the asymptotic solution has the form:

$n(x', y', z') = e^{\lambda_L z'} V(x,y,z)$

so

$\nabla^2 V = \lambda_L^2 V$