Solution details

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

[math]\displaystyle{ (\frac {\partial^2{}}{\partial{x^2}} +\frac {\partial^2{}}{\partial{x^2}}) }[/math]n + [math]\displaystyle{ D_L \frac {\partial^2{}}{\partial{z^2}} }[/math] - [math]\displaystyle{ W \frac {\partial{}}{\partial{z}} }[/math] n = 0

Steps to solve Boltzmann equation

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

[math]\displaystyle{ n(x', y', z') = e^{\lambda_L z'} V(x,y,z) }[/math]

so

[math]\displaystyle{ \nabla'^2 V = \lambda_L^2 V }[/math]

[math]\displaystyle{ \nabla'^2 V = \frac {\partial^2{}}{\partial{x'^2}} + \frac {\partial^2{}}{\partial{y'^2}} + \frac {\partial^2{}}{\partial{z^2}} }[/math]