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 <ref name="Huxley"> Huxley, L. G. H. Leonard George Holden, The diffusion and drift of electrons in gases, John Wiley and sons, 1974 , call number QC793.5.E628 H89 </ref>

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]

where

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

and

[math]\displaystyle{ x' = \frac {D_L}{D} x }[/math] [math]\displaystyle{ y' = \frac {D_L}{D} y }[/math]

In spherical coordinates:

[math]\displaystyle{ \frac {1}{r'^2} \frac{\partial{}}{\partial{r'}}r'^2 \frac{\partial{V}}{\partial{r'}} + \frac {1}{r'^2 sin\theta'} \frac{\partial{}}{\partial{\theta}} sin\theta \frac{\partial{V}}{\partial{\theta}} = \lambda_L^2 V }[/math] which is symmetric in [math]\displaystyle{ \phi }[/math] direction.

Assuming [math]\displaystyle{ V(r',\theta) = R_k(r')P_k(\mu) }[/math]the solution of the zenith angle direction is the Legendre polynomial, and can be written as:

[math]\displaystyle{ \frac {1}{r'sin\theta} \frac{\partial{}}{\partial{\theta}} sin\theta\frac{\partial{V}}{\partial{\theta}} = R_k(r') \frac{d}{d \mu} \left [ (1- \mu^2) \frac{d{P_k(\mu)}}{d{\mu}} \right] }[/math]

and

[math]\displaystyle{ \frac{d}{d \mu} \left [ (1- \mu^2) \frac{d{P_k(\mu)}}{d{\mu}} \right]= -k(k+1) P_k(\mu) }[/math]

so,

[math]\displaystyle{ \frac {1}{r'^2} \frac{d}{dr'}\left (r'^2 \frac{dR_k}{dr'}\right) - \left [ \frac{k(k+1)}{r'^2} +\lambda_L^2 \right]R_k = \frac{d^2 R_k}{dr'^2} +\frac{2}{r'} \frac{dR_k}{dr'}-\left [ \frac{k(k+1)}{r'^2} +\lambda_L^2 \right]R_k = 0 }[/math]

The modified Bessel functions, first and second kind, are the solutions for the previous equation but the boundary conditions determines which one to use, in this case [math]\displaystyle{ r'\rightarrow 0 }[/math], [math]\displaystyle{ n \rightarrow \infty }[/math], and [math]\displaystyle{ n \rightarrow 0 }[/math] as [math]\displaystyle{ r'\rightarrow \infty }[/math]. so only the modified Bessel of second kind K_k are the non-zaro terms. so the the general solution for the equation can be written as :

[math]\displaystyle{ V= R_k (r') Pk(\mu) = \exp{(\lambda_L z)}\sum_{k=0}^{\infty} A_k r'^{-1/2} K_{k+1/2} (\lambda_L r') P_k(\mu) }[/math]

Solution Analysis

The general form of the previous equation and its solution are defined as the following:

[math]\displaystyle{ \frac{d^2 y}{dx} +\frac{1-2\alpha}{x} \frac{dy}{dx}-\left [ \frac{\nu^2 \gamma^2 - \alpha^2}{x^2} + (\beta \gamma x^{\gamma - 1})^2 \right]y = 0 }[/math]

and

[math]\displaystyle{ y = x^{\alpha} I_{\nu} (\beta x^{\gamma}) }[/math] and

[math]\displaystyle{ y = x^{\alpha} K_{\nu} (\beta x^{\gamma}) }[/math]

where [math]\displaystyle{ I_{\nu} }[/math] and [math]\displaystyle{ K_{\nu} }[/math] are the modified Bessel of the first and the second kind.

The previous solution bcan be written as

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