Introduction

Consider a system of free electrons or of electrons in a semiconductor within the effective mass approximation. Since the density of states for free electrons, $\rho(E)\propto E^{1/2}$, the density of electrons, $n$, below a chemical potential $\mu$ is $n\propto\mu^{3/2}$. If an electrical potential energy, $V$ is added while keeping the electrochemical potential energy (or Fermi level), $E_F = \mu + V$, constant, then the electron density changes to $n\propto (E_F - V)^{3/2}$.

In the Thomas-Fermi approximation this charge density is assumed to be a local quantity. Poisson's equation for the electrical potential then takes the form

$${\d^2 V\over\d x^2} \propto (E_F - V)^{3/2}\quad,$$ (5.1)

which can be simplified into the form

$${\d^2 \phi\over\d x^2} = A \phi^{3/2}\quad.$$ (5.2)

This is a second order differential equation, so the general solution must contain 2 unknowns. One solution is known, namely

$$\phi = {400\over A^2(x - x_0)^4}\quad,$$ (5.3)

which contains the single unknown $x_0$.

The object of the exercise here is to find a more general solution, or at least something which may be used as such (i.e. a series expansion).