2.5.5.3 Screening Theory

The screening effect is the most important manifestation of the electron-electron interaction. In this work only linear screening is considered. Nonlinear effects which require the solution of the nonlinear Poisson equation either analytically [46,47] or numerically [48] are neglected here.

A positively charged particle introduced into the electron gas will create an excess of negative charge in its vicinity which screen its electric field^2.35. The bare potential (2.142) satisfies the Poisson equation:

$$\varepsilon\nabla^{2}V_{0}(\vec{r})=-\rho_{0}(\vec{r}),$$ (2.144)

where $\rho_{0}(\vec{r})=Ze[\delta(\vec{r})+\delta(\vec{r}-\vec{R})]$ is the volume density of the particle charge^2.36. The total physical potential $V_{t}(\vec{r})$, which is produced both by the particle and the electron cloud, is described by the Poisson equation of the form:

$$\varepsilon\nabla^{2}V_{t}(\vec{r})=-\rho(\vec{r}),$$ (2.145)

where $\rho(\vec{r})$ is the total volume charge density:

$$\rho(\vec{r})=\rho_{0}(\vec{r})+\rho_\mathrm{ind}(\vec{r}).$$ (2.146)

Here $\rho_\mathrm{ind}(\vec{r})$ denotes the volume density of the charge induced in the electron gas by the positive particle. From the electrodynamics of continuous media [49] it is known that $V_{0}(\vec{r})$ and $V_{t}(\vec{r})$ are linearly related to each other:

$$V_{0}(\vec{r})=\int \varepsilon(\vec{r},\vec{r}^{'})V_{t}(\vec{r}^{'})\,d\vec{r}^{'}.$$ (2.147)

For the homogeneous electron gas function $\varepsilon(\vec{r},\vec{r}^{'})$ can only depend on the distance between the two points $\vec{r}$ and $\vec{r}^{'}$:

$$\varepsilon(\vec{r},\vec{r}^{'})=\varepsilon(\vec{r}-\vec{r}^{'}).$$ (2.148)

This gives the following relation between the Fourier components of $V_{0}(\vec{r})$ and $V_{t}(\vec{r})$:

$$V_{t}(\vec{q})=\frac{V_{0}(\vec{q})}{\varepsilon(\vec{q})}.$$ (2.149)

In the linear screening theory it is assumed that $\rho_\mathrm{ind}(\vec{r})$ and $V_{t}(\vec{r})$ are linearly related to each other^2.37 which implies for Fourier components:

$$\rho_\mathrm{ind}(\vec{q})=\chi(\vec{q})V_{t}(\vec{q}).$$ (2.150)

The Fourier transform of the Poisson equations (2.144) and (2.145) together with (2.150) leads to the relation:

$$\varepsilon(\vec{q})=1-\frac{1}{\vec{q}^{2}}\chi(\vec{q}).$$ (2.151)

To calculate the quantity $\chi(\vec{q})$ some approximations are necessary. In this work we use the so called Lindhard screening theory^2.38. Within this model the expression for the dielectric function [50] is:

$$\varepsilon(\vec{q},0)=1+\frac{\beta^{2}_{s}}{q^{2}}G\left(\xi,\eta\right),$$ (2.152)

with the inverse screening length given by the Thomas-Fermi theory:

$$\beta^{2}_{s}=\frac{e^{2}n}{\varepsilon k_{B}T_{L}} \frac{\mathcal{F}_{-1/2}\left(\eta\right)}{\mathcal{F}_{1/2}\left(\eta\right)}.$$ (2.153)

Here $\mathcal{F}_{i}$ stands for the Fermi integral of the $i$-th order, and $\eta$ is the reduced Fermi energy:

$$\eta=\frac{E_{F}-E_{C}}{k_{B}T_{L}}.$$ (2.154)

$G$ represents the screening function given by the following expression:

$$G\left(\xi,\eta\right)=\frac{1}{\mathcal{F}_{-1/2}\left(\eta\righ... ...1+\exp\left(x^{2}-\eta\right)}\ln\biggl\vert\frac{x+\xi}{x-\xi}\biggr\vert\,dx,$$ (2.155)

where the argument $\xi$ is:

$$\xi^{2}=\frac{\hbar^{2}q^{2}}{8m^{*}k_{B}T_{L}}.$$ (2.156)

The corrections described above can be taken into account within the first Born approximation. In this approximation the scattering amplitude is given as:

$$f\left(\vec{q}\right)=-\frac{1}{4\pi}U\left(\vec{q}\right),$$ (2.157)

where $U(\vec{q})$ is defined by

$$U(\vec{q})=-\frac{2eV_{t}(\vec{q})m^{*}}{\hbar^{2}}.$$ (2.158)

The differential cross section is defined as:

$$\sigma(\vec{q})=\frac{(2\pi\hbar)^{3}}{m^{*2}\vert\vec{v}(\vec{k})\vert}\vert f(\vec{q})\vert^{2}\rho(\epsilon).$$ (2.159)

Now the total scattering rate is obtained through

$$\lambda(\vec{k})=N_{p}\vert\vec{v}(\vec{k})\vert\sigma_\mathrm{tot}(\vec{k}),$$ (2.160)

where $N_{p}=N_{I}/2$ is the density of scattering pairs and $\sigma_\mathrm{tot}(\vec{k})$ is equal to:

$$\sigma_\mathrm{tot}(\vec{k})=\frac{2\pi}{\vec{k}^{2}}\int\sigma(\vec{q})q\,dq.$$ (2.161)

The simple calculations give the final total ionized impurity scattering rate taking into account the momentum dependent screening and the two-ion correction:

$$\lambda(\vec{k})=C\left(k\right) \int_{0}^{2k}\frac{1}{(q^{2}+\be... ...}G\left(\xi,\eta\right))^{2}}\left(1+\frac{\sin\left(qR\right)}{qR}\right)\,dq,$$

$$C\left(k\right)=\frac{N_{i}Z^{2}e^{4}}{2\pi\hbar^{2}\varepsilon^{2}\vert\vec{v}(\vec{k})\vert}.$$ (2.162)

S. Smirnov: