## Electron screening

Mobile electrons will be attracted to positive ions in a solid. This negative charge compensates for some of the positive charge and reduces the electric field in the region around the positive ion. The phenomena is called screening.

Consider Gauss's law for a positive point charge located at the position $\vec{r}'$,

$$\nabla \cdot \vec{E} = \frac{e \delta(\vec{r}-\vec{r}')}{\epsilon_0}.$$

The relation between the electric field and the electrostatic potential $V$ is

$$\vec{E} = -\nabla V.$$

Combining these equations resuts in the Poisson equation,

$$\nabla^2 V = - \frac{e \delta(\vec{r}-\vec{r}')}{\epsilon_0}.$$

The solution of this equation is the Coulomb potential of a positive point charge,

$$V = \frac{e}{4\pi\epsilon_0|\vec{r}-\vec{r}'| }.$$

If a point charge is put in a metal, electrons will move towards it and screen it. Mathematically this effect is described by adding an induced charge density $\rho_{ind}$ to the point charge. The Poisson equation changes to

$$\nabla^2 V = - \frac{e \delta(\vec{r}-\vec{r}')}{\epsilon_0} - \frac{\rho_{ind}}{\epsilon_0}.$$

For small displacements of the induced charge density, the leading order term would be linear. It is mathematically convenient to just keep the linear term and describe the induced charge density as,

$$\frac{\rho_{ind}}{\epsilon_0} = -k_s^2V$$

where the constant $k_s$ is called the screening parameter.

Equation (5) can then be rewritten as the 3-dimensional Helmholtz equation,

$$\nabla^2 V - k_s^2V = - \frac{e \delta(\vec{r}-\vec{r}')}{\epsilon_0}.$$

The solution of the Helmholtz equation for a point charge gives a screened Coulomb potential, where the screening parameter $k_s$ specifies the strength of the screening effect,

$$V = \frac{e}{4\pi\epsilon_0|\vec{r}-\vec{r}'| } \exp(-k_s|\vec{r}-\vec{r}'|).$$

The exponential function reduces the range of the Coulomb potential, i.e. the Coulomb potential gets narrower.

 $V(r)$ eV $r$ Å

$k_s=$0.2 Å-1

$n$ $\Large =\frac{\pi^4\hbar^6\epsilon_0^3k_s^6}{2m^3e^6}=$ m-3

Here $n$ is the electron density of the metal using the model described below.

### Thomas-Fermi screening length

To calculate the screening parameter $k_s$ it is necessary to find a relation between the induced charge density $\rho_{ind}$ and the potential $V$. To find such a relation, consider a free electron gas with its typical quadratic dispersion relation. Since the electrostatic potential changes as a function of position, the free electron parabola moves up and down with the potential. The fermi energy on the other hand has the same value for any position. This leads to a smaller electron concentration for a parabola that was moved up (if the parabola is moved up, it is narrower at the fermi energy).

If the free electron parabola was moved up by an energy $eV$ the change of the electron concentration is

$$\Delta n = -D(E_F)eV.$$

The induced charge density is then given by the charge times the electron concentration,

$$\rho_{ind} = e \Delta n = -e^2 D(E_F) V.$$

The density of states at the fermi energy of a three dimensional free electron gas is given by

$$D(E_F) = \frac{3n}{2E_F}$$

with the fermi energy

$$E_F = \frac{\hbar^2}{2m}(3\pi^2n)^\frac{2}{3}.$$

Using this induced charge density the Helmholtz equation can be written as

$$\nabla^2 V - \frac{3e^2n}{2\epsilon_0 E_F}V = - \frac{e \delta(\vec{r}-\vec{r}')}{\epsilon_0}.$$

Comparing this equation to equation (7) yields

$$k_s^2 = \frac{3e^2n}{2\epsilon_0 E_F} = \frac{3^{1/3}me^2n^{1/3}}{\epsilon_0\hbar^2\pi^{4/3}}.$$

The inverse of the screening parameter $\frac{1}{k_s}$ is called Thomas-Fermi screening length. The screening length strongly depends on the electron density. For high electron density, the Thomas-Fermi screening length is short, indicating strong electron screening. For electron densities approaching zero also the screening parameter goes to zero. This makes the exponential factor in equation (8) vanish, leading to an unscreened Coulomb potential.

### References

1. Charles Kittel: Introduction to Solid State Physics (8th edition), Chapter 14