Figure 84: Unscreened and screened potential.So the resulting potential isV =
−exp (−k|r−r′|)
4 πε|r−r′|. (182)
Fig. 84 shows the unscreened and the screened Coulomb potential. The distance how far the screened
potential can reach, the screening length, is determined byks. The Thomas-Fermi approximation is
one possible way to calculate it, where
k^2 s=
3 e^2 n
ε 0 EF. (183)
The so-called Thomas-Fermi screening length depends mostly on the electron densitynin the metal.
If the electron density is large,ksbecomes large and the screening length is small and vice versa. This
means that a metal with a large electron density can screen the potential of the extra electron better
than a metal with a small electron density.
The actual screening of a point charge looks different to an exponential decay. For example fig. 85
shows an STM picture of GaAs doped with Si on Ga-sites. Around the dopants there are the screening
clouds of electrons. But the charge goes up and down and makes these oscillations which are called
Friedel oscillations. The reason for these oscillations is that the electrons that shield the point charge
(δ-distribution) have a certain wave numberk. But the ones with the highest wave number (shortest
wavelength) are aroundkF. Therefore you can’t build structures with electron waves that are smaller
than this wavelength and so the reaction distribution oscillates around the point charge.