Set of supplementary notes.

2.5. SCREENING AND THOMAS-FERMI THEORY 29
µ
δn(r)
E (r)
F
−e V (r) ext
−e V (r)
tot
Figure 2.2:
Thomas-Fermi approximation
metal is neutral. An external potential will, however, cause a redistribution of charge, leading
to local accumulation of positive or negative charge, which will tend to screen the external
potential. The net effect will be that the total potential seen by an individual electron in
the Schrödinger equation is less than the external potential. We wish to calculate the charge
density induced by such an external potential ρ ind ([V ext (r)]).
Jellium. The potential in the problem is the total potential (external plus induced, V tot =
V ext + δV ) produced by the added charge and by the non-uniform screening cloud (see Fig. 2.2)
− h¯ 2
2m ∇2 ψ(r) + (−e)(δV (r) + V ext (r))ψ(r) = Eψ(r) . (2.26)
Slowly varying potential. Assume that the induced potential is slowly varying enough
that the energy eigenvalues of (2.26) are still indexed by momentum, but just shifted by the
potential as a function of position:
E(k, r) = E 0 (k) − eV tot (r)) , (2.27)
where E 0 (k) follows the free electron, parabolic dispersion h¯2k 2
. This only makes sense in
2m
terms of wavepackets, but provided the potential varies slowly enough on the scale of the Fermi
wavelength 2π/k F , this approximation is reasonable.
Constant chemical potential. Keeping the electron states filled up to a constant energy
µ requires that we adjust the local Fermi energy E F (r) (as measured from the bottom of the
band) such that 2 .
µ = E F (r) − eV tot (r) , (2.28)
Local density approximation. We assume that E F just depends on the local electron
number density n via the density of states per unit volume g V (E):
∫ EF
gV (E)dE = n . (2.29)
2 One is often sloppy about using E F and µ interchangeably; here is a place to take care