My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
z = s
z = 0
z
vacuum
metal
vacuum
x
Figure 8.1: The metal slab.
The appropriate versions of Maxwell’s equations are
∇ · E = ρ ɛ 0
∇ · B = 0
∇ × E = − ∂B
∂t
(8.1)
∇ × B = µ 0 J + 1 c 2 ∂E
∂t . (8.2)
8.1.1 The metal conductivity
In the jellium model of a bulk metal, the (smoothed-out) charge density is zero in the
steady state: the electron and background charges cancel each other out completely.
In equation (8.1), ρ refers to the net charge.
Any departure of ρ from zero will be caused by motion of electrons, because the
background charge is not free to move; the same is true for any current density. The
current density at the point r is therefore given by
J(r) = −n(r, t) e v(r, t) (8.3)
where n(r, t) is the electron density and v(r, t) is the velocity of the electrons at r.
For small amplitude oscillations, it is reasonable to make the linearising approximation
1 n(r, t) = n 0 (r), where n 0 is the steady-state electron density.
1 The formal approach is to treat any time-dependent quantity as a small perturbation. In fact,
n(r, t) = n 0 (r)− ρ(r,t)
e
. In the expression for the current density, the second-order term ρ(r, t)v(r, t)
is then discarded.
121