My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
These solutions are transverse, because ∇ · E = 0 (except at the boundaries).
Equations (8.14a), (8.14b) and (8.14c) make it clear that ω = ω p constitutes
a special case.
The oscillations for which this condition is satisfied are the bulk
plasmons; note that there are no bulk plasmon solutions in the vacuum. 3
Equation (8.11) shows that there is no B-field at this frequency; then, from
equation (8.10), ∇ × E = 0, which implies that the E-field is longitudinal. The only
restriction on the functional form of the field is that
ik dE x
dz + k2 E z = 0. (8.22)
The remaining special case is when ω 2 = ω 2 p +k 2 c 2 (or in vacuum, ω = kc). Then
E x = E 0
E z = ikzE 0 + E 1
(8.23a)
(8.23b)
where E 0 and E 1 are constants.
8.1.3 Boundary conditions
At any plane interface, Maxwell’s equations may be used to demonstrate that the
component of the E-field parallel to the interface is continuous, while there is a
discontinuity in the component perpendicular to the interface equal to the surface
charge density.
The same free-electron model which was used to estimate the conductivity is
useful here. For the interface illustrated in figure 8.2, the surface charge density at
the interface is −n 0 eq, where q represents the displacement (in the direction normal
to the interface) of each electron at the boundary from its equilibrium position:
E (v)
z
− E (m)
z
= − n 0eq
ɛ 0
. (8.24)
3 The equivalent of a bulk plasmon in vacuum is the zero-frequency solution: the electric field
generated by a charge density which is constant in time.
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