My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
This reproduces equation (7.62). In the slab and the vacuum, ω 2 p is constant; the
equation of motion in each of these regions becomes
∇ 2 ¨φ = −ω
2
p ∇ 2 φ. (8.75)
Searching for a travelling-wave solution of the form
φ(r, t) = φ z (z)e i(ωt−kx) (8.76)
leads to the following equation for φ z :
(
ω 2 k 2 φ z
)
− d2 φ z
= ω 2
dz 2 p
(
)
k 2 φ z − d2 φ z
. (8.77)
dz 2
As noted above, the bulk plasmons (with ω = ω p ) are solutions of this equation.
When ω ≠ ω p , the solutions must satisfy Laplace’s equation:
d 2 φ z
dz 2 − k2 φ z = 0. (8.78)
In the case of the slab, the solutions take the simple form
⎧
Ae
⎪⎨
kz when z < 0
φ z = Be kz + Ce −kz when 0 < z < s
⎪⎩
De −kz otherwise.
(8.79)
This ensures that φ → 0 as z → ±∞, as long as the trivial solution with k = 0 is
excluded. The other boundary conditions (as derived in section 8.1.3) are:
∂φ (m)
( )
∂x
1 − ω2 p ∂φ
(m)
ω 2 ∂z
= ∂φ(v)
∂x
(8.80)
= ∂φ(v)
∂z . (8.81)
Applying these conditions to equation (8.79) leads to the dispersion relation
√
ω 1 ± e
−ks
=
ω p 2
(8.82)
which is plotted in figure 8.5. The relationship between the amplitudes A, B, C,
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