My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
the interface:
⎧
⎪⎨ ik
K
ρ s = ɛ
m
(−E 1 + E 2 ) − ik
K v
E 3 at z = 0
0
⎪ ( ⎩ − ik
K v
E 0 e −Kvs − ik −E1
K m
e −Kms + E 2 e Kms) at z = s.
(8.44)
The dependence on x and t is exactly as for the E-field. Using equations (8.34),
(8.36) and (8.38), the various field amplitudes may be related to each other and thus
eliminated (except for one):
E 0 = ±e Kvs E 3
E 1 = 1 − R E 3
2
E 2 = 1 + R E 3 .
2
(8.45a)
(8.45b)
(8.45c)
The surface charge density is then
ρ s = ± ɛ 0ik
K v
ω 2 p
E
ω 2 − ωp
2 3 . (8.46)
The positive sign always applies at z = 0. For symmetric oscillation, it is also taken
at z = s: the surface charge density variations on the two interfaces are then in
phase. The negative sign applies at z = s in the case of antisymmetric oscillation,
so that the two surface charge densities are in antiphase. The lower branch of the
dispersion graph corresponds to symmetric oscillation.
The electric field created by the surface charge is given by the solution of equation
(8.1), with a charge density of
ρ(x, z, t) = ρ 0 (ω) ( δ(z) ± δ(z − s) ) e i(ωt−kx) . (8.47)
This longitudinal (electrostatic) field may be expressed as the gradient of a scalar
potential, φ. Assuming a solution of the form
φ(x, z, t) = φ z (z)e i(ωt−kx) (8.48)
means that the scalar potential must satisfy the equation
)
(−k 2 + d2
φ
dz 2 z (z) = − ρ 0
(δ(z) ± δ(z − s)) . (8.49)
ɛ 0
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