My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
of the boundary conditions at infinity is that K must be real if any field is to exist
in the vacuum; equation (8.21) then implies that
This condition also ensures that K is real in the metal. 4
with
k > ω/c. (8.29)
The surface plasmon solutions (ω ≠ ω p ) for E x and E z are therefore
⎧
E 0 e
⎪⎨
−Kvz
when z > s
E x = E 1 e −Kmz + E 2 e Kmz when 0 < z < s
⎪⎩
E 3 e Kvz
otherwise
⎧
−
⎪⎨
ik
K v
E 0 e −Kvz
when z > s
(
E z = ik −E1
K m
e −Kmz + E 2 e Kmz) when 0 < z < s
⎪⎩ ik
K v
E 3 e Kvz
otherwise
√
(8.30)
(8.31)
K v = k 2 − ω2
(8.32)
√
c 2
K m = k 2 − ω2
c + ω2 p
2 c . (8.33)
2
The y-component of the field, as was mentioned previously, is not coupled to the
others. This component is transverse everywhere, because ∇ · E = 0, even at the
interfaces. Thus, E y is not related to any change in the charge density, and will not
be dealt with further.
The boundary conditions at z = 0 and z = s impose the relations:
E 0 e −Kvs = E 1 e −Kms + E 2 e Kms
− ik
K v
E 0 e −Kvs =
E 3 = E 1 + E 2
(
1 − ω2 p
ik
K v
E 3 =
) ik (
−E1 e −Kms + E
ω 2 2 e Kms)
K m
( )
1 − ω2 p ik
(−E
ω 2 1 + E 2 ) .
K m
(8.34a)
(8.34b)
(8.34c)
(8.34d)
4 This is not required, merely a consequence of equation (8.29).
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