My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
vacuum
q
z
metal
Figure 8.2: The surface charge density.
To relate this to the previous calculation, note that v z = ˙q. It then follows that
E (v)
z
− E (m)
z
= − n 0ev z
iɛ 0 ω
(8.25)
= σ
iɛ 0 ω E(m) z (8.26)
= − ω2 p
ω 2 E(m) z . (8.27)
To summarise, the boundary conditions at an interface between metal and vacuum
are:
E (m)
x
E (m)
y
= E (v)
x
= E (v)
y
(8.28a)
(8.28b)
( )
1 − ω2 p
E (m)
ω 2 z = E z (v) . (8.28c)
Note that from now on, ω p is being used exclusively to refer to the plasma frequency
of the metal.
In the limit z → ±∞, all field components are required to tend to zero. This excludes
both the non-physical (constant or increasing) and the radiative (oscillatory)
solutions.
8.1.4 Surface plasmons
In order to maintain the relationship specified by equation (8.28) at the interfaces,
k and ω must take the same values in the metal and the vacuum. One consequence
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