My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
8.2.1 Approximate analytic solution for an infinite slab
It is possible to obtain an approximation to u pl by taking the slab width s and the
cell size L to infinity, in which case the sums become integrals:
∑
→ s π
k z>0
∫ ∞
0
dk z (8.94)
∑
∫ ∞ ∫ ∞
→ L2
dk
(2π) 2 x dk y (8.95)
k
−∞ 0
‖
The system is no longer a slab but a single surface (at z = 0) of infinite extent in
the x- and y-directions.
Note that a further approximation has also been introduced here: the k-space
cut-off described in the previous section has been neglected in order to enable the
integrals to be solved analytically.
In this limit, the bulk term becomes
∫
u ∞ bulk(r, r ′ ) =
e2 4
ω p ɛ 0 V k k cos k 2 ‖ · (r ‖ − r ′ ‖) sin k z z sin k z z ′ V d 3 k
π(2π) 2 Θ(z)Θ(z′ )
e 2 ∫ ∞
)
=
dk
2π 3 z
(cos k z (z − z ′ ) − cos k z (z + z ′ )
ɛ 0 ω p
×
∫ ∞
0
dk ‖
0
k ‖
k 2 ‖ + k2 z
∫ π
0
dθ cos ( k ‖ ∆r ‖ cos(θ − φ) ) Θ(z)Θ(z ′ ) (8.96)
where ∆r ‖ = r ‖ − r ′ ‖ and φ = tan−1 (∆y/∆x). Performing the integration gives
u ∞ bulk(r, r ′ ) =
=
e 2 ∫ ∞
2π 3 ɛ 0 ω p 0
∫ ∞
k ‖
×
0
e 2 ∫ ∞
2π 2 ɛ 0 ω p
0
)
dk z
(cos k z (z − z ′ ) − cos k z (z + z ′ ) Θ(z)Θ(z ′ )
( )
dk ‖
k‖ 2 + πJ 0 k‖ ∆r ‖ Θ(z)Θ(z ′ )
k2 z
( )
dk z
(cos k z (z − z ′ ) − cos k z (z + z ′ )
)K 0 kz ∆r ‖
× Θ(z)Θ(z ′ )
(
)
e 2
π
=
2π 2 ɛ 0 ω p 2 √ (z − z ′ ) 2 + (∆r ‖ ) − π
2 2 √ Θ(z)Θ(z ′ )
(z + z ′ ) 2 + (∆r ‖ ) 2
( √ )
e 2
1
=
1 −
Θ(z)Θ(z ′ ). (8.97)
4πɛ 0 ω p |r − r ′ | 1 + 4zz′
|r−r ′ | 2
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