My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
z z ′ F k‖ (z, z ′ )
< 0 < 0 e k ‖(z+z ′ −s) ( −A k‖ C k‖ + B k‖ D k‖
)
)
< 0 0 → s e k ‖(z+z ′ −s)
(−A k‖ + B k‖
< 0 > s e k ‖(z−z ′ ) ( A k‖ C k‖ + B k‖ D k‖
)
)
+ e k ‖(z−z ′ )
(A k‖ + B k‖
0 → s 0 → s 2 cosh [ k ‖ (z + z ′ − s) ] ( −A k‖ /C k‖ + B k‖ /D k‖
)
+2 cosh [ k ‖ (z − z ′ ) ] ( A k‖ /C k‖ + B k‖ /D k‖
)
]
]
0 → s > s e k ‖(s−z−z ′ )
[−A k‖ + B k‖ + e k ‖(z−z ′ )
[A k‖ + B k‖
> s > s e k ‖(s−z−z ′ ) ( −A k‖ C k‖ + B k‖ D k‖
)
Table 8.1: The function F k‖ (z, z ′ ). This contains all the z- and z ′ -dependence of the part of u pl
arising from the surface plasmon contribution. The constants A k‖ , B k‖ , C k‖ and D k‖ are defined
in equation (8.91).
Combining the bulk and surface terms gives the full plasmon two-body function:
u pl (r, r ′ ) = u bulk (r, r ′ ) + u surf (r, r ′ )
= e2 ∑
(
cos k ‖ · (r ‖ − r ′
ω p ɛ 0 V
‖) F k‖ (z, z ′ )+
k ‖
+ ∑ k z
4
k 2 sin k zz sin k z z ′ Θ(z)Θ(s − z)Θ(z ′ )Θ(s − z ′ )
)
. (8.93)
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