My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
Unlike the bulk plasmons, the electrostatic surface plasmon modes (equations
(8.84a) and (8.84b)) are not degenerate in frequency; the dispersion relation is given
in equation (8.82), and plotted in figure 8.5. The same restrictions on k ‖ apply as
in the bulk case, 6 with the additional constraint that k ‖ ≠ 0. The surface plasmon
contribution to the two-body function is therefore
u surf (r, r ′ ) =
e2 ∑
(√
2
[
]
φ +
ω p ɛ 0 1 + e −k ‖s 1k ‖
(r)φ + 1k ‖
(r ′ ) + φ + 2k ‖
(r)φ + 2k ‖
(r ′ )
k ‖
√
2
[
] )
+
φ −
1 − e −k ‖s 1k ‖
(r)φ − 1k ‖
(r ′ ) + φ − 2k ‖
(r)φ − 2k ‖
(r ′ )
= e2 ∑
cos k ‖ · (r )
‖ − r ′ ‖ Fk‖ (z, z ′ ) (8.90)
ω p ɛ 0 V
k ‖
where F k‖
is defined in table 8.1. The constants used in this definition are
A k‖ =
B k‖ =
s √ 2
√
2k ‖ 1 + e
−k ‖ s
s √ 2
√
2k ‖ 1 − e
−k ‖ s
C k‖ = 1 − e k ‖s
(8.91)
D k‖ = 1 + e k ‖s .
There is no surface plasmon contribution to the one-body term; the integral
∫
φ ± jk ‖
(r ′ )¯n(z ′ ) d 3 r ′ (8.92)
is zero for all surface plasmon modes φ ± jk ‖
.
V
6 The surface plasmon dispersion curve tends to ω p / √ 2 when k ≫ 1/s; it should therefore meet
the electron-hole continuum earlier than the bulk plasmon dispersion curve (which is flat, lying at
ω = ω p ). It could be argued that a smaller cut-off should be used for the surface plasmon modes,
but because the cut-off is only expected to be correct to within an order of magnitude, this seems
unnecessary.
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