My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
Eliminating E 0 and E 3 gives
E 1 e −Kms + E 2 e Kms = 1 R
(
E1 e −Kms − E 2 e Kms) (8.35a)
−E 1 − E 2 = 1 R (E 1 − E 2 ) .
(8.35b)
where
A final rearrangement gives
R =
K m ω
( 2
). (8.36)
K v ω2 − ωp
2
E 1 (1 − R)e −Kms = E 2 (1 + R)e Kms
E 1 (1 + R) = E 2 (1 − R)
(8.37a)
(8.37b)
so that the dispersion relation is
(1 − R) 2
(1 + R) 2 = e2Kms . (8.38)
This may be solved numerically: results are shown in figure 8.3. To obtain an
approximate analytical solution, equation (8.38) is first expanded:
k = ω √ [tanh (K ms/2)] ±2 + ω 2 / ( ωp 2 − ω 2)
c [tanh (K m s/2)] ±2 − ω 4 / ( ωp 2 − ω 2) 2
(8.39)
Note that the right-hand side contains K m , which depends on k. However, equation
(8.33) shows that for small k (and therefore small ω) K m ≈ k p , where k p = ω p /c.
More precisely, let
∆ =
( k
k p
) 2
−
( ω
ω p
) 2
. (8.40)
Because k > ω/c, ∆ is always positive and is of order (k/k p ) 2 at most; when k ≪ k p ,
∆ ≪ 1 and an expansion in terms of ∆ is reasonable. Substituting for K m gives
( ) ( ) ( ) [ ( )] 2 Km s
kp s kp s kp s
tanh = tanh + ∆ sech + O[∆ 2 ]. (8.41)
2
2 4
2
Inserting this expression into equation (8.39) and expanding then leads to the following
approximate form of the dispersion relation:
√
k
= ω ( ) [
ω
1 + [tanh (k p s/2)] ∓2 2
( ) ] [ 7 ( ) ] 3
+ 2 ω4 ω ω
+ O + O ∆ 2 .
k p ω p ωp
2 ωp
4 ω p ω p
(8.42)
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