My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
This solution includes the case k ‖ = 0 discussed previously. Note that k z is restricted
to positive values.
Any electric field within the slab may be expressed as a sum of the normal
modes described in equation (8.65). However, any physical field is real; it is useful
to convert the set of complex modes to an equivalent set of real modes by forming
appropriate linear combinations. 5
The symmetry relation
E k‖ k z
= −E ∗ (−k ‖ )k z
(8.66)
which follows from equation (8.65) shows that real modes may be obtained by taking
the combinations
E 1k (r) =
√ i
)
(E k‖ k z
(r) + E (−k‖ )k z
(r)
2
= 2
k √ V
(
k‖ sin k z z sin k ‖ · r ‖ − k z cos k z z cos k ‖ · r ‖
)
E 2k (r) = 1 √
2
(E k‖ k z
(r) − E (−k‖ )k z
(r)
= 2
k √ V
)
(8.67a)
(
k‖ sin k z z cos k ‖ · r ‖ + k z cos k z z sin k ‖ · r ‖
)
. (8.67b)
There are now two modes for each k-vector, so the number of labels required is
reduced by half. The restriction applies to the xy-plane: if (k ‖ + k z ) is a valid label
for a mode, then (−k ‖ + k z ) is not. With this restriction, the modes described in
equation (8.67) form a complete orthonormal set, in the sense that
∫
E jk (r) · E j ′ k ′(r) d3 r = δ jj ′δ kk ′. (8.68)
V
The special case k ‖ = 0 is included in equation (8.67), with the understanding that
there is no type 2 mode.
The potential associated with these normal modes may be obtained by integra-
5 When quantising a Hamiltonian expressed in terms of the normal mode coordinates, it is convenient
for them to represent real quantities, and thus to be associated with Hermitian operators.
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