Surface magneto-plasmons in magnetic multilayers - Walther ...

Section 2.2
Dispersion relation 9
In metals the electrons can be treated as a free electron gas. The dielectric function
for a free electron gas can be calculated with the Drude formula[28, 5], giving
ε ′(1) (ω) = 1 − ne2
ε0m∗ω2 = 1 − ω2 p
ω
2 , (2.17)
with the electron density n, the elementary charge e, ε0 being here the vacuum per-
meability and m ∗ the effective mass of the electrons, and ωp is the plasma frequency
of metals,
ωp =
ne2 . (2.18)
ε0m∗ With nAu = 5, 9 × 10 28 m −3 [29] the plasma frequency becomes ωp = 1.36×10 16 rad/s.
Equations (2.17) and (2.15) can thus be rewritten
k ′ x = ω
c
ε′(2)
1 − ω2
p
ω
ε ′(2)
+ 1 − ω2 (2.19)
p
ω
This is the approximate dispersion relation for surface plasmons at a metal-dielectric
interface, where the complex part of the metal’s permeability is neglected.
In Fig. 2.2 the dispersion relation for an air-gold interface (ε ′(2) = 1) is plotted (black
line). It splits up into two branches with a forbidden gap in between. The upper
branch becomes for kx → 0 equal to the plasma frequency ωp (upper green line) in
Fig. 2.2(a) and increases for kx → ∞. Because ωp can be regarded as the plasma
frequency of volume plasmons the upper branch describes the dispersion relation for
volume plasmons [30]. The lower branch becomes zero for kx → 0. For kx → ∞ the
dispersion relation ω approaches the surface plasmon frequency (lower green line).
ωsp =
ωp
√ 1 + ε ′(2)
(2.20)
This is the limiting case when the real part of the metal’s dielectric function ap-
proaches the permeability of the dielectric ε ′(1) → −ε ′(2) . Or in other words the
damping part of the metal ε ′′(1) can be neglected. With Eq. (2.18) and E = ω the
energy of surface plasmons is in the order of Esp = 4.8 eV
So in the case of kx → ∞ the group velocity (∂ω/∂kx) and the phase velocity (ω/kx)
goto zero. Therefore, surface plasmon are localised longitudinal oscillations of the