11.6 Surface Physical Properties
347
Fig. 11.12 Sketc.h of
surface plasmon field
distribution. Adapted
from [1077]
Fig. 11.13 Surface
plasmon polariton
dispersion for ZnO:Ga/air
for three different carrier
concentrations as labelled,
experimental data
(symbols) and theory
(lines). Adapted
from [1078]
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Energy (eV)
0
2
4
6
8
Wave vector (10 m )
6
- 1
ZnO:Ga
Wavelength (µm)
2
3
4
5
1.5
10
n=1.2 10 cm
20
-3
n=3.2 10 cm
20
-3
n=7.2 10 cm
20
-3
The surface plasmon (polariton) (SPP) is a wave localized at the surface with evanescent parts into
the metal and the dielectric. The dispersion of the surface plasmon (polariton) is given by
k SPP =
ω
c
1 2
1 + 2
.
(11.3)
For large k, the limiting frequency is the SPP frequency (from 1 = 2 )
ω SPP =
ω p
√
1 + d / r
< ω p ,
(11.4)
which is smaller than the plasma frequency. For a metal against vacuum, ω SPP = ω p /
√
2. The SPP
dispersion for a ZnO:Ga/air interface is depicted in Fig. 11.13 for three different doping concentrations.
11.6.3 Electronic Surface States
The bulk band structure is given by the energy eigenvalues E n (k). The surface adds its own states
E
surf
n (k || ), many of them in the gap. The calculation of gap states has been already briefly discussed
in Sect. 6.2.3. The nature of bulk states is oscillatory in the bulk and exponentially decaying in the
outside (Fig. 11.14a), the nature of surface states is localized at the surface and decaying both towards