346
11 Surfaces
0
1
2
3
4
0.0
0.5
1.0
Frequency (ω
max
)
C surface (C 1 )
M =M , C =2C
2
1
2
1
C 2
Fig. 11.10 One-dimensional model calculation of surface vibrational state frequency (in units of the maximum bulk
optical phonon frequency ω m ) as a function of the spring constant C surface of the surface atom replacing C 1 (blue curve)
or C 2 (black curves). As model parameters we use equal masses and C 2 = 2 C 1 (γ = 0.943 from (5.17)). The grey areas
represent the energy range of the acoustic and optical phonon bulk bands
J‘
J
Energy (eV)
40
30
20
10
0
GaAs (110)
Fig. 11.11 Surface phonon dispersion for GaAs(110) (solid lines) with the surface projected bulk band structure (light
grey area). The symboles are experimental data from two different methods. The dark grey areas at about 10 meV indicate
regions where the A 1 -peak is present in the calculated scattering cross-section. Adapted from [1063]
bulk band structure (Fig. 11.11). This concept pertains to phonon dispersion as well as to electronic
states.
11.6.2 Surface Plasmons
Free-carrier oscillations in the bulk had been discussed in Sect. 9.9.1. A surface plasmon is the quantum
of a surface-bound plasma oscillation. Such effect has been discussed in [1074] and reviewed in
[1075, 1076]. A metal (or conductive semiconductor) with dielectric constant according to (9.77)
1 = m = r (ω p /ω)
2 and a dielectric (or vacuum) with 2 = d are assumed (Fig. 11.12).