66 3 Lattice Dynamics: Phonons
from those of GaAs and KI by convergence toward a degeneracy with the
acoustic branches, e.g., at the X point, while these two groups of branches
do not interpenetrate for GaAs and are even well separated by a gap for KI.
This behavior can be understood from the solution for Problem 3.1, the linear
chain with two atoms in the unit cell: If the two atoms in the unit cell have
different masses, the dispersion curves show a gap at the boundary of the
Brillouin zone (as for GaAs and KI); this gap closes, if the masses are equal
(as for Si). Even the difference between these gaps of GaAs and KI can be
explained within this model, it increases with increasing mass difference. The
other striking difference is the splitting of the optical phonon branches at the Γ
point for GaAs and KI – the longitudinal–transverse (or LT) splitting – while
these branches are threefold degenerate in Si. The splitting is a consequence
of the macroscopic p olarization inherent with a longitudinal optical mode in
a binary compound solid (such as GaAs and KI) that gives rise to a stronger
restoring force than for the transverse mo de, i.e., ω
L
>ω
T
, as discussed in
Sect.
3.5, and is connected with the Reststrahlen band. The rather flat optical
phonon branches lead to a pronounced peak in the density of states (DOS),
shown for GaAs in Fig.
3.13. Nevertheless, the phonon dispersion cur ves for Si
and GaAs resemble each other much more than those of GaAs and KI. This is
due to the similarity of the diamond and zinc blende structure and the small
mass difference between Ga and As. On the other hand, the characteristic fre-
quencies are higher in Si than in GaAs, which can be ascribed to a weakening
of the covalent binding in GaAs (which becomes partially ionic) and to the
larger masses of Ga and As as compared to the mass of Si.
With these aspects of selected phonon curves in mind, it is not difficult to
make an excursion to other materials maintaining the same crystal structure
but replacing the atoms. As an example, we may consider AlAs and GaP, both
in the zinc blende structure. Compared to GaAs, we expect a more pronounced
separation of acoustic and optical branches in the phonon spectrum of both
materials, because the mass difference of the two atoms in the unit cell, taken
from different rows of the period i c table, has increased. Moreover, due to the
lighter masses of Al and P, compared to those of Ga and As, respectively,
the characteristic phonon frequencies are higher than those of GaAs. These
features are found in the phonon dispersion curves of AlAs and GaP (see [
94]).
The lattice dynamics pr esented in this chapter is designed for the extended
solid without regarding its surface. Surface atoms experience a different struc-
tural surrounding and forces, which differ from those acting on the bulk
atoms. Consequently, they have their own dynamics, which, in a simplifi ed
two-dimensional model, has been treated already in Problem 3.2. For a more
detailed description we refer to [
85–87]. A simple example is the phonons of
a Cu(100) surface as shown in Fig.
3.14. Calculations of surface phonons are
usually performed for slab configurations where the outermost atomic layers
experience the modified environment of the surface, while the central layers
reproduce the bulk situation. The results are plotted for wave vectors in the
first BZ of the 2D periodic surface structure. The shaded area in Fig.
3.14