108
5 Mechanical Properties
Fig. 5.15 Optical phonon
frequencies (TO: filled
squares, LO: empty
squares) for a number of
III–V compounds with
different lattice constant
a 0 . 1 meV corresponds to
8.065 wave numbers (or
cm −1 ). Adapted from [380]
Using (5.8), we find for one polarization (E m = ω m )
N (E) =
2N
π
arcsin
E
E m
.
(5.33)
The DOS D(E) is given by
D(E) =
dN (E)
dE
=
2 N
π E m
1
1 − (E/E m ) 2
.
(5.34)
Often the density of states is scaled by the (irrelevant) system size and given per atom (D/N ) or per
volume (D/L
3 ), per area (D/L
2 ) or per length (D/L) for three-, two- or one-dimensional systems,
respectively.
In the diatomic linear chain model, additionally the optical phonons contribute to the density of
states. In Fig. 5.16 the phonon density of states is shown for γ = 0.9 and for comparison for γ = 1
(gapless phonon dispersion). For small wavevector, the density of states is 4N /(π E m ).
2 Within the
gap the density of states vanishes. At the edges of the band gap the density of states is enhanced. The
total number of states for both dispersions is the same.
In a three-dimensional solid the total number of modes is 3 p N (N 1, p is the number of atoms
in the base). Then (5.32) is modified to (for three degenerate polarizations)
N (E
) =
4π
3
3
(2π/L) 3 k
,
(5.35)
taking into account all states within a sphere in k-space of radius k
. Assuming a linear dispersion
ω = v s k (for sufficiently small wave vector), we obtain
2 The factor 2 compared to (5.34) stems from the folded Brillouin zone compared to the monoatomic chain model.