5.2 Lattice Vibrations
107
Fig. 5.13 Displacement of atoms for various phonon modes in wurtzite crystals. Adapted from [378]
(a)
(b)
Fig. 5.14 a Raman spectra of GaAs with different isotope content as labeled. b Energy of optical phonons in GaAs with
different isotope content [using the Raman spectra shown in (a)]. Reprinted with permission from [379], c
1999 APS
The dependence of the phonon frequency on the mass of the atoms (∝ 1/
√
M) can be demonstrated
with the isotope effect, visualized for GaAs in Fig. 5.14. The dependence of the phonon frequencies
on the stiffness of the spring can be seen from Fig. 5.15; the smaller lattice constant provides the stiffer
spring.
5.2.5 Density of States
The density of states (DOS) tells how many of the total number of modes are in a given energy interval.
The states are spaced equally in k-space but not on the energy scale (see also Sect. 6.13).
For the monoatomic linear chain model, the number of states N (E
) from E = 0 up to E = ω =
E
(k
) for the dispersion of the acoustic phonons (5.8) is given as
N (E
) = k
N
π/a
=
L
π
k
.
(5.32)