5.2 Lattice Vibrations
105
Fig. 5.10 Phonon dispersion in a GaP and b GaAs. Experimental data (symbols) and theory (solid lines, 14-parameter
shell model). ‘L’ and ‘T’ refer to longitudinal and transverse modes, respectively. ‘I’ and ‘II’ (along [ζ, ζ, 0]) are modes
whose polarization is in the (1 ¯
10) plane. The grey area in (a) denotes the gap between acoustical and optical states. a
Adapted from [375], b Adapted from [373]
atoms, there are also 3 acoustic branches and 3( p − 1) optical branches. For a diatomic base (as in the
zincblende structure) there are 3 optical phonon branches (1 LO phonon mode and 2 TO phonon modes).
The total number of modes is 3 p. The dispersion ω(k) now has to be calculated for all directions of k.
In Figs. 5.9 and 5.10, the phonon dispersion in silicon, GaAs and GaP is shown along particular
lines in the Brillouin zone (cf. Fig. 3.38). A detailed treatment can be found in [372]. The degeneracy of
the LO and TO energies at the point for the covalent group-IV semiconductor is lifted for the III–V
semiconductors due to the ionic character of the bond and the macroscopic electric field connected
with the long-wavelength LO phonon (see Sect. 9.5). Comparing GaAs [373] and GaP [374], the quite
different mass of Ga- and P-atoms (M × /M + ≈ 0.92) leads to the formation of a clear gap between the
acoustic and optical branches, while for GaAs M × /M + ≈ 0.9994 is close to 1 and no gap forms.
We note that the degeneracy of the TA phonon is lifted for propagation along the 110 directions
() because the two transverse directions 001 and
1 ¯
10
are not equivalent.
In boron nitride the masses of the two constituents are so similar that no gap exists between acoustical
and optical branches (Fig. 5.11). Also the density of states (averaged over the entire Brillouin zone) is
depicted (see the next chapter).
The displacement of atoms is shown in Fig. 5.12 for the different phonon modes present in zincblende
crystals. In Fig. 5.13 the optical phonon modes for wurtzite crystals are depicted. The E-modes with
105
Fig. 5.10 Phonon dispersion in a GaP and b GaAs. Experimental data (symbols) and theory (solid lines, 14-parameter
shell model). ‘L’ and ‘T’ refer to longitudinal and transverse modes, respectively. ‘I’ and ‘II’ (along [ζ, ζ, 0]) are modes
whose polarization is in the (1 ¯
10) plane. The grey area in (a) denotes the gap between acoustical and optical states. a
Adapted from [375], b Adapted from [373]
atoms, there are also 3 acoustic branches and 3( p − 1) optical branches. For a diatomic base (as in the
zincblende structure) there are 3 optical phonon branches (1 LO phonon mode and 2 TO phonon modes).
The total number of modes is 3 p. The dispersion ω(k) now has to be calculated for all directions of k.
In Figs. 5.9 and 5.10, the phonon dispersion in silicon, GaAs and GaP is shown along particular
lines in the Brillouin zone (cf. Fig. 3.38). A detailed treatment can be found in [372]. The degeneracy of
the LO and TO energies at the point for the covalent group-IV semiconductor is lifted for the III–V
semiconductors due to the ionic character of the bond and the macroscopic electric field connected
with the long-wavelength LO phonon (see Sect. 9.5). Comparing GaAs [373] and GaP [374], the quite
different mass of Ga- and P-atoms (M × /M + ≈ 0.92) leads to the formation of a clear gap between the
acoustic and optical branches, while for GaAs M × /M + ≈ 0.9994 is close to 1 and no gap forms.
We note that the degeneracy of the TA phonon is lifted for propagation along the 110 directions
() because the two transverse directions 001 and
1 ¯
10
are not equivalent.
In boron nitride the masses of the two constituents are so similar that no gap exists between acoustical
and optical branches (Fig. 5.11). Also the density of states (averaged over the entire Brillouin zone) is
depicted (see the next chapter).
The displacement of atoms is shown in Fig. 5.12 for the different phonon modes present in zincblende
crystals. In Fig. 5.13 the optical phonon modes for wurtzite crystals are depicted. The E-modes with