Selection rules and phonon modes

When photons from a laser are scattered from a crystal with emission or absorption of phonons, the energy shifts of the photons are small, but can be measured by interferometric techniques. Usually, the phonon wave vectors are very small compared to the size of the Brillouin zone so that the interactions are only with zone center phonon. Thus, one can have interaction with either the zone center acoustic phonons or the optical phonons. The interaction with acoustic phonons is called Brillouin scattering while the interaction with optical phonons is call Raman scattering. All of the Raman mode frequencies, intensities, line-shapes, and line-widths, as well as polarizations can be used to characterize the lattices and impurities. The intensity gives information on crystallinity. The line-width increases when a material is damaged or disordered, because damage or disorder occurs in a material will increase phonon damping rate or relax the rules for momentum conservation in Raman process.

The different long-wavelength phonon branches in a given crystal correspond to

different symmetries of vibration of the atoms in the unit cell and are characterized by irreducible representations of the space group of the crystal lattice. If the wavelengths of the Raman phonons are assumed to be effectively infinite, then the crystal point group can be used in classifying the phonon symmetries. This infinite wavelength assumption is not valid for Raman-active phonons which are also infrared active, and this type of vibration will be discussed separately in the following section.

The selection rules for Raman-active phonons can be determined by standard group-theoretical methods and the calculation is described in some detail by Heine, who based his work on the polarizability derivative theory of Born and Bradburn.

The result of this approach is that a phonon can participate in a first-order Raman transition if and only if its irreducible representation is the same as one of the irreducible representations which occur in the reduction of the representation of the polarizability tensor. The irreducible representations by which the components of the polarizability tensor transform are conveniently listed by Herzberg and Wilson et al. for the set of molecular point groups, which includes the 32 crystal point groups.

The intensity of the Raman-scattered radiation depends in general on the directions of observation and illumination relative to the principal axes of the crystal.

The angular variation of the scattering gives information about the symmetry of the lattice vibration responsible. The anisotropy of the scattering can be predicted for a vibration of any given symmetry by standard group-theoretical methods.

GaN-, AlN- and InN-based materials are highly stable in the hexagonal wurtzite structure although they can be grown in the zinc blende phase and unintentional phase separation and coexistence may occur. The wurtzite crystal structure belongs to the space group C_6v⁴ and group theory predicts zone-center optical modes are A , 2B , E1 ^B1 1

and 2E . The A and E modes and the two E modes are Raman active while the B 2 1 1 2

modes are silent. The A and E modes are polar, resulting in a splitting of the LO and the transverse (TO) modes. The Raman tensors for the wurtzite structure are as follows:

Here x, y or z in brackets after an irreducible representation indicates that the vibration is also infra-red active and has the direction of polarization indicated. Such vibrations occur only in piezo-electric crystals (i.e. crystals with no center of inversion symmetry). In crystals which do have a center of inversion symmetry, only even-parity vibrations, whose representations have a subscript g, can be Raman active and only odd-parity (subscript u) vibrations can be infrared active. This fact leads to the important complementary nature of infra-red absorption and Raman effect measurements. Directly above each irreducible representation is a matrix which gives the non-vanishing components of the Raman tensor, i.e. of αρσ,μ or R_σρ^μ . The different elements of the matrices are the nine components of the tensor obtained by allowing both ρ and σ to take on the values x, y and z. Here x, y, and z are the crystal principal axes chosen to be identical with the principal axes x1, x and x2 3 defined for all the crystal classes by Nye. The component μ of the phonon polarization for the

case of infrared-active vibrations is the quantity given in brackets after the irreducible representation symbol.

Fig. 2-2 Phonon dispersion curves for ZnO crystal of wurtzite structure. (after Calleja et al. [4])

The vibrational modes in ZnO wurtzite structures are given in Figure 2-2. At the Γ point of the Brillouin zone, it can be seen that the existence of the following optic phonon modes: A +2B +E1 ^B1 1+2E ; A2 1 and E1 modes are both Raman and infrared active; and B1^B (low) and B^B1(high) modes are silent. However, the nonpolar E2 modes are Raman active and havetwo frequencies: E2(high) is related to the vibration of oxygen atoms and E2(low) is associated with the Zn sublattice. The displacement vectors of the phonon normal modes are illustrated in Fig. 2-3. For the lattice vibrations with A1 and E1 symmetry, the atoms move parallel and perpendicular to the c-axis, respectively. Table 2-1 summarizes a list of observed zone-center optical-phonon wave numbers along with those calculated for wurtzite ZnO.

Fig. 2-3 Displacement vectors of the phonon modes in ZnO wurtzite structure. (after Jephcoat et al. [5])

Table 2-1 Phonon mode frequencies of wurtzite ZnO at the center of the Brillouin zone obtained from infrared spectroscopic ellipsometry and Raman scattering measurements in comparison with theoretical predictions. [1]