In order to calculate the vibrational modes we need the force-constants, which are second derivatives of the total energy versus the atomic displacements. Unlike forces, which can be obtained via the force theorem from the charge density of the unperturbed system, this requires knowledge of the first-order corrections to the wavefunctions. The reason why the unperturbed density can be used to estimate the forces is that errors in the charge density lead only to second-order errors in the total energy, and forces are first-order changes in the energy. The appropriate theoretical framework for calculating force constants is density-functional perturbation theory (DFPT), or linear-response theory. This method differs from “textbook” perturbation theory in the sense that the first-order corrections of the wavefunction are calculated self-consistently with the first-order changes in charge density which they produce within the framework of density-functional theory. Once the charge density has only second-order errors, the corresponding total energy has only fourth-order errors, and hence, up to third-order derivatives of the total energy can be computed. This is known as the 2n + 1-theorem (97). The beauty of linear-response theory is that if we impose a first-order change in the external potential, say a displacement wave with wavevector q, then the

Table 15.11 Parameters of effective Hamiltonian: inverse mass parameters ,Di) in h^{2}/2m_{e}, parameters in e^{2}/2, and energy splittings A_{ci}

(meV), i =1, 2.

parameter

ZnSiN2

ZnGeN2

ZnSnN2

CdGeN2

GaN

Ai

-4.71

-6.39

-8.23

-6.3

-5.98

A2

4.11

5.96

7.77

5.90

5.44

A3

-0.26

0.02

0.02

0.05

Bi

0.73

-0.51

-0.49

-0.35

-0.58

B2

-1.93

-2.01

-2.80

-1.94

-2.46

B3

0.04

0.05

0.11

0.25

Ci

-0.52

-0.01

-0.03

-0.13

C2

-1.26

-0.01

-0.05

-0.05

C3

-0.32

2.15

2.80

1.95

2.53

Di

-0.64

4.3

5.7

3.89

5.06

D2

-5.41

4.1

5.5

4.90

-2.19

D3

-5.41

4.1

5.5

3.2

-2.19

Ei

0

0.015

0

0.007

0.021

E2

0

-0.005

-0.005

-0.005

-0.021

Aic

160

115

82

97

12

Д2С

-20

-14

-94

-28

0

response to first order also involves only Fourier components of wavevector q. By rewriting the perturbation theory in terms of the periodic parts of the Bloch functions, one can obtain the relevant force constants for any arbitrary q from calculations of the standard primitive cell. In other words, there is no need to construct supercells with sizes commensurate with the imposed perturbation wavevector periodicity, as in the frozen-phonon methods (98, 99).

The long wavelength limit corresponding to q ^ 0 presents its own special problems for compounds with partially ionic bonding. As is well known, there are LO-TO splittings related to the long-range Coulomb forces. These are treated analytically in the DFPT by considering separately the response to a static electric field. The latter are treated as a separate type of perturbation. The derivative of the total energy versus the electric field is the polarization, and although it also breaks the crystal periodicity, it can be calculated using the modern theory of polarization (100, 101) in terms of a Berry phase. The polarization itself is not uniquely defined, but adiabatic changes in polarization, which maintain the insulating phase along the entire adiabatic transition, are. We may then consider various mixed second-derivatives:

Here, т_{R} = т + R is the position of atom т in a unit cell, with lattice vector R and т_{а} indicating its Cartesian components. F_{Ta} is the long-range force on atom т in the a direction, and is the polarization in the в direction. The first of these equations gives the short-range contribution to the force constants, the second represents the Born-effective charge tensor, and the third is the high-frequency electronic susceptibility. The latter corresponds to below-band-gap frequencies, because we include only static electric fields, but do not include the phonon contributions. For a general wavevector q, we form the Bloch sum,

and find the phonon frequencies from the generalized eigenvalue problem:
with normalization

Summation over repeated indices is understood here. For q = 0 this gives the transverse phonon frequencies u_{n} and eigenvectors иП_{а} = иПа^{=0}. Using the Born effective charges and the eigenvectors of the above diagonalization problem, the dielectric function can be written:

where ш_{п}, Г_{п} are the mode frequencies and damping factors respectively, and H is the volume of the unit cell. The damping factors are added empirically. The oscillator strengths are given by

Here we consider that in view of the orthorhombic symmetry, the effective charge tensor is diagonal in the Cartesian indices. The zeros of this dielectric function give the longitudinal modes, while the poles give the transverse modes. Thus the LO-TO splittings of each mode that is subject to long-range forces can be calculated.

As a byproduct, we obtain the dielectric constant tensor at high frequency, ^{?}aa, from which we can obtain the indices of diffraction, discussed in Section 15.9. Furthermore, the imaginary part of the above dielectric function gives us explicitly the infrared absorption spectrum due to the phonons. Using the real part, we can also obtain the infrared reflectivity and the loss function, A{— e^{—}a(a)}, whose peaks correspond to the LO modes. As already mentioned, we can also obtain the phonons for finite wavevector q and hence the full phonon spectrum, as well as the phonon density of states. By taking various integrals of these functions one obtains the lattice-dynamical contributions to the thermal properties, such as the specific heat, and the vibrational entropy.

Finally, we turn to Raman spectroscopy. The Raman intensity is proportional to |е* • R^{m} • e_{0}^{2} where e* and e_{0} are the incident and scattered light polarization vectors and R is the second rank Raman susceptibility tensor for mode m, given by

This involves the third-rank tensor,

calculated from a third derivative of the total energy. As already mentioned, this can still be obtained from the first-order corrections to the wavefunctions, because of the 2n +1 theorem.

Now we turn to some specific group-theoretical considerations for the materials at hand. The modes subject to LO-TO splitting are those that transform like the components of a vector, i.e. bi, b_{2}, ai for electric fields along x, y, z or a, b, c crystallographic axes. These modes are infrared-active. So, the only mode which is not infrared-active is a_{2}.

All modes are Raman active but still different selection rules apply. For ai symmetry the tensor is diagonal:

which means that incident and scattered light must have parallel polarization but different values will be obtained for the x, y and z direction. The scattering geometry is fully specified by giving k* (e*e_{0})k_{0}. If the scattering wavevector q = k_{0} — к* is parallel to the polarization of the mode, then the longitudinal modes are excited, otherwise the transverse ones. For example, x(yy)x will measure the b component of the a_{iT} modes, while z(yy)z will measure the b for the a_{iL} modes. The remaining modes correspond to off-diagonal matrix elements of the Raman tensor, as follows: