Lattice Dynamics

The thermal properties of the diamond, BC8 and ST12 structures are extracted from phonon calculations. In this section the method of calculating phonon dispersion relations from an empirical potential will be presented. It is, in general, very complicated to include anharmonic terms in such calculations. All phonon calculations presented in this chapter will be found using the harmonic approximation. The validity of this approximation is examined more fully in Chapter 6 when self-interstitial configurations in the diamond structure are examined. In that case it is found that anharmonic effects are negligible and therefore it is assumed here that they can also be neglected.

Phonon dispersion relations are calculated by looking for wavelike solutions to the classical equations of motion of atoms under a small displacement from their equilibrium sites. Firstly, a crystal is composed of an infinite number of primitive unit cells in three dimensions labelled  where  are integers. Each primitive unit cell is defined by three linearly independent vectors  and  forming a parallelipiped. The origin of the  unit cell can then be defined as

relative to the origin l=(0,0,0) This unit cell defines the periodic structure of the crystal by repeating its primitive unit cell throughout space.
Each unit cell contains n atoms which are labelled by k=1,...,n each of which are at positions  relative to the origin of each cell  , therefore the position of any atom lk is given by .
Vibrations occur when the atoms are displaced from their equilibrium positions. If the atoms are allowed to move from their equilibrium positions by an amount then the actual position of any atom under the influence of a vibration (thermal fluctuation) is  .
The potential energy,  , of the crystal is obtained from a potential which itself is a function of the instantaneous coordinates,  , of the atoms.
If the displacements are assumed to be small compared to interatomic distances (which is generally true for stable crystals at a temperature well below its melting point), then the potential function can be expanded about the equilibrium positions:

where  and  vary over all x, y and z coordinates and the coefficients  and  are

and

The subscript 0 means that the coefficients are evaluated at the equilibrium positions,  .
The first term in the series is constant, that is, it is independent of the displacements of the atoms from their equilibrium sites and so can be taken as the zero of the potential energy. The force on the atom is given by

which is the coefficient of the second term in the series. Provided the derivative is evaluated at the equilibrium position, this term will be zero since at equilibrium the total force on any atom is zero. In the harmonic approximation all terms in cubic or higher order are assumed negligible, therefore the potential can be expressed simply as the third term in the expansion.
That term is not defined for self interactions, that is, it allows only the  terms. To define the self interaction term consider a rigid translation of the entire crystal. This can be defined by replacing all the displacements  by a constant value  . The equations of motion of the atoms is then

which must be equal to zero because such a translation clearly cause the forces,  to vanish. But the  are arbitrary hence the sum over the coefficients  must be zero. Therefore in addition to the definition of  given above, the diagonal terms must be given by

Now that all the elastic constants  are well defined it is possible to solve the equations of motion

by looking for the wavelike solutions

where c.c. stands for the complex conjugate of the first term and  and  are amplitudes of the wave. Note here that  is used for the wavevector here so as not to confuse it withk which is the index of the atoms within the primitive unit cell.
On this substitution into the equations of motion the following dispersion relation is found:
  
where D is known as the dynamical matrix. The elements of D are given by

The dimension of the dynamical matrix in 3n. Therefore using the symmetry of the crystal by expressing it in terms of repeating primitive unit cells replaced an infinite set of equations of motion of an infinite number of atoms by a set of 3n linear homogeneous equations. The condition that the homogeneous system of equations have a non-trivial solution is that the determinant of the coefficients in Equation (5.10) vanish, that is to say

therefore the eigenvalues of the dynamical matrix give the allowed (squares of) phonon frequencies for a given  . The dynamical matrix is Hermitian therefore the eigenvalues, are real.
This lattice dynamics method has been coded for the massively parallel CM200 Connection Machine and is described elsewhere[76]. Although, in the present case, using such a large computer is unnecessary, it becomes so when very large dynamical matrices are required as in Chapter 6 where eigenvectors of matrices up to dimension  are needed. The parallelisation strategy is straightforward, with the elements of the dynamical matrix spread across the  processors. This data-driven parallelisation is an inevitable consequence of the `single instruction multiple data' architecture of this machine. The matrix diagonalisations are done using the Jacobi rotation method.

Source: http://cmt.dur.ac.uk/sjc/thesis/thesis/node52.html