Some topics in solid state physics

The subject of this thesis is a theoretical study of some of the optical properties of crystals of CaF2 and Si which have had atoms of different mass and chemical properties substituted for a small percentage of the host atoms. The low concentration of the defects enables each to be considered independently as a single foreign atom substituted into the crystal. The work divides into two parts, distinguished by the different structures of the crystals and by the different phenomena to be explained in either system.
In Si, with B- or F+ substituted in as defects, the problem of interest is the mechanism and intensity of optical absorption by the crystal. Since Si forms a covalent bond with itself., the pure crystal is homopolar, and does not absorb light. A charged defect will absorb in the normal way. However, in homopolar crystals such as Si, theoretical attempts to describe neutron scattering and other experiments have led to the rejection of the rigid-atom model in favour of the Shell model, which allows each atom a shell of outer electrons not rigidly attached to the nucleus. The effect of this is that the atoms are polarizable, and have an extra three decrees of freedom, associated with their electric dipole moments. In the case of a charged defect, there are now three absorption mechanisms: one due to the dipole of the oscillating charge itself; one due to the dipoles created throughout the crystal by the effect on the polarizable atoms of the changed polarizability of the defect; and a final mechanism whereby the charge on the defect sets up a static electric field which induces dipoles on the polarizable atoms through an anharmonic effect.
The significance of this last mechanism relative to the others depends on whether or not the Lorentz local field correction is to be used. Using a version of the Shell model in which changes in the defect charge and polarizability can be included explicitly in the absorption, we show that this correction factor is always presents although it may be modified by changes in the defect polarizability. We are able to fit the experimental frequencies and intensities for the localized modes due to boron, and also for those due to carbon, using these changes as parameters.
Our calculation is based on the simple relationships between the absorption, the susceptibility and the frequency-dependent displacement- displacement Green's functions. The parameters used in the Shell model for the perfect silicon lattice are taken from experimental fits to neutron data for the dispersion relations.
In CaF2, H- and D- ions have been substituted into F- sites. These ions differ from F- mainly in mass, and can be observed experimentally by absorption of infra-red light at 1.849 x 1014 ra/sec (H-) and 1.323 x 1014 ra/sec (D-). These figures indicate that the effect of the change of mass is reduced by a weakening of the force constants which bind the defect to the other ions in the lattice. We explain the changes in the above fundamental frequencies, and those of the harmonics, which appear when the crystal is subjected to uniaxial stress. The stress causes the lattice to strain, and the effect of this strain on the anharmonic forces present is to produce new terms in the harmonic forces, which modify the frequencies. In particular the weakened force constants at the defect allow additional relaxation of its neighbours, which makes the change in the frequency of the localized node more pronounced.
We find the relaxation by a Green's function technique which inverts the force constant matrix, Φ, of the perfect lattice.
The six force constants which constitute Φ out to 3rd neighbours have been evaluated as parameters in a rigid-ion model which fits the elastic constants of CaF2 and other experimental data. The changes in the central force constants coupling the defect with its 1st and 2nd neighbours are found by fitting the vibrational frequencies to the experimental data, and are respectively - 65% and - 22%. (It is also necessary to reduce the effective charge on the defect by 46% to get this fit.) Using the relaxation obtained, assuming that the defect vibrates in a static environment, and taking as parameters the central anharmonic forces coupling the defect with its 1st and 2nd neighbours, we derive expressions for the parameters of the phenomenological model which describes the frequency changes of the defect vibrations under stress. We thus evaluate some of the anharmonic force constants, and can show that a Lennard-Jones potential describes the behaviour of the defect to a fair approximation.
A useful sideline of this work is a derivation of the changes in the macroscopic elastic constants of a crystal which contains a small concentration of defects.