1.3 Theoretical Developments
1.3.2 Models of Lattice Vibration
Here, we illustrate some of the very fundamental theoretical models which have been de- veloped over the years to describe lattice vibrations in a crystal. We present the underlying assumptions and the associated limitations for various approaches.
1.3. Theoretical Developments 11 1.3.2.1 Einstein Model
In this model [27], the atoms in a crystal are treated as independent quantum harmonic oscilla- tors with no correlation between the motion of different atoms. This leads to atomic vibrations which remain entirely unaffected by the motion of the neighbors. The model also makes the simplifying assumption that all the atoms vibrate with the same fundamental frequency, i.e.,
ω(q)=ωE (1.18)
The Einstein model has achieved reasonable success; its results are in good agreement with the experiments over most of the temperature range. The theory accounts satisfactorily for the breakdown of equipartition of energy at low temperatures and predicts correctly the specific heat value to be zero at T=0 K. However, the decay of specific heat with decreasing temperature is faster than what is seen in experiments. The observed specific heat at very low temperatures approaches zero as T3rather than exponentially as predicted by the model. This basic deficiency of the Einstein model is a result of the over simplified assumption that the oscillators do not interact. If the atoms do not interact with one another, sound waves would not propagate through solids.
1.3.2.2 Debye Model
An improved model [28] with correlated atomic motion was brought about by Debye in 1912 to deal with the low temperature discrepancy in Einstein’s model. The proposed model known as the elastic isotropic continuum model was the most simplistic view of a solid in which the atomic crystal structure was smeared out and treated as a continuous elastic medium. Within this approximation, the phonon problem is solved in the acoustic limit with the dispersion relations for the three acoustic branches are assumed to be the same and represented by
ω(q)=vsq, (1.19)
where vsis mean sound velocity.
12 Chapter 1. Introduction Based on such a simple approximation of continuous medium, the model is surprisingly effective to capture the essential physics behind many important features, including the T3- dependence of specific heat at low temperatures. In spite of its manifold successes, the model, however, becomes inadequate at higher temperatures. The assumption of ignoring the dis- creteness of the lattice is expected to hold well at low temperatures as modes of only low frequency or long wavelength for which the consequences of discreteness are irrelevant, are involved there. But at high temperature, when the wavelength is short enough to be compara- ble to inter-atomic spacing, the calculation of the vibrational properties requires a knowledge of the crystal structure and the Debye approximation certainly breaks down. An important consequence of this failure is that the model of Debye can not be used to calculate the phase diagrams correctly. While computing the vibrational free energy differences between different compounds, the high frequency portion of the phonon densities of states plays a significant role which Debye model describes incorrectly. Among other deficiencies, the optical modes which become relevant in polyatomic solids are completely ignored within this model. Also from the theoretical point of view, the model is unrealistic due to the existence of singularities in the density of states.
1.3.2.3 Born-von Karman Model
Around the same time as Debye worked out his theory of specific heats, Max Born and Theodore von Karman proposed another model [29, 30] in which the atomic structures of solids entered explicitly. Within this model which considers interactions of an atom with its neighbors, the restoring force on an atom is determined not by the displacement of the atom from its equilibrium position, but by its displacement relative to its neighbors. This was initially proposed to model the inter-atomic forces between nearest neighbors with a central and non-central component. Later, the model was modified to incorporate the interactions between higher neighbors. The effective crystal potential energy which was proposed for the second order term V2is given by
V2 = 1 2
Xp
lb
Xp
l′b′
X3 i
X3 j
Φi j(lb; l′b′)ui(lb)uj(l′b′), (1.20)
1.3. Theoretical Developments 13 where p is the total number of atoms in the lattice crystal structure andΦi j are the force constants between atoms lb and l′b′. These inter-atomic force constants are considered as parameters which are fitted to the experimental data for the phonon modes. The corresponding vibrational frequencies ω(q) are then derived from the eigenvalues of the dynamical matrix D.
1.3.2.4 Ab-initio Methods
Although the Born-von Karman model provides a vast improvement over the previously avail- able models, it, however, has certain limitations which need to be addressed. For the com- putation of lattice dynamics in a completely random alloy, the model assumes an average lattice and neglects any fluctuations due to the presence of specie with different chemical properties. Therefore, this formalism never provides a correct picture of the complex nature of inter-atomic forces in a substitutionally disordered alloy. Moreover, the force constants are considered as fitting parameters for the experimentally obtained normal mode frequencies, possibly leading to several sets of force constants which may describe the frequencies equally well. Such deficiencies are completely done away within the first-principles electronic struc- ture methods which provide an accurate and parameterless calculation of microscopic electron response to lattice vibrations. The first such microscopic theory of lattice dynamics was for- mulated in terms of dielectric matrices by Martin et al. in 1970 [31]. However, the constraint that the electron-ion interaction must be described by a local potential had limited its utility [32, 33] for the study of vibrational properties. Modern day calculations mainly rely on two approaches which include the direct method [34–36] and the linear response method [37, 38].
While the heavy computational cost associated with the direct method has limited its applica- bility to zone-center and selected zone-boundary phonon modes in relatively simple materials, the latter approach in contrast is most efficient for the calculation of full phonon dispersion curves and the vibrational density of states. A brief account of the linear response approach will be presented in Chapter 3.
14 Chapter 1. Introduction