Figure 2.3: Structural subunit of the YAG crystal struc- ture.
model to be used depends strongly on the nature of the binding forces between atoms inside the material under study.
For ionic materials a model often used, with well-studied force constant laws and empirical potentials with tested parameters is the Rigid Ion Model (RIM). Its core assumption is that atom cores are treated as point charges, of charge zk, (which is not necessarily equal to the ion’s formal valence charge). Ions interact with each other mainly through long- range Coulomb forces. Short-range central potentials are also featuring in this scheme, that approximately model the behaviour of core electrons, such as a repulsive Born-Mayer potential, which models the repulsion between overlapping electron densities, due to the Pauli Exclusion Principle at close distances. Besides, we shall bear in mind Earnshaw’s Theorem [17], which states that no system of point charges can be maintained in a stable stationary equilibrium configuration solely by electrostatic forces. These short-ranged potentials act, therefore, towards further stabilizing the system.
The RIM has been extremely accurate in reproducing experimental data and numerous lattice dynamical and thermodynamical studies similar to the present one have been pub- lished in the literature (see for example Papagelis [31], Monteseguro [29], etc). However, it has two central deficiencies that stem from its core assumptions ([4]):
• As it only features central forces, it predicts the Cauchy relationsC12=C44. These are only approximately satisfied for most solids, while for YAG, they are not satisfied at all. (see Section 3.2.6 for discussion)
• It completely disregards the electronic polarizability, which gives rise to absorption phenomena at optical and near-UV frequencies, as well as to Raman scattering.
The latter not only requires the existence of electronic polarizability, but also its variation with ionic displacement. Calculation of the Raman Tensor in the RIM gives exactly zero, while experimentally measured Raman active modes are clearly present (see Chapter 4).
These deficiencies are not present in the Shell Model introduced by Dick and Overhauser [8]. It is an extension of the RIM, which takes account of the electronic polarizability in a very simplistic way. Its core principle is that apart from the existence of long-range Coulomb and short-ranged forces, we can think of each ion as consisting of a rigid ion core, on which a spherical electronic shell is coupled by a simple harmonic spring of constant k0. Here, ricore andrjshell are the instantaneous core and shell positions:
Uijshell= 1
2k0(ricore−rshellj )2 (2.1)
The electronic shell has its own charge, which is not necessarily negative. It also has zero mass, corresponding to the requirements of the Adiabatic Approximation, as we have discussed in the previous chapter. A schematic representation of the model, due to Br¨uesch [4] depicted in figure 2.4. As is implied by the figure, short-range core-shell interactions are restricted to only within the same ion.
Figure 2.4: Pictorial representation of the Shell Model.
Figure due to [4].
In other words, the long-range Coulomb forces can act between all pairs of atoms species, whether core or shell, while short-range forces act only between shells. Within an ion, there is an additional core-shell interaction given by eq. 2.1 . There is also an extended form of the Shell Model due to Cochran et al. [6], in which short-range core-core and- core-shell interactions are allowed.
In this work we make use of the Shell Model to describe the interatomic forces inside YAG.
In our model, only the O2− anions possess shells, as they are the most polarizable. First, there is the regular electrostatic term, which includes core-core, core-shell and shell-shell interactions, given by Coulomb’s law:
UijCoulomb= e2zizj 4πε0rij
(2.2) where theziare the charges of the respective atom species. They are part of our parameter set.
As a short-range potential, we have used a Buckingham potential, which is composed of a Born-Mayer potential together with an additional Van der Waals r−6 term that arises
due to the mutual polarization of the atoms. It only acts between the cations and the oxygen shells:
UijBuckingham =Ae−(
rij ρi+ρj)
− C(6)
r6ij (2.3)
Here A, and C(6) and ρij are parameters to be determined. ρij =ρi+ρj is thought to be the sum of the ionic radii of the interacting ions, which essentially provides an effective radius within which the exponential term has significant value. It acts between cation cores and oxygen shells, as well as between oxygen shells. In figure 2.5, the Buckingham potential is plotted with an arbitrary set of parameters. As is evident, for large r-values, both terms tend to zero, while until a certain threshold, the exponential term dominates, and the potential is repulsive. The position and the height of this threshold depends on the ratio
C(6)·(ρi+ρj)6 A
Below this threshold, ther−6term dominates and the potential becomes steeply attractive, tending towards −∞ at zero. This behavior is obviously unphysical, since the VDW r−6 term originates from perturbation theory; it doesn’t contain electron-electron repulsion interactions which become significant at smaller distances and it is obviously not valid for small internuclear distances. Therefore, in a good implementation of the model, the parameters are set in such a way that internuclear distances don’t fall below the threshold of infinite attraction.
Finally, in our model we have also included an additional three-body potential term, not regularly used in shell models. This was motivated strongly by the fact that YAG does not obey the Cauchy relations C12 = C44. This term represents repulsion between bond pairs. Although we are dealing here with an ionic compound, Al-O bonds are known to be partly covalent (for example inAl203, in a van Arkel diagram [2]). For simplicity, we have chosen a harmonic form that penalizes deviations from bond angles that are expected of a certain coordination environment. The three-body term is thus given by
Uijkthree = 1
2k1(θijk−θ0)2 (2.4)
i.e for octahedral coordination θ0 = 90◦, for tetrahedralθ0 = 109.5◦ and for dodecahedral coordinationθ0 = 70◦. In our model, the three-body interaction also applies only between metal cores and oxygen shells. It is noted that we chose to impose such an interaction for Y-O bonds as well. Therefore, the total internal energy, after taking care of double and triple counting of terms is given by:
Utotalinternal = 1 2!
X
i,j
(UijCoulomb+Uijshell+UijBuckingham) + 1 3!
X
i,j,k
Uijkthree (2.5)
Figure 2.5: The Buckingham potential, plotted with an arbitrary set of parameters. Note that close to zero it tends to −∞
It is obvious that all potentials must have a finite range in order to be calculable. This is certainly not the case for the Coulomb and the short-range potentials. The issue of convergence of the Coulomb term is an old problem and a standard solution for it has been developed, namely the Ewald summation method, which we will discuss in the Section 4. As for the short-range potentials, the natural way to truncate them is by specifying a spherical cut-off radius beyond which no short-range potential acts. In the present work, we have chosen rcutof f = 3.5 ˚A, in order to include interactions between up to fourth neighbours. It shall be noted, though, that at the cut-off boundaries, energy discontinuities are introduced, which can affect out calculations when searching for energy minima. For a further discussion, see Section 4.
In our model there is a total of 19 parameters, as we have only included short-range interactions between metal cation cores and oxygen anion shells. These parameters are listed in table 1.4. The parameters were chosen as to be close to the ones used in a previous RIM study by Papagelis et al. [31]. Also, the parameters used for the Born-Mayer part of the Buckingham potential 2.3 were among those reported by Milanese et al. [27].
Harmonic Three-Body k1 (eV / degree2) Oshell−Aloct−Oshell 2.0930 Oshell−Altet−Oshell 4.093
Oshell−Y −Oshell 2.093
Oshell−Ocore−Oshell -
Oshell−Oshell−Oshell -
Shell k0 (eV /˚A2) Aloct - Altet -
Y -
Ocore -
Oshell 60
Buckingham ρij(˚A) A (eV) C(6)(eV /˚A6
Aloct−Oshell 0.3118 1114.9 -
Aloct−Oshell 0.3205 1200 10.654
Y −Oshell 0.3491 1345.1 10
Ocore−Oshell - - -
Oshell−Oshell 0.149 22,764 27.88
Coulomb z Aloct +2.9 Altet +2.9
Y +3
Ocore +0.91 Oshell -2.86 Table 2.4: The Shell Model parameters used in this work