next up previous
Next: Effect of Pressure on Up: Complex Structures: Empirical Treatment Previous: Lattice Dynamics

The Empirical Potential

Much effort has been invested in deriving simple potential models for covalent materials such as silicon. The intuitive picture of a covalent bond suggests that pairwise interactions should dominate the structural properties, and yet no standard short-ranged pair-potential stabilises the diamond structure. Various methods have been introduced to circumvent this: several authors have introduced potentials with a strong angular dependence[77, 78, 79]. Pettifor[80] arrived at a similar solution from approximate tight-binding ideas. Models based on the notion of a bond charge[81], or on a function of the local coordination(see for example [82]) have also been tried. Most of these models were successful in the regime for which they were parameterised, but showed a lack of transferability. The fourth version of the model of Tersoff perhaps comes closest to being transferable, but at the price of considerable functional complexity and 13 fitted parameters. A recent survey of six such potentials[83] concluded that each has strengths and limitations, none appearing clearly superior to the others, and none being fully transferable.

For the present study we require a potential which gives a reasonable treatment of high-pressure phases and phonons, and can give an insight into the different behaviours of silicon and germanium. Recently, it has been shown that a very simple model, based on analytic pair potentials representing one bond per electron, gives good results in studies of phonons, high-pressure phases, surface reconstruction, defects and cluster formation. An unusual aspect of this potential is its simplicity (three free parameters, two more to define length and energy scales) and its transferability.

In its parameterised form, this potential is written as follows:

  equation1679

The function A term represents the short-ranged repulsion due to core overlap. B represents the covalent bonds themselves, the sum is interpreted as being over electrons but can be written as a sum over atom pairs (limited here to four neighbours per atom, though double bonds tex2html_wrap_inline6690 could be incorporated in to the model). C represents the repulsion between adjacent bonds as the angle between them is reduced, due mainly to distortions in the orbitals to preserve orthogonality. Again, although the final term is interpreted as a sum over pairs of bonding electrons sharing a common atom, it can be written as a sum of pairwise interactions between pairs of neighbours of a given atom. Thus, the rather complicated notation in which tex2html_wrap_inline6692 means the label of the mth bonded neighbour of atom i.

The chosen functional forms of each term of the potential are

   

The parameters used in the following calculations are as follows: A=330816.2, B=36.39820, tex2html_wrap_inline6702 , tex2html_wrap_inline6704 and C=1.0.

In general there is no unique way to select the four `bonded' neighbours (choosing the four nearest neighbours may not satisfy the requirement that if i is bonded to j then j must be bonded to i). In the present case, however, the structures which will be considered (diamond, BC8 and ST12) are fourfold co-ordinated crystal structures so this difficulty does not arise. The extension to non-fourfold structures is probably invalid.

The electronic structure calculations also provide strong evidence that this model will not be suitable for modeling BC8 carbon, because that appears to be more like a molecular crystal. This is similar to the situation found in clusters[75], where small silicon and germanium clusters form distorted structures maximizing their number of bonds (up to four). By contrast, rather than distort bond angles, small carbon clusters form rings and chains containing double bonds. This preference for double bonding rather than massively distorted bond angles is a qualitative difference between silicon and carbon. It is difficult for a single potential formalism to describe both behaviours.

Within the current model, the physical picture which explains the difference is that the bond-bending arises primarily from orthogonalization within the atom core (which is why it runs over pairs of bonds from the same atom). In silicon the tex2html_wrap_inline6716 valence electrons are kept away from the centre by orthogonality to the tex2html_wrap_inline6718 (this effect is even more pronounced in germanium), consequently their overlap and attendant orthogonality contribution to bond angle distortions are smaller. The consequence of this is that in Si and Ge, distorted tetrahedra are favoured over double bonding.

This argument can be developed further to explain the differences between Ge and Si found in the previous chapter. ST12 has bondlengths clustered more closely around the `ideal' (diamond) value than does BC8, but at the expense of a much wider spread of bond angles. Consequently, in germanium where smaller overlap allows for easier bond angle distortion, the ST12 structure is favoured. In silicon the bond-bending is more costly, so the BC8 structure is found. Finally in carbon, with no core p electrons, the orthogonalization cost of bond angle distortion is too great for either ST12 or BC8 to form with true fourfold coordination.

Thus the model of equation 5.13 appears to meet the criteria for the current study.


next up previous
Next: Effect of Pressure on Up: Complex Structures: Empirical Treatment Previous: Lattice Dynamics

Stewart Clark
Thu Oct 31 19:32:00 GMT 1996