Method of development of the potential

The main tool in the developm ent o f the potential param eters w as the com puter program GULP, w ritten by Julian Gale[15]. G ulp enables the fitting o f potential param eters by m eans o f an iterative process o f m inim isation and com parison, based upon a least squares algorithm , using a “figure or m erit” function[16]. In the GULP program, the core least squares algorithm is known as “Squares” . In each iteration of the developm ent cycle, a three stage approach was used. To begin, the potentials are given an initial value. The first stage, controlled w ith the keyw ord “fit” does not

C h apter 6 : ZnO po ten tia ls fittin g

Squares routine, and alters the variable portions o f the potentials in order to m inim ise these forces, or first derivatives o f the energy equations. The second stage is controlled w ith the keyw ord “sim ul” which repeats the procedure o f the first stage but allow s the shells to relax within the m odel along w ith the potential param eters. Again, the forces are used by the Squares routine to m inim ise the system. The final stage, “relax” , allows all of the atomic co-ordinates to relax, as well as the param eters, and judges convergence by passing the co-ordinates to the Squares function (i.e. it calculates the least squares o f the relaxed co-ordinates against the original co-ordinates). The three stages are em ployed in order to bring the system gently to convergence, as sim ply letting all o f the param eters and co-ordinates vary from the start w ould probably produce an unstable calculation w ith very large and very small variables resulting as particular variables dom inate the system.

The cycle will continue until the differences betw een the observed and calculated values fall below a specified value. Such a convergence criterion how ever does not necessarily m ean that a suitable solution has been found as it m ay include larger deviations from the m ore im portant param eters and sm aller deviations from the less im portant ones. In such cases, the output of the calculation may be taken as a starting point, and m odifications m ade to the variables and the process repeated. In order to ensure that priority is given to a good agreem ent w ith the observed properties which are considered to be m ost im portant, a w eighting system m ay be used as it is com m on to give high w eighting to structural param eters.

W eightings can be applied as appropriate, to give precedence to particular experim ental param eters. The cycle is term inated w hen the calculated physical properties are suitably converged with the m easured properties, although again this type convergence point is determ ined manually.

A typical input file fo r a cycle of this fitting procedure is show n in Table 6.3.1, which lists actual input inform ation and a key. T he values in the table were taken from the initial fitting cycle. The items with labels containing a “w ” correspond to data for the w urtzite phase and those with labels containing a “c” correspond to data

C h apter 6 : ZnO po ten tia ls fitting

for the cubic or rock salt phase. The labels containing a “p” belong to the items which are the potentials which w ere being fitted.

Label

Input type

Value(s)

w l cell 3.249600 3.249600 5.204200 90.000000 90.000000 120.000000 w2 Zn core 1/3 2/3 0.0000000 2.00000000 1.00000 0.00000 w3 0 core 1/3 2/3 0.3800000 0.92290116 1.00000 0.00000 w4 0 shell 1/3 2/3 0.3800000-2.9229011 1.00000 0.00000 w5 Sdlc 1 1 9.2600 w6 Sdlc 3 3 11.0000 w7 Hfdlc 1 1 4.1170 w8 Hfdlc 3 3 4.0480 w9 Elastic 1 1 20.9700 wlO Elastic 1 2 12.1100 w l l Elastic 1 3 10.5100 w l2 Elastic 3 3 21.0900 w l3 Elastic 6 6 4.4300 w l4 K points 1 0.0001 0.0000 0.0000 1.0000 c l Cell 4.271000 4.271000 4.271000 90.00 90.00 90.00 c2 Zn core 0.0000000 0.0000000 0.0000000 2.00000000 1.00000 0.00000 c3 0 core 0.5000000 0.0000000 0.0000000 0.92290116 1.00000 0.00000 c4 0 shell 0.5000000 0.0000000 0.0000000 -2.9229011 1.00000 0.00000

pi buck Zn core 0 shel 650.758220 0.341410 0.00000 0.000 10.000 p2 buck 0 shelO shel 21240.235000 0.226095 32.00000 0.000

10.000

P3 spring 0 28.117974 2879.677000

Table.6.3.1. Input param eters for an optim isation cycle in the fitting procedure. The values which w ere initially allowed to vary are m ade bold.

For the w urtzite structure, the cell param eters and the core and shell pair o f the oxygen ion (in the direction of the long axis of the cell) w ere allow ed to relax. In particular, it was desirable to ensure that the w urtzite structure rem ained as a m inim um energy configuration, to avoid the final potentials adopting other structures. Relaxation in the other directions was not incorporated in the first cycle as the m ajor polarisation effects w ould be expected to occur in the direction which exhibits a dipole. Items w5 through to w l3 show the dielectric and elastic constants included as observable data.

C h apter 6 : ZnO po ten tia ls fittin g

potential param eters to be fitted were lim ited to the A and p values o f the Z n -0 potential and the harmonic term o f the spring potential. The other param eters were allow ed to relax in subsequent cycles, but the initial cycle was conducted in this m anner as a coarse fitting, varying only the param eters w hich w ere likely to change most.

Further, in subsequent cycles, it was found that the harm onic term o f the spring potential alone was not sufficient to reproduce the polarisation o f the oxygen core and shell pair in all circumstances, which was determ ined by creating a surface model o f the [0001] face o f the crystal and introducing a single ad-atom to the surface. The ad-atom was an oxygen ion placed upon the zinc term inated (0001) surface and the test was done to ensure that the relaxation o f this low coordination site w ould match with predictions made by electronic calculations. The surface model was geom etry optim ised in both the M ARV IN program m e and by electronic m ethods using D ensity Functional Theory (DFT)[17J; the bond distances between the ad-atom and its bonded surface neighbour were com pared, the result from the electronic calculation being taken as a standard. W ith only a harm onic term in the spring potential, all o f the trial fittings o f potentials yielded optim ised surface structures with the ad-atom much closer to the surface than the standard. However, with an extra term, the quartic term (see equation 3.4 in C hapter 3), incorporated into the fitting procedure, an optim ised geom etry was obtained w hich exhibited an ad-atom distance 0.05Â shorter than the electronic calculation predicted by the D FT calculation. As it has been docum ented that D FT m ethods can result in bond lengths which are slightly too long[18], and with a lack o f suitable other m ethods to corroborate the test o f the surface ad-atom, the spring quartic term was included in the final param eterisation. Com parisons between the bulk structures calculated with and w ithout the quartic term for the spring potential show ed that inclusion o f the latter did not affect the relaxation of the bulk structure. In a physical sense, the harmonic com ponent o f a spring potential reproduces the linear effects o f the crystal field upon the polarisation of the ion, w hereas the quartic term reproduces the non-linear effects of the changing polarisability o f the ion as a function o f the crystal field. As the intended applications of the potential included surface studies, the

C h apter 6 : ZnO poten tials fittin g

inclusion o f the quartic term was essential to allow for the variability o f the crystal field in the surface region.

D uring the fitting procedure, it becam e apparent that a suitable solution was unlikely to be found: every cycle produced a fit which had som e unacceptable discrepancy w ith the observable data. The main cause o f these problem s was found to be that no particular Buckingham potential could adequately describe the zinc to oxygen shell interaction at all distances. In particular, the w urtzite structure allow s for three particular types o f zinc-oxygen interaction depending on Z n -0 separation.

The three different types o f interaction can be readily understood: the first refers to the bonded ions, which represent the nearest neighbours o f each ion. Taking a zinc ion as a reference point, the tetrahedral distribution o f ions m eans four nearest neighbour bonded oxygen ions. The next nearest neighbour oxygen ions are located three chem ical bonds away, but crucially the bond angles m ean that they lie with no ions screening the interaction between the pair. The distance betw een these next nearest pairs in the bulk structure is 3.2Â; again there are four o f these interactions. The third region contains all of the other zinc-oxygen pairs, as these interactions are all shielded by interm ediate ions they are considered together.

In view o f the three different regions o f interaction, three different Buckingham potentials w ere introduced to the fitting procedure, rather than a single one. Suitable ranges were applied to the three Buckingham potentials : 0.0 - 2.2Â for the bonded ions, 3.1 - 3.3Â for the non-bonded and non-shielded ion pairs and 3.6 - 12.0Â for the non bonded, but shielded ion pairs. The A, p and C values for each o f the three Buckingham potentials were fitted during subsequent cycles o f the fitting procedure. To supplem ent the shortest range o f B uckingham potential, a Lennard-Jones potential was co-fitted in the 0.0 - 2.2Â range. This extra potential was found to be required to make the total interaction stronger in the bonding region, giving a m ore realistic profile to the com bined potential. Including a Lennard-Jones potential also has the advantage o f ensuring that the potential does not start to becom e attractive at

C h apter 6 : ZnO poten tials fittin g

The three B uckingham regions o f the Z n -0 potential used during the fitting procedure were, obviously, not continuous with regard to the ranges, hence splining functions were used to link together the different Buckingham regions. The splining potentials had to give as near as possible a continuous function w ith continuous first and second derivatives. This requirem ent is to account for a case w here the potential could be applied in a situation where the zinc-oxygen shell distance w ould lie outside o f the three ranges. Such a situation may occur in a vacancy structure for exam ple, w here the lattice relaxes to com pensate for the defect. It is im portant that the transition between regions of the potential should be sm ooth, as any discontinuities or spurious points of inflection could cause a m inim isation or m olecular dynamics calculation to produce spurious results ow ing to sudden shifts in energy.

A polynom ial potential was selected to do the jo b o f bridging the ranges between the Buckingham potentials as polynom ial functions are easily m anipulated and may be controlled by incorporating extra terms and coefficients. A Fortran program was written to create and modify the polynom ial potentials[19] and a spreadsheet program was used to visualise them; it was found that six term s w ere required to m ake the functions flexible enough to m eet the sm ooth gradients criteria. The form o f the polynom ial function is :

£■ = Co + Cii? +

+ C

R* + C^R^

W here R represents the distance between the ions.