Defects in the (lOlO) surface structure

D efect energies in this section have been calculated using the follow ing equation :

E v a c

=

^ *

where E'^^^£ is the energy of the fully occupied surface, EfJ^^ace is the total energy o f the defective surface in question, Edmer'i^ the energy o f a ZnO pair w hich is included in a bulk lattice and n is the num ber o f such pairs w hich m atch the difference in ZnO dim ers between the fully and partially occupied surface m odels.

O ur reported defect energies are therefore a m easure o f the energetic difference between a perfect surface and a defective surface with additional bulk dim ers equal to the num ber o f vacancies. The effect is to analyse m odels with the same num bers o f ions, although we do not have to actually add the additional bulk dim ers into our com puter model.

7.2.4.1 Simulation details

Sim ulations to investigate defective surfaces were undertaken w ith the same basic model as for the fully occupied surface, but w ithout specific ions in order to introduce vacant sites. To avoid finding local m inim a associated w ith the m inim ised fully occupied structure, each calculation on defective surfaces was initiated with the ions positioned at their projected bulk co-ordinates. This approach is com putationally m ore expensive than using the m inim ised surface from the fully occupied m odel, but is less likely to give a result which is biased tow ards that fully occupied structure.

Several types o f defects could be im agined to occur at crystal surfaces, including vacant sites, ions occupying interstitial sites and the presence o f either im purities included in the lattice, o r contam inant ions adsorbed onto the surface. Only vacancy defects are considered here. O ther types o f surface defects have not been reported in

C h apter 7 : ZnO surface studies

previous studies; no reports o f interstitial sites have been m ade and the sam ple preparation for experim ental studies typically rem oves contam inants to below detectable levels. For this particular surface, the m ost recent experim ental study of Jedrecy et al.[47] finds vacancies in the upperm ost layers o f the crystal, as m entioned above. The study, how ever gives no inform ation on the distribution o f the vacancies or on the stoichiom etry o f the vacant sites. O xide m aterials are often reported to exist with a net lack o f oxygen ions[48], suggesting that charged defects (along with neutral defects) can somehow be accom m odated w ithin the structure. However, investigating charged vacancies w ith M A RV IN is not yet possible ow ing to the requirem ent of electroneutrality. Furtherm ore, selecting the ratio o f anion to cation vacancies w ould never the less have to be conducted on an arbitrary basis due to the lack o f experim ental evidence at the surface. Thus, for these tw o reasons, we im pose charge neutrality by an equal concentration o f zinc and oxygen vacancies.

Tw o further sim plifications have been built into the investigation: the first is the assum ption that vacant sites o f opposite charge will be located adjacent to each other; the second that both o f the adjacent vacancies will be located in the same layer. Thus all the vacancies considered will be o f the form o f a “dim er vacancy” as illustrated in Figure 7.2.5. To justify the first assum ption, a com parison was made using a surface m odel containing one anion and one cation vacancy. The vacancies were arranged so that they w ould be adjacent in one m odel and in the second m odel, the anion vacancy was placed 2 unit cells in each surface direction away from the cation vacancy (using a surface supercell o f 4x4 unit cells). The energy o f the adjacent vacancies is 8.1 eV m ore favourable than the non-adjacent vacancies. It would o f course be expected that in a real crystal, non-adjacent vacancies w ould indeed appear, in a sim ilar m anner to the occurrence o f Schottky defects in the bulk. However, inclusion o f a range o f defects in a single m odel w ould require an enorm ous surface repeat unit so we concentrate on adjacent vacancies.

The reason for the unfavourable energetics o f isolated vacancies in our m odel can readily be understood by exam ining the environm ent o f the ion adjacent to the

C h apter 7 : ZnO surface studies

ions being energetically less stable com pared to bulk ions. W hen a surface ion is adjacent to a vacancy, the co-ordination of that ion is further reduced to tw o w ith the consequent reduction in the energy o f that ion. O f course, each surface vacancy reduces the co-ordination of three ions, not ju st the adjacent one; the other tw o ions reside in the layer below and their co-ordination is altered from four to three. In the case w here two adjacent surface ion sites are vacant, each vacant site leaves two underlying ions with a reduced co-ordination. A nother w ay to think about this effect is to consider the vacancies as charged defects, the oxygen vacancy w ould be a positively charged defect, for exam ple. A pair o f such defects will have an interaction energy which is inversely proportional to the separation. As such, a zinc vacancy and an oxygen vacancy will have a favourable interaction energy w hich is stronger when they are close together. These two situations are illustrated in Figure 7.2.6, w hich shows that adjacent vacancies low er the co-ordination o f a total o f four ions, w hilst non-adjacent vacancies low er the co-ordination o f a total o f six ions, including two ions which are left with a co-ordination o f two. The ionic properties may not be drastically perturbed by changes in co-ordination; how ever, such changes will not be fully represented by m odelling m ethods based on atom istic potentials as the shell model description is unable to resolve the subtleties o f changes in the electron polarisation o f the surface ions. N evertheless, the differences are expected to be small as the shell model does represent the polarisations to an extent, and atomistic m ethods are sufficient for our present purposes.

The assum ption that the vacancy pairs will be located in the plane parallel to the surface is m ade for sim ilar reasons. Inserting a neutral defect w hich involved a cation from layer one and a bonded anion (or vica versa) from layer tw o w ould again cause the ion adjacent to the layer one ion to be co-ordinated to only tw o ions. A gain, this type o f defect is excluded from the investigation because it is likely to have a high energy and because the interatom ic potential m ethod is not flexible enough to give a proper treatm ent to the ions with co-ordination o f two.

To avoid verbose description of the type o f vacancies in the discussion, subsequent sections o f this chapter will use the term “vacancy” to m ean a single layer dim er

Chapter 7 ; ZnO surface studies

Figure 7.2.5 : Dimer vacancy. The positions of the vacant sites are represented by spheres and the lattice is represented by the wire frame. Vacancies considered as mono-layer, di-ion vacancies and are presumed always to occur with the vacant zinc and oxygen sites to be adjacent.

Figure 7.2.6 : Difference between adjacent and non-adjacent vacancies. Again the vacant sites are marked with spheres. The upper framework shows a pair of vacancies as a vacancy dimer, whilst the lower framework shows a separated pair of vacancies; a configuration not investigated here.

C h apter 7 : ZnO surface studies