Fixed Region
T.2.4.2 Single layer defects
Steps on the (1 0 Î0 ) surface appear to have a height o f about 2.9Â [49], w hich is consistent with the upper tw o layers o f ZnO dimers. Such structures are expected as they leave the sm aller interlayer spacing between layers one and tw o, w hereas a one-layer step w ould leave an unfavourable term ination for the low er terrace. H owever, in the investigation o f the terrace areas o f the surface, Jedrecy et al. [50] reported occupancy factors for layer one that were significantly low er than those in layer two, suggesting that vacancy defects occur in the terraced regions o f the surface which may contain stabilised single layer vacancies. (There is a possibility that a larger num ber o f layer one vacancies may be stabilised by a sm aller num ber o f layer two vacancies, which is investigated in the next section.)
As no inform ation was available from the literature concerning the distribution o f the vacancy defects, a w ide variety o f different distributions have been investigated. U sing the 4x4 supercell, it was possible to allow for occupancy factors which varied by 0.0625 (1/16). Thus, introducing one dim er vacancy into the surface m odel gives an occupancy factor o f 0.9375; a second vacancy corresponds to 0.875; a third gives 0.8125, and four vacancies naturally gives an occupancy o f 0.75 (the reported experim ental value w as 0.77 ± 0.02[51]). M odels can be m ade for each occupancy factor (apart from 0, 0.0625, 0.9375 and 1) w hich exhibit different vacancy “patterns” . T he occupancy factors w hich are com parable to the reported occupancies have several such patterns and we investigated the single vacancy case as a trial and the four vacancy case as the case w hich is closest to the reported experim ental value. W hile it is likely that the configuration with the m axim um vacancy-vacancy separations has the low est energy, the problem is not trivial and requires investigation. Thus, tw o m ain factors are investigated w ith regard to single layer vacancies : occupation and distribution.
For the case with one vacancy per supercell (occupation 0.9375), there are o f course no com plications related to distribution. The energy cost for the vacancy was found to be 1.44 eV. The relaxation o f the surface around the vacancy was different from
C h apter 7 : ZnO surface studies
layer tw o dim ers which are exposed and under co-ordinated will relax in a sim ilar m anner to layer one dimers, rather than in the same fashion as the fully co-ordinated layer tw o dimers. Figure 7.2.7 shows the relaxation around the single vacancy. M ore fully; the vacant oxygen site leaves two layer tw o zinc ions under co-ordinated and they both relax away from the surface and aw ay from each other in the direction perpendicular to the dimers. The vacant zinc site leaves tw o layer tw o oxygen ions under co-ordinated which has the effect o f exaggerating the upw ards relaxation of the associated layer two zinc ions and also, facilitating the underlying layer three zinc ions, to relax upwards, as if they were layer two ions. T he under co-ordinated oxygen ions them selves do not relax to a significant extent, m uch like fully co-ordinated oxygen ions, but their under co-ordination influences the nearest neighbour zinc ions. N ot surprisingly, these relaxations serve to shorten the bond lengths between the under co-ordinated oxygen and the neighbouring zinc ions. In the under co-ordinated case the bond lengths are 1.848, 1.889 and 1.901 A ngstrom s w hilst for a fully co-ordinated layer two oxygen, the bond lengths are 1.915, 1.915, 2.008 and 2.091 A ngstrom s.
Finally, for the models with four vacancies (occupation 0.75), five different vacancy patterns were investigated. The patterns include the m ajor defect arrangem ents which are possible on a 4x4 surface supercell. All other vacancies can be thought o f as being com binations o f these five patterns and w ould be unlikely to yield m ore favourable vacancy energies because they w ould necessarily contain aspects o f the patterns which turned out to be least favourable. Figure 7.2.8 show s the arrangem ent of vacancies in each pattern and Table 7.2.3 details the vacancy energy found for each arrangem ent. Tw o patterns o f vacancies are significantly m ore favourable than the others; both were linear. The result that the linear arrangem ents w ere on the w hole better than the scattered arrangem ents is not surprising, as they concentrate and m inim ise the disruption to the perfect surface. H ow ever, we note that a line of vacancies in the [O il] direction does not form an unbroken groove, as it w ould do in the [010] and [001] directions, rather it results in a series o f vacant layer one dim ers, separated by occupied layer two dimers. This feature occurs because layer one and layer tw o dim ers alternate in the [O il] direction, w hereas in the [010] and [001]
Chapter 7 ; ZnO surface studies
Figure 7.2.7 : Layer two dimers relax as layer one dimers when exposed.