We analysed the off-diagonal density m atrix elements to flnd the bond order for the long and short addimers for our flat and buckled addimer dimerised structures. We do not present results for the undimerised structure th at is obtained from the ground-state (7x4) system as its electronic properties are essentially identical to those of the (7x2) unit cell. We found th at there are much stronger a bonds and tt bonds for the short addimer of the line than for the long addimer. The ss overlap in particular (0.388 for the short addimer vs 0.282 for the long addimer) is much stronger for the short addimer case (see Figure 5.13), and there is also a larger tt{jpxPx) overlap (0.214 vs 0.145) for the
short addimer (see Figure 5.14).
We have also evaluated the occupation of the electronic states on the short and long addimers of the line to see if there is a charge density wave present in the second-level addimers. We found th at there is a différence in the occupation of states on atoms in the long addimer when compared with atoms of the short addimer. We found th a t the for the flat addimer system of the metastable structure the long addimers have « 0.1 more of an electron than the short addimers; for the buckled addimer system of the
10
5
0
Eigenvalue (n) (Hartrees)
Figure 5.13; Plot of the COOP for all occupied states as a function of eigenvalue n for the overlap of the s orbitals for the second-level addimers of the line for a (7x8) dimerised unit cell.
-0 . 6 - 0.6 - 0.4 - 0.2 Eigenvalue (n) (In H anrees)
Figure 5.14: Plot of the COOP for all occupied states as a function of eigenvalue n for the overlap of the Px orbitals for the second-level addimers of the line for a (7x8) dimerised unit cell. See Figure 5.13 for colour coding.
metastable structure there is % 0.4 more of an electron on the long addimers than on the short addimers. In both cases this additional contribution for the long addimers fills the antibonding part of the ss and pxPx orbitals. We think that the reason there appears to be more charge difference between the long and short addimers for the buckled addimer of the metastable case is that the flat top addimer case involves a lot of charge transfer, with the electron transferred from the top addimer localising on the second-level addimers. This extra charge is found predominantly on the shorter second level addimers.
When we looked at which atoms of the surface of the unit cell had a large overlap with the HOMO state for the flat addimer system, we found that the largest overlap is on the adatoms that form the silicon string. The up adatoms of the silicon string have a larger overlap with the HOMO than the down atoms of the silicon string. There is also a large overlap of the HOMO state with the short (% 2.38 Â ) addimers of the line (see Figure 5.10). The HOMO state is tightly bound to structures that are oriented along the x4 direction as it was in the (7x2) structure, this is why the HOMO state is still quasi-one-dimensional. The LUMO state is tightly bound to the top addimers of the line
for the flat addimer structure. Any electrons or holes th a t are injected into this system would propagate along the string (or line).
For the buckled addimer case we find th at the HOMO state has a strong overlap with the long (% 2.60 Â ) second level addimers of the line, the silicon atoms of the surface th at are between these addimers, one of the top level addimers of the line and the flat silicon string of the surface (see Figure 5.12). Similarly the LUMO state has strong overlaps with the surface addimers. This HOMO (or LUMO) state is much more diffuse than the tightly localised HOMO (or LUMO) state of the flat addimer system, with a very large overlap across the line. This is why the HOMO (or LUMO) state has a much larger dispersion in the x7 direction than across the x4 direction. Any charges th at are injected into this system would disperse across the string or the line.
5.6
C onclusions
We have tested our tight-binding parameterisation against DFT results and found th at our relaxed structures are in general agreement with the DFT structures. We have used the thermodynamically favoured models of the (3x2) and c(4x2) unit cells as the structural components of the (nx2) series of reconstructions. We have found th a t we could obtain two classes of structure, a ground-state structure with a well defined and buckled c(4x2) surface, and a metastable structure where the surface has a partial (2x1) character. The ground-state structures were found to be wide bandgap semiconductors with a nearly isotropic and weakly dispersive HOMO state which had a slightly larger dispersion along the x n than the x2 direction. The LUMO state was found to have a strong dispersion along the x2 direction and a weak dispersion along the x n direction. The metastable structures are narrow bandgap semiconductors with an anisotropic and more strongly dispersive HOMO state. For these metastable states a structural transition occurs when n > 7; the top addimer of the line becomes flat. The HOMO state then has a large dispersion along the x2 direction and a small dispersion along the x7 direction and so is quasi-one-dimensional. Similarly the LUMO state has a weak dispersion along the x n direction and a stronger dispersion along the x2 direction. A tentative explanation of the structural transition is proposed on the basis th at, as we increase n, we increase the separation between the HOMO states of neighbouring unit cells th at are localised on the silicon string. This makes this state interact more strongly with the nearest silicon
line, and makes it energetically possible for the top addimer to flatten.
We then use our ground-state and metastable structures to construct and relax a (7x4) unit cell. The results depend on whether the ground-state or m etastable (7x2) structure is used. We found th at the ground-state (7x2) unit cell does not dimerise and the (7x4) unit cell has a structure composed of two (7x2) unit cells. We also found th at the metastable structure does dimerise to form a (7x4) unit cell. This dimerised unit cell has a lower total energy than the (7x4) unit cell derived from the (7x2) ground-state. Thus the dimerised (7x4) unit cell is thermodynamically favoured. This dimérisation also increases the bandgap, implying that it is a Peierls-like transition. The HOMO state is still quasi-one-dimensional and is confined to the silicon string and the silicon line.
Clearly the (nx2) series of reconstructions is a complex series with a rich variety of behaviours depending upon the details of the surface structure. However are we modelling the surface accurately? We are aware th at the charge transfer which is involved in the m etastable surface structure could be inaccurately treated, as the tight-binding method we use is a non-charge-self-consistent scheme. Currently we have been unable to test our
structures using a more accurate ab initio formalism.
The silicon string (the surface addimer) and the silicon line (the line addimers) are codependent in the metastable system. The presence of the line causes a silicon string to form which then causes the structural transition in the silicon line. If we strongly buckle the silicon string then we buckle the top addimer of the line. Is it reasonable th at the string-like structure forms in the first place on the c(4x2) part of the surface? As stated in the last chapter, the c(4x2) surface is only formed in defect-poor regions of the silicon rich (001) face of cubic silicon carbide. The silicon lines do constitute defects
on this surface. As n increases the surface between the lines does become more c(4x2)-
like. However the surface near the silicon lines should not be c(4x2)-like but (2xl)-like (as shown in section 4.1.4 of Chapter 4). We found this is what happens. Therefore we propose that the silicon string formation, which makes the silicon line symmetric, may occur on the surface. There is some experimental evidence th at the silicon lines are symmetric [106, 240, 216]; STM scans along and across the lines show th at the line is composed of symmetric units. However compared to the various surfaces we have modelled there is a paucity of experimental evidence on the silicon lines. There is no experimental evidence th at the surface near the lines has a (2xl)-like character, but STM
scans near the line may be strongly affected by the presence of the line.
We have found a (7x4) unit cell with two line-like features (the silicon string and the silicon line) which appears to have undergone a Peierls-like transition and has a quasi-one-dimensional electronic character with a narrow bandgap. Is this system a good candidate atomic scale wire? To find out we need to calculate the transport properties of this system. This is the subject of the next chapter, which concerns polaron formation and the effects on the electronic structure of the system.