Electronic structure

As has already been mentioned we cannot comment in detail on the electronic structure of the Bunk model as our implementation of PAW is limited to sampling only four k -

calculate the properties of excited states because of the LDA. At the four A:-points sam­ pled we found that the gap (defined as the difference in K-S eigenvalues) at select parts of the Brillouin zone varies from 0.18 eV at the AT-point to 0.96 eV at the A'-point. This indicates that the Bunk model has a narrow bandgap with a large range and may be metallic. 0.0003 0.0002 .è

I

Cell height (Bohrs)

Figure 7.12: Plot of the charge density of the HOMO state as a function of cell height. Left side is bottom of the unit cell. Dashed line is position of the top of the unit cell.

Our simulations of the Bunk model show that there are two surface state bands separated from the valence band, with another band which is partially separated from the valence band; this is in agreement with angle-resolved photoemission experiments [263, 265]. The distribution of these states in the unit cell as a function of cell height is shown in Figures 7.12-7.14. The three highest occupied states all have quasi-one-dimensional dispersions with large dispersions along the x l direction (1.16 eV, 1.18 eV and 1.23 eV from HOMO to 3rd HOMO state) and much smaller dispersions along the x4 direction (0.4 eV, 0.23 eV and 0.21 eV from HOMO to 3rd HOMO state). These dispersions are the differences in Kohn-Sham eigenvalues at different points of the Brillouin zone.

We found that the highest occupied state has a large overlap (we define the overlap by the sum of the squares of the projectors pi for the s,px,Py and p^ orbitals used in (2.20)) with the indium adatoms (see Figure 7.4 for layout). The HOMO state has a very strong overlap with the middle pair of indium adatoms (0.8909 for the left middle In adatom, 0.4806 for the right middle In adatom) and the indium adatom on the left (0.2355). The third HOMO state also has a large overlap with these three indium adatoms, with an overlap of 0.3294 on the left middle indium adatom, 0.1820 on the right middle indium adatom and an overlap of 0.1569 on the top left indium adatom. The second highest state does not have a strong overlap with any of the adatoms on the surface. The silicon

20 40

C e» height (Bohrs)

Figure 7.13: Plot of the charge density of the second HOMO state as a function of cell height. See Figure 7.12 for layout.

chain does not have a strong overlap with any of these states.

0.0003

0.0002

0.0001

C e» height (Bohrs)

Figure 7.14: Plot of the charge density of the third HOMO state as a function of cell height. See Figure 7.12 for layout.

A plot of the charge density of the HOMO and 3rd HOMO states also shows that they are strongly associated with the middle pair of indium adatoms and that there is a very strong connection from one (4x1) unit cell to the next along the x l direction. These charge density plots are shown in Figure 7.15. Along with the dispersion characteristics of these states and the large surface components these states possess this strongly implies that if a hole is injected into these indium adatoms it would diffuse easily along the x 1 direction and would be confined to the indium atomic chain. The Bunk model would make a good atomic scale wire.

We cannot compare in detail our electronic structure with the theoretical electronic structural results that have been published in the literature because we sample only a small set of A:-points, whereas other theoretical studies have used a large set of A;-points [276, 277]. Our electronic structure results agree in general with the results published

xl I ] ABOVE r ~ ~ l 0,0006- I I 0,0 0 0 6- 0 0 0 0 6- xl x4

Figure 7.15: Plot of the charge density of the HOMO state (top) and 3rd HOMO state (bottom). This is a slice parallel to the surface at a point 3 Â above the surface.

in the literature, where both Miwa et al. and Nakamura et al. finding that the three highest occupied states possess quasi-one-dimensional behaviour and are confined to the indium atomic chains.

7.5

Simulation of the Saranin model

7.5.1 Physical structure

The relaxed structure of the Saranin model is shown in Figure 7.16. The surface is locally stable. We found that the silicon atomic chain is 0.5081 Â higher than the indium atomic chains. The silicon atomic chain is strongly buckled by 0.4501 Â. The indium atomic chain is also strongly buckled with the average buckle distance 0.6045 Â, with the two outer indium adatoms higher than the inside indium adatom. The separation between the two outer indium adatoms is 6.2715 Â, which is not in good agreement with the inferred separation between the atomic rows [275]. The indium atomic chain is not

equidistant between two silicon atomic chains with the separation from one side 2.5632 Â and from the other side the separation is 4.9649 Â. This asymmetry is not observed in STM images. The silicon substrate is in a (7665... ) arrangement. The details of the structure are shown in Table 7.2. Unlike the Bunk model, we cannot compare our atomic and electronic structures with results published in the literature as no simulations of the Saranin model have been presented. This also means that this work features the first analysis of the relative thermodynamic stability of the two competing models of the (4x1) surface reconstruction.

Figure 7.16: Relaxed structure of the Saranin model. See Figure 7.11 for layout.

7.5.2

Electronic structure

Although we cannot comment in detail on the electronic structure of the Saranin model we found that this system has a very small bandgap at all four A;-points in the Brillouin zone. The bandgap varies from 0.14 eV at the F-point to 0.32 eV at the X ' point. This indicates that the bandgap for the Saranin model is small and that the unit cell may be metallic.

Our simulations of the Saranin model show that there are four surface state bands sep­ arated from the valence band and there is another band which is partially separated from the valence band. This is in disagreement with angle-resolved photoemission experiments [263, 265], which shows th at there are only three surface state bands. The distribution of the three highest states in the unit cell as a function of cell height is shown in Figures

0.0003

0.0002

0.0001

Ce» height (Bohrs)

Figure 7.17: Plot of the charge density of the HOMO state as a function of cell height. See Figure 7.12 for layout.

7.17-7.19. The five highest occupied states all have quasi-one-dirnensional dispersions with large dispersions along the x l direction (0.68 eV, 0.97 eV, 1.18 eV, 1.24 eV and 1.25 eV from HOMO to 5th HOMO state) and much smaller dispersions along the x4 direction (0.39 eV, 0.17 eV, 0.22 eV, 0.23 eV and 0.20 eV from HOMO to 5rd HOMO state). 0.0004 0.0003 0.0002 0.0001 40

Cot! height (Bohrs)

Figure 7.18: Plot of the charge density of the second HOMO state as a function of cell height. See Figure 7.12 for layout.

We found that the highest occupied state has a large overlap on all three indium adatoms (see Figure 7.4 for layout). The HOMO state has a very strong overlap with the middle indium adatom (0.9137), and weaker overlaps with the other two indium adatoms (0.6391 for the right In adatom and 0.1887 for the left In adatom). The second HOMO state has a very large surface component, with very large overlaps on the two outer indium adatoms (1.1054 on the left In adatom and 0.7455 on the right In adatom). There is also a large overlap of this state with the middle In adatom (0.5067) and on the silicon atomic chain, with overlaps of 0.1515 on the higher Si atom and 0.1882 on the