Self-consistent pseudo-potential calculations

In principle, surface atomic and electronic structures are both available from self-consistent calculations of the electronic energy and surface potential. Until recently, however, such calculations were rather unrealistic, being based on a one-dimensional model using a square well crystal potential, with a semi-infinite lattice of pseudo-ions added by first-order perturbation theory. This treatment could not adequately describe dangling bond surface bands. Fortunately, the situation has improved enormously as the result of an approach due to Appelbaum and Hamann (see ref. 70 and references cited therein), which is based on the following concepts. 

Since both bulk and surface states are molecular in character, the wave functions of atoms in both types of position can be calculated by the same method. Appelbaum and Hamann [70] assume two-dimensional periodicity along the surface and make the same Fourier expansion of the pseudo-wave function as for the bulk, except that at each of a set of discrete surface normal co-ordinates a different set of expansion coefficients is used. These sets can be integrated from outside the surface into the bulk. Well inside the bulk, these wave functions are matched to bulk states of similar lateral symmetry and the matching condition determines energies and wave functions. 

The result of such calculations show that, on a clean semiconductor, surface atomic sites in equilibrium always differ substantially from those of a semi-infinite lattice and there is an inward force on these surface atoms, since the presence of a dangling bond on a surface atom strengthens the back bonds to atoms in the second layer. This means that the back bonds assume some double bond character (i.e. the bond order becomes greater than unity). The consequent change in bond length leads to so-called surface relaxation (see Sect. 3.3). 

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Self-consistent pseudo-potential calculations, based on a relaxed surface demonstrating this movement, were carried out by Chalikowsky et al. [191], while Mele and Joannopoulos [188], Calandra et al. [192] and Pandey et al. [167] used a tight-binding approach and obtained similar results, but in general there were still intrinsic gap states, contrary to experimental data. 

Concomitant with the experimental work described above, several attempts have been made to calculate the surface electronic structure of 110 GaAs, and we can now consider how the theoretical models compare with and explain the experimental results. The first point to make is that where calculations consider only an unrelaxed, i.e. ideal, surface, one or two bands of intergap states are predicted, from either a tight binding approach [167, 188, 189] or a self-consistent pseudo-potential treatment [190]. There is an occupied As dangling bond band and an unoccupied band corresponding to Ga dangling bonds. 

The most detailed calculations, in relation to the different surface structure models, have been performed very recently by Chadi [193, 194] with the tight-binding method, and Chalikowsky and Cohen [195] using self-consistent pseudo-potentials. The first important point is that both calculations show a complete absence of intergap states, both filled and empty, for a preferred, although not unequivocally established surface structure in which the angle of inward rotation of the Ga atoms is... 

In addition, recent calculations by Ho et al. [213] using self-consistent pseudo-potentials basically confirm the existence of the trihydride phase. They suggest that relatively minor discrepancies between the structural model and the calculations can be accounted for by strong hydrogen-hydrogen interactions, which become important in the dense hydrogen monolayer of the trihydride phase. 

Chlorine is the only other gas whose interaction with silicon surfaces has been studied in a reasonably detailed and systematic manner. Florio and Robertson [246] reported on the kinetics of the reaction of Cl2 with Si 111 7x7 over a decade ago and, more recently, several groups [247—250] have used various types of photoelectron spectroscopy to evaluate the chemical binding sites and electronic states which occur when a saturation coverage of Cl2 is chemisorbed on Si lll] 2 x l,Si lll 7x7 and Si 100 2x1. These results, relating to Cl-induced surface energy bands, have been compared with self-consistent pseudo-potential [247, 250] and tight-binding [249] calculations. 

FIGURE 6.15 Fermi surfaces of LiC6 calculated using empirical pseudo-potentials and a self-consistent determination of the charge transfer the Fermi surfaces for the lower (a) and upper (b) bands. (From Ohno, T., J. Phys. Soc. Jpn. 49(Suppl. A), 899, 1980. With permission.)... 

The core electrons of all atoms were treated via ultra-soft pseudo potentials [10,11] with a cut-off of 25 Ry for wave function, and 240 for electronic density. The PBE gradient-corrected exchange-correlation function was used in self-consistent DFT calculations. The geometry optimization was performed using a lxlxl /c-point mesh. Because of the natural paired electron occupancies of the adsorbates, spin polarization effects were not considered to be important and were not treated explicitly in this study. 

This does not mean that the LCAO approach of the type we have used is incorrect or not useful. Recent applications of LCAO theory, based only upon electron orbitals that are occupied in the free atom, have been made to the study of simple metals (Smith and Gay, 1975), noble-metal surfaces (Gay, Smith, and Arlinghaus, 1977), and transition metals (Rath and Callaway, 1973). In fact, the LCAO approach seems a particularly effective way to obtain self-consistent calculations. The difficulty from the point of view taken in this book is that, as with many other band-calculational techniques, LCAO theory has not provided a means for the elementary calculations of properties emphasized here, but pseudo-potentials have. 

Fig. A1.2 Ga and As atoms and pseudo-atoms radial parts of the wavefunctions (solid lines) ane pseudo-wavefunctions (broken lines), as calculated for valence electrons from the self-consistent potentials and pseudopotentials of Fig. Al.l. 

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