7.1 Structural and electronic effects	229

up an energy gap (valence±conduction, equivalent to bonding±antibonding) within the sp3 band, largest in C (diamond) at 5.5 eV, and 1.1, 0.7 and 0.1 eV for Si, Ge and (gray) Sn respectively. Sn has two structures; the semi-conducting low temperature form, alpha or gray tin (with the diamond structure), and the metallic room temperature form, beta or white tin (body centered tetragonal, space group I41/amd).

The question of phase transitions in Si as a function of pressure is also a fascinating test-bed for studies of bonding (Yin & Cohen 1982, Sutton 1994). Even at normal pressure, there is some discussion of bonding in these group IV elements, especially in the liquid state (Jank & Hafner 1990, Stich et al. 1991). For example, liquid Si is denser than solid Si at the melting point, and interstitial defects are present in solid Si at high temperature. In this state, the bonding is not uniquely sp3, but is moving towards s2p2. Pb has basically this con®guration, but, as a heavy element, has strong spin-orbit splitting. This relativistic eVect is also important in Ge, being the cause of the diVerence between light and heavy holes in the valence band. You can see that all these topics are fascinating: the only danger is that if we pursue them much further here, we will never get back to surface processes!

7.1.2Simple concepts versus detailed computations

Simple concepts start from the idea of sp3 hybrids as the basic explanation of the diamond structure. These hybrids are linear combinations of one s and three p electrons. Their energy is the lowest amongst the other possibilities, but as seen in the arguments given by Pettifor, Sutton and others, it can be a close run thing. The hybrids give the directed bond structure along the diVerent K111L directions in the diamond structure, so that

c[111]51/2 {s1px1py			Å Å	51/2 {s1px		2py	2pz}
			1pz}, c[111]
Å Å	51/2 {s2px	2py		Å Å	51/2 {s	2px	1py2pz},	(7.2)
c[111]			1pz}, and c[111]

which has a highly transparent matrix structure, exploited in the tight binding and other detailed calculations. The key point is that these bonds are directed at the tetrahedral angle, 109° 28Â. This is the angle preferred by the group IV elements, not only in solids and at surfaces, but also in (aliphatic) organic chemistry (i.e. from CH4 onwards).

We can contrast this with the planar arrangement in graphite, where three electrons take up the sp2 hybridization, leaving the fourth in a pz orbital, perpendicular to the basal (0001) plane. The in-plane angle of the graphite hexagons is now 120°, with a strong covalent bond, similar to that in benzene (C6H6) and other aromatic compounds, and weak bonding perpendicular to these planes. The binding energies of carbon as diamond and graphite are almost identical (7.35 eV/atom), but the surface energies are very diVerent ± basal plane graphite very low, and diamond very high. The combination of sixand ®ve-membered rings that make up the soccer-ball shaped Buckminster-fullerene, the object of the 1996 Nobel prize for chemistry to Curl, Kroto and Smalley, is also strongly bound at ,6.95 eV/atom. All these are fascinating aspects of bonding to explore further.

230 7 Semiconductor surfaces and interfaces

The next level of complexity occurs in the III±V compounds, of which the archetype is GaAs. This is similar to the diamond structure (which consists of two interpenetrating f.c.c. lattices), but is strictly a f.c.c. crystal with Ga on one diamond site and As on the other; with the transfer of one electron from As to Ga, both elements adopt the sp3 hybrid form of the valence band, and so GaAs resembles Ge. However, there are diVer-

Å

ences due to the lack of a center of symmetry (space group 43m), which we explore in relation to surface structure in the next section. In addition, many such III±V and II±VI compounds have the wurtzite structure, which is related to the diamond structure as h.c.p. is to f.c.c. These two structures often have comparable cohesive energy, leading to stacking faults and polytypism, as in a- and b-GaN, which are wide-band gap semiconductors of interest in connection with blue light-emitting diodes and high power/ high temperature applications, as described in section 7.3.4.

7.1.3Tight-binding pseudopotential and ab initio models

Professional calculations of surface structure and energies of semiconductors typically consider four valence electrons/atom in the potential ®eld of the corresponding ion, in which the orthogonalization with the ion core is taken into account via a pseudopotential. This yields potentials which are speci®c to s-, p-, d-symmetry, but which are much weaker than the original electron±nucleus potential, owing to cancellation of potential and kinetic energy terms. All the bonding is concentrated outside the core region, so the calculation is carried through explicitly for the pseudo-wavefunction of the valence electrons only, which have no, or few nodes;1 overlaps with at most a few neighbors are included. There are many diVerent computational procedures, and there is strong competition to develop the most eYcient codes, which enable larger numbers of atoms to be included. In particular, the Car±Parinello method (Car & Parinello 1985), which allows ®nite temperature and vibrational eVects to be included as well, has been widely used. This method is reviewed by Remler & Madden (1990), while Payne et al. (1992) give a review of this and other ab initio methods.

The tight binding method is described in all standard textbooks (Ashcroft & Mermin 1976); a particularly thorough account is given by Yu & Cardona (1996). Tight binding takes into account electrons hopping from one site to the next and back again in second order perturbation theory, which produces a band structure energy which is a sum of cosine-like terms, as can be explored via problem 7.2. Zangwill (1988) applies this method in outline to surfaces; he shows that the local density of states (LDOS) is characterized by the second moment of the electron energy distribution. As a result, the second moment of r(E) is proportional to the number of nearest neighbors Z, and to the square of the hopping, or overlap, integral (h or b). At the surface, the number of neighbors is reduced, and so the bandwidth is narrowed as Z1/2. But this simple argument on

1The pseudo-wavefunction, being `smoother', requires fewer (plane wave) coeYcients to compute energies to a given accuracy. However, this can lead to diYculties in comparing diVerent situations, e.g. between solids and atoms, which may require diVerent numbers of terms. Some of these points are mentioned in Appendices J and K.

7.1 Structural and electronic effects	231

Table 7.1. Lattice constants, binding energies and vibrational frequencies of Si and Ge

Material			Spacing (nm)	Energy (eV)		Frequency (THz)
1.	Bulk Si	calc.	0.545c, 0.550e, 0.537g	4.67c, 5.03e, 4.64g		15.16c,15.6e
		expt.	0.5430	4.63	6 0.04	15.53c
2.	Si	calc.	0.225b, 0.227e, 0.228f	4.18b, 3.62f		15.0b, 15.9e, 14.4f
	2	expt.	0.224e	3.21	6 0.13	15.3b,e
3.	Bulk Ge calc.		0.556c, 0.558g	4.02c, 3.86g		8.90c
		expt.	0.5658	3.83	6 0.02	9.12c
4.	Ge2	calc.	0.234a, 0.242d, 0.2326±0.2385h	4.14a, 2.50±2.67h		8.57a, 8.48d, 8.42±8.82h
		expt.		2.70	6 0.07d

References: (a) Northrup & Cohen (1983); (b) Northrup et al. (1983); (c) Yin & Cohen (1982), Cohen (1984); (d) Kingcade et al. (1986); (e) Sankey & Niklewski (1989); (f) Fournier et al. (1992); (g) Krüger & Pollmann (1994, 1995); (h) Deutsch et al. (1997). Where not referenced, experimental values are from table 1.1; others can be traced via the papers cited.

its own is not suYcient to reproduce the band structure of Si and Ge in any detail, either in bulk or at the surface. The various approximations for bulk band structures are discussed by Harrison (1980), Kelly (1995), Yu & Cardona (1996) and Davies (1998).

One of the ®rst workers to pioneer tight binding methods was D.J. Chadi (for a short review see Chadi 1989), but there are many others who have made realistic calculations on semiconductor surfaces, and atoms adsorbed on such surfaces (e.g. in alpha-order C.T. Chan, M.L. Cohen, C.B. Duke, R.W. Godby, K.M. Ho, J. Joannopoulos, E. Kaxiras, P. Krüger, R.J. Needs, J. Northrup, M.C. Payne, J. Pollmann, O. Sankey, M. ScheZer, G.P. Srivastava, D. Vanderbilt, A. Zunger, to mention only a few). The most frequent use of tight binding methods is as an interpolation scheme, ®tting ab initio LDA/DFT methods of the type discussed in chapter 6 or more chemical multicon®g- uration calculations, but computationally much faster.

Examples of the level of agreement with lattice constants, dimer binding energies and vibrational frequencies from the ab initio work are given in table 7.1. Note that the spacing is the lattice spacing of the solid, or the internuclear distance in the dimer. The energy represents the sublimation energy at 0 K, including the zero point energy, for the bulk solid; for the dimer it is the dissociation energy of the molecule in its ground state. The frequency is the optical phonon or stretching frequency. The argument is that if one gets both bulk Si (Ge) and the dimer Si2 (Ge2) correct, then surfaces and small clusters, which are in between, must be more or less right. If tight binding schemes can bridge this gap, then large calculations can be done with more con®dence. One may note from table 7.1 that the early ab initio LDA calculations tended to be overbound, sometimes by as much as 1 eV, but this improved over time. Many more details of tight binding methods in the context of surfaces are explained by Desjonquères & Spanjaard (1996); however, work is still proceeding on schemes which really can span the range of con®gurations which are encountered in molecules, in solids and at surfaces (Wang & Ho 1996, Lenosky et al. 1997, Turchi et al. 1998).