Lecture notes by John A. Venables. Lecture given 16 Apr 96. Notes updated 4 Nov 96.
If you are not familiar with semiconductors and their structures, you will need a book which describes the diamond, wurtzite and graphite structures, and which also descibes the bulk band structures. It is also very helpful to have some prior knowledge of the terms used in covalent bonding, such as s and p bands, sp2 and sp3 hybridisation. Books that I have found very helpful for both metals and semiconductors are D. Pettifor, ‘Bonding and Structure of Molecules and Solids’ (1995), and A.P. Sutton, ‘Electronic Structure of Materials’ (1994), both published by Oxford University Press. These books are aimed at materials science students, but are useful much more widely. The standard semiconductor book aimed primarily at electronic engineers is S.M. Sze, ‘Semiconductor Devices: Physics and Technology’, Wiley (1985).
The first thing to realise is that the reconstructions of semiconductor surfaces are not, in general, simple. In section 1.4 we introduced reconstructions via the (relatively simple) Si(100) 2x1 surface. This aimed to instill ideas of symmetry lowering at the surface, domains, and the association of domains with surface steps. At the atomic cell level we saw the formation of dimers, organised into dimer rows; since then we have seen the beautiful STM pictures illustrating all these features. If all this can happen on the simplest semiconductor surface, what can we expect on more complex surfaces? More importantly, how can we begin to make sense of it all? This is a topic which is still very much at the research stage. But enough has been done to try to describe how workers are going about the search for understanding, which is what I attempt here.
However, there is much more to it than that; the extent to which we can go back to first principles is limited only by everyone’s time. In the lectures I went through a 2-page handout culled from Pettifor, Chap 3, page 54, and Chap 7, pages 198-201. This connects bonding and anti-bonding orbitals in homonuclear diatomic molecules with the overlap, or bonding integral, h; for heteronuclear diatomic molecules with h and the splitting of levels between the molecules A and B, wAB, combining as
This leads to ideas, and scales, of electronegativity/ ionicity, based loosely on the value of (wAB/h): for group 4 molecules this is zero, increasing towards 3-5’s 2-6’s etc; these scales try to establish the relevant mixture of covalent and ionic bonding in the particular cases: 3-5’s are partly ionic, and 2-6’s are clearly more so. If this material is unfamiliar or rusty, this is a good time to look at it again.
In the diamond structure solids, the tetrahedral bonding does indeed come from sp3 hybridization, but it is not obvious that this will produce a semiconductor, and the question of the size of the band gap, and whether this is direct or indirect, is much more subtle. The s-p level separation in the free atoms is about 7 eV, but the bonding integrals are large enough to enforce the s-p mixing and to open up an energy gap (valence-conduction, equivalent to bonding-anti-bonding) within the sp3 band, largest in C (diamond) at 5.5 eV, and 1.1, 0.7 and 0.1eV for Si, Ge and (grey) Sn respectively.
Sn has two structures; the semiconducting low temperature form, alpha or grey tin, (with the diamond structure) and the metallic room temperature form, beta or white tin, (body centered teragonal, 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 (M.T. Yin and M.L. Cohen, Phys. Rev. B26 (1982) 5668, see Sutton, chap 11, pages 209-212). Even at normal pressure, there is some discussion of bonding in these group 4 elements, especially in the liquid state (see e.g. W. Jank and J. Hafner, Phys. Rev. B41 (1990) 1497). For example, liquid Si is denser than solid Si at the melting point, and interstitial defects are present in solid Si at high T. In this state, the bonding is not uniquely sp3, but is moving towards s2p2. Pb has basically this configuration, but, as a heavy element, has strong spin-orbit splitting. This relativistic effect is also important in Ge, being the cause of the difference between light and heavy holes in the conduction band (see e.g. C. Kittel, Quantum Theory of Solids, Wiley 1987, pages 268-271 and Ashcroft and Mermin, pages 564-570). You can see that all these topics are fascinating: the only danger is that if we pursue them further here, we will never get back to surfaces!
We can contrast this with the planar arrangement in graphite, where three electrons take up the sp2 hybidization, leaving the fourth in a pz orbital, perpendicular to the basal (0001) plane. The in-plane angle of the graphite hexagons is now 120 deg, 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), but the surface energies are very different- basal plane graphite very low, and diamond very high. The combination of 6- and 5-membered rings which 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.
The next level of complexity occurs in the 3-5 compounds, of which the archetype is GaAs. This is similar to the diamond structure (which consists of two interpenetrating fcc lattices), but is strictly a fcc 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 resembles Ge. However, there are differences due to the lack of a centre of symmetry (space group bar43m), which we explore in relation to surface structure in the next section. A useful point to note is that a Ga atom, being tri-valent, would prefer sp2 bonding, which has the 120 degree angle, but that the penta-valent As atom prefers s2p3 bonding. This has an inter-bond angle of 94 deg. When the atoms are at the surface, they have some freedom to move in directions which changes their bond angles, and do indeed move in directions consistent with the above arguments.
The tight binding method is described in all standard Solid State textbooks (e.g. Ashcroft and Mermin, chapter 10). Zangwill applies this to surfaces in chapter 4; there, he shows that the local density of states (LDOS) is characterised by the second moment of the electron energy distribution, which can be illustrated as in Diagram B1. Tight binding takes into account electrons hopping from one site to the next and back again in second order perturbation theory. As a result, the second moment rho(E) is proportional to the number of nearest neighbors Z, and to the square of the hopping, or overlap, integral, beta. At the surface, the number of neighbors is reduced, and so the bandwidth is narrowed. Many more details of the tight binding methods in the context of surfaces are given by Desjonqueres and Spanjaard, Chap 5.5, pages 254-283.
One of the first workers to pioneer these methods was D.J. Chadi (see e.g. Ultramicroscopy 31 (1989) 1), but there are many other authors who have made realistic calculations on semiconductor surfaces, and atoms adsorbed on such surfaces (e.g. in alphabetical order M.L. Cohen, C.B. Duke, J. Joanopoulos, R. Godby, E. Kaxiras, R.J. Needs, J. Northrup, M.C. Payne, J. Pollmann, M. Scheffler, R.D. Vanderbilt, A. Zunger, to mention only a few).). An example of the level of agreement with lattice constants, binding energies and vibrational frequencies is provided by the table B1.
Recently, some of these authors have spent time in establishing principles by which such surfaces can be ‘understood’. This is possible because a large data base of solved structures now exists; one can therefore discuss trends, and the reasons for such trends. In particular, Duke has enunciated 5 principles in several articles, which helps us understand the following examples (C.B. Duke, Chemical Reviews (1996) in press; Scanning Microscopy 8 (1994) 753-764; Appl. Surf. Sci. 65 (1993) 543-552. Zunger has looked at ‘structural motifs’ which occur at 3-5 surfaces, regarding surfaces as special arrangements of these motifs (S.B. Zhang and A. Zunger, Phys. Rev B53 (1996) 1343). For earlier reviews on similar topics see Chapters 2 and 3 of ‘Surface Properties of Electronic Materials’ (Eds D.A King and D.P. Woodruff, Elsevier, 1988), i.e. Chemical Physics of Solid Surfaces and Heterogeneous Cataylsis, vol 5.
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