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We can start with a 2-page handout, abstracted from Pettifor's book. This connects bonding and
anti-bonding orbitals in s-bonded homonuclear diatomic molecules with the overlap, or bonding
integral, h. (Note that h is not Planck’s constant, and the symbol often used for overlap
integral is b). For heteronuclear diatomic molecules where
DE is the energy difference of levels between the molecules
A and B, the splitting of the levels w_{AB} combines as

_{AB} = (4h^{2} +DE^{2})^{1/2}
(7.1) |

Several of you have already explored this equation via problem 1.4.3. Equation 7.1 leads to ideas, and scales, of electronegativity/ ionicity, based on the relevant value of (DE/h): for group IV molecules this is zero, increasing towards III-V’s, II-VI’s etc, roman numerals being the convention for the different columns of the periodic table; 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.

In the diamond structure solids, the tetrahedral bonding does indeed come from sp^{3}
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, as
indicated in figure 7.1. The s-p level separation in the
free atoms is about 7-8 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-antibonding) within the sp^{3} band,
largest in C (diamond) at 5.5 eV, and 1.1, 0.7 and 0.1eV 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).

Figure 7.1: a) Hybridization gap in due to sp^{3} bonding in diamond, Si,
Ge and gray Sn; b) stages in the establishment of the valence and conduction bands
via s-p mixing, involving DE_{sp} and the overlap
integral h (after Harrison 1980 and Pettifor 1995).

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), as described by Sutton,
pages 209-214. 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 sp^{3}, but is moving towards
s^{2}p^{2}. 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 valence band.

_{x} + p_{y} + p_{z}},
y[ 1-1-1] = 1/2{s + p_{x} - p_{y} - p_{z}},
y[-1-1 1] = 1/2{s - p _{x} - p_{y} + p_{z}},
y[-1 1-1] = 1/2{s - p_{x} + p_{y} - p_{z}}.
(7.2) |

The above equation (7.2) 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^{o} 28’. This is the angle preferred by the group IV
elements, not only in solids and clusters, but also in (aliphatic) organic
chemistry (i.e. from CH_{4} onwards).

We can contrast this with the planar arrangement in graphite, where three electrons
take up the sp^{2} hybridization, leaving the fourth in a p_{z}
orbital, perpendicular to the basal (0001) plane. The in-plane angle of the
graphite hexagons is now 120^{o}, with a strong covalent bond, similar to
that in benzene (C_{6}H_{6}) 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 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 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 sp^{3} hybrid form of the valence band,
and so GaAs resembles Ge. However, there are differences due to the lack of a
center of symmetry (space group bar43m). 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.

There are many different computational procedures, and there is strong competition to
develop the most efficient codes, which enable larger numbers of atoms to be included.
In particular, the Car-Parinello method (Car & Parinello 1985), which allows finite
temperature and vibrational effects 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), and has been developed here in the previous lecture; a particularly thorough account, with worked-out examples for silicon, 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 7.4.2.

The most frequent use of tight binding methods is as an interpolation scheme, fitting ab-initio LDA/DFT methods of the type discussed in lecture 5 or more chemical multiconfiguration 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.

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

The argument is that if one gets both bulk Si (Ge) and the dimer Si_{2}
(Ge_{2}) 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 confidence. 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. However, work is still proceeding on schemes which really can
span the range of configurations which are encountered in molecules, in solids and
at surfaces (Wang & Ho 1996, Lenosky *et al.* 1997, Turchi *et al.* 1998).

a) the existence of a direct versus an indirect band gap, and the position of the conduction band minimum;

b) spin-orbit splitting and the differences between light and heavy holes;

c) the removal of degeneracy by stress in compressed thin films, e.g. of Ge on Si(001).

a) that 8 electrons are required to describe the system, resulting in the need to diagonalize an 8x8 matrix to solve for the band structure as a function of the wavevector

b) that a possible approach to this problem is to use sp

c) that you can construct and solve for the energy bands of Si and or Ge using one or other of these basis sets, and particular values of the matrix elements, using a matrix diagonalization package, display your results graphically, and compare your results to the literature.

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