QMMS Lecture #7 (Venables/Heggie)

Notes for QMMS Lecture 7 (Venables/Heggie)

Lecture notes by John A. Venables and Edward Hernández. Lecture scheduled for 29 Nov 05. Latest version 28th October 2005.

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7. Diamond structure semiconductors: the case of Silicon

This lecture describes semiconductor structures, band structures, and the models used to explain them. If you are not familiar with semiconductors and their structures, you will need access to sources which describe the diamond, wurtzite and graphite structures, and which also describe the bulk band structures. It is also helpful to have some prior knowledge of the terms used in covalent bonding, such as s and p bands, sp2 and sp3 hybridization. However, these terms will be discussed in the lecture and can be explored via problem 7.4.1.

7.1 Bonding in diamond, graphite, Si, Ge, GaAs, etc.

In the case of the group 4 elements, there is a progression from C (diamond, with 4 nearest neighbors), through Si and Ge with the same crystal structure, then on to Sn and Pb. The last two elements are metallic at room temperature, Pb having the ‘normal’ f.c.c structure with 12 nearest neighbors. We might well ask what is giving rise to this progression, and where do Si, Ge, GaAs, etc fit on the relevant scale. A frequent answer is to say something about sp3 hybrids, assume that is all there is to say, and move on. However, there is much more to it than that; the extent to which one can go back to first principles is limited only by everyone’s time, as explained in the reference list.

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 wAB combines as


wAB = (4h2 +DE2)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 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, 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 sp3 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 sp3 bonding in diamond, Si, Ge and gray Sn; b) stages in the establishment of the valence and conduction bands via s-p mixing, involving DEsp 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 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 valence band.

7.2 Simple 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 different <111> directions in the diamond structure, so that


y[ 1 1 1] = 1/2{s + px + py + pz},    y[ 1-1-1] = 1/2{s + px - py - pz},
y[-1-1 1] = 1/2{s - px - py + pz},    y[-1 1-1] = 1/2{s - px + py - pz}.     (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, 109o 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 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 120o, 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 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 sp3 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.

7.3 Tight-binding pseudopotential and ab-initio models of semiconductors

Professional calculations of surface structure and energies of semiconductors typically consider 4 valence electrons/atom in the potential field of the corresponding ion, in which the orthogonalization with the ion core is taken into account via a pseudopotential. This yields potentials which are specific to s-, p-, d- symmetry, but which are much weaker than the original electron-nucleus potential, due 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; overlaps with at most a few neighbors are included.

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 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 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).

7.4 Problems and projects relating to this topic

7.4.1: Band structures of Si, Ge and GaAs

Look up the calculated band structures of bulk Si, Ge and GaAs, and explain the meaning of the following terms in relation to these three solids.
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).

7.4.2 Band structures in tight binding models

Tight-binding pseudopotential models of tetrahedral semiconductors such as Si or Ge treat the valence s- and p-electrons as moving in the potential field of the nuclei plus closed shell electrons. Consult selected references for this lecture, and show one of more of the following:
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 k.
b) that a possible approach to this problem is to use sp3 hybrid states as the basis set, formed by linear combinations of 1 s- and 3 p-electrons as in equation (7.2). Find the relations between the overlap (or hopping) integrals expressed in the s, px, py, pz system and the sp3 system. What is the potential advantage of the sp3 basis?
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.

7.4.3 The 2x1 reconstruction at the surface of Si or Ge(001)

The simplest model of band structure associated with the 2x1 reconstruction on Si or Ge(001) involves fixing the atoms below the surface plane in their bulk positions, and constructing a matrix as a function of wavevectors kx, ky in the plane of the surface, and kz perpendicular to the surface, again involving an 8x8 matrix, but now with some of the matrix elements set equal to zero. What is needed in addition to calculate the equilibrium position of the surface atoms and the resulting surface band structure and surface states?


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