Homopolar semiconductors

The nature of bonding can be seen immediately from a consideration of the effect on electron density of symmetric and antisymmetric linear combinations of the wave functions gi and g2. Consider for simplicity the interaction of two s-orbitals. (The sp hybrids behave essentially the same way but with more complex geometry.) 

Wave function amplitudes (probability of finding an electron) 

The energy difference between the bonding or antibonding states and the hybrid molecular orbital energy is referred to as the homopolar energy, V2 and the bondingantibonding orbital energy difference is 2V2. V2 can be shown [see Harrison, Ref. 1 for example] to depend approximately upon the inverse square of the interatomic distance, d, as  

---

At the same time, calling this a fit to the bands is very much understating the accomplishment. The set of four parameters in Table 2-1 and the term values in Table 2-2 (all in the Solid State Table) allow calculation of energy bands for any of the homopolar semiconductors or any of the zincblcndc-structure compounds, as simply for one as for the other, without computers, with consistent accuracy, and without need for a previous accurate calculation for that compound. Only in first-row compounds is there indication of significant uncertainty in the results. Furthermore, as we noted in Table 2-1, the theoretical matrix elements are very nearly equal to the ones obtained by fitting bands thus, if we had plotted bands in Fig. 3-8,a that were based upon purely theoretical parameters, the curves would have been hardly distinguishable. 

Harrison, 1973c). To sec this, look first at homopolar semiconductors, for which Eq. (3-40) reduces to... 

Values can be obtained from the exact definition for the homopolar semiconductors (Eq. 3-41), by using the metallic energies of Table 2-2 and the Solid State Table. The results are given in Table 3-2. The predicted critical value of unity is only approximate. We will see in Chapter 6 that the gap Eq in grey tin is zero or slightly negative. In Chapter 6 the quantitative values for Eq will also be discussed. 

Metallic energy and covalent energy (in cV), and metallicity for the homopolar semiconductors. 

We might also expect the matrix elements of djdx to scale inversely with d among the homopolar semiconductors (and correspondingly, for the matrix elements of X to scale with d) and this is in fact predicted by the pseudopotential theory of Chapter 18. However, that does not describe the trends in Xi(0) well, and in Section 4-C, we shall allow the proportionality constant to vary from row to row in the Periodic Table. 

Notice first that the minimum gap between the valence and conduction bands in Fig. 4-4 occurs at F this is the minimum gap for the entire Brillouin Zone. The corresponding energy difference, in this case less than 1 eV, is called Eq and, as indicated in E ig. 4-3, is the minimum energy at which absorption occurs. It is the same Eq evaluated in Eq. (3-39) and discussed there. The situation has a complication in the homopolar semiconductors in that the minimum energy in the conduction band does not occur at F where the valence-band maximum is. The difference is called an indirect gap and absorption cannot occur at that energy in the absence of other perturbations such as thermal vibrations. For this reason we ghose InAs as better suited for discussion than silicon. 

For homopolar semiconductors, we see from Eq. (4-17) that it is twice the covalent energy -in contrast to the hybrid covalent energy of Eq. (3-6)- defined by... 

The observation made earlier, that the 2 values in compounds vary similarly with variation in bond length would in general suggest a value of d In x/d In d of order unity for compounds as well as homopolar semiconductors (Kastner, 1972, lists 1.8 and 3.6 for GaP and GaAs, respectively). 

Notice that the gap vanishes for a homopolar semiconductor, which is true also for the exact bands, and if V were equal to V , it would simply be equal to times the predicted band width, p4K,. Thus, qualitatively, the gap is in very simple correspondence with the polarity of the system. The observed splittings are from 30 percent to 45 percent lower than those predicted in the K,-only theory by Eq. (6-12). The value is not modified as we add additional matrix elements within the Bond Orbital Approximation (Pantelides and Harrison, 1975). P rom Eqs. (6-3) and (6-4) we see that the situation is greatly complicated if the Bond Orbital Approximation is not u.scd (that is, bonding antibonding matrix elements are added), though of course the predicted gaps do go to zero as the polarity goes to zero in any case. 

We may summarize the LCAO interpretation of the energy bands. Accurate bands were displayed initially in Fig. 6-1. The energy difference between the upper valence bands and the conduction bands that run parallel to them was associated with twice the covalent energy for homopolar semiconductors, or twice the bonding energy 2 Vl -1- in hetcropolar semiconductors. The broadening of those... 

A.31h / md ) and Fig. 6-5 to write F as 0.66h / md ), and then used Eq, (4-16) to write h / md ) as F2/2.I6. This result can be compared with that resulting from Eq. (6-19), which is written in the form 0 = 3.6OF2 — 4.44F, for the homopolar semiconductors. Neither is very accurate, but both correctly reflect a bondingantibonding splitting from the covalent energy, reduced by the band-broadening... 

Wc turn finally to materials that are not direct-gap semiconductors. The conduction bands of semiconductors have an analogy in the conjugation of verbs those encountered oftenesl have the exceptional forms. Let us first consider the four homopolar semiconductors diamond. Si, Gc, and Sn. In the text following... 

Eq. (6-18), we gave the energy difference between the conduction band levels at F as 2Bi -h 2B. For homopolar semiconductors it takes the simple form... 

The cohesive energy per bond (commonly called bond energy) for the homopolar semiconductors and compounds isoelectronic with them, plotted against covalcncy. The lines represent an extremely simple empirical rule. 

The same effect can be seen in the zig-zag chain of Fig.. 3-11. It is remarkable that we can compute the angular force constant in that model exactly, as well as in the Bond Orbital Approximation (see Problems 8-1 and 8-2). The results turn out to be identical for the homopolar semiconductors, but for polar semiconductors, the exact solution has a, replaced by . Sokel has shown that the result is not so simple for the tetrahedral solid, but turns out quantitatively to be very close to an dependence. We will also find an ot dependence when we treat tetrahedral solids in terms of the chemical grip in Section I9-F. This suggests the approximation to the full calculation,... 

Unreconstructed surfaces in a homopolar semiconductor. The crystal is viewed along a [ 110]... 

Although (111) surfaces have lowest energy, it is possible experimentally to obtain other surfaces on the homopolar semiconductors, notably (110) and (100) surfaces. In considering reconstruction on these surfaces we will assume that the dehybridization energy dominates the distortion, as we found to be so on the (111) surface. Thus we can guess the forms of reconstruction without further calculation. 

We can also expect reconstruction on the surfaces of polar semiconductors. In particular, the distortions on the (110) cleavage plane may be expected to be of the same form as on the (110) surfaces of homopolar semiconductors, as shown in Fig. 10-5, with the nonmetallic atom displaced outward since its hybrid is doubly occupied the metallic atom is displaced inward with its purely p-like hybrid unoccupied. This is the distortion proposed by MacRac and Gobcli (1966) for essentially the same reasons described here. This appears to have been confirmed by recent analysis of LEED data (Lubinsky, Duke, Lee, and Mark, 1976). 

Nearest-neighbor LCAO bands for the homopolar semiconductors, found by using interatomic parameters predicted as in Fig. 18-1 (and listed in Table 2-1) and term values from the Solid Stale Table. Energies arc in electron volts. Notice that the vertical scale is reduced for the carbon bands. [After Froyen and Harrison, 1979.]... 

Making first a comparison of the covalent energy, notice that in homopolar semiconductors, Wy, becomes simply w, The various geometrical factors in the empty-core pseudopotential may be directly evaluated. Then, the pseudopotential matrix element becomes... 

The values for the homopolar semiconductors are listed, along with V2, in Table... 

Pscudopotential matrix elements for the homopolar semiconductors compared with Pj, all in cV. Values of ky for the diamond structure appear in the Solid State Table. 

Change in the bands as a homopolar semiconductor is made increasingly polar, and then as the two atom types are made more alike without broadening the levels. 

---