Physics of Strained Quantum Well Lasers pp 57-126 | Cite as

# Bulk Semiconductor Bandstructure

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## Abstract

In perfect crystalline solids, Bloch’s theorem asserts that the electronic energies and wavefunctions are described by energy *bands*. Many important optoelectronic material properties, such as electron and hole effective masses, conduction and valence band densities of state, bandgaps, and even the ideas of electrons and holes themselves are embedded in the band theory of solids. Therefore, to calculate the electronic properties of semiconductors we must begin with a proper calculation of semiconductor bands, and in this chapter we present two techniques—the tight binding and k · p methods—that can be used for this.

## Keywords

Matrix Element Valence Band Tight Binding Heavy Hole Valence Band Edge## Preview

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## Notes

- 1.In particular,
*Solid State Physics*by N. W. Ashcroft and N. D. Mermin, (Saundcrs College, Philadelphia, 1976) provides an excellent discussion of the fundamental ideas.Google Scholar - 2.Note that the exclusion principle cannot even be
*formulated*without making the one-electron approximation, and, in fact, the exclusion principle is an artifice to force the one-electron approximation to be*consistent with*the more exact requirement (2.1).Google Scholar - 3.More detailed discussions of Bloch’s theorem, energy bands, Brillouin zones, and the like may be found in introductory solid state physics texts such as Ashcroft and Mermin.Google Scholar
- 4.We use parentheses to label
*points*in**k**-space, such as Γ = (0,0,0), and angle brackets to denote*directions*in**k**-space, such a.s Λ = 〈1,1,1〉.Google Scholar - 5.The reader is expected to be familiar with the concepts of electrons, holes, and the Fermi level. Further details may be found in Ashcroft and Mermin.Google Scholar
- 6.It is regrettable that the same symbol Δ is used to represent both the spin-orbit splitting energy
*and*the**k**-space direction 〈1,0,0〉.Google Scholar - 7.The factor of is inserted to provide a convenient, normalization for the functions xαk. Technically, we should also restrict the set of wavevectors
**k**by applying periodic boundary conditions over a macroscopic volume shaped like a giant unit cell. See Ashcroft and Mermin, Appendix D, for details.Google Scholar - 8.In principle, we should also show that the set of functions xαk is
*complete*. In fact it is, provided that it includes Bloch sums for a complete set of atomic Orbitals ø_{α}and for all wavevectors**k**in the first Brillouin zone.Google Scholar - 9.Actually, this relation holds only when the wavevectors k are restricted to the values allowed by enforcing periodic boundary conditions on the solid. See Ashcroft and Mermin, Appendix F, for details.Google Scholar
- 10.It is possible to perform tight binding calculations without this assumption, but there is little to gain. In practice, it is much easier to incorporate the effects of non-orthonormality by including more distant neighbors in the
*Hamiltonian*matrix elements (2.20).Google Scholar - 11.Many tight, binding presentations assume from the start that
*H*is diagonal in k, and begin with the expansion iustead of (2.16).Google Scholar - 12.This phenomenological application of the tight hinding model was first IHcscnted by J. C. Slater and G. F. Koster, “Simplified LCAO:\lethod for the Periodic Potential Problem,” Phys. Rev.
**94**, 1498 (1954).Google Scholar - 13.The EBOM was first presented by Y. C. Chang, “Bond-Orbital model for superlattices,” Phys. Rev. B
**37**, 8215 (1988). It was modified to fit the*X*-point gaps by J. P. Loehr, “Improved effective-bond-orbital model for superlattices,” Phys. Rev. B**50**, 5429 (1994).ADSCrossRefGoogle Scholar - 14.Because of the γ
_{kk}′ in (2.20), we suppress the**k**dependence of the matrix elements in (2.23) for brevity.Google Scholar - l5.Molecular orbital theory, very popular and successful in quantum chemistry, relies heavily on σ and π bonding/antibonding combinations. Tight binding theory is essentially molecular orbital theory for solids. Sec W. A. Harrison,
*Electronic Structure, and the Properties of Solids*, (Dover, New York, 1989) for an extensive discussion.Google Scholar - 16.Throughout the text, we dellote
*measured*critical-poillt energy values by calligraphic notation, such as ctc.Google Scholar - 17.Strictly speaking, these energies represent spin-averaged values at the
*X*point.Google Scholar - 18.When we encounter the k-p theory in Section 2.G, we will derive the same matrix. Thus, we denote the matrix in (2.50) by
*H*_{k.p.}Google Scholar - 20.Note that (2.62) is expressed in a different basis from (2.51). Thus we are solving for a different set of expansion coefficients dα(k), as discussed briefly in Chapter 1. This is a minor detail for now. We will account for the eigenvector correction when we present the k·p model.Google Scholar
- 21.Note that (2.66), (2.68), and (2.G9) are linear in
*v*^{2}; we get by taking the positive square root of (2.54).Google Scholar - 22.We ignore the other relativistic corrections to the Haniiltonian, assuming that they are accounted for in the average potential
*V(r)*.Google Scholar - 23.Our final results, however, will hold under more general circumstances than might be expected.Google Scholar
- 24.It might seem that, following our discussion in Chapter 1, we should start with total angular momentum eigenstates instead of those in (2.80). But this would force us to recalculate all our tight binding matrix elements using very geometrically-complex basis states. In fact it is much easier to begin with directed orbital states, and transform to the total angular momentum basis when necessary.Google Scholar
- 25.For brevity, we suppress the argument k = 0 to the coefticnts cα in (2.84) and (2.85).Google Scholar
- 26.To fully describe these states, we should add a
*radial*index and write*n,l,s,j,m*,_{j}), as we did in Chapter 1. But since the spin-orbit term affects only the*p*orbitals, and since all of these have the same radial dependence in (2.22), we can safely omit*n*.Google Scholar - 27.
*Much*algebra is required to generate this result. The best approach is to use symbolic algebra software.Google Scholar - 28.For convenience, we write a positive effective mass in (2.108). The split-off band
*energies*Google Scholar - 29.Remember that we are ignoring spin for the moment and are discounting the possibility of accidental degeneracies.Google Scholar
- 30.For example, G. F. Koster, J. O. Dhnmock, R. C. Wheeler, and H. Statz,
*Properties of the Thirty-Two Point Groups*. (MIT Press, Cambridge, 1963); or, better yet, C. Kittel,*Quantum Theory of Solids*, (Wiley, New York, 1987), Chapter 10.Google Scholar - 31.Recall that,
*parity*refers to the behavior of a function under the inversion operation r → —r. Functions*f*of even parity satisfy*f*(—r) =*f*(r) while those of*odd*parity satisfy*f*(—r) = —*f*(r). Recognize that any*odd*function*f*integrates to zero, i.e.*f*dr*f*(r) = 0, where the integral is carried out. over all space. Although not all functions have definite parity, any function can be written as a sum of even and odd functions via*f*(r) = 1/2 [*f*(r) +*f*(—r)] + 1/2 [*f*(r) —*f*(—r)]. We can also refer to the parity of a function with respect to a particular coordinate inversion. For example,*xy*^{2}is odd with respect to*x*but even in*y*and*z*.Google Scholar - 32.This information is ascertained by spectroscopic methods, just as in atomic and molecular physics. Small perturbations are applied to the crystal, and by observing the number and character of split levels the representations are determined.Google Scholar
- 33.This represents a notational simplification of a vastly more complicated process. First, group theory requires that functions belonging to a representation Γ
_{j}only*behave*like those in Table 2.1 under symmetry operations belonging to the point, group: they need not actually*be*these functions. Thus the actual zone-center basis functions*u*_{n}in the solid can (and do) look considerably different from the simple coordinate functions*xyz, xy*, etc., although they behave identically under the symmetry operations of the zincblende crystal structure. Second, any useful wavefunction should be square-integrable. Therefore, when considering concrete representations of these basis functions we should imagine each one being multiplied by a rotationally-symmetric function ø(r) vanishing rapidly at infinity. For example,*yz*→ ø(*r*) sin θ cos θ cos ø, etc. Third, the zone-center wavefunctions*u*_{n}must also have the periodicity of the lattice, and we must imagine centering a given function from Table 2.1 on each lattice site.Google Scholar - 34.With this normalization convention for
*u*_{n}〉, the full Bloch functions as written in (2.121) are*not*normalized over the entire crystal. We will remedy this in Chapter 3 by rewriting (2.121), and by forcing*u*_{n}) to satisfy a different normalization condition. We omit these complications here.Google Scholar - 35.Do not confuse the parity under a full coordinate inversion r → —r with even or odd behavior under reversal of a
*single*coordinate, such as*x*→ —*x*.Google Scholar - 36.Note that these transformation properties are exactly the
*opposite*of the “simpler” states.Google Scholar - 37.While this is rigorously true in diamond structures, it is only an approximation in zincblende ones. Recall that the
*true*zincblende conduction and valence baud edge wavefunctions have mixed parity, and this allows a small second-order contributionGoogle Scholar - 38.This argument can be made rigorous as follows. The proper way to include spin is to describe the crystal by the appropriate
*double group*, which accounts for the additional symmetry operations possible when treating spinors. For example, spinors acquire a phase factor of —1 when rotated by 360°. These double group representations have exactly the degeneracies and symmetry properties we arrived at when solving (2.94). Thus, our description is consistent with the more rigorous approach. This argument also furnishes an*a-posteriori*justification of the approximations we made to derive the matrix elements of*H*_{so}between tight binding wavefunctions.Google Scholar - 39.For an excellent account of tensors and their applications to crystals, see J. F. Nye, “
*Physical properties of crystals: their representation by tensors and matrices*” (Clarendon Press, Oxford, 1985).Google Scholar - 40.This is not. obvious. See Nyc for details.Google Scholar
- 41.See Nye for details.Google Scholar
- 42.The elastic stiffness constants do not appear explicitly in the deformation potential Hamiltonian (2.230), but are required in (2.174) to compute the elements ∈
_{ij}of the strain tensor.Google Scholar - 43.For semiconductors with small bandgaps, such as InSb and GaSb, or small split-off gaps, such as Si and Ge, this coupling between blocks is important, and an 8 × 8 or 6 × 6 (valence band only) model must be used.Google Scholar
- 44.The ø-dependent factors in
*U*are not strictly necessary, since ξ and η will depend on ø, but in hindsight we find them convenient.Google Scholar - 45.It is more common to see a straight arithmetic average, but it makes more sense to take the average value. In practice it makes little difference.Google Scholar
- 46.See B. Zhu and K. Huang, “Effect of valence-band hybridization on the exciton spectra in GaAs-Ga
_{1}-_{x}Al_{x}As quantum wells,”*Phys. Rev.*B**36**, 8102 (1987); B. Zhu, “Oscillator strength and optical selection rule of excitons in quantum wells,”*Phys. Rev.*B**37**, 4689 (1988); and A. Pasquarello, L. C. Andreani, and R. Buczko, “Binding energies of excited shallow acceptor states in GaAs/Ga_{1-x}Al_{x}As quantum wells,”*Phys. Rev.*B**40**, 5602 (1989).ADSCrossRefGoogle Scholar