Patterson, Bailey - Solid State Physics Introduction to theory

184 3 Electrons in Periodic Potentials

to next-nearest neighbors, more interpolation parameters would be introduced and hence greater accuracy would be achieved.

Results for the nearly free-electron approximation, the tight binding approximation, and the Kronig–Penny model are summarized in Table 3.3.

The Wigner–Seitz method (1933) (B)

The Wigner–Seitz method [3.57] was perhaps the first genuine effort to solve the Schrödinger wave equation and produce useful band-structure results for solids. This technique is generally applied to the valence electrons of alkali metals. It will also help us to understand their binding. We can partition space with polyhedra. These polyhedra are constructed by drawing planes that bisect the lines joining each atom to its nearest neighbors (or further neighbors if necessary). The polyhedra so constructed are called the Wigner–Seitz cells.

Sodium is a typical solid for which this construction has been used (as in the original Wigner–Seitz work, see [3.57]), and the Na+ ions are located at the center of each polyhedron. In a reasonable approximation, the potential can be assumed to be spherically symmetric inside each polyhedron.

Let us first consider Bloch wave functions for which k = 0 and deal with only s-band wave functions.

The symmetry and periodicity of this wave function imply that the normal derivative of it must vanish on the surface of each boundary plane. This boundary condition would be somewhat cumbersome to apply, so the atomic polyhedra are replaced by spheres of equal volume having radius r0. In this case the boundary condition is simply written as

With k = 0 and a spherically symmetric potential, the wave equation that must be solved is simply

subject to the boundary condition (3.257). The simultaneous solution of (3.257) and (3.258) gives both the eigenfunction ψ0 and the eigenvalue E.

The biggest problem remaining is the usual problem that confronts one in making band-structure calculations. This is the problem of selecting the correct ion core potential in each polyhedra. We select V(r) that gives a best fit to the electronic energy levels of the isolated atom or ion. Note that this does not imply that the eigenvalue E of (3.258) will be a free-ion eigenvalue, because we use boundary condition (3.257) on the wave function rather than the boundary condition that the wave function must vanish at infinity. The solution of (3.258) may be obtained by numerically integrating this radial equation.

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a linear combination of spherical solutions. Such a procedure gives an augmented plane wave for one Wigner–Seitz cell. (As already mentioned, Wigner–Seitz cells are constructed in direct space in the same way first Brillouin zones are constructed in reciprocal space.) We can extend the definition of the augmented plane wave to all points in space by requiring that the extension satisfy the Bloch condition. Then we use a linear combination of augmented plane waves in a variational calculation of the energy. The use of symmetry is quite useful in this calculation.

Fig. 3.16. The “muffin tin” potential of the augmented plane wave method

Before a small mathematical development of the augmented plane method is made, it is convenient to summarize a few more facts about it. First, the exact crystalline potential is never either exactly constant or precisely spherically symmetric in any region. Second, a real strength of early augmented plane wave methods lay in the fact that the boundary conditions are applied over a sphere (where it is relatively easy to satisfy them) rather than over the boundaries of the Wigner–Seitz cell where it is relatively hard to impose and satisfy reasonable boundary conditions. The best linear combination of augmented plane waves greatly reduces the discontinuity in normal derivative of any single plane wave. As will be indicated later, it is only at points of high symmetry in the Brillouin zone that the APW calculation goes through well. However, nowadays with huge computing power, this is not as big a problem as it used to be. The augmented plane wave has also shed light on why the nearly free-electron approximation appears to work for the alkali metals such as sodium. In those cases where the nearly free-electron approximation works, it turns out that just one augmented plane wave is a good approximation to the actual crystalline wave function.

The APW method has a strength that has not yet been emphasized. The potential is relatively flat in the region between ion cores and the augmented plane wave method takes this flatness into account. Furthermore, the crystalline potential is essentially identical to an atomic potential when one is near an atom. The augmented plane wave method takes this into account also.

The augmented plane wave method is not completely rigorous, since there are certain adjustable parameters (depending on the approximation) involved in its

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The coefficients Kk+Gn, are determined by a variational calculation of the energy. This calculation also gives E(k). The calculation is not completely straightforward, however. This is because of the E(k) dependence that is implied in the Rl(r,E) when the dlm are determined by fitting spherical solutions to plane waves. Because of this, and other obvious complications, the augmented plane wave method is practical to use only with a digital computer, which nowadays is not much of a restriction. The great merit of the augmented plane wave method is that if one works hard enough on it, one gets good results.

There is yet another way in which symmetry can be used in the augmented plane wave method. By the use of group theory we can also take into account some rotational symmetry of the crystal. In the APW method (as well as the OPW method, which will be discussed) group theory may be used to find relations among the coefficients Kk+Gn. The most accurate values for E(k) can be obtained at the points of highest symmetry in the zone. The ideas should be much clearer after reasoning from Fig. 3.17, which is a picture of a two-dimensional reciprocal space with a very simple symmetry.

For the APW (or OPW) expansions, the expansions are of the form

Fig. 3.18. Self-consistent energy bands in ferromagnetic Ni along the three principal symmetry directions. The letters along the horizontal axis refer to different symmetry points in the Brillouin zone [refer to Bouckaert LP, Smoluchowski R, and Wigner E, Physical Review, 50, 58 (1936) for notation]. [Reprinted by permission from Connolly JWD, Physical Review, 159(2), 415 (1967). Copyright 1967 by the American Physical Society.]

3.2 One-Electron Models 189

10

20

β

30

40

Energy, Ry

Fig. 3.19. Density of states for up (α) and down (β) spins in ferromagnetic Ni. [Reprinted by permission from Connolly JWD, Physical Review, 159(2), 415 (1967). Copyright 1967 by the American Physical Society.]

Suppose it is assumed that only G1 through G8 need to be included in the expansions. Further assume we are interested in computing E(k ) for a k on the symmetry axis. Then due to the fact that the calculation cannot be affected by appropriate rotations in reciprocal space, we must have

Kk −G2 = Kk −G8 , Kk −G3 = Kk −G7 , Kk −G4 = Kk −G6 ,

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and so we have only five independent coefficients rather than eight (in three dimensions there would be more coefficients and more relations). Complete details for applying group theory in this way are available.17 At a general point k in reciprocal space, there will be no relations among the coefficients.

Figure 3.18 illustrates the complexity of results obtained by an APW calculation of several electronic energy bands in Ni. The letters along the horizontal axis refer to different symmetry points in the Brillouin zone. For a more precise definition of terms, the paper by Connolly can be consulted. One rydberg (Ry) of energy equals approximately 13.6 eV. Results for the density of states (on Ni) using the APW method are shown in Fig. 3.19. Note that in Connolly’s calculations, the fact that different spins may give different energies is taken into account. This leads to the concept of spin-dependent bands. This is tied directly to the fact that Ni is ferromagnetic.

The Orthogonalized Plane Wave Method (A)

The orthogonalized plane wave method was developed by C. Herring in 1940.18 The orthogonalized plane wave (OPW) method is fairly similar to the

augmented plane wave method, but it does not seem to be as much used. Both methods address themselves to the same problem, namely, how to have wave functions wiggle like an atomic function near the cores but behave as a plane wave in regions far from the core. Both are improvements over the nearly freeelectron method and the tight binding method. The nearly free-electron model will not work well when the wiggles of the wave function near the core are important because it requires too many plane waves to correctly reproduce these wiggles. Similarly, the tight binding method does not work when the plane-wave behavior far from the cores is important because it takes too many core wave functions to reproduce correctly the plane-wave behavior.

The basic assumption of the OPW method is that the wiggles of the conduction-band wave functions near the atomic cores can be represented by terms that cause the conduction-band wave function to be orthogonal to the coreband wave functions. We will see how (in the Section The Pseudopotential Method) this idea led to the idea of the pseudopotential. The OPW method can be stated fairly simply. To each plane wave we add on a sum of (Bloch sums of) atomic core wave functions. The functions formed in the previous sentence are orthogonal to Bloch sums of atomic wave functions. The resulting wave functions are called the OPWs and are used to construct trial wave functions in a variational calculation of the energy. The OPW method uses the tight binding approximation for the core wave functions.

Let us be a little more explicit about the technical details of the OPW method. Let Ctk(r) be the crystalline atomic core wave functions (where t labels different core bands). The conduction band states ψk should look very much like plane waves between the atoms and like core wave functions near the atoms. A good

17See Bouckaert et al [3.7].

18See [3.21, 3.22].

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error. But we must always keep in mind that neglect of electron–phonon interactions and other interactions may importantly affect the electronic density of states. In particular this may lead to errors in predicting some of the optical properties. We should also remember that we do not do a completely selfconsistent calculation.

The exchange-correlation term in the density functional approximation is difficult to treat exactly so it can be approximated by the free-electron-like Slater ρ1/3 term [88] or the related local density approximation. However, density functional techniques suggest some factor19 other than the one Slater suggests should multiply the ρ1/3 term. In the treatment below we will not concern ourselves with this problem. We shall just assume that the effects of exchange (and correlation) are somehow lumped approximately into an ordinary crystalline potential.

This latter comment brings up what is perhaps the crux of an energy-band calculation. Just how is the “ordinary crystalline potential” selected? We don’t want to do an energy-band calculation for all electrons in a solid. We want only to calculate the energy bands of the outer or valence electrons. The inner or core electrons are usually assumed to be the same in a free atom as in an atom that is in a solid. We never rigorously prove this assumption.

Not all electrons in a solid can be thought of as being nonrelativistic. For this reason it is sometimes necessary to put in relativistic corrections.20

Before we discuss other techniques of band-structure calculations, it is convenient to discuss a few features that would be common to any method.

For any crystal and for any method of energy-band calculation we always start with a Hamiltonian. The Hamiltonian may not be very well known but it always is invariant to all the symmetry operations of the crystal. In particular the crystal always has translational symmetry. The single-electron Hamiltonian satisfies the equation,

for any Rl.

This property allows us to use Bloch’s theorem that we have already discussed (see Appendix C). The eigenfunctions ψnk (n labeling a band, k labeling a wave vector) of H can always be chosen so that

Three possible Hamiltonians can be listed,21 depending on whether we want to do (a) a completely nonrelativistic calculation, (b) a nonrelativistic calculation

19See Kohn and Sham [3.29].

20See Loucks [3.32].

21See Blount [3.6].

3.2 One-Electron Models 193

with some relativistic corrections, or (c) a completely relativistic calculation, or at least one with more relativistic corrections than (b) has.

a) Schrödinger Hamiltonian:

where m0 is the rest mass and the third term is the spin-orbit coupling term (see Appendix F). (More comments will be made about spin-orbit coupling later in this chapter).

c) Dirac Hamiltonian:

where α and β are the Dirac matrices (see Appendix F).

Finally, two more general comments will be made on energy-band calculations. The first is in the frontier area of electron–electron interactions. Some related general comments have already been made in Sect. 3.1.4. Here we should note that no completely accurate method has been found for computing electronic correlations for metallic densities that actually occur [78], although the density functional technique [3.27] provides, at least in principle, an exact approach for dealing with ground-state many-body effects. Another comment has to do with Bloch’s theorem and core electrons. There appears to be a paradox here. We think of core electrons as having well-localized wave functions but Bloch’s theorem tells us that we can always choose the crystalline wave functions to be not localized. There is no paradox. It can be shown for infinitesimally narrow energy bands that either localized or nonlocalized wave functions are possible because a large energy degeneracy implies many possible descriptions [95, p. 160], [87, Vol. II, p. 154ff]. Core electrons have narrow energy bands and so core electronic wave functions can be thought of as approximately localized. This can always be done. For narrow energy bands, the localized wave functions are also good approximations to energy eigenfunctions.22

22For further details on band structure calculations, see Slater [88, 89, 90] and Jones and March [3.26, Chap. 1].