Electron Energy Bands in Solids

Electron Energy Bands in Solids JOSEPHCALLAWAY Department of Physics. University of Miami. Coral Gables. Florida

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I Introduction and General Theory., ................................... 1. The Energy Band ................................................ 2. The Many-Electron Problem ...................................... 3. The Hartree-Fock Equations., .................................... 4. The Crystal Potential ............................................ 5. Symmetry Properties ............................................. 6. Behavior of a Band near B Degeneracy ............................. 7 The Density of States ............................................ 8. Normal Form of an Energy Band .................................. 9. Methods of Calculation ........................................... I1. The Alkali Metals .................................................. 10 General Discussion............................................... 11. Determination of the Fermi Energy ................................ 12. Crystal Potentials for the Alkali Metals ............................ 13. Core Polarization ................................................ 14. Relativistic Effects ............................................... 15. Lithium ........................................................ 16. Sodium ......................................................... 17. Potassium ...................................................... 18. Rubidium and Cesium ............................................ 19. The Knight Shift ................................................ 20. Metallic Hydrogen ............................................... 21 General Survey and Conclusions................................... I11. Metals of Groups I1 and I11......................................... 22. General Diacussion ............................................... 23 Beryllium ....................................................... 24 Magnesium ..................................................... 25. Calcium ........................................................ 26. Aluminum ...................................................... 27. Solid Helium .................................................... IV Elements of Group IV and Related Semiconductors..................... 28. General Considerations........................................... 29. Graphite ........................................................ 30. Diamond ....................................................... 31. Silicon .......................................................... 32. Germanium ..................................................... 33. Tin, Indium Antimonide, and Indium Arsenide ...................... 34 Gallium Arsenide................................................

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100 100 101 104 106 108 111 113 115 118 119 119 121 123 125 126 128 133 136 140 143 145 146 147 147 148 151 154 154 158 158 158 159 164 168 172 176 179

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JOSEPH CALLAWAY

V. The Transition Metals.. ............................................ 35. General Considerations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36. Experimental Information. . 37. Titanium................. 38. Chromium........................ 39. Iron ...................... 40. Nickel.. . . . ........... ........... 41. Copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Other Substances. .... .............................. ....................... 42. Silver and Gold. ... ....................... 43. Tungsten..................... 44. Bismuth .......... .............................. 45. Selenium and Tellurium.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46. Alkali Halides: Sodium Chloride.. ..................... 47. Potassium Chloride. ............... .............. 48. Lithium Rydride and Lithium Fluoride.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49. Lead Sulfide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 50. Zinc Sulfide . . . . . . . . . . . . . . . . 51. Barium Oxide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52. Miscellaneous Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Conclusions and Prospects.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

180 180

193 19i 197 198 198

206 205 208 209 210 210

1. Introduction and General Theory I

1. THEENERGY BAND A convenient assumption to make in beginning the discussion of the behavior of electrons in crystals is that we may consider the motion of only one particle in a periodic potential. This is an oversimplification, for one really has a many-electron problem. Moreover, real crystals are not perfectly periodic but contain imperfections. Some of the complications which arise because of the many-particle nature of the problem will be discussed in subsequent sections; a general treatment of the problem of imperfections is beyond the scope of this review. With this simplifying assumption mentioned above, Bloch proved that solutions of Schrodinger's equation have the form' in which uk(r) is periodic in r with the periodicity of the potential. This theorem is a general consequence of translational symmetry. The wave functions are characterized by the wave number k which may be thought of as equivalent t o the crystal momentum. The energy of a state g k depends in a reasonably continuous fashion on the wave number. The relation between energy and wave number characterizes an energy band. The simplifying assumptions made above could lead to the false im1

F. Bloch, 2.Physilc 62, 555 (1928).

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101

pression that energy bands are found only in periodic structures. It is still useful to talk about energy bands in a disordered material provided the atoms are sufficiently close that the atomic wave functions of atoms on neighboring sites overlap. In such a case, one part of the material is, on the average, just like any other part, so that wave functions and energy values can still be characterized by a reasonably continuous dependence of energy on momentum. We will not expect however, that Eq. (1.1) will hold in such circumstances. It should be apparent that the E(k) relation is one of themmostimportant properties of a material. It is involved in a basic way in-any calculation of the electric, optical, or thermal characteristics of the substance. The calculation of E(k) relations for actual materials is extremely complicated. Many considerations of electrical properties, for example, are based on extremely simplified models of the real band structure. Progress has been made, however, in determining energy bands in a number of the simpler metals and semiconductors. We shall describe, first of all, some of the general features of energy bands in periodic lattices, and will then discuss both theoretical and experimental results in materials which have been subject to detailed study. There will be no discussions of artificial models, such as the Kronig-Penney model, or of methods of calculation. For a survey of these topics, the reader is referred to the reviews by Reita and by Slater.2 2. THEMANY-ELECTRON PROBLEM

The enormous number of particles involved in a typical solid makes direct solution of the Schrodinger equation completely impossible, and leads naturally to the development of a one-electron approximation. It is necessary, however, to relate the one-electron theory to the actual manyelectron situation, and to estimate the corrections to the results of the one-body approximation. One might well wonder at first how a one-particle approximation can have any validity in the case of electrons which interact with reasonably strong and very long-range forces. Experience does convince us, however, that it is quite useful. We will examine the reason for this. Suppose we have a set of one-particle wave functions believed to be appropriate to the problem. The best choice of these functions will be discussed in the next section. An antisymmetric N-body wave function can be found by taking N of the one-particle functions and forming a Slater determinant in the standard manner. Such an approximate wave function for the system will be called a model function in this article. If * J. R. Reitz, Solid State Physics 1, 1 (1955);J. C.Slater in “Handbuch der Physik,” Vol. 19, p. 1. Springer, Berlin, 1956.

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JOSEPH CALLAWAY

we construct all the possible N by N determinants out of the one-body functions, the result is a complete set of antisymmetric N-body functions in terms of which the real wave function of the system can be expanded. It usually is convenient to choose the model functions to be eigenfunctions of a partial Hamiltonian which either does not include the interactions between particles at all, or includes only the average of these interactions. The model functions can then be considered as approximate, or unperturbed wave functions, for the system and the difference between the actual and model Hamiltonians can be treated as a perturbation. The actual machinery of this perturbation theory is quite complex and will not be discussed in detail here for the reason that the Coulomb interaction in metals (as well as many of the interactions of interest in nuclear physics) apparently leads to infinite results. Procedures have been devised which circumvent this difficulty.3~~ The question of principal concern to us is the relation between standard energy band theory and the many-body problem. It appears that much of the basic language of the theory is unchanged. W. Kohn has shown that the behavior of an extra electron or hole in an insulator can be described in some circumstances by an effective mass e q ~ a t i o nA . ~more general approach would seem to be afforded by the work of Bethe, GellMann, and Brueckner, etc.*s4The one-particle functions which are combined to form the model wave function of standard energy band theory satisfy Bloch's theorem and can be designated by the appropriate wave vector and band index. A model wave function can be characterized by specifying the one-particle states which are occupied. From a given model state, a real state can be constructed. The actual energy of the system can be associated with the model state from which the real state is derived. It may not be possible, however, to obtain all the real states in this way. In the model wave function of lowest actual energy Eo, the wave vectors of the occupied one-electron states will lie within some surface in k space. The excited states of interest to band theory are specified, in relation to the ground-state model function, by listing the wave vectors and spins of the previously occupied states which are now vacant (kl)and the previously empty states now filled (hi"). (Here i = 1, . . . Y, where Y is the number of excited particles or holes.) Let the real energy of an

* For a treatment of the general theory of the many-body problem see, for instance, 4

H. Bethe, Phys. Rev. 103, 1353 (1956), and the references contained therein. The Coulomb interaction is treated by M. Gell-Mann and K. A. Brueckner, Phys. Rev. 106, 364 (1957); M. Gell-Mann, Phys. Reu. 106, 369 (1957); K. Sawada, Phys. Rev. 106, 372 (1957). W. Kohn, Phys. Rev. 106, 509 (1957). For a general'discussionof the many-body problem in solids see H. Haken, 2. Nuturforsch. Qa, 228 (1954).

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excited state be E. Then we write (following Gell-Mann*):

E

=

Eo

+ Zi[W(ki’) - W(ki’)]+ O(U/N).

(2.1)

The quantity W(ki”) is the difference in actual energy between the ground state (Eo)and a state containing N 1 particles, the additional particle being (as far as the model wave function is concerned) in the previously unoccupied state ki”. Here W(ki”) may be thought of as representing the complete interaction energy of one particle in the state k4’ with the rest of the system, but it differs quantitatively from a similar quantity calculated in the Hartree-Fock approximation (see Section 3) because of the more accurate inclusion of the correlation between particle motions in this procedure. Similarly W(ki’) is the difference in energy between the ground state and a state with N-1 particles, one particle being removed from the previously occupied state ki’. In addition to the quantities W(k), the energy of the excited N-body state must include terms containing the interaction between excited particles and between excited particles and holes. This interaction is of the order u/N, and can be neglected if v / N is small. In more physical terms, the many-body system is capable of singleparticle excitations, and also of other excitations, such as collective oscillations,6which are not described easily in a single-particle model and in which the interaction between the excited particles is important. The interesting single-particle excitations are those whose energy is low compared to a collective oscillation. An energy band is specified by the function W(k).A description of the many-body system in terms of energy bands will be meaningful formally when the excitation energy is not too large (less than is required to excite a collective oscillation) and as long 88 we are not concerned with the interactions between excited particles and between excited particles and holes. The energy bands defined in this way are the same as those obtained from the Hartree-Fock equations (3.1) when the more detailed treatment of the particle interactions is neglected. In general, however, there are quantitative differences arising from the more detailed treatment of particle interactions in W(k). Experimental results seem to indicate that the basic ideas of the oneparticle approximation actually are applicable to metals.’ A well-defined Fermi surface (the surface enclosing the occupied region of k space for the one-particle functions) exists in metals and its characteristics can be

+

D. Pines, Sotid State Physics 1, 367 (1955). In a recent paper Noaihres and Pines discussed the nature of the elementary excitations in solids. P. Nozi&resand D. Pines, Phys. Rev. 109, 1062 (1958). 7 N. F. Mott, Nature 178, 1205 (1956).

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JOSEPH CALLAWAY

determined. It also appears that many excitations of low-energy can be described conveniently in terms of a one-particle model. This is quite consistent with the point of view expressed in the foregoing. It should be possible, and it is a basic objective of band theory to determine the shape of the Fermi surface and to account quantitatively for the one-particle excitations by computing the function W ( k ) . Similar successes of the one-particle model are observed in semiconductors. The reasons for success in this case are perhaps more obvious. I n the first place, the number of holes and electrons involved in transport processes is usually very small, so that it is quite reasonable to neglect terms of the order v / N . Second, because of the existence of a finite energy gap between filled and vacant states in the one-particle model, ordinary perturbation theory of the electron interaction is convergenLs 3. THE HARTREE-FOCK EQUATIONS Although a theory for finding the exact energy to be associated with a model state exists, it has not yet been possible to make calculations for real systems. All existing calculations of energy bands are based on the use of one model function. The question then presents itself: what is the best choice of a model function? This can be determined by use of the variational principle. The best model function is that for which the expectation value of the energy is a minimum, subject to the conditions that the one-particle functions which compose it be orthonormal. We are lead in this way to an equation for the oneparticle functions,2 namely, the The Hartree-Fock equations. Let such a one-particle function be ~(zj). equation is:

In the second term on the left, riL is the distance between electron i and the nucIeus L (assumed fixed). The sum runs over all the nuclei of the system. The third term expresses the average electrostatic potential of all the electrons of the system and the fourth term is the exchange interaction. In the approximation of Koopmans’ t h e ~ r e mthe , ~ energy parameter ei measures, the energy required to remove an electron in state i from the system. In other words it is the quantity W(k)for the Hartree-Fock

* C. W.Ufford, Phys. Rev. 69, 598 (1941). 9

T.Koopmans, Physica 1, 104 (1933).

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approximation. The interpretation of this equation is discussed in the reviews by Reitz and Slater.2 The calculation of energy bands is based, in principle, on the HartreeFock equations. These equations form a quite complicated nonlinear integro-differential system. It is clear that numerical methods of solution must be employed. The usual procedure is the method of self-consistent fields. At the outset, values for the functions ui are assumed. The integrals appearing in the equations are calculated with the use of the starting functions and the equations (which now have the standard Schrodinger form for a single particle) are solved for the eigenvalues ei and the function ui. If the solutions agree, within assigned limits, with the functions assumed, the work is completed; if not, the process is repeated until such time as the results of the nth and n 1 stages agree. The procedure is quite lengthy, and has been applied only to some of the simpler free atoms.l0 In the case of a solid, there are two types of difficulties which stand in the way of obtaining a self-consistent field calculation. Assume that a choice of starting functions has been made, and the appropriate integrals have been evaluated. Then Eq. (2.1) can be written effectively as

+

where V . and Vex are ordinary and exchange potentials respectively. Vexmay depend on the state i under consideration. (See Section 4.) It is then necessary to solve (3.1) for a sufficiently large number of states that a reasonable idea of the band structure and wave functions can be obtained. This problem is more difficult than for free atoms because central symmetry does not prevail in solids. Thus the wave equation, in general, cannot be separated. Certain methods have been devised which will give good results, at least for some states. These methods are discussed in the review article of R e i k 2 Once the equations have been solved, the integrals must be re-evaluated and the process repeated until self-consistency has been achieved. This is quite difficult, in general, because it involves summing the charge distribution over all occupied states. I n fact, the problem is so difficult that self-consistent calculations have not been completed for solids except possibly in the case of the alkali metals. For the results to be meaningful, it is necessary that the potentials employed in (3.1) be reasonably close to those which would result from a self-consistent calculation. Physical arguments must be used. 10

D. R. Hartree, “The Calculation of Atomic Structures.” Wiley, New York, 1957.

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JOSEPH CALLAWAY

It is apparent that the principal physical problem in a calculation band structure is the choice of an appropriate crystal potential. Some of the common approximations will be discussed in the next section. It is obvious immediately that those results which depend critically on the details of the potential used in a particular calculation must be viewed with some caution. Moreover it is important to determine the features of a calculation which are likely to be sensitive to the potential.

4. THE CRYSTALPOTENTIAL Since the construction of a crystal potential is of crucial importance in a band calculation, it is desirable to discuss the various approximations in more detail. In Eq. (3.2) the crystal potential was separated into two parts; namely, an ordinary potential which is the sum of the nuclear attractions and the averaged electronic repulsions, and an exchange potential. The symmetry of the crystal permits us to calculate the potential within a single atomic cell. Consider first the ordinary potential. It is convenient to separate this into two parts; namely, the potential arising from the charge distribution in the particular cell, and that arising from all the other cells. Since the computation of these requires a knowledge of the charge distribution in the crystal, and hence the self-consistent solution of the problem, certain approximations are made. Consider first the case of a monatomic crystal. Each cell is electrically neutral, so that we need to consider only one atom. In the solid state problem we usually are concerned only with the valence electrons. The core electron distributions on different atoms overlap very little, so that bands of negligible width are formed. Thus it is legitimate to regard the core electron distribution as the same in the solid and in the free atom. For this reason any information we may have about the distribution of the core electrons in the free atom is also relevant to the solid. Consider as examples the alkali metals, in which there is one electron outside a core of compact closed shells. Information regarding the core electron distribution may be obtained either from a self-consistent field calculation for the free ion, or from the spectroscopic data for the free atom. In the latter case, either an empirical potential can be constructed (see Section 12) or spectroscopic information can be used almost directly and explicit construction of a potential function can be avoided. For substances more complicated than the alkali metals, one generally must rely on self-consistent field calculations for the free atom in determining the core electron distribution. It is very unfortunate that self-consistent fields are available for less than half the atoms in the periodic table and that only a small number of the calculations that have been made include exchange.ll As a result of the assumptions which must be made concerning the 11

Existing SCF calculations are listed by R. S. %ox,

Solid State Phys. 4,413 (1957).

ELECTRON ENERGY BANDS IN SOLIDS

107

valence electron distribution, the construction of the crystal potential is uncertain. In the alkali metals, the Wigner-Seitz approximation (Section 10) allows us to avoid this difficulty by using the ionic potential. For more complicated materials, some assumption must be made explicitly. It is probably adequate in many metals to assume that the external electrons are uniformly distributed. This is not allowable for d electrons in the transition elements. In ionic crystals, one may assume free ions, and use the charge distribution appropriate to them. There is no clearly satisfactory procedure in valence semiconductors. If the crystal being considered is not monatomic, there is the possibility of ionic character, for each atomic cell need not be electrically neutral. The potential within a particular cell will not be determined by the charge within the particular cell alone under these conditions. The rest of the lattice must be taken into account. This contribution can be estimated if the charges of the ions are known. The constant term in this potential can be determined easily from the lattice parameter, effective charge, and the Madelung constant. This "Madelung" potential is not actually constant in the unit cell; however, it has the symmetry of the lattice, so that it is reasonable to neglect all but the constant part for cubic crystals. The determination of the exchange potential is still more uncertain. Fortunately, this potential usually is reasonably small compared with the ordinary potential and need not be determined with extreme accuracy. Actually, an exchange potential is defined only in reference to a particular state, and varies from state to state. For the function ui we define:

The sum runs only over states having the same spin as ui. It usually is not practical to obtain different exchange potentials for all the electronic states of interest in a band calculation. Slater has proposed certain simplifying approximations which can be used to obtain an average exchange potential for all states.I2A comparison of approximate exchange potentials has been given by Herman, Callaway, and Acton.Ia The most celebrated of these approximations replaces (4.1) by the exchange potential which would exist in a free electron gas having the same density, namely:

1) 1*

J. C. Raterl Plcys. Rev. 81,385 (1951). F. Herman, J. Callaway, and F. S. Acton, Phys. Rev. 96,371 (1954);see also V. W. Maslen, Proc. Phys. SOC.(London) A M , 734 (1956), for further discussion.

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JOSEPH CALLAWAY

Here p is the density of electrons having the same spin as the state considered and is a function of position. This expression seems to give a reasonably faithful qualitative rendition of the results of (4.1) for many states, but it is inaccurate quantitatively. [It may be noted here that we must include j = i in (4.1).] It seems to over-estimate the effect of exchange by as much as 20 to 30% in some cases." Slater's more involved average12does not seem to be appreciably better.18 A somewhat better approximation probably can be obtained by assuming that the exchange potential is the same for all states having the same angular momentum. I* States are not characterized by an angular momentum quantum number in a solid; however, in discussing states at symmetry points of the Brillouin zone in simple metals, it will often be possible to characterize the states by the predominant angular momentum i'n the expansion of the wave function in terms of spherical harmonics. It is probably better not to average the exchange potential over all the states of the free atom having the angular momentum considered, as proposed in ref. 12, since such states are likely to be more tightly bound than those of interest in the solid. It is more reasonable to construct approximate wave functions and determine an exchange potential from them. One drawback in the procedure of introducing an exchange potential is that V,, will have infinities if the function ui in (4.1) has nodes. These infinities actually contribute nothing to the energy, but must be removed by some smoothing or averaging process if an exchange potential is to be used like an ordinary potential. The reader should be aware of the very considerable uncertainty that is involved in a crystal potential. It is likely that this difficulty will not be resolved satisfactorily until such time as it is possible to calculate a selfconsistent field for a solid on the basis of Eq. (3.1). Even in cases in which the questions of the ordinary and exchange potentials within an individual cell are settled, there remains the question of the interactions of the free electrons in different cells with each other. This interaction depends on the wave vector k and consequently needs to be included in an energy band calculation. There is, however, no quantitative way of doing this a t present. 5. SYMMETRY PROPERTIES Because uncertainties exist in constructing a crystal potential, it is desirable to employ symmetry considerations whenever possible to aid in determining wave functions and energy levels. The classification of crystal wave functions according to their symmetry properties is, of course, independent of the detailed nature of the crystal potential. Such classification I4J.

Callaway, Phys. Rev. 99, 500 (1955).

ELECTRON ENERGY BANDS IN SOLIDS

109

will usually be found to be of considerable value in the calculational process. Often symmetry considerations can be combined with experimental data and simple theoretical inferences to derive quite reasonable ideas of the bands in a particular crystal. It is assumed that the reader is familiar with the basic ideas concerning the symmetry properties of a crystal, in particular the Brillouin zone, as discussed in standard texts.16 In this section the principles underlying the symmetry classification of wave functions will be presented, based on the work of Bouckaert, Smoluchowski, and Wigner.lBA detailed review of these considerations has been given by Koster. l7 In subsequent sections, the behavior of an energy band near a degeneracy will be discussed ( 6 ) , the application of topological results to the determination of the density of states will be considered (7),and finally a normal form will be proposed for separate bands in simple lattices (8). In the Hartree-Fock approximation, each one-electron wave function must transform according to some representation of the space group of the crystal. Consider first the invariant subgroup formed by the translations. Since all the translations commute with each other and with the Hamiltonian, the energy eigenfunctions are also eigenfunctions of the translation operators. This statement is equivalent to Bloch’s theorem and implies that the functions are characterized by a wave vector k. In addition to the translations, there are other operations which are rotations and reflections in the simplest cases. If such an operation is applied to a wave function for a particular k, the latter will be transformed into a function having a different wave vector k’, which arises from k by the operation considered. All the k’formed from a general k,will be different; however, the energy will be the same for each. For some values of k, there will be symmetry operations which carry k into itself or into vector k‘ = k -!- K for which K is a reciprocal lattice vector. Such operations form a group, which is called the group of the wave vector. The wave function for such a k will transform, under the group of k, as an irreducible representation of this group, called a small representation. The small representations and their connections have been worked out for the simple cubic, body-centered cubic, and face-centered cubic lattices in ref. 15. The notation of this paper will be used throughout. The hexagonal lattice and the diamond lattice have been considered by Herring. ls The zincblende structure has been studied by Parmenterlg and F. Seitr, “The Modern Theory of Solids.” McGraw-Hill, New York, 1940. P. Bouckaert, R. Smoluchowski, and E. Wigner, Phys. Rev. 60, 58 (1936). 1’ G. F. Koster, Solid State Phys. 6, 173 (1957). 18 C. Herring, J . Franklin Inst. 233, 525 (1940); see also W. Doring and V. Zehler, Ann. Phys. 18, 214 (1953). 19 R. H. Parmenter, Phys. Rev. 100, 573 (1955). 16

l6L.

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JOSEPH CALLAWAY

Dresselhaus.20Elliotta1has discussed the double group (spin included), as have Parmenter and Dresselhaus. The reader is referred to these sources for character tables. A general review of this material is given by Koster." The utility of classifying the wave functions according to symmetry in a band calculation is that the Hamiltonian has matrix elements only between those functions which belong to the same row of the same irreducible representation. Consequently, use of symmetrized functions is of great value in diagonalizing the Hamiltonian. Modern methods of calculating energy bands employ expansions based on symmetrized functions. The determination of the matrix elements of various operators is also facilitated. Symmetry considerations also furnish information concerning the form and connections of the various bands in the Brillouin zone. Suppose energy levels have been determined for a particular wave vector ko and is desired to obtain the energies at a nearby wave vector ko s. As is well known, the energies of states at ko s can be found from these at koby determining the effect of the perturbation 2is V (see Section 7). Symmetry considerations are not particularly helpful if kois a general point in the zone. If ko is a symmetry point, and (ko s) is a general point, all the degeneracy which may have been present a t ko is removed. If the group of (ko s) is a subgroup of the group of ko, but contains more than the identity, as occurs on going away from k = 0 along a symmetry axis, the wave functions a t (ko s) transform according to the subgroup. If the appropriate representation of the group of ko is reducible as a representation of the subgroup at (ko s), the degeneracy at kowill be removed at least in part. Of course, if the groups at ko and ko s are the same, the degeneracy will be the same. The compatibility relations summarize this information and express the way in which levels at symmetry points may connect with the bands along symmetry axes. These relations are obtained from the representations as indicated above, and are given in the references already listed. A degeneracy, or sticking together of energy bands can be required by symmetry only when the wave vector ko is invariant under some symmetry operations. The problem of accidental degeneracy of energy bands (degeneraciesnot required by symmetry) has been examined by Herring.22 It is almost impossible for bands having the same symmetry to cross. For example, at a general point, two wave functions having the same wave vector never have the same energy, so that energy bands do not cross at a general point. Similarly, on a symmetry axis, bands of the same sym20 G.Dreeselhaus, Phys. Rev. 100, 580 (1955). 21 R.J. Elliott, Phy8. Reu. 96, 380 (1954).

+

+

+

+

+

+

** C. Herring, Phys. Rev. 62,365

(1937).

+

111

ELECTRON ENERGY BANDS IN SOLIDS

metry do not cross. Bands of unlike symmetry may cross, however. These rules are important in determining bands away from symmetry points, and when taken in conjunction with the known compatibility relations and calculated values of energy levels at symmetry points, enable us in many cases to derive qualitative ideas as to the form of energy bands in a particular crystal. 6. BEHAVIORO F A BANDNEAR A DEGENERACY Near a point of degeneracy in the band structure, a simple Taylor series expansion of E(k) is not possible. The first investigation of this question appears to have been made by Bowers.28The first published account is that of Sh~ckIey.~* The one-electron Schrodinger equation for a state $k, (including spin orbit coupling but neglecting an explicit exchange interaction and relativistic effects other than spin orbit coupling) is:

The wave functions have the Bloch form #ko = eilr'J%,,. satisfies the equation

The equation for k = ko

The function uk,,

+ s is

The term

can be treated as a perturbation. The solution of (6.3) according to ordinary second-order perturbation *a 24

W. A. Bowers, Ph.D. Thais, Cornell University, 1943 (unpublished). W. Shockley, Phys. Rev. 78, 173 (1950).

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JOSEPH CALLAWAY

theory is

Here The superscript 0 indicates the state which is being perturbed and the superscript y indicates any other state, including the core states for the given ko.The operator

transforms a8 a vector. The solution (6.4) is valid only as long as Uk: is not a member of a degenerate set; Eq. (6.4) is equivalent to a Taylor series expansion of the energy about k. The first term in (6.4) gives rise to a linear dependence of energy on wave vector going away from k. This h e a r term vanishes for certain states a t symmetry points. In these cases &(k) is quadratic in s near ko:and an analytic critical point (Section 7) is formed. I n order for $uk;*H’uk: dT to be different from zero, it is necessary that the integrand be a scalar. More precisely, the direct product roX ro X r0 must contain rl, where I’l is the completely symmetric representation of the group of k, r. is the representation of a vector, and J?o is the representation to which Uk: belongs. All representations are the same at a general point of the zone, and the linear dependence will generally exist. The linear term may vanish at symmetry points. For examples V 8 = 0 for all representations at r, H , and N in the body-centered cubic lattice. If u k ; is a member of a degenerate set, the perturbation removes the degeneracy. In the most important cases, the degeneracy is removed in second order, and the perturbation theory appropriate for this case must be used.26The energy is obtained as the solution of a secular equation, of which the general element is

The indices i and j run over the members of the degenerate set and the index y runs over all other states. The order of the equation is the degree aSL.

I. Schiff, “Quantum Meeh&nics,”2nd ed., p. 156. McGraw-Hill, New York, 1949.

ELECTRON ENERGY BANDS I N SOLIDS

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of degeneracy of the set. The dependence of E on s, although generally quadratic, is more complicated than given by (6.4) and can lead to complicated, warped surfaces and nonanalytic critical points. The precise forms of the energy surfaces which are degenerate at k = 0 are important in the analysis of certain experiments on semiconductors, notably p-type silicon, germanium, and indium antimonide. Semiempirical analyses of the bands near k = 0 in these substances have been based on (6.5). (See Sections 31, 32, and 33.) 7. THE DENSITY OF STATES

It is useful in very many applications to introduce the density of states G(E), which we define to be the fractional number of states between E and E dE. If the energy bands have been specified completely, the density of states can be determined from the expression

+

in which is the volume of the unit cell. The integral is taken over a surface of constant energy. I n this section, the implication of the periodicity and symmetry of a crystal for the density of states will be examined. The general theory of the density of states has been discussed by Van Hove26and Phillips27who have considered the bands of lattice vibrational frequencies. The discussion below follows their work. The reader is referred to their papers for more detailed discussion and for proofs of the results quoted. Of particular concern in a discussion of the density of states is the question of singularities in the function and its derivatives. This problem arises naturally because of the form of Eq. (7.1). Crystal symmetry requires that some or all of the components of the gradient vanish a t various points and along certain lines of symmetry in the Brillouin zone. We will call a point where VkE = 0 a critical point. It is not difficult to locate the critical points required by symmetry. There may exist, however, critical points in addition to those required by symmetry. To determine the first set we examine the expectation value of k - P in accord with Section 6, since this determines the linear term in E(k). If it vanishes, a critical point exists. Near a critical point, it may happen that the energy possesses a simple expansion in powers of k:

E(k)

=

E(ko)

+ Zi-i*c~i(k - ko)i2.

(7.2)

Such a point will be called an analytic critical point. These are the only 26

2'

L. Van Hove, Phys. Rev. 89, 1189 (1953). J. C.Phillips, Phys. Rev. 104, 1263 (1956).

114

JOSEPH CALLAWAY

kinds of critical points considered by Van Hove. If the state #kt belongs to a degenerate set, the expansion (7.2) is not valid and E(k) is more complicated. Such cases will be called fluted critical points. In addition there are singular critical points at which one or more components of the QkE’ change sign discontinuously whereas the remaining components vanish. It is possible to classify the critical points according to the form of the B(k) surfaces in the vicinity in such a way that the numbers of the various kinds of critical points are related. In the case in which the simple analytic behavior of (7.2) occurs four cases are enumerated: aa,all negative (maximum) PI: a1, a2, negative; positive (saddle point) P2: (111’ negative, at; aa, positive (saddle point) Pa:a1, a2,at, all positive (minimum).

PO:a ~a,2 ,

Let Pj indicate an analytic critical point of index j: (j = 0, 1,2,3) and let Nj be the number of such points. Morse has shown that there exist certain relations among the Nj.” These are:

Phillips has proved that Morse’s relations are also valid for the nonanalytic critical points caused by the existence of degeneracies. Each nonanalytic point can be assigned an index j and a topological weight which determines the number of times it is counted in entering into the relations (7.3). These rules are given in Phillips’ paper. The significance of Morse’s relations is this: the critical points produced by crystal symmetry imply the existence of others, at a minimum at least as many as are required to satisfy the relations (7.3). The extra critical points required by the relations (7.3)may or may not be all those which actually exist. An interesting and pictorial analysis of the form of the energy surfaces ~ ~ analytic near a critical point hm been given by R o s e n s t ~ c k .Only critical points are considered. The behavior of the function describing the density of states can be determined in the vicinity of each critical point. The analytic critical points discussed introduce singularities into the first derivative of the density of states. More precisely, the density of states varies as d m i M. Morse, “Functional Topology and Abstract Variational Theory,” Memorial Sciences Mathematiques, Fascicule 92. Gauthier-Villars, Paris, 1938. 40 H. B. Rosenstock, P h p . Rev. 97, 290 (1955); J . Phys. Chem. Solids 2, 44 (1957). 48

ELECTRON ENERGY BANDS IN SOLIDS

115

in the vicinity of the critical point. Other types of critical points introduce different behaviors. Once the various critical points have been located, the form of the density of states is determined in their neighborhood. Comparatively simple procedures should suffice to give a reasonable idea of the remainder of the function. Detailed applications have not yet been made to electronic energy bands. (See, however, Walker’s treatment of lattice vibrations in aluminum.a0)Nevertheless, it can hardly be doubted that these conditions will be important in future studies.

8. NORMAL FORM OF AN ENERGY BAND It waa pointed out in Section 4 that the uncertainties involved in the choice of a crystal potential raise the problem of how strongly the results of a band calculation depend on the choice of potential. It appears from that the shape of an recent work, principally on the alkali insolated band of a particular type, such as an s, p , d band, is reasonably independent of the details of the potential, whereas the relation of the various bands to each other is more strongly dependent on the potential. For this reason, it seems useful to define a normal or standard form for a particular band which is characteristic of a given crystal lattice. The concept of a normal band form may be useful in metals in which a single band may be of interest for many problems. The usefulness of this approach may be somewhat less in semiconductors. By a normal band form we mean the following: At symmetry points of the Brillouin zone the normal order of levels of a particular type is that of the corresponding free electron band in an empty lattice. The type of level is determined by the lowest spherical harmonic in the expansion of the wave function.*2As an example, consider the levels at the center of the zone in cubic lattices: I’l is called an s level, r 1 6 a p level, r12and r26r are d levels, r25 and r2. areflevels, etc. In an “empty” lattice of the bodycentered cubic type, the lowest level would be formed by a plane wave for which k = 0. This would be of the l’l type. The next levels are those formed from plane waves of the type (Z?r/a) (1,1,0>.From these, functions of the type rl, r16,r12,I’w, and I’w can be formed. At the corner point H, in the same lattice, the lowest levels are those formed from waves (%/a) (1, 0,0).These waves can be combined to derive functions of the type Hl, H12, and H16.If the possible levels are examined at these and the other symmetry points (Pand N) of the zone, and if the levels are classified according to type using the results of Bell,a2a normal level order can be constructed. This order is given in Table I for body-centered and faceC. B. Walker, Phys. Rev. 103, 547 (1956). J. Callaway, Phys. Rev. 103, 1219 (1956). 8 ) D. G.Bell, Revs. Modem Phys. 26,311 (1954).

30

11

116

JOSEPH CALLAWAY

centered cubic lattices. The Brillouin zones for these lattices are shown in Figs. 1 and 2. In both the body-centered and the face-centered lattices, an isolated S band would be quite simple. The minimum of the band is at the center TABLEI. NOR-

FORM OF ENERGY BANDS*

I. eband bcc

fcc

rl

rl L1

N1

XI

P I

K1

HI

W1

11. p band

bcc

fcc

LV

N 1 1

P4 "'}degenerate Nd

Xd

Ka

in empty lattice

(W z f ,Wa) degenerate in empty lattice

xsr

r16

LO

rls

K4 111. dband

bcc

fcc

xo

HI2

Lo

N2

rrz6p}degeneratein empty lattice

N4 Pa

r26'

K2

( X , X s ) degenerate in empty lattice

Hesf

Na

IV. f band

r12

rz6 P6

H6

Nz' H2s

rZi

W1.

rot

XI.

W2 XZ'

rzs

Lf

* Levels are listed in order of increasing energy. of the zone and the maxima occur a t the corners. In the body-centered lattice, the N1state contains an admixture of d, and the PIan admixture off functions, so that N1and PImay sometimes be thought of as belonging to the d and f bands, respectively. The band is flat a t PIin spite of the fact that the group of P does not contain the inversion. In the face-cen-

ELECTRON ENERGY BANDS IN SOLIDS

117

tered lattice, K 1also contains p functions so that it also belongs to the p band. L1and X1contain d functions and W1 also mixes f functions. The form of the p band is quite different from that of the s band. In the body-centered lattice, we find that the maximum of the band is at

FIO. 1. Brillouin zone for the body-centered cubic lattice. Points and lines of symmetry are indicated.

FIG.2. Brillouin zone for the face-centered cubic lattice. Points and lines of symmetry are indicated. This zone also applies to the diamond lattice.

the center of the zone, whereas the minimum occurs on the faces. The level Pq should be close to the bottom of the band, as should H16.The situation is similar in the face-centered lattice. The level at the center of the zone is close to the top of the band; the minimum occurs at the center

118

JOSEPH CALLAWAY

of a hexagonal face. The levels Wi and W 3are degenerate in the empty lattice. All the p levels also contain f functions. In addition, P I , W Y ,W 3 , K3, and K1 contain d functions. In the d band, the levels at the center of the zone I’26t and I’12 are near the middle of the band. In the body-centered structure, these levels are degenerate in the empty lattice, so that they tend to be relatively close together in the crystal. The separation of the split d levels at the center of the zone should be considerably smaller than at the corners and faces of the zone. The maximum of the d band occurs on the face. If N Iis classed as belonging t o the d band, the minimum will also occur there. The lowest level which is “really d” is Hlz. The order of the d levels a t the center of the zone when they are split usually seems to be the reverse of the order at Hi: namely I’zst is below r12. Consequently there will be accidental degeneracies on the 100 axis. I n the face-centered structure, the d levels at the center of the zone are not degenerate in the empty lattice and should be well separated in real materials, I’ZW being well below r12.The minimum of the band occurs at the center of a square face or at the corner W if Wzrand W 3are considered to belong to the d band. The j band has not been studied as extensively as the others. It is not so well defined since all the p levels can also belong to the f band, Of the levels listed in Table I, it is interesting to note that the minimum occurs at the center of the zone, however, the symmetry of this minimum is different in the body-centered and face-centered structure. These ideas are in reasonable agreement with those contained in the discussions of Wigner and S e W 3although there are differences in detail. It is difficult to make predictions concerning the relation of the levels belonging to different band types since this appears to be quite sensitive to the crystal potential. 9. METHODS OF CALCULATION Once a crystal potential has been chosen, there remains the problem of solving the simplified Hartree-Fock equations for enough states to obtain a reasonable idea of the form of the bands. This is a difficult problem, but it is one that is becoming tractable because of the development and widespread availability of electronic computing equipment. No attempt will be made in this paper to study the methods of calculation in detail. The attention of the reader is called to ref. 2 and the- citations therein for such information. It should be realized, however, that there is no method which is really practical for a general point in the Brillouin zone. All the methods in common use involve expansion of the wave function in some set of func*a E. Wigner and F. Seitz, Solid State Phy8. 1, 97 (1955).

ELECTRON ENERGlY BANDS IN SOLIDS

119

tions. These expansions must be terminated after a finite, and relatively small, number of terms. It usually happens that some of the terms at symmetry points of the zone, or along symmetry axes can be combined so that the effective size of the expansion is reduced. For instance, if the OPW (orthogonal plane wave) method is used to determine the state rl in the body-centered cubic lattice, 135 plane waves can be included in an expansion involving only eight independent coefficients. It would require 135 independent terms to obtain the same accuracy for a general point. Continued development of electronic computing equipment may ultimately solve the problem of handling such complex cases. Until then, calculations will be concerned with points and lines of symmetry. Fortunately, these are of greatest interest, since the maxima and minima of the bands are most likely to occur there. Judged by criteria of reliability and widespread use, the principal methods of calculating energy bands are the cellular method and the method of orthogonalized plane waves. In the cellular method, the solid state wave function is expanded in products of radial functions and spherical harmonics. The orthogonalised plane wave method employs symmetriaed combinations of plane waves which have been made orthogonal to the states of core electrons. The cellular method solves the wave equation directly. Eigenvalues are determined by requiring that the solutiom satisfy boundary conditions obtained from Bloch’s theorem on the surface of the atomic cell. The principal complication of the cellular method lies in the difsculty of satisfying the boundary conditions exactly. Several techniques are available, however. The orthogonalised plane wave method is basically a variational approach. It determines the function of lowest energy which is orthogonal to a given set of functions. Determination of the set of core functions required is the most serious problem. A detailed discussion of the OPW method has been given by WoodruEa4 A third approach, namely the tight binding method, assumes the wave function in the solid can be obtained aa a linear combination of wave functions for free atoms. Difficulties regarding overlap integrals are so serious that the results obtained are seldom quantitative. The apparent simplicity of the method haa led to very wide-spread applications. It is probably at its best when regarded as an interpolation scheme. II. The Alkali Metals

10. GENERALDISCUSSION

The alkali metals are such simple physical systems compared to other metals that the attention given them has been quite out of proportion to 14

T. 0. W o o M , Solid S W Phys. 4,367 (1957).

120

JOSEPH CALLAWAY

their practical importance. They are by far the best understood of all the metals. Calculations of energy bands in the alkali metals go back to the early 1930’s. The physical picture proposed by Wigner and Seitz has been extended to some extent but has not been revised radically. It is well summarized in a recent review.38 Much of the work has been concerned with the cohesive energy and related properties such as the equilibrium lattice constant and the compressibility. The essential ideas of Wigner and Seitz regarding the cohesive energy of the alkalis are as follows. The cohesive energy is principally the difference of two quantities, the boundary correction and the Fermi energy. The boundary correction is the difference in energy between the energy of the lowest state of a valence electron in the solid and in the free atom. It derives its name from the fact that the boundary condition on the solid state function employed for the lowest level in the cellular method requires the function to have zero radial derivative on the atomic sphere. In contrast, the derivative is negative in the free atom at that distance. Consequently, the kinetic energy of the electron is less in the solid and the energy is lower. Only two of the electrons can be in each state, however, and account must be taken of the distribution of electrons among states of higher energy. The correction is the Fermi energy, which may be regarded as a repulsive term. It can be calculated if the density of states is known. The problem of determining the Fermi energy will be discussed in more detail subsequently. In addition to these effects, which are the major ones, the Coulomb interaction of the valence electrons must be also included. It might be thought that the Coulomb interaction of the valence electrons would yield a very substantial effect. This apparently is not the case if the correlation energy is included with the use of Wigner’s formula.36The most recent analysis of this matter is that of Bohm and Pines, who have investigated the screening effect of the rest of the electron distribution on the field of any single electron,6JSwith the use of the collective coordinate method. *sE.Wigner, Phys. Rev. 46, 1002 (1934); D. Pines’ has discovered an error in the

rmg*

-0.88 low density limit of Wigner’s formula. The expression should be E, = *ED. Bohm and D. Pines, Phys. Reu. 92, 609 (1953); D. Pines, Phys. Rev. 92, 626 (1955).

The formula for the correlation energy given by Pines (E, = 0.0313 In r. - 0.114) agrees rather closely with the result of Wigner. There is a significant omission in Pines’ work480 that a better expression for the correlation energy is E , = 0.0622 In r. - 0.096. This disagrees very seriously with Wigner’s work for moderately low densities, r. > 1. However, the expression Gell-Mann and Brueckner is based on an expansion good only for high densities r. < 1. There is no really adequate treatment for moderate densities. See, however, E. Wigner, Trans. Furaday SOC.84,678 (1938), for a discussion of the lowdensity limit. See also J. Hubbard, Proc. Roy. SOC. (London) OUA, 336 (1957).

ELECTRON ENERGY BANDS I N SOLIDS

121

The preceding paragraph summarizes the theory of the cohesive energy in a qualitative fashion. The mathematical theory can be found in ref. 15. A detailed comparison of theory with experiment will be given subsequently. It is sufficiently good that one can feel confident of the basic ideas presented. 11. DETERMINATION OF THE FERMI ENERGY

The energy must have cubic symmetry (rl)when expressed as a function of wave number. Since the lattice constants are sufficiently large that kj < 1 for all the alkalis (k, is the wave number on the Fermi surface), it is reasonable to expand the energy as follows: E(k) = Eo

+ E2k2 + E4(l)k4+ 5E4(2)(k,2ky2+ kw2kz2+ kz2kzz- *k4).

(11.1)

Here EOis the energy of the lowest state of a valence electron, which is determined by solving the wave equation subject to the boundary condition d$/dr = 0 for r = rI. The expansion (11.1) breaks down near a symmetry point at which E(k) is required to have zero slope (except, of can be computed if the actual course at k = 0). The quantities Ez and E4c1) atomic cell is approximated by a sphere. Rather than use the perturbation expansion (6.4) which involves matrix elements, we shall find it is easier to solve the differential equation of perturbation t h e ~ r y . ~Let ~ . us ~* write $k = @*fur m

uk

knun.

=

(11.2)

n=O

Silverman's results are (to order k2)

u1 = ipl(e)up- T U ~ ) (11.3)

where

He has also given expressions for u3 and u4. The functions f p and fd are the radial parts of the p and d solutions of the equation (- Vz V - Eo)$ = 0.

+

17 88

J. Bardeen, J . Chem. Phys. 6,367 (1938). R. A. Silverman, Phys. Rev. 86, 227 (1952).

122

JOSEPH CALLAWAY

The equation satisfied by auo/aE is (11.4)

Silverman also obtains (11.5)

and

where (11.7)

also (11.8)

and P = rf,. When (11.7) and (11.8) are used, it is apparent that both E2 and E4 can be obtained if the functions uo,jp, jd, and their derivatives are known on the boundary of the sphere r = T.. It is not necessary to determine the functions for all values of T . This result is important in the application of the quantum defect method. Brooks has given a different formula for El(') particularly suitable for use in the quantum defect method.gBIt is

where

The Fermi energy is found to be (11.9)

It is not possible to obtain E42 in this way. The energy surfaces will be spherical as long as the approximation of replacing the atomic cell by a *9

H. Brooks, Varenna Lectures, 1957 (unpublished).

ELECTRON ENERGY BANDS .IN SOLIDS

123

sphere is maintained. However, a large value of El(') probably implies that Ed2 is also substantial. Brooks has discussed the corrections to the coefficients Eo, E2, and E4(I)caused by the approximate nature of the boundary conditions in the spherical approximation. He found that for the alkali metals the corrections to EOand EZare negligible and can be neglected, but that for E4the corrections may be significant. 12. CRYBTAL POTENTIALS FOR THE ALKALIMETALS In computing the potential seen by an electron in an alkali metal, the assumption is commonly made that there is only one electron in each cell. This is not a result of the Hartree-Fock approximation. This approximation reduces the density of electrons whose spins are parallel to that of the one considered producing the Fermi hole, but does not effect the density of electrons of antiparallel spin in a corresponding manner. It is assumed on physical grounds that a reduction also occurs in the second density. It is also assumed that the electron whose wave function is being calculated is localized in one cell. The potential seen by this electron consists of its Coulomb and exchange interaction with the charge in that cell (the ion core field) plus the Coulomb potential produced by the charge in other cells.*O Since the other cells are electrically neutral and nearly spherical, the latter contribution is essentially zero. Wigner and Seitz (see ref. 15) have estimated the errors resulting from the replacement of the polyhedron by the sphere of equal volume, the use of free electron functions to evaluate Coulomb and exchange integrals, 'and similar factors. Such effects appear to contribute less than 1 kcal/mol to the cohesive energy in sodium. There are residual uncertainties particularly involving correlation which may amount to as much as 5 kcal/mol. Agreement of a simple model with experiment within such a factor will be considered satisfactory, Let ua consider the problem of determining the ion core field. For lithium and sodium, it is possible to construct empirical potentials which account rather accurately for the observed spectroscopic energy levels.41*42 These potentials include more than is contained in the Hartree-Fock equations. For example, part of the core polarization potential is included. It is not possible to find adequate empirical potentials for the heavier alkalis, because the exchange potential varies from state to 40 The use of an ionic potential is, of course, only an approximation. The exchange and correlation holm are not localized in a particular cell, but follow the electron around. 41 F. Seitz, Phye. Rev. 47, 400 (1935). An error exists in the printed version of the potential (for lithium) used by Seitz. The correct form of the potential used is given by W. Kohn and N. Rostoker, Phys. Rev. 94, 1111 (1954). 42 W. Prokofjew, 2.Physik 68,255 (1929).

124

JOSEPH CALLAWAY

~ t a t e . ' ~Since . ~ ~ an approach which utilizes the observed spectroscopic data is more accurate in many respects than one based on a self-consistent field calculation, there has been considerable interest in the development of a method for using this information directly. Such a method is the Quantum Defect Method, frequently abbreviated by the designation QDM, introduced by Kuhn and Van V l e ~ k and , ~ ~extended later by Brooks46 and Ham.47 In this method, which is explained in detail in ref. 47, the spectroscopic information is used almost directly and the construction of a potential is avoided. The quantum defect method depends on the circumstance that the ion core in an atom of an alkali metal is compact and the lattice in the solid is quite large. The ion core can be assumed to be unaltered in going from the atom to the solid. Moreover, the field in the outer region of the atomic cell is just the Coulomb field of the positive ion. The wave function in this region must be a linear combination of the independent solutions (regular and irregular functions) of the wave equation for a Coulomb potential. It turns out that the ratio of the coefficients of the functions is determined as a function of energy, by a quantity which, for eigenvalues of the free atom, is closely related to the spectroscopic quantum defect. This function can be computed for the eigenvalue of the free atom and must be extrapolated to the energies of interest. Once this function is obtained, the wave function is determined, apart from normalization, in the outer part of the atomic cell. This is sufficient to determine both boundary connection and the Fermi energy, as was shown in Section 11. This approach is extremely powerful. It includes the exchange interaction, and portions of the correlation interaction between the valence and core electrons, as well as the average electrostatic interaction in the interior. Relativistic effects are included for s states and with a slight extension may also be included for other states. It does not include effects which alter the field at large r, such as the polarization, which varies as l/r4 for large r. It should be mentioned, however, that the empirical potentials usually do not contain such large T effects. It appears to be possible to correct the quantum defect method to include the polarization. It is probable that one could h d , however, a set of empirical potentials, one for each angular momentum. 44 A simplified version of the empirical field approach was devised by H. Hellmann and W. Kassatotschkin, J . Chem.Phys. 4,324(1936);ActaPhysiwchim. 6,23 (1936), and applied with good success to the calculation of the cohesive energietl of Ns, K, Rb, and Cs. 46T. S. Kuhn and J. H. Van Vleck, Phys. Rev. 79,382 (1950). 46 H.Brooks, Phys. Rev. 91, 1027 (19531,and unpublished work. ' 47 F. 6. Ham, SoZid State Phys. 1, 127 (1955). 48

ELECTRON ENERGY BANDS IN SOLIDS

125

The principal difEculty encountered by the quantum defect method probably is the practical one of extrapolating the crucial function of energy [Ham’s 7(E)]from the spectroscopic eigenvalues to the range of energies of interest in the solid. There is no completely satisfactory method of doing this. Inclusion of approximate core eigenvalues, as proposed by Ham4’ does not appear to be of much help. It remains to be seen how much uncertainty is introduced into the method by the difficulties of extrapolation. Possibly such uncertainties also are implicit in the case of empirical potentials. 13. COREPOLARIZATION An alternative to the use of an empirical potential or the quantum defect method is to use a potential obtained from a self-consistent field. Even when the ordinary potential is supplemented by an explicit exchange interaction between valence and core electrons, the eigenvalues obtained are usually higher than those obtained by the other methods. This probably is principally the result of the neglect of the correlation between valence and core electrons. It is possible, however, to include some of the effect of this correlation in the form of an ordinary potential which may be added to a self-consistent field calculation. The polarization of the ionic core by the valence electron is one form of correlation. In &st approximation, the ion is spherically symmetric so that the electric field external to it is characterized by the potential energy -ee/r. The external electron distorts the core, however, and induces in it a dipole moment proportional to the polarizability of the ion. The field of the distorted core then reacts back upon the valence electron, decreasing its potential energy by an amount proportional t o e2/2r4at large r. Higher electropole moments are also induced. If the T-4 dependence of the polarization potential persisted to T = 0, an infinite contribution to the energy would result. Actually it does not, and a more elaborate treatment is req~ired.~8 Such a treatment of polarization has been given using the HartreeFock wave functions for the core electrons.4gUnder the influence of the field of the valence electron, the one-electron wave functions of the core electrons depend parametrically on the coordinates of the valence electron. Consequently the energy of the core electrons is also a function of the position of the valence electron. This coordinate-dependent energy Some attempts to determine approximate polarization potentials are discussed by D. R. Hartree, “The Calculation of Atomic Structures,” p. 162. Wiley, New York, 1957. 4s J. Callaway, Phys. Rev. 106,868 (1957).

40

126

JOSEPH CALLAWAY

then serves as a potential energy function for the valence electrons. After a number of approximations a polarization potential is derived :

4Zi

v,(r>= Tt.

ui(o)(rl)rlcos e ui(l)(rl,ro) drl.

(13.1)

The function ui(*) is the unperturbed core function for the state i, whereas UP)is the perturbed function which was obtained from the work-of Sterr~heimer.~~ Explicit polarization potentials were obtained for lithium, sodium, and potassium.4sThe potentials probably tend to overestimate the effect since Sternheimer’spolarisabilities appear to be somewhat large. In the case of potassium, for example, the calculated polarization potential lowers the 4s energy level below the experimentally observed value. The polarization potential calculated according to (13.1) would make a very substantial contribution to the cohesive energies of potassium, rubidium, and cesium. This contribution would spoil the rather good agreement between theoretical and experimental values of the cohesive energies for these elements obtained by both the quantum defect method and the self-consistent field approach, when polarization is neglected. Brooks has suggested that the polarization potential is reduced in the solid compared to the free atom.s1 The polarization interaction may be screened by the valence electron distribution so that it is nearly zero on the boundary of the atomic cell. The screening of the polarization potential in the solid is one of the most important problems remaining in the study of the cohesive energies of the alkali metals. (See also Section 17.) 14. RELATIVISTIC EFFECTS It is pos&ible to formulate the theory of the cohesive energies of the alkali metals on the basis of the relativistic self-consistent field equation.62 This equation, developed by S w i r l e ~considers ,~~ the relativistic motiqa of one electron in the average electrostatic field of the other electrons, but does not consider the relativistic interaction of the electrons. The source of the relativistic effects is the rapid motion of the electron in the strong attractive field near a nucleus. It is known that such effects are not negligible in solids. For example, there is a substantial splitting of the levels at the top of the valence band in germanium as a result of the spin orbit coupling. 60

R. M. Sternheimer,Phys. Rev. 96,951

(1954).

H. Brooks, Private communication. 6 1 J. Callaway, R. D. Woods, and V. Sirounian, Phys. Rev. 107,934 (1967). 6) B. Swirles, Proc. Roy. SOC.(London) A162, 625 (1935). 61

ELECTRON ENERGY BANDS IN SOLID8

127

Unfortunately, existing self-consistent fields for free atoms usually do not include reIativistic effects. The corrections arising from the relativistic motion of an electron in the nonrelativistic self-consistent potential can be calculated by perturbation theory. These are reasonably substantial for cesium, being at leaat 10% of the cohesive energy, and tend to increase the binding. There is something else to be considered, hotvever. The relativistic effects lead to increases in the binding of the core electrons relative to the energies obtained in a nonrelativistic calculation, so that the wave functions are drawn further in toward the nucleus. Thus the screening of the nucleus by the core electrons is more effective. This in turn, reduces the depth of the potential seen by a valence electron and so tends to cancel the extra binding. Both effects can be included in a calculation usin@; a quantum defect method. The boundary correction is obtained in the same way as in the ordinary theory. The Dirac equation or, more properly, the relativistic self-consistent field equation is to be solved, subject to the condition that the large component have a zero normal derivative on the atomic sphere. The calculation of the effective mam can be made with the method of Silverman.**Bloch’s functions #k = e*%k are used. The function tkk is expanded in powera of k. The first- and second-order perturbed functions u;satisfy the relations: (-ca - p

(-ca * p

+ B ~ c *+ V - Eo)ul = h a + 0 ~ +~V 9- E&z = h a

*

kuo kul+ Ezuo.

(14.1)

The result of the calculation is that (14.2)

where

The functionsfa*, fpr, and f& are solutions of the Dirac equation for states s and p states of j = 8 and j = # (all have mj = 4). The prime indicates differentiation with respect to r. This equation is the relativistic extension of (11.5). A relativistic expression for E P can also be obtained.64 A quantum defect calculation has been carried out for cesium using (14.2).62 The boundary correction given by the ordinary QDM contains all important relativistic effects. It was necessary to obtain quantum defects separately for p states of j = 8 and j = 3. There was considerable uncertainty in extrapolating the defects. There are indications, however, 64

F.S. Ham, Private communication.

128

JOSEPH CALLAWAY

that the net relativistic effect is small, of the order of 1% of the cohesive energy. This implies that there is almost exact cancellation between the extra binding and the stronger screening. The relativistic effects doubtless would be more important in a calculation of the magnitude of the wave function at the nucleus, which is involved in the theory of the Knight shift.

15. LITHIUM There have been many studies of energy levels in lithium.a7-39~41~60~6s-e3 Most of the early work is summarized in the book of Seitz,16and will not be discussed in detail here. It is convenient to consider first the calculations pertaining to the lowest band.67~68~60~e2~*a These calculations employ the empirical potential of S e i t ~ . ~The ' results obtained for the band parameters of Section 11 are presented in Table I1 (for the observed TABLE11. BANDPARAMETERS FOB LITHIUM Silverman and Kohn and Brooks Brown and Kohn, Silverman &stoker (r, = 3.20) Glaaser and Callaway h m h a n s l

Eo

Es

El(" El(')

-0.6832 0.727 -0.031

-0.6832 0.723 +0.039 -0.033

-0.6865 0.7305 -0.0303

-0.6863

-0.6827 0.755

lattice constant ra = 3.21 except a noted). The cohesive energy of lithium obtained by Silverman and KohnS7and Silverman is given in Table 111. The quantities EB and EI are given. Es is the sum of the boundary correction and the Fermi energy.

EB= Eo

- Ea + 2.21E2/ra2+ 5.81E4/ra4.

(15.1)

Here Ea is the energy of the valence electron in the lowest state of the free atom; in this case E,, = -0.3906 ry. J . Millman, Phys. Rev. 47, 286 (1935). C. Herring, Phys. Rev. 69, 598 (A) (1939). 67 R. A. Silverman and W. Kohn, Phys. Rev. 80,912 (1950); 83, 283 (1951). 68 C. Herring, Phys. Rev. 83, 282 (1951). 60 R. H. Parmenter, Phys. Rev. 86, 552 (1952). 60 W. Kohn and N. Rostoker, Phys. Rev. 94, 1111 (1954). 6 1 B. Schiff, Proc. Phys. Soc. (London) A67, 2 (1954). 6) M. L. Glwer and J. Callaway,Phys. Rev. 109, 1541 (1958). 68 E. Brown and J. A. Krumhansl, Phys. Rev. 109,30 (1958). 55

56

ELECTRON ENERGY BANDS IN SOLIDS

129

EX is the energy of electron interaction calculated with Wigner’s formula for the correlation energy. (15.2)

The cohesive energy is -E, = EB

+ EX.

TABLE111. COHESIVEENEEQY OF LITHIUM

EB = -0.1384 (ry) = -43.2 kcal/mole 2.7 kc4/mole EI = +0.0085 (ry) E, = 0.1299 (ry) = 40.6 kcd/mole (theoretical) E C = 36.5 kcal/mole (experimental) =i

The QDM calculation of BrooksBQ leads to a value of -0.1399 ry for EB and thus to a cohesive energy of 41.1 kcal/mole. The agreement between theory and experiment, while reasonably satisfactory, is not remarkable. The principal uncertainties in the results quoted probably lie in the electron interaction energy (perhaps as much as 5 kcal/mole), and in the polarimtion correction. I n a previow calculation, based on an earlier form of the quantum defect method, Brooks found the lattice constant of lithium to be 3.40 A,46.47 which is to be compared with the experimental value of 3.43 A. He also estimated the theoretical pressure corresponding to the compression which is experimentally observed at 10,OOO atmospheres and obtained a value of 10,500 atmospheres. These results are less sensitive to uncertainties in the correlation energy than is the cohesive energy. It is interesting to note that lithium is the only alkali metal for which the effective mass at the bottom of the band found by calculation is greater than the free electron mass. This can be interpreted with the use of the formulas of perturbation theory (6.4) to be a consequence of the absence of a p state in the core. Herring attempted an orthogonalized plane wave (OPW) calculation of energy levels in lithium56but the results were not published in any detail. Parmenter has also made an OPW calculation of energy bands in lithium.sQHis potential was adapted from a self-consistent field calculation and exchange was included by Slater’s averaging process. His potential is probably not as good a representation of the state of affairs as Seitz’s potential is. There are, in addition, several unfortunate features in regard to the use of the OPW method in Parmenter’s calculation. In the first place Parmenter represented the 1 s core function as an exponential with a variationally determined constant. This is likely to be inadequate because it does not represent properly the tail of the wave function, which

130

JOSEPH CALLAWAY

makes a significant contribution to the orthogonality coefficients. Also, it is not evident that Parmenter has included enough plane waves to insure convergence of the expansions. Parmenter believes that a spurious term in his potential renders it impossible for him to determine Eo, however, he believes that the value for Ez, 0.808, is significant. Schiff has made a calculation of energy levels in lithium using the cellular method. Unfortunately, the correct Seitz potential was not available to him. He has determined a semiempirical potential according to the manner of Prokofjew, but his potential does not bind as tightly as that of Seitz and is probably somewhat less accurate. Energy values were calculated for the states rl, NI', NI, HI, H12, Hla. Wainwright and Parzen have attempted a calculation of energy levels .~~ in lithium using a variational method based on Wannier f u n ~ t i o n sThe (corrected) Seitz potential was employed. The results do not compare well with those derived by the cellular or the OPW method using this potential. The trouble appears to be in the determination of the 1sfunction, which has to be found by numerical integration both in their method and in the OPW procedure. They obtained an energy EIS= -5.35 ry for this function whereas Glwer and Callaway6zfound EIS = -3.765. Brown and Krumhans16*have recently calculated energy levels in the lowest band of lithium by a variational procedure closely related to the OPW method. The Seitz potential, in corrected form, was employed. Energies were determined for states at the center of the zone and with wave vectors along the 100 and 111 axes. The results for levels below the Fermi surface are in fairly good agreement with those of Kohn and Rostoker,60and the energy of P I at the zone corner is close to that determined by Glasser and Callaway.6z Glasser and Callaway have recently completed an OPW calculation for lithium."*The Seitz potential was used. A 1s core function was determined by numerical integration for this potential as is required by the OPW method. The principal object was to determine the positions of higher states at the symmetry points of the Brillouin zone. The energy of the lowest state I'l is in good agreement with the results of the cellular method. Calculations were made for 24 irreducible representations at the four symmetry points I', H , P, and N. The convergence of the OPW method was very good for s-like states and reasonable for p and d states. The results are given in Table IV and shown graphically in Fig. 3. The order of levels in the isolated 8, p , and d bands is completely in accord with the normal level order for the body centered cubic lattice. Callaway and Glasser have recently shown that inclusion of the 04 T. Wainwright and G.Parzen, Phys. Rev. 92, 1129 (1963).

131

ELECTRON ENERGY BANDS I N SOLIDS

potential of the valence electron distribution cannot appreciably affect the energies of the levels in lithium.'" The crystal potential is required by symmetry, however, to have a vanishing normal derivative on the surface of the atomic polyhedron. The Seitz potential does not have this TABLE IV. ENERGY LEVELSIN Stab

m '

Lithium

N1 P1 HI

N1.

PI

HI5

Nt N1,

rls

dike

0.0

0.5 0.75 1

-0.6863 -0.176 +O .330 +O .571

0.5 0.75 1 1.5 1.5 2

-0.4042 -0.1887 -0.0920 +O .174 +O .475 +O .617

1 1.5 2 2 2.5 2.75 3 3.5

+O .227 +O .524 +O .854 +1.146 + 148 l. +1.60 +1.83 +2.21

"he quantity m*,* = )k:(

p-like

Potassium

sodium ~~

rl

ALKALIMETALS(OPW)METHOD

TEE

~

Cesium

~~

-0.5974 -0.263 -0.041 +O .193

-0.4304 -0.224 +O .054 +O. 265

-0.5286 -0.285 -0.118 +O. 144

-0.316 -0.162 -0.015 +O .301 +0.310 +o .632

-0.190 -0.126 +O .062 +O. 176 +O .356 +O .699

-0.342 -0.259 -0.173 -0.009 +O .069 +O .269

-0.132 +O .055 +o .220 +O .264 +O. 436 +O .486 +O. 537 +O .694

-0.208 -0.167 -0.116 +O .042 +o .099 +O .098 +O. 130 +o. 242

d-like

where k is the wave vector of the lowest OPW in the

expansion for the particult& state.

property, as is also the case for the potentials used in the calculations of energy levels in potassium and cesium. This probably does not affect r, signxcantly, but may dter the other levels by f0.05 rydbergs. It is to be noticed that the lowest band is bounded by states of p-like character, namely HIS,PI,and NI',in agreement with the results of S C M .There ~ ~ is no evidence to support Mott's conjecture that states on the boundaries of the first Brillouin zone are of S character." This p r e J. Callaway and M. L. Glaaser, to be published. N. F. Mott, Phil. Mag. [7] 44, 187 (1953).

640

6s

132

JOSEPH CALLAWAY

+I.!

+I.(

n

cn

2 +.! (3

m

*a

P

CI

5 a W

z t

W

.*

,I

-1.c

FIG.3. Energy level diagram for lithium according to Glaaser and Callaway.6* .Lev& at four symmetry points up to 1.5 rydbergs are shown.

posal waa introduced in an attempt to explain the soft x-ray emission spectrum of lithium.66The spectrum shows a falling off in intensity before the Fermi surface is reached. Since transitions are observed from the con66 D. E. Bedo and D. H. Tomboulian, Phys. Rev. 109, 35 (1958); J. Callaway, Bull. Am. Phys. Soc. [2] 3, 29 (1958).

ELECTRON ENERGY BANDS IN SOLIDS

133

duction band to the 1s level, it was supposed that S-like states were prominent on the Fermi surface. 16. SODIUM Calculations of energy levels in sodium have been made by many workers.z8~8s~41~4~41~01-~1Q The early papers of Wigner and Seitz listed here0?@ form the foundation of the theory of cohesive energy. The results are summarized adequately in Seitz’s bookl5 and will not be discussed further. The original band calculation by Slater08suffers, as do many similar calculations, from insufficient accuracy in the application of the cellular method boundary condition^.'^ Von der Lage and Bethe presented a method whereby much of the diEculty could be overcome. The problem cannot be said to be solved completely however. The parameters pertaining to the bottom of the band are presented in Table V. The observed lattice spacing corresponds to T. = 3.94 at 0°K. The value of E P given in Table V was determined by Callaway at a somewhat larger lattice spacing, T, = 4.07. TABLE V. BANDP m m ~ FOB s SODIUM

EO E2

E p Q

b

Bardeena (r. = 3.96)

(r. = 4.0)

Brooksb

-0.610 1.069 -0.051

-0.6011 1.022 -0.0096

J. Bardeen, J . Chem. Phys. 6, 367 (1938). H. Brooks, F’rivate communication. The valuea attributed to Brooks have been found by quadratic interpolation in the published results. Equation (15.2) was used to compute EI.

The cohesive energy is given in Table VI according to the band parameters of Bardeen. The QDM calculations of Brooksaalead to a value of -0.0824 for EB and to a cohesiveenergyof 26.8kcal/mole for T. = 4.0. The agreement with experiment is quite respectable. The theoretical uncertainties, particularly those concerning the correlation energy, are sizeable. In this respect, E. Wigner and F. Seitz, Phys. Rev. 45,804 (1933). J. C. Slater, Phys. Rev. 46, 794 (1934). 6oE. Wigner and F. Seitz, Phys. Rev. 46,509 (1934). 70 F. C. Von der Lage and H. A. Bethe, Phys. Rev. 71, 612 (1947). 71 D. J. Howarth and H. Jones, Proc. Phys. SOC.(London)A66, 355 (1952) noJ. Callaway, Phys. Rev. to be published. 7) W. Shockley, Phys. Rev. 51, 129 (1937). 67

68

134

JOSEPH CALLAWAY

it is curious that the net effect of the Coulomb interaction is an increase in the binding energy. The equilibrium lattice constant obtained by Brooks is 4.26 A which is to be compared with the experimental value of 4.26 A. The theoretical pressure corresponding to the compression which is experimentally observed at 10,OOO atmospheres is 9OOO atmospheres. Uwdin haa calculated the cohesive energy of sodium by a method quite Merent from those ordinarily used for metals.7aHis method is an adoption of. the tight binding approximation and considers overlap integrals for quite distant neighbors. He obtained values for the cohesive TABLEVI. Comsm ENEEGY OF SODIUM EB = -0.0814 (ry) = -25.5 kcal/mole EI = -0.0032 (ry) = - 1.0 kcal/mole E, = +0.0846 (ry) = 26.5 kcal/mole EO = 26.0 kcal/mole (experimental)

energy and lattice spacing (24.7 kcal/mole and a lattice constant of 3.67 A aa compared with 4.22 A), but did not include any correction for the correlation energy. This calculation haa been criticized by Raime~7~ on the ground that his I'l wave function is a poor approximation to the solution of the Hartree-Fock equation and is not orthogonal to the core states. For this reason, the agreement with experiment probably is accidental. Calculations of higher energy levels in sodium are not as complete as for lithium. The cellular method has been used to study a few of the The calculations of Von der low-lying levels at r, H , P, and N.28*47J0s71 Lage and Bethe, Bowers, and Howarth and Jones were based on the Prokofjew potential. The results are presented in Fig. 4. In the case of conflict, the results of Howarth and Jones are given. It may be noted that the bands are not as wide as in lithium. In particular the d levels have been lowered with respect to 8 and p states. The lowest band appears to be quite accurately parabolic, with an effective mass ratio close to unity. The lowest level at the point N , which is the symmetry point nearest the Fermi surface, is N 1 which is s-like. Ham,47however, has found in his QDM calculation, using a slightly different lattice constant, that N1' is lower than N1. At any rate, the states N1 and N1'are much closer than in lithium. Not enough levels have been computed to make a good comparison with the normal order of levels. '8 P.0.Liiwdin, J . Chm. Phys. 19, 1570, 1579 (1951); Advances in Phys. 6, 1 (1956). 74

S. Raimea, Proc. Phys. Soc. (Londrm)A67, 52 (1954).

135

ELECTRON ENERGY BANDS IN SOLIDS

The relative closeness of N Iand Nr' suggests that the proportion of s-Ue states on the Fermi surface is greater in sodium than in lithium. This is supported by magnetic resonance measurements see Section 20)76 and is also consistent with the soft x-ray emission data. 1.00

+ .6 +.5 +.4

+.3 +.2

+.I

"HI

--H

Q

-.I

c,.

12

E.

J'HIII

-.2

-.3 -.4 -5

-.6 E

r

H

P

N

FIQ.4. Energy level diagram for eodium according to r d t e of Von der Lsge and

Bethe70 and Howarth and authom am shown.

In ewe of disagreement, the rerJulta of the latter

The simplified empirical field approach to the determination of the crystal potential originated by Hellmann and Kassatots~hkin4~ has been applied to the calculation of energy levels of predominately s and p symmetries in sodium.71aThe potential in one cell is written as (16.1)

The constants A and B were adjusted so that agreement is obtained with 16 H. Jones and B. SchitT, PTOC. Phys. SW. (London)A67.217 (1964).

136

JOSEPH CALLAWAY

the lowest s and p levels of the valence electron in the free atom. The values obtained were A = 24.578 and 13 = 2.101. The second term in (16.1) includes the effect of orthogonality of the valence electron wave function on those of the core electrons as a repulsive potential, so that the lowest bound state in this potential pertains to the valence electron. A plane wave method was applied to calculate energy levels in this potential. The results are included in Table IV. They are in good, though not exact, agreement with those of Ham. 17. POTASSIUM

There have been several calculations of energy levels in potassium.81~46~46J6J7 In the first calculation, Gorin78 found it impossible to construct an empirical potential which would account for all the spectroscopic data. His calculation of the cohesive energy was based on a selfconsistent field. The results are summarized and discussed in ref. 15. Sufice it to say here that he did not obtain reasonable agreement with experiment, but found a cohesive energy which is much too small and a lattice constant much too large. Gorin also had to make a rather arbitrary assumption of increased correlation energy in the solid as compared with the free atom. The failure of Gorin’s work probably stimulated the development of the quantum defect method. It has been shown, however, that inclusion of an exchange interaction between valence and core electrons makes it possible to obtain a reasonable value for the cohesive energy.T7 The parameters pertaining to the bottom of the band are listed in Table VII. TABLEVII. BANDPARAMETERS FOR POTASSIUM Callawaya 4.84)

Brooksb (r. = 4.86)

-0.4525 1.168 -0.4

-0.4876 1.149 -0.3

(7‘

Eo

E2

Ed(’) ~

a

b

~~~~~~~~~

J. Callaway, Phys. Rev. lOS, 1219 (1956);S. Berman, J. Callaway, and R. Woods, Phys. Rev. 101, 1467 (1956). A numerical error was discovered in the calculation of El reported in ref. 36. The correct value is given here. H. Brooks, Phys. Rat. 91, 1027 (1953);unpublished work; private communication.

The cohesive energy of potassium can be computed on the basis of the foregoing parameters. The results are given in Table VIII (for r8 = 4.84). 76 E. Gorin, Phys. 2.sourjetunia 9,328 (1936). 77 5. Berman, J. Callaway, and R. D.Woods,Phys. Rev. 101, 1467 (1956).

ELECTRON ENERGY BANDS IN SOLIDS

137

Since an empirical potential is not used for potassium, a polarization correction is important in a calculation based on the use of a self-consistent field. The correction computed in ref. 37 amounts to 8.7 kcal/mole and spoils the agreement between theory and experiment obtained in its absence, however, the numerical value obtained in that paper is open to question for two reasons. The first of these is that Sternheimer’s polarisabilities probably are too large. The second is the likelihood of screening of the polarization field by the valence electron sea. These corrections cannot be calculated on the basis of any theory. A crude estimate would TABLEVIII. COHESIVE ENERGY OF POTASSIUM Ee = - 0 . 0 5 5 ~ Er = -0.011 ry E, = +0.066 ry Correction (polarization) E. (including polarization) E C

-

= 17.2 kcal/mole = 3.4 kcal/moIe = 20.6 kcal/mole (simple model) = 2.6 kcal/mole = 23.2 kcal/mole = 22.6 kcal/mole (experimental)

be to scale the polarization potential on the basis of the values of Van Vleck,’S and to take account of the screening by adding an energy equal to the magnitude of the polarization potential on the boundary of the sphere.s1 When this is done, the polarization connection is reduced to 2.6 kcal/mole. This result gives a very reasonable cohesive e11ergy.7~ Brooks’ values for the band parameters quoted in Table VII already include a polarization correction, but not the constant factor added to account for the screening. The cohesive energy obtained in this way amounts to 20.4 kcal/mole. An earlier calculation of Brooks lead to a maximum cohesive energy of 22.2 kcal/mole for a lattice constant of 5.16 A as compared to the experimental value of 5.20 A.46 Callaway investigated higher energy levels in potassium in an OPW calculation.81 Crystal potentials including exchange were calculated separately for s-, p-, and d-like states with the use of approximate wave functions. Core states were not determined for s and p states in this particular potential, however, the solutions of Hartree’s self-consistent field with exchanges0 for the free K+ ion were used. Since exchange was 78

J. H. Van Weck, “The Theory of Electric and Magnetic Susceptibilities.” Oxford,

’9

Scaling the polarization potential reduces the correction from 8.7 to 5.7 kcal/mole.

1932.

The polarization potential on the sphere surface amounts to 3.1 kcal/mole. The net effect is then 2.6 kcal/mole. (0 D. R. Hartree and W. Hartree, Proc. Roy. Soc. (London) A166, 450 (1936).

138

JOSEPH CALLAWAY

included in this calculation, thk error may not be too serious. Energy values were computed for twenty-f our irreducible representations at four symmetry points of the Brillouin zone: r, H,P, and N. Although the convergence of the OPW method was not as good as that obtained in lithium, most levels are probably convergent to 0.04ry. This accuracy is H

HI--

H15--

N

p4

-

Nz*-N4-

-rNz --t44'

HP--

N3-p5-

NI

--

Nt--

"26'--

p3N4-N3'-Hy-

NI-N4*--

"IS-'

PI-

H12--

p4

N2--

N I ' 1 S

NI

H P N FIG.5. Energy level d h g ~ r t mfor potassium according to Callsway.*l Levels at

four symmetry points up to 1 rydberg are shown.

sufficient to answer most questions regarding the order of levels. Some of the results are shown in Fig. 5 and in Table IV. Comparison of the results for sodium and lithium reveals a marked lowering of the d levels in potassium relative to the s and p levels, This

139

ELECTRON ENERGY BANDS IN SOLIDS

would be expected from the position of potassium in the periodic table. The separated bands conform to the normal order of levels for the bodycentered cubic lattice. Experimental information pertaining to the effective masses of electrons on the Fermi surface in potassium and in other alkali metals comes from optical measurements by Ives and Briggs.81 The alkali metals become transparent in the near ultraviolet. This phenomena can be explained simply on the basis of a free electron theory and can be used to determine an effective mass on the Fermi surface. More precisely, the optical effective mass is (if interband transitions can be neglected)B2

l/mo = - VkEdsp =

1

3(2~)~N ~

Vk2Ed31C.

(17.1)

The surface integral in (17.1)extends over the Fermi surface, the volume integral over the occupied part of the Brillouin zone. If the bands are spherical, the result is the ordinary effective mass on the Fermi surface. Cohen has made a least squares fit to the dielectric susceptibility determined from the index of refraction and absorption measurements of Ives and Briggs with a function of the form A - B/u2 where w is the circular frequency of the light.82The optical effective mass can easily be determined from the value of B, and the result for A can be related to the core polarizability. Cohen finds the following values: sodium, mo/m = 1.01 f 0.02; potassium, 1.08 f 0.02; rubidium, 1.08 f 0.03; and cesium, 1.02 f 0.02.Since the effective masses a t the bottom of the ) less than unity for potassium, rubidium, and cesium band ( ~ / E sare according to the band calculations, the results indicate significant negative Ed terms in these elements. Detailed calculations of mo would require evaluation of the integral (17.1). The soft x-ray emission spectrum of potassium has been observed.8a Transitions between the conduction band and the 3p levels were studied. The shape of the spectrum is consistent with a dependence of the density on a factor EI which is to be expected from s-p transitions. The band width is about 1.9ev, which is somewhat smaller than the value of 2.4 ev predicted from the data of Table VIII. The relatively large value of Edindicates that the electrons in potassium should not be considered as free. It also suggests that there may be significant departures of the Fermi surface from spherical symmetry. H.E.Ives and H.B. Briggs, J . Opt. Soc. Am. 26, Na; 27, 395 (1937): Rb and Cs. ** M.H. Cohen, Phil. Mag. [S]8, 762 (1958). 81

8*R. H. Kingaton, Phys. Rev. 84,944 (1951).

238 (1936):

K;27,

181 (1937):

140

JOSEPH CALLAWAY

18. RU~IDIUM AND CESIUM

The heavier alkali metals have not been studied as extensively as the lighter ones. Less work has been done on r ~ b i d i ~ m ~than ~ .on ~ ~ + 8 ~ ~ cesium.46.46.84-66 The early work of Gombas,s4 who considered sodium, potassium, rubidium, and cesium, was based on a method in which perturbation theory was applied to free electron wave functions. He used a semiempirical potential. In the lowest order, he replaced the actual potential in he exterior of the atomic cell by a constant, its average value. The

TABLE IX. BANDPARAMETERS FOR RB AND Cs Rubidium Brooks

(r. = 5.21)

Eo Ep

El(')

= i

-0.462 1.10

Cesium

Callaway and Callaway and Morgan Sternheimer Haase (rr == 5.21) (re = 5.64) (r, = 5.735) -0.4442 1.181 -1.28

-0.4258 1.336

-0.4156 1.393 -3.64

Brooks

(r, = 5.64) -0.4274 1.197

difference between the potential and its average was treated as a perturbation. This procedure may be reasonably valid since the potential varies slowly over most of the volume of the cell. The energy of the ground state so obtained is in reasonable agreement with other determinations. Unfortunately, incorrect boundary conditions were used for the state believed to lie a t the top of the band. Callaway and Morgan calculated the cohesive energy of rubidium using a potential obtained from a self-consistent field for Rb+, supplemented by an exchange potential.Sh The exchange potential was constructed from the Hartree-Fock equations using a trial wave function. The values of the band parameters and the cohesive energy are given in Tables IX and X respectively. The calculation of S t e r ~ ~ h e i r n was e r ~ ~designed to test the hypothesis that an observed discontinuity in the compressibility of cesium at high pressures is the result of a crossing of 6s and 5d bands. He tried to locate the positions of these bands as a function of interatomic distance. A O4

*6

P. Gombh, Z. Ph.ysik llS, 150 (1939). J. Callaway and D. F. Morgan, Phgs. Rev., to be published. R. M. Sternheimer, Phys. Rev. 78, 235 (1950). J. Callaway and E. L. Haase, Phys. Rev. 108, 217 (1957).

141

ELECTRON ENERGY BANDS IN SOLIDS

potential based on a Hartree field was used; however, exchange was included and the result was adjusted in order t o agree with the energy of the lowest state of a valence electron in the free atom. Unfortunately, Sternheimer used inappropriate boundary conditions for the d band, which probably has a shape quite different from the simple parabolic form he predicted. This error makes much of his calculation invalid. However, his results for the s band are reasonably good in the vicinity of the observed lattice constant. Callaway and Haasess made a calculation of the energy bands in cesium. The bottom of the s band was studied by the cellular method and the OPW method was applied to the higher states. The parameters pertaining to the bottom of the band in Rb and Cs are presented in Table IX. The cohesive energy computed from these data is given below in Table X. The values attributed to Brooks were found by interpolation TABLEX. COHESIVEENERGIES OF RUBIDIUM A N D CESIUM Rb EB = -0.0651 EI = -0.0132 E, = +0.0781 Ee (exp.1

= 20.4 kcal/mole = 4.1 kcal/mole = 24.5 kcal/mole

= 18.9

CB (Brooks)

EB = -0.0580 ry = 18.1 kcal/mole El = -0.0152 = 4.8 kcal/mole E , = -0.0721 = 22.9 kcal/mole

Rb (Callaway and Morgan) EB = -0.053 ry = 16.6 kcal/mole EI = -0.013 ry = 4.1 kcal/mole E, = -0.066 ry = 20.7 kcal/mole Cs (Callaway and Haase)

EB = -0.0556 ry EI = -0.0156 ry E, = -0.071 ry Ei (exp.)

= 17.4 kcal/mole

= 4.9 kcal/mole

= 22.3 kcal/mole

= 19.7

between the published values. Polarization effects are not included in these results. For rubidium, Brooks obtained a lattice constant of 5.45 A, which can be compared with the experimental value 5.56 A. For cesium, he found the lattice constant to be 5.93 A. The experimental value is 5.92 A. The pressure corresponding to the observed compression a t 10,000 atmos was found to be 7200 and 13,000 atmos in rubidium and cesium, respectively. The crystal potential used in the calculation of Callaway and Haase was adopted from that of Sternheimer and reproduces the experimental value for the lowest energy state of the valence electron in the free atom approximately. This potential was used for all states. An error which may be considerable in an atom as heavy as cesium is introduced. Core states were found by numerical integration using this potential.

142

JOSEPH CALLAWAY

The OPW method appears to converge quite satisfactorily. There is an uncertainty of the order of 0.05 rydberg in regard to the position of the levels. A level diagram is shown in Fig. 6. Some energy values are given in Table IV. The calculated bands in cesium exhibit two striking

*0

+.7-

-

+.8-

-

+.3-

-

CI

c Y

m

-= a

I-

+.I--

*0

B "t

-.\--

-.3-

-

-.6-

r

n

P

N

FIG.6. Energy level diagram for cesium according to Callaway and Haase.86 Levels at four symmetry points up to 0.75 rydberg are shown.

features. First, there is a general tendency for the bands in cesium to be narrower than in the other alkali metals. Something of this nature would be expected because of the larger lattice constant. It is interesting to note that, in spite of this tendency, the separation of the split d bands at the center of the zone is larger than in potassium. Second, the p levels tend to be rather low compared to s levels. This is probably sensitive to details

ELECTRON ENERGY BANDS I N SOLIDS

143

of the potential, but is reasonably consistent with a substantial negative value of EJ.~' The higher levels conform reasonably well to the normal order of levels for the body-centered cubic lattice, with the exception of the interchangeof Psand N4. Thisis probably the result of incomplete convergence. There is little experimental information pertaining to the Fermi surface. The effective mass suggested by the optical absorption is consistent with a negative Ed. The low position of the p levels at N suggests there may be a substantial departure of the Fermi surface from spherical symmetry. 19. THEKNIQHTSHIFT

The shift of the magnetic resonance line of a nucleus in the solid relative to that in a nonmetallic form, has been measured in four of the five The small value of the nuclear magalkali metals: Li, Na, Rb, and CS.~* netic moment in potassium makes observation of a resonance difficult in that material. The shift yields a value of the quantity < I # F ( O ) ~ ~ > where I,bF is the wave function of an electron on the Fermi surface and < > designates the average taken over the Fermi surface.89 (19.1)

In Eq. (19.1), X P is the spin paramagnetic susceptibility (per unit volume) and 52 is the atomic volume. Uncertainty concerning the value of X p renders the experimental determination of < I#p(0)I*> uncertain for the heavier metals. It often is convenient to compare < J # F ( O ) ) ~ > with J#A(O)~~ where #A is the wave function of the valence electron in the free atom. This quantity can be obtained from the observed hyperfine splitting with use of the formula of Fermi.Qo (19.2)

Here, AW is the hyperfine splitting in energy units, p is the nuclear magnetic moment in units of the nuclear magneton, pn is the nuclear magneton, and P B is the Bohr magneton. The reason for introducing (I,bA(0)la H. Brooks (private communication) also reports substantial negative values for E4 in rubidium and cesium. 88 W. D. Knight, Solid State Phys. 2, 93 (1956). 89 C.H.Town-, C. Herring, and W. D. Knight, Phys. Rev. 70, 852 (1950). 87

90

E.Fermi, Z.Physik 60,320 (1930).

144

JOSEPH CALLAWAY

is that the ratio (19.3)

may be predicted more accurately with the use of the theory than either quantity separately. Many of the defects in the fields may be the same for both. #F and $A are reasonably accurately known in the lighter alkali metals, however, the relativistic effects are of importance in cesium. A comparison of experimental and theoretical values for < (#F(O)I~>and I # A ( o ) [ z is given in Table XI. TABLEXI.



Li 0.110," 0.110' 0.223," 0.242' 6th 0.49,' 0.455' < I $ P ( O ) J * > ~ ~ ~ 0 . 1 0 + 0.05d I$A(o) I*~XP 0.231' &xp 0 . 4 3 k 0.02d th

Id'A(0)l'th

AND ] $ A ( o ) l '

Na 0.555' 0.685" 0.8lc 0.53k 0.751' 0.7055

(ATOMIC UNITS) K

Rb j

cs

0.664d 0.786" 1.76' 3.0/ 2.47' 0.840' 0.676# 2.18' 2.69 2.97' 0.790b 1.28 0.81' 1.21 0.832' 0.05d 2 . 32d 4.39d.h 1.11' 2.34' 3.88' 0.7d 0 . 993h 1 .13deh

W. Kohn, Phys. Rev. 96,590 (1954). Unpublished calculations of H. Brooks, quoted by G. B. Benedek and T. Kushida, Phys. Rev. to be published. T. Kjeldaas and W. Kohn, Phys. Rev. 101, 66 (1956). d G. B. Benedek and T. Kushida, Phys. Rev. to be published. J. Callaway, unpublished. f J. Callaway and E. L. Haase, Phys. Rev. 108,217 (1957). 0 R. M. Sternheimer, private communication. h Deduced from experiment using theoretical values of the paramagnetic susceptibility according t o the work of D. Pines, Solid State Phys. 1,367 (1955). i Determined from hyperfine structure measurements listed by W. D. Knight, Solid State Phys. 2, 93 (1956). i J. Callaway and D. F.Morgan, Phys. Rev., to be published found these values for Rb: th, 2.16: ]$A(0)1'th, 1.96: Eth, 1.10a

b

0

Measured values of the spin paramagnetic susceptibility are available for lithium and sodium, and make possible an unambiguous determination of < l # ~ ( O ) 1 ~ > in these metals. For rubidium and cesium, X , has t o be obtained from the theoretical calculation of PinesSson the basis of the collective electron picture. The agreement between the theoretical and experimental values of X , for lithium and sodium is good. Kohn has computed values of $A(O) and $P(O) for lithium and sod i ~ mThe . ~ values ~ for lithium were obtained using Seitz's potential and a variational method; those for sodium were obtained from the wave 91

For source references, see the notes to Table XI.

ELECTRON ENERQY BANDS IN SOLIDS

145

function of Von der Lage.g2The wave function obtained by a perturbation theory expansion of $k (Section 2) also gives good results for in these cases (4 = 0.46 for Li). The values computed by Callaway for potassium, rubidium, and cesium were obtained using perturbation theory. The values of Brooks were obtained through use of the quantum defect method. Relativistic effects are probably responsible for the disagreement between theoretical and experimental values of gA(0) and # p ( O ) for rubidium and cesium. It appears that the ratio 5 is more accurately given. Benedek and Kushida have studied the pressure dependence of the Knight shift.g1They determined the dependence of X , on volume from the formula of Pines and unpublished calculations of Brooks. From this, the volume dependence of < I$p(0)12>could be computed. For lithium, appears to decrease with increasing volume. For the other alkali metals, it increases with decreasing volume. The results appear to be in good agreement with unpublished caIculations of Brooks except in the case of cesium. In that element, the quantity < I $ p ( O ) l * > shows an anomalously strong volume dependence: It is observed to increase by about 50% for a change in volume of about 22%. The calculations of Kjeldaas and Kohn for the volume dependence in sodium do not agree with experiment as well as those of Brooks. 20. METALLIC HYDROGEN

Although metallic hydrogen does not exist a t pressures obtainable in the laboratory, it is of some interest as a theoretical exercise. In the Wigner-Seitz approximation, the field within a single cell is Coulombic, so that an analytic solution of the Schrodinger equation is possible. Wigner and Huntington calculated the cohesive energy of metallic hydrogen.9aRecently the problem has been taken up by Baltenspergerg4and Stern and TalIeyg6in connection with the problem of the impurity band. There has also been some interest because of possible geophysical application. Stern and Talley found, EO = -2.041, EL?= 0.963 for T. = 1.58.96If the cohesive energy is computed using the standard theory with these parameters, one finds E, = 31.9 kcal/mole. It appears that significant corrections must be made to the free electron exchange and correlation expressions. Taking these into account, Wigner and Huntington obtained F. C. Von der Lage, Ph.D. thesis, Cornell University, 1943 (unpublished). E. Wigner and H. B. Huntington, J . Chem. Phys. 3, 764 (1935). 94 W. Baltensperger, Phil. Mag. [7] 44, 1955 (1953). 96 F. Stern and R. M.Talley, Phys. Rev. 100, 1638 (1955). 96 R. Kronig, J. De Boer, and J. Korringa, Physica 12, 245 (1946).

92 9)

146

JOSEPH CALLAWAY

a maximum cohesive energy of 10.6 kcal/mole for r, = 1.5.9sThe energy of the molecular form is much greater than this a t a much lower density (52.4 kcal/mole a t a density 0.087 or r, = 2.8). However, the metallic form might be stable under extreme pressures. The energy of the lowest state increases, that is becomes less negative, with increasing r, and approaches EO= -1 as r, -+ 00. The effective mass is always equal to or greater than the free electron mass since there are no p states in the core. Accurate calculations of higher levels have not been made, but crude estimates based on simplified boundary conditions show that the bottom of the 2 p band is below the top of the 1s band a t small lattice constants. For larger lattice constants, the 1s band disengages itself from the p band. This is analogous to the situation in lithium. 21. GENERALSURVEY

AND

CONCLUSIONS

The alkali metals have been emphasized in this review because the theory is in a rather well-developed state. This is principally the result of the simple physical situation, which allows one to determine a potential (either explicitly or implicitly as in the quantum defect method) without detailed knowledge of the distribution of conduction electrons. In regard to the cohesive energy, which is perhaps the most fundamental quantity, it is seen that theory is able to account for the experimental results within reasonable error. The precision of the experimental determinations is somewhat difficult to estimate. The cohesive energies are reproduced about equally well by the methods involving explicit potentials and by those based on the quantum defect method. The principal uncertainties in the calculations of cohesive energy arise from features not included in a one-electron approximation, namely the correlation between core and valence electrons, as expressed in the polarizability problem, and the correlation energy in the free electron sea. In regard to the polarization problem, it would be very helpful if experimentally reliable determinations of the polarizability were available. There is also the serious problem of estimating the screening of the polarization potential. It was mentioned previously that the lack of an adequate expression for the energy of a free electron gas a t low density is a great impediment. Even if Wigner’s formula for the correlation energy is valid to within 20% for cesium, there is still an uncertainty of f4 kcal/mole which is greater than 20% of the cohesive energy. Perhaps it is worth repeating that the recent results of Bohm and Pines, Gell-Mann, and Brueckner do not apply at the densities actually found in the alkali metals. Since the correlation energy as given by Wigner’s formula varies

ELECTRON ENERGY BANDS IN SOLIDS

147

slowly with r8,a calculation of the lattice constant should be less sensitive to correlation than is the cohesive energy. The agreement between the theoretical calculation of Brooks and experiment in this respect is rather good. It is apparent that more experimental information concerning the nature of the Fermi surface would be very desirable. Measurements of the electron specific heat would give the density of states on the Fermi surface. The measurements are complicated for lithium and sodium by the phase transitions which occurg7at low temperature. The explanation of these transitions is not known. Measurements should be possible for potassium, rubidium, and cesium, however. More precise measurements of the optical properties would be welcome, particularly in the infrared region. Lithium and cesium need particular consideration. Finally, experiments, such as the anomalous skin effect which can reveal the anisotropy of the Fermi surface,g*would be very useful in the alkali metals. 111. Metals of Groups II and 111

22. GENERALDISCUSSION

Metals of group I1 possess two s electrons. According to the simplest ideas concerning bands, the lowest band should be full and the materials should be insulators. Moreover, the cohesive energy should be small compared with that for the alkali metals. Actually, these substances are conductors and the cohesive energies are larger than those of the neighboring alkali metals. This behavior is explained by the fact that there is a very considerable overlapping of bands (s and p or s and d) in the region of the observed lattice spacing so that conduction can occur. The problem of filling out this simple picture with quantitative detail is quite difficult. The difficulty is principally related to the problem of obtaining a crystal potential and treating correlations. In the first place the potential seen by a valence electron within one cell is not the same in the free atom and in the solid. The charge distribution of the other valence electron is altered in going from the free atom to the solid. This accentuates the difficulty of obtaining self-consistency. Some form of the valence electron charge distribution must be assumed in order to start. Secondly, there is the question of treating the interaction-of the valence electrons within a single cell. These electrons can be assumed to have opposite spin because of the presence of the Fermi hole, however, the correlation between the electrons is important and may be different in the atom and in p7

C. S. Bsrrett and 0. R. Trautz, Trans. A.I.M.E. 176, 579 (1948).

@*R. G. Chambera, Cun. J . Phys. 12, 1395 (1956).

148

JOSEPH CALLAWAY

the solid. We may note in passing that in the case of a trivalent atom it is also necessary to consider the exchange interaction of the valence electrons. The divalent metals generally tend to have closepacked structures : beryllium and magnesium are hexagonal, calcium and strontium are facecentered cubic. Barium, however, is body-centered. The cohesive energy is not expected to depend critically on crystal symmetry as is made evident by the multitude of allotropic modification which exist in metallic systems.aaThere is calculational evidence in support of this assertion. Among the trivalent metals, aluminum has received most attention. It has a larger cohesive energy and a higher electrical conductivity than magnesium. There is considerable experimental information concerning the band structure. Some experimental information also exists on gallium, but its complicated crystal structure hinders calculation. The theoretical problems are essentially the same as for the divalent elements. 23. BERYLLIUM Studies of the cohesive energy and energy bands have been made by Herring and Hillggand by Donovan.loo The calculation of Donovan is considerably simpler and less involved than that of Herring, and will be discussed first. Donovan’s approach is an extension of the theory of the cohesive energy considered in Part I1 (see also Section 24). It is based on the spherical approximation, the atomic cell being replaced by a sphere, so that no detailed account is taken of the crystal symmetry. A boundary correction and a Fermi energy are determined; free electron values of the exchange and correlation in the valence electron distribution are used. The crystal potential was taken from a self-consistent field calculation; exchange between the valence and core electrons was neglected. For r, = 2.37, Donovan obtained E O = -0.90, Ef = +0.616. Computation of the cohesive energy is uncertain because of the effect of correlation of the valence electrons in the free atoms. If such correlation is neglected, the energy in the free atom on the Hartree theory can be used. This gives 96 kcal/molelO1for the cohesive energy. If the experimental value of the energy in the atom is used, exchange and correlation will have been included completely in the free atom but not entirely for the solid. The cohesive energy is then decreased by 45 kcal/mole to 51 kcal/mole. The C. Herring and A. G. Hill, Phys. Rev. 58, 132 (1940). B. Donovan, Phil. Mag. [7] 45, 868 (1952). 101Donovan’s figures have been altered by taking the correct version of Wigner’s correlation expression (see footnote 35). The difference amounts to 5.3 kcal/mole for r, = 2.37.

0s

100

ELECTRON ENERGY B A N D S IN SOLIDS

149

experimental value is about 75 kcal/mole. The agreement is quite reasonable when the uncertainties of the correlation energy are included. Donovan also obtains good results for the lattice constant and the compressibility. Donovan’s calculation can be criticized in several respects. No account has been taken of departures of E(k) from the simple parabolic form. Exchange between valence and core electrons has not been included. It is a little surprising that the cohesive energy is given as closely as this. The calculation of Herring and Hill on the other hand, is carried out extremely carefully. The potential used is almost the same as Donovan’s. A crude kind of self-consistency is obtained in both calculations. Donovan’s calculation is self-consistent with respect to the wave function of the lowest state. It is assumed that the charge distribution in a cell can be represented adequately by taking it to be the same as that of the two electrons in the lowest state. The calculation is continued until the computed wave function of the lowest state $0 is the same as the $0 assumed. Herring and Hill did essentially the same thing except that they assumed the charge in one cell should be represented by one s and one p electron. The accuracy of such procedures is open to question, however: the charge distribution of the valence electrons probably is very nearly uniform so that the errors probably are not of major significance. The cellular method was applied to the bottom of the band and the OPW method was used for higher states, in the vicinity of the Fermi surface. Calculations were made for three values of the lattice parameters c and a, corresponding to r, = 2.07, 2.37, and 2.67. (The ratio c/a was assumed to be 1.63 throughout.) The effective mass, which was assumed to be isotropic (this is not required by hexagonal symmetry), is in good agreement with the results of Donovan. Herring and Hill neglected the exchange interaction between valence and core electrons in solving the wave equation. The interaction was included, however, in the calculation of the cohesive energy. The variation of the exchange integrals with k was computed in this procedure. Corrections for the lack of orthogonality of the Hartree functions were also made. In computing the variation of the core valence exchange with k, the wave function was obtained to second order in k. The departure of the exchange energy of the valence electron sea from the free electron value was also estimated and was found to be about 6% lower. The uncertainty in the correlation may be considerably greater. The first Brillouin zone for the hexagonal close-packed lattice is shown in Fig. 7, and the energies are shown for certain directions in Fig. 8. Herring and Hill constructed a function giving the density of states. The density is shown in Fig. 9, in which it is compared with an equivalent

150

JOSEPH CALLAWAY

FIO.7. Brillouin zone for hexagonal close-packed lattice, according to Herring.18 Points and lines of symmetry are shown.

0.8

0.6 r)

0.4

0.2

0.0

FIG.8. Energy bands in Be parallel to the hexagonal (k,) axis, according to Herring and H i l l . 9 9 The function n(k) represents the approximate ionization energy of a state.

free electron density. The effective mass a t the bottom of the band, including exchange interaction is 1.40. It is greater than 1since there are no p states in the core. The energy on the Fermi,surface is 0.865 ry above the bottom of the band. The band width is 11.6 ev, compared to a free electron value of 10 ev if m = 1.4 and 14 ev if m = 1. The cohesive energy calculated on this basis turns out to be 53 kcal/ mole at r, = 2.37 or 58 kcal/mole if one uses the correct form of Wigner's

151

ELECTRON ENERGY BANDS I N SOLIDS

correlation expression. This result is obtained employing the observed energy of the neutral Be atom and can be regarded as an underestimate. The Fermi energy is greater than in Donovan's calculation, but this is compensated by the more accurate calculation of the total energy.

2.0

-

n(q)

I.0-

I

.50

I .oo 3

FIQ.9. Density of states in.Be according to Herring and Hill.Qo The parabolic curve is the density of states for free electrons with the calculated effective mass. The dotted line is the Fermi energy. 9 is the ionization energy.

24. MAGNESIUM Energy levels for magnesium have been calculated by Raimes102 and Trlifai.lo* The work of Raimes, which preceded that of Donovan on Betloois very similar to the latter. The calculation is of the Hartree type, exchange between core and valence electrons being neglected. The potential is taken from a selfconsistent field. A crude kind of self-consistency was obtained, as discussed in the preceding section. The zone structure is neglected and the band is assumed to be parabolic throughout. For T, = 3.35, Raimes obtained Eo = -0.67, Ez = 1.29. The difficulty, found in beryllium in regard to the cohesive energy, is also present in this case. A value of the energy in the free atom must be chosen; this is uncertain because of the correlation energy. If the value obtained from a Hartree calculation is used, the cohesive energy should be overestimated. When this figure is used, a cohesive energy of -8.7 kcal/mole (no binding) is obtained for T, = 3.35. The experimental cohesive energy is 38 kcal/mole. The dislo* 101

S. Raimes, Phil. Mag. [7] 41, 568 (1950). M. Trlifaj, CzechosZov. J . Phy8. 1, 110 (1952).

152

JOSEPH CALLAWAY

crepancy must be ascribed to the defects of the method, that is to the neglect of exchange and the assumption of a parabolic band. Raimes does obtain a cohesive energy of 25 kcal/mole for rS = 4.16. His calculated compressibility is reasonable. It would appear that the simple method of estimating the cohesive energy should not be extended beyond beryllium. Raimes has extended his treatment to all the divalent rnetals.lo4 Additional approximations are introduced. The experimental value of the second ionization potential of the free neutral atom is used in a manner similar to that employed by FrohlichIo6 to determine the energy of the lowest state. The equilibrium lattice constant and the cohesive energy state are obtained, based on this information. An explicit potential is not constructed. All the bands are assumed to be parabolic and to have the mass ratio Ez = 1. The calculated values of lattice constants and compressibilities are fair agreement with experiment, but the lattice constant tends to be too large. The cohesive energies are rather poor. It is difficult to single out the major source of error, but it probably is the use of the free electron Fermi energy for a parabolic band. It is unlikely that the band has nearly parabolic form, particularly in the heavier alkaline earths and in the metals of group IIb. Trlifaj lo* applied to magnesium a method developed by Matyaslo6 which combines an early form of the augmented plane wave method of Slaterlo7and the statistical approach of Gombis.los A group theoretic analysis was used to determine the symmetries of wave functions at various points in the ~ 0 n e . The l ~ ~analysis is based on the use of a zone twice the size of that employed by Herring. A simple semiempirical potential of analytic form was employed. In the approach of Gombis, it is assumed that the valence electrons are distributed uniformly. The energy of interaction of this distribution with the ions is calculated and a potential is included to represent the resistance to intrusion of the valence electrons into the electron shells of the ions. This is done to include the effects of orthogonalizing the valence electron wave functions to those of the core. The ordinary potential of the ion is included, of course, either empirically or with the use of Hartree’s calculations. Exchange and correlation interactions in the free electron distribution are also included. I04 S . 106

Raimes, Phil. Mag. [7] 4S, 327 (1952).

H.Frohlich, Proc. Roy. SOC.(London)Al68, 97 (1937).

2. Matyaa, Czechoslov.J . Phys. 1, 3 (1952). J. C . Slater, Phys. Rev. 61, 846 (1937). 108 P. Gombh, “Die statistische Theorie des Atoms und ihre Anwendungen.” Springer, Vienna, 1949. 109 E. Antoncik and M. Trlifaj, Czechoslov.J . Phys. 1, 97 (1952). 106

lo7

ELECTRON ENERQY BANDS IN SOLIDS

153

Trlifaj calculated the energy associated with iifteen irreducible representations at ten symmetry points of the zone. The results were used to construct a density of states. The result shows a close qualitative resemblance to those obtained from the soft x-ray emission. It is illustrated in Fig. 10. A total band width of 6.3 volts, which is in good agreement with the experimental value of about 6.5 ev, was obtained. A rise in the x-ray spectrum is seen to occur a t an energy which corresponds closely with the predicted onset of transitions from the second zone. A maximum which corresponds to the highest level density in the first zone may also

6

4

2 0 ENERGY (RYDBERGS)

Fro. 10. Density of states for Mg according to Trlifaj.'oa

be noted. The cohesive energy calculated on this basis, namely 44 kcall mole, is in good accord with the experimental result. The agreement with experiment is very satisfying. The correctness of the calculation is a little di5cult to assess. There is no reason, other than one based on comparison of results, for believing that Gombas' approach really is applicable to divalent and trivalent metals. Moreover, Trliiaj does not give enough evidence to judge the convergence of the perturbation method he employed. A calculation for a simple metal, such as an alkali, in which comparison with other work would be possible, would be very desirable. Matyas has promised such a paper, but apparently it has not appeared. Finally, it is not clear how carefully the density of states was constructed. The significance of agreement between the shape of a calculated density of states and one obtained from the soft x-ray emission curve is also somewhat uncertain, since there are unresolved discrepancies even in the simple case of lithium.

154

JOSEPH CALLAWAY

25. CALCIUM

Energy bands in calcium were studied by Manning and Krutter in an early calculation."O The potential was based on a self-consistent field without exchange. The potential of a valence electron whose charge density was normalized to the atomic volume was added to the potential of the ion core. Exchange interactions between core and valence electrons were not included. The calculation was made in accord with the early form of the cellular method. The boundary conditions were not fitted with sufficient accuracy. The results have little quantitative significance. In particular, the fivefold degeneracy of the d levels that obtains in the free atom has not been removed a t the center of the zone. The entire shape of the d band is probably quite different from that obtained by Manning and Krutter. The authors found a small overlap between the s and d band, which accounted for the metallic properties of calcium. An approximate density of states which has been copied in several standard references was constructed. The rapid rise in this function at the presumed bottom of the d band is in part a consequence of the fact that the fivefold degeneracy was not removed, and should not be taken too seriously. 26. ALUMINUM

The energy bands in aluminum have been studied by several authors. ln-l10 Considerable experimental information is also available. The calculation of Matyaslll was based on a tight binding approximation using one s function and three p functions. Potential integrals with first neighbors alone were included. The effects arising from the lack of orthogonality of wave functions on different atoms were not considered. The interaction integrals were not calculated directly, however, some were estimated from x-ray data. An approximate density of states was proposed which is in qualitative accord with that obtained from the x-ray emission spectrum. The calculation cannot be expected to yield quantitative information because of the magnitude of the approximations involved. Gasparlla and Antoncik"' have applied methods based on the work of Gombasgsto aluminum. Different forms of the potential representing the effective repulsion of the core electron shells for the valence electron were M. F. Manning and H. M. Krutter, Phys. Rev. 61, 761 (1937). 2. Matyas, Phil. Mug. [7] 59, 429 (1948). 11* R. Gaapar, Acta Phys. Acad. Sn'. Hung. 2, 31 (1952). l l * S. Raimes, Proc. Phys. Sx.(London) 00,949 (1953). 114 E. Antoncik, Czechosloy. J . Phys. 2, 18, 31 (1953). 116 V . Heine, Proc. Roy. SOC.(London) A240, 340, 354, 363 (1957).

ELECTRON ENERGY BANDS I N SOLIDS

155

used. The results agree reasonably well with experiment in both cases. s ~determine ~ energy Antoncik haa also applied the method of M ~ t y a to levels for some of the lowest states at five symmetry points of the zone. A function for the density of states was also constructed and the results were compared with those obtained from the soft x-ray emission. A band 12.2 volts, in width, was obtained. Most of the comments made in Section 25 regarding the calculation of Trlifaj also apply here. Raimes extended his semiempirical treatment of the divalent meta1s104 to a calculation of the compressibility of metallic aluminum. The equilibrium lattice constant and energy of the ground state are found from the third ionization potential of the free neutral atom. The Fermi energy and Coulomb interaction were computed assuming the electrons are free. The agreement of the cohesive energy and lattice constant with experiment is rather poor; the relation between the compression and pressure is rather satisfactory. The use of the semiempirical method for obtaining r, does not seem to be well justified. Heine has reported a very careful study of the energy bands in aluminum in a series of three papers. The results of two calculations in which different potentials are used, are presented. He has also been abIe to interpret the experimental data in terms of a detailed model of the band structure. Following Heine, the experimental information is considered first. This comes principally from measurements of the de Haas-van Alphen effect,lI6 and is supplemented by measurements of the anomalous skin effect"' and the low-temperature specific heat.ll8 The period of the oscillation of the diamagnetic susceptibility, when regarded as a function of 1/H, determines the area in k space of an appropriate cross section through the part of the Fermi surface responsible for the effect. It is possible to determine the shape of this portion of the Fermi surface by varying the orientation of the field with respect to the crystal axes. Two sets of oscillations have been observed in aluminum: high-frequency oscillations arising from sections of the surface which contain 0.6 X lo-* electron or hoIes per atom, and low-frequency oscillation associated with sections containing 5 X loT6electron per atom. The effect of the main part of the Fermi surface has not been observed. The anomalous skin effect, which has been studied only for polycrystalline samples, gives the total area of the Fermi surface. The low-temperature specific heat yields the density of states at the Fermi level. If the free electron model were exact, the first Brillouin zone would E. M. Gunnersen, Phil. Trans. Roy. SOC.A249, 299 (1957). T. E. Faber and A. B. Pippard, Proc. Roy. SOC.(London)A2S1,336 (1955). ll*D. H. Howling, E. Mendoza, and J. E. Zimmerman, Proc. Roy. Soc. (London) 116

117

A229,86 (1955).

156

JOSEPH CALLAWAY

be full and the second half full. The energy surfaces would be caps of spheres having their convex sides toward the center. Heine interprets the observations to indicate that the surface is deformed around the zone corners so that pockets of holes and electron exist in the vicinity. The area of the Fermi surface tends to be larger than would otherwise be expected because of this deformation. The shape of the pockets of holes, which give rise to the high-frequency oscillations mentioned previously, has been determined from experiment and is consistent with the situation expected in the neighborhood of the square face centers X or the corners W. As will be seen, there are theoretical reasons for preferring W . The location of the electrons which produce the low-frequency oscillation has not been determined experimentally, but it probably is also close to W, perhaps along the line W X . The first of the band calculations reported by Heine was made using the OPW method and a potential obtained from a self-consistent field without exchange for AP+. Core valence exchange was not included in this calculation. Levels were calculated for 15 irreducible representations a t r, L, X , W, and K . In addition, 140 more general points were studied. The results indicate that E(k) is very close (within 0.01 ry) to the free electron values except close to the zone surface. This calculation indicates that the first zone should be completely filled, and the third and fourth zones almost unoccupied. The more accurate band calculation was based on a crystal potential which was constructed very carefully. The contribution of the AP+ cores was obtained from a recent calculation which included exchange. A correction made for correlation effects among the core electrons was determined by analogy with a calculation carried out by Bernal and Boys for sodi~m.11~ Separate core-valence exchange potentials were computed for s and p states. The contribution of the valence electrons to the crystal potential was calculated assuming that the wave function for the valence electrons are single OPW’s. This potential resembles that of a uniform charge distribution but departs from it by small but possibly significant amounts. Comparison of the assumed potential with that computed from the wave function finally obtained indicates that the work is very nearly self-consistent. A correction was applied to take account of the departure of the potential from that of a uniform spherical model, since the actual charge distribution in the metal does not consist of overlapping spheres of charge which drop to zero at the sphere radius. Exchange among the valence electrons was included by employing the results of Bohm and Pines. This energy was determined as a function of k, and the variation with r was also determined for states on the Fermi surface. The resulting 119 M.J. M.Ekrnal and S. F. Boys, Phit. Trans. Rw. Soc. A246, 139 (1952).

157

ELECTRON ENERGY BANDS IN SOLIDS

crystal potential appears to be the most carefully constructed one used for a multivalent atom. Appropriate core functions were found by numerical integration for this potential. The OPW calculation appeared to be convergent within 0.02 ry. Except in the vicinity of the Fermi surface, E(k) is close to the free electron values for the case in which the effective mass m* = 1.03. The results do not agree in detail with the model deduced from experiment, in the sense that the levels at the corner in the first zone appear to be TABLEXII. ENERGY LEVELSIN ALUMINUM,ACCORDING TO HEINE State

rl

LI

x1

Ki Wl

L2*

Xt

KaI Wa w21

Fermi level

Calculation I

Calculation I1 (position relative to 8

-0.637 0.086 0.288 0.311, 0.441 0.613 0.047 0.230 0.313 0.399 0.404 0.48 (approx) 1.44

band

rl only)

0.000 0.929 0.966, 1.298 1.182

p band 0.806 0.925 1.012 1.063 1.083

d band

slightly (0.07 ry) below the Fermi surface instead of above it. The calculation does not predict the existence of pockets of holes. The levels at the center of a square face are substantially below the Fermi surface, so that if they do exist, it is likely that such pockets are located around the corners. The effective masses and the length of axes of the pockets can be calculated if it is assumed arbitrarily that pockets of holes do exist there. The results agree with experiment in order of magnitude but not in quantitative detail. The effective masses are small because the levels at the corner are close. This is as required by experiment. It is worth mentioning that, in both calculations, the relative order of levels in the s and p bands agrees with the normal level order for the face-centered lattice. The results are presented in Table XII. The total band width is 14.7 volts which is somewhat greater than the value of 10.6 volts obtained from soft x-ray emission spectra.

158

JOSEPH CALLAWAY

It is not known whether the quantitative discrepancies with experiment can be explained in terms of residual defects in the crystal potential, such as the neglect of a core polarization potential, or whether correlation effects among the valence electrons must also be considered in detail. 27. SOLID HELIUM Ten Seldam has estimated the interatomic distance at which a model of solid helium would become a metal.lZ0He assumed that solid helium would have the face-centered cubic structure at high pressures. The goal of the analysis was to determine the distance a t which the bands would overlap so that some levels in the second zone would be below those in the first. The calculation was made by expanding the wave function in symmetrized combinations of plane waves. There are no core states to give trouble in orthogonalization. A potential for a neutral lattice of helium atoms was taken from a self-consistent field. Various methods of selecting the valence electron exchange were tried. Energy values were obtained for representations WI and LZlusing determinants to 10th order, and the results for infinite order were estimated. Calculations were made for three values of the atomic radius; W1 belongs to the s band and is at the top of the first zone; L+ is a t the bottom of the second. The difference in energy of the levels is quite sensitive to the choice of exchange potential, Using the potential believed to be most realistic, it appears that a pressure of 207 mega-atmospheres would be required to cause the energy of Lr to be lower than that of WI and so produce metallic behavior. The results are quite sensitive to the exchange interaction. IV. Elements o f Group IV and Related Semiconductors

28. GENERALCONSIDERATIONS

There has been a considerable effort, both experimental and theoretical, to determine of the band structure of group IV elements. This effort is, in the main, a result of the development of semiconductor devices. A great deal of progress has been made, and many detailed features of the band structures of germanium and silicon have been determined. Graphite has also been studied rather intensively, as have some of the 111-V compounds. It has been possible to understand many of the electrical and optical properties of these materials in terms of specialized band models. The experiments are mutually consistent. This success of a one-particle model in a very gratifying confirmation of the theory of energy bands. The rea120

C. A. Ten Seldam, Ptuc. Phys. Soc. (London) A70,97,529

(1957).

ELECTRON ENERGY BANDS I N SOLIDS

159

sons for expecting a one-particle theory to work under these circumstances were mentioned briefly in Section 2. It is worth restating that semiconductors and insulators are intrinsically simpler than metals theoretically because a small number of particles participate in conduction processes and there is an energy gap between the vacant and the occupied oneelectron states. This gap, which provides a nonvanishing minimum energy denominator in the formulas of perturbation theory, reduces the effect of the electron interaction. For this reason, the quantitative application of the theory of energy bands should meet its greatest success for these materials. One must confess, however, that the predictions of theoretical calculation have been somewhat disappointing when compared quantitatively with experiment, although there have been marked qualitative successes. Much additional theoretical work of the most careful kind will be required to determine whether the one-electron theory can really yield accurate quantitative results. Most of the present difficulties appear to be connected with the problem of obtaining self-consistency : a band calculation must be based on an assumption concerning the distribution of the valence electrons which are under study. In this respect, the simple approximation based on the use of a uniform distribution of valence electrons within one cell is much less valid than in the monovalent, divalent, and trivalent metals studied, for the binding is of covaIent rather than metallic character. 29. GRAPHITE There have been a number of calculations of energy bands in graphite,121-12s which crystallizes in layers. The atoms are arranged in hexagons in each layer. The spacing between layers is considerably greater than that between atoms in a given layer. Many of the electrical properties, for example, the conductivity, exhibit large anisotropies. A two-dimensional model of a single layer has often been the object of study. Calculations based on such a model may give qualitatively reasonable explanations of the experimental facts. There is covalent bonding between the atoms in each layer. In fact a layer may be imagined to be a large aromatic molecule, whereas the binding between planes is of van der Waals P. R. Wallace, Phys. Rev. 71,622 (1947). C. A. Coulson, Nature 169, 2651 (1957). 1 2 1 C. A. Coulson and R. Taylor, Proc. Phys. SOC.(London) A M , 815 (1952). I f 4 J. L. Carter and J. A. &umhansl, J . Chem. Phys. 21, 2238 (1953). 125 J. C. Slonczewski and P. R. Weiss, Phys. Rev. 99, 636 (1955); 109, 272 (1958). 126 W. M. Lomer, Proc. Roy. SOC.(Lmdon)A227, 330 (1955). 127 D. F. Johnston, Proc. Roy. Soe. (London) A227, 349, 359 (1955); A237,48 (1956). I** F. J. Corbato, Ph.D. Thesis, Massachusetts Institute of Technology, 1956 (unpublished). 121

160

JOSEPH CALLAWAY

character. The calculations generally have employed the tight binding approximation. The early work of Wallace,121although primitive in some respects, furnishes a simple model in terms of which many experiments can be interpreted. The basis of his analysis is that only a single layer need be considered. The unit cell for such a layer contains two nonequivalent atoms, and the Brillouin zone is the hexagon shown in Fig. 11. Points and lines of symmetry are indicated according to the notation of Lomer,126 who has analyzed the symmetry properties of the wave functions in detail

Q

P

FIQ.11. Brillouin zone for a graphite layer after Lomer.1*6

The carbon atoms are considered to be in the configurationpa.The orbitals which are formed from functions of symmetry 8, p,, and pv, lie in the plane (u). Such orbitals are fully occupied and do not contribute to conduction processes. The fourth electron has a wave function with symmetry p.; these orbitals are perpendicular to the layer (a).The 1 electrons are responsible for conduction; it will be seen that both holes and electrons exist in the band. Wallace considered only potential integrals between a-electron wave functions on nearest neighbors of both kinds, which he carried as undetermined parameters. He neglected the nonorthogonality of functions centered on different atoms. In this way, one obtains a parameterization of the band structure of graphite which, although crude, indicates some of the most important features. The basic point is that there is degeneracy at the point P between the filled and the empty a bands. This is a consequence of the symmetry of the layer model. At absolute zero, the lowest r band would be full, the upper one empty; however, there is no energy gap. The energy depends linearly on (k-k,l in the vicinity of P.

ELECTRON ENERGY BANDS IN BOLIDS

161

The most detailed theoretical calculation based on the single layer model is that of Corbato.lZ8The other calculations employ rather unjustified approximations in the interest of simplifying the tight binding scheme to permit easier calculation. In Corbato's calculation, Bloch functions were formed from Is, 29,and 2 p orbitals taken from a self-consistent field for the spa configuration of the free carbon atoms. The crystal potential was taken to be the sum of spherically symmetric potentials on each atom; the potentials were derived from the self-consistent field. Exchange effects were not included. Two-center overlap and potential integrals were included through ninth neighbors; three-center potential integrals were included through fourth neighbors. The entire calculation, including the computation of the integrals and the solving of the secular equation, was programmed for a digital computer. Solutions were found along the symmetry lines p, q, and T . There is a minimum gap of about three-fourths of a rydberg unit between the bands formed by the u electrons. The r bands tend to lie in this gap (the degeneracy at P is contained in it), but overlap the u bands in the vicinity of the center of the zone. Omission of the 1s Bloch functions was found to induce important changes, particularly in the u bands. This raises a question concerning the effect which would be encountered if excited wave functions (39,3 p ) were included. It is worth remarking that the large static diamagnetic susceptibility of graphite can readily be explained in terms of the layer model as originating in the large gradient of the energy in the vicinity of P. Calculations which take account of the real three-dimensional nature of the graphite crystal have not been performed in as much detail as the work on layers. In this case, there are four atoms in the unit cell, two associated with each layer. The Brillouin zone is a thin hexagonal prism shown in Fig. 12.It has been standard to consider the four bands formed from 2P, orbitals on the four atoms in the unit cell. The interaction between r and u orbitals is neglected, except in the most recent calculation of Johnston.127This approximation may be adequate for a treatment of the bands near the Fermi surface, since, according to Corbato, this is located in the gap between the u bands. It cannot be expected to give a good description of the bands throughout the zone. The interaction between layers removes some of the degeneracy predicted by the layer model for the edges HKH and H'K'H'. The splittings are small; of the order of tenths of an electron volt; but this is large compared to kT and consequently vital in any detailed considerations of transport properties. Wallace's three-dimensional calculation,121which took only nearest neighbor interactions into account, yielded a conduction band which was degenerate with one valence band along a zone edge.

162

JOSEPH CALLAWAY

He found no overlap of conduction and valence bands. Calculations by Johnston127which included more distant neighbors, revealed additional degeneracies near the zone edge, a small overlap between valence and conduction bands slightly off the edge, and determined the dependence of E on kz along the edge. A tight binding study of the band structure of a rhombohedra1 form of graphite has recently been reported by Haering.lZ8”

I

/

H‘ FIQ.12. Brillouin zone for graphite (three-dimensional).

Much recent work has been based on a model developed by Slonczewski and Weiss.lzs Since the interesting part of the zone extends only 1% of the way from the edge of the zone to the center, perturbation theory is useful. The Hamiltonian is expressed to first order in the parameter x, the shortest vector from the edge of the zone to the point of interest through the use of k * p perturbation theory (see Section 6). The k, dependence is taken into account by a Fourier series, which is equivalent to the tight binding approximation for this direction. The resulting Hamiltonian contains six parameters, of which yo, the only one that appears in a layer calculation, is much larger than the others. The parameters are to be l~~ determined empirically to fit the results of experiment. M c C l ~ r e and Noziereslso have applied the Slonczewski-Weiss model to detailed considerations of the de Haas-van Alphenlal effect and cyclotron resonance,1a2Ja8respectively. These studies are similar in many respects, R. R. Haering, Can. J . Phys. 36, 352 (1958). J. W. McClure, Phys. Rev. 104, 666 (1956); 108, 612 (1957).

l**a 1*9

P. Nozibes, Phys. Rev. 109, 1510 (1958). D. Shoenberg, Phil. Trans. Roy. SOC.246, 1 (1952). J. K. Galt, W. A. Yager, and H. W. Dail, Jr., Phys. Rev. 103, 1586 (1956). I** B. Lax and H. J. Zeiger, Phys. Rev. 106, 1466 (1957). 180

111

ELECTRON ENERGY BANDS IN SOLIDS

163

although there are some significant differences. The de Haas-van Alphen effect measurements reveal the existence of both holes and electrons in the band structure, the effective masses being 0.070 mo for holes and 0.036 mo for electrons. The observed masses are very anisotropic. The

FIG.13. Energy bands in graphite along the line KH parallel to the k, axis, according to Nozieres.lSo The band Ea is doubly degenerate; the bands E l and E2are nondegenerate. The separation between E I and EZa t K is about 0.6 ev; that between Ea and El and E2a t H is about 0.025 ev. The dashed line Ep marks the position of the Fermi level. El is about 0.02 ev below the Fermi level a t H and about 0.01 ev above it at K.

mass perpendicular to the kz axis is much smaller than that parallel to it (m,/m,, = 0.005 or less for both holes and electrons.) The number of carriers effective at low temperatures is very small, of the order of 10-6 per atom. The energy bands calculated by McClure and Nosieres exhibit a simple sinusoidal dependence of the energy on k, along a zone edge (see Fig. 13). Two bands are degenerate along this edge. Away from the edge, the degeneracy is removed, and an overlap between valence and conduc-

164

JOSEPH CALLAWAY

tion bands is produced so that holes and electrons can exist a t T = 0. The energy surfaces near the edge are warped and anisotropic. As an illustration of the complicated behavior near the edge, a model of the Fermi surface proposed by McClure is shown in Fig. 14. Nozieres predicts a continuous distribution of electron effective masses between 0.054 mo and 0, and a distribution of effective masses for holes ranging from 0.066 mo to 0.054 ma. He h d s that this distribution is consistent with the de Haas-van Alphen effect measurements and is able to explain all the lines in the cyclotron resonance data. It is likely that further detailed analysis of cyclotron resonance, the de Haasvan Alphen effect, and the static diamagnetic susceptibiIity will ultimately yield a consistent characterization of the energy bands based on the Slonczewski-Weiss Hamiltonian. Haering and Wallace have studied the electrical and magnetic properties of graphite using a three-parameter model derived from the tight binding approximation.18aa The principal features of their model are a relatively large number of electrons per atom at low temperature (10-4) implying a relatively large Fermi energy (about FIG. 14. Fermi surface in 0.06 ev), a large value for 70 (about -2.5 or graphite according to -2.6 ev) and a small out-of-plane exchange McClure. integral. This model is in good agreement with the high-temperature magnetic susceptibility, the anisotropy of the conductivity, and the low-temperature Hall coefficient. It does not, however, predict the presence of holes in addition to electrons. 30. DIAMOND There have been many calculations of energy bands in diamond. 134-141 Interest in diamond originates not only in its interesting mechanical, R. R. Haering and P. R. Wallace, Phys. Chem. Solids 5, 253 (1957). G. E. Kimbal, J . Chem. Phys. S, 560 (1935). I S S F . Hund and B. Mrowka, Ber. Verhandl. s&h. Akad. Wiss. Leipzig, Math.natumiss. KZ. 87, 185 (1935). 18%A. Monita, Sn'. Rpts. T6hoku Univ., First Ser. 85, 92 (1949). 1117 F. Herman,Phys. Rev. 88, 1210 (1952); 95, 1214 (1954). 1 a s L . A. Schmid, Phys. Rev. 92,1373 (1953); Am. J . Phys. 22, 255 (1954). 119 G. G. Hall, Phil. Mag. t71 43,338 (1952). 140 J. C. Slater and G. F. Koeter, Phys. Rev. 94, 1498 (1954). 141 V. %Her, Ann. Phgeik [S]lS, 229 (1953).

1Ja 1x4

ELECTRON ENERQY BANDS I N SOLIDS

165

electrical, and optical properties, but also in the circumstance that the semiconductors of principal importance, silicon and germanium, have the same crystal structure. Most of the earlier work13c13sis only of historical interest. Schmidl38 has calculated the cohesive energy of diamond. He started by defining two-electron functions with which to represent a bonding lobe. The two-electron wave function contains two parts. One part describes the case in which an electron is on each atom of the bonded pair. In the other, both electrons are on the same atom, one atom being ionized. The complete crystal wave function is an antisymmetric combination of such functions. The energy of the crystal is calculated with these functions. An arbitrary constant A giving the ratio of the two parts of the two-electron functions is varied so as to minimize the energy of the system. The effect of a very complicated configuration interaction process is obtained in this way. The basic orbitals which compose the two electron functions were taken from a self-consistent field calculation. They were combined to have the proper directed character, and adjusted so that functions on different atoms were orthogonal to a good approximation. If the constant A is set equal to zero, no binding is obtained, as is expected from work on the hydrogen molecule. 14* The process of minimization yields a value A = 0.82 f 0.06 and a cohesive energy, relative to the ground state of free carbon, of 0.28 f 0.30 rydberg per atom for the system. Much of the uncertainty of the calculation arises from the orthogonality correction. The experimental value of the cohesive energy is rather uncertain, but lies between 0.24 and 0.54 rydberg per atom (75 and 169 kcal/mole). Schmid's calculation does not include consideration of the energies of the electronic states. Herman has made a rather detailed investigation of the energy bands in diamond, utilizing the OPW method. The crystal potential was determined as follows. Neutral carbon atoms, in the (2s)l(2p)* 6S configuration, were arranged in a diamond lattice with the observed lattice constant. The crystal charge density was then taken to be the spatial sum of the atomic charge densities, the latter being obtained from a self-consistent field. An approximate exchange potential was included by using Slater's free electron average.'* The core states employed in the orthogonalization procedure were also taken from the self-consistent field. Energy values were calculated at the three symmetry points: I', X, and L. The Brillouin zone is the same as for the face-centered cubic lattice. The effective masses associated with some of the states at the center of the zone were computed. In the tight binding approximation, the levels at the center of the zone may be represented as bonding or antibonding combinations of a and p 14*J.

c. Slater, J . Chem. Phy8. 19, 220 (1961).

166

JOSEPH CALLAWAY

orbitals on the two face-centered-cubic lattices that compose the diamond lattice. It was found that the convergence of the OPW expansion was much more rapid for those states (rl,I'i,etc.) which must be made orthogonal to the core states, whereas the p states, which are orthogonal to the core states MIa result of symmetry, did not converge very well. There is a question concerning the consistency of the potentials used for valence and core electrons. Herman neglected this problem; however, this neglect does not seem to have had serious consequences. The minimum gap between valence and conduction bands was found to be about 6 volts. The valence band has a width of 22 volts. The results of calculation are reasonably consistent with the experimental ones; however, exact values are not known. It is interesting to note that one of the conduction bands has a negative curvature in the 100 direction a t k = 0, so that the minimum of the conduction band is not a t the origin, but very possibly along the 100 axis. The maximum of the valence band lies a t k = 0; the curvature there is negative and greater than that of the conduction band, so that the minimum vertical separation between the bands probably is at k = 0. Such a band structure has not been verified experimentally for diamond, but is found in silicon. Slater and Koster have applied a tight binding interpolation scheme to determine the energy bands along the 100 and 111 axes in diamond. The parameters required were obtained from Herman's OPW calculation. The analysis leads to effective masses in disagreement with those calculated by Herman in the two directions. In addition, their work suggests that the lowest valence band has negative curvature a t the center of the zone, whereas the effective mass there probably is quite close to 1. It may be concluded that the tight binding interpolation scheme proposed by Slater and Koster does not yield quantitatively accurate results for this case. The description of the wave functions for the diamond lattice that is given by a simple tight binding approximation is very useful for qualitative purposes and will be summarized here, following Slater and Koster. 140 The arguments apply to germanium and silicon as well, with very little change. Four orbitals, one s, and three p functions are considered for each of the two atoms in the unit cell. Eight bands can be formed from them. At the center of the zone, we find that the lowest level is rl, which is a combination of s functions that is symmetric about the midpoint of a line joining the two atoms. I'Z5t, which is a symmetric combination of the p orbitals on the two atoms, is a t the top of the valence band. The lowest state of the conduction band is r15,which is an antisymmetric combination of the same p orbitals. The highest is I'r, which is an anti-

167

ELECTRON ENERGY BANDS IN SOLIDS

symmetric combination of the 5 orbitals. Away from the center of the zone, 5 and p functions are mixed on the different lattices. On the 100 axis (direction k,), the function A 1combines the s and p , functions (separately) on the two atoms in a symmetric way, whereas A21 combines the same functions in an antisymmetric manner. The ratio of the coeficients of the s and p combinations is, of course, a function of k and has to be determined by solving the band problem. The other band that is formed on this axis is A5, which is doubly degenerate. This function contains a combination of the p , and p . orbitals on a given atom (py p , or p , - p.) admixed with similar function obtained from its neighbor in the unit cell. At the face center X(lOO), all wave functions are doubly degenerate. In X1 an s function on one atom is combined with a p, function on another. The A6 band goes into Xd. At the point L, the state LI is composed of a combination of s functions and p functions having the symmetry x y z which is symmetric between the two atoms. LZIis a similar combination which is antisymmetric. There also are the doubly degenerate states Lt and Lag, which are formed respectively as symmetric and antisymmetric combinations of functions on the two atoms having the symmetry x - y and z - 8 ( x y). Zehler141has studied the energy bands in diamond with both the cellular method and a variational technique involving plane waves. He found rather large errors in the empty lattice test even when complicated expansions involving Kubic harmonics were employed. The crystal potential was obtained from a self-consistent field; however, a uniform distribution of the valence electrons was assumed. Energy levels were determined a t the center of the zone and at the point X. The order of levels a t these points is essentially in agreement with that of Herman. Minima along the 100 axis were not obtained. A band width of 18.5 ev and a band gap of 5.9 ev were found. The large errors in the empty lattice test render the results somewhat uncertain, especially for the point X. The variational technique band on plane waves, which was used for the states delineating the band gap, should be equivalent to Herman’s procedure. Casella has considered the energy bands in a hypothetical carbon metal of the face-centered cubic structure.142aThe lattice constant was chosen so that the hypothetical material would have the same average electron density as diamond, and consequently is smaller than for diamond by a factor of 2t. The crystal potential was taken from a selfconsistent field for the ( 2 ~ ) ~ ( 2 paP ) ~configuration of the free atom, and exchange was included according to Slater’s free electron approximation. The OPW method was used to compute energy levels at the center of the

+

+ +

+

14fo

R.

c. Casella, Phys. Rev. 109, 54 (1958).

168

JOSEPH CALLAWAY

zone r, the face centers X and L, and the midpoint of the 100 axis A. The 1s core wave function was determined variationally for the crystal potential used. The convergence of the OPW expansions seems to be quite satisfactory for most states. The order of the s and p levels is consistent with the normal level order in the face-centered cubic lattice. The bands appear to be nearly parabolic with an average effective mass m* = 1.13. The energy discontinuities across the faces of the first Brillouin zone are small. An attempt to compare the cohesive energy for this structure with that for diamond (based on Herman's calculations) did not lead to a definite result. 31. SILICON There have been a number of calculations of energy bands in siliC O ~ . A ~ substantial ~ ~ - ~ ~amount ~ of experimental information which gives a reasonably detailed picture of the band structure has been accumulated. A number of basic papers are listed, with no attempt a t completeness.lS1-l 67 It probably will be convenient for the reader to keep the experimental band structure information in mind and to use it as a guide in evaluating the results of theory: the width of the valence band is about 17 volts.1Ss The lowest state in this band is rl, and the highest (in the absence of spin orbit coupling) is I'45t. Actually spin orbit coupling2' splits the sixfold degenerate level (including the spin degeneracy) into a fourfold degenerate state rg+ which is at the top of the band and a doubly degenerate state r7+which is depressed by about 0.04 ev. At points J. F. Mullaney, Phys. Rev. 66, 326 (1944). D. I(. Holmea, Phys. Rev. 87,782 (1952). 14sE.Yamaka and T. Sugita, Phys. Rev. 90,992 (1953). 146T.0. Woodruff, phys. Rev. 98, 1741 (1955); 108, 1159 (1956). 147D. P. Jenkins, Proc. Phys. Soc. (London)A69, 548 (1956); see also D. G. Bell, R. Heneman, D. P. Jenkins, and L. Pmcherle, Proc. Phys. Soc. (Lorrdon) A67, 562 148

144

(1954).

140

F. Bassani, phys. Rev. 108, 263 (1957).

Proc. Z.R.E. 11, 1703 (1955). E. 0. Kane, J . Phys. Chem. Solids 1, 83 (1956). 161 G. Dreaselhaus, A. F. Kip, and C. Kittel, Phys. Rev. 98,368 (1954). 161 G. L. Pearson and C. Herring, Physica 20,975 (1954). 168 A. H.Kahn, phys. Rev. 97, 1647 (1955). 164 G. G. Macfarlane and V. Roberts, Phys. Rev. 98, 1865 (1955). 166 H. Y. Fan, Solid Stde phys. 1, 283 (1955). 166 D. H. Tomboulisn and D. E. Bedo, phy8. Rev. 104,590 (1956). 167 W. C. Dash and R. Newman, Phys. Rev. 99, 1151 (1955). 149F. Herman, 150

169

ELECTRON ENERGY BANDS IN SOLIDS

sufficiently close to the top, the E(k) relation can be written: h2 2m0

E1,2(k) = -[Ak2 f (B21C'

+ C2(Ckt2kv2+ kv2kz2+

kz2kz2])fj

(31.1)

where mo is the free electron mass. For silicon we have158 *A h2

2m0 = 4.0 _+ O.l,--B fi2

2m0 f 0.4,-C h2 A = 0.04 ev. = 1.1

=

4.1 f 0.4

The minimum of the conduction band in silicon occurs on the 100 axis, not on the face. This means that the energy surfaces are six ellipsoids of revolution. If a coordinate system is chosen along the axis of each ellipsoid, (31.2)

The transverse and longitudinal effective masses are:

! !?!

mo

=

(0.98 _+ 0.04);- mt = (0.19 f 0.01). mo

The minimum gap between valence and conduction bands, which does not occur vertically in a band diagram in k space, is 1.14 ev at 0°K. The minima probably are located about 85% of the way toward the zone face.160 At the center of the zone, the lowest conduction band level probably is r16although rzP is not ruled out completely. The (vertical) gap there is believed to be about 2.5 ev.lS7 A diagram in which the experimental results and the indications of the latest theoretical calculations are combined is presented in Fig. 15. The parameters A, B, and C can be related to certain sums of matrix elements defined by Kittel, through the use of perturbation the0ry.~50.~~l Let
170

JOSEPH CALLAWAY

The sum runs over all the states belonging to r21,the states being characterized by the index I ; Eo is the energy of I ' 2 6 4 0 ) . Moreover, let (31.3b)

It may be possible to neglect H2,since the states of symmetry r2&* are +12-

+8-

- v)

!i4 0 > z

z

k 0

0-

W -I

-

w-4-

> a (3

-

w- 8 -

z w

-

-12

-

-16

-

FIG.15. Energy bands in silicon along the 100 and 111 axes according to the experimental information and the calculations of Woodruff,1" and Baasani.148 Levels determined by experiment are circled. high above the valence band. If this is done, F, G, and H I can be determined from A, B, and C. The results are:

F = -5.0h2/2m,

G = -1.1h2/2m,

and

H I = -3.9h2/2m. (31.3e)

The results given in the foregoing are different from those given in ref. 151, where a different choice of sign of the cyclotron resonance constants is made. Dresselhaus, Kip, and Kittel take the relation B' = - B and

ELECTRON ENERGY BANDS IN SOLIDS

171

obtain the values

F'

=

-1.2h2/2m,

G' = -0.4h2/2m,

Hi

=

-6.7fi2/2m.

(31.3f)

Kane gives some reasons for preferring the choice (31.3e).lK0 Results of the band calculations are only roughly in agreement with experiment. The early calculations of M ~ l l a n e y land ~ ~ H01mes'~~ using the cellular method are unsatisfactory because the boundary conditions were not satisfied adequately. The same appears to be true of the calculation of Yamaka and Sugita."' In this respect, it should be observed that the cellular method is rather difficult to apply correctly in the diamond lattice because the polyhedral cell is complicated.141Jenkins14' has also applied the cellular method to silicon, using a variational method to satisfy the boundary conditions. The potential employed in his work and in the calculations previously mentioned was that constructed by Mullaney from a self-consistent field. Exchange interactions were not included. Jenkins obtains reasonable values for the band width a t the center of the zone, a gap which is somewhat too small, and what is presumed to be the correct order of levels. Unfortunately, calculations on the 100 axis show that the minimum of the conduction band along that axis falls below the maximum of the valence band a t the center of the zone; thus the calculation predicts metallic behavior. This result quite probably is a consequence of the potential and not a result of the use of incorrect boundary conditions. Woodruff has applied the OPW method to ~ i l i c 0 n . He I ~ ~constructed a crystal potential using approximate analytical wave functions of the type developed by Slater assuming a (3~)'(3p)~ configuration. An exchange interaction was included by means of Slater's free electron approximation. l2 Analytic approximations to the core eigenfunctions were found for this potential. Woodruff calculated energies of four states a t the center of the zone. The convergence of the OPW expansion appears The order of levels obtained by to be reasonable for all states except rZv. Woodruff is the same as that shown in Fig. 14. The width of the valence band at k = 0 was found to be 8.4 ev and the band gap waa 3.6 ev at k = 0. Bassani has extended Woodruff's calculations to the point X a t the extremity of the 100 axis.14*The order of levels obtained by Bassani was used in constructing Fig. 15. The lowest level of the conduction band a t X is XI, which has an energy rather close to that of rlo. Bassani used his results and those of Woodruff in combination with the Slater-Koster tight binding interpolation scheme140 to determine the energy bands along the 100 axis. The conduction band does have a minimum along the 100 axis, in agreement with experiment. There is evidence, however,

172

JOSEPH CALLAWAY

from other features of this calculation that the interpolation scheme is not adequate. The valence band maximum appears to be well out along the 100 axis and the effective mass at the bottom of the valence band (rJ is negative. Phillips has recently constructed an interpolation scheme based on the OPW methodlSgwhich appears to be more accurate and to require fewer parameters than the method of Slater and K 0 ~ t e r . The l ~ ~ Fourier coefficients of the crystal potential are treated as parameters to be determined from experimental data or from the results of accurate calculations a t symmetry points. Phillips has applied this approach to diamond. silicon, and germanium. Three parameters are used for diamond and silicon, four for germanium. I n the cases of silicon and germanium the parameters are determined from experimental results. Spin orbit coupling is not included. Bands are studied along the 100 and 111 axes. The results are consistent with all experimental data on these elements. The superiority of Phillips interpolation scheme over that based on the tight binding approximation indicates that the electron wave functions most important for the band structure in these elements are closer to modulated plane waves than to free atom functions. The success of semi-empirical techniques such as those of Kane and Phillips in accounting for a variety of experimental results indicates that band theory gives a t least good phenomenology. It remains to be seen whether self-consistent field calculations can yield results in agreement with experiment. More attention must be given to the choice of the crystal potential and more elaborate calculations must be performed for both germanium and silicon. Brief mention should be made of a short report of an energy band calculation for silicon carbidelBOin which the tight binding method was used. The overlapping of first and second neighbors was considered. A band gap of 6 volts and a band width of 22 volts a t k = 0 were found. Solutions were obtained along the 100, 110, and 111 axes. 32.

GERMANIUM

The experimental information concerning the band structure of germanium is more detailed than for any other element. Theoretical studies also have been ~ n d e r t a k e n . ~Many ~ ~ ~of~ the ~ ~experimental -~~~ papers Iisted in the section on silicon (31) are also concerned with germanium. A few others are listed below. Again no attempt at completeness 160

S. Kobayasi, J . Phys. SOC.Jupun 11, 175 (1956).

F. Herman and J. Callaway, Phys. Rev. 89,518 (1953). F. Herman, Physicu 20, 801 (1954). lo* F. Herman, Phys. Reu. 96, 847 (1954). 161 162

ELECTRON ENERGY BANDS IN SOLIDS

173

is It is convenient to begin by summarizing the experimental information. The valence band in germanium has a width of 7.0 volts.166As in silicon and carbon, the lowest state almost certainly is rl,and the highest r26,. The spin orbit splitting at the top of the valence band is substantially larger than in silicon, being about 0.29 ev.lS6The energy surfaces for holes are a set of warped spheres which are characterized by the cons t a n t ~ [see ' ~ ~Eqs. (31.1) and (31.3)]. A = 13.1 f 0.4h2/2mo,B = 8.3 f 0.6h2/2mj C = 12.5 f 0.5h2/2m F = -28.4h2/2mo, G = -1.2h2/2mo, HI = -5.8h2/2mo.

These results imply that there are a band of heavy holes and a band of light holes. The minimum of the conduction band occurs'at the center of a hexagonal face (point L),16' and presumably corresponds to the state L1. The energy surfaces for electrons have the form of a set of somewhat needleshaped ellipsoids of revolution. The effective masses168(31.2) associated with them are

2 = 0.0819 f 0.0003

mo

mz- 1.64 & 0.03. mo

The minimum (not vertical) energy gap between valence and conduction bands is 0.65 ev a t room temperature"' and 0.744 ev at T = 0. The lowest rather than r16 because state at the center of the zone is believed to be of the size of F relative to HI described in the foregoing. The vertical band gap at k = 0 is 0.803 ev at room temperature166 and about 0.88 ev a t 77OK.lS7The effective mass a t k = 0 in the state normally unoccupied has been obtained from the oscillatory magnetoabsorption associated with the direct optical transition,lB6and is m(k = O)/mo = 0.036. It is inferred from magnetoresistance measurements on germanium-silicon alloys that the conduction band has six minima along the 100 axis in ml is nearly the germanium as well as silicon. The effective mass ratio mt same as in silicon namely about 5.1e8The band structure of germanium is shown, in Fig. 16, in accordance with the best current experimental and theoretical ideas. There is considerable experimental information on germanium silicon r21

G. G. Macfarlane and V. Roberts, Phys. Rev. 97, 1714 (1955). B. Abeles and S. Meiboom, Phys. Rev. 96, 31 (1954). S. Zwerdling, B. Lax, and L. M. Roth, Phys. Rev. 108, 1402 (1957). 167 J. H. Crawford, H. C. Schweinler, and D. K. Stevens, Phys. Rev. 99, 1330 (1955). 188 M. Glicksman and S. M. Christian, Phys. Rev. 104, 1278 (1956). 164

166

174

JOSEPH CALLAWAY

Herman has constructed a model which fits the data quite well, although it is not a rigorous consequence of theory.16aConduction minima of the conduction occurs by migration of electrons in the 111 (L1) band in germanium. Although the 100 minima exist, they are too high above the 111 minima to be populated. As silicon is added, the effective potential becomes less attractive and the bands tend to rise. In particular, +6

v

+4

3

)

'

3+2.

z

0

a + 0

0'

W J

'

5-2

a

>. (3

a

W

z -4

L2'

W

-6

FIG.16. Energy bands in germanium along the 100 and 111 axes according to the experimental information and the calculations of Hermsn.161-16* Spin orbit coupling is neglected. Levels determined by experiment are circled.

s states rise faster than p states, so that r21 rises faster than and presumably crosses rls.The 111 minima rise faster than the 100 minima. In the range of composition in which between 8 and 20% silicon is present, conduction takes place as a result of carriers present in both the 100 and 111 minima. For higher concentrations (20 to 100% Si), only the 100 minima are populated. Thus there is a continuous transition between the energy bands in germanium and those in silicon. It is no doubt quite significant for the theory of alloys that these simple ideas seem to be successful. The OPW method was applied to germanium in the band calculation of Herman and Callaway.161A crystal potential was formed from the 169 170

E. R. Johnston and S. M. Christian, Phys. Rev. 96,560 (1954). G. Dresselhaus, A. F. Kip, H. Y. Ku, and G. Wagoner, Phys. Rev. 100,1218 (1955).

ELECTRON ENERGY BANDS I N SOLIDS

175

charge distribution obtained from a self-consistent solution for isolated atoms in the (4~)’(4p)~ state. Exchange effects were not included. Energy levels were calculated at the point r at the center of the zone and at the point X a t the center of a square face. Effective masses were found at the center of the zone. The crucial point L was not investigated. The width of the valence band was found to be 13.3 volts, and the magnitude of the gap 1.45 ev. Heavy and light holes were found. The lowest level in the conduction band at the point k = 0 was predicted to be rlSrather than r2,.A conduction band minimum along the 100 axis was indicated. In Herman’s16z subsequent, more elaborate calculation, exchange interactions were included by selecting an average of two of Slater’s averaging techniques. The core functions were chosen as an orthonormal set of functions constructed from the Hartree fields. The energies of these functions, also required in the OPW method, were adapted from x-ray data. Unfortunately, this procedure is inconsistent with the requirements of the OPW method, in which the appropriate core wave functions are eigenfunctions of the crystal Hamiltonian.171 This inconsistency may spoil the quantitative value of the calculation. This probably is the reason why some of Herman’s OPW expansions are poorly convergent. Energy levels were calculated for thirteen nonequivalent k in the zone: r, X g, W, and L, and certain additional points along symmetry axes (the 100, 111, and 110 axes). Herman found that the width of the valence band was about 11 volts and that the minimum, vertical energy gap waa 1.5 volts. The lowest conduction band state at the center of the zone was FZt, in agreement with experiment. This was also predicted to be the lowest conduction band level, which is not in agreement with experiment. Although L1was found to be the lowest state of the conduction band at the point L, itis about 1.5 volts above rr. It is apparent that more detailed calculation will be required before the extent of quantitative agreement between band theory and experimental results can be determined. Kane has taken a semiempirical point of view in the analysis of band structures which seems to be quite fruitful. He has analyzed the shapes of the bands near the origin in p-type germanium, taking account of the spin orbit At the center of the zone, in the absence of spin orbit coupling, the state rw at the top of the valence band is sixfold degenerate (including spin). Both the spin orbit coupling and the k p perturbation remove this degeneracy as one goes away from the center. The bands away from the center have shapes which differ from the shapes they would have in the absence of spin orbit coupling. The difference

-

171

J. Calkway, Phys. Reu. 97, 933 (1955).

176

JOSEPH CALLAWAY

is more radical than a constant factor for the split-off band. Kane constructed the secular equation (6.5) using the six constituent functions belonging to I'w which diagonalize the spin orbit coupling as a basis. The term k (d X V V ) in the perturbation is neglected since it is quite small. The constants required are taken from cyclotron resonance experiments. The resulting sixth-order equation factors into two identical cubics. All bands are doubly degenerate. Kane has solved this cubic equation along certain axis (100,110,and 111).He finds that the bands along these lines are not strictly parabolic because of the presence of the spin orbit coupling. The deviations from parabolic behavior occur in the range in which the energy from the top of the band is of the order of the spin orbit splitting of the level. A similar calculation was performed for silicon. Departures of the bands from parabolic form are also caused by higher order mixing of valence and conduction band levels. These effects however, are smaller than the ones included. The probability of absorption of light by an electron is determined by the matrix elements of the operator A * p (A is the vector potential). Since the wave functions at k = 0 in the valence band have a well-defined parity, the interband transitions that are observed in p-type gerrnaniumlsa depend on the firstorder correction to the wave function brought about by the k * p interaction. The transition probabilities can be computed as a function of k from the cyclotron resonance constants. Kane has done this forgermanium, and finds fair agreement with experiment. 33. TIN, INDIUM ANTIMONIDE, AND INDIUMAFEENIDE Tin exist in two forms: a tetragonal form (8) which is stable a t room temperatures and above, and a cubic form having the diamond lattice which is stable below room temperature. No calculations of the bands have been made for either form. Observations of cyclotron resonance ~ in white tin have been reported by Fawcett and by Kip et ~ 1 . " Values of the effective mass ranging from 0.23 mo to 0.43 mo were reported by Fawcett, and a range from 0.2 to 3 mo was noted by Kip. A spread of effective masses is to be expected for a complicated crystal structure. Gray tin is a semiconductor having a very small band gap, namely about 0.08 ev. It would be an interesting element to study extensively; however, the difficulty of obtaining single crystals has discouraged measurements. The effective mass should be quite small, of the order of 0.01 or less, if the band structure is similar to that of indium antimonide, which is probably the case. Indium antimonide is of considerable interest because of its small 171 E. Fawcett, Phys. Rev. 103, 1582 (1956); A. F. Kip, D. N. Langenberg, B. Rosenblum, and G. Wagoner, Phys. Rev. 108, 494 (1957).

ELECTRON ENERGY BANDS IN SOLIDS

177

energy gap (0.23 ev at 0°K) and high electron mobility. KO calculation of the band structure has been made from first principles. Considerable experimental information exists, and a semiempirical analysis of the band structure has been made.17aThe symmetry properties of the structure and their implications for the energy bands have been a n a l y ~ e d . ~A~ . * ~ few of the experimental papers are listed below. 174-178 The experimental situation is not as clear for InSb as it is for Gel but the model proposed by Kane17aseems to be well established. I n order to understand the calculations and experiments concerning the 111-V compounds, a brief discussion of the results of a study of the symmetry properties is in order. The principal differences between the symmetry properties associated with energy bands in the diamond structure and in the zincblende structure result from the lack of inversion symmetry of the latter. From Kramer’s theorem, one still has the relation E(k) = E ( - k ) ; however, a twofold degeneracy of spin states throughout the Brillouin zone is not required. It is worth noting that the spin orbit coupling plays a very vital role in InSb particularly, in consequence of the high atomic number of the atoms involved. As a result of the existence of the spin orbit interaction in the perturbing Hamiltonian for the cell periodic functions, it is not necessary that all the bands be flat a t k = 0. In particular, representations formed from I’s, which is situated a t the top of the valence band, will not have zero slope. The width of the valence band in InSb is not known. The maximum of the band apparently occurs in the vicinity of the center of the zone. The linear terms give a maximum in the valence band away from k = 0 in the 111 direction. Kane estimates that this maximum has an energy about ev above the energy at k = 0, and is located about 0.3% of the distance to the zone edge. There is evidence that both heavy holes and Iight holes exist; however, the form of the energy surfaces is not known as well as in germanium and silicon. The average effective masses are about 0.18 mofor the heavy holes and 0.04 mo for the light holes. The conduction band is centered at k = 0. The energy surfaces are spherical, and are characterized by an effective mass equal to 0.013 mi. As a result of the very small gap, it is not likely that this curvature is maintained away from k = 0. The small effective mass implies that the E. 0. Kane, J . Phys. Chem. Soh& 1, 249 (1956). G. Dresselhaus, A. F. Kip, C. Kittel, and G. Wagoner, Phys. Rev. 98,556 (1955). 175 H. Y. Fan and G . W. Gobeli, Bull. Am. Phys. SOC. 6, 298 (1956). 178

174

177

V. Fbberts and J. E. Quarrington, J . Electronics 1, 152 (1955). H. J. Hmstowski, F. J. Morin, T. H. Geballe, and G. H. Wheatley, Phys. Rev. 100,

179

E. Burstein and G. S. Pious, Phys. Rev. 106, 1123 (1947).

176

1672 (1955).

178

JOSEPH CALLAWAY

density of states in the conduction band is quite low. Consequently the conduction band becomes filled quite readily and the conduction electrons soon become degenerate. There is evidence for an increase in the effective mass with increasing energy, that is, as the conduction band is filled. I n third order of k, the conduction band is split into two singly degenerate bands except along the 100 and 111 directions. This splitting has not yet been observed. ~ his earlier work on The semiempirical analysis of K ~ n e ' ?resembles germanium and silicon.'$*In this case, the interactions of the conduction band, the two valence bands which are degenerate at k = 0, and the split-off valence band which occurs as a result of the k - p and the k-independent spin-orbit coupling interactions are treated exactly. The interactions with other bands can be treated as perturbations. The bands are parabolic at k = 0 in this approximation. The terms linear in k originate in the k (d X VV) interaction and in combinations of the k p and the k-independent spin orbit interactions. The matrix element of k * p between conduction and valence bands can be derived from the effective mass for electrons determined by cyclotron resonance with reasonable accuracy, since it is likely that this interaction dominates as a result of the small gap. Kane finds:

-

-

P s = - I 12 h2

m2

= 0.44

(in atomic units).

The k independent spin orbit splitting is estimated to be 0.9 ev. The matrix element P determines the probability of the fundamental optical transition between valence and conduction bands at k = 0. The wave functions for values of k leaving the center can be found to a good approximation as the eigenvectors of the Hamiltonian matrix diagonalized previously. This makes it possible to calculate the absorption constant associated with direct transitions in the vicinity of k = 0. The results agree fairly well with the experiments of Fan and G0be1i.l~~ The compound indium arsenide has also been the subject of some experimental study. Lax has found an energy gap 0.360 f 0.002 ev at room temperature. The effective mass of electrons is about 0.03 mo. This probably implies that the conduction band minimum is at the center of the zone. Stern and T a l l e ~ have ' ~ ~ observed the room temperature infrared absorption in p-type indium arsenide, and have carried out an analysis of this absorption which is similar to the work of Kane on indium antim ~ n i d e . " The ~ valence band maximum is assumed to be at the center 179

F. Stern and R. M. Talley, Phys. Rev. 108, 158 (1957).

ELECTRON ENERGY BANDS I N SOLIDS

179

of the zone.'*O From the analysis, the k-independent spin orbit splitting is estimated to be 0.46 ev and the mean heavy hole mass is 0.4 ma. Light holes, whose effective mass would be of the order of the electron mass, should also exist. 34. GALLIUM AFSENIDE The labor involved in an accurate band calculation is so large that it is scarcely feasible to make self-consistent calculations for all substances of interest. For this reason, it is desirable to develop perturbation procedures which will make it possible to relate the bands in several materials. The 111-V and 11-VI semiconductors are particularly interesting. They crystallize in the zincblende structure, which is the binary analog of the diamond lattice consisting of two interpenetrating face-centered cubic lattices. Thus the Brillouin zones are the same as for the group IV elements just considered. Furthermore, the lattice constants in a horizontal sequence, that is, compounds formed of elements lying in the same row of the periodic table, are nearly equal. Herrnanls1 has made a general survey of horizontal sequences and diagonal sequences of these compounds. The latter are formed of elements in different rows of the periodic table. He finds that the increase in the energy gap within a sequence as the atoms become more unlike can be accounted for approximately by perturbation theory. In the case of a horizontal sequence, the principal perturbating potential is antisymmetric with respect to the midpoint of the line joining the two atoms in the unit cell. Consequently, the principal effects occur in second-order perturbation theory. The change in the energy gap between GaAs and ZnSe is three or four times that between GaAs and Ge, which is about what one would expect from simple reasoning. Callaway has considered this perturbation technique in more detail for gallium arsenide182in an effort to relate the energy bands in that substance to those in germanium. A perturbating potential was calculated from the self-consistent fields for neutral atoms of gallium, germanium, and arsenic, and was supplemented by introduction of approximate experimental effective charges on the atoms. The potential was expressed as the sum of symmetric and antisymmetric parts centered about the 18%

As in the case of indium antimonide, there should be a linear term in E(k) near k = 0 which will displace the maximum slightly from k = 0. This linear term may

be somewhat more important in InAs than InSb because of the greater difference in atomic number between the constituents. Stern and Talley believe the linear term will markedly affect the shape of the absorption a t low temperature. 181 F. Herman, J. Electronics 1, 103 (1955). 18) J. Callaway, J. Electronics 2, 330 (1957).

180

JOSEPH CALLAWAY

midpoint of the lines joining the neighboring gallium and arsenic atoms. The antisymmetric part appears to be considerably the larger. The effects of the antisymmetric perturbing potentials can be studied qualitatively by determining from symmetry arguments which matrix elements exist. Combined with an order of magnitude estimate of the size of the matrix elements and reasonable guesses concerning the energy denominators, the displacement of the levels under the influence of the perturbation can be determined. The essential results are as follows. There is a general tendency for the valence band to move to lower energies relative to germanium whereas the levels at the bottom of the conduction band are relatively stationary. The minimum of the conduction band probably shifts from the point L at the center of a hexagonal face to the center of the zone. An increase in the optical energy gap of the observed order of magnitude is found. There is experimental evidence to indicate that the effective mass in gallium arsenide is small, about 0.04 mo, and of the order of that reported for the r2l band in germanium.lS8 This suggests that the conduction band minimum is indeed at the center of the zone. V. The Transition Metals

35. GENERALCONSIDERATIONS Under the title of transition metals, we will consider only those elements of the fourth period of the periodic table in which the free atoms possess incomplete d shelIs. It is convenient to add copper to this list, since there have been several attempts to draw inferences concerning the bands in the transition metals proper from calculations of energy bands in copper. The transition metals are particularly interesting because of their magnetic properties. Iron, cobalt, and nickel are ferromagnetic; chromium and manganese are antiferromagnetic. Many of the simple compounds of these elements are antiferromagnetic. The magnetic properties are naturally ascribed to the d electrons, and it then becomes a pqncipal task of band theory to give a satisfactory quantummechanical amount of these properties. Detailed discussions of the band theory of ferromagnetism and antiferromagnetism can be found in papers by Slater.2J84.186 This is not the place for a thorough account of these theories. It will suffice to state that it has not been possible to give a quantitative explanation of the existence of the origin of either ferromagnetism or antiferrornagnetism on the band L. Barcus, A. Perlmutter, and J. Callaway, Bull. Am. Phys. SOC.[2] 3, 30 (1958). J. C. Slater, Revs. Modern Phys. 26, 199 (1952). 186 J. C. Slster, Phys. Rev. 82, 538 (1951). I**

184

ELECTRON ENERGY BANDS IN SOLIDS

181

picture. Some of the difficulties involved should become apparent in the course of this discussion. Experimental evidence, particularly on the magnetic behavior (atomic moments) of these elements and their alloys, suggests a rather smooth and gradual variation of electronic structure from element to element, regardless of rather substantial changes in crystal forms. Thus it seems reasonable to try to construct a general model of the d bands in the transition elements; there have been several attempts at t h i ~ . ' 8 ~ ~The *0 earliest proposal was that of Mott. The basic ideas of Mottlg6were that there is a narrow d band of width less than 1ev which is overlapped by a wide 4s band. A narrow d band is necessary in the band theory of ferromagnetism so that the decrease in energy arising from increased exchange upon spin alignment will outweigh the increase in energy resulting from promotion of the electrons to higher band states. The magnetic properties are determined by the d band while the cohesion and the conductivity are produced by a relatively small number of s electrons, less than one per atom. Scattering of s electrons into vacant d band levels, where the density of states is high, accounts for the relatively large resistivity. On the other hand, Pauling assumes that the d electron (two in iron) responsible for the magnetic properties form a band of essentially zero width.189 The atomic model is adequate for these electrons. The other electrons, s and d in the free atom, form a very broad band based on hybridization of s,p, and d orbitals. Ferromagnetism is believed to be produced by the exchange coupling of the atomic d electrons with the conduction band, in a manner similar to the proposal of Zener.191 A small number of the conduction electrons contribute to the net magnetic moment. A function giving the density of states which was calculated by Slater1g2-1g4 has been used for the interpretation of a number of experiments on the properties of the transition elements. Slater derived this function from a calculation of the energy bands in copper made previously by K r ~ t t e r Characteristic .~~~ features of the Slater curve are the high density of states a t the bottom and top of the band and a minimum near the middle of the band. Slater used this density of states to estimate the 188 N. F. Mott, Proc. Phys. SOC.(London) 47, 571 (1935). 187 N. F. Mott, Proc. Phys. SOC.(London) A62, 416 (1949). Pauliig, Phys. Rev. 64, 899 (1938). Pauling, Proc. NuU. Acad. Sci. U.S.39, 551 (1953). 1ooF. Bader, K. Ganzhorn, and U. Dehlinger, 2.Physik 137, 190 (1954). 191 C. Zener and R. R. Heikes, Revs. Modem Phys. 26, 191 (1953). 199 J. C. Slater, Phys. Rev. 49, 537 (1936). 19) H. M.Kmtter, Phg8. Rev. 48,664 (1935). 194 E. Rudberg and J. C. Slater, Phys. Reu. 60, 160 (1936). ISSL.

189L.

182

JOSEPH CALLAWAY

Curie temperature of nickel, but it has been applied to the entire series of transition elements. It is therefore worth emphasizing here that the errors in the basic band calculation resulting from faulty application of the cellular method are so serious that the density of states derived from it has little validity. For example, the fivefold degeneracy of the d bands was not removed at the center of the zone, and this leads to a quite spurious peak in the density of states a t the bottom of the band. Rudberg and Slater extended the calculated density of states to higher energies in a study of the inelastic scattering of electrons by c0pper.1~4 Quantitative calculations of the energy bands in the transition metals are unusually difficult. The problem of attaining self-consistency is present in a very aggravated form. Not only is it more difficult to obtain self-consistency than in the other systems, but the consequences of a lack of self-consistency are more serious. There may be a large number of d electrons present on each atom. An appropriate charge distribution must be assumed from them. There are two basic difficulties: (1) A uniform distribution will, very probably, be far from self-consistent, (2) It is known from calculations of wave functions in free atoms that the wave functions and energies of d electrons are rather sensitive to small changes in the potential. Adequate self-consistent field calculations for free atoms of these elements do not exist. In addition, if application is made to the ferromagnetism problem, self-consistency must also be achieved with respect to the number of electrons of each spin present. Recent d band calculations have not been sufficiently quantitative to lead to a precise theoretical model. However, a number of comments can be made which are reasonably well-founded theoretically, and which point the way to something more sophisticated than the models of Mott, Pauling, and Slater. The extent to which experimental evidence supports these ideas is not completely clear. I n agreement with the early work of Mott, the d band is quite narrow, but not as narrow as he proposed. It would seem that the band width must be greater than 1.5 ev. A figure of about 3 ev seems reasonable. The shape of this band should be in substantial agreement with the normal level order for an isolated d band in the appropriate lattice. In particular there is little evidence for a split of the d band into bonding and antibonding sub-bands as proposed by Bader et al.lgoThe d electrons must be assumed to contribute substantially t o the cohesive energy. The 4p levels appear to lie relatively well above the 3d band so that mixing of p and d levels should not be too important. The band model outlined above is in a sense a synthesis of the results of band calculations which have been performed. Recently Mott and

ELECTRON ENERGY BANDS IN SOLIDS

183

Stevens1s4ahave constructed a model of the d bands in the transition metals according to which there is a very significant difference between the nature of the bands in the closed-packed and in the body-centered transition metals. In the latter, Mott and Stevens propose that the d band is split into two parts, one formed from functions of I'26' symmetry and associated with rather diffuse wave functions, and the other part formed from functions of r12symmetry with compact wave functions. The conventional band structure obtains in the closed-packed metals. In chromium, the r12 band is believed to be above the Fermi surface and is unoccupied; in iron, it contains the two electrons per atom which are responsible for the magnetic properties. Electrons in this band contribute nothing to the conductivity. The conduction electrons are in the r2v band. This model is based on interpretation of experimental data for these metals; in particular, the measurements of R. J. Weiss on x-ray scattering which are discussed in subsequent sections. It is not consistent with existing band calculations or with the normal level order in the body-centered cubic lattice. A few remarks should be made concerning the general applicability of the energy band picture to the d electrons in the transition elements, particularly in regard to the question of ferromagnetism. It is well known that the energy band theory erroneously predicts ferromagnetism in the limit of infinite atomic separation. This is a result of the fact that the energy band theory overestimates the energy of the nonmagnetic state in this limit: polar states of the individual atoms in which some have too many or too few electrons are predicted in large number. The energy of the ferromagnetic state does go to the correct limit as the atomic distance becomes large because the Pauli principle effectively prevents such polar states. The trouble is an aspect of the general problem of treating the electron interaction in more detail than is possible in the Hartree-Fock approximation. The basic difficulty facing the energy band theory in the application to actual ferromagnetic metals is this: if the band is narrow, the neglect of correlations leads to a serious overestimate of the tendency to ferromagnetism; if the band is wide, the promotion energy inhibits ferromagnetism. It is difficult to arrive at a quantitative estimate of the error induced by the neglect of correlations. Van Vleck has proposed abandoning the standard band theory in favor of a generalization of the Heisenberg model in which there is a nonintegral number of spins per atom, and the spins are constantly being redistributed among the lattice He finds that this model may give a lower energy. Callaway has N. F. Mott and K. W. H. Stevens, Phil. Mag. [S] 2, 1364 (1957). J. H. Van Vleck, Revs. Modern Phys. 25, 220 (1957).

1940 106

184

JOSEPH CALLAWAY

made a crude estimate of the effects of correlation in opposing ferromagnetism for iron a t the observed lattice spacing1gswith the result that the tendency toward ferromagnetism calculated by energy band theory should be reduced by about 0.25 ev. A self-consistent energy band calculation for d electrons is required for further progress. 36. EXPERIMENTAL INFORMATION Experimental information pertaining to energy bands in the transition metals is far less precise and detailed than in the case of semiconductors. Much of the existing material pertains to the density of states. Measurements of the low-temperature specific heats of the transition elements have been analyzed by Horowitz and and by Daunt.1vB The electronic specific heat is directly proportional to the density of states on the Fermi surface. Horowitz and Daunt interpret the data in terms of a common density of states for all the transition elements. There is no fundamental justification for such a procedure, but if it is accepted, a pronounced minimum is indicated a t the position of the Fermi surface in chromium, a maximum in the vicinity of vanadium, and a strong maximum toward the end of the series. A high peak may exist in manganese. The temperature variation of the magnetic susceptibility of the paramagnetic transition elements has been discussed by Kreissman and C ~ d l e n . ’The ~ ~ value of the magnetic susceptibility is proportional to the average value of the density of states N ( E ) in an interval of a few kT in width centered a t the Fermi energy. The temperature variation of the susceptibility depends on the first and second derivatives of the density of states, and is usually negative. A positive temperature coefficient can be obtained in the vicinity of a minimum. The data then indicate that the Fermi energy in titanium and chromium comes near a minimum in the density of states. It may be useful to comment in regard to the proposed interpretation of these properties that it is quite legitimate to regard the data as indicating that the density of states in chromium a t the Fermi energy is much smaller than in nickel, for example, a t its Fermi energy, and probably that the Fermi energy in chromium is near a minimum of its density of states. It requires much more faith in the simplicity of d bands than m J. Callaway, PbD. Thesis, Princeton University, 1955 (unpublished). 19’ M. Horowitz and J. G. Daunt, Phys. Rev. 91, 1099 (1953). lBsJ. G. Daunt in “Progress in Low Temperature Physics” (C. J. Gorter, ed.), Vol. 1, p. 202. Interscience, New York, 1955. 1~ C. J. Kreissman and H. B. Callen, Phys. Rev. 94, 837 (1954).

ELECTRON ENERGY BANDS IN SOLIDS

185

experience justifies to believe that the density of states of nickel will show a minimum at a position corresponding to the Fermi level in chromium. The soft x-ray emission spectra of the transition elements have been studied by a number of worker^.^^^-^*^ The emission intensity is proportional to the product of the transition probability and the density of states. The observed emission cannot be interpreted directly in terms of a density of states unless particular assumptions about the transition probabilities are made. A determination of the total band width should be more reliable. Both 4s and 3d electrons may contribute to the observed emission, but it is likely that the d electrons will dominate because they are present in greater number. Unfortunately there are serious disagreements between the results of the various authors, and further work, both experimental and theoretical, would seem to be required. According to the results of Skinner,202the observed band widths are in the vicinity of 6 ev for all the transition metals, but are apparently somewhat greater for chromium, vanadium, and copper than for the others. The band widths reported by Gyorgy and Harvey are in reasonable agreement with those of Skinner except in the case of iron where a smaller band width of 3.7 f 1.0 ev204was reported. Any interpretation of the results in terms of a density of states would be premature, except that it may be noted that there is no indication of detailed structure. Beeman and Friedman have attempted an interpretation of the structure beyond the K absorption edge especially in copper and nicke1206in terms of the density of states calculated by Rudberg and Slater.lg4 Measurements of the Hall effect in the transition elements indicate that conduction occurs predominantly by means of holes in vanadium, chromium, manganese, and iron; by electrons in cobalt and n i ~ k e l . ~ ~ ~ * ~ O ' Since it is unlikely, although not impossible, to find holes in an s band, it is probable that carriers in the d band dominate the conduction process in those elements where hole conduction is observed. The effective masses of carriers in the d band should be substantially higher than of those in the s band, so if there were any appreciable number of s electrons available, the latter would have the predominant effect. One concludes that the number of effective s electrons is quite small. J. Farineau, Ann. phys. 10, 20 (1938). Y. Cauchois, Phit. Mug. [7] 44, 173 (1953). zo* H. W. B. Skinner, T. G. Bullen, and J. E. Johnston,Phil. Mug.[7] 40,1070 (1954). *oa E. M. Gyargy and G. G. Harvey, Phys. Rev. 87, 861 (1952). 204 E. M. Gy6rgy and G. G. Harvey, Phys. Rev. 95,365 (1954). )06 W. W. Beeman and H. Friedman, Phys. Rev. 06,392 (1939). S. Foner and E. M. Pugh, Phy8. Rev. 91,20 (1953). 107 S. Foner, Phys. Rev. 107, 1513 (1957). 901

186

JOSEPH CALLAWAY

Weiss has recently presented measurements on the form factors for x-ray scattering which challenge the generally accepted concepts of the d band in the transition metals in a very serious way.2o8The observed intensity of a Bragg reflection depends on a form factor which is the appropriate Fourier coefficient of the charge density. The form factor is rather difficult t o determine experimentally since corrections must be made to allow for extinction. An experimental form factor can be compared with theory if electron wave functions are available. In computing the form factors, Weiss used wave functions based on a self-consistent field (with exchange) for the core electrons. The remaining charge distribution required to account for the results is a measure of the number of d electrons. The 4s electrons will not contribute to the form factor, since it is not sensitive to a uniform charge distribution. It is assumed that the distribution of d electrons is rather compact. Weiss’ results for the number of d electrons are as follows: Chromium 0.2 & 0.4 Iron 2 . 3 & 0.3 Cobalt 8.4 & 0.3

Nickel 9.5 & 0.3 Copper 9.8 f 0.3

The discrepancy between the results for the body-centered elements, chromium and iron, and the rest which have close-packed structures is not accounted for by any theory. If these results are verified there is apparently a serious disagreement between theory and experiment (see Section 39 for further discussion). We now begin a detailed consideration of individual band calculations.

37. TITANIUM The energy bands in titanium have been studied by Schiff.209,210 Both the hexagonal and the body-centered cubic forms were investigated. A rough semiempirical potential was found by the method of P r o k o f j e ~ ~ ~ to represent the Ti+4ion. The potential of a uniform distribution of three valence electrons was added to this. Only the states at the center of the zone were investigated for the hexagonal form. Later an error was discovered in some of the expansions and the corresponding boundary conditions employed for the hexagonal case. For the body-centered form, the energies of five states at the points r and H were found using the cellular method and the boundary condiR. J. We& and J. De Marco, Rev. Mod. Phys. SO, 59 (1958). B.Schiff, Proc. Phys. Soc. (London)A68, 686 (1955). *lo B. Schiff, Proe. Phys. Soc. (London) A69, 185 (1956);see also S.L. Altmann, Proc. Phys. 8oc. ondo don) A69, 184 (1956). 208

209

ELECTRON ENERGY BANDS IN SOLIDS

187

tions of Howarth and Jones.71 The principal results of this calculation are that the r12d state at the center of the zone lies above the lowest s state I’l, the H12d state a t the zone corner is lower than r12as is required by the normal form of the energy bands in the body-centered lattice, and that one of the lowest p band states, Hl5, lies quite well above the d states investigated. Recently Altmann and Cohan have reinvestigated the states a t the center of the zone in hexagonal titanium.210aThe cellular method was applied. The calculation was made possible by the development of a computer program for the least squares fitting of boundary conditions.210b Calculations were made for each of three potentials: (a) the potential obtained by SchitT,***(b) a potential obtained from a self-consistent field with exchange for Ti+4to which was added the potential of a uniform distribution of three electrons, and (c) the same Ti+* potential as in (b), but with the valence electrons in atomic orbitals for a 3d24s configuration. The most accurate calculations employed potential (a). Eight spherical harmonics were employed in the expansion of the wave function, and boundary conditions were applied a t sixteen point on the surface of the polyhedron. The energies of twelve representations were obtained. The Iowest states were found to be, in order of increasing energy (Herring’s notationes) rl+,rr-,r5+,r6+,and r6-.These states are predominantly s and d states with the s state lowest. The latter three states are doubly degenerate. Comparison of the results obtained with the different potentials showed that the relative position of the four lowest levels in (a) and (b) agreed to within 0.06 rydberg, but that the results of (c) were in serious disagreement with those of (a) and (b). Thus, the sensitivity of the d electron states to the distribution of valence electrons is emphasized. Altmann has also applied the cellular method to a calculation of energy levels at the center of the zone in zirconium. The calculation is similar to that previously discussed for titanium. The potential was obtained from a self-consistent field for Zr+4. The potential of the conduction electrons was assumed to be that of a uniform charge distribution. The order of the levels is the same as found for titanium. There are four mainly d states above the ground state, which is s-like. 38. CHROMIUM No detailed calculations of energy bands in chromium have been published. However, Slater and Koster have applied a tight binding interpolation scheme to this element. They have taken values for the required two center potential integrals from the calculations of Fletcher

L. Altmann and N . V . Cohan, Prac. Phys. Sac. (London) 71, 383 (1958). S.L. Altmann, Proc. Roy. Sac. (London) A244, 141, 153 (1958).

210aS.

*lo*

188

JOSEPH CALLAWAY

and Wohlfarth for nickel.211.212 The distance between nearest neighbor atoms differs by only 0.25% in chromium and nickel at 20°C. Slater and Koster set up the matrix components of energy for the body-centered cubic lattice, inserted the parameters (first neighbor interactions only and neglecting orthogonality corrections) and solved the resulting fifthorder determinantal equation at many points through the Brillouin zone. From these results they computed a density of states by numerical methods. Sufficient care was taken to insure that the density of states would be an accurate expression of the calculated band structure. It is asserted to be characteristic of the body-centered cubic transition metals. Unfortunately, the underlying assumptions are not adequate. The calculated density of states turns out to be symmetric about its midpoint. This is so only in consequence of the small number (three) parameters used in the tight binding approximation. This symmetry definitely would be removed in a more precise calculation. Moreover, the energy bands determined by Slater and Koster disagree very seriously with the normal form of energy bands in the body-centered lattice. They predict, for instance that the separation of the d bands at the center of the zone is about the same as it is at the corner H . Also they find that HW lies below r12.There are other discrepancies as well. It is probable that these results are a consequence of the unjustified use of the parameters determined by an inadequate calculation for nickel for another element. Slater and Koster proceed to discuss the antiferromagnetism of chromium in terms of this model of the band structure. Chromium is believed, as the result of neutron diffraction, to be weakly antiferromagnetic.218 Atoms on each of the two interpenetrating simple cubic lattices presumably have oppositely-directed moments. It is necessary to use a cubic cell containing two atoms in a band calculation for such a system. The Brillouin zone is now a simple cube which has half the volume of the ordinary zone for the body-centered lattice. The potential energy of an electron of upward spin is different on the two atoms in the unit cell. In the simple tight binding approximation employed, such a difference in potential energy would tend to produce a gap in the d bands, coming at the middle of the density of states. As the d band is presumably half full (see, however, ref. 208), the density of states at the Fermi level would be quite low, being determined by the overlapping s band. This model is consistent with the experimental evidence. Slater believes that such a model would also be self-consistent theoretically. It is not clear 211 G. C. Fletcher and E. P. Wohlfarth, Phil. Mag. [?I 42, 106 (1951). *I* *I*

G. C. Fletcher, Ptoc. PhyS. Soc. (Lmrdon) A66, 192 (1952). C. G. Shull and M. F. Wilkinson, Reus. Modern Phys. 26, 100 (1953).

189

ELECTRON ENERGY BAND8 IN SOLIDS

to this author just how much of this viewpoint can be maintained in view of the inadequate band calculation. 39. IRON There have been several calculations of energy bands in ir0n.~14-~18 The cellular calculations of Manning214for body-centered iron and of Greene and Manning for face-centered2I5iron have the defect of inadequate satisfaction of the boundary conditions, in common with most early band studies. This work can hardly be said to be of quantitative value. Callaway has made a study of energy bands in bcc iron using the method of orthogonalized plane waves, and has applied the results to a calculation of the tendency to ferromagnetism. 216 The original potential was taken from a self-consistent field without exchange for the free iron atom. A configuration d6s2 for an iron atom waS assumed to exist in the solid initially. Exchange interactions were included on the basis of $later’s free electron approximation12 at first. Subsequently, a correction to this was calculated from the Hartree-Fock equations. The results showed that the initial potential was not self-consistent in the sense that the d band was predicted to be quite far below the s band. Thus a configuration d8s0was predicted in place of the d6s2 assumed. This result is of some interest in view of the experiment of Weiss208 concerning the effective number of d electrons in iron. Although no band calculations have been made on the basis of a crystal potential which is consistent with the work of Weiss, it would appear that a charge distribution substantially more uniform than of atomic iron cannot be selfconsistent. If the charge distribution conforms to the Weiss model, the potential at reasonable distances from a nucleus (from to 8 of the distance to the atomic cell boundary) will be substantially more attractive than in the free atom because of the reduced screening. This kind of attractive potential will tend to bind the d electrons quite strongly and should result in a d band which is well below all the s-p levels. These d functions will be even more tightly bound than the atomic functions. Consequently, if the results of Weiss are correct, a fundamental disagreement between theory and experiment apparently exists. On the basis of Callaway’s calculation, the shape of the d band con-

+

M. F. Manning, Phys. Rw. 88, 190 (1943). J. B. Greene and M. F. Manning, Phys. Rev. 88, 203 (1943). 316 J. Callaway, Phys. R w . 99, 500 (1955). *nF. Stem,Ph.D. Thesis, Princeton University, 1955 (unpublished). M.$&%yneki, Ac& phy.9. P o h . 14, 493 (1955); 15, 111 (1956). a14 116

190

JOSEPH CALLAWAY

forms fairly well to the normal order of levels for the body-centered lattice. In particular the separation of the d band levels at the center of the zone is much less than at the corners and faces of the zone. The width of the d band is approximately 2 ev. I n view of the fact that the assumed potential probably binds the d electrons too strongly, the calculated band width should be regarded as a lower limit. The order of the levels at r, H,and N is shown in Fig. 17. -.550

-

m.600 a a W m P

&650

Y

> a W 2300 W 0

t750 ~

H

FIG 17. Level diagram for iron according to Callaway.*l6The N 1levels were obtained using an interpolation scheme and are less reliably placed. A lower limit on the band width is of interest in connection with Zener’s theory of the origin of ferromagneti~m.~~~ Zener described the 4s electrons in the transition elements by band wave functions and the 3d electrons by localized atomic functions. He assumed that there is one 4s electron per atom. He calculated the decrease in energy obtained by aligning the spin of the d shells and reversing the spins of some of the conduction electrons so as to be parallel to the d electrons. Thus ferromagnetism is assumed to arises from the s-d exchange coupling. The decrease in energy upon alignment is estimated to be about 0.05 ev per atom in iron if the direct exchange interaction between the d shells on adjacent atoms is neglected. One difficulty with this theory is that the finite width of the d band is neglected. With a band about 2 volts wide, 110

C. Zener, P h p . Rev. 81, 440 (1951);83, 299 (1951).

ELECTRON ENERGY BANDS IN SOLIDS

191

the increase in the Fermi kinetic energy is far too large to permit the exchange coupling between the s-d electrons to cause ferromagnetism by itself. The promotion energy was estimated to be about 0.7 ev per atom on the basis of the narrow band obtained in the work of ref. 216. A crystal potential calculated on the basis of a d7s configuration was also investigated. The difference between this potential, having exchange integrals calculated from the Hartree-Fock equations, and the one used previously was computed and treated by first-order perturbation theory. The bottom of the s band was found to be below the d band, so that the calculation is more nearly self-consistent. The splitting of the d and s bands when the spin of one d electron per atom was reversed in order to produce a magnetic state was computed using two sets of d band wave functions. The splitting was found to be between 2.46 and 3.16 ev for the d band. The tendency to ferromagnetism was calculated to be between 0.89 and 0.54 ev per atom. This result, which is much too large, confirms the view that the calculated band width is too small. The low 4p level H16was found to lie quite far (about 0.7 ry) above the d hand. This result suggests that 4p functions do not play an important role in the band structure. Suffczynski has calculated the two-center potential integrals for iron218in the tight binding approximation, using wave functions with exchange and a simple screened Coulomb form for the potential. First and second nearest neighbors were considered. The overlap integrals of wave functions on different atoms were not calculated. The band structure implied by these integrals has not been computed in detail. Stern has attempted a computation of the cohesive energy of iron.*17 His method is a modification of the tight binding approximation. A preliminary calculation using wave functions and potentials based on the configuration dss2 gave results in agreement with those of Callaway for this configuration: namely a very narrow d band which lies entirely below the 4s band. Stern then calculated self-consistent wave functions without exchange for higher configurations of atomic iron: d7s1and d8.220 The energy of the 3d electrons is raised quite substantially in going from the des2 configuration to the d7s1and ds configurations. The eigenvalues are -0.79 ry, -0.50 ry, and -0.34 ry, respectively. The wave functions become more diffuse in the higher energy configurations; such functions yield wider bands. A calculation based on wave functions from the d8 configuration gave a band width of about 0.5ry, which is probably too large, and placed the bottom of the 4s level below the 3d band. Stern believes that an approximately self-consistent configuration in the solid would be d7Js0.8.The order of levels calculated by Stern for the 3d band **O

F. Stern, Phys. Rev. 104,684 (1956).

192

JOSEPH CALLAWAY

is in good agreement with the normal order for the body-centered-cubic lattice. Evidence concerning the wave functions of magnetically active electrons in iron comes from experiments on the polarization of slow neutrons. These experiments have been interpreted by Steinberger and Wick.221 The experiments involve the standard form factor for the distribution of magnetically active electrons. This form factor was calculated by Steinberger and Wick on the basis of a d7s1 configuration. Exchange integrals from the Hartree-Fock equations were included. Although the boundary conditions used are not appropriate for d electron wave functions in solids, the results indicate that the wave function for these electrons must be rather more compact than those given by the Hartree field without exchange for the free atom. The d electron wave function obtained by Callaway also exhibits this compactness. The alignment of the 3d electrons tends to polarize the 4s band through the s-d exchange coupling. The amount. of this polarization has been calculated by Callaway216and by Pratt.222Callaway obtained a magnetization of the s band of 0.07 P B per atom, and Pratt, by a more detailed calculation, found about 0.2 C(B per atom. The difference is principally a result of the larger number of s electrons (one per atom) assumed by Pratt. It is worth remarking that recent work, which explains the negligible polarization of the s band in dilute Cu-Mn does not require that this polarization be absent in a ferromagnetic metal in which long-range magnetic order prevails. A substantial screening of the s-d exchange interaction does not seem to be required either.

40. NICKEL The energy bands in nickel have been studied by Slater192 and by Fletcher and Wohlfarth.211.212 Slater obtained a density of states for nickel from Krutter's calculation for copper,lgaas has been discussed in Section 35. He proceeded to estimate the tendency to ferromagnetism of nickel. He obtained empirical values for the appropriate exchange integrals from an analysis of the atomic spectra. This work led to a value of 0.99 ev per atom for the decrease in energy as a result of the increased exchange when spins are aligned. This result is lower than the value ranging from 1.23 to 1.58 ev obtained for iron in the work of ref. 216. The difference originates in the use of a different method of averaging the exchange interactions and of slightly different values for the appropriate integrals. Slater calculated the tendency to ferromagnetism and obtained reasonable 1%'

J. Steinberger and G. C. Wick, Phys. Rev. 76, 994 (1949).

*** G. Pratt, Phys. Rev. 106, 53 (1957). *** J. Owen, M.Browne, W. D. Knight, and C. Kittel, Phys. Rev. 102, 1501 (1956). ss4E. W. Hart, Phys. Rev. 106, 467 (1957). 116 K . Yosida, Phys. Reu. 106, 893 (1957).

ELECTRON ENERGY BANDS I N SOLIDS

193

agreement with the observed Curie temperature. It is likely, however, that this good agreement is related to use of too great a band width, and that a more accurate calculation would give a larger tendency to f erromagnetism. The tight binding calculations of Fletcherz1*were made with the use of wave functions for the Cu+ ion. The potential was taken as a simple screened Coulomb interaction. Potential integrals were calculated with first neighbors only. Overlap integrals involving wave functions on different atoms were not included. The band width obtained was 2.7 ev. A curve giving the density of states was computed graphically for the upper part of the band and good agreement with experiment was obtained for the electronic specific heat. It is interesting that the computed band width is about half that used by Slater.lgaFletcher has not classified the bands according to the notation of Bouckaert et at.,Is but it can be deduced from the degeneracies and from the compatibility relations that the results are in reasonable agreement with the normal level order for the face-centered cubic lattice. In particular, the separation of the d levels at the center of the zone is considerably less than on the faces of thezone. The calcuhtion of Fletcher cannot be regarded to be quantitative. The particularly serious approximations are the neglect of mixing of s and d bands, the use of the screened Coulomb potential, and the neglect of overlap and second-neighbor potential integrals. The neglect of s-d mixing may lead to erroneous band shapes. Koster used the integrals caIculated by Fletcher to determine numerically a density of states for nickel which covers the entire band.22s Energy levels were found at 680 points within one forty-eighth of the Brillouin zone. The density of states is, no doubt, an accurate representation of the Fletcher-Wohlfarth band structure. The over-all accuracy depends on the validity of the original calculation, which is somewhat in doubt. It is interesting that the density of states shows the characteristic dip which is approximately in the middle of the band, discussed in Section 35. 41. COPPER

There have been several studies of energy levels in ~opper.19***~~-288 The cohesive energy of copper has been calculated by F u ~ h s * ~and 7 by G. F. Roster, Phy8. Rw. 98, 901 (1955). K.Fuche, Proc. Roy. Soc. (Landon)A l l l , 585 (1935); AlSS, 622 (1936). *** M.I. Chodorow, Phys. Rev. 66, 675 (1939). r)o S. R. Tibbs, Proc. Cambridge Phil. Soc. 84, 89 (1938). *so D. J. Howarth, Proc. Roy. Soc. (London)A220, 513 (1953). K. Kambe, Phy8. Rev. 99, 419 (1955). D. J . Howarth, Phy8. Rev. 99,469 (1955). la*M. E’ukuehi, Prop. Themet. Phys. (Kyoto) 16,222 (1956). )*I

*s2

194

JOSEPH CALLAWAY

Kambe.2a1The early calculation of Fuchs employed the cellular method in a manner quite analogous to the original calculation of Wigner and Seitz for sodium’? except that the potential was taken from a selfconsistent field suplemented by an exchange interaction. The effective mass was assumed to be unity, and the interactions of the valence electrons were neglected. In order to obtain reasonable values for the compressibility, Fuchs found it necessary to take account of the repulsive energy between the ions, which he estimated by a statistical method. In this way he obtained a cohesive energy of 34 kcal/mole and a lattice constant of 4.2 A in comparison with the experimental values of 81 kcal/ mole and 3.6 A. Kambe used the quantum defect method4? to calculate the cohesive energy of copper. Because the atomic cell is small, the potential in the vicinity of a cell boundary is not completely hydrogenic. Corrections for departure of the potential from the simple Coulomb form at the cell boundary were made on the basis of a self-consistent field. The quantum defects were somewhat more difficult to extrapolate than in the case of the alkali metals because of the presence of configuration interaction in the atomic spectra. Neglecting the interactions between the ions, Kambe obtained a cohesive energy of 62.4 kcal/mole at the observed lattice spacing.234The effective mass at the bottom of the band is m/mo = 1/Ez = 1.01. A part of the discrepancy between theory and experiment in regard to the cohesive energy may be associated with the neglect of k4 terms in the Fermi energy, but most is probably the result of rather strong van der Waals attractions between the ions.66.236 These interactions contribute perhaps 25 kcal/mole to the cohesive energy. Their strength can be related to the optical absorption. The short-range exchange repulsion of the ions is also important, particularly in the calculation of the lattice constant and the compressibility. There should also be a correction for the Coulomb interaction with the conduction electrons of the slightly interpenetrating d shells.236 Howarth’s two calculations are concerned with the 3d band and higher levels in the conduction band.230.2a2 The first calculation employed the cellular method. Boundary conditions were determined and satisfied in much the same way as in the work of Howarth and Jones.’’ The second calculation employed the augmented plane wave method of Saff ern and Slater.2a7Two potentials were used for the conduction band in each Kambe’s published result has been modified by using the corrected form of Wigner’s correlation formula. The correction amounts to 3.1 kcal/mole. 285 J. Friedel, Proc. Phys. SOC. (London)BBS, 769 (1952). 2a6 Y . C. Hsu, Phys. Rat. 89, 975 (1951). 237 M. M.Saffern and J. C. Slater, Phys. Rat. 92, 1126 (1953). 284

ELECTRON ENERGY BANDS IN SOLIDS

195

calculation: One was based on a self-consistent field in which exchange was excluded and the other on a self-consistent field which included exchange. Exchange interactions between valence and core electrons were not taken into account. The contribution of one 4s electron was added to the potential in calculating the 3d band whereas that of one 3d electron was subtracted. In using the augmented plane wave method, it is necessary to assume that the potential is constant outside an inscribed sphere. The results of the cellular method and the augmented plane wave calculation differ very substantially. For example, the order of the levels a t the center of the zone is reversed. It is improbable that all the difference can be ascribed to the assumption that the potential is constant outside the inscribed sphere. Since the order of levels in the d band obtained from the cellular calculation agrees quite well with the normal order in the face-centered lattice, whereas the results obtained from the augmented plane wave treatment differ violently, it seems likely that a mistake exists somewhere in the latter. In consequence, only the results of the cellular calculation will be reported. TABLEXIII. EIGENVALUES OF THE d BANDFOR COPPER ACCORDING TO HOWARTA (IN RYDBERGS) CELLULAR CALCULATION Xs

L1

rls* rl,

-0.344 -0.248 -0.235 -0.195

Xs

XI Xs

-0.13 -0.12 -0.088

TABLEXIV. EIGENVALUES OF THE CONDUCTION BANDFOR COPPER (IN RYDBERGS)

rl

L$ XI, L1 KI XI

Howarth*s"

Fukuchi

-0.596 0.18 0.20 0.055

-1.282 -0.696 -0.499 -0.393 -0.344, -0.260 -0.108

0.24

The d band eigenvalues are given in Table XIII.2aoThe width of the d band is 3.46 ev. In Table XIV, the results for the conduction band of copper are given both in accordance with the Hartree-Fock potential used by H o ~ a r t h , ~and ~ O as computed by Fukuchi.2aaThe results are valid for the observed lattice constant. Since different potentials were used in Howarth's calculation for the s and d bands, a separate calculaition is required to relate the eigenvalues. The result 'placecthe top of

196

JOSEPH CALLAWAY

the 3d band 3.7 ev below the Fermi level in the 4s band (the latter is 7.1 ev above the bottom of the 4s band). The bottom of the d band ( X , ) is 0.1 ev below the bottom of the 4s band (rl). Fukuchi has made an OPW calculation of the conduction band of copper."* His potential differs from the Hartree-Fock potential used by Howarth by the addition of an exchange interaction obtained with the use of Slater's free electron average12and of the contribution of a uniform charge density of conduction electrons. It is not clear that Fukuchi found the appropriate core functions for this potential. It is also not clear that enough plane waves were included to obtain convergence for all the states. The effective mass appears to be rather less than 1; however, the question of convergence is important here. Mott has proposed that states at the top of the first Brillouin zone in This is suggested by the optical properties (the color copper are ~-1ike.6~ presumably is a result of transitions between the 3d band and the conduction band), and by the x-ray emission spectrum. The states a t the bottom of the second Brillouin zone would then be s-like. This conjecture is supported by the calculation of Fukuchi, but not by that of Howarth. The effective mass on the Fermi surface in copper has been determined to be m/mo = 1.45 from optical measurements by Schulz.2as Since the effective mass at the bottom of the band is very close to 1.0, the result suggests that substantial negative k4 terms are present in E(k). The situation is somewhat similar to that prevailing in the alkali metals. The electronic specific heat of copper has also been me&sured.2aB The effective mass determined from these measurements is 1.38, in reasonable agreement with the optical results. Pippard has made a detailed study of the anomalous skin effect in single crystals of copper and has constructed a model of the Fermi surface which agrees with his The surface constructed differs from spherical in that it is extended in the direction of L (refer to Fig. 2) ; is depressed a bit at K where the curvature is anisotropic and almost zero along the line KL;has a rudimentary ridge along the line XL with a saddle point where the curvature becomes slightly concave. The surface is approximately described by the equation. kz2

+ ku2 + kz2 - akz2kV2kz2 + b(kz2ku2+ kU2kz2+ kz2kz2) = 1

with constants a = 5.6, b = 0.3. It is probable that the surface touches the zone face a t L. The electron specific heat calculated with this model

*** L. G. Schulz, Advancea in Phys. 6, 101 (1957).

*uW. S. Corak, M. P. Garfunkel, C. B. Satterthwaite, and A. Wexler, Phys. Rev. 98, 1699 (1955). Read&are given in terms of an effective mass by Schdz.*a* *40 A. B. pippard, Phil. Tram. Roy. Soc. AIM), 325 (1957).

ELECTRON ENERGY BANDS IN SOLiDd

i97

of the Fermi surface is characterized by a thermal effective mass 1.87, which is significantly larger than the experimental result.2agIt is possible that the discrepancy could be removed by a correlation correction to the density of states as suggested by Pines.*6Further indications of significant deviations of the Fermi surface from spherical form originate in measurements of the specific heat of a-brasses a t low temperatures.241Rayne noted a rapid variation in y, the coefficient of the linear term in the lowtemperature specific heat, with the concentration of zinc at low concentrations. The sharp initial rise is not consistent with a parabolic form for the density of states in copper. The measurements of Rayne can be interpreted qualitatively in terms of a crude model of the density of states in copper proposed by Jones.242The approach of the Fermi surface to the boundary of the Brillouin zone in the vicinity of the point L causes a sharp increase of the density of states in the vicinity of one s electron per atom, which is then followed by a linear decrease. The thermoelectric power of the noble metals can also be explained in terms of this model.24aHowever, no indication of a maximum in the curve for the density of states just beyond the Fermi surface for copper have been revealed from measurements of the magnetic susceptibilities of a-bras~es.~'~ VI. Other Substances

42. SILVER AND GOLD The cohesive energies of silver and gold have been studied recently by Kambe.lal These metals have the face-centered cubic structure. The quantum defect method was employed in the calculation, as in the case of copper (Section 41). The corrections for departures of the potential from a simple Coulomb form near the cell boundary were determined from the results for copper, and may be somewhat less accurate. For silver and gold, Kambe obtained a cohesive energy of 59.0 and 52.2 kcall mole respectively at the observed lattice constant (r, = 2.99 in each case). The experimental values are 68.0 and 92.0 kcal/mole. The calculated effective masses a t the bottom of the bands are 0.992 and 0.994, respectively. The principal discrepancies between theory and experiment are presumably due to the van der Waals attractions of the ions, but the other corrections mentioned in Section 41 may also be important. As in other Tibbs has calculated energy bands in copper and ~ilver.~~V J. A. Rayne, Phy8. Rev. 108, 22 (1957). H. Jones, Proc. Phys. Soc. (London)49,250 (1937). *aH. Jones, PYOC. phy8. Soc. (London)AM,1191 (1955). 244 W. G. Henry and J. L. Rogers, Phil. Mug. [S] 1, 237 (1956).

241 242

198

JOSEPH CALLAWAY

early cellular method calculations, the boundary conditions are satisfied so inadequately that the results are not quantitative. Effective masses of electrons on the Fermi surface have been determined by optical absorptionz3* and thermal measurements (electron specific heat). The results are given in Table XV. TABLEXV

Metal

Optical m/mo

Thermal m/mo

Ag

0.97 0.98

0.96 1.16

AU

The discrepancy between the optical and thermal effective masses for gold is probably outside the experimental error. If true, it implies a rather warped Fermi surface. The agreement between the results for silver and that computed for the bottom of the band may possibly imply absence of k4 terms in the band structure, but this seems somewhat unlikely in view of indications that such terms are important in copper and gold. There would seem to be an opportunity here for more detailed band calculations to determine whether the Fermi surface is distorted from spherical symmetry and whether it actually does touch the Brillouin zone. In the absence of self-consistent fields with exchange for free atoms of silver and gold, the quantum defect method could be extended t o the higher states. Further theoretical studies are important because these metals are easy to work with experimentally.

43. TUNGSTEN Energy bands in metallic tungsten (bcc) have been studied by Manning and C h o d o r ~ wThe . ~ ~calculation ~ involved the cellular method and was based on a potential derived from a self-consistent field. The same comment applies to this calculation as to many other prewar cellular method calculations : the boundary conditions are satisfied sufficiently inadequately that the results have no quantitative value. The shape of the d band and the density of states are probably quite erroneous. The cohesive energy of tungsten has been studied by HsuZa6on the basis of Zener’s theories.21g 44. BISMUTH

Bismuth has a rhombohedra1 structure. The structure is perhaps thought of most easily as a distorted simple cubic lattice. A simple cubic 146

M. F. Manning and M. I. Chodorow, Phys. Rar. 66,787 (1939).

ELECTRON ENERGY BANDS IN SOLIDS

199

lattice is composed of two interpenetrating face-centered cubic structures. If one of the component face-centered lattices is displaced with the atoms moving away from their normal places, the bismuth structure is produced. Since the free atom contains five electrons, the Brillouin zone shown in Fig. 18, which holds five electrons per atom, is often It is not the smallest possible zone: the reduced zone resembles that for the face-centered cubic lattice, Fig. 2, quite closely except that it is skewed somewhat, the three A axes being not quite perpendi~ular.~~7

f I

/

/

A.

7

FIQ.18. Brillouin zone for bismuth according to Jones.246 This zone can contain five electrons per atom. Overlapping occurs across the faces of the zone.

Bismuth is a semimetal. The bands can be separated into valence and conduction bands which overlap very slightly. As in the case of graphite, small numbers of electrons and holes will be present at absolute zero. No detailed calculation of energy bands in bismuth has been made. Morita considered the bands in the tight binding approximation including only nearest neighbor potential integrals and neglecting overlap integrals. 247 S. Mase also applied the tight binding approximation to His work differs from that of Morita in that spin orbit coupling is included. Mase constructed character tables for the double group of H. Jones, Proc. Roy. SOC.(London) A141, 396 (1934). A. Morita, Sci. Repts. TBhku Univ. First Ser. 38, 144 (1949). t470 s. Maw,J . Phys. SOC.(Japan) 13, 434 (1958).

200

JOSEPH CALLAWAY

bismuth. He considers bismuth to be in the configuration s2pa,and includes only the p electrons. The two-center approximation is made and nearest neighbors alone are included. With these approximations, a quantitative calculation is impossible, but qualitative aspects may be correct. The slight overlapping of the bands containing small numbers of free electrons and holes characterized by small effective masses can be regarded as due to the removal of degeneracies which would be present in a cubic structure. The effect of this rhombohedra1 distortion is small compared to the spin-orbit coupling. The interesting points are those corresponding to L in the Brillouin zone of the face-centered cubic structure. In bismuth, however, not all these points are equivalent. There are two types: (A and C in the notation of Mase). Levels at these points will be slightly displaced with respect to each other: also, there is a small splitting of levels a t these points which would otherwise be degenerate. These ideas may also be applicable t o antimony and arsenic. A large amount of experimental information has accumulated on energy bands in bismuth. Measurements of the De Haas-van Alphen effect, cyclotron resonance, and galvanomagnetic effects can be interpreted in terms of a simple model of the energy surfaces originally proposed by S h ~ e n b e r gA. ~few ~ ~ papers are listed belo~.249-~5” Aubrey and Chambers interpret the data in the following way.268a The electrons occupy three ellipsoids in k space. If k, and k, are chosen along the binary and trigonal axes, one ellipsoid can be represented as

E = alkr2

+ u2kw2+ aYkz2+ 2a4k,ks.

The two other ellipsoids can be obtained by rotations through %/3 about the trigonal axis. The coefficients ai are: a1 = 168, uz = 2.0, = 100, u4 = - 10. Referred to the principal axes, the effective masses are 0.0060, 0.330, and 0.0101. Holes occupy a single ellipsoid:

where bl = 0.71 and ba = 0.19. The overlap between the bands of holes and electrons is EI = 0.0184 ev. The Fermi level is at Ep = 0.017, ev D. Shoenberg, Proc. Roy. Soc. (Lon&on)A170,341 (1939). V. Heine, Proc. Phys. Soc. (London)A69,513 (1956). *so B. Abeles and 8. Meiboom, Phys. Rev. 101, 544 (1956). *I* M. Tinkham, Phye. Rev. 101,902 (1956). 161 B. Lax,K.J. Button, H. J. Zeiger, and L. M. Roth, Phys. Rev. 102, 715 (1956). *I* J. S. Dhillon and D. Shoenberg, Phil. Tram. Roy. Soc. A248, 1 (1955). *ma J. &. Aubrey and R. 4 Chambers, J . Phyu. C h m . SoZidu 8, 128 (1957).

848

249

ELECTRON ENERQY BANDS I N SOLIDS

201

a t 4°K.At that temperature there are 0.86 X hole or electron per atom present. The ratio of the density of states for electrons to that of holes is 1 :27. 45. SELENIUM AND TELLURIUM Selenium and tellurium are semiconductors with unusual crystal structure. A spiral chain which runs along the c axis of the crystal is located at the center and each of the six corners of a hexagon. The interatbmic spacing along a chain is much smaller than the separation between chains. Each atom forms covalent bands with its two neighbors in the chain; the chains are probably held together by van der Waals forces. The bond angle in a chain is close to 90" (actually somewhat larger). Studies of energy bands in these elements have been made by Callen264 and by Reitz.2ss Callen replaced the actual hexagonal structure by a related tetragonal structure with simpler symmetry properties. Reitz asserts, however, that this model will yield metallic conductivity. A group theoretical analysis of the actual selenium structure has been given by Asendorf.266 Reitz constructed a model in which only the atoms along a single chain are considered, somewhat analogous to the layer model of graphite. He has applied the tight binding approximation, including only nearest neighbor potential integrals with atoms along the chain. The band structure is given in terms of these integrals as undetermined parameters [ ( p p )and (ppu) in the notation of Slater and K ~ s t e r l ~Overlap ~]. integrals of wave functions on different atoms are ignored. Spin orbit coupling is neglected. This approximation may be quite inappropriate for tellurium. As a first approximation, a model in which the bond angle is 90" was studied. In this simple model, the bands are triply degenerate. Increasing the bond angle removes the degeneracy. The p bands split into three groups separated by gaps, the lower two being occupied normally. The d bands were also considered. Matrix elements for optical transitions were determined. The absorption limit for direct transactions at long wavelengths occurs at k, = u/c and depends on the polarization of the light parallel or perpendicular to the c axis of the crystal. From the order of the levels, it is predicted that the absorption edge for light polarized perpendicularly to the c axis is at longer wavelength (independently of whether the transitions are p-p or p-d), in agreement with experiments for tellurium.2b1Increasing the bond angle decreases the energy gap.

H.B. Callen, J . C h . Phy8. 22, 518 (1954). J. R. Reite, Phys. Rev. 106, 1233 (1957). 166 R. H.Aaendorf, J . C h . Phys. 27, 11 (1957). 157 J. J. Loferski, Phys. Rev. BS, 707 (1954). *64

166

202

JOSEPH CALLAWAY

Since such a decrease occurs in tellurium, the results are capable of explaining the change in the band gap under pressure. It is quite likely, of course, that more elaborate calculations will be required to interpret experiments in a quantitative way. The optical band gaps reported by LoferskizS7are 0.32 ev for light polarized perpendicularly to the c axis, and 0.37 ev for parallel polarized light. The transmission beyond the edge also depends on the state of polarization. Choyke and Patrick have measured the absorption of light in selenium.268They find a n energy gap of 1.79 k 0.01 ev at room temperature and a large decrease (9 X 15-4 ev/”K) of the gap with temperature. They also interpret the absorption near the edge as the result of indirect transitions, which is not predicted in the calculation of Reitz.266 46. ALKALIHALIDES:SODIUM CHLORIDE

The alkali halides have been studied intensively for many years. The information pertaining to energy bands in these substances is, however, relatively meager. Much attention has been given to the problems of determining the nature of classes of imperfections in these crystals, and to understanding the effects of the strong coupling between electrons and the vibrating lattice. Such subjects are, however, outside the scope of this review. The binding energy of the alkali halides is principally a result of the electrostatic interaction of the ions. A reasonably satisfactory account of this energy can be given on the basis of a simple semiclassical theory which is described in standard texts. Quantum-mechanical treatments of the cohesive energies have been given by Landshoff 259 and Liiwdin.260 The calculations of Lowdin are the most recent and will be discussed briefly here. Since the individual ions have closed shells, the HeitlerLondon and Bloch wave functions are equivalent except for normalization.261 No attempt is made to calculate the energy bands. The cohesive energy is derived by calculating the expectation value of the Hamiltonian relative to the theoretical energies of the ions in the Hartree-Fock approximation. Lijwdin determined the correction to the atomic wave functions so that functions on different atoms are orthogonal. The cohesive energy can be broken into three parts: the electrostatic energy W. J. Choyke and L. Patrick, Phys. Rev. 108, 25 (1957). R.Landshoff, 2.Phyaik 102, 201 (1936); Phys. Rev. 62, 246 (1937). 280 P. 0. LGwdin, Arkiu Mat. Astron. Fysik SbA, Nos. 9, 30 (1947); see also Advuncea in Phys. 6, 1 (1956). 261 See F. Seitz, “The Modern Theory of Solids,” p. 301. McGraw-Hill, New York, 1940. 268 269

ELECTRON ENERGY BANDS IN SOLIDS

203

of the undistorted charge distribution, the exchange energy of the undistorted wave functions, and the S-energy, which contains all the effects of distortion of the wave functions through overlap. The latter contribution gives rise to the repulsive energy between the ions, which could not be treated from first principles in the classical theory. Only the interaction between nearest neighbors is considered, Wave functions were obtained from self-consistent fields including exchange. Inclusion of exchange in treating the ionic wave functions is important because the functions which do not include exchange are too diffuse, and therefore lead to an overestimate of the repulsive energy. The agreement between theory and experiment is quite good, especially for NaCl. The calculated cohesive energy and lattice parameters for the substance are 183.2 kcal/mole and 5.50 A; the empirical values are 182.8 kcal/moIe and 5.58 A, respectively. The agreement with experiment generally is good for the chlorides LiCl and KCL; the agreement is reasonable for the fluorides LiF and NaF, but is not quite so close. There seems to be a general tendency for the theoretical calculations to overestimate the compressibility. There have been some calculations of energy bands in NaC1.262-264 The calculations of Shockley and Casella are concerned with the highest filled bands which are formed from the 3p wave functions on the C1- ions. Shock1ey26*chose a crystal potential in the following way. An effective charge distribution was taken from a self-consistent field without exchange for C1- which was normalized so that all eight valence electrons were contained in a sphere whose volume is equal to the difference between the volume of the unit cell in NaCl and the volume of a spherical Na+ ion. The potential of seven such electrons produces the field acting on any one. The field of a Na+ ion was assumed to be the same as that in sodium metal. A constant potential (the Madelung potential) was added to represent the fact that the lattice is not neutral. In addition, the potential on a sodium ion was modified by allowing one electron to move off into the six surrounding chlorine ions. Various methods of applying the boundary conditions were investigated. None of the methods used is likely to be very accurate much aa in the case of many of the early calculations using the cellular method. The band form appears to be in agreement with the normal level order in the facecentered cubic lattice if boundary conditions are satisfied under conditions in which the sodium atoms are neglected entirely, or if both sodium and chlorine atoms are included in the boundary conditions. The band

*(* W. Shockley, Phys. Rev. 60, 754 (1936).

S. R. Tibbs, Trans. Faraday SOC.56, 1471 (1939). Is4R. C. Gasella, Phys. Rev. 104, 1260 (1956).

204

JOSEPH CALLAWAY

width in the latter, more accurate, case is about four volts. This is likely to be an overestimate because exchange was not included and the wave functions are, in consequence, too diffuse. C a ~ e l l has a~~ estimated ~ the minimum width of the Cl- 3p band, using the tight binding approximation. The sodium atoms were neglected. Wave functions were taken from a self-consistent field with exchange for neutral argon. Such wave functions are likely to be too compact,

X FIG.19. Qualitative sketch of the chlorine 3 p band in NaCl according to Shockley:*(*and Casella.*"

because of the greater atomic number, so that the band width obtained should be a lower limit. The ionic potential was also taken from the selfconsistent field. Exchange was added with the use of Slater's free electron approximation.12 This method should lead to an overestimate of the binding and to a lower limit for the band width. Potential integrals which included only nearest neighboring chlorine atoms were considered. Overlap integrals were neglected. The band width was found to be about 1 ev; the computed bands agree with the normal level order. The width is estimated to be accurate within 30%. The qualitative form of the bands scheme agrees with that of Shockley and is probably correct. It is illustrated in Fig. 19. Tibbs tried to estimate the effective mass of the electrons in the con-

ELECTRON ENERGY BANDS I N SOLIDS

205

duction band in sodium chloride.260 Proceeding in a way similar to Shockley, he calculated conduction band levels at I’ and X. From the difference of these, he obtained a band width and an effective mass close to unity. The validity of this calculation is open to criticism on several grounds. Any comparison with experiment would have to include the effect of the interaction between electrons and lattice vibrations. 47. POTASSIUM CHLORIDE

Liiwdin and Howland have studied the lattice energy and energy bands in potassium ~hloride.266,~66 Howland’s calculation of the lattice energy is similar to that of Lijwdin260except that the interactions between nearest neighboring chlorine ions, which are second neighbors in the lattice, were included. The result of including the second neighboring chlorine ions is to increase the repulsion due to overlap substantially, so that the cohesive energy is lower than that of Liiwdin, but is in reasonable agreement with experiment. The overlap energy amounts to about 15% of the cohesive energy. The lattice energy calculated by Howland is slightly too high, that is, the binding is too small. The discrepancy is ascribed to correlation between electrons on different ions. In the band calculation, an eighth-order secular equation containing the interaction between 3 s and 3P levels on both the potassium and chlorine ions was solved. Chlorine-chlorine overlap and potential integrals were included as mentioned above. Energy values were obtained for twenty-two independent values of k. The 3 p band of chlorine has a width of 1.52 electron volts. This is considerably lower than the result obtained by Shockley for NaCl, but is in reasonable agreement with the lower limit obtained by Casella. The lower bands are narrower and well separated. Measurements of the K x-ray absorption and the soft x-ray emission have been reported and discussed by Parratt and There are several possible interpretations of these experiments, but it seems quite likely that the chlorine 3P band is much narrower than the 4 ev obtained by Shockley for NaCl. The band width would seem to be in the vicinity of 1 or 2 ev, in reasonable agreement with the calculations of C i ~ e l l aand ~ ~ Ho~land.~66 ~ The gap between valence and conduction bands is estimated to be about 9.5 ev on the basis of ultraviolet absorption.26R ZssL. P.Howland, Phys. Rat. 109, 1927 (1958). 966L. G. Parratt and E. L. Jossem, Phys. Rat. 97, 916 (1955). 167 L. G. Parratt and E. L. Jossem, J . Phys. Chem. Solids 2, 67 (1957).

*(* N. F. Mott and R. W.

Gunney, “Electronic Processes in Ionic Crystals.” Oxford University Press, London & New York, 1948, quoted by Parratt and Jossem.*6@

206

JOSEPH CALLAWAY

48. LITHIUMHYDRIDE AND LITHIUMFLUORIDE

The cohesive energies of these (fcc) crystals have been studied by Lowdin and L ~ n d q v i s t ~ 6according ~ . * ~ ~ to the procedures mentioned in Section 46. The agreement with experiment is quite reasonable, but there are unsettled theoretical problems regarding LiH. Further information may be found in the review by Lowdin.2uoA calculation has recently been made by Morita and T a k a h a ~ h i . ~ ~ ~ ~ Ewing and Seitz have calculated energy bands in these sub~tances.~~O A form of self-consistency was obtained with respect to the states at the center of the zone. The calculations were continued until the charge distribution of the valence electrons computed from the wave functions of occupied states at the center of the zone was the same as that used in computing them. The distribution of the 1s electrons in the fluorine ion was taken from a self-consistent field without exchange. The ionic potential previously computed by Seitz4I was used for lithium. Correction for the Madelung potential was made in a way depending on the charge in each cell. Exchange interactions were not included in these calculations. The wave equation was solved with the aid of the cellular method. Unlike many other cellular method calculations of the prewar period, proper account was taken of crystal symmetry in the choice of spherical harmonics for the expansion of the wave function and in the satisfaction of boundary conditions. It is not entirely clear that enough of the proper terms were included in the expansion of the wave function to satisfy all the boundary conditions satisfactory. Later work has indicated that quite high 1 values may be required and has shown a strong dependence of the energy on the points chosen for the application of the boundary conditions. However, the calculation was far in advance of others of that time. Energy values in LiF were found for fifteen symmetry types at r, X, L, K, and the midpoint of the 100 axis A. The bands are qualitatively similar to those shown in Fig. 19 for NaCl, except that the 2s valence band and the conduction band also were studied. The results are in reasonable, although not perfect, agreement with the normal order of levels for the face-centered lattice. Some of the eigenvalues for LiF are given in Table XVI. According to the criterion of self-consistency used, it was found that the lithium, fluorine, and hydrogen atoms are only very weakly ionic in contradiction t o the classical picture. This result might be altered somewhat if exchange and correlation effects *m S. 0. Lundqvist, Arkiu Fysik 8, 177 (1954).

A. Morita and K. Takahashi, Progr. Theoret. Phys. (Kyoto) 19, 257 (1958). D. H. Ewing and F. Seitz, Phys. Rev. 60, 760 (1936).

*6Qa 370

ELECTRON ENERGY BANDS IN SOLIDS

207

were included, or if a more completely self-consistent calculation were carried out. The principal reason for the small effective charge probably is that cells of equal volume were taken for the two ions. The charges would be given more properly if the cell volumes were determined from the ionic radii. The highest valence band (LiF) was found to have a width of about 5 ev. The valence band maximum occurs at the point K; and the (vertical) gap between valence and conduction bands is about 7 ev. TABLEXVI. ENERGY LEVELSIN LIF s-like states

rl =

-0.77,0.7 LI -0.76, 0.82 XI = -0.73 K I = -0.68. 0.01 (contains TI aa well) X I

L2

= -0.11

KI = x 4

La

=

= rls= Kc =

plike states

0.03 0.05 0.10 0.12 0.18 0.27

In lithium hydride, the valence band is found to have a width of about 2.5 ev whereas the energy gap is about 6 ev. Experimental values of the energy gaps quoted by Ewing and Seitz are 10 ev and 6.5 ev in LiF and LiH, respectively. 49. LEADSULFIDE A rather elaborate calculation of energy levels in lead sulfide (fcc) has been made.271A crystal potential was constructed from self-consistent fields for the core electrons of sulfur and lead. These were obtained by scaling the Hartree fields for Ar, C1,and Hg. As in the case treated by Ewing and S e i t ~ , *self-consistency '~ with respect to the charge distribution of occupied valence band states at k = 0 was achieved in this calculation. The 6s and 6 p electrons of Pb and the 3 p electrons of S (eight in all) were included in the self-consistent field calculation. An appropriate correction w a ~added for the constant part of the Madelung potential. The cellular method was applied. The appropriate Kubic harmonics were obtained for the expansion of wave functions. Considerable care D. G. Bell, D. M. Hum, L. Pincherle, D. W. Sciams,and P. M. Woodward, Ptoc.

Rag. Soe. (London) A217, 71 (1953).

208

JOSEPH CALLAWAY

was taken so that the matching of boundary conditions would be free from arbitrariness. Errors in the empty lattice test were of the order of 2% for states of high symmetry. A method involving matching of Fourier coefficients was used for calculation of E(k) curves. Terms up to I = 3 were included in the expansion of the wave function for all states, and higher 1 values were included for states at k = 0. The self-consistent procedure mentioned above leads to an effective ionic charge of 1.4. This figure was obtained under the assumption that the distribution of charge of the states at k = 0 is representative of the entire zone. It was found, however, that the charge distribution depends significantly on k. Energy levels were calculated at the center of the zone and along the 100 and 110 axis. The 111 axis was not considered. From the states calculated, PbS was found to be a semiconductor with a minimum energy gap (not vertical) of 0.3 ev and a minimum vertical gap of 1.3 ev. The maximum of the valence band was found to occur along the 110 axis about 39 of the distance to the face, and the minimum of the conduction band at the point K on the face (state KI). Calculations of the bands which define the gap were carried out for a smaller lattice constant, but with the same potential, in order to estimate the change in the gap with ev/deg. temperature. The result was equivalent to a shift of +2 X The experimental situation in regard to PbS is somewhat obscure, and much work remains to be done. Indirect transitions have been observed in the infrared absorption with an energy gap of 0.37 ev; direct transitions have a gap of 0.41 ev.272 50. ZINC SULFIDE

Zinc sulfide exhibits two crystal forms: zincblende (which is facecentered cubic) and wurtzite (hexagonal). There are two interesting studies of energy levels in zinc sulfide, both of which emphasize the cubic form.273-274 Asano and Tomishima calculated the cohesive energy of zinc sulfide according to the method developed by Schmidlas for diamond. No attempt was made t o determine the energy bands. Two electron functions are defined, each of which contains two parts. One part represents the ionic state in which both the 4s electrons of zinc have been transferred to the sulfur, and the other part represents a covalent state in which both the zinc and the sulfur atom are in s'pa states. An arbitrary constant A is induced giving the ratio of the amplitudes of the covalent

W. W.Scadon, Phys. Reu. 109, 47 (1958).

S. Asano and Y. Tomishima, J . Phys. Soc. Japan 11,044 (1956).

274

J. Birman, Phys. Rev. 109, 810 (1958); C. Shakin and J. Birman, Zbid. 109, 818 (1958).

ELECTRON ENERGY BANDS IN SOLIDS

209

to the atomic state. The energy of the crystal is calculated as a function of the parameter A , which is varied to minimize the energy. The wave functions for sulfur and the 4P functions for zinc were obtained by interpolation from SCF calculations for C1- and As+, respectively. The minimum of the energy occurs for A = 0.22 and gives a cohesive energy of 2.92 rydbergs per ion pair. This can be compared with the experimental result of 2.72 rydbergs per ion pair. The effective charge was determined from the parameter A to be f1.66 for zinc and sulfur, respectively. Birman studied in detail the problems involved in the calculation of a crystal potential for use in a band calculation for ZnS.274He discussed the problem of the effective charge to be assumed and concluded that charges of f0.5 for zinc and sulfur are most reasonable. The crystal potential is expressed as the sum of three parts: (1) The spherically symmetric Coulomb potential of the charge distribution within the equivalent volume sphere; (2) the potential within one cell due to the other ions in the crystal; and (3) the exchange potential, computed according to Slater's free electron approximation. The second part is not necessarily spherically symmetric. Birman studied this potential along three crystallographic directions, and chose that for the 111 direction to use in the band calculation, multiplied by a constant so that the sum of parts one and two would be zero midway between nearest neighbors. The cellular method was applied in the band calculation of Shakin and B i r m a ~ Each ~.~~ of ~the states considered were expanded in a fourterm series of products of radial functions and lattice harmonics. Boundary conditions were applied a t discrete points on the bounding surface between zinc and sulfur polyhedron. Eigenvalues were found at three symmetry points: l?, L,and X . From the preliminary results reported, it appears that the valence band maximum and conduction band minimum are both a t k = 0 (states r,) and "1, respectively), and that the minimum band gap (at k = 0) is about 6 to 8 ev. 51. BARIUM OXIDE A cellular method calculation of energy bands in barium oxide has been made by Morita and H ~ r i e The . ~ ~calculation ~ is similar in many respects to that of Ewing and Seitz for LiF.270 The crystal potential was taken from self-consistent field calculations of the 1s wave functions for 0- and for the 1s through 4d levels of Cs+. Exchange interactions were not included in the band calculation, but an approximate correction for exchange was computed a t the end. The Madelung potential was coma75

A. Morita and C. Horie, Sci. Repts. T8hoku Univ., First Ser. S6, 259 (1952).

2 10

JOSEPH CALLAWAY

puted according to the ionic model with an effective charge of 2. Energy levels in the bands formed from the 2s and 2p levels of oxygen and the 5p of barium were studied. It is not clear that enough spherical harmonics were included to satisfy the boundary conditions adequately. The 2p and 5p bands overlap, but corresponding levels in the 2p band are higher than in the 5p. The order of levels in these bands are in agreement with the normal level order in the face-centered cubic lattice. The 2p band was found to be about six volts wide, and to have its highest energy at -7.2 ev. The gap between the 2p band and the conduction band was found to be 10.4 ev after all corrections. Sproull in the A.I.P. Handbook estimates the experimental gap to be 4.2 e ~ . ~ 7 6 52. MISCELLANEOUS MATERIALS

Yamazaki, has reported tight binding calculations of energy levels in metallic b o r i d e ~and ~ ~in~ boron c ~ rb i d e . ~Rather ~B drastic assumptions concerning the symmetry had to be made in order to simplify the symmetry so calculations would be possible in these complicated structures. Very crude approximations were made to estimate potential integrals involving direct and nearest neighbors. It was found that the borides, an example of which is CaBs, should be semiconductors if the metal atoms have two valence electrons. Solutions of the 25 X 25 determinental equation were examined a t symmetry points and along symmetry axes of a simplified Brillouin zone. Numerical estimates for CaB6 gave an energy gap of 1.5 ev. I n boron carbide, B&, the crystal structure is built up of icosahedra containing twelve boron atoms and linear chains of three carbon atoms, all arranged in a rhombohedra1 giant lattice. Since there are fifteen atoms per unit cell, the secular determinant is 60 X 60. The symmetry was simplified by considering the carbon atoms in a chain en bloc and the boron atoms in an icosahedra en bloc. The secular equation was solved for two points within the simplified Brillouin zone. Consideration of the compatibility relations showed that it was possible to obtain an energy gap between the completely full valence bands and empty conduction bands. VII. Conclusions and Prospects

The survey of existing calculations of energy bands indicates that tremendous progress has been made in determining the level spectrum associated with a given potential. The contrast between present-day applications of the cellular or the OPW method and the prewar work is 2’6 R. L. Sproull in “American Institute of Physics Handbook” (D. E. Gray, ed.), *77

2’8

pp. 5-158. McGraw-Hill, New York, 1957. M. Yamazaki, J . Phys. SOC.Japan 12, 1 (1957). M.Yamazaki, J . Chem. Phys. 27, 746 (1957).

ELECTRON ENERQY BANDS IN SOLIDS

211

very striking. Agreement between theory and experiment is now becoming a valid test of the initial assumptions, particularly of the adequacy of a given potential. Much work remains to be done before calculations for systems other than the simplest metals are reasonably self-consistent. This problem is particularly important for semiconductors and the transition metals. It is desirable, of course, to do more than test the adequacy of an assumed potential. One wishes to ascertain the ability of the one-electron approximation to predict the results of experiments. A real answer to this question cannot be given until more progress has been made in achieving self-consistency. There are, however, three points which stand out prominently a t the present time: (1) The language of the theory is adequate to describe a large number of phenomena in semiconductors and in metals. (2) There is a large region of qualitative agreement between theory and experiment. (3) Quantitatively, the results tend to be somewhat disappointing. The ability of the one-electron approximation to furnish a framework for the description of experiments is most apparent in semiconductors. A large range of electrical, optical, and thermal properties can be explained in terms of a band model whose basic parameters, the effective masses, are determined by direct experiment. This is a very impressive success; the reasons for it are understood qualitatively. The theory is also reasonably successful in this respect in metals. X-ray experiments demonstrate the existence of a well-defined Fermi surface; other experiments, such as the de Haas-van Alphen effect and related experiments reveal many of its features. There are, however, other phenomena in metals for which the situation is less clear. The situation is quite confused in the alkali halides, because crystal imperfections are important and the interaction between electrons and the lattice vibrations is strong. Phenomena which are described more readily in terms of an atomic model rather than a band theory do seem to exist. Calculations of energy bands usually have a measure of qualitative agreement with experiment. In germanium, for example, the theory predicts that the top of the valence band is a doubly degenerate state (if spin is not counted) and predicts the existence of heavy and light holes. However, quantitative results are not achieved. The energy gaps and the effective masses are not given accurately, nor is the minimum of the conduction band located properly. It will be extremely interesting to see to what extent more accurate calculations improve the situation. The situation is similar in metals. In aluminum, for example, Heine found that pockets of holes should lie in the neighborhood of W , if they actually exist. This prediction is indicated by the experiments, however, Heine was not able to demonstrate their existence unambiguously from

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JOSEPH CALLAWAY

the theory. The situation seems to be most favorable in the alkali metals. Good results are obtained for the cohesive energy and related quantities although it must be admitted that agreement is attained only after the correlation energy term is included. The Knight shift seems to be predicted reasonably well. There are indications that it will be possible to account for the effective masses on the Fermi surface. We may emphasize again that in studying the properties of metals it is never adequate to treat the interaction between the valence electrons entirely in terms of the one-electron theory alone. Those calculations which strive for a greater degree of self-consistency will have special merit in the future from the theoretical point of view. Moreover, there seems to be an opportunity for semiempirical studies analogous to the work of Kane and of Phillips. Progress in our knowledge of electron interaction is most urgently needed. The role this interaction plays in ferromagnetism must be determined. Experimentally, more effort should be made to obtain quantitative information concerning the Fermi surface in metals. This is particularly true for the alkali metals in which the theory stands the best chance of explaining experimental observations. Measurements of the electronic specific heat, the optical properties, the anomalous skin effect, and cyclotron resonance should be extended. It is recognized, of course, that serious practical problems arise in carrying through such experiments; however, the possibility of detailed comparison between theory and experiments should be a stimulus to the effort. A detailed way of determining features of the band distribution well above the Fermi surface is still lacking. Measurements of x-ray absorption and electron energy loss may be useful in this respect. There still are theoretical problems to be overcome before the measurements provide useful tools in a quantitative sense. The role played by the hole in the inner atomic shells in determining the x-ray spectra will have to be ascertained; careful calculations of transition probabilities will have to be made. The effect of the plasma oscillations in determining the energy loss of electrons must be studied further. A careful calculation of the energy loss spectrum for a simple metal on the basis of interband transitions would help considerably in this respect. The author wishes to thank Dr. J. Goldman and the staff and management of the scientific laboratory of the Ford Motor Company for their hospitality during the summer of 1957, when this review was begun. The work of the author a t the University of Miami has been supported by the Office of Naval Research and the Air Force Office of Scientific Research. Dr. Conyers Herring kindly read the manuscript prior to publication.