RELATIVISTIC ELECTRONIC BAND STRUCTURE OF THE HEAVY METALS AND THEIR INTERMETALLIC COMPOUNDS

HAL Id: jpa-00215043

https://hal.archives-ouvertes.fr/jpa-00215043

Submitted on 1 Jan 1972

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

RELATIVISTIC ELECTRONIC BAND STRUCTURE OF THE HEAVY METALS AND THEIR

INTERMETALLIC COMPOUNDS

A. Freeman, D. Koelling

To cite this version:

A. Freeman, D. Koelling. RELATIVISTIC ELECTRONIC BAND STRUCTURE OF THE HEAVY

METALS AND THEIR INTERMETALLIC COMPOUNDS. Journal de Physique Colloques, 1972, 33

(C3), pp.C3-57-C3-72. �10.1051/jphyscol:1972309�. �jpa-00215043�

JOURNAL DE PHYSIQUE

Colloque C3, suppliment au no 5-6, Tome 33, Mai-Juin 1972, page C3-57

RELATIVISTIC ELECTRONIC BAND STRUCTURE OF THE HEAVY METALS AND THEIR INTERMETALLIC COMPOUNDS (*)

A. J. FREEMAN

Physics Department, Northwestern University,

Evanston, Illinois 60201, and Argonne National Laboratory, Argonne, Illinois 60439 and D. D. KOELLING

Magnetic Theory Group, Physics Department, Northwestern University, Evanston, Illinois 60201

RBsumB. - Les metaux lourds et leurs composCs inter-metalliques presentent beaucoup de proprieth interessantes mais ma1 comprises. C'est surtout vrai des actinides et de leurs structures de bande dont on connait peu de choses pour interpreter leurs proprietes magnetiques, optiques et Blectriques connues. Cette etude analysera les travaux recents sur ces mCtaux lourds (de Th a Bk) et sur plusieurs composes inter-mCtalliques des << terres rares

>>

(LaSn3 et LaIn3) et decrira les conclusions que l'on peut tirer de cette theorie des bandes 1'6gard des mecanismes qui expli- quent certains phenomenes observes. I1 sera indispensable de considerer les effets de la relativite pour obtenir la structure de bande exacte des actinides. Tandis que les electrons 5 f dans les acti- nides inferieurs pourront Btre decrits comme des Btats de bande (ou etats itinerants), les actinides superieurs prksenteront une augmentation marquee de la localisation des orbitales

>>

5 f, et le modkle de bandes ne s'appliquera plus a ces &tats.

On s'attachera surtout aux complications diverses que l'on rencontre dans la determination des structures de bandes de ces systkmes et a quelques-unes des techniques employees pour obtenir des solutions. Parmi ces dernihes techniques se trouvent

:

1) la nCcessit6 (bien Cvidente) d'inclure I'effet de relativite, 2) les problkmes de convergence, causes par la presence des Ctats de moment angulaire plus eleve et de symetrie reduite, 3) le grand nombre de bandes qu'on trouve, 4) I'incerti- tude dans la configuration electronique et 5) le r81e de I'approximation dite du

<<

moule a made- leines ,> (mufin-tin approximation).

Abstract. -The heavy metals and their intermetallic compounds exhibit a large number of interesting but not well understood properties. This is especially true of the actinides for which little is known about their band structures with which to interpret their observed magnetic, optical and electric properties. This paper describes recent work on these heavy metals (Th through Bk) and some intermetallic compounds of the rare-earths (LaSn3 and LaIn3) and the understanding derived from band theory of the mechanisms responsible for various observed phenomena. Rela- tivistic effects are found to be indispensable for obtaining the correct band structure of the actinide metals. While the 5 f electrons in the lower actinides may be described as band (or itinerant) states, the higher actinides show a marked increase of the localization of the 5 f orbitals and the band description is no longer applicable for these states.

Particular attention is paid to the various complications which are faced in determining the band structures of these systems and some of the techniques used in obtaining solutions. Among these are the (obvious) necessity of including relativistic effects, the convergence problems due to the presence of the higher angular momentum states and lowered symmetry, the large number of bands present, the uncertainty in the electronic configuration. and the role of the muffin-tin approximation.

I. Introduction. - The energy band method has become a n increasingly powerful and sophiscated tool for studying theoretically the multitudinous properties of solids [I]-[3]. The proliferation of methods and their applications to a large number of problems attests t o its current popularity. I t is important a t this confe- rence, with its emphasis on perspectives for the future, t o recall that the manifold successes we have come t o accept are all the more remarkable when only some 15 years ago very little confidence o r physical meaning

(*)

Supported by the AFOSR, the AEC and by ARPA through the Northwestern University Materials Research Center.

was attributed t o ab initio energy band structures.

In the very near future emphasis will shift from band structure determinations of the simpler systems (metals and compounds with simple crystallographic structure) t o those of more complex structures. More importantly, the focus of these efforts will be towards obtaining wave functions and, from these, expectation values of operators in order t o compare predictions of calculated observables with experiment and to assess the relative magnitude and importance of many-body effects. Our own work is directed towards these ends and some progress has already been made - in part because we were faced with certain complexities which

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972309

C3-58

A. J. FREEMAN AND

D. D.

KOELLING demanded new solutions. We here describe some aspects

of our recent efforts using the actinide metals and seve- ral rare-earth compounds as illustrative examples.

The actinide metals and the rare-earth compounds (important but not well understood classes of mate- rials) exhibit a wide variety of interesting phenomena ranging from magnetism to superconductivity. In order to obtain a better understanding of the funda- mental interaction mechanisms responsible for the observed properties of these materials we have deter- mined the electronic band structure of the actinides metals and several rare-earth (RX,) compounds (where R

=

La and X

⁼

Sn or In). In this paper we present some results obtained on the electronic band structure and then discuss the properties of these materials in the light of our present understanding.

Further, in keeping with the purposes of this confe- rence, particular attention is paid to (1) the various complications which we have had to face in deter- mining the band structure of these systems and (2) some of the techniques we have used in obtaining solutions.

The actinide metals are expected because of the unu- sual nature of their atomic structure to have conduction band structures which are more compIicated than that of either the transition metals or the rare-earth metals.

The electronic structure of the actinide atoms consists of a radon core (with 86 electrons), a partially filled 5 f shell (zero in Th, three in U, six in Pu, seven each in Am and Cm and eight in Bk), and two to four valence electrons in the atomic 6 d and 7 s states (e. g., 6 d2 7 s2for Th, 6 dl 7 s2 for U, Cm and Bk and 7 s2 for Pu and Am). In the rare-earth metals, an open 4 f shell in the rare earth forms a narrow localized band located well below the d and s bands, which are exactly like those of transition metals. As we shall see, the 5 f electrons in the actinides are not so well localized in the first part of the series as the 4 f orbitals in the rare-earths and so their itinerant nature makes them hybridize strongly with both the 6 d and 7 s bands 141, [5]. This important difference accounts for one of the major difficulties encountered in theoretical deter- minations and interpretations of the electronic band structure and resultant properties of the actinide metals [6], [7].

The rare-earth intermetallic compounds, RX,, have the Cu,Au structure shown in figure 1, with the X atoms occupying the cube faces. We should note that the Lax, compounds are an important starting point for this general type of investigation for several reasons. Since the free atom configuration of the outer electrons of lanthanum is 5 dl 6 s2 the La configuration in the Lax, compounds (to first order) would be expected to involve electrons of s and d character, but not of f character. The lack of f cha- racter in the bands is particularly useful because La compounds allow some evaluation of the effects of the 5 d band on the first part of the rare-earth series. Without the f band, the band structure calculations are simpler and the magnetic and super-

FIG.

1. -

Cubic cell

of

the C U ~ A U structure.

conducting interactions in the d bands are not domi- nated or quenched by 4 f electron effects.

However, the Lax, compounds are of great interest in themselves precisely because of the absence (or possibly the presence of a very small admixture) of f electron behavior. In particular, the large temperature dependent (and possibly exchange enhanced) para- magnetic susceptibilities in the normal state and high superconducting transition temperatures which cer- tain alloys exhibit makes these systems very interest- ing [8]-[lo]. In LaSn,, T,

=

6.45 OK, essentially that of hexagonal La metal. For LaIn,, T, has dropped to 0.71 OK. There is also an interesting and large variation of the magnetic susceptibility. For example,

x for LaSn, follows a Curie-Weiss law with an effective moment of one Bohr magneton unit cell. This is a surprisingly result in view of the high superconducting transition temperature of this compound. LaIn,, on the other hand has a x which is larger than that of LaSn,, but does not follow a Curie-Weiss law.

This paper is organized as follows : the next section, section 11, discusses complications and methodology ; section 111 presents results and discusses the actinide metals ; section IV does the same for the Lax, systems.

11. Complications and methodology.

-

There are a

number of complications which are met when one

studies the electronic band structure of the heavy

metals and their compounds which do not arise for

other materials and some complications which are a

result of the APW method itself. A number of these

also form part of several other papers presented at this

conference. Aside from the (obvious) necessity of

including relativistic effects (which remove degenera-

cies and so introduce additional energy band splittings

and raise the number of bands in the energy range

considered) there are the following problems to be

considered : (1) convergence of the computed eigen-

values due to the presence of the higher angular

momentum states and lowered symmetry, (2) muffin-tin

approximation, (3) question of configuration assumed

for the starting potential, (4) localized versus itinerant

states and their proper theoretical descriptions, (5)

RELATIVISTIC ELECTRONIC BAND STRUCTURE OF THE HEAVY METALS C3-59 dense number of bands and their determination using In these expressions, f2 is the volume of the unit cell, the APW method with its energy dependent matrix R is the radius of the APW spheres, t is the position elements, (6) the so-called

^ct

asymptote

⁾⁾

problem of the atoms within the unit cell, ^gK is the large- and (7) the near linear dependence of the basis component radial function, and

^fK

is the small-compo-

functions. nent function (and

K

is the relativistic quantum

A. SYMMETRIZED

RELATIVISTIC

APW

METHOD. -

In our calculations we take full account of relativistic effects by means of the symmetrized relativistic APW (SRAPW) method [ I l l developed using a modified, but equivalent, form of the RAPW matrix elements as given by Loucks [12]. This symmetrization is greatly facilitated by the realization that the same RAPW technique can be obtained by using Pauli spinors in the region outside the muffin-tin spheres in the Foldy-Wouthusen transformed Dirac formalism [I 31.

The SRAPW method is based on the use of the full crystal symmetry to form the basis set in a relativistic APW calculation. To formulate the procedure, it is convenient to start with a modified form of the unsym- metrized relativistic APW matrix elements obtained by Loucks for a muffin-tin potential and then add a nonflat potential outside the APW spheres to correct the muffin-tin approximation. (The

cc

muffin-tin

^>)

potential consists of a spherically symmetric potential inside the APW spheres and a constant between them.) The modified form of the matrix elements for the muffin-tin potential problem can be obtained by a Bessel function identity and some recombining of the K-sum terms to get a sum on (I). The matrix elements then take the form

M(kl s' ; ks)

^E

< ^{k' st I H}

^-

^{E I ks} >

⁼

=

< S ' I S > M ~ ~ ( ~ ' ; ~ ) + < S ' I M ~ ~ ( ~ ' ; ~ ) I S > , (1)

MNR(k' ; k)

=

= ( k f . k - E ) U ( k - k t ) + j4

^-

nnR2) ^S(k

^-

^{kt) x}

S(k

-

k')

=

exp[i(k

-

k')] . t , ₍₄₎

t

number [13]). he expression is written for more than one atom per unit cell of the same species and muffin- tin environment. Thus the c, and y, are the same for each site, and only the structure factor S appears.

Thus, for one atom per unit cell, S

=

1.

Writing the matrix elements in this form has a number of advantages. First, the non-relativistic limit is immediately obvious since MNR is exactly the non-relativistic matrix element as c

-+ co,

because 5,

becomes (2 1 + 1) times the logarithmic derivative (it is just a j-weighted sum of the spin-up and spin- down logarithmic derivatives). For c finite, MNR contains all the relativistic effects except the spin- orbit coupling obtained in the second term. Second, the evaluation of the matrix elements is easier since it has reduced the number of quantities that one must have the computer calculate. The most important result for symmetrization is that the spin factors y, are within the noise of the numerical integration for I > 4, even for uranium, resulting in a great savings of array space when symmetry elements are used.

Further, the 5, can be well represented by a linear fit

for 1 > 4, since the radial functions are very nearly the Bessel functions, j,[r(E + v ~ ) ' / ~ ] in their small argument region [ll], [14]. In fact, the value of 5 ,

for a Bessel function,

could be used directly. The ability to use a linear fit for <, means that the symmetry elements need not be stored for I > 4. (The angular momentum sum is usually carried out to 1

=

12, although 1

=

6 is probably sufficient, because it requires almost no extra computer time.)

Using standard projection operator techniques, one obtains the symmetrized matrix element for the representation

as a linear combination of RAPW matrix elements

z(akl)

⁼

exp ia. (ak'

-

k,,,) .

In these expressions, y j is the ray group representation of the double point group [15] of the reduced vector

5

C3-60

A .

J.

FREEMAN A?

rfD

D. D. KOELLING k,,, but the sum need only be carried out over the

single point group operations. (As our origin is normally chosen at an inversion site, the non-primitive translation phase factor

z

can only be + 1.) This is precisely the same procedure as used in forming the non-relativistic symmetrized APW (SAPW) method [I 61 except for the spin coordinate. In practice this addi- tional coordinate creates some difficulty in choosing adequate basis sets. Another complication however, is that the time reversal symmetry can enter in one of three ways. When the

^y

can be made real, one has the same problem as at a general point and the Soven procedure [17] should be used. This occurs sufficiently rarely at symmetry lines and points that one usually just tolerates the degeneracy. In the other two cases [(a)

y"

equivalent to

y

but necessarily complex, or (b) y* inequivalent to y] the symmetry assumed by the Soven procedure has been already

^<(

projected out

H.

As with any APW method [18]-[23] the SRAPW method is easily extended to include a more general

<(

warped muffin-tin

⁾⁾

potential approximation which

requires only that one spherically average the model potential out to the muffin-tin sphere radius. For the case of a more general potential it is useful to write the potential as [19]

where V1 and V, denote the difference between the potential V and its muffin-tin approximation VMT.

This difference has been broken into two parts by requiring V1 to be non-zero only outside the muffin-tin spheres and V, to be non-zero only inside the spheres.

This is done because it is appropriate to expand these two quantities differently

-

Vl as a Fourier series and V2 in spherical harmonics. Furthermore, since the effects of V, are much larger [24] than the effects of V,, it is possible to consider the inclusion of V, alone. This intermediate approximation will be denoted as the warped muffin-tin (WMT). Because the basis functions are plane wave in the region where V, is non-zero, the WMT approximation only requires augmentation of the muffin-tin secular equation with the Fourier components of Vl,

< ks I H~~~

-

E I k' S' >

⁼

< ^{k s} I HMT

-

E I k'

sf

>

+ < s l s ' > Vl(k' - k ) . (11) (Note that this is not perturbation theory.) We have not discussed the inclusion of V2 as it is a much smaller correction. While it has been included in a number of non-relativistic calculations [19], [20], [24], it would be more complicated to include in a relativistic calcu- lation.

To summarize, the major advantages of the SRAPW method are : (i) it permits a reduction in the size of the secular matrix (and hence in computing time) at symmetry points and along symmetry directions, and (ii) it allows identification of various computed eigenstates (particularly important for dense bands arising from so many different atomic states) (iii)

it provides better convergence than the RAPW scheme and (iv) it is easily generalized to include the ((warp- ing

⁾⁾

of the muffin tin potential.

B. MODIFICATIONS

^{OF THE}

APW METHOD.

-

The APW method has a number of (well known) distinct drawbacks which arise from its formulation as an energy dependent matrix element scheme, i. e., the matrix elements depend on the trial energy in a rather complicated way. Thus, one must repeatedly compute and plot the values of the determinant as a function'of trial energy and search for its zeroes. (Or, equivalently, solve for a self-consistent eigenvalue.) Two of the APW methods [25], [26] have the interesting property that the basis functions are constructed for each trial energy by expanding the function inside the muffin-tin spheres by performing a

⁽⁽

pre-variational variation

^P.

Obviously, to obtain energy-independent matrix elements, it is necessary only to include this variation into the final variation to actually obtain the eigen- values [27], [28].

Instead of insisting that each basis function be continuous across the muffin-tin sphere surface, one instead chooses to use very discontinuous functions realizing that the variational formalism applied will yield continuous results. This is done by separating out the augmenting components for the lower I values and including them as additional basis functions.

(One achieves several methods based on the types of

augmenting function chosen.) Then, one straight-

forwardly applies a variational formalism. One

advantage of this formulation comes from the fact

that there is no longer any problem with asymptotes

thus making the calculations less susceptible to nume-

rical errors. (Asymptotes arise in the APW scheme

whenever the radial wave function has its zero at the

sphere radius in which case the corresponding loga-

rithmic derivative goes through a singularity.) For the

lower atomic number (Z) metals only few bands have

been found which cross the asymptote regions ; in the

higher Z metals, the density of asymptotes in the

energy region of interest increases. For compounds,

it becomes more likely that the bands due to one

element will cross the asymptote region of another

thus leading to possible numerical errors if the regu-

lar APW scheme were followed. The method offers

other advantages as well. (1) The normalization of the

trial functions is more easily obtained ; this facilitates

the use of k . p approaches and the calculation of

matrix elements. (2) Uncertainties in starting potential

are more easily adjusted because the input data

appears as only a few numbers instead of the tabulated

set of logarithmic derivatives usually employed. (3)

The secular equation needs to be solved only once per

k-point. (4) Non-spherical terms in the potential may

be easily included. The one major drawback is that all

this has been obtained at the price of considerably

increasing the matrix size. However, should one

prefer the smaller matrices with energy dependent

RELATIVISTIC ELECTRONIC BAND STRUCTURE OF THE

HEAVY

METALS C3-61 matrix elements, this procedure reduces to the Modifi-

ed APW (MAPW) [25] or Alternative APW (AAPW) methods [26].

We have also investigated another APW formulation (LAPW) which also overcomes the problems of dealing with energy-dependent matrices for some problems.

If the basis functions are sufficiently slowly varying, the logarithmic derivatives, r,(E), can be represented as linearized functions of the energy,

over a restricted range of energy about Eo. Since the Hamiltonian is a linear functional of the logarithmic derivative, one obtains an energy independent matrix equation.

Ho C

=

E(S

$-

S,) C (13) which is readily solved using standard orthogonali- zation and diagonalization techniques. This formula- tion has a number of distinctive features : 1) One obtains directly the normalized wave functions. 2) The linearized APW formalism is easily seen to be a pseudopotential method since the various terms in the resulting expressions are the same as those introduced into pseudopotential calculations to include non-local and spin-orbit effects (thus, it adapts easily to empirical adjustment). 3) By changing the continuity requirement at the APW sphere surface from the value of the basis function to its derivative for the 1

=

2 partial wave, one can use a linearized form for the d-bands as well. 4) Finally, one can also use this scheme to study the enkrgy band structure in the energy region of the asymptotes.

C. WAVE

FUNCTIONS FROM A COMBINED BASIS FUNCTION METHOD.

-AS discussed in the introduction, the very success of energy band theory has created new areas where current techniques have not been fully adequate. One of these is the determination of various expectation values of operators such as generalized susceptibilities, oscillator strengths and Coulomb interaction, i. e., of matrix elements which depend strongly on the wave functions themselves. Likewise calculations of thermodynamic quantities such as total energies and pressure-volume relations based on self-consistent band calculations require extensive knowledge of wave functions throughout the Brillouin zone. Such calculations have only been published for a few simple systems [29]. The main factor slowing down the progress of further applications of energy band results is the cost, both in time and money, of calculating eigenvalues and eigenvectors at sufficiently many points in the Brillouin zone. The two factors which determine the computational effort of a (varia- tionally based [30]) energy band method are 1) the ease with which the matrix elements can be constructed and 2) the number of basis functions (and hence the size of the matrix to be solved) necessary to obtain eigen- values and eigenvectors with the required precision.

For an optimally programmed RAPW calculation, most of the calculational time is spent evaluating determinants (roughly 90 %). Since the time required to evaluate a determinant varies as the matrix size to the n th power (in practice, 2 < n < 3), an obvious way to reduce the computational time is to decrease the size of the matrix. To do so, a more efficient basis set than the individual RAPW's is needed.

We have investigated the possibilities of using linear combinations of the RAPW basis functions to create more efficient basis function sets [31]. Although the main objective is to cut down the size of the matrix needed at a general k-space point and thus reduce the computational time, there are many other advantages arising from the flexibility of such a procedure. For example, by the proper choice of prescription for one's linear combinations, one can perform a sym- metrized calculation, ((telescope

⁾⁾

a partially converged solution to convergence, or obtain local (in k-space) eigenvalues with small matrices to avoid interpolation schemes.

The basis procedure is the familiar one of defining a trial function

as a linear combination of the basis functions

@ j .

The coefficients bnj and eigenvalue En(k) are then to be determined by the Rayleigh-Ritz variational method.

In this method we choose to expand the basis functions as linear combinations of RAPWs [12].

k,

=

k + K,.

Here the P functions are RAPWs and the coefficients cy define the linear combination being used. Thus, the cj" are not varied as a part of the Rayleigh-Ritz varia- tion. Km is a reciprocal lattice vector and sm the spin- coordinate.

Our efforts have been directed primarily at finding efficient basis sets. From the knowledge of the wave- functions at a few points within the Brillouin zone, we conclude that it is possible to perform rapid calcu- lations of wavefunctions and eigenvalues at other points. In fact, for fcc praesodymium, we were able to obtain both these quantities well over an order of magnitude faster than the fully optimized RAPW code yields eigenvalues alone. To find the optimum mesh of initial k-points to assure well converged results in the rest of the zone for various structures and types of materials requires considerably more testing than we have done so far. From our results on praesodymium and the experiences gained in previous k.p-type calcu- lations, we have reasons to believe that a sparse grid of initial points will be sufficient in most cases.

It cannot be overemphasized that this is a wave

function based method. One is constantly concerned

C3-62

A.

J.

FREEMAN AX

_:D

D. D.

KOELLING

with the wavefunction at every step and has them available at all times to perform wavefunction depen- dent calculations. Hence, we feel that this is the one area where the utilization of the RAPW expanded basis functions has its greatest potential for future applications. The other area where we anticipate the method to be advantageous is in performing band calculations for heavy compounds and materials of open structure such as those discussed below.

111. Electronic band structure of the actinide metals.

- The actinide metals have a number of unique physi- cal properties which arise from their unusual electronic structure. A key question has been the nature (loca- lized vs. itinerant) and the role of the open shell of 5 f electrons in accounting for the observed (and often times anomalous) electrical properties (resistivity thermal conductivity, superconductivity, Hall coeffi- cient, magneto-resistance and, in the case of Th, de Haas-van Alphen effect), magnetic properties (magnetic susceptibility and Mossbauer effect) and certain thermodynamic properties

[31a].

In the absence of theoretical knowledge about their electronic band structure, the actinides have been variously treated by analogy with their

⁽⁽

sister

^>)

elements, the rare-earth metals with their open shell of localized 4 f electrons.

We shall show by means of fully relativistic APW calculations that these models are incorrect for the first part of the series where the 5 f orbitals, because of their extensive overlap with neighboring atoms, form wide bands which hybridize strongly with the << 6 d

⁾⁾

and << 7 s-p

⁾⁾

bands, and must be considered in the itinerant (or band) picture. With increasing atomic number, the 5 f orbitals are found to localize (i. e.

greatly decreased overlap increased coulomb correla- tion and rapid narrowing of their band widths) and to resemble more closely the 4 f electrons of the rare-earth metals. This separation of the metals into these two classes appears to give a qualitative understanding of many of these metals.

A. ELECTRONIC

STRUCTURE OF THE FREE ATOMS. -

It is instructive to review some of the free atom characteristics and properties of the actinide elements before discussing the band calculations and our specific results for some of the metals. The two are clearly related and allow us to anticipate (and later understand) some of the band results. The relative positions in energy of these outer electrons states are shown in figure 2 for atomic uranium [32]. Since there is some uncertainty as to the <<correct confi- guration

))

in the metal (at least as regards the atomic charge densities to be used in making up the crystal potential) we compare three configurations in figure 2.

The striking features of this figure include the large separation in energy between the closed 6 s and 6 p shells, the large relativistic (spin-orbit) splitting of these states and the close relative positioning of the three different valence shells. This behavior is in sharp contrast with the positioning of the valence electrons

ATOMIC

LEVELS

OF URANIUM

---%c .---

^{6 p 3}

d

FIG.

2. - Atomic energy levels for three electronic configurations of uranium. Energies are given in Rydborgs. Note that a different

energy scale has been used for the 6 s and 6 p states.

in the rare-earths where the atomic 5 d and 6 s states lie close to one another in energy and are widely separated from the 4 f shell. The crossing of the 6 d and 7 s levels by the 5 f, as the number of such elec- trons is decreased, reflects the importance of coulomb correlation effects (direct coulomb and exchange) for these atomic electrons. From figure 2 one expects the 5 f electrons to contribute to the conduction processes in the metal along with the 6 d and 7 s electrons. The large spin-orbit splittings demonstrate the importance of relativistic effects which reflects the situation actually found for the metals. (The 6 p3/,-6 p,,, splitting is so large that in certain cases the 6 ~ - 3 , ~ . bands are found to overlap the bottom of the conduction bands.)

In figure 3 we show the relative outer radial extent of the atomic electrons for plutonium determined from relativistic Hartree-Fock-Slater atomic calcula- tions [32]. Note the position of the maximum of the 5 f density just inside the maxima of the closed 6 s, 6 p shells (and its long tail) and the great radial spread of the 6 d and 7 s densities. (The oscillatory structure of the densities at small radius from the nucleus has been omitted for the sake of clarity.)

Figure 4 compares the radial extent of the sama- rium 4 f, the iron 3 d and the plutonium 5 f orbitals.

(Here the inner node of the 5 f was included to empha-

size that neither the 3 d nor the 4 f have such an

orthogonality oscillation.) While such a comparison

correctly compares the spatial extent of these three

important types of orbitals, we should actually use a

radial scale which is normalized by the observed lattice

spacings for their metals. This would then reorder the

overlaps of the orbitals on their nearest neighbor sites

as 3 d, 5 f and 4 f in decreasing order. If one carries

out this procedure it becomes clear that unlike the

RELATIVISTIC ELECTRONIC BAND STRUCTURE OF THE HEAVY METALS

C3-63

FIG.

3.

- Radial charge densities for Pu as a function of radius (in Bohr units). The configuration is (fs dl 9).

FIG.

4.

- Comparison of the radial charge densities of

a)

a

3

d electron in Fe, b) a

4

f electron in Srn, and

c )

a 5 f electron in Pu.

case of 4 f electrons (which are tightly bound to the atom, do not overlap neighbors appreciably and hence form a very narrow energy band in the solid), the 5 f orbitals in the lower actinides must be considered from the itinerant or band point of view (since they do overlap strongly with their neighbors and hence are expected to form a fairly broad band). Thus whereas the 4 f electrons may not be treated as band electrons because of the large errors (some 10 to 15 electrons volts) introduced by neglect of the intra- atomic Coulomb correlation terms, the 5 f electrons appear to be very much part of the conduction band

structure formed together with the 6 d and 7 s electrons.

Since the latter two overlap their neighbors very strongly, they will form an s-d band of considerable width in much the same way as a typical transition and rare-earth metal.

B. CRYSTAL

POTENTIALS : EXCHANGE AND WARPING OF THE MUFFIN-TIN.

- The band calculations were carried out by means of the SRAPW method and hence all relativistic effects were included. As in all ab initio calculations, however, the starting potential plays a decisive role unless the calculations are carried out to self-consistency. [For the actinides, the uncer- tainty as to the nature of the 5 f electrons (localized vs.

itinerant) was thought to raise the question as to whether they should or should not be included in the band calculations and made questionable the meaning of self-consistency itself.] As is generally done, the crystal potential is constructed from a charge density which is a sum of free-atom charge densities centered about each lattice site. The Coulomb potential produced by such a charge distribution is easily obtained and the exchange interactions are approximated using the local statistical p1I3 exchange approximation [33].

While it is amazing that this model potential should work so well (neglecting as it does all distortions caused by inserting, the atoms into the crystals), this prescription has considerable respectability in light of its past successes and so provides the natural starting point for a study of the actinides.

We have already indicated the uncertainty as to the appropriate atomic configuration i. e., relative number of 5 f, 6 d, and 7 s electrons. For this reason and in order to estimate the possible importance of self- consistency we have done, in several instances a number of band calculations in which different atomic configurations [e. g., (f4 s2), (f3 d l s2) and (f2 d2 s2) in U, (f7 do s2) and (f6 d l s2) for Am and (d2 s2) and (f' d1 sl) for Th] were used to generate starting crystal potentials. While there is a depen- dence of the computed band structure on the number of 5 f electrons in the assumed potential it is found to be somewhat smaller than the dependence of the bands on the exchange approximation for the lower actinides.

The commonly used statistical exchange approxi- mation suffers from uncertainties as to the best form to be used. This uncertainty is expressed in the para- meter a which multiplies the p1I3 term of the exchange operator

:

a

=

1 is known as Slater's value of exchange and frequently yields better eigenvalues than the more formally correct term for which a

=

213 as taken from the work of Gaspdr [34] and Kohn and Sham [35].

In many instances a is now taken to be a variational parameter (213 < a < 1)

-

as described at length by Slater and associates in other papers in this volume.

The higher values for a normally occur in the more

inhomogeneous and lower Z (atomic number) systems

[36] where the assumptions of the statistical approxi-

mations are not valid.

C3-64

A.

J.

FREEMAN A N D D. D.

KOELLXNG

This form of exchange approximation is admittedly We discuss first the special case of Th metal in some crude and is acceptable in the lighter elements because detail as it illustrates many of the problems described of a fortunate coincidence. In the worst case, that of a above and sets the stage for discussing the other metals.

transition metal band structure, the plane wave (or

<(

s-like

^D)

bands are fairly insensitive to exchange and

so the position of the more sensitive d bands can be properly placed by adjusting the value of a ; the resulting band structures usually agree well with experiment. For the rare-earth metals, the unfilled 4 f states are localized (atomic) in character and so do not belong in a conduction electron band structure which then resembles closely that of a transition metal.

[Further, the 5 d and

⁽⁽

6 s-p

⁾⁾

bands of the rare- earths apparently hybridize so strongly that the variation of a only shifts the positions of the bands in the same direction (up or down in energy) rather than relative to each other.] In the actinide metals, the 6 d bands are found to be very broad and less sensitive (but not insensitive) to the exchange approxi- mation. The 5 f states, however, are sensitive to a with their relative position changing by about 0.25 Rydberg in going from a

=

213 to a

=

1.

Finally, it is standard practice in an APW or KKR -calculation to approximate this model potential by one which has the form of a muffin-tin [37], [38]. That is, one which is 1) spherically symmetric within non- overlapping spheres about each lattice site and 2) constant (flat) between the spheres. This approxima- tion is a good one for an fcc or bcc lattice where very little electronic charge is found outside the muffin-tin spheres. However, as the atomic number increases, more and more charge is found in this

((

outside region >). Further, since a study of the actinides involves other structures besides the fcc and bcc structures, we must consider using a less stringent approximation.

The calculations reported here have used a warped muffin-tin approximation [23] which removes the second requirement of the muffin-tin approximation

:

the potential between the spheres need not be constant.

That is, the only approximation made to the model potential is to spherically average within non-overlap- ping spheres. The effects of the

^<(

warping

⁾⁾

terms are small

-

but significant

-

in the fcc and bcc actinides and become as large as 314 eV. in alpha uranium. The remaining (first) approximation is much smaller in size and need not be included for the fcc and bcc structures. We will merely have to see how much of an approximation it is for the lower-symmetry structures before deciding whether to account for any non-spheri- cal terms inside the APW spheres.

C. ENERGY

BAND STRUCTURE. -

We may now present some results of our extensive studies of the electronic properties of the actinide metals using the SRAPW method. Except for [39]-[42] Th, other band structure calculations for the actinides were done in the non-relativistic approximation [7]. As we shall see, the important relativistic splittings and shifts makes questionable any results obtained by a calculation in which they are absent [4], [6], 1431, [44].

1. Thorium metal.

-

In their RAPW calculation on thorium metal, Keeton and Loucks [40] used the full Slater exchange value (a

=

1) and found energy bands (believed by them to be s-d bands) which did not correlate well with experiment because (as pointed out by Waber) their 5 f levels turned out to be in the middle of these bands. To deal with this difficulty Gupta and Loucks [41] artificially removed the 5 f bands from their calculations (by removing the asymptote-like structure from the f-state logarithmic derivatives) in order to obtain a conduction band structure which would correlate well with de Haas- van Alphen measurements 1451. They justified this procedure by arguing that, in analogy with the case of the 4 f electrons, the localized nature of the 5 f electrons makes their position in energy very sensitive to the potential and to intra-atomic exchange interac- tions not included in the energy band model. As discussed above, it is not at all certain that this situa- tion prevails in the case of Th. Indeed, the overlap arguments given earlier in this section tended towards the opposite view which is also supported by the fact that Keeton and Loucks mistook the f bands for d bands but would certainly not have done so in the rare-earths (as the 4 f bands are far too narrow).

I n our Symmetrized RAPW calculations [42], we have used several different starting potentials in order to examine the dependence of the calculated energy bands on the assumed configuration (occupation number). The potentials were determined from atomic charge densities which are the solutions of the Dirac- Slater equations [32] for the 6 d2 7 s2 and 5 f1 6 d1 7 s2 configuration of Th. We used the Gaspar-Kohn- Sham [34], [35] value for exchange (213 of the Slater exchange) and a warped muffin-tin approximation to the crystal potentials. The resulting energy bands in the (100) direction (T-X) for the two potentials are shown in figure 5. Two things should be noted from the figure

:

1) The f-states (which can be identified by their negative parity at T) shift by only about .05 Ryd. when going from the d2 s2 to the f1 d1 s2 configuration. This is far from the large sensitivity observed in the rare-earths.

2) The f-states lie considerably above the Fermi energy (which is between 0.55 and 0.6 Ryd). This is a direct consequence of our choice of the reduced exchange.

(The cross-hatched area marked

^<(

f-asymptotes

⁾⁾

is the energy range where APW calculations are inaccurate. The size of this region could be reduced by exercising more care in this region but it is not necessary for our purpose.)

The full band structure obtained with the d2 s2

configuration is given in figure 6. One can see that

the three Fermi surface structures discussed by Gupta

and Loucks [41] (the Superegg, the Lungs, and the

Dumbells) will also appear in this calculation. In

addition there is an ellipsoid along the C (1 10) direction

RELATIVISTIC ELECTRONIC BAND STRUCTURE OF THE HEAVY METALS

C3-65

(experimental difficulty) and to the assignment of the Fermi energy (theoretical difficulties). The N(E,) obtained from straightforward but crude histogram techniques using a n/4 a mesh (the so-called 89 points mesh) is 17

$.

5 atom-' Ryd-l. A detailed mapping of the Fermi surface has been made. However, it now appears that the current experimental data is insuffi- cient to distinguish between our results and those of Gupta and Loucks 1411. To do so from Fermi surface data alone would require greater accuracy in the experimental data and a more careful examination of the exchange approximations used in the calcula- tions. It is therefore important that optical experi- ments be done on this metal to resolve the differences.

2. Uranium metal. - Unlike thorium, many of the actinide metals crystallize in low symmetry forms (tetragonal, orthorhombic or monoclinic) at lower temperatures. This lowered symmetry causes great difficulties both in performing accurate band calcula- tions and in interpreting the results obtained. We have therefore focused our efforts on the high symme- try forms that do occur (fcc and bcc) in order to develop methods and to be able to assess and evaluate FIG.

5. -

Energy bands in the (100) direction for the (d2 sz) and the results without the added complexities introduced the (fl dl

^{s 2 )} configuration

of

fcc

Thorium. The a

= 213

exchange the low 'yrnmetry (convergence

approximation has been used. large matrices and resulting high cost of compu- tations, etc).

We have done a number of detailed calculations on and a

⁽⁽

button

^D

near the W point. These two features bcc uranium, its high temperature structure, using would quickly disappear if the Fermi surface were various choices of atomic configurations (cf. Fig. 2) moved up slightly. Thus they are sensitive to impurities to make up the crystal potentials in the warped

fcc THORIUM (d2 <)

a z 2 / 3

WMT

7+

FIG.

6. -

The energy bands are shown for

all

symmetry directions in fcc Thorium.

The (dz s2) configuration has been used with

cc = 213

exchange.

C3-66

A.

J.

FREEMAN AND D. D. KOELLING muffin-tin (WMT) approximation, and have varied a,

the exchange constant.

The effect of changing the atomic configuration on the resulting band structure is shown in figure 7 for the three atomic configurations represented earlier in figure 2, namely (f4 s2), (f3 dl s2) and (f2 d2 s2).

Here the bands along the high symmetry direction (T-H) are shown for the choice a

=

213 and (as in the case of Th) the region of 5 f asymptotes is shown shaded for (f2 d2 s2) configuration. The strong hybri- dization of the 5 f bands with the very broad

^(<

7 s-p

⁾⁾

and broad 6 d bands is clearly evident as is the upward movement of the 5 f bands as the number of 5 f elec- trons in the assumed atomic potential (2, 3 or 4) is increased. The relative insensitivity of the three bands structures confirms the applicability of the band model to the lower actinides. Note that the position of the Fermi energy is given for the (f3 dl s2) potential as this is the configuration considered to be the

^<(

best

))

choice for the starting potential. The band structure for this potential along the major symmetry directions is shown in figure 8. Not surprisingly the computed density of states, N(E), shows considerable structure and reflects the strong s-d-f hybridization of the energy bands. The density of states at the Fermi energy, N(EF), is found to be fairly high - in agreement with expectations derived from specific heat measurements on a-uranium and other metals. Full details of these results and the caIculated Fermi surface will be presented elsewhere, as they lie outside the scope and purpose of this paper.

3. Plutonium metal. - The first calculation of the energy band structure of fcc Pu metal was done by

Kmetko and Waber [7] using the non-relativistic APW method. This is also a metal which has been studied in some detail in our work using the SRAPW scheme. We use these results to illustrate the effect of varying a, the exchange parameter, on the computed energy band structure, i. e., we hold the atomic configuration in the assumed potential constant (f5 d l s2). As before, we present results using the WMT approximation.

The energy bands along the high symmetry direc- tion T-H are compared in figure 9 for the full Slater exchange (a

=

1) and the Kohn-Sham-Gaspar value (a

=

213). While these two band structures look at first sight to be very different, a detailed comparison shows that they are very similar except in one impor- tant way. (This comparison holds for the bands along all the major symmetry directions and not just along r-X.) The effect of changing a from 1 to 213 has been to change the position in energy of the 5 f bands relative to the underlying << 7 s >>-6 d band structure and to change the spatial character of the 5 f orbi- tals and hence their local vs. itinerant character. If one examines the a

=

1 bands one sees the f bands as a set of rather narrow, flat relativistically and crystal field split bands, overlapping (and therefore hybridizing) with the s-d bands. We see that with such bands one is in the gray region separating the local versus itinerant descriptions (it is also a preccur- sor of what we may expect, and actually find, with increasing atomic number). Further, the electronic configuration estimated from the bands is far from the assumed starting configurations.

The shifting upward of the f bands found for a

=

213

FIG.

7. -

Energy bands for

bcc

uranium in the

(100)

direction with three configurations. From left to right, these are

the (f4

do

s2), the (f3

dl

s2), and the (f2 d2

^sZ)

configurations. The exchange is

^cr = 213.

(Note the change in scales.)

RELATIVISTIC ELECTRONIC BAND STRUCTURE O F THE HEAVY METALS

bcc

URANIUM (f3d's2) a = 2 / 3 W M T

FIG.

8. - Energy band structure for (f! dl s2) bcc uranium in all symmetry directions.

configuration and (as in Th metal) give an overall more satisfying physical picture.

As a test of our results and because it provides a

1 .o

better understanding of the effects of the 5 f states,

0.9

we have calculated the band structure which would

exist in fcc Pu if the f bands were absent. The results

0.8

of this unrealistic calculation are presented in figure 10.

Comparing these results with the bands of a high

0.7

atomic number fcc transition metal, such as platinum

0.6

[46], we see that the bands of figure 10 are those of a

typical transition metal. There is the free-electron band

0.5

which begins with the r6+ state and rises rapidly. Cutting

across this band - and hybridizing with it - are the

0 4

d-bands consisting of the r8+-r: spin-orbit split

0.3

states and the higher r8+ state. These are broad

d-bands with the descending bands coming down to the

0.2

energy region of the I'l. Several bands have interacted

r a

x

via the spin-orbit coupling to form anti-crossings where

0. I

r

^A

x there would have been crossings in the non-relativistic

FIG. 9.

- Comparison of a = 213 (on right) and cc = 1 (on left)

bands. The real band structure, with the 5 f states

exchange for fcc Pu. Configuration is (fs dl 9).

included as in figure 9, is now more readily understood as that of a transition metal with 5 f bands superposed onto the s-d bands and strongly hybridized with them.

leaves the

⁽⁽

s-d

⁾⁾

bands relatively undisturbed and 4. The higher actinide metals.

-

The heavier results in (1) greater overlap of the f orbitals and (2) actinide metals, americium, curium and berkelium [47]

stronger hybridization (especially with the d bands) have also been studied by means of the SRAPW

and hence effectively wider

^<(

f

⁾⁾

bands than in the method in the WMT approximation. These metals

case of the a

=

1 band. These band results appear crystallize in the double hcp structure at low tempera-

close to self-consistency with the assumed starting ture and in an fcc structure at higher temperatures

RELATIVISTIC ELECTRONIC BAND STRUCTURE OF THE HEAVY METALS

FIG. 11.

- Energy band structure for fcc Am using the full Slater exchange with a (f6 dl ^s2) configuration.

non-relativistic results in a number of significant ways of which the most important are

:

the states are split in energy by the spin-orbit interaction by an amount which is usually larger than the crystal field splittings for the 5 f electrons ; and the orbitals of lower angular momentum are shifted downward relative to the higher angular momentum states. This latter effect is graphi- cally shown in figure 1 of Lehman [43] and Table I of Boyd, Larson and Waber [44]. To further illustrate this effect, we have compared the relativistic [32] and non-relativistic [50] atomic eigenvalues for uranium in the 5 f 3 d l s2 configuration using KSG exchange.

In the non-relativistic calculation the 7 s states are 0.4 Ryd above the 5 f state while in the relativistic calculation they are 0.1 Ryd below the 5 f energy.

Similarly, the 7 s are (barely) above the 6 d in the non-relativistic calculation and about 0.15 Ryd below in the relativistic one. Such an effect is certainly present for Pu as pointed out by Kmetko and Waber [7].

Furthermore, the relativistic 5 f orbitals are expand- ed spatially relative to the non-relativistic ones by roughly 20 % due to the relativistic contraction of the core states causing increased screening of the nuclear charge [51], [52].

I t is therefore not surprising that non-relativistic energy band calculations will only give a crude repre- sentation of the band structure of the actinide metals.

Comparing our SRAPW results with the non-relativis- tic [7] calculations which have been reported recently shows that, as expected [43], [44], the non-relativistic bands are incorrect in a number of ways. The ordering and separations of the bands is incorrect ; except for actinium, the bottom of the 7 s-like bands (their T ,

state) lies above the 5 f bands and the lowest state at point H (in the bcc metals) is found to be far below their r , state. The large spin-orbit splitting of the bands is missing and degeneracies have not been removed causing crossing of bands rather than anti- crossings. Finally, the relativistic effect causing 5 f orbital expansion is absent and hence estimates of 5 f band widths or electronic charge within the APW spheres are doubtful. All these differences make questionable the value of any non-relativistic band result for the actinide metals.

IV. Relativistic electronic band structures of some rare-earth compounds.

-

We here discuss results of our SRAPW studies of the band structure of several RX, compounds with a view towards illus- trating some of the problems which arise in these types of systems (cf. section 11). For this reason only some brief indication of results for LaSn, and LaIn, will be given here. Comparison will be made with a recent non-relativistic modified orthogonalized plane wave (MOPW) calculation for LaSn, in which relativistic effects on the band states were included by means of perturbation theory [53].

The RX, compounds crystallize into the Cu3Au structure shown in figure 1 with the unit cell construct- ed from one rare-earth (R) and three non - transition atoms (X) such as Sn or In. The lattice can be easily viewed as an fcc type lattice with the rare-earth on the corner sites and the X atoms on the face-centered sites.

The resulting unit cell is then a simple cube with four

times the volume, and the Brillouin zone (BZ) is also a

simple cube with one fourth the volume of the fcc BZ.

C3-70

A.

J.

FREEMAN

AND D. D. KOELLING This mapping of the fcc homonuclear BZ into the

smaller simple cubic BZ maps four points into one as has been illustrated in Table I for the high symmetry directions. This implies that the number of RAPW's required per point will be roughly four times as large as for the fcc crystal. Further, one has four times as many bands for the homonuclear system treated as a simple cubic system. (We are helped here by the fact that the X atoms are not transition elements.) Clearly this simple modification of our crystal has greatly increased the computational effort.

Folding of the.fcc symmetry direction into the Cu,Au Brillouin zone. The symmetry labels are standard :

r ^(ooo),

~ C C - x

(loo), K (314 314 01, w (1 112 01, -c-X (112 0 0), M (1 /2 112 O), R(1/2 112 112).

Simple cubic fcc

-

( r Z ) (r-A)

(X-A)

(X- W) (twice) V-Z)

(X-U/K-C) (X-C) (twice) V-L)

(X-L) (thrice)

The potentials were constructed from a superposi- tion of relativistic atomic charge densities derived from the configurations (5 d l 6 s2) for La and (5 s2 5 p2 for Sn and (5 s2 5 pl) for In. In the results reported here the exchange contribution to the potential was calculated using a

=

213 and the muffin-tin approxima- tion was made. In our initial calculations we used equal APW sphere radii but this caused the Sn s asymptote to fall just below the calculated d bands very near the Fermi energy. We then determined the sphere radii so as to maximize the atomic charge inside each sphere and this resulted in moving the Sn s asymptote to just barely above the d bands and well above the Fermi energy. It also reduced the rms error in the interstitial region from 0.2 Ryd to 0.08 Ryd.

The band structures LaSn, and LaIn, shown in figure 12 consist of a set of s-p plane wave type bands (which are really very appropriate to a pseudopotential type calculation [54]) plus some very narrow d-bands.

The lowest three bands (tentatively identified as the X s-bands) are remarkably similar in LaSn, and LaIn,.

The upper plane wave bands are lower in LaIn, and compressed relative to the LaSn, bands. The d-bands are quite narrow in these structures (cf. the r8+-r:-r,'

combination near the top of the LaIn, bands) with the crystal field splitting at r just slightly larger than the spin-orbit splitting (0.01 Ryd). These d-bands move down in going from LaSn, to LaIn, (one fewer electrons) as they must if La is to maintain any d-character.

L a In, ( a = 2 1 3 )

L a Sn, ( a =*/,)

FIG. 12. -Band structure of LaSn3 and LaIn3 for the (110) direction. The

a =

213 exchange has been used.

We have not yet performed sufficient variations on the crystal potential to test for sensitivity. However, by comparing the LaIn, and LaSn, results, one can

((

predict >> that the plane-wave band structure will be

relatively insensitive to changes of configuration or

exchange. But the relative position (and to a far lesser

extent, the band shape) of the d-bands wiH be quite