Z. PhysikB 34, 29 35 (1979)

•Zeitschrift for Physik

B

© by Spriiager-Verlag 1979

Theory of Itinerant Ferromagnetism in the 3-d Transition Metals H. Capellmann Institut ftir Theoretische Physik, Rheinisch-Westf~ilische Technische Hochschule Aachen, Aachen, Federal Republic of Germany Received February 21, 1979 A description of itinerant ferromagnetism is presented, which allows for a local exchange splitting without long range order. The five-fold degeneracy of the d-bands is contained in the model, the parameters of which may be determined from first principles by fitting a given bandstructure. The theory is applied to derive the magnetic transition temperature. Numerical results are presented for Tc of Iron and Nickel in good agreement with experiment. The occurence of antiferromagnetism in Mn and Cr can also be understood in the framework of the theory presented here.

1. Introduction

The theoretical understanding of the ferromagnetic transition metals Fe, Co, Ni is yet incomplete. One might argue, that the ground state properties can be understood by Stonerlike one-electron theories. Working within the spin-density functional formalism Gunnarsson [1] calculates the ground state properties of V, Fe, Co, Ni, Pd, Pt and (having no adjustable parameters) he finds ferromagnetism in Fe, Co, Ni and paramagnetism in V, Pd, Pt, as it should be. Similarly Callaway and Wang 1-2] calculate the bandstructure of Ni and Fe, and the values obtained for the magnetization, Fermi surface and spin-wave stiffness constant are reasonable. The understanding of finite temperature properties and of what happens at the magnetic transition temperature T~ is much worse. Stoner-like theories fail. There is no experimental evidence for a vanishing exchange splitting at T~ [3], and the direct calculation of T~ [1] yields values too high by a factor of 5. The persistance of spin waves above Tc also cannot possibly be explained within a Stoner-like theory. The qualitative ideas developed over the years to account for the behaviour of ferromagnetic transition metals near Tc are, that even above T~ a local exchange splitting still exists and thereby a local magnetization (or local moments), but that this local magnetization lacks long range order. Only at Tc the long range order develops. Here transverse fluc-

tuations of the magnetization (which are neglected completely in Stoner theory) will determine T~. This qualitative picture differs drastically from Stoner theory which has the exchange splitting itself vanish at T~. There are several attempts to formalize these qualitative ideas. Starting with an itinerant model this author I-4] derived an equivalent Heisenberg-like Hamiltonian which governs the magnetic properties. This procedure allows for a local exchange splitting above T~ without long range magnetic order. A more detailed discussion of these ideas is given in a series of papers by Korenman and Prange 1-5, 6]. Recently, these authors, too, derived an equivalent classical Heisenberg Hamiltonian [6] from an itinerant model. They make a crude fit of the effective exchange interaction appearing in this Hamiltonian via the experimentally determined spinwave stiffness constant. The T~ calculated in this way is in good agreement with experiment. Although this success is very encouraging, one might have serious objections against the theories developed in Refs. 4 6, arguing that the model used as a starting point is oversimplified and possibly inadequate. Both approaches use the Hubbard model, thereby neglecting the fivefold degeneracy of the 3-d bands. In this paper the theory is extended to take into account this fivefold degeneracy. An equivalent

0340-224X/79/0034/0029/$01.40

30

H. Capellmann: Theory of Itinerant Ferromagnetism

Heisenberg-like Hamiltonian governing the magnetic properties will be obtained, which will be applied to calculate the transition temperature. It is the purpose of this paper, to give a full description of the theory and the formalism used, whereas the numerical results obtained for T~ of Fe and Ni are also published separately in Letter form.

2. Hamiltonian and Description of Method The discussion is based on a model Hamiltonian which will be restricted to the essentials in order to make the calculations as transparent as possible. For the 3d-series it is necessary to retain all five d-bands. A rough estimate yields that the hybridization with the broad s band is of minor importance for T~ (I expect corrections due to s - d hybridization to be of the order of 10 %). Therefore the s band is left out of the description. The electron-electron interactions will be restricted to Wannier-states on the same atom, interatomic interactions will be neglected. This will result in a k-independent exchange splitting. To include its k-dependence will be left to future publications. The Hamiltonian for the d-band electrons in the Wannier-representation is H = H ° + H w = ~' t~..c~, t c ~. -zd - i s --~S _I_1 ~ ' l/U/~vp/~t prt t p~ ,~v

2 z . , " ,¢ ~is ~is, ~i~' ~i~.

(1)

The c*(c) are creation (destruction) operators. The greek indices distinguish between the different dbands and run from 1 to 5, s is a spin index, and i, j indicate lattice points. I have chosen the representation which diagonalizes H ° with respect to #. For the interaction only matrix elements for electrons on the same lattice point are retained. To simplify the model as much as possible, the matrix elements are restricted to the direct Coulomb interactions W,"~= U,v and exchange interactions W y / = J ~ , and we take U , v = U and Juv=J for # ~ v and W~","=U +J. All these simplifications still leave us with a model having enough structure to describe the essential features of itinerant magnetism. Following [4], where a similar programm was carried out for the Hubbard model, the Hamiltonian (1) is treated in a general Hartree-Fock method with broken symmetries: Spin symmetry may be broken

M~__-Z

u 4=0, <%.* ,rs,,cis,>

(2)

Ss'

where a has the three Pauli-matrices as its components. The brackets ( ) indicate the selfconsistent

expectation value in the Hartree-Fock state under consideration. Furthermore it is of central importance to discuss states which break translational symmetry: M~ :t:M~

(3)

in general for i,i=j. The reason for considering the states described above is the following: If we go back to Stoner-theory we can state, that there the finite temperature properties are calculated based on an ensemble with the following properties: a) All states of the ensemble are single Slater determinants; b) Every single Slater determinant in the ensemble is made up of Bloch functions, having the full translational symmetry of the crystal. This second restriction b) is responsable for the failure of Stoner theory for the transition metals. From the usual theory of Heisenberg-ferromagnets we know that there the states which break the translational symmetry of the crystal carry more and more weight in the thermal average when the temperature is raised; they finally dominate above T~, destroying the long range magnetic order. In any case, the therm a l averages are translationally invariant, of course. I argue, that also in itinerant ferromagnetism thermal properties have to be calculated with an ensemble containing states breaking the translational symmetry. These symmetry breaking states may still possess a local magnetization, the direction of the magnetization may vary in space, however. The task of calculating thermal properties (in this paper only the magnetic transition temperature T~ will be discussed) now breaks up into two parts: First: symmetry breaking states have to be described and their energies have to be calculated. (This shall be done here in a general Hartree-Fock method). The major difficulty arizing in this step is to account for the loss of translational symmetry in a band picture. Second: The thermal averages have to be carried out. Both steps will be attacked in a rather elementary way in this paper. For the first problem, I shall restrict the discussion to the case, where the translational symmetry is broken only such that short range order still exists. As we shall see, for this case there exists an appropriate expansion parameter, which allows a solution of the general Hartree-Fock problem. It will be argued, that the states thereby described are the dominant ones in determining the magnetic transition temperature T~. The thermal average will then be carried out for a calculation of T~ using a m e a n f i e l d theory. We shall see that these simple methods already yield excellent agreement with experiment when applied to the transition temperatures of Nickel and Iron.

H. Capellmann: Theory of Itinerant Ferromagnetism

31

3. General Hartree-Foek Problem

involves the matrix elements

To carry out the first step of our program (the solution of the general Hartree-Fock equations) a Green's function method is used:

tij(M~ - M j).

#v

Gss'= - i ( T [ c ~ s ( t ) C~st,(0)l),

ij

(4)

where t is the time argument, T is the usual time ordering operator, and i is the imaginary unit. The general Hartree-Fock problem now reduces to find the selfconsistent solutions for the selfenergy £ de)

Sij =/~i (~ij= i(~ij ~ ~

{ U Gii(fO)

- U3-e Gu(cO) + J / e u Gii( c ° ) - J / e s Gu(c°)}

(5)

G(co) is the Green's function which is Fouriertransformed with respect to time t, 6~j is the Kronecker symbol. The X~, G~i are matrices with respect to orbital indices (#, v) and spin indices. 9-e indicates the traces with respect to orbital and spin indices; ~'~u(s) is the trace with respect to orbital (spin) indices alone. The G's on the right hand side of (5) contain the S to be determined selfconsistently. As pointed out in Chap. 2, it is essential to find solutions to (5) for broken spin symmetry (2) and broken translational symmetry (3). Whereas the solution of (5) for the groundstate - where translational symmetry is conserved and all magnetic moments Mi have the same direction - is straightforward, the solution for varying directions of M~ is not so easy. As already pointed out above, those states will dominate the thermal averages near T~. Here it is argued, that the most general solution to this problem, (which involves the difficulties of Anderson-localization and mobility-edge) is not headed to determine Tc. It is possible here to restrict the discussion to states, where the direction of M~ varies slowly in space, i.e. short range order still exists. Several authors [4-8] have argued that short range magnetic order is present in the transition metals up to temperatures above T~, an extensive discussion is to be found in [51, the most direct experimental proof being the existence of spin-waves even well above T~ [9, 101. For states with short range order it is advantageous to write the Green's function as G=

1 o - H ° --Z

1 = (¢o - U ° + Z) ic° _ HO)2 _ Z2 _ [nO, El"

(6)

The commutator K K = [ H °, N] = H ° S - X H °

(7)

For the transition metals, the hopping matrix elements are short ranged, therefore the commutator K is an appropriate expansion parameter for states possessing short range magnetic order. In the following, an expansion of G up to second order in K will be used. The advantage of this expansion is that for states having the same magnitude IM~[ for every lattice point i, ~2 is translationally invariant, even for varying directions of M~. As we know from Ell that the characteristic temperature for fluctuations in the magnitude IM~'[ is about 5 times Tc for the magnetic transition metals, it is reasonable to neglect these magnitude fluctuations in a first approximation for T~. The same expansion was already used for the single band Hubbard model [41 . For the case of five d-bands it is seen from (1), (5), and (6), that for H ° diagonal in orbital space, also 2 and G may be assumed to be diagonal in orbital space. If we expand the diagonal elements Z~ (which are still matrices in spin space) in the 2 by 2 unit matrix a o and the Pauli-matrices a j, (j = 1, 2, 3),

~ = 9 ~ o + D~. ~,

(8)

Eq. (5) may be written as

D•-!rTMu+½JMI; i -- 2 ....

i

:DU=(½U + J)nU.

M i -- -~

i,

M/Z

(9a)

¢t

(9b)

n u is number of electrons in band #,/]u is the shift in the chemical potential in band # (which turns out to be proportional to nU), and the vector D~ is proportional to the magnetic moment. Through the exchange integral J, D~ is coupled to the overall local moment Mi, whereas the direct Coulomb interaction couples D~' only to the moment in the same subband #. This allows for the possibility of different exchange splittings in the different d-bands. To avoid confusion: (9a, b) is just a short hand formulation of the general Hartree-Fock equations (5), which enables an interpretation of the parameters appearing in Z by physical quantities. The solution of (5), which determines the parameters D~ has still to be found. The solution is straightforward, however, using the expansion of G in the commutator K up to second order. From (6) we see that the poles cok of the Green's function are obtained easily by Fouriertransformation: ( D ~ +_-- e

k# + D~ # +_DU

(10)

32

H. Capellmann: Theory of Itinerant Ferromagnetism

where e~ are the fouriertransformed t~j and D u is the magnitude ID~[ which is independent of i. The restriction to states having short range magnetic order makes possible the definition of effective bands and of a "local exchange splitting" A": A u =2D".

(11)

This aspect is called "Local Band Theory" by Korenman and Prange [5]. The solution of (5) now is easy (although somewhat tedious) carrying out the cointegrals using the residue method. To simplify the result we neglect interband (optical) magnons. This amounts to the assumption that all moments M~ for identical i have the same direction for all #. For different i the directions might vary of course. This condition favours states which satisfy Hund's rule. E.g. for the case that only majority spin electrons are occupied in the ground state (Ni, if we speak of holes instead of electrons), the condition mentioned above guarantees in a natural way that Hund's rule remains satisfied also at finite temperatures. Even using a band description we thus are able to respect Hund's rule. Using the solution of (5) it is possible to calculate the total energy according to i .de) = - ~ s ~ j ~ - (co +/_/o) ~(co)

(12)

where 5~/z is the trace over all indices. In the G under the integral, the expansion with respect to K is used again up to second order, and the poles cok of G as obtained by the solution of (5) are inserted. For each Hartree-Fock state, characterized by a different arrangement of the moments M~, the respective energy is then obtained. Up to constant terms, which do not depend upon the relative arrangements of the moments on different sites, this energy turns out to be E = 2 JijMz "Mr, i+j

(13)

4. The Thermal Average: The Transition Temperature Tc

If the exchange split bandstructure of the magnetic transition metal under consideration is known, the coupling constants (14) may be calculated without any open parameters. It is then possible to calculate thermal properties. One way to proceed is to consider (13) as a classical Hamiltonian governing the magnetic properties and to carry out the necessary thermal averages in a purely classical way. It turns out that this way fails when applied to Nickel, the Tc calculated turns out to be consistently too low by a factor of 3 to 5. Quantum corrections are very important, a purely classical calculation is insufficient. This is not surprising considering the fact, that also for a Heisenberg spin 1/2 system quantum corrections lead to a factor of 3. For Ni the effective moment is even smaller and quantum corrections turn out to be even more important. If T~ is calculated in mean field theory, the way to include quantum corrections is quite simple. If we look for the divergence in the magnetic susceptibility )/to find T~ in mean field theory, we obtain 1 (2#B)2 2)~o(T~) - ~ F(q, k)= 1

where J~j = ~ exp {i(q - k)(R i - R j)}-F(q, k),

subspace possess short range magnetic order. Long range magnetic order is not necessary, however. The second subspace contains all other states. It was argued above, that for the magnetic transition metals it is possible to restrict the considerations to the first subspace alone, if thermal properties up to Tc and slightly above are calculated. All other states have too large an energy to carry much weight in the thermal averages. The energies of the states in the first subspace are given by (13-15); we may therefore consider (13) to give the eigenvalues of an effective Hamiltonian which is restricted to this subspace, and which determines the magnetic properties of the system up to and even somewhat above T~.

(14)

q,k

and V ~ A~(e~-eq)f(coq-) A~(e2-eq)f(coq+)~ (15~ r(q,k):?[4/2(A v ~ ~ ~M~(~ j . , ,

The f ' s are Fermi functions. The significance of (1315) is the following: The complete Hilbert space spanned by all eigenstates of (1) has been divided into two subspaces: The first only contains states with nonvanishing values of M~ such that the magnitude [M~I is independent of i. Furthermore all states in this

q,k

(16)

where )~o is the susceptibility of the noninteracting system. Quantum corrections only enter through the determination of Zo. To calculate Zo, we have to know ((M~))o (here the double brackets indicate the thermal average and the index 0 indicates the noninteracting system) in the presence of a magnetic field B. The calculation of ((M~))o is somewhat unusual: I recall that (13) is applicable only in a subspace described above, which is characterized by the spin-split bandstructure (10). When calculating g0, we of course

H. Capellmann: Theory of Itinerant Ferromagnetism

33

have to restrict the partition function to states in this subspace. First I demonstrate how this is accomplished if there is only one single d-band with exchange splitting A. As in the calculation for ((Mi))0 I have to neglect the explicit coupling (13) the "Hamiltonian" is /4= -~pi.B;

pi=#B.Si

(17)

where #s is the Bohr magneton, and Si is:

(is) ss"

For the noninteracting system the partition function factorizes for different sites i, I only consider the contribution of one particular site. The first question to answer is which statistical ensemble to use. We have to allow for the fact that the bandstructure is metallic and that the number of electrons varies on a given site. The average number is controlled by a Lagrange-parameter which we call ~. This is not the only constraint to be put on the ensemble, however. We also have to take into account the fact, that the effective bandstructure is split by a local exchange splitting, thereby reducing the probability of finding two electrons on the same site. For a given bandstructure this average double occupancy (which we call d) is given by

d =N--~ ~ f (o.)k_) . ~q f (Coq+).

(19)

This is taken into account by a Lagrange-parameter called ~'. From these constraints on the statistical ensemble we arrive at a partition function Z=Tre

~r~+~% +,~)+~,,,.,~

This result is quite plausible: W1/2=(n-2d ) is the probability of finding exactly one electron on a given site and C1/2 is the Curieconstant for a spin 1/2. When the procedure of calculating )(o is generalized to the case of five different d-bands, we have to allow for an additional constraint put on the states contributing to the partition function: As stated in Chap. 3, Eq. (13) was derived assuming that all moments M~ are parallel for identical i but different #. For the case of a strong ferromagnet this means that a state, having e.g. exactly two electrons from different subbands on a given site, is restricted to having these two spins parallel. Similar constraints apply for 3, 4, 5 electrons from different subbands. We obtain a Curie law Ceff

)(o- T

(25)

where the effective Curie-constant is given by 5

Cefe = ~ Win~2 Cm/2. rn=l

(26)

Here Wm/2 is the probability of finding m electrons from different subbands on a given site, which are not compensated by spins of electrons from the same subbands. For a given bandstructure coT,+ these probabilities are calculated as trivial generalization of (19). Cm/2 is the Curie-constant for a total spin m/2. Finally the transition temperature in mean field theory is given by

Tc = 2 -Cef - f L F(q, k). C1/2 q,k

(27)

(20) 5. Tc of Iron and N i c k e l

where fi = (kBT ) - 1 and G = c~ % The Lagrange parameters c~ and ~' are fixed by ((n~ + n ~ ) > o = n -

DlnZ 0c~

(21)

and < > o =

d = 0 aIn Z ~'

'

(22)

It is now straightforward to calculate ((M~>>o, and the non-interacting susceptibility Zo turns out to be C X0 = ~ -

(23)

where the Curie-constant C is given by C _ N (2#B) 2 (n - 2d) = CI/2 • W1/2. V 4k B

(24)

We are now able to attack the numerical problem of calculating Tc of the ferromagnetic transition metals. Equation (27) gives T~ dependent only upon parameters which can in principle be determined by bandstructure calculations. In the case of Nickel, however, bandstructure calculations [2] based on the spin density functional, although giving excellent results for the magneton number, disagree with experiment [3, 11, 123 on the value of the exchange splitting A as well as on the total band width. The disagreement on A may not be so surprising, because the spin density functional formalism by concept is made to produce good total energies, densities, and spin densities and results for those quantities are excellent [1, 2]. As far as the bandstructure itself is concerned, however, it is not clear that the one electron energies obtained from

34 the spin density functional are better than those obtained from a true Hartree-Fock calculation. In a single particle picture the magnetic effects (exchange splitting) are caused by the exchange term in the corresponding Hartree Fock equations. In the density functional formalism this exchange term is replaced by a potential. While it is true, that the total energy and the densities may be exact if this potential is chosen cleverly, there is no such statement for the bandstructure itself, if it is interpreted as giving the single particle energies. Therefore one should not be too surprised, if the calculated A for Ni differs by about a factor of 2 from the measured one. This does not speak against the usefulness of the spin density functional, one rather has to treat certain quantities produced as being subjected to error bars. Even for Nickel these error bars are rather small in absolute energies. I expect the r e l a t i v e error to be smaller in Iron, because there the correctly reproduced magneton number will ensure that the average A is of the correct order. This is not the case for Nickel, which has all majority electron states filled, because there A may be changed without changing the magneton number. Photoemission experiments [13, 14] of Fe seem to give rough agreement with theory, although a detailed measurement of the k-dependence of the energy bands (which is available for Ni [3]) is still missing for Iron. The results which are presented here are based on the canonical bandstructure of Andersen and Jepsen [15], which give the unhybridized densities of states of gg and T2g symmetry. This is particularly well suited for the theory presented here, which neglects s - d hybridization. It is also clear from (27) that to calculate T~ only the densities of states for the different d-bands are needed. This is due to the fact that mean field theory was applied in the thermal average to obtain T~. For I r o n a band width of 6 eV is used. The exchange splitting A is adjusted such that the experimental magneton number 2.12 (a g factor of 2.09 is included to account for the observed magnetization) is reproduced. This leads to a A of the order 1.5 eV. Using the densities of states of Ref. 15, the only "parameter" left is the total number of d-electrons n. Varying n between 6.8 and 7.2, the corresponding T~ varies approximately linearly with n going from 800 K (for n = 6.8) to 1,300 K (for n = 7.2). The experimental T~ is around 1,040 K, and the paramagnetic Curie-temperature Tp (the extrapolation of Z i(T) to zero coming from high temperatures) is around 1,090K, which agrees well with a number of d-electrons of the order of 7. We may state that agreement between theory and experiment is satisfactory. The trend-visible in the data presented above for Fe -

H. Capellmann: Theory of Itinerant Ferromagnetism that T c drops with decreasing n was checked further going to n ~ 6 and smaller, which corresponds to Manganese and Chromium. These densities yield a negativ T~, which in our formalism signals that ferromagnetism is unstable and a n t i f e r r o m a g n e t i s m is preferred. This does not depend upon the specific form of the densities of states used, even constant densities of states yield the same trend towards antiferromagnetism in the center of the 3-d series. For N i c k e l again the canonical unhybridized bands [15] were used and I took the experimental value [3] of 3.4eV for the bandwidth and of 0.19 eV for d at T~. The number of d-holes nh in Nickel at low temperatures equals the magneton number 0.56 (all majority electron states are filled), and ! take that number n h as being independent of temperature. These values of the parameters lead to a situation such that the lower of the exchange split bands no longer is completely below the Fermi-energy E r at T~, in contrast to low temperatures (there A is around 0.31eV [3]). Equation (19) yields a T~ for Ni of the order of 600 K. Changing the densities of states slightly, such that the lower of the exchange split bands is completely below E F even at T,. (this was accomplished by putting more weight into the T2g peak near EF) yielded a transition temperature of the order of 800 K. The experimental value of T~ is 630 K, the paramagnetic Curie temperature is 650 K.

6. Discussion

A theory was developed for the ferromagnetic transition metals which allows the calculation of finite temperature properties. The underlying picture, already discussed qualitatively before [4-6], is based on the existence of short range magnetic order up to and even somewhat higher than the magnetic transition temperature Tc. The exchange splitting in this picture does not vanish at Tc in contrast to usual Stoner theory. Here the transverse fluctuations of the magnetization determine T~, similar to the qualitative picture of a Heisenberg ferromagnet. The itinerant model used as a starting point in this paper contains all five d-bands, but so far neglects the hybridization with the broad s-band. The theory is applied to derive the magnetic transition temperature Tc. If we assume the bandstructure of the five d-bands to be given, the expression for T~ derived does not contain any further parameters. Using a crude bandstructure, the numerical results obtained for the transition temperatures of Iron and Nickel art in good agreement with experiment. The trend towards antiferromagnetism in the middle of the 3-d series (Mn, Cr) is contained in the theory

H. Capellmann: Theory of Itinerant Ferromagnetism

presented here, which puts the boundary of ferromagnetism to antiferromagnetism correctly between Iron and Manganese. The quantitative results obtained so far are as good as can be expected from the present day knowledge of independent particle properties of the ferromagnetic transition metals. This first successful calculation of To, which is based upon a microscopic theory, gives considerable support to the qualitative picture of "local band theory of itinerant ferromagnetism". It gives a microscopic basis for the somewhat phenomenological theory existing so far [5], which also showed that Tc may be understood qualitatively within the framework of "local band theory". It suggests that also other finite temperature properties, such as the properties of spin waves in the vicinity of To, might be understoood quantitatively. This still necessitates considerable work, but the general success of "local band theory" should also lead to a quantitative understanding of finite temperature properties other than Tc. To improve the microscopic theory presented here one may include hybridization and the/c-dependence of the exchange splitting. Also further experimental investigations, such as angle resolved photoemission, will be very helpful to determine the bandstructure among other things, which serves as an input into the approach presented here.

35

References 1. Gunnarson, O.: J. Phys. F6, 587 (1976) 2. Wang, C.S., Callaway, J.: Phys. Rev. B15, 298 (1977), and Phys. Rev. B16, 2095 (1977) 3. Eastman, D.E., Hirnpsel, F.J., Knapp, J.A.: Phys. Rev. Let. 40, 1514 (1978) 4. Capellmann, H.: J. Phys. F4, 1466 (1974) 5. Korenmann, V., Murray, J.L., Prange, R.E.: Phys. Rev. B 16, 4032, 4048, 4058 (1977) 6. Prange, R.E., Korenman, V.: to be published 7. Liu, S.H.: Phys. Rev. B13, 2979 (1976) 8. Edwards, D.M.: J. Phys. F6, L289 (1976) 9. Mook, H.A., Lynn, J.W., Nicklow, R.M.: Phys. Rev. Let. 30, 556 (1973) 10. Lynn, J.W.: Phys. Rev. Bll, 2624 (1975) 11. Heimann, P., Neddermeyer, H.: J. Phys. F6, L257 (1976) 12. Dietz, E., Gerhardt, U., Maetz, C.J.: Phys. Rev. Let. 40, 892 (1978) 13. Heimann, P., Neddermeyer, H.: Phys. Rev. B18, 3537 (1978) 14. Kevan, S.D., Wehner, P.S., Shirley, D.A.: Sol. St. Comm. 28, 517 (1978) 15. Andersen, O.K., Jepsen, O.: Physica 91B, 317 (1977)

H. Capellmann Institut ftir Theoretische Physik Rheinisch-West f~ilische Technische Hochschule Aachen TH Erweiterungsgeliinde Seffent/Melaten D-5100 Aachen Federal Republic of Germany