Band structure calculations for f-electron systems

ELSEVIER

Physica B 206 & 207 (1995) 1-7

Band structure calculations for f-electron systems M.S.S. Brooks European Commission, Joint Research Centre, European Institute for Transuranium elements, Postfach 2340, D-76125 Karlsruhe, Germany

Abstract

Attempts to calculate the crystal structures and contributions to the spin and orbital magnetization densities of transition metals, rare earths and actinide from first principles are reviewed. Emphasis is placed upon crystal structures and orbital contributions to the magnetic moments.

1. Introduction

Density functional (DF) theory [1] is used for nearly all electronic structure calculations in solids. Density functional (DF) theory is a ground state theory. The eigenvalues that are obtained in this theory are the functional derivatives of the total energy with respect to the density or spin density. They are not in general simply related to measured quantities, although the temptation to compare the two is rarely resisted. D F theory contains an unknown functional for the exchange and correlation energy into which most of our lack of knowledge of the contribution of many body interactions to the total energy is placed. In practice, the functional is usually replaced by an approximate one which has been derived for a homogeneous electron gas with constant density. A t each point in space, the functional and its functional derivatives, having been used to obtain the potential in terms of the density, the constant density is then replaced by the real density, the local density or spin density approximation ( L D A or LSDA) [1,2]. The primitive nature of this approximation would not lead to the expectation that a quantitatively accurate theory would result if it were not for the fact that computed ground state properties of solids, obtained in this manner, often agree very well with measurements, even for narrow band metals such as Fe, Co and Ni and for the rare earth metals where the conduction

electrons provide the coupling between localized electrons [3,4] and provide the Fermi surfaces for which existing theory and experiment are in good agreement [5]. One of the areas where genuine problems arise is with narrow band systems, in particular the 4f bands in cerium or the 5f bands in actinides. This is partially because many of the properties of interest in narrow band materials are related to the eigenvalues or quasiparticle energies which L S D A does not yield particularly well. Properties that involve the total energy such as crystal structure are quite well described by LSDA. Ground state magnetic moments also depend upon total energies but here an additional difficulty arises. The total energies of localized magnetic states depend upon all of the degrees of freedom of the total atomic moments (in the sense that multiplets are formed), whereas the total energy in L S D A depends only upon the average value of the moment. Nevertheless, in many cases, L S D A yields satisfactory results. One simple example of where practical quantities can be calculated using L S D A is intra- and inter-atomic exchange coupling in rare earth transition metal intermetallics [6,7]. Although the 4f states are localized, it is possible to calculate their spin density as part of the total spin density of the solid. However, the requirement that the total number of 4f electrons and the total 4f spin be fixed is made, whereas the partial conduction electron occupation is determined

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M.S.S. Brooks / Physica B 206 & 207 (1995) 1-7

self-consistently by Fermi statistics. The interatomic exchange coupling can be obtained from the difference in the total energies of ferrimagnetic and ferromagnetic states.

2. Crystal structures The relationship between crystal structure and band structure of the transition metals was first studied by Pettifor [8] who related the HCP--->BCC---> HCP---> FCC sequence as a function of pressure to the change in d occupation number across the transition metal series. For the rare earths, the structure sequence HCP---> Sm-type---~ DHCP--> FCC was shown by Duthie and Pettifor [9] to depend upon the 5d occupation number. In fact, the same structure sequence is found for Y, the early 5d transition metals and the heavy actinides which are 6d transition metals at ambient pressure. Subsequent treatment of the band structure problem for the rare earths by Skriver [10] added accuracy to the computed structural energy differences. The heavy actinide metals Am, Cm, Bk and Cf all have DHCP structures at ambient pressure and the analogy with the rare earth metals is inescapable. Eriksson et al. [11] used the force theorem to compare the energies of the HCP and DHCP crystal structures with those of the FCC structure for the heavy actinides A m - C f . The DHCP structure has the lowest energy but at reduced volumes the FCC structure becomes more stable. Similar calculations have been made by Skriver [12] for the light actinide metals where the HCP, BCC and FCC structures were compared. Nevertheless, the problem of low temperature structural stability of the light actinides is a difficult problem since Pa is BCT, U is orthorhombic with 2 atoms/cell, Np is orthorhombic with 8 atoms/cell and Pu is monoclinic with 16 atoms/ cell. Wills and Eriksson [13] have made full potential LMTO calculations for Th, Pa and U and S6derlind et al. [14] have extended the theory to Np. The total energies of the three elements were calculated in three structures: FCC, BCT and the orthorhombic tx-U structure. The experimentally observed structures were found to have the lowest energies. Wills and Eriksson [13] argued that the preference of materials to form open structures is due to non-sphericity in the charge densities. The energy gain from covalent bonding is at the expense of the electrostatic Madelung energy. In many metals, the Madelung contribution dominates and high symmetry structures are formed. When a complex of energy bands, such as the 5f

derived bands in the actinide metals, cross the Fermi energy, the covalent bonding energy gain is particularly large and low symmetry (or open) structures become stable. Evidently, there is an analogy with the formation of Peierls distortions in the sense that complex systems have at least one contribution to their Hamiltonian that tends to reduce the symmetry of the ground state. That charge density waves have also been observed in actinides that also have low symmetry structures [15,16] is suggestive, and we would suggest that the underlying mechanism is the same. S6derlind et al. [17] argue that there is a unified structure sequence for both d and f transition metals. They have shown, using total energy calculations, that transition metals have asymmetric structures at larger volumes and symmetric structures at lower volumes and that the same applies to the actinides. The difference is that, at ambient pressures, the transition metals are at small enough volumes to already be in symmetric phases and would become asymmetric upon expansion. The actinides are in asymmetric phases at ambient pressures but become symmetric upon compression. Thus, a - U is predicted to undergo transitions to BCT and BCC phases under pressure.

3. Magnetic moments The narrow 5f bands and the large spin-orbit interaction in actinides produces the ideal situation for itinerant electrons to support the strong orbital magnetism, which is one of the remarkable features of actinide magnetism. On the left hand side of Fig. 1, we have resolved the spin contributions to the magnetic moments of late transition metals into local and diffuse parts (the orbital moments belong almost entirely to the 3d or local electrons). Here, in accordance with common usage, by local part we mean the 3d contribution and by diffuse part we mean the sp contribution, which is also that part of the moment whose density lies mainly in the interstitial region of the crystal and is not detected in neutron diffraction experiments under normal circumstances. Both the exchange interaction and hybridization between local and diffuse electrons influence the relative sign of the local and diffuse moments. Exchange interactions between the local and diffuse moments are always positive in L S D A and lead to parallel polarization in the absence of other influences. In Fe, Co and Ni, the spin moments are small and the exchange interactions between local and

M.S.S. Brooks / Physica B 206 & 207 (1995) 1 - 7

(a)

pz loc

z

JlJdiff

nloc>2t, l

T Z .UlZ oc ,IJdiff _

_

ntoc< 2t,1

(b)

JJlZc

nloc > 2I*1

]Jc~iff

L

T ~c~iff

ntoc< 2t* 1

Fig. 1. Possible contributions to the total magnetic moments from local and diffuse spin densities and the orbital densities. The top and bottom halves are for orbital moments less than and greater than the spin moments, respectively. The left and right panels correspond to the narrow bands being greater and less than half-filled, respectively. Top left corresponds to Fe, Co or Ni, whereas bottom right corresponds to the actinide site in an actinide compound.

diffuse moments small enough that hybridization dominates. The origin of the relative signs of the diffuse and local moments is therefore hybridization between the 3d and sp electrons. The 3d bands are more than half-filled and the Fermi energy lies close to the bottom of the broad, free electron like, sp bands. The hybridization is therefore similar to that between an early transition metal (the sp conduction band system) and a late transition metal (the 3d band system) and results in the relative sign of the local and diffuse moments being negative. [18-20]. In Fe, Co

3

and Ni, the diffuse part of the moment is antiparallel to the local part. The light actinides with itinerant 5f electrons are early 5f transition metals. The diffuse electron density is dominated by the 6d density and also constitutes an early transition metal system. The 5f moments then polarize parallel to the 6d moments. In the actinides, therefore, both exchange and hybridization lead to parallel polarization of the local and diffuse moments, as shown on the right hand side of Fig. l(a). Calculations for Fe, Co and Ni [21] yielded orbital contributions to the moments of 0.08 P-a, 0.14/z a and 0.05 p~, respectively. The orbital contributions to the moments are parallel to the spin contributions of 2.13/~.a, 1.52/z a and 0.57/z B for Fe, Co and Ni, respectively, since the 3d bands are more than halffilled. Such a situation is shown graphically on the left hand side of Fig. l(a). But light actinides have a less than half-filled 5f band, therefore the induced orbital moment is antiparallel to the 5f spin moment as shown on the right hand side of Fig. l(a). Thus there are two sign changes occurring between the right and left hand sides of the figure (the transition metal and actinide), for both diffuse and orbital moments. The diffuse spin moment is actually antiparallel to the total moment in uranium compounds as is verified by comparison of the measured relative magnitudes of 5f and total moments in neutron scattering and magnetization experiments [22,23], from the interpretation of spin polarized photoemission experiments on uranium chalcogenides [24] and deduced from magneto-optical spectroscopy [25]. But the calculated diffuse moment (right hand side of Fig. l(a)) is always parallel to the total spin moment in actinide NaCl-type compounds. Only if the orbital component of the moment at the uranium site is larger than, and antiparallel to, the spin component can theory and experiment be consistent. This situation, which is consistent with all known data, is shown graphically on the right hand side of Fig. l(b). Several relativistic energy band calculations have yielded orbital moments which are larger than the spin moments in compounds containing actinides [26-28].

4. Magnetization densities The magnetization density is responsible for the magnetic scattering of neutrons [29]. The magnetic form factor is given by [29,30]

F(g) = [ (Jo), m: + (Jo + j ~ ) , m ; ] / m " ,

(1)

4

M.S.S. Brooks / Physica B 206 & 207 (1995) 1-7

where the total ground state m o m e n t is the integral of the m o m e n t density given by rnZ(r) =/xB[ff(r ) + 2sZ(r)] in terms of the orbital angular m o m e n t u m and spin densities. Here ( J, ) , = 4~rlS Ji(Or)r2n, (r) dr]/m~ where a = s, 1 denotes the spin or orbital density and m o m e n t . Typical radial integrals are plotted in Fig. 2. Clearly ( j z ) contributes only when there is an orbital contribution to the moment. Furthermore the way in which (J2) contributes depends critically upon the relative signs and magnitudes of the spin and orbital moments. We illustrate this schematically in Fig. 2(b), (c) and (d) where we have drawn form factors for m~ and m~ parallel, m~ < m~ but m~ and m~ antiparallel, z z z and m~ > rn t with m, and rn~ again antiparallel. It is relatively difficult to extract the orbital m o m e n t when the spin and orbital contributions are parallel, as in a

2

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,

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'

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Fig. 3. The calculated actinide moments for part of the AnFe 2 series in several approximations. The dotted line labelled 'spin only' is a conventional spin polarized calculation. The dashed curve indicates the results when spin-orbit interaction is added, producing an orbital contribution to the moment, and the full curve indicates the results when orbital polarization is included in the calculations. The measured moments are the full squares. After Eriksson et al. [26].

I 0.5 sinelx (A"~1

0.5

sinel~ (A 41 Fig. 2. Schematic illustration of the spin and orbital contributions to the form factors obtained by adding the Bessel functions (a) in various ways depending upon whether the orbital and spin contributions are parallel or anti-parallel.

heavy rare earth or late transition metal, since the tail in the measured form factor arising from the dependence of (j2) upon the scattering vector, appears only as a b u m p at large scattering vectors. But when orbital and spin moments are antiparallel, especially if they are almost cancelling, the tail in the form factor arising from the orbital contribution to the magnetization density develops into a prominent b u m p which has been observed in several actinide compounds [30]. In the presence of both spin polarization and s p i n orbit interaction, there is a net current, and it is from

M.S.S. Brooks I Physica B 206 & 207 (1995) 1 - 7

this current that the orbital m o m e n t arises. The orbital m o m e n t density may be calculated from the orbital angular m o m e n t u m density, a readily computable quantity [26,21]. Calculations [26] for the uranium monochalcogenides, which are ferromagnets, showed that a large orbital contribution to the m o m e n t density is a c o m m o n feature in u r a n i u m compounds. In subsequent calculations for actinide compounds, the orbital contribution at the actinide site has been found to be large [26-28]. In particular, a series of actinide c o m p o u n d s where a thorough analysis has been made is the A n F e 2 series. The results are shown in Fig. 3. Here, as in the other compounds, the induced orbital m o m e n t s are very large compared with those obtained for 3d transition metals. However, although the calculated orbital moments are very large in actinides, it seems to be generally true that they are still too small compared with experiment [31] as may be seen from Fig. 3, especially for Pu compounds. Fig. 4 shows the calculated equation of state and site resolved magnetic moments in U F e 2 as a function of v o l u m e [32]. The measured uranium form factors of both U F e 2 and UNi 2 [33] showed that the cancellation between spin and orbital moments was almost corn-

5

plete. Other detailed studies have been made for UNi z [28] where there is appreciable scattering only at the uranium site [33]. The total m o m e n t is very small but the c o m p o u n d is nevertheless magnetic. The calculated and measured magnetic amplitudes are shown in Fig. 5. Comparison with experiment shows that the calculated absolute values of the orbital m o m e n t s are almost always too small [26,34]. This also seems to be true in Fe, Co and Ni [21,35], although the larger discrepancies for the actinides are more obvious. O n e factor that is missing in L S D A is interaction b e t w e e n the orbital moments. Such an interaction is well known to be important in atoms where it is responsible for H u n d ' s second rule. O n e way to approximate orbital interactions, which has had some success, has been suggested [26]. The functional dependence of the energy upon occupation n u m b e r in H a r t r e e - F o c k

0.15

l''''l

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1

/" TI.\

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"\,

0.10

~.00 ttic

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paromognmtic

---~.r,

f~

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E

T

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0.5 •

,

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0~

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I

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82 v

'

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]

(A3)

Fig. 4. The calculated equation of state (top panel) and site resolved magnetic moments as a function of pressure (bottom panel) for UFe 2. The curves labelled 'Pauli' are from spin polarized calculations and those labelled 'Pauli + so' are from

calculations with spin-orbit coupling included.

Fig. 5. A comparison of measured and calculated 5f magnetic amplitudes for UNi 2. The thick full curve is a fit to the experimental results [33]. The dot-dashed curve is from a self-consistent calculation with the uranium spin moment constrained to be equal to 0.47/xn yielding a total uranium moment of 0.082/xa. The dashed curve corresponds to a slightly different constraint on the uranium moment, such that it is equal to 0.07/~. After Severin et al.

6

M.S.S. Brooks / Physica B 206 & 207 (1995) 1-7

theory was approximated quite well by - ( 1 / 2 ) E 3 L ~ where E 2 is a Racah parameter (a linear combination of Slater C o u l o m b integrals). The differential of the orbital polarization energy with respect to occupation n u m b e r leads to different orbital energy levels when there is an orbital moment. The results for the A n F e 2 series are shown in Fig. 3. Applications to non-actinides such as Fe, Co and Ni and some cobalt compounds have also improved agreement with experiment for the orbital moments [21,35-37]. N o r m a n [37] has applied this and similar approximations to the transition metal oxides. More recently, the enhanced orbital m o m e n t s at Co sites in C o / P d multilayers have b e e n studied by Wu et al. [38] using magnetic circular X-ray dichroism. Comparison between the measurements and first principles calculations by Daalderop et al. [39] indicate that the orbital polarization correction is essential if agreement for the moments at the Co sites is to be obtained.

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[35] H.J.F. Jansen, J. Appl. Phys. 67 (1990) 4555; R. Coehoorn and G.H.O. Daalderop, J. Magn. Magn. Mater. 104-107 (1992) 1081. [36] G.H.O. Daalderop, EJ. Kelly and M.F.H. Schuurrnans, J. Magn. Magn. Mater. 104-107 (1992) 737. [37] M.R. Norman, Phys. Rev. Lett. 64 (1990) 1162. [38] Y. Wu, St6hr, B.D. Hermsmeier, M.G. Samant and D. Weller, Phys. Rev. Lett. 69 (1992) 2307.

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[39] G.H.O. Daalderop, P.J. Kelly and M.F.H. Schuurmans, Phys. Rev. B 44 (1992) 12054. [40] A. Hjelm, O. Eriksson and B. Johansson, Phys. Rev. Lett. 71 (1993) 1459. [41] R.C. Maglic, G.H. Lander, M.H. Mueller and R. Kleb, Phys. Rev. B 17 (1978) 308.