Valence degeneracy in cerium systems

Valence degeneracy in cerium systems

Journal of Magnetism and Magnetic Materials 177-181 (1998) 373-374 ~ ,~ ELSEVIER Journalof magnetism and magnetic materials Valence degeneracy in ...

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Journal of Magnetism and Magnetic Materials 177-181 (1998) 373-374

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Journalof magnetism and magnetic materials

Valence degeneracy in cerium systems Anna Delin*, B6rje Johansson Physics Department, Uppsala University, Box 530, S-751 21 Uppsala, Sweden

Abstract We have calculated the generalized cohesive energies for Ce-systems, using a full-potential linear muffin-tin orbital method and a generalized gradient-corrected density functional. Together with atomic coupling energies for cerium, the generalized cohesive energies are used to study the valence configuration degeneracy in CeA12 and CeX (X = N, P, As, Sb, Bi). We find that in all these systems, the fl and fo configurations are far from being degenerate. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Density functional calculations; Linear muffin-tin orbitals; Rare earth, intermediate valence; Rare earth pnictides; Rare earth - intermetallic compounds

The concept of intermediate valence is of central importance in the physics of rare-earth materials. Many cerium systems, e.g. a-Ce, CeN and CeA12, are often classified as being intermediate valence systems. These systems exhibit anomalies in e.g. volume, compressibility, thermal expansion, and magnetic susceptibility. The origin of this anomalous behaviour is often discussed theoretically in terms of a degeneracy between the trivalent fl[spd]3 configuration and the tetravalent f°[spd]4 configuration of the Ce atom. In this model, the 4f level is very close to the Fermi level and has acquired a small width due to hybridization with the conduction electrons. As a result, the number of 4f electrons will be less than one, hence the term 'intermediate valence'. However, the 4f state is still mainly localized. Nowadays, it seems that the term intermediate valence has, quite misleadingly, also come to include all cases of 4f hybridization, even the case when the hybridization is so strong that the electronic structure can be excellently described using an itinerant picture of the 4f-states, like e.g. in ~-Ce [1] and CeN [2]. In this paper, we calculate the energy difference between the two possible valence configurations for several Ce-systems, assuming that in the trivalent case, the 4f electron is completely localized, and in the tetravalent case, the 4f electron is completely promoted into the

conduction [spd]-band. This calculation cannot be done completely ab initio, since atomic coupling energies, for which the current approximations to the true density functional are known to give relatively inaccurate results, play a decisive rote in the rather delicate energy balance. However, we realize that these energies should be the same irrespective of the Ce-system due to the atomic nature of the localized trivalent 4f configuration, and therefore, it is still possible to calculate this energy balance, once an internal energy scale is set. This scale is provided by the trivalent and tetravalent states for Cemetal, where it is established that the energy difference is 2.0 eV [3, 4]. Fig. 1 illustrates the idea behind our calculation. The two generalized cohesive energies (this concept is defined below) E*(f 1) and E*(f °) can be calculated

*Corresponding author. Fax: +46 18 471 3524; e-mail: [email protected].

Fig. 1. Schematicenergy-balance diagram for Ce. The symbols are defined in the text.

TRIVALENT

TETRAVALENT -I----?~V

ATOM~

0304-8853/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved Pll S 0 3 0 4 - 88 5 3 ( 9 7 ) 0 0 6 9 3 - 8

................. t~d's2 [E'(fJ)

.....

E,(fo .................. f°[spd} '

374

A. Delin, B. Johansson / Journal of Magnetism and Magnetic Materials 177-181 (1998) 373 374

ab initio. E, which is the total energy difference between the tri- and tetravalent bulk states is the quantity to be determined. The full-drawn lines refer to the total energies including coupling energies within the 4f shell and between this shell and the 5d shell in Ce. The dashed lines refer to the total energies when the coupling energies, Ecoupling, are not taken into account. Thus, the involved Ecoupling terms are constant irrespective of Cesystem and are represented by the distances between the full lines and the above-lying dashed lines. Note that the size of each individual Ecouv~i,g remains undetermined. Only their total effect on the energy balance is relevant. Above, we introduced the generalized cohesive energy, E*. This quantity is the cohesive energy from which is subtracted the promotion energy caused by the valence change during condensation and the coupling energies within and between the open 4f and 5d shells in the atom [3, 4]. The generalized cohesive energy is easily calculated ab initio as the difference between the total energy of the spin-degenerate atomic and bulk ground states. The bulk calculations were performed using the fullpotential linear muffin-tin method developed by Wills [5]. In this method, no sha,pe approximation of the potential, wave functions or charge density is made. For the density functional, we used the generalized gradientcorrected functional by Perdew and Wang [6, 7]. The calculated energy differences are presented in Table 1. First, we notice that both CeN and CeAI2 are closer to degeneracy than the Ce metal itself. However, the energy difference is still almost 2 eV, which, of course, must be considered as being far from degeneracy. CeP is seen to have about the same energy difference as Ce metal. For the heavier Ce-pnictides, the energy difference

Table 1 Total energy difference E System

E (eV)

Ce metal CeA12 CeN CeP CeAs CeSb CeBi

2.0 1.61 1.71 1.98 2.47 2.78 2.99

increases with atomic number, reaching its highest value, 3 eV for CeBi. We thank J.M. Wills for letting us use his code. This work was financed by the Swedish Research Council for Engineering Sciences and the Swedish Natural Science Foundation.

References [1] [2] [3] [4] [5] [6] [7]

P. S6derlind et al., Phys. Rev. B 51 (1995) 4618. A. Delin et al., Phys. Rev. B 55 (1997) R10173. B. Johansson, Phys. Rev. B 20 (1979) 1315. B. Johansson, J. Phys. F 7 (1977) 877. J. M. Wills, unpublished. J.P. Perdew, Y. Wang, Phys. Rev. B 40 (1989) 3399. J.P. Perdew, in: P. Ziesche, H. Eschrig (Eds.), Electronic Structure of Solids 1991, vol. 1l, Akademie Verlag, Berlin, 1991.