FERROMAGNETIC EXCITATIONS IN HEXAGONAL COBALT

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FERROMAGNETIC EXCITATIONS IN HEXAGONAL COBALT

P. George, E. Thompson

To cite this version:

P. George, E. Thompson. FERROMAGNETIC EXCITATIONS IN HEXAGONAL COBALT. Journal

de Physique Colloques, 1971, 32 (C1), pp.C1-820-C1-821. �10.1051/jphyscol:19711290�. �jpa-00214320�

JOURNAL DE PHYSIQUE Colloque C I , supplkment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1 - ⁸²⁰

FERROMAGNETIC EXCITATIONS IN HEXAGONAL COBALT (*) P. K. GEORGE (**) and E. D. THOMPSON

Electrical Sciences and Applied Physics Division and Center for the Study of Materials Case Western Reserve University, Cleveland, Ohio 44106

RBsumB. - Sont presentes ici les resultats de calculs de l'knergie des magnons et des excitations de Stoner pour le cobalt hexagonal. 11s sont fond& sur une representation & liaison forte de la structure de la banded ; l'influence del'inter- action des vecteurs d'onde des electrons dans la determination de l'intensite de I'integrale dyechange est kgalement dis- cutke. Une limite supkieure de la contribution intra-atomique a l'intensite d'kchange ne conduit qu'8 un resultat trks inferieur a la valeur experimentale ; il est necessaire d'introduire les contributions interatomiques pour obtenir un bon accord. La dispersion des magnons et les excitations de Stoner le long des axes principaux sont presentees pour un dedou- blement de 0,91 eV et une Cnergie d'echange interatomique de 0,016 eV.

Abstract. - The results of a multiple band calculation of the magnon energy and Stoner excitations for hcp cobalt, based upon a tight-binding representation of the d-band structure, are presented and the role of a wave vector dependent electron interaction in determining the exchange stiffness of cobalt is discussed. An upper limit on the intra-atomic contri- bution to the exchange stiffness is found to lie significantly below the experimental value and accordingly it is found neces- sary to include interatomic exchange to obtain agreement with experiment. The magnon dispersion relationand Stoner excitations along the principal axis directions are presented for a band splitting of .91 eV and interatomic (intercellular) exchange of 16 meV.

Spin wave resonance and inelastic neutron scatter- ing experiments have provided valuable experimental information on the excited spin states of the ferro- magnetic 3 d transition metals. With the increase in understanding of the electronic structure of these materials, it has become possible to attempt theoretical calculations of the low energy portions of these states.

Various forms of the itinerant electron model have been taken as the basis of studies of fcc nickel 11-41 and bcc iron [5]. We report here some interesting results obtained in an extension of this work to include cobalt.

The band structure of hcp cobalt [6-91 is inherently more complicated than that of iron or nickel due to its two atoms per unit cell. Since each atom gives rise to five 3 d states the band structure for Co contains ten 3 d bands. For a calculation of the magnon energy (and Stoner excitations) it is necessary either to determine these bands from first ~ r i n c i ~ l e s or to obtain a good

facsimile by fitting to an existing calculation. In our treatment we fitted the APW energy bands of Hodges and Ehrenreich [7] using the Slater-Koster interpola- tion scheme with nearest-neighbor interactions [lo, 111. Emphasis was placed on getting a good fit at the higher energy states and as is implied band hybridiza- tion was neglected. We assumed that the majority spin 3 d bands (10) were entirely filled (5 electrons per atom) while the minority spin bands (10) contained 1.56 holes per atom. It was determined that five hole bands were involved. The energy bands were then split by an amount A such that the configuration for the ferro- magnetic specie was as is shown in figure 1. A was treated as an arbitrary parameter-limited only by the condition that the ferromagnet be strong.

Our original calculations of the magnon energy hw(q) were made using the random phase approxima-

(*) Research sponsored by the Air Force Office of Scientific Research. Office of Aerosoace Research. U. S. A. F.. under AFOSR 'grant 68-1484.

(**) Present address : Technische Hoaeschool Delft. Delft.

he Netherlands. -

DENSITY OF STATES [PER ATOM PER RYDBERG)

10 30 20 10 0 ID 20 P O LO

1

1 HEXAGONAL

FERMl LEVEL FERROMAGNETIC COBALT

---a 4

-

MINORITY SPINS

$

'-%

4 - Z

3 -

FIG. 1. - Tight-Binding Density of States for Ferromagnetic hcp Cobalt. The majority spin electrons fill all states in majority 3 d bands while minority spins allow for 1.56 holes per atom.

Five hole bands are involved in determining the calculated magnetic properties. The exchange splitting A is treated as a

parameter.

tion (neglecting interband Stoner excitations) follow- ing the procedure described by Thompson and Mook [5] in an earlier paper on iron-with the excep- tion that interatomic exchange was set equal to zero.

The summation over the Brillouin zone involved in evaluating the dispersion relation was performed using 16,800 points situated on two interpenetrating rectangular lattices. The results of this calculation in the long wavelength limit can be summarized by

where D[001] is the exchange stiffness determined for the [001] direction,

is the Fermi energy measured from the top of the d bands (0.72 eV) and A is the exchange splitting measured in eV. In our preliminary calculations we took the interatomic exchange J to be zero. In view of the fact that the experimentally determined exchange stiffness D[001] is in the neigh- bourhood of 560 meV at T = 0 OK [I21 i t is difficult t o reconcile eq. (1) with a purely intra-atomic description.

It is true that we have ignored interband transitions, however, examination of the form of the results for the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711290

FERROMAGNETIC EXCITATIONS IN HEXAGONAL COBALT acoustic mode [4, 131 (long wavelength limit with

only intra-atomic interactions) indicates that this will not affect the general form of the solution for D. We would expect that the result obtained in eq. (1) for A = oo would place an upper bound on the exchange stiffness due to intra-atomic interactions. The effect of correlation is expected to enhance the negative term in D[001] and further restrict the magnitude of the exchange stiffness.

The question arises whether these and other previous results depend critically upon the choice of energy bands. While no one has investigated this aspect thoroughly for any of the 3d transition metals there are indications that they do not. The recent work on nickel by Callaway and Zhang [4] appears to be in good agreement with the earlier work of Thompson and Myers [I] insofar as the relationship between exchange splitting and exchange stiffness goes although the results are somewhat dependent upon the mixing parameter B. In their susceptibility work on nickel Windsor and al. [14] indicate that they obtain best absolute agreement with experiment when they include a wave vector dependence in Ieff corresponding to a nearest-neighbour exchange parameter of. 025 eV.

This is in remarkably good accord with that found by Mook and al. [3]. Considering the diverse origins of the above information the overall situation seems to be encouraging. With regard to the magnitude of the exchange stiffness in eq. (1) we might comment that a similar calculatioil for iron (in which A is fixed by moment considerations) yields a value of D = - 106 meV. a2 [5], in considerable disagreement with the experimental value of 270 meV. A2.

As a remedy to the above situation we added a wave vector dependence to the effective electron interaction in the form of an interatomic exchange term J(q) defined in terms of lattice sites (as opposed t o atomic sites as in the case for the Heisenberg model). The two independent parameters associated with lattice transla- tions in and normal to the basal plane were related by assuming that J(q) was isotropic for small wave vectors.

The effect of adding interatomic exchange is reflected in the 28.6 J term in eq. (1). Here J is the value asso- ciated with nearest neighbour lattice translations R in the basal plane.

From eq. (1) it is evident that for reasonable values of the exchange stiffness [I 51 the dominant contribution to D[001] arises from the interatomic component. For a band splitting of A = .91 eV it is necessary to include 16 meV of intercellular exchange to reproduce the experimental data. In figure 2 we show the correspond-

r

NORMALIZED WAVE VECTOR q c / 2 r

FIG. 2. - Magnon Dispersion Relation and Stoner Excitations along Principal Axis Directions. Intercellular exchange has been included in calculating the dispersion relation in order to obtain agreement between theory and experiment along the [001]

direction. The notation is for a basis in which the two vectors in the basal plane are at an angle of 60°.

ing results for the magnon dispersion relation and Stoner excitations throughout the zone along the three principal axis directions (basis vectors at 600 in basal plane). From these results it appears unlikely that the acoustical branch along the [OOl] direction will ever intersect the continuum. We expect, however, inter- sections in the basal plane along the [I101 and [I001 directions at about qc/n ⁼ 1. The exact position of the intersection is dependent upon the value of the band splitting. I n a more general treatment we expect to find optical modes in addition to the acoustical mode presented here. These higher energy modes are absent in our treatment due to our assumption of band independent interactions which effectively converts our problem into that for a single band.

rences

THOMPSON (E. D.) and MYERS (J. J.), Phys. Rev., [8] WONG ( K . C.), Solid State Phys., 1970, 3 C , 378.

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MOOK (H. A.), NICKLOW (R. M.), THOMPSON ^@. D.) [ l o ] SLATER (J. C.) and KOSTER (G. F.), Phys. Rev., 1954, and WILK~NSON (M. K.), J. Appl. Phys., 1969, 94, 1498.

40, 1450. [ l l ] MIASK (M.), Phys. Rev., 1957, 107, 92.

CALLAWAY (J.) and ZHANG (H. M.), Phys. Rev., 1970, [12] SHIRANE (G.), MINKIEWICZ ^(V. J.) and NATHANS (R.),

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