The electronic structure of cubic laves phases: ZrZn2

Solid State Communications Vol. 9, pp. 2039—2043, 1971.

Pergamon Press.

Printed in Great Britain

THE ELECTRONIC STRUCTURE OF CUBIC LAVES PHASES: ZrZn~ D.D. Koelling Department of Physics, Northwestern University, Evanston, Illinois 60201 David Linton Johnsont and S. Kirkpatrick James Franck Institute, University of Chicago, Chicago, Illinois 60637 F.M. Mueller Argonne National Laboratory, Argonne, Illinois 60439 and Department of Physics, Northern Illinois University, DeKaib, Illinois 60115

(Received 4 August 1971 by R.H. Silsbee)

A simplified model for the calculation of the electronic structure of Laves phase materials is developed that focuses on the effects of d bonds between tetrahedrally coordinated sites. This model is shown to be in good agreement with all of the existing data on ZrZn 2. Indirect evidence for the validity of the model in other systems is sketched.

CHEMICAL bonding in tetrahedrally coordinated, semiconducting materials has been of interest to solid-state1 has physicists for many Much been devoted to years. deriving the recent effort charge density anisotropy and structural stability

such system, the cubic Laves phase compound ZrZn~,in which we have neglected the scattering produced the Zn ions andco-ordinated focus on thetransscattering due toby the tetrahedrally ition metal Zr ions.

commonly thought of as ‘bonding’, from the electronic energy band structure. Transition metals, with incomplete d shells, should also show bonding effects in a tetrahedral environment. This occurs at metallic packing densities in the Laves phases2 of intermetallic AB 2 compounds in which the A’s B’s occupy are an tetrahedrally interpenetrating coordinated, lattice and that the fills

Early interest on four properties: heat,3 the nuclear neutron diffraction

in this material had centred the susceptibility,4 the specific magnetic resonance,3 and a study of the spin density.5

All of these measurements complicated by 6 A were related series of commetallurgical pounds, ZrCu~Al problems. 2.~, eemsshould, to be free of suggest these 6 sand as we preparation problems below, have an electronic structure similar to that of ZrZn 2. We urge further examination of these

the space. this note we report of theone firstremaining calculation of theInelectronic structure

Supported by the United States Atomic Energy Commission, Air Force Office of Scientific Research, Advanced Research Projects Administration, and Army Research Office (Durham). ~ Fannie and John Hertz Foundation Fellow,

and similar materials with a view to the study of d bonding.

*

.

.

The tendency to itinerant ferromagnetism may be understood completely from the fact that the 2039

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ELECTRONIC STRUCTURE OF CUBIC LAVES PHASES

Fermi level lies at the top of a sharp narrow dband peak. Experimentally this peak has a height of about 2.7 states/eV/Zr atom/spin and a width of roughly 0.2eV in ZrCu 11 Al09, and a height of 5.4 states/eV/Zr atom/spin and a width of 0.15eV in ZrZn2. (The experimental results have removed enhancement effects, approximately.) Although the total spin moment is small (~= 0.18 ,LLB/atom), the spin density is highly delocalized and, in fact, has a maximum between nearest neighbor zirconium ions. Also, indirect thermochemical evidence exists for d bonding in the Laves phases. From the heats of formation7 available some one finds for that, if T20isLaves-phase a transition compounds, metal ion, and M is a simple metal ion (i.e., Mg, Cd Li and K) (AH)TM>>

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Vol. 9, No. 23

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FIG. 1. Energy of ZrZri2. The the symmetry labelsband havestructure been assigned by using compatability relations together with our knowledge of the double group representations at and the band structure of the group IV semicon-

r

(—AH)~T 2> (—AH)M~

,

where (—LVI) is the heat of formation (enthalpy). cubicOur TMcalculation is greatly simplified for a 2 compound such as ZrZn2 by the observation that the most important features of the electronic structure are essentially unaffected by scattering from the sublattice of simple metal ions, which we shall proceed to neglect. The reason is twofold: First, the d states are contributed to by the transition metal ions only, whereas each simple metal site will scatter as a weak pseudopotential~ Second, the full space group of the AB2 crystal is identical to that of the A sublattice alone. Thus scattering from the B sublattice of pseudopotentials may, through its zero wavevector component, shift the planewavelike part of the A sublattice band structure with respect to the d-band portion, but neither the dbands nor the states hybridizing with them are otherwise strongly affected. Scattering by higher plane waves has little effect on the occupied portion of the band structure, To describe ZrZn2, therefore, we have calculated a band structure for zirconium in a diamond lattice of unit cell edge 7.4 A. By neglecting the zinc potentials, we reduce the difficult six atom/unit cell calculation to a well-known case with two atoms/unit cell. The results of 9the nonrelativistic augmented plane wave (APW)

ductors. calculations are shown in Fig. 1. The muffin-tin potential used was derived a zirconium 25s2 configuration,10 and from Slater-free-electron 4d exchange (a = 1) was used. The lowest 11 bands were found at 89 equally spaced points in 1/48th of the f.c.c. Brillouin zone (82). These were used in conjunction with a quadratic interpolation scheme (QUAD)~ to construct a density-of-states histogram with 0.OO2Ry resolution, shown in Fig. 2, through Monte Carlo integration at 64,000 random points in 1/48th of the BZ. The many flat bends seen clustered near the Fermi level in Fig. 1 yield the dramatic peak at EF shown in Fig. 2. From the symmetries of the bands plotted in Fig. 1 we can conclude that this large peak consists entirely of states of d-like bonding character. The lower peak near —0.7eV consists of hybridized states of s, p, and d character, whereas the peaks seen in Fig. 2 above the Fermi level are all essentially of d character. Thus we expect on the basis of both the strong critical point structure exhibited in Fig. 2 and the existence of allowed p to d direct transitions, that optical measurements should rather easily confirm or deny our one-electron model band structure for the TM2 Laves phases.

Vol. 9, No. 23

ELECThONIC STI~UCTUREOF CUBIC LAVES PHASES

2041

of 0.005 Ry. This fit requires full, three-centred z

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Zr Zn 5

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ENERGY IN ELECTRON VOLTS

d-overlap terms, but only the first two zirconium neighbor shells were included. From the success of the model, we conclude that the d-state overand b.c.c. iron,’5 for which lap in the Laves-phase3”4 ferromagnetic is comparable to an that accurate in f.c.c. tight-binding nickel’ description of the dbands is possible in terms of overlap with the twelve or eight, respectively, nearest neighbors Oflly.

The eigenvectors of the model Hamiltonian support our conclusions about the d-bonding nature of the large peak in the density of states at EF. The flat bands near EF in Fig.1 are cornpletely unhybridized along symmetry lines, and were present for any reasonable choice of parameters of the model Hamiltonian. In general, the

FIG. 2. The density of electron states of ZrZn 2. As explained in the text, the weak scattering by the Zn ions have been ignored in our model calculations but should introduce only minor modifications to the large peak structure exhibited here.

lowest four bands were found to consist of two that hybridize over most of the BZ and two rather flat bands that do not, 12 in close analogy to a typical semiconductor valence band. We conclude

The peak at EF in Fig. 2 has a maximum value of 5.9 states/eV/Zr atom/spin, a width of 0.2eV and contains 1.0 states/Zr atom/spin. ~ll three numbers are in remarkable agreement with the experimental estimates. This value of V(E~), used with the specific-heat data cited in reference 3 for ZrZn ~, in which the Fermi level is assumed to lie at the peak of the density of states, implies an enhancement of 40%, only slightly less than that of common transition metals such as Pd. The agreement of the widths must be regarded as fortuitous for two reasons. First, 6 is somewhat dependent on the the estimated shape assumed value for the model density of states. Second, disorder present in the ZrZn,• 9 samples, due to either vacancies or excess atoms peaks takinginzinc sites, should broadenzirconium the calculated

Zr atom, are a consequence of thL critical point structure imposed by the diamond lattice, and will also be observed when other transition metals are substituted for zirconium. This affords the possibility of understanding the experimental data of HfZn 2 and related compounds 17 within the general picture advanced here for ZrZn2.

Fig. 2 somewhat.

that of Pickart et a!., is that we find that the largely density-of-states peak is dominated by d character, whereas the previous workers assumed

We have also analyzed the zirconium bands by a model Hamiltonian with a mixed basis of 10 tight-binding d states and 6 plane wave-like OPW states, 12,13 the results of which will be considered briefly. (Other than this increase in basis functions the model Hamiltonian is similar to those of reference 12 and 13.) This model has, at present, fit the lowest 7 bands of Fig. 1 with an rrns accuracy

therefore, that the lowest two peaks in the density of states of Fig. 2, containing almc~tfour states/

If we then assume that the large peak near the Fermi energy consists entirely of d-bonding orbitals and split the bands by means of a weak molecular field M of 1500kG (or /iBM =total 0.017eV) the experimentally observed momenttoofobtain O.l8~iB/Zr atom, we obtain a net spin density that rather closely approximates the observedatoms. spin 5 with a peak between the zirconium density, The essential difference between our model and

that about one-half the total moment was the resuit of (s-like) conduction electron polarization. Parenthetically we note that the radial charge density of our / = 2 radial wavefunction calculated in the APW at the Fermi energy and the atomic zirconium d-state wavefunction of Herman and Skiilrnan, for a configuration 4d25s2, match to

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ELECTRONIC STRUCTURE OF SUBIC LAVES PHASES

about 3%, with our charge density being slightly more spread out and larger near the muffin-tin radius of 2.99 atomic units. We conclude that the states near the Fermi energy of ZrZn 2 are welldescribed by atomic tight-binding functions. 8 of these functions is Corrections to the tails’ such as to make the actual bonding even bigger than we estimate here. We hope, through further and more accurate calculations 19 of the charge and spin densities to place our conclusion re-

Vol. 9, No. 23

garding the bonding model on an even firmer basis. We believe, however that the major features of the model presented here based on strong d-band bonds will remain. Acknowledgement — We wish to acknowledge useful conversations with W. Brinkman, M.H. Cohen, J. Darby, S. Doniach, G.S. Knapp, B. Veal; S. Katilavas for programming assistance, and the computation center of Argonne National Laboratory for excellent service.

REFERENCES

23, p.23 (1970); WALTER J.P. and COHEN M.H., Phys. Rev. Leit.

1.

PHILLIPS J.C., Physics Today 26, 17 (1971).

2.

See, for example, the book Inter-metallic Compounds (edited by WESTBROOK J.H.), Wiley, New York (1967).

3.

KNAPP G.S., FRADIN F.Y. and CULBERT H.V., J. app!. Phys. 42, 1341 (1971) and references therein; see also WOHLFARTH E.P., J. appi. Phys. 39, 1061 (1965).

4.

KNAPP G.S., CORENZWIT E. and COOPER A.S., Phys. Rev. (to be published).

5.

SHIRANE G., NATHANS R., PICKART S.J. and ALPERIN H.A., 1964, p.223.

6.

DARBY J. and KNAPP G.S., private communications.

7.

ROBINSON P.M. and BEVER M.B., intermetallic Compounds, ibid., p.52.

8.

HARRISON W.A., Pseudopotentials in the Theory of Metals, Benjamin, New York, (1960), and references contained therein. -

9.

SLATER J.C., Phys. Rev. 51, 846 (1937).

mt.

Conf. of Magnetism, Nottingham,

10.

HERMAN F. and SKILLMAN S., Atomic Structure Calculations, Prentice—Hall, Englewood Cliffs, N.J., (1963).

11.

MUELLER F.M., GARLAND J.W., COHEN M.H. and BENNEMANN K.H., Ann. Phys. to appear.

12.

JOHNSON D.L., (unpublished).

13.

HODGES L., EHRENREICH H. and LANG N.D., Phys. Rev. 152, 505 (1966); MUELLER F.M., Phys. Rev. 153, 659 (1967).

14.

ZORNBERG E.I., Phys. Rev. 81, 244 (1970).

15.

CORNWELL J.F., HUM D.M. and WONG K.C., Phys. Lett. 26A, 365 (1968); MAGLIC R. and MUELLER F.M., mt. J. Magnetism 1, 610 (1971); DUFF K.J. and DAS T.P., Phys. Rev. B3, 192, 2294 (1970).

16.

KNAPP G.S., VEAL B.W. and CULBERT H.V.,

17.

OGAWA S., Phys. Lett. 25A, 516 (1967).

18.

HEINE V., Phys. Rev. 153, 673 (1967).

19.

JOHNSON D.L., KIRKPATRICK S., KOELLING D.D. and MUELLER F.M., (unpublished).

mt.

J. Magnetism, to be published.

Vol. 9, No. 23

ELECTRONIC STRUCTURE OF CUBIC LAVES PHASES

vereinfachtes Modell ztw Berechnung der elektronischen Struktur von Laves Phasen wurde entwickelt mit Betonung des Binflusses von Em

d-Bindunge ri zwischen tetrahedrisch koord inierten Plaetzen. Es wird gezeigt, dass dieses Model! mit allen vorhandenen Daten von ZrZn 2 gut uebereinstimmt. Es werden Hinweise fuer die Gueltigkeit des Modells in anderen Systemen diskutiert.

2043