VOLUME 71, NUMBER 13 PHYSICAL REVIEW LETTERS 27 SEPTEMBER 1993

Comment on "State-Tracking First-PrinciplesDetermination of Magnetocrystalline Anisotropy"

In a recent Letter by Wang, Wu, and Freeman(WWF) [1] where a new "state-tracking" method is pro-posed for calculating magnetocrystalline anisotropy ener-gies (MAE), a use of the force theorem [2] is proposedwhich is substantially incorrect and violates Fermi-Diracstatistics. It is the purpose of this Comment to point outthat this drastic step is quite unjustified and indeed un-necessary; standard Brillouin zone (BZ) integrationschemes yield comparable stability [3] while treating con-tributions to the MAE arising from the lifting of degen-eracies close to the Fermi energy [4] on the same footingas contributions arising from the coupling of states far re-moved from the Fermi energy. We also wish to rectify anumber of other incorrect and misleading statementsmade by WWF.

Correct application of the force theorem requires thatthe energy bands be occupied "blindly" according toFermi-Dirac statistics. That the blind band filling neednot present any problem (such as "spikes") is illustratedin Fig. 1 where we show the results of ab initio calcula-tions of the MAE for a free standing [111]Co monolayerfor three diFerent Fermi levels as a function of the num-ber of BZ sampling points. Very satisfactory convergencewith only 24X24 points (for 9.4 electrons, 48X48) isfound. The small number of points required merely

reAects the simple electronic structure of a free standingtwo-dimensional (2D) monolayer. For 3D systems suchas Coi/A'2 magnetic multilayers [3], more samplingpoints are required because of the dispersion in the zdirection. Well converged anisotropy energies were ob-tained for these multilayers using 1176 (or 3600) sam-pling points in the full 3D BZ. This sampling corre-sponds to a total of only 196 (or 400) points in the basalplane which should be compared with the 100 and 421points used by WWF for their two-dimensional BZ.

In Ref. [5] we did not conclude that only 10 2 or evenfewer electrons at the Fermi energy contribute to MAEas stated by WWF; states far removed from the Fermienergy also contribute. Contrary to the claim made byWWF, the MAE can be changed drastically by dopingwith less than a few percent of impurities. This is welldocumented for doping of Ni with Ru, Rh, and Cu [6].Finally, the claim that lifting of degeneracies is a smalleH'ect is at variance with our experience with the Co/Xmultilayers where we are able to identify the lifting of adouble degeneracy at the K point as the single most im-portant contribution to the perpendicular anisotropy ofthese multilayers [3,7].

6. H. O. Daalderop, P. J. Kelly,and M. F. H. Schuurmans

Philips Research LaboratoriesProf. Holstlaan 45656 AA Eindhoven, The Netherlands

T T

144 96T T T

48 24 12

8.2 e

Received 17 May 1993PACS numbers: 75.30.Pd, 75.70.Ak

0.7 I I I I I I II

—1.0

9.0 e

-2.0 I I I IIIII I I I I IIIII I I I I IIII

-3.0

—3.5

9.4 e

aI IVI llil I I I I I IIII I I I I I IIII I I t I I III

10 10 10

2S (SBZ)

FIG. 1. Convergence of the anisotropy energy of a Co mono-layer with lattice constant a 2.51 A at three band fillings, 8.2,9.0, and 9.4 electrons, as a function of the area, s, of the tri-angular surface element used to perform the two-dimensionalBz integral. s is given as a fraction of the area of the 20 Bz,Sgz. The number of divisions, N, of the reciprocal lattice vec-tors corresponding to each surface element is indicated at thetop of the figure. 2s% Sgz. The horizontal lines were ob-tained by a least squares fit through the data points using aweight 1/s for each data point and denote the converged value.A negative value of the MAE corresponds to in-plane magneti-zation.

[1] D. Wang, R. Wu, and A. J. Freeman, Phys. Rev. Lett. 70,869 (1993).

[2] A. R. Mackintosh and O. K. Andersen, in Electrons atthe Fermi Surface, edited by M. Springford (CambridgeUniv. Press, Cambridge, 1980); M. Weinert, R. E. Wat-son, and J. W. Davenport, Phys. Rev. B 32, 1215 (1985).

[3] G. H. O. Daalderop, P. J. Kelley, and M. F. H. Schuur-mans, Phys. Rev. B 42, 7270 (1990); in Science andTechnology of Nanostructured Magnetic Materials, edit-ed by G. C. Hadjipanayis and G. A. Prinz, NATO Ad-vanced Study Institutes Series (Plenum, New York,1991), p. 185; G. H. O. Daalderop, P. J. Kelly, andF. J. A. den Broeder, Phys. Rev. Lett. 6$, 682 (1992).

[4] The surface pair coupling (SPC) contribution whichWWF claim may be neglected [1].

[5] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuur-mans, Phys. Rev. B 41, 11919 (1990).

[6] 1Vumerical Data and Functional Relationships in Scienceand Technology, edited by H. P. J. %'ijn, Landolt-Bornstein, New Series, Volumes III/19a and III/19b(Springer, Berlin, 1988).

[7] G. H. O. Daalderop, P. J. Kelly, and M. F. Schuurmans(to be published).

0031-9007/93/71 (13)/2165 (1)$06.001993 The American Physical Society