V O L U M E 71, N U M B E R 13 P H Y S I C A L R E V I E W L E T T E R S 27 SEPTEMBER 1993

Wang, Wu, and Freeman Reply: Daalderop, Kelly, and Schuurmans [1] (DKS) made a number of factual and physical errors in describing our work [2] as part of their defense of the "blind Fermi filling" (BFF) procedure that was widely used in first principles calculations of the magnetocrystalline anisotropy (MCA). We attempt to clarify and correct these errors and misunderstandings concerning the state-tracking (ST) method we developed to overcome the problem with BFF.

Instabilities exist in MCA calculations with BFF. — This was well known and well documented for bulk, surface, and interface systems by (a) the "curious" be­havior (described by Gay and Richter [3]) of the conver­gence of MCA with respect to the number of k points and its "anomalous" deviations at some k points due to "unidentified" reasons: A similar example was also given by DKS [4] themselves for bulk Ni; (b) the sharp "spikes" ( — ± 4 eV) in the MCA contribution in the Brillouin zone (BZ) for even the simplest monolayer films; and (c) the vigorously oscillating behavior of the theoretical band filling dependence of the MCA in the rigid band approximation. DKS [4] results for bulk Ni gave one example that the calculated MCA changes sign 3 times when the number of valence electrons iq) in­creases from 10.0 to 10.6; even more vigorous oscillations were shown [5] for bcc Fe where the MCA changes sign 4 times over a range of only 0.07 electron (see Fig. 6 in Ref. [5]).

Nevertheless, DKS assert in their Comment that BFF does "not present any problem," give the special example of the C o d 11) monolayer, and neglect all the above cited results (including their own) that were obtained since the first surface MCA studies [3]. This is strange, since they specifically stated "for some values of q it seems to be possible to extrapolate the calculated value of AE^iq) to an infinitely dense mesh whereas for other values of q, and in particular for q = 10.0, the convergence is not very systematic" [4], but gave no explanation and did not know the origin of this instability.

Origin of the instability and justification of the ST approach. — We [2] pointed out the origin of this instabil­ity as arising from the determination of the perturbed oc­cupied states by a BFF procedure when the force theorem was adopted for calculating the change of the total ener­gy arising from the spin-orbit coupling (SOC) perturba­tion. The ST approach properly identifies the perturbed occupied states in non-self-consistent calculations by referring to the wave function properties. In this pro­cedure, we first determine the occupied states according to Fermi-Dirac statistics (FDS) and then calculate the effect of the perturbation on this set of states. It is by no means a replacement for nor a violation of FDS (in 3D transition metal systems, the SOC is much weaker than the crystalline field).

ST gives stable MCA results.—The stable MCA re­sults obtained with ST [2] are due to the elimination of this randomness. Thus, the correct (smooth) distribution

2166

of MCA over the BZ is achieved and the instability listed above disappears; such a smooth MCA distribution in the BZ [6] has never been obtained with BFF. Contrary to DKS's assertion, the stability shown [2] is not a result of the simplicity of the electronic structure of a monolayer film: The strong fluctuations shown previously very clear­ly on the same system (Fe monolayer) [3,7,8] disap­peared in our ST calculation [6]. Moreover, we have also obtained stable results for interface A'2/Coi/A'2 sandwich systems (A' = Cu and Pd) with the same small number of k points as for the monolayer; this again shows the ad­vantage of using the ST, in that, contrary to previous be­liefs, much smaller numbers of k points give good MCA results even for many layer slabs.

Inconsistency between previous theoretical band filling dependence and experiments.— By contrast to the sensi­tive and drastic behavior of the previously calculated band filling dependence of the MCA, we pointed out that "the MCA is a short-ranged intrinsic property of fer­romagnetic materials and it can hardly be aff'ected by a small amount (a few percent) of impurities" [2]. DSK quoted the MCA decrease of Ni-Ru (Rh,Cu) alloys with respect to the increase of Ru(Rh,Cu) content as evidence in defense of BFF but neglected experimental results for a large variety of other alloys; they also failed to note that even in the cases they quoted, the rapid MCA de­crease is due to the decrease of the magnetic moments with doping as tabulated on almost the same pages in the Landolt-Bornstein reference they cited. Moreover, even in these cases, MCA is also a smooth function of the composition—unlike the rapid oscillations with BFF.

Finally, the statement quoted in our paper [2] that "the dominant contribution to the anisotropy energy came from doubly degenerate bands close to the Fermi energy" was made by Strange et al. [5] in reviewing previous pub­lications, not in Ref. [4]. However, the sensitivity of the MCA results was certainly shown in both papers [4,5] as stated by us [2].

Ding-sheng Wang, Ru-qian Wu, and A. J. Freeman Department of Physics and Astronomy Northwestern University, Evanston, Illinois 60208

Received 9 June 1993 PACS numbers: 75.30.Pd, 75.70.Ak

[1] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuur­mans, preceding Comment, Phys. Rev. Lett. 71, 2165 (1993).

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

[3] J. G. Gay and R. Richter, Phys. Rev. Lett. 56, 2728 (1986); J. Appl. Phys. 61, 3362 (1987).

[4] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuur­mans, Phys. Rev. B 41, 11 919 (1990).

[5] P. Strange et al., Physica (Amsterdam) 172B, 51 (1991), [6] D. S. Wang, R. Wu, and A. J. Freeman, Phys. Rev. B 47,

14932 (1993). [7] R. Richter and J. G. Gay, Mater. Res. Soc. Symp. Proc.

151, 3 (1989). [8] C. Li et al., Phys. Rev. B 42, 5433 (1990).