327

MAGNETIC ANISOTROPY IN SPIN DENSITY WAVE ANTIFERROMAGNETS

J.W. A L L E N Xerox Palo Alto Research Center, Palo Alto, California 94304, USA

and

C.Y. Y O U N G * Northeastern University, Boston, Massachusetts, USA

A theory of uniaxial magnetic anisotropy due to the spin-orbit and dipole-dipole interactions in an incommensurate spin density wave antiferromagnet like chromium has been constructed for a simplified band model. The temperature dependence of the anisotropy from both mechanisms is that of the antiferromagnetic gap squared, and therefore, will not explain the occurrence of the spin-flip transition in chromium. This indicates that either the detailed band structure must be taken into account, or an additional mechanism is operative.

Below T N = 312 K chromium metal is antifer- romagneticai ly ordered with an incommensura te linear spin density wave (SDW) [1]. As the tempera ture is lowered through Ts~= 123.5 K, the SDW polarization changes f rom transverse (AF1 phase) to longitudinal (AF2 phase) [2]. The fact that the magnetization, but not the SDW Q-vector , changes direction shows that in the single-Q state chromium has a uniaxial magnetic anisotropy with respect to the Q-direction, which is along one of the original cubic axes. For the commensura te -Q, simple ant iferromag- netic ordering, the magnetic anisotropy would have cubic symmetry . The spin-flip transition can be described phenomenological ly by as- suming a uniaxial anisotropy with a tem- pera ture-dependent coefficient that changes sign at Tsf [3]. A microscopic theory of the spin-flip transition then entails a microscopic theory of the anisotropy coefficient and its temperature dependence. Although there is considerable literature on magnetic anisotropy for itinerant fe r romagnets (not yet entirely successful), there is no previous work for itinerant ant i ferromag- nets. This short paper summarizes an SDW anisotropy theory the details of which will be presented elsewhere.

The theory is pat terned after that for fer- romagnets [4]. There it is assumed that the magnetic state can be described by an ex- change-split band structure which neglects sp in- orbit coupling. The anisotropy calculation then accounts for the changes in the free energy due to the effect of spin-orbi t coupling on the ex-

*Present address: Q.E.I. Company, Bedford, Mass., USA.

change-split band structure. Per turbat ion theory is used to find the energy shift of orbitally nondegenerate occupied states due to mixing with unoccupied states. It appears that this contribution to the anisotropy has a magnitude too small and a tempera ture dependence too slow to accord with experiment . For orbitally degenerate states at the Fermi level a secular equation must be solved, and the energy change is larger and very sensitive to the position of the Fermi level. Since different portions of the Fermi surface make contributions of opposi te sign to the anisotropy, and have somewhat different tempera ture dependences , the net anisotropy is very sensitive to the position of the Fermi level, and hence to the exchange splitting, so that a fast tempera ture dependence can result. The implications for this theory of recent results suggesting a temperature-in- dependent exchange splitting in several fer- romagnetic metals [5] have not been explored.

In the present calculation the incommensura te ant iferromagnetic state is modeled by three bands, ak, bk + Q, b k - Q, as in the work [6] of Young and Sokoloff (YS) on SDW harmonics. These three bands near the Fermi level are coupled by the spin-orbi t interaction to three higher bands, ck, dk ± Q. All states are taken to be orbitally nondegenerate for simplicity and because this appears to be true for the nesting bands in chromium. By including ant iferromag- netic state couplings (gaps) only among the up- per bands and lower bands, but not between them, it is possible to utilize the results of YS directly to obtain the 6-band Green ' s function matrix. If Eg, the energy separat ion of upper

Physica 86-88B (1977) 327-328 © North-Holland

328

and lower bands, is large compared to kT, or the gaps, the gap equations reduce to the ones ob- tained for the lower bands by YS, and ones that can be solved algebraically to give the upper band gaps in terms of the lower band gaps. Thus, using the results of YS, a solution of the 6-band problem can be obtained.

The anisotropy energy due to spin orbit cou- pling VLS is then found by computing the cor- rections to the free energy t l with diagrammatic perturbation theory, taking the 6-band Green ' s function described above as unperturbed. Se- cond order perturbat ion loops involving bands (acd+b+) and (acd_b_) contribute to the uniaxial anisotropy. The loop (b b+d d+) does not con- tribute because the gaps involved are charge density waves , as shown by YS, and hence do not produce a spin polarization. It can be proved that the resulting expression, I)LS, has uniaxial anisotropy only if Q is incommensura te for paramagnet ic bands having cubic symmetry , as in chromium. Essentially the SDW band structure is uniaxial due to the incommensura te Q, and the magnetization " fee l s" this through the spin-orbi t interaction. Utilizing the gap equation to define an average, OLS can be ex- pressed in the following form:

[A(T)]2 [A(O)]2vn C ~-~Ls(T) = 4 L A--(-~-J L Van J

x I v " ( v + vn -)l 0)2, (1)

where A(T) is the tempera ture-dependent SDW gap, Vab is a combinat ion of Colomb matrix elements defined as v in YS, vnc is defined analogously to V,,b, the V ii term is an average of spin-orbi t matrix elements over k-space and is a unit vector in the direction of the mag- netization. For chromium the various parameters can be est imated as A(0)~ 10 -13 erg, Van ~--- 10 -35 erg cm 3, Vbc ----- 10 -1 V,,b, VOlEs ~-- 0.04, leading to I)LS(0) ~ 1.6 × 10 +5 erg cm 3, about ten times the anisotropy deduced f rom the phenomenological analysis.

The tempera ture dependence of IlLS is that of the SDW gap squared, so it goes monotonical ly to zero as T increases to TN, and cannot pass

through zero at Tsf. This type of temperature dependence is typical of the anisotropy due to a single source and spin flip transitions in other materials are usually explained by a competi t ion between two sources of similar magnitude, op- posite sign, and different temperature depen- dence. One example is competi t ion between two different microscopic mechanisms, like spin- orbit coupling and the magnetic dipole-dipole interaction for the Morin transition in Fe203 [7]. Another example mentioned above is com- petition between the anisotropy due to orbitally degenerate states at different parts of the Fermi surface in itinerant ferromagnets . Such a com- petition must be found for chromium, but nei- ther of the two types just mentioned appear to work. The anisotropy contribution due to the dipole-dipole interaction can also be evaluated diagrammatical ly within the above model and the temperature dependence is again A(T) 2. Examining chromium band structure cal- culations indicates that the strongly paired states at the Fermi level are orbital singlets. It has been pointed out that pairing of states away f rom the Fermi level does contribute to the magnetization without directly affecting the energy of magnetic ordering [8]. Such pairings, which are not included in the present theory, might be important for the anisotropy energy and might lead to a new tempera ture depen- dence. The present calculation demonst ra tes a mechanism of sufficient magnitude to be useful in competi t ion with another contribution and serves to bet ter define the task ahead.

Relerences

[1] R. Street, B.C. Munday, B. Window and J.R. Williams, J. Appl. Phys. 39 (1968) 1050.

[2] M.O. Steintz, L.H. Schwartz, J.A. Marcus, E. Fawcett and W.A. Reed, Phys. Rev. Lett. 23 (1969) 979.

[3] J.W. Allen and C.Y. Young, AlP Conf. Proc. No. 24 (1975) 410.

[4] E.J. Kondorsky, IEEE Trans. on Magnetics, MAG-10 (1974) 132.

[51 J.B. Sokoloff, J. Phys. F: Metal Phys. 5 (1975) 528. [6] C.Y. Young and J.B. Sokoloff, J. Phys. F: Metal Phys. 4

(1974) 1304. [7] J.O. Artman, J.C. Murphy and S. Foner, Phys. Rev. 138

(1965) A912. [8] C.G. Windsor, J. Phys. F: Metal Phys. 2 (1972) 742.