Spin-polarized relativistic electronic structure calculations for disordered alloys using the CPA: application to FexCo1−x and CoxPt1−x

Journal of Magnetism and Magnetic Materials 104-107 (1992) 733-734 North-Holland

Spin-polarized relativistic electronic structure calculations for disordered alloys using the CPA: application to FexCo l_x and CoxPtl_ x H. E b e r t a, B. D r i t t l e r b a n d H. A k a i c a Siemens AG, Central Research Laboratories, ZFE ME TPH 11, Postfach 3220, W-8520 Erlangen, Germany b Inst. fiir Festk6rperforschung der KFA Jiilich, Postfach 1913, W-5170 Jiilich, Germany ¢ Department of Physics, Nara Medical UniL,ersity Kashihara, Nara 634, Japan The Coherent Potential Approximation (CPA) approach to deal with disordered alloys has been combined with the spin-polarized version of relativistic multiple scattering theory (SPRKKR-CPA). This formalism allows to calculate the electronic structure of magnetic alloys treating all relativistic effects in a rigorous manner. By this way one gets access to a number of interesting spin-orbit induced properties as for example: orbital contributions to the magnetic moments and the hyperfine fields, magnetic X-ray dichroism or magneto-resistivity. First results are presented for the systems FexCo ~ x and COxPt~ x together with a comparison to experimental data. During the last ten years the K K R - C P A (Korring a - K o h n - R o s t o k e r Coherent Potential Approximation) has proven to be a very reliable method to study the electronic properties of random alloys [1]. Originally the K K R - C P A has been derived in a non-relativistic manner, which allowed a straightforward application to spin-polarized systems. For alloys containing heavy elements, this approach is however not adequate. For this reason a corresponding fully relativistic version of the KKR-CPA-formalism has been derived by Staunton et al. [2], giving access to paramagnetic alloys. A further generalization of the K K R - C P A , which allows to deal with ferromagnetic alloys in a fully relativistic way, became possible recently by a corresponding extension of multiple scattering theory, i.e. of the KKR-formalism, to its spin-polarized relativistic version ( S P R K K R ) [3-5]. The combination of the SPRKKR-formalism with the CPA, presented here for the first time, allows not only to calculate the electronic structure of magnetic alloys containing heavy elements, but also - even more importantly - to study the various physical consequences of the simultanous occurrence of spin-orbit-coupling and spin-polarization in random alloys. Such effects, as, for example, the magneto-optical Kerr-effect, are not only of scientific but also of technical interest - especially for this important class of materials. The aim of the C P A alloy theory is to give access to the configurationally averaged properties of random alloys by supplying a corresponding reference medium. Within the K K R - C P A this medium is described by the corresponding single site t-matrix "QQ' ,CeA and the scattering path operator TQQ,CPA,which are in turn found by solving the so-called K K R - C P A - e q u a t i o n s [1]:

TQQ, cPA

= x~o

' + (1 - x ) ~ o , ,

QQ' d z BZ JJ2BZ

(1)

"r~Q,=[(t") = A, B,

t--(tCr'A)-1+(~'CeA)

1] '

QQ,' (3)

where G(k, E) is the KKR-structure constants matrix. In eqs. (1)-(3) a binary system A x B 1 x with one atom per unit cell has been assumed. This is described by the corresponding single site t-matrices t~Q, and component projected scattering path operators rOQ, (~ = A, B), from which more or less all electronic properties of interest can be derived [1]. Obviously, the CPA-equations (1)-(3) can be used without any modifications within the SPRKKR-forrealism [5]. The only difference from the paramagnetic B case is that the single site t-matrices t~Q,, and tQQ,, respectively, have to be determined by solving the Dirac equation for a spin-dependent potential. In contrast to the non-relativistic and paramagnetic relativistic case, this gives rise to a non-diagonal form of t~Q, and t~Q,, caused by the non-spherical symmetry of the potential in spin-space. We have implemented the S P R K K R - C P A formalism working throughout in the (K, p.)-representation, i.e. the combined index Q in eqs. (1)-(3) stands for the spin-orbit and magnetic quantum numbers, K and ~ [2]. Using the algorithm suggested by Butler to solve the KKR-CPA-equations, these can be achieved in most cases within 4 - 1 2 iterations. The following first results are obtained by the S P R K K R - C P A using spin-dependent potentials which stem from self-consistent scalar-relativistic K K R - C P A calculations [6]. In fig. 1 the density of states of Fe and Co in bcc-Fe0.7Co0. 3 are shown. As in former non-relativistic investigations of this alloy system, no pronounced alloying effects are found. As can be seen, the density of states curve for Fe in Fe0.TCo0. 3 does scarcely differ from that for Fe in pure bcc-Fe. This obviously comes from the very similar scattering behaviour of the two components. Nevertheless both components bear a

0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

734

H. Ebert et al. / SPRKKR-CPA applied to Fe~Co I

f''

Co "-""'t'~ Fe - ~ pure Fe -.....~'[~

30n(E) 20 (1/Ry)

• 10 0 -11

-9

-7

-5

,

1.5-

and Co~Pt~

,

~X'x'x-:)¢.,. ......

/i

l.xlx.

1

(~tB) 0.5-

-3

-1

i

1

J

20

energy (eV)

i

i

T

40 at

Fig. 1. Density of states for Fe (full line) and Co (long dashed line) in bcc-Fe0.vCo().3 and for Fe in bcc-Fe (short dashed line). To keep the numerical effort to calculate these curves low, the calculations have been done for an imaginary part of the energy of 0.01 Ry. This broadens the curves accordingly. different local spin magnetic m o m e n t which changes with concentration. Again these m o m e n t s agree within some few percent with the results of non-relativistic calculations. However, in c o n t r a s t to thesc the S P R K K R - c a l c u l a t i o n s give access to the s p i n - o r b i t induced orbital c o n t r i b u t i o n to the m a g n e t i c m o m e n t , which a m o u n t s - just as for the pure c o m p o n e n t s [7] to a r o u n d 5 and 10% of the spin m o m e n t for Fe and Co, respectively. As a second example of an application of the S P R K K R - m e t h o d , the spin-resolved density of states for Co a n d Pt in fcc-Co05Ptl).5 are shown in fig. 2. Strictly speaking these curves r e p r e s e n t the energy d e p e n d e n c y of the expectation value of ½(1 4- o-:). Obviously the alloying p a r t n e r s are very different, leading to a r a t h e r structureless d - b a n d complex for Pt. Only in its u p p e r part t h e r e is some hybridisation with the r a t h e r narrow C o - d - b a n d complex. This in turn is still similar in width and s h a p e to that of pure fcc-Co. F r o m the various curves in fig. 2 one can expect that Co in fcc-Co0.sPt0. 5 will have a spin m a g n e t i c m o m e n t comparable to that of p u r e Co, while a small m o m e n t will be induced on the Pt-site by hybridisation. T h e total m o m e n t s ~ that were found by our calculations are

i

60

r

i

80

i

100

% Pt

Fig. 3. Magnetic moments ~ (in #B) in fcc-Co.~Pt I ,: SPRKKR-CPA ( I total), scalar relativistic KKR-CPA ( • spin-only) and experimental ( x total). shown in fig. 3 t o g e t h e r with c o r r e s p o n d i n g s c a l a r - r e l ativistic and experimental data. As one can see, b o t h sets of theoretical results are in r a t h e r good a g r e e m e n t with experiment. However, one has to note that the relativistic m o m e n t s include the orbital c o n t r i b u t i o n which again a m o u n t s to a r o u n d 10% of the spin magnetic m o m e n t . In contrast to the case of Fe~Co~ ~ the difference in the spin magnetic m o m e n t b e t w e e n a fully relativistic and a scalar-relativistic calculation is found to be quite p r o n o u n c e d . As one would expect, this applies primarily to the Pt-rich side of the system, where differences of up to 15% are found. Interestingly, this does apply to Pt as well as to Co, for which one might expect relativistic effects to be of m i n o r importance. In summary, a new a p p r o a c h - called S P R K K R C P A - to calculate the electronic structure of r a n d o m magnetic alloys in a fully relativistic way has b e e n presented. Results for the spin magnetic m o m e n t s in the alloy systems F e , C o ] , are in full accordance with previous non-relativistic investigations. For Co~Pt~ .~, on the o t h e r hand, a p r o n o u n c e d influence on the spin magnetic m o m e n t s by the inclusion of s p i n - o r b i t - c o u pling has b e e n found. In b o t h alloy systems the s p i n orbit induced orbital c o n t r i b u t i o n to the magnetic mom e n t has b e e n calculated, which t u r n e d out to be 5 - 1 0 % of the spin magnetic m o m e n t .

30n(E)

References

-

H\ C° A

20(1/Ry)

Pt

-

10

0

i

1

i

-9

i

~

-7

f

i

-5

J

i

-3

i

J

i

i

i

-1

energy (eV)

Fig. 2. Spin-resolved density of states for Co and Pt in fcc-Co0.sPt0. 5 (majority/minority spin: dashed/full line). The calculations have been done for an imaginary part of the energy of 0.01 Ry.

[1] G.M. Stocks and H. Winter, in: The Electronic Structure of Complex Systems, eds. P. Phariseau and W.M. Ternmerman (Plenum Press, New York, 1984) p. 463. [2] J.B. Staunton, B.L. Gyorffy and P. Weinberger, J. Phys. F 10 (1980) 2665. [3] R. Feder, F. Rosicky and B. Ackermann, Z. Phys. B 52 (1983) 31. [4] P. Strange, J.B. Staunton and B.L. Gyorffy, J. Phys. C 17 (1984) 3355. [5] P. Strange, H. Ebert, J.B. Staunton and B.L. Gyorffy, J. Phys.: Condens. Matter 1 (1989) 2959. [6] H. Akai, J. Phys.: Condens. Mater 1 (1989) 8045. [7] H. Ebert, P. Strange and B.L. Gyorffy, J. Phys. F 18 (1988) L135.