s-d–type exchange interactions in inhomogeneous ferromagnets A. Rebei, 1, * W. N. G. Hitchon, 2 and G. J. Parker 3 1 Seagate Research Center, Pittsburgh, Pennsylvania 15222, USA 2 Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA 3 GE Global Research, One Research Circle, Niskayuna, New York 12309, USA Received 7 February 2005; published 4 August 2005 Motivated by a need to understand spin-momentum transport in CPP current perpendicular to the plane structures, a quantum field theoretical treatment of spin-spin interactions in ferromagnets is presented. The s-d interaction of the conduction electrons and the magnetic medium is treated nonperturbatively from first principles in real space. The localized magnetic moments also interact with each other through a Heisenberg exchange potential. To take into account correlation effects, a second quantized formulation is used. The semiclassical limit is taken by using a coherent-state path-integral technique which also allows us to go beyond a linear-response approach. We derive a set of coupled equations of motion for the nonuniform magnetization, the spin current, and the two-point correlation functions of the magnetization. The rate of change of the magnetization is shown to obey a generalized Landau-Lifshitz equation that takes into account interaction with the conduction electrons. Within the relaxation time approximation it is shown that the polarization of the conduction electrons obeys a generalized diffusion equation. The diffusion tensor, which has off-diagonal terms due to the s-d exchange interaction, is now explicitly dependent on the magnetization of the medium. We also show that the magnetization fluctuations satisfy a diffusion-type equation. The derived equations are used in two illustrative examples. DOI: 10.1103/PhysRevB.72.064408 PACS numbers: 05.60.-k I. INTRODUCTION Spin-spin interactions in ferromagnetic metals are of para- mount importance in today’s giant magnetoresistance GMR recording heads. There is also currently great interest in the magnetic recording industry in using spin currents, instead of magnetic fields, to switch the magnetization in a writer de- vice. In this case a polarized electronic current is needed, such that the net spin of the polarization becomes effectively another magnetic source which induces an interaction with the magnetic moments of the medium. One widely used ap- proximation is to separate the degrees of freedom of the cur- rent from those of the local magnetic moment. This latter separation is not justified in conducting metals but it never- theless produces reasonable results in some cases. 1 This pa- per explores in some detail the consequences for the spin accumulation problem in ferromagnets of assuming that the interaction between the conduction electrons and the local moments is of the s-d exchange type. This interaction can give rise to what is now known as spin-momentum transfer SMT in magnetic multilayers. This latter mechanism has been predicted by Berger 1 and Slonczewski 2 and later veri- fied experimentally by various groups. 3,4 Other interaction mechanisms between the conduction electrons and the mag- netization vector have been proposed since the Berger- Slonczewski work. 5–9 In previous work, the interaction of the polarized current with the magnetization has not been treated self-consistently. In fact the equations of motion were based on those of a similar system, that of a current interacting with magnetic impurities. 10 We believe that this approach is not suitable for transition metals. In this work, we start from a microscopic description of the conduction electrons and the ferromagnetic medium and then take the semiclassical limit to derive equations for mac- roscopic quantities of physical significance to experiment and to other phenomenological approaches. Although the derivations are somewhat complex, one can go to the main results e.g., Eqs. 10 and 28 which are generalized Boltzmann-type equations and see that the correct physics is contained in them. Our results extend those of Ref. 8 and are in general rel- evant to problems of spin momentum transfer, domain walls, and spin-wave excitations. 11 We use many-body field theo- retical methods to describe the system of magnetic moments plus conduction electrons. Even though only a single particle picture is needed, the methods we use permit us to treat the magnetic part of the problem and the conduction electrons on the same formal level. This allows us to derive transport equations for the conduction electrons and the local magnetic moments and include relaxation effects without recourse to more phenomenological modeling. Exchange effects, which are important in transition metals, are also included self- consistently. Finite-temperature properties are naturally in- cluded through the use of a path-integral formulation of the problem. 12 Including spin-dependent interactions in a trans- port problem means that we have to deal with many indexes. Path integrals are helpful with bookkeeping and hence sim- plify the discussion considerably as compared to the alterna- tive approach of Ref. 10. Finally a path-integral representa- tion helps in making consistent approximations to the derived transport equations. Our treatment is nonperturbative and applies to systems far from equilibrium. One of the important results is that we are able to give an equation for the correlation functions of the magnetization from which a nonequilibrium fluctuation- dissipation result follows. PHYSICAL REVIEW B 72, 064408 2005 1098-0121/2005/726/06440812/$23.00 ©2005 The American Physical Society 064408-1