Spin Torque and Waviness in Magnetic Multilayers: A Bridge between Valet-Fert Theory and Quantum Approaches

Abstract

We develop a simple theoretical framework for transport in magnetic multilayers, based on Landauer-Buttiker scattering formalism and Random Matrix Theory. A simple transformation allows one to go from the scattering point of view to theories expressed in terms of local currents and electrochemical potential. In particular, our theory can be mapped onto the well established classical Valet Fert theory for collinear systems. For non collinear systems, in the absence of spin-flip scattering, our theory can be mapped onto the generalized circuit theory. We apply our theory to the angular dependance of spin accumulation and spin torque in non-collinear spin valves. 72.25.Ba; The discovery of Giant Magnetoresistance (GMR) [1] and its subsequent counterpart, the current induced spin torque [2], was at the origin of a new field which aims at controlling the magnetization dynamics of small metallic devices through standard electronics. The theory of transport in those systems is now well developed and includes a number of approaches that range from classical Valet Fert (VF) theory in the diffusive regime [3] , Boltzmann equation [4] to quantum approaches (original [5] and generalized [6]) circuit theory, random matrix theory (RMT) for the scattering matrix [7], and ab-initio based models [8,9]. Many connections exist between these different approaches. One popular route [10] starts from the Keldysh Green's function formalism. In the quasiclassical approximation it yields the Boltzmann equation and the (VF) diffusive equation. A different strategy where local equilibrium is assumed only in certain points leads to circuit theory. An alternative route is the Landauer-Büttiker formalism which expresses the problem of transport inside a quantum conductor as a scattering problem where electrons undergo various transmission/reflection events as they go through the system. This approach is well suited for coherent system and is equivalent to the Keldysh approach. However, the classical concepts of chemical potential or local equilibrium do not arise naturally in the scattering approach so that classical intuitions do not easily transfer into its language.In this letter we take the scattering formalism as our starting point and develop a theory which fully captures VF and (generalized) circuit theory. Our theory (here after referred as C-RMT for Continuous Random Matrix Theory) can be tabulated by the same set of (experimentally accessible) parameters as VF [11]. On the other hand, it properly includes Sharvin resistance and allows for non collinear and even a one-dimensional texture of magnetization (i.e. domain walls). We apply C-RMT to the discussion of the angular dependance of spin torque ("waviness" [12,13]) in asymmetric spin valves. Two quantum conductors in series. Before introducing C-RMT, we start with an instructive example of transport in a non-magnetic metal. C-RMT is a mere extension of the concepts developed below to properly take into account the electronic spins. Classically, the electronic transport in a syste...

Multiscale approach to spin transport in magnetic multilayers

“…[77][78][79][80] for an extensive discussion of waviness where neither the parallel nor the antiparallel magnetic configuration is stable). The corresponding CRMT calculation is close, but below, to the wavy instability threshold 7,29,81 . This small discrepancy indicates a weakness of the effective TB approach as its physics is fairly sensitive to the choice of TB parameters.…”

“…Many different approaches have been developed, ranging from purely quantum 16,17 to semi-classical Boltzmann equation [18][19][20] , (generalized) circuit theory [21][22][23][24][25][26][27] or Random Matrix Theory (RMT) 28,29 or full ab-initio (density functional theory) calculations [30][31][32][33] . A good understanding has now been reached of the respective domains of applicability and links between the various approaches 26,29 . On the other hand it is becoming increasingly clear that the magnetization dynamics of real devices cannot be properly captured by simple (analytically tractable) models and that numerical simulations that treat the magnetic and transport degrees of freedom on an equal footing need to be developed.…”

Control of spin current by a magnetic YIG substrate in NiFe/Al nonlocal spin valves

“…To quantify our results we performed threedimensional finite element simulations using COMSOL Multiphysics (3D-FEM) [19,20] that uses a set of equations that are equivalent to the continuous random matrix theory in 3 dimensions (CRMT3D) [21]. The charge current j α c ( r) and spin current j α s ( r), (where α ∈ x, y, z), are linked to their corresponding driving forces via the electrical conductivity as…”

Charge pumping and the colored thermal voltage noise in spin valves

“…2 is skewed by g i similar to for the spin diode effect discussed by Kupferschmidt et al 48 and Kovalev et al 32 For asymmetric spin valves, a nonmonotonic angular dependence of the magnetoresistance and a vanishing torkance at a noncollinear magnetization configuration has been demonstrated. [49][50][51][52][53] A sign change in torkance leads to a sign change in the charge pumping voltage. The magnetic contribution to the thermal noise vanishes at the zero torkance point.…”