Theory of spin-transfer torques in magnetic nanostructures

Magnetic multilayers that exhibit giant magnetoresistance for current flowing perpendicular to the interfaces can also exhibit an effect called spin transfer. When the magnetizations of two adjacent layers are not parallel to each other, the current, which has been spin polarized by passing through one layer, can exert a torque on the next layer. These torques lead to a variety of behaviors in different conditions. In some situations, the torques can switch the magnetization between two stable configurations. In these situations, the directness of these torques makes them potentially useful for switching bits in magnetic random access memory (MRAM) cells. In other situations these torques can lead to very stable magnetic precession, which is potentially useful to make high frequency, current controlled oscillators. Despite all of the research done on these effects, there still remains much to be understood before they are incorporated into applications.

Figure 1. Magnetic multilayer structure with non-colinear magnetizations

A full description of the spin transfer torques and their consequences depends on a hierarchy of models. At the most basic level, quantum mechanical calculations describe the behavior of spins when they scatter from interfaces. For these calculations, we have used the spin-dependent reflection amplitudes from our first principles calculations to compute the behavior of electrons that have spins that are not collinear with the magnetization of the ferromagnet. When electrons in a non-magnet scatter from an interface with a ferromagnet, the two spinor components of the electron's spin along the magnetization separate spatially because the reflection amplitudes are different for the majority and minority electrons. The total angular momentum along the magnetization in the incoming state is conserved in the outgoing state. However, the loss of interference due to spatial separation of the two components leads (generally) to a reduction in the component of the spin that is transverse to the magnetization. In the process, angular momentum is effectively transferred from the electron spin to the ferromagnetic magnetization. This transfer of angular momentum can be described as a torque from the electron spin on the magnetization. This process is illustrated in the figure below for the simple limit of perfect transmission for majority electrons and perfect reflection for minority electrons. In this figure, any component of the spin that is transverse to the magnetization is absorbed by the magnetization giving an effective torque. In the full calculations, we find this simple result holds; to a good approximation, any transverse spin current incident on a ferromagnet is absorbed.

Figure 2. Schematic behavior of a spin scattering from an interface with a
ferromagnet in a simple limit of ideal transmission and reflection.

The result of the quantum mechanical calculations, that the transverse spin current is absorbed by the ferromagnetic magnetization at the interface, enters into semiclassical transport calculations as a boundary condition. The former calculation determines the direction of the torque, both together determine its magnitude. We have used the Boltzmann equation and the drift-diffusion equation to compute the torque for various system geometries and magnetic configurations. The figure below shows the spatial variation of two components of the spin current for a magnetic multilayer with perpendicular magnetizations. The cartoon at the bottom shows the directions of the torques on each of the magnetizations resulting from the discontinuities in the spin currents. These semiclassical calculations highlight the importance of spin flip scattering in determining the polarization of the currents and hence the resulting torques. Analytic approximations we have developed are useful guides for determing how to vary geometrical and material properties to control the torque.

Figure 3. Spin currents in a magnetic multilayer with perpendicular magnetizations.

The semiclassical calculations give the torques as a function of device geometry and magnetic configuration. Since electronic time scales are much shorter than magnetic time scales, these torques, computed for static magnetizations, can be used as input into classical simulations of the magnetization dynamics. From the calculated torques, we have computed the behavior of the magnetization in typical devices as a function of the current flowing though them, the external field applied to them, and their temperature. These calculations show that the models can qualitatively describe the behavior seen in experiment, but quantitative disagreements remain. These disagreements point toward the future research necessary to completely understand these effects and use them in future applications. The figure below shows the calculated behavior as a function of applied field and current for two different temperatures. The multilayer can be found with the magnetizations parallel (P), antiparallel (AP), precessing with an in-plane precession axis (IPP), or an out of plane axis (OPP). In some regions the system can be stable in two different configurations; these are denoted P/IPP, for example.

Figure 4. Calculated phase diagrams for the behavior of a magnetic multilayer
as a function of temperature and applied magnetic field.

We have also studied the effects of non-uniform magnetizations in the ferromagnetic layer. An interesting consequence of non-uniform magnetization is that a current passing through a single ferromagnetic layer can cause a precessional instability of that layer. Such instabilities have been observed experimentally in point contacts and nanopillar multilayers.

Calculation of torques:
Anatomy of Spin-Transfer Torque

Transport calculations:
A Numerical Method to Solve the Boltzmann Equation for a Spin Valve
Boltzmann Test of Slonczewski's Theory of Spin Transfer Torque
Non-collinear spin transfer in Co/Cu/Co multilayers

Dynamics:
Macrospin Models of Spin Transfer Dynamics
Magnetic Normal Modes of Nano-Elements
Phenomenological Theory of Current-Induced Magnetization Precession

Review Articles:
Spin Transfer Torque and Dynamics
Spin Transfer Torques



Mark D. Stiles - NIST
Christian Heiliger - University of Maryland
Paul M. Haney - NIST


Andy Zangwill - Georgia Institute of Technology
Jiang Xiao - Delft University of Technology
Tom Silva - NIST
Mark Hoefer - NIST
C.-L. Chien - Johns Hopkins University
Tingyong Chien - Johns Hopkins University
Jack Bass - Michigan State University
Daniel Ralph - Cornell University
Jacques Miltat - University of Paris Sud

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Online: February 2002
Last Updated: February 2008

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