Spin-transfer torque

Spin-transfer torque is a torque that exerts on a magnetization by conduction electron spins, i.e. the angular momentum transferred from spins to magnetic moment. This concept was first introduced by John Slonczewski and Luc Berger in magnetic multilayers in 1996. [1,2]

The transferring of angular momentum from the conduction electrons spins to local magnetic moment can be achieved by two different mechanisms: 1) spin-dependent interfacial scattering which causes spin rotation and filtering of the spin current (similar optical polarizers, see above), and 2) spin-dephasing which causes the self-cancellation of the transverse spin component for the ensemble of electrons. [3] Because both mechanisms are due to the nature of the magnetic moment which acts as an internal field for electron spins and causes the spin-dependent scattering and the precessing of electron spins that is not collinear, thus the angular momentum loss due to both mechanisms is transferred the the local magnetic moments, therefore a torque is exerted on the magnetic moment. The spin-transfer torque is found to be the transverse component (relative to the magnetization) of the injecting spin current. [3]

Spin-transfer torque in spin valves

The spin-transfer torque requires a spin current, thus the torque exerted by spin-up and spin-down electrons do not cancel. The effect of spin-transfer torque is first demonstrated in a spin valve structure (see above), where the spin current created by the thicker (fixed) FM layer exerts a torque on the magnetization in the thinner (free) FM layer. The direction of the torque depends on the applied current direction (see above). When the current is large enough and has the proper polarity, the torque destabilize the magnetization in the free layer and causes magnetization switching. In spin valves, the spin-transfer torque acting on the free layer can be generally written in the following form:

${\bf N}_{st} = \eta(\theta){\hbar J\over 2e}({\bf m}\times{\bf m}_0)\times{\bf m}$ where J is the current density, and are the magnetization in the fixed layer and free layer, and η(θ) is angular-dependent the current polarization factor with be the angle between m₀ and m.

This switching manifests as a change in the magneto-resistance of the spin valve, where the parallel (anti-parallel) configuration has low (high) resistance. This has been demonstrated in experiment, [4] where the resistance of a spin valve changes from low to high when the current exceeds a critical value in one direction and changes back when the current polarity reverses and exceeds another critical value (see below). The current hysteresis loop is a indicator of spin-transfer torque effect.

In spin valves, the direction of the torque is in-plane, i.e. lies in the plane spanned by the magnetizations in the two FM layers, and the magnitude of the torque depending on the relative angle between the magnetizations in the fixed and free layer. The angular dependence is asymmetric about the angle (see below), i.e. the torque near parallel (P) and anti-parallel (AP) is different, which is the reason why the critical currents are different for switchings from P to AP and from AP to P. The torque in spin valve is calculated by Slonczewski first based on a very simple theory that combines the simplified Boltzmann treatment and circuit theory. [1,5] This simple theory has been proved to be very accurate by comparing to a full Boltzmann calculation. [6]

Spin-transfer torque in magnetic domain walls

Spin-transfer effect also exists in other type of magnetic structures, in a magnetic domain wall for example, where the magnetization varies continuously over space. [7,8] In general, the spin-transfer torque in domain walls can be written as:

${\bf N}_{st} = c_1{\partial{\bf m}\over\partial x} + c_2{\bf m}\times{\partial{\bf m}\over\partial x}$

where the first term is the in-plane component, and the second term is the out-of-plane. In comparison, out-of-plane torque contributes little in spin valves. An interesting topic in domain walls is to determine how important is the out-of-plane component. In the beginning, the out-of-plane torque is believed to come from the misalignment between electron spins and local magnetic moment (this is why this component is also called non-adiabatic). However, we showed that the non-adiabaticity contribution is quite small and can be ignored except for very narrow domain walls. [8] Currently, the debate about the out-of-plane torque is still going on, many people have tried to calculate this component based on different mechanisms. The relative importance between out-of-plane and in-plane torque, or the β=c2/c1 factor, ranges from zero to the order of magnetic damping parameter – α. [8-13]

Spin-transfer torque in magnetic tunnel junctions

A ‘spin valve’ like structure with the metallic spacer replaced by an insulating tunnel barrier is called magnetic tunnel junction (MTJ). Similar to spin valves, there is also spin-transfer torque in MTJs. [14-19] Different from the torque in spin valves, but similar to that in domain walls, the torque in MTJs has both the in-plane and out-of-plane components. And the torque has the following form:

${\bf N}_{st} = V[\tau_{ip}{\bf (m\times m_0) \times m} + \tau_{op}{\bf m_0\times m}]$

where τ_ip and τ_op are called in-plane and out-of-plane torkance (derivative of torque over bias), which are angular-independent. Similar form of torque can be written for spin valves as well, in which τ_ip and τ_op becomes angular-dependent and the out-of-plane term contributes little. The absence of out-of-plane torque in spin valves is due to the multiple scattering of electrons in the metallic spacer, which averages out the out-of-plane component. However, because of the insulating spacer, most electrons only tunnel through the spacer once and never come back again, thus the out-of-plane component is carried through. The theoretical calculations show that both torques have parabolic shape on the bias dependence, [14, 18, 19] the in-plane torque is a shifted parabolic curve and the out-of-plane torque is a symmetric parabolic curve (See below for a comparison between theory [19] and experiment [16]).

References:

[1]. Slonczewski, J.C. Current-driven excitation of magnetic multilayers. Journal of Magnetism and Magnetic Materials 159, L1 (1996).
[2]. Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54, 9353 (1996).
[3]. Stiles, M.D. & Zangwill, A. Anatomy of spin-transfer torque. Phys. Rev. B 66, 014407 (2002).
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[7]. Zhang, S. & Li, Z. Roles of Nonequilibrium Conduction Electrons on the Magnetization Dynamics of Ferromagnets. Phys. Rev. Lett. 93, 127204 (2004).
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[11]. Duine, R.A., Nunez, A.S., Sinova, J. & MacDonald, A.H. Functional Keldysh theory of spin torques. Phys. Rev. B 75, 214420-13 (2007).
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[13]. Garate, I., Gilmore, K., Stiles, M.D. & MacDonald, A.H. Nonadiabatic spin-transfer torque in real materials. Phys. Rev. B 79, 104416-15 (2009).
[14]. Theodonis, I., Kioussis, N., Kalitsov, A., Chshiev, M. & Butler, W.H. Anomalous Bias Dependence of Spin Torque in Magnetic Tunnel Junctions. Phys. Rev. Lett. 97, 237205-4 (2006).
[15]. Slonczewski, J. & Sun, J. Theory of voltage-driven current and torque in magnetic tunnel junctions. Journal of Magnetism and Magnetic Materials 310, 169-175 (2007).
[16]. Kubota, H. et al. Quantitative measurement of voltage dependence of spin-transfer torque in MgO-based magnetic tunnel junctions. Nat Phys 4, 37-41 (2008).
[17]. Sankey, J.C. et al. Measurement of the spin-transfer-torque vector in magnetic tunnel junctions. Nat Phys 4, 67-71 (2008).
[18]. Heiliger, C. & Stiles, M.D. Ab Initio Studies of the Spin-Transfer Torque in Magnetic Tunnel Junctions. Phys. Rev. Lett. 100, 186805-4 (2008).
[19]. Xiao, J., Bauer, G.E.W. & Brataas, A. Spin-transfer torque in magnetic tunnel junctions: Scattering theory. Phys. Rev. B 77, 224419-9 (2008).