Spin transfer torque in continuous textures: Semiclassical Boltzmann approach

Abstract

We consider a microscopic model of itinerant electrons coupled via ferromagnetic exchange to a local magnetization whose direction vector n͑r , t͒ varies in space and time. We assume that to first order in the spatial gradients and time derivative of n͑r , t͒, the magnetization distribution function f͑p , r , t͒ of itinerant electrons with momentum p at position r and time t has the ansatz form f͑p , r , t͒ = f ʈ ͑p͒n͑r , t͒ + f 1r ͑p͒n ϫ ٌ r n + f 2r ͑pٌ͒ r n + f 1t ͑p͒n ϫ ‫ץ‬ t n + f 2t ͑p͒‫ץ‬ t n. Using the Landau-Sillin equations of motion approach ͑Zh. Eksp. Teor. Fiz. 33, 1227 ͑1957͒ ͓Sov. Phys. JETP 6, 945 ͑1958͔͒͒, we derive explicit forms for the components f ʈ ͑p͒, f 1r ͑p͒, f 2r ͑p͒, f 1t ͑p͒, and f 2t ͑p͒ in "equilibrium" and in out-of-equilibrium situations for ͑i͒ no scattering by impurities, ͑ii͒ spin-conserving scattering, and ͑iii͒ spin-nonconserving scattering. The back action on the localized electron magnetization from the out-of-equilibrium part of the two components f 1r and f 2r constitutes the two spin transfer torque terms.

Spin pumping by a field-driven domain wall

“…This is because both the Gilbert damping parameter ␣ G and the dissipative spin transfer torque parameter ␤ arise from processes in the microscopic Hamiltonian that do not conserve spin. [26][27][28]30 Hence, for the case that ␣ G = ␤ = 0, our results agree with the results of Barnes and Maekawa. 40 The remainder of this paper is organized as follows.…”

“…For ␤ / ␣ G =1, on the other hand, the domain wall moves with velocity v s . Although theoretical studies indicate that generically ␤ ␣ G , [26][27][28]30 it is not well understood what the relative importance of spin-dependent disorder and spin-orbit effects in the bandstructure is, and a precise theoretical prediction of ␤ / ␣ G for a specific material has not been attempted yet. Moreover, the determination of the ratio ␤ / ␣ G from experiments on the current-driven domain-wall motion has turned out to be hard because of the extrinsic pinning of the domain and nonzero-temperature 29,38 effects.…”

“…11 The opposite effect, i.e., the manipulation of magnetization with spin current, is called spin transfer. [12][13][14][15] Recently, the possibility of manipulating with current the position of a magnetic domain wall via spin transfer torques has attracted a great deal of theoretical [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] and experimental [31][32][33][34][35][36][37][38] interest. Although the subject is still controversial, 18,21 it is by now established that in the longwavelength limit, the equation of motion for the magnetization direction ⍀, which in the absence of current describes damped precession around the effective field −␦E MM ͓⍀͔ / ͑ប␦⍀͒, is given by…”

Functional Keldysh theory of spin torques

“…Spin transfer torques, which lead to current-driven nanomagnet reversal and to domain wall motion in narrow wires, have been at the center of this activity 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28 . In spin-transfer-torque theory, the usual LandauLifschitz-Gilbert (LLG) equation of motion for the magnetization directionΩ acquires terms corresponding to so-called adiabatic and non-adiabatic spin transfer torques which are both proportional to current.…”

Spin torques due to diffusive spin current in magnetic texture

“…The idea was first applied by Berger to the current-induced domain wall motion [3][4][5] , where the electron spin can follow the local magnetization when traversing the domain wall. Since the theoretical details of the current-induced spin torques were exposed by phenomenological 6,7 and microscopic 8 analyses, it is well-known that there are reactive torque (also known as spin-transfer torque) [9][10][11][12] and dissipative torque (also called β term) 6,7,[13][14][15][16] in the presence of spin relaxation of conduction electrons. They are expressed by the first and the second terms of…”