How spin orbit coupling torque is happening in either crystalline or amorphous nature of perpendicular magnetic anisotropy multilayer thin films ?
Spin-Orbit coupling is a relativistic effect, which may be neglected in light atoms or systems consisting of such atoms, described in almost all good textbooks. You may for instance consider the book Lecture Notes on Electron Correlation and Magnetism, by P Fazekas (World Scientific, Singapore, 2003), or The Theory of Magnetism. I. Statics and Dynamics, by DC Mattis (Springer, Berlin, 1981). You may also consider the more basic book as Modern Quantum Mechanics, revised edition, by JJ Sakurai (Addison-Wesley, New York, 1994). There is an important effect, referred to as Thomas Precession, which has been discussed in detail in Sec. 11.8 of the book Classical Electromagnetism, 3rd edition, by JD Jackson (Wiley, New Jersey, 1999).
There are also some specialised books dealing with the spin-orbit coupling, such as Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems, by R Winkler (Springer, Berlin, 2003), and Spin Physics in Semiconductors, edited by MI Dyakonov (Springer, Berlin, 2008).
A relevant classic paper is by JB Goodenough, Spin-Orbit Effects in Transition-Metal Compounds (Phys. Rev. 171, 466 (1968)), the link to which I present below.
http://journals.aps.org/pr/abstract/10.1103/PhysRev.171.466
http://journals.aps.org/pr/abstract/10.1103/PhysRev.171.466
Spin orbit torque is a torque exerted on the local magnetic moments of the considered material when the non-equilibrium spin density is not collinear to the local magnetic moments (magnetization). It differs from the spin transfer torque, as the non-equilibrium spin density in this case is originating from the spin orbit coupling (say Rashba spin orbit coupling for example), where it originates from the polarization by a ferromagnetic layer (spin injection) in the other case (spin transfer torque). Spin Hall effect torque is another kind of spin orbit torque where the non-equilibrium spin density originates due to the presence of spin Hall effect.
Consider for example a Rashba spin orbit coupling due to the inversion asymmetry (can be favored by considering an interface between a ferromagnet and a paramagnet), this interaction is associated with a magnetic field, hence the incoming electrons will have their spins precessing around this field resulting in a non-equilibrium spin density mostly given by the direction of this field. Hence the spin orbit torque will be given by the cross product between this non-equilibrium spin density and the local magnetic moments,
tau=s \times m
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