# What is the theory behind spin-transfer torque?

I would like to get a layman's understanding of STT (Spin-transfer torque). By that I mean I don't have time to understand the mathematical and exact physical theory, but I would still very much like to get at least some understanding of it. I study physics though, so layman only in the sense that I don't know anything about STT theory.

1. Is there a basic equation/model that can describe it?

2. In which circumstances can this model be applied?

3. What governs the direction of the angular momentum transfer at an interface?

I listed three (sub) questions to indicate what kind of answer I am looking for.

• This is an accessible and fairly complete explanation which seems to answer your sub-questions: arxiv.org/pdf/cond-mat/0202397.pdf. The mathematics are fairly basic, though it requires some commitment to get through, and it may be what you are looking for. – Ernie Jun 20 '15 at 14:04

A good overview of STT is given in the book "Handbook of Spin Transport and Magnetism" by Evgeny Y. Tsymbal, Igor Zutic. Based on chapter 8 of the book, I will try to explain the concept of STT below:

The principle behind STT is conservation of total angular momentum of the system. The equation that describes spin dynamics is called the LLG equation, named after Landau, Lifshitz and Gilbert. I will try to illustrate its physical meaning by using a general example. Consider an s-d model, where the sp-band electrons are itinerant and thus contribute to conductivity, whereas the d-band electrons are more localized and contribute to the magnetization of the material. Therefore, the total Hamiltonian of the system will contain three terms: one corresponding to the sp-band electrons, one corresponding to d-band electrons and one modelling the interaction between those two:

$H_{tot}=H^d + H^{sp} + H_{ex}$

The hamiltonian for the sp-band electrons, $H^{sp}$ contains a kinetic energy term, a potential term and a spin-orbit coupling term (in general, though for simplicity in this case can be ignored). The hamiltonian for the d-band electrons contains a term that describes the interaction of the d-band spins with the effective magnetic field that they feel, which comes from anisotropy, exchange fields, dipolar interactions and of course, any additional external field imposed. Finally, the interaction term contains the dot product of the two kinds of spins (d-band and sp-band). Consequently, the total hamiltonian can be written as following:

$H_{tot}= - \frac{g \mu_b}{2} \sum S^{d} \cdot H_{eff} + \frac{\mathbf{p}^2}{2m}+V(\mathbf{r}) + H_{soc} -J_{sd}\sum S^{d}_i \cdot S^{sp}$

where the first term corresponds to the d-band hamiltonian, the second to the sp-band and the last one to the interaction between the two kinds of spins. $S$ is the spin operator and the superscript indicates which band it corresponds to. $J_{sd}$ is the energy of interaction. In order to derive the spin dynamics, we can apply Ehrenfest's theorem. According to it,

$\frac{d}{dt} <A> = -\frac{i}{\hbar} [A,H] + <\frac{dA}{dt}>$

Thus, for a time-independent operator, we only need to compute its commutation relation with the system's Hamiltonian in order to see how its observable will change in time. In our case, the observables are the expectation value of the spin operators $S^{d}$ and $S^{sp}$. Thus, we need their commutation relation with the Hamiltonian. Leaving out the mathematics and the SOC (it is good however to try it on your own), we get for the magnetization densities the following results:

$\frac{d}{dt}\mathbf{M^d}=-\gamma \mathbf{M^d}\times\mathbf{H}_{eff} + \frac{a}{M_s}\mathbf{M^d}\times\frac{d\mathbf{M^d}}{dt} -2\frac{J_{sd}}{\hbar}\mathbf{M^d}\times{M^{sp}}$

and

$\frac{d}{dt}\mathbf{M^{sp}}= -\nabla \cdot \mathbf{J}_s -\frac{\delta\mathbf{M^{sp}}}{\tau_{sf}} + \frac{2SJ_{sd}}{\hbar M_s}\mathbf{M^d}\times\mathbf{M^{sp}}$

where, $\gamma=g\mu_b/2$ is the gyromagnetic ratio, $M_s$ is the saturation magnetization of the material, $\mathbf{J}_s$ is the spin-current tensor.

The first equation from those two, has a similar form to the LLG equation. Its first term corresponds to a precessional term. It comes simply from the fact that the spins when being in a magnetic field will start to precess around the axis of the field. However, experiments have deduced that after a sufficient period of time the spins, or equivalently the material's magnetization will align with the external field. This is modelled by the phenomenological factor $a$ which describes damping. The last term of this equation is the STT term. Furthermore, attention should be given to the term $\frac{\delta\mathbf{M^{sp}}}{\tau_{sf}}$. This term models spin-flip processes, where the electrons lose their spin, i.e. spin orientation. $\delta \mathbf{M^{sp}}$ corresponds to the difference between the equilibrium spins and spins lost'' by spin-flip processes. Solving the equations for the magnetization dynamics simultaneously will give you a description for the itinerant and localized spin dynamics. When you have reached a steady state in your system, this means that the derivative with respect to time is zero and you can easily conclude from these two equations that the STT term is given by

$\mathbf{T}=-\frac{2SJ_{sd}}{\hbar M_s} \mathbf{M^d}\times \mathbf{M^{sp}} = -\nabla \cdot \mathbf{J}_s -\frac{\delta M^{sp}}{\tau_{sf}}$

That is the torque that is felt by the itinerant spins. It simply is the divergence of the spin current plus an additional term due to spin-flip. The direction of change of the magnetization will change accordingly to the orientation of the spins from the spin current. To give you a practical example, consider two materials where one can change its magnetization easily (soft-magnet) and the other is much harder to do so (hard-magnet or pinned). By passing a current through the hard-magnet the electrons become polarized, i.e. their spins are oriented according to the magnetization of the hard-magnet. When they enter the soft-magnet layer, they interact with the local d-band spins according to the above equations. Thus, they exert a torque on the local spins and also the local spins exert a torque on them. If the spin current density is sufficiently high, then we can change the magnetization of the soft-magnet layer electrically. That is also the basic principle behind STT-RAM.

The principle of STT can be applied to a wide variety of circumstances, basically wherever there exists interaction between local spins and spins passing by. However, the exact models also vary from case to case as this depends on the kind of materials and the processes you take into account. For example in 3d-transition metals, the s-d model is not a good description. Also, in other cases you can also consider contribution from Spin-orbit coupling and thus you need to include the corresponding term in the equations also.

I tried to keep the math to a minimum and be as descriptive as possible. I hope my answer satisfies your curiosity about STT. In the meantime, I am preparing also some images to update my post and make the principle more understandable.