LLG Equation + Spin Torque, A simple derivation

[sokrates]

Hi everybody:

I have been working on a "SIMPLE" derivation (for a long time now) and I 'll go insane if I don't get this right. I am trying to do a similar research problem - but before I attempt to change anything I'd like to make clear that I can do this right. The formulas look complicated but this is a simple perturbative analysis - and I am just not clear on some of the "definitions" the author uses. Maybe you could help me on clarifying what he means

If you have PRB access the paper is J.Z. Sun - "Spin-Current Interaction with a monodomain magnetic body : A model study" -(2001)

However, I'll repeat all the crucial steps here and present my confusion over some overly simplified derivations by Sun.

Up until Equation (11) in the paper, he writes down the LLG equation in spherical coordinates with subject to relevant anisotropic forces and the spin current.

He arrives at the following ODE system:

[tex]
\frac{d \phi}{dt}=-1-{\it hp}\, \left( \cos \left( \phi \right) -\alpha\,\sin \left(
\phi \right) \right) \cos \left( \phi \right) -h+{\it hs}\,\alpha
[/tex][tex]
\frac{d\theta}{dt}= \theta \left( -\alpha-{\it hp}\,\cos \left( \phi \right) \sin \left( \phi
\right) -{\it hp}\, \left( \cos \left( \phi \right) \right) ^{2}
\alpha-h\alpha-{\it hs} \right)
[/tex]

Then he solves these sets of equations for hs=0, alpha=0 to find the "unperturbed solution".
He gives these unperturbed solutions as:

[tex]
\phi(t)= arctan \left[\left(\frac{\epsilon+1}{\epsilon}\right)^{1/2} cot(w_p t)\right]
[/tex]

[tex]
\theta(t)= \theta_0\left(\frac{2 \epsilon + 1 + cos(2 w_p t)}{2\epsilon + 1}\right)^{1/2}
[/tex]

where he defines:

[tex]
\epsilon = \frac{1+h}{hp} \ \ and \ \ w_p = hp \sqrt{\epsilon (1+\epsilon)}
[/tex]

There's a redundant relation here -- but maybe useful:
[tex]
\theta (t) = \frac{\theta_0^2 \epsilon}{\epsilon + cos^2(\phi(t))}
[/tex]in implicit form. (theta_0 is the initial condition)

To apply the perturbative analysis and see how the system reacts to the perturbation, he uses the energy function which is defined as:

[tex]
U(\theta,\phi)={\theta}^{2}+{\it hp}\,{\theta}^{2} \left( \cos \left( \phi \right)
\right) ^{2}-2\,h
[/tex]

Now; treating hs and alpha terms as perturbations to the original equation, uses all equations above to SOMEHOW get to the following: (which is a mystery to me!)

[tex]
\left<\frac{dU}{dt}\right> = -(2 hp \epsilon \theta_0^2) hs - (2 hp \epsilon \theta_0^2) \alpha hp \left[-\epsilon (1+\epsilon) A + (1+ 2\epsilon) + \frac{hs}{hp^2} B \right]
[/tex]

where A and B are:

[tex]
A = \left<\frac{1}{\epsilon + cos^2(\phi)}\right> = \frac{2 \epsilon}{2\epsilon(1+\epsilon)}
[/tex]

[tex]
B = \left<\frac{sin(\phi)cos(\phi)}{\epsilon+cos^2(\phi}\right> = 0
[/tex]

--> How is the averaging over U done here? He says nothing about it in the paper apart from saying it's an average. If it's a time-average over, say one, period - then it becomes extremely messy because of the way theta and phi depend on time?
In other words, how are A and B evaluated?

--> How can you arrive at the dU/dt term based on what I wrote above? I tried the obvious: take derivative of U wrt time and substitute all the terms using the theta, phi relations but it got very messy and I came nowhere close to what he has...

But since I have no idea on how he takes the average, maybe I am missing something obvious?... Even the slightest hint would be greatly appreciated!

 

[sokrates]

Could it be related to my misunderstanding of perturbation theory?...

All I did was to take the derivative of U wrt t:

[tex]
\frac{dU}{dt}=2\,\theta \left( t \right) {\frac {d}{dt}}\theta \left( t \right) +2\,
{\it hp}\,\theta \left( t \right) \left( \cos \left( \phi \left( t
\right) \right) \right) ^{2}{\frac {d}{dt}}\theta \left( t \right)
-2\,{\it hp}\, \left( \theta \left( t \right) \right) ^{2}\cos
\left( \phi \left( t \right) \right) \sin \left( \phi \left( t
\right) \right) {\frac {d}{dt}}\phi \left( t \right)
[/tex]

Then substitute what I have given above for theta(t) and phi(t)
before taking any average. But as I said, doing this without the average results in a very complicated equation...

But given that theta and phi depend on time, how else can one do this?.. Is he leaping over steps or am I not seeing something?

Of course, he doesn't do this.. He just takes the average, but the average involving some sort of integral will have - dt- terms in it. Maybe these are cancelling the derivative's denominators, leaving d theta and so forth... but what are the boundaries of the integral now?... (even before, what are they?)

 

[yuanyuan5220]

Hope the following link useful to you
http://docs.google.com/fileview?id=...DYtMzUwOC00YWZlLThmMzgtNzk1ZDUxYTExNzAz&hl=en
Eq.(57)-Eq.(60)

 

[sokrates]

I have figured this out, and did the hard part myself. It wasn't necessary to re-do everything else this author did. You just need to keep track of all the weird new variables he comes up with.

But now I have another problem, which will be the topic of another thread.

Edit : To answer my own question that might be of interest:
The averaging is done over one period by the usual:

[tex]
1/ T \int_0^T f(\tau) d\tau
[/tex]

The reason for averaging is to get a meaningful estimate on the change of dU/dt -- because the unperturbed equations are periodic ; a single point on the solution wouldn't mean much. The steady (average) increase or decrease is important here.

U is sometimes called (apparently, I just learned this) the Liapunov Function, and checking whether it's negative or positive is the Lyapunov stability condition.

It basically says that " IF the rate of change of ENERGY of a system, near an equilibrium point is DECREASING, then the solution is evolving towards an equilibrium point (where the energy of the system is defined to be zero), otherwise solution (near this equilibrium point is not stable".In this particular problem, I am trying to find the INSTABILITY condition because I am seeking the threshold for a dramatic global change (magnetization reversal) due to a spin-polarized current injection into the magnet.

The 21st century way of reversing magnets is to shoot spins towards it whereas the old-fashioned way is simply to apply a magnetic field in the other direction. This effect is called the spin-torque effect, because the injected current is really, physically exerting a TORQUE on a soft thin-film magnet.

The effect was predicted in 1996 by the theorists John Slonczewski of IBM and by Luc Berger of Carneige Mellon (independently) and first demonstrated in 2004 or so...

 

[yuanyuan5220]

recommend one paper on stability analysis of LLG equation with spin-polarized current
prb, 76, 054414 (2007)

 

[sokrates]

Looks very interesting, thank you.