Extended equations
===================

Stochastic LLG equation
---------------------------------------
The implemented equation including the thermal fluctuation effects is

.. math::
   \frac{\partial \vec{m}}{\partial t} = - \gamma \vec{m} \times (\vec{H}+\vec{\xi}) + \alpha \vec{m} \times  \frac{\partial \vec{m}}{\partial t}

where :math:`\vec{\xi}` is the thermal fluctuation field and is assumed to have the following
properties,

.. math::
   \left< \vec{\xi} \right> = 0, \;\;\; \left< \vec{\xi}_i^u,\vec{\xi}_j^v \right> = 2 D \delta_{ij} \delta_{uv}

and 

.. math::
   D = \frac{\alpha k_B T}{\gamma \mu_s}

.. Including the susceptibility 
.. -------------------------------
.. The dynamic equation we used including the susceptibility is,

.. .. math::
..    \frac{\partial \vec{S}_i}{\partial t} = - \gamma \vec{S}_i \times \vec{H}_i + \alpha \gamma \vec{H}_i + \frac{1}{\chi} (1-S_i^2)\vec{S}_i
..    :label: eq_llg_s

.. where :math:`\vec{S}_i` is defined by the ratio of magnetic moment :math:`\vec{\mu}_i` and the equalibrium magnitude of the effective magnetic moment :math:`\mu_s=M_0 a^3`, 

.. .. math::
..    \vec{S}_i=\frac{\vec{\mu}_i}{\mu_s}.

.. The effective field is defined by,

.. .. math::
..    \vec{H}_i = - \frac{1}{\mu_s}\frac{\partial \mathcal{H}}{\partial \vec{S}_i}

.. Therefore, the corresponding fields are,

.. .. math::
..    \vec{H}_{i,ex} =\frac{J}{\mu_s} \sum_{<i,j>} \vec{S}_j

.. .. math::
..    \vec{H}_{i,an} = \frac{2 D}{\mu_s} S_{x,i} \vec{e}_x



Spin transfer torque (Zhang-Li model)
---------------------------------------
The extended equation with current is,

.. math::
   \frac{\partial \vec{m}}{\partial t} = - \gamma \vec{m} \times \vec{H} + \alpha \vec{m} \times  \frac{\partial \vec{m}}{\partial t}   + u_0 (\vec{j}_s \cdot \nabla) \vec{m} - \beta u_0 [\vec{m}\times (\vec{j}_s \cdot \nabla)\vec{m}]

Where 

.. math::
   u_0=\frac{p g \mu_B}{2 |e| M_s}=\frac{p g \mu_B a^3}{2 |e| \mu_s}

and :math:`\mu_B=|e|\hbar/(2m)` is the Bohr magneton. Notice that :math:`\partial_x \vec{m} \cdot \vec{m}=0` so :math:`u_0 (\vec{j}_s \cdot \nabla) \vec{m}=  - u_0 \vec{m}\times[\vec{m}\times (\vec{j}_s \cdot \nabla)\vec{m}]`. Besides,
we can change the equation to atomistic one by introducing :math:`\vec{s}=-\vec{S}` where :math:`\vec{S}` is the local spin such that

.. math::
   \vec{M}=-\frac{g \mu_B}{a^3}\vec{S} =\frac{g \mu_B}{a^3}\vec{s}


so  :math:`u_0=p a^3/(2|e|s)`, furthermore,

.. math::
   \frac{\partial \vec{s}}{\partial t} = - \gamma \vec{s} \times \vec{H} + \frac{\alpha}{s} \vec{s} \times  \frac{\partial \vec{s}}{\partial t}   + \frac{p a^3}{2|e|s} (\vec{j}_s \cdot \nabla) \vec{s} -  \frac{p a^3 \beta}{2|e|s^2} [\vec{s}\times (\vec{j}_s \cdot \nabla)\vec{s}]

 
However, we perfer the normalised equaiton here, after changing it to LL form, we obtain,

.. math::
   (1+\alpha^2)\frac{\partial \vec{m}}{\partial t} = - \gamma \vec{m} \times \vec{H} - \alpha \gamma \vec{m} \times (\vec{m} \times \vec{H}) + (1+\alpha\beta) u_0 (\vec{j}_s \cdot \nabla) \vec{m} - (\beta-\alpha) u_0 [\vec{m}\times (\vec{j}_s \cdot \nabla)\vec{m}]

although in principle :math:`\partial_x \vec{m}` is always perpendicular to :math:`\vec{m}`, it's better to take an extra step to remove its longitudinal component, therefore, the real equation written in codes is,

.. math::
   (1+\alpha^2)\frac{\partial \vec{m}}{\partial t} = - \gamma \vec{m} \times \vec{H}_{\perp} + \alpha \gamma \vec{H}_{\perp}   + (1+\alpha \beta) u_0 \vec{\tau}_{\perp} - (\beta-\alpha) u_0 (\vec{m}\times  \vec{\tau}_{\perp})

where :math:`\vec{\tau}=(\vec{j}_s \cdot \nabla)\vec{m}` is the effective torque generated by current.


Nonlocal spin transfer torque (full version of Zhang-li model)
---------------------------------------------------------------
The LLG equation with STT is given by,

.. math::
   \frac{\partial \vec{m}}{\partial t} = - \gamma \vec{m} \times \vec{H} + \alpha \vec{m} \times  \frac{\partial \vec{m}}{\partial t} + \vec{T}

where :math:`\vec{T}` is the spin transfer torque. In the local form the STT is given by,

.. math::
   \vec{T}_{loc} = u_0 (\vec{j}_s \cdot \nabla) \vec{m} - \beta u_0 [\vec{m}\times (\vec{j}_s \cdot \nabla)\vec{m}]

And in general case, the spin transfer torque could be computed by,

.. math::
   \vec{T}=-\frac{\vec{m} \times \delta \vec{m}}{\tau_{sd}}

where :math:`\tau_{sd}` is the s-d exchange time and :math:`\delta \vec{m}` is the nonequilibrium spin density governed by

.. math::
    \frac{\partial \delta \vec{m}}{\partial t} = D \nabla^2 \delta \vec{m} + \frac{\vec{m} \times \delta \vec{m}}{\tau_{sd}} - \frac{\delta \vec{m}}{\tau_{sf}} +u_0 (\vec{j}_s \cdot \nabla) \vec{m} 

By changing the LLG equation to LL form, we obtain,

.. math::
   (1+\alpha^2)\frac{\partial \vec{m}}{\partial t} = - \gamma \vec{m} \times \vec{H} - \alpha \gamma \vec{m} \times (\vec{m} \times \vec{H}) + \vec{T} + \alpha \vec{m} \times \vec{T}
   
i.e.,

.. math::
   (1+\alpha^2)\frac{\partial \vec{m}}{\partial t} = - \gamma \vec{m} \times \vec{H} - \alpha \gamma \vec{H}_{\perp}  - \frac{\vec{m} \times \delta \vec{m}}{\tau_{sd}} + \alpha \frac{\delta \vec{m}}{\tau_{sd}}


Spin transfer torque (current-perpendicular-to-plane, CPP)
-------------------------------------------------------------------------------------------
In this case (current-perpendicular-to-plane, CPP), there are two types of torques can be 
added to the orginal LLG equation. One is the so called Slonczewski torque 
:math:`\vec{\tau}_s=-a_J \vec{m} \times (\vec{m} \times \vec{p})`, and the other is 
a fieldlike torque :math:`\vec{\tau}_f=- b_J(\vec{m} \times \vec{p})` [PRL 102 037206 (2009)]. 
So the full LLG equation is
 
.. math::
   \frac{\partial \vec{m}}{\partial t} = - \gamma \vec{m} \times \vec{H} + \alpha \vec{m} \times  \frac{\partial \vec{m}}{\partial t} -a_J \vec{m} \times (\vec{m} \times \vec{p}) - b_J (\vec{m} \times \vec{p})

where :math:`\vec{p}` is the unit vector of the spin polarization. The parameter :math:`b_J = \beta a_J` and 

.. math::
    a_J = \frac{\hbar \gamma J P}{2 |e| d M_s}

As we can see, this equation is exactly the same as the one used in Zhang-Li case if we take :math:`\vec{p} = (\vec{j}_s \cdot \nabla)\vec{m}`. So the implemented equation in the code is,

.. math::
   (1+\alpha^2)\frac{\partial \vec{m}}{\partial t} = - \gamma \vec{m} \times \vec{H}_{\perp} + \alpha \gamma \vec{H}_{\perp}   + (1+\alpha \beta) a_J \vec{p}_{\perp} - (\beta-\alpha) a_J (\vec{m}\times  \vec{p}_{\perp})

where :math:`\vec{p}_\perp=\vec{p}-(\vec{m}\cdot\vec{p})\vec{m}`.