Introduction

In this section we will walk through the basics of the LLG equation and the transformation to the LL-form of the LLG equation. The LLG equation is a proper equation in physical sense (derived from actual mechanics), whereas LL is simply a transformed approximation of the LLG equation that allows us to solve it numerically (no implicit term on \(\frac{dm}{dt}\)).

Landau Lifshitz form of Landau Lifshitz-Gilbert equation

Standard form of the LLG-SOT equation:

\[\frac{d\textbf{m}}{dt} = -\gamma \textbf{m} \times \textbf{H}_{\mathrm{eff}} + \alpha \textbf{m}\times \frac{d\textbf{m}}{dt} -\gamma|\tau_{fl}|\textbf{m} \times \phi-\gamma|\tau_{dl}|\textbf{m}\times\textbf{m}\times \phi\]

Multiply that equation \(\times m\):

\[ \begin{gathered} \textbf{m} \times \frac{d\textbf{m}}{dt} = -\gamma \textbf{m} \times\textbf{m} \times \textbf{H}_{\mathrm{eff}} + \alpha \textbf{m} \times \textbf{m}\times \frac{d\textbf{m}}{dt} - \gamma|\tau_{fl}| \textbf{m} \times \textbf{m} \times \phi- \gamma|\tau_{dl}|\textbf{m} \times \textbf{m}\times\textbf{m}\times \phi \\ \textbf{m} \times \frac{d\textbf{m}}{dt} = -\gamma \textbf{m} \times\textbf{m} \times \textbf{H}_{\mathrm{eff}} - \alpha \frac{d\textbf{m}}{dt} - \gamma|\tau_{fl}| (\textbf{m}(\textbf{m}\cdot\phi) - \phi) + \gamma|\tau_{dl}|\textbf{m}\times \phi\end{gathered} \]

Substitute RHS of the derived equation above into \(\textbf{m}\times \frac{d\textbf{m}}{dt}\):

\[ \begin{gathered} \frac{d\textbf{m}}{dt} = -\gamma \textbf{m} \times \textbf{H}_{\mathrm{eff}} + \alpha [-\gamma \textbf{m} \times\textbf{m} \times \textbf{H}_{\mathrm{eff}} - \alpha \frac{d\textbf{m}}{dt} - \gamma|\tau_{fl}| (\textbf{m}(\textbf{m}\cdot\phi) - \phi) + \gamma|\tau_{dl}|\textbf{m}\times \phi] \\ - \gamma|\tau_{fl}|\textbf{m} \times \phi \\ - \gamma|\tau_{dl}|\textbf{m}\times\textbf{m}\times \phi \\ \frac{d\textbf{m}}{dt}(1 + \alpha^2) = -\gamma \textbf{m} \times \textbf{H}_{\mathrm{eff}} - \alpha\gamma\textbf{m}\times\textbf{m}\times\textbf{H}_{\mathrm{eff}} \\ - \gamma|\tau_{fl}|[\textbf{m} \times \phi + \alpha (\textbf{m}(\textbf{m}\cdot\phi) - \phi) ] \\ - \gamma|\tau_{dl}|[\textbf{m}\times\textbf{m}\times \phi - \alpha\textbf{m}\times \phi]\end{gathered} \]

Rearranging the terms gives:

\[ \begin{aligned} \frac{d\textbf{m}}{dt} = \frac{-\gamma}{1 + \alpha^2}[\textbf{m} \times \textbf{H}_{\mathrm{eff}} + \alpha\textbf{m}\times\textbf{m}\times\textbf{H}_{\mathrm{eff}} &\\ + |\tau_{fl}|[\textbf{m} \times \phi + \alpha (\textbf{m}(\textbf{m}\cdot\phi) - \phi) ] &\\ + |\tau_{dl}|[\textbf{m}\times\textbf{m}\times \phi - \alpha\textbf{m}\times \phi]]\end{aligned} \]

In this form, \(\gamma\) is the gyromagnetic ratio and is equal to \(\gamma \approx 2.2e5 \frac{m}{As}\). The last part can be re-arranged to:

\[ \begin{aligned} (...) + |\tau_{fl}|[\textbf{m} \times \phi + \alpha (\textbf{m}(\textbf{m}\cdot\phi) - \phi) ] + |\tau_{dl}|[\textbf{m}\times\textbf{m}\times\phi - \alpha\textbf{m}\times \phi]]&\\ = \textbf{m} \times \phi(|\tau_{fl}| - \alpha|\tau_{dl}|) + \textbf{m}\times\textbf{m}\times\phi(|\tau_{dl}| + \alpha|\tau_{fl}|)\end{aligned} \]

What is evident in this form of LL form of the LLG equation is the mixing of the torques with damping as the scaling factor (the field-like term for instance now becomes \(|\tau_{fl}| - \alpha|\tau_{dl}|\)). Proper LL-form of the LLGS equation is:

\[ (...)+ \frac{-\gamma}{1 + \alpha^2}[ \tau'_1 \textbf{m}\times(\textbf{m}\times \phi) + \tau'_2 \textbf{m}\times \phi] \]

STT interaction

The origin of STT is different, thus, we use a different set of quantities:

\[ \begin{gathered} |H_{dl}| = \beta |H_{fl}|, \quad \beta \in [0, 1] \\ a_j = \frac{\hbar j_e}{eM_s t_{FM}} \\ \eta = \frac{\sigma \Lambda^2}{\Lambda^2 + 1 + (\Lambda^2 -1)\textbf{m}\cdot\sigma}\end{gathered} \]

Given those two terms above, the non-torque part remains the same:

\[... + a_j\eta\beta\textbf{m} \times \sigma + a_j\eta\textbf{m}\times\textbf{m}\times \sigma\]

Then, equation only changes the coefficients scaling the damping-like and field-like torques.

The LL form of the STT equation is:

\[ \begin{aligned} \frac{d\textbf{m}}{dt} = \frac{-\gamma}{1 + \alpha^2}[\textbf{m} \times \textbf{H}_{\mathrm{eff}} + \alpha\textbf{m}\times\textbf{m}\times\textbf{H}_{\mathrm{eff}} &\\ + (-a_j (\mathbf{m}\times\mathbf{p}) + a_j\beta (\mathbf{m}\times\mathbf{m}\times\mathbf{p})] \end{aligned} \]

where \(\(a_j = \gamma_0 \eta \frac{\hbar j}{e M_\mathrm{s} t_\mathrm{FM}}\)\)

Stochastic LLGS

Stratonovich formulation of the s-LLGS SDE

A stochastic formulation of LLGS will take the form of a Stratonovich SDE:

\[\mathrm{d}X_t = f(X_t, t)dt + g(X_t, t)\circ \mathrm{d}W_t\sqrt{\Delta t}\]

where \(f(X_t, t)\) is the deterministic part of the equation and \(g(X_t, t)\) is the stochastic part of the equation. \(\mathrm{d}W\) is \"derivative-like\" of the Brownian motion. The symbol \(\circ\) denotes the Stratonovich product which distinguishes it from Ito's SDE. By assuming that the effective field contains thermal fluctuations \(\mathbf{H}_{\mathrm{eff}} \rightarrow \mathbf{H}_{\mathrm{eff}} + \mathbf{H}_{\mathrm{T}}\) we transform the standard LLGS equation into the form that fits Stratonovich SDE. The thermal fluctuations have zero mean and a preset standard deviation:

\[\sigma(t) = \sqrt{\frac{4\alpha k_bT(t)}{\mu_0 M_s V\gamma_0}} \quad (1)\]

where \(V\) is the volume of the cell (layer), and \(k_bT(t)\) is the thermal energy of the system. As a result, \(\sigma(t)\) should be dimensionless.

To convince ourselves that this is the correct form, one can take a look at the units. In the standard LLG, let's take a term \(\frac{dm}{dt} = -\gamma\mathbf{m}\times\mathbf{H}_{\mathrm{eff}}\). We take \(\mathbf{m}\) to be unit and \(\mathbf{H}_{\mathrm{eff}}\) to be in units of A/m. Then, we have"

\[ \left[\frac{1}{s}\right] = \left[\frac{m}{As}\right]\left[\frac{A}{m}\right] \]

and thus we multiply by the time step \(\Delta t\) to get the unit again.

We now take a look at the equation \((1)\). We have (we take the sqrt off for now):

\[ \frac{ \left[J\right] }{ \left[\frac{N}{A^2}\right] \left[\frac{A}{m}\right] \left[m^3\right] \left[\frac{m}{A s}\right] } = \frac{ \left[Nm\right] }{ \left[\frac{N}{A^2}\right] \left[m^2\right] \left[\frac{1}{s}\right] } = \frac{ \left[A^2\right] }{ \left[m^2\right] \left[\frac{1}{s}\right] } = \frac{ \left[A^2 s\right] }{ \left[m^2\right] } \]

So, instead of \(\frac{A}{m}\) we get \(\frac{A s^{1/2}}{m}\) after taking the square root. But, in the end for stochastic torque we have:

\[ RHS = \left[\frac{m}{As}\right]\left[\frac{A}{m}\right]\left[\sqrt{s}\right] = \left[\frac{\sqrt{s}}{s}\right] \]

but we multiply the RHS by \(\sqrt{\Delta t}\) to get the unit back \(\rightarrow \left[\frac{\sqrt{s}}{s}\right] [\sqrt{s}] = 1\).

Finally, we set \(\mathbf{f}(\mathbf{m}_t, t)\) to LL form where \(\mathbf{H}_{\mathrm{eff}}\) contains no stochastic (thermal) parts and the \(g\), the stochastic part, to the following:

\[ \mathbf{g}(\mathbf{m}_t, t)\circ\mathrm{d}W = - \frac{\sigma\gamma}{1+\alpha^2}[\mathbf{m}\times\mathrm{d}W + \alpha\mathbf{m}\times(\mathbf{m}\times\mathrm{d}W)] \]

with \(\mathrm{d}W \in \mathbf{R}^3 \sim \sqrt{t}\mathcal{N}(0, 1)\), a multinomial Gaussian distributed random vector (here we make a transition from \(W\) being a generalised Brownian process to a Wiener process). For numerical solutions, we have have \(\Delta W\) instead of \(\mathrm{d}W\). \(\Delta W(t) = W(t + \Delta t) - W(t)\), where the stochastic vector is being drrawn from a normal distriubtion, with zero mean and unit variance: \(\xi_t \in \mathbf{R}^3 \sim \mathcal{N}(0, 1)\). The form above follows from the distributive properties of cross-product over addition. Furthermore, there is some evidence that the second term in that equation should be skipped if the noise is sufficiently small which seems to be the case for up to room temperature experiments.

Numerical solutions

We generally solve the stochastic model by either with Euler-Heun or Heun method.

Euler-Heun method

This is in fact first order-method in the limit of 0 K. Euler-Heun method is suitable for Stratonovich SDEs as Euler-Maruyama can only be applied to Ito's SDEs. The update of the step is:

\[Y_{n+1} = Y_n + f_n \Delta t + \frac{1}{2}[g_n + g_n(\hat{Y}_n)]\Delta W_n\sqrt{\Delta t}\]

where \(\hat{Y}_n = Y_n + g_n\Delta W_n\). Contrary to the Milstein method, it is easier to the user the Euler-Heun due to the lack of quadratic terms of \(\Delta W_n\). The cost is in the convergence order which is 0.5 for strong convergence and 1 for weak convergence. For the solution, we substitute \(Y_n = \mathbf{m_t}\), \(f_n = \mathbf{f}_n(\mathbf{m_t}, t)\), \(g_n= \mathbf{g}_n(\mathbf{m_t}, t)\).

Heun method

Now preferred method to solve stochastic form of the LLG equation is the Heun method. It introduces second order correction to the non-stochastic part as well and therefore is deemed a better method.

\[Y_{n+1} = Y_n + \frac{1}{2}\left[f_n(\hat{Y}_{n+1}, t_{n+1}) + f_n(Y_n, t_n)\right] + \frac{1}{2}\left[g_n(\hat{Y}_{n+1}, g_{n+1}) + g_n(Y_n, t_n)\right]\Delta W_n\]

where \(\hat{Y}_{n+1} = Y_n + f_n(Y_n, t_n)\Delta t + g_n(Y_n, t_n)\Delta W_n\sqrt{\Delta t}\).

References

Numerical Integration of SDEs: A Short Tutorial, Thomas Schaffter, January 19, 2010