extended_eqs.rst

Extended equations

Stochastic LLG equation

The implemented equation including the thermal fluctuation effects is

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

where ξ⃗ is the thermal fluctuation field and is assumed to have the following properties,

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

and

$$D = \frac{\alpha k_B T}{\gamma \mu_s}$$

Spin transfer torque (Zhang-Li model)

The extended equation with current is,

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

Where

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

and μ_B = |e|ℏ/(2m) is the Bohr magneton. Notice that ∂_xm⃗ ⋅ m⃗ = 0 so u₀(j⃗_s ⋅ ∇)m⃗ =  − u₀m⃗ × [m⃗ × (j⃗_s ⋅ ∇)m⃗]. Besides, we can change the equation to atomistic one by introducing s⃗ =  − S⃗ where S⃗ is the local spin such that

$$\vec{M}=-\frac{g \mu_B}{a^3}\vec{S} =\frac{g \mu_B}{a^3}\vec{s}$$

so u₀ = pa³/(2|e|s), furthermore,

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

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

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

although in principle ∂_xm⃗ is always perpendicular to m⃗, it's better to take an extra step to remove its longitudinal component, therefore, the real equation written in codes is,

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

where τ⃗ = (j⃗_s ⋅ ∇)m⃗ 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,

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

where T⃗ is the spin transfer torque. In the local form the STT is given by,

T⃗_loc = u₀(j⃗_s ⋅ ∇)m⃗ − βu₀[m⃗ × (j⃗_s ⋅ ∇)m⃗]

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

$$\vec{T}=-\frac{\vec{m} \times \delta \vec{m}}{\tau_{sd}}$$

where τ_sd is the s-d exchange time and δm⃗ is the nonequilibrium spin density governed by

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

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

$$(1+\alpha^2)\frac{\partial \vec{m}}{\partial t} = - \gamma \vec{m} \times \vec{H} - \alpha \gamma \vec{m} \times (\vec{m} \times \vec{H}) + \vec{T} + \alpha \vec{m} \times \vec{T}$$

i.e.,

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

Spin transfer torque (Slonczewski type)

We consider the LLG equation with a Slonczewski-type extension [PRB 91 064423 (2015)],

$$\frac{\partial \vec{m}}{\partial t} = - \gamma \vec{m} \times \vec{H} + \alpha \vec{m} \times \frac{\partial \vec{m}}{\partial t} + u \vec{m} \times (\vec{p}\times \vec{m})$$

where p⃗ is the unit vector in the direction of the spin polarization. Similar to the Zhang-Li case, the implemented equation in the code is,

$$(1+\alpha^2)\frac{\partial \vec{m}}{\partial t} = - \gamma \vec{m} \times \vec{H}_{\perp} + \alpha \gamma \vec{H}_{\perp} + u \vec{p}_{\perp} + \alpha u \vec{m} \times \vec{p}_{\perp}$$

where p⃗_⊥ = p⃗ − (m⃗ ⋅ p⃗)m⃗.