2.6.1 Landau-Lifshitz-Gilbert equation with Slonczewski
spin-transfer torque term

In order to introduce a model equation for magnetization dynamics in presence of spin polarized currents, let us first consider the model derived by Slonczewski in Ref. [44]. In his paper, a five layers structure is considered. In this structure, the first, the third and the fifth layers are constituted by paramagnetic conductors and the second and the fourth layers are ferromagnetic conductors (it is a three layers structure as the one mentioned in the introduction with paramagnetic conductors as spacer and contacts). The multilayers system is traversed by electric current normal to the layers plane. The electron spins, polarized by the fixed ferromagnetic layer (the second layer) are injected by passing through the paramagnetic spacer into the free ferromagnetic layer (the forth layer) where the interaction between spin polarized current and magnetization takes place. The magnetic state of the ferromagnetic layers is described by two vectors $\mathbf{S}_1$ and $\mathbf{S}_2$ representing macroscopic (total) spin orientation per unit area of the fixed and the free ferromagnetic layers, respectively. The connection of this two vectors with the total spin momenta $\textbf{L}_1$ and $\textbf{L}_2$ (which have the dimension of angular momenta) is given by the equations $\textbf{L}_1= \hbar\textbf{S}_1 A$, $\textbf{L}_2= \hbar\textbf{S}_2 A$, where $A$ is the cross-sectional area of the multilayers structure. By using a semiclassical approach to treat spin transfer between the two ferromagnetic layers, Slonczewski derived the following generalized LLG equation (see Eq.(15) in [44]):

$$\frac{d\textbf{S}_2}{dt}= \textbf{s}_2 \times \left( \gamma H_u \... ...c{d\textbf{S}_2}{dt} + \frac{I_e g}{e} \textbf{s}_1 \times \textbf{s}_2 \right)$$ (2.74)

where $\textbf{s}_1$, $\textbf{s}_2$ are the unit-vectors along $\textbf{S}_1$, $\textbf{S}_2$, $\gamma$ is the absolute value of the gyromagnetic ratio, $H_u$ is the anisotropy field constant, $\textbf{c}$ is the unit vector along the anisotropy axis (in-plane anisotropy), $\alpha$ the Gilbert damping constant, $I_e$ the current density (electric current per unit surface), $e$ is the absolute value of the electron charge, $g$ a scalar function given by the following expression

$$g(\textbf{s}_1 \cdot \textbf{s}_2)=\left[ -4 +(1+P)^3 \frac{(3+ \textbf{s}_1 \cdot \textbf{s}_2 )}{4P^{3/2}} \right]^{-1}$$ (2.75)

and $P$ is the spin polarizing factor of the incident current which gives the percent amount of electrons that are polarized in the $\textbf{p}$ direction (see Ref. [44] for details). The current $I_e$ in Eq. (2.74) is assumed to be positive when the charges move from the fixed to the free layer. Let us notice that in Eq. (2.74) the ferromagnetic body is assumed to be uniaxial with anisotropy axis along $\textbf{c}$. In the sequel, we will remove this simplifying assumption by taking into account the effect of the strong demagnetizing field normal to the plane of the layer in order to consider the thin-film geometrical nature of the free layer. Our next purpose is to derive from Eq. (2.74) an equation for magnetization dynamics. We will carry out this derivation by using slightly different notation and translating all the quantities in practical MKSA units. First of all, let us introduce a system of cartesian unit vectors $\mathbf{e}_x$, $\mathbf{e}_y$ $\mathbf{e}_z$, where $\mathbf{e}_z$ is normal to the film plane and pointing in the direction of the fixed layer, and $\mathbf{e}_x$ is along the in-plane easy axis (in the Slonczewski notation $\mathbf{e}_x=\textbf{c}$). The current density will be denoted by $J_e$ (instead of $I_e$ as in Eq. (2.74)), the anisotropy field as $H_$an and the function $g(\textbf{s}_1\cdot \textbf{s}_2)$ in Eq. (2.75) will be denoted with $b$ to avoid confusion with the free energy and the Landé factor $g_e$ that will be used in the following. In this reference frame the current density vector is $\mathbf{J}=J_e \mathbf{e}_z$, which means that when $J_e > 0$ the electrons travel in the direction opposite to $\mathbf{e}_z$, namely from the fixed to the free layer. By using these modified notations and by including the effects of the demagnetizing field $\mathbf{H}_m$ and the applied field $\mathbf{H}_a$ , Eq. (2.74) becomes

$$\frac{d\textbf{S}_2}{dt}= \textbf{s}_2 \times \left( \gamma H_\te... ...d\textbf{S}_2}{dt} + \frac{J_e b}{e} \textbf{s}_1 \times \textbf{s}_2 \right) ,$$ (2.76)

where S$_2=\vert\textbf{S}_2\vert$. The sum of the demagnetizing field $\mathbf{H}_m$ and the applied field $\mathbf{H}_a$ will be indicated in he following by $\mathbf{H}$ to shorten the notation, i.e.

$$\mathbf{H}=\mathbf{H}_m+\mathbf{H}_a .$$ (2.77)

In order to check the correctness of the signs of precessional terms in Eq. (2.76), let us transform this equation in a slightly different form, by factoring out from the parenthesis the constant S$_2$. For the sake of simplicity, we will carry out this derivation in the case $\alpha=0$ and $J_e=0$. We have then the following equation

$$\frac{d\textbf{S}_2}{dt}= \textbf{S}_2 \times \left( \gamma H_\te... ...dot \textbf{s}_2) \mathbf{e}_x - \gamma (\textbf{H}_m +\mathbf{H}_a) \right) .$$ (2.78)

We observe now that what is generally defined as effective anisotropy field is given by

$$\mathbf{H}_ =-H_ (\mathbf{e}_x \cdot \textbf{s}_2) \mathbf{e}_x .$$ (2.79)

The minus sign in this equation is due to the fact that the direction of $\textbf{s}_2$ is opposite to the direction of magnetization. This issue will be discussed below. By substituting Eq. (2.79) into Eq. (2.78) we obtain

$$\frac{d\textbf{S}_2}{dt}= - \gamma \textbf{S}_2 \times \left( \ma... ...}_m +\mathbf{H}_a \right)= - \gamma \textbf{S}_2 \times \mathbf{H}_\text{eff},$$ (2.80)

where

$$\mathbf{H}_ = \mathbf{H}_ +\textbf{H}_m +\mathbf{H}_a= -H_ (\mathbf{e}_x \cdot \textbf{s}_2) \mathbf{e}_x +\textbf{H}_m +\mathbf{H}_a .$$ (2.81)

Equation (2.80) is the correct precession equation for the spin vector dynamics. Next, we want to derive the dynamical equation for the magnetization vector $\textbf{M}$ associated with the free layer. In this respect, we have first to consider the relation between $\textbf{S}_2$ and $\textbf{M}$. The total magnetic moment $\vec{\mu}$ associated with the free layer is given by

$$\vec{\mu} = -\gamma \textbf{L}_2= -\gamma \hbar \textbf{S}_2 A = -\gamma \hbar \textbf{S}_2 A ,$$ (2.82)

where $A$ has been introduced above and coincides with the area of the surface of the film. The magnetization $\textbf{M}$ is obtained by dividing the total magnetic moment by the volume of the film V:

$$\textbf{M}=\frac{\vec{\mu}}{V}=\frac{\vec{\mu}}{A d}$$ (2.83)

where $d$ is the free layer thickness, thus we obtain the following relations

$$\textbf{M}=\frac{-\gamma \hbar \textbf{S}_2 A}{A d}=\frac{-\gamma \hbar \textbf{S}_2 }{d}= \frac{-g_e \mu_\text{B}}{d}\textbf{S}_2$$ (2.84)

where $g_e$ is the Landè factor for electrons, $\mu_$B is the Bhor magneton, and the relation $\gamma=g_e \mu_$B$/ \hbar$ has been used. Let us notice that, as a consequence of Eq. (2.84), we have

$$\textbf{s}_2=-\textbf{{m}}=-\frac{\textbf{M}}{\text{M}_s}$$ (2.85)

where M$_s=\vert\textbf{M}\vert$ is the saturation magnetization and $\textbf{{m}}$ is the unit vector along $\textbf{M}$. By multiplying both sides of Eq. (2.76) by the factor ${-g_e \mu_\text{B}}/{d}$ and taking into account Eq. (2.85), one ends up with the following equation

$$\frac{d\textbf{M}}{dt}= -\textbf{{m}}\times \left( \gamma H_\text... ...rac{g_e \mu_\text{B}J_e b}{e d} \textbf{s}_1 \times \textbf{{m}}\right) ,$$ (2.86)

which can be further normalized by dividing both sides by M$_s$, leading to

$$\frac{d\textbf{{m}}}{dt}-\alpha \textbf{{m}}\times \frac{d\textbf... ...\text{B}J_e b}{e d \text{M}_s } \textbf{s}_1 \times \textbf{{m}}\right) .$$ (2.87)

In order to derive a time normalized form of the equation, we factor out from the parenthesis the term $\gamma M_s$ which has the dimension of a frequency, and thus we have

$$\frac{d\textbf{{m}}}{dt}-\alpha \textbf{{m}}\times \frac{d\textbf... ...\text{B}}{e d \text{M}_s } J_e b \textbf{s}_1 \times \textbf{{m}}\right) ,$$ (2.88)

where

$$\kappa_ = \frac{H_\text{an}}{\text{M}_s}, \qquad \textbf{h}=\frac{\textbf{H}_m + \textbf{H}_a}{\text{M}_s} = \textbf{h}_\text{m}+\textbf{h}_\text{a} ,$$ (2.89)

where $\textbf{h}_$a$=\mathbf{H}_$a$/M_s$, $\textbf{h}_$m$={\mathbf{H}_\text{m}}/M_s$. Finally, let us denote the direction of magnetization in the fixed layer by $\textbf{p}$. According to the previous reasoning, this direction is opposite to $\textbf{s}_1$, i.e. $\textbf{s}_1=-\textbf{p}$. We also define the following constant which has the physical dimension of a current density:

$$J_p={\gamma M_s}\frac{e \text{M}_s d}{g_e \mu_\text{B}} .$$ (2.90)

By using the notations defined above, we arrive to the following form of Eq. (2.88):

$$\frac{d\textbf{{m}}}{dt}-\alpha \textbf{{m}}\times \frac{d\textbf... ... + \textbf{h} +\frac{J_e}{J_p} b \textbf{{m}}\times \textbf{p} \right) ,$$ (2.91)

where the scalar (and dimensionless) function $b$, in the new notations, is

$$b=b(\textbf{{m}})=\left[ -4 +(1+P)^3 \frac{(3+ \textbf{{m}}\cdot \textbf{p} )}{4P^{3/2}} \right]^{-1} .$$ (2.92)

By measuring the time $t$ in units of $(\gamma M_s)^{-1}$, and introducing the following definitions,

$$\textbf{h}_{\text{eff}}= \kappa_\text{an} (\mathbf{e}_x \cdot \te... ...ext{a} , \qquad \beta=\beta(\textbf{{m}})=\frac{J_e}{J_p} b(\textbf{{m}}) ,$$ (2.93)

equation (2.91) can be written in the compact form

$$\frac{d\textbf{{m}}}{dt}-\alpha \textbf{{m}}\times \frac{d\textbf... ...eft( \textbf{h}_{\text{eff}}+\beta \textbf{{m}}\times \textbf{p}\right) .$$ (2.94)

In the following, we will find convenient to recast LLG equation in the following compact form:

$$\frac{d\textbf{{m}}}{dt}-\alpha \textbf{{m}}\times \frac{d\textbf{{m}}}{dt}= - \textbf{{m}}\times {\cal H}_ ,$$ (2.95)

where

$${\cal H}_ (\textbf{{m}})=\textbf{h}_{\text{eff}}(\textbf{{m}}) +\beta \textbf{{m}}\times \textbf{p} .$$ (2.96)

Equation (2.95) is formally identical to LLG when there are no current-driven torque term. With the definition of the generalized effective field ${\cal H}_$eff we included the current-driven torque inside the effective field. The micromagnetic equilibria including spin torque effect are now related to the following equations similar to Eqs. (2.22):

$$\left\{\begin{aligned}&\textbf{{m}}\times {\cal H}_\text{eff}(\... ...\quad, & \vert\textbf{{m}}\vert=1 \quad. \end{aligned}\right.$$

The basic difference between the ordinary effective field $\textbf{h}_{\text{eff}}(\textbf{{m}})$ and the generalized effective field ${\cal H}_\text{eff}(\textbf{{m}})$ is that the first one can be derived by the gradient of a free energy, while the second one cannot.