2.6.1 Landau-Lifshitz-Gilbert equation with Slonczewski

spin-transfer torque term

where , are the unit-vectors along , , is the absolute value of the gyromagnetic ratio, is the anisotropy field constant, is the unit vector along the anisotropy axis (in-plane anisotropy), the Gilbert damping constant, the current density (electric current per unit surface), is the absolute value of the electron charge, a scalar function given by the following expression

and is the spin polarizing factor of the incident current which gives the percent amount of electrons that are polarized in the direction (see Ref. [44] for details). The current 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 . 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 , , where is normal to the film plane and pointing in the direction of the fixed layer, and is along the in-plane easy axis (in the Slonczewski notation ). The current density will be denoted by (instead of as in Eq. (2.74)), the anisotropy field as

where

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

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

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

where

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 associated with the free layer. In this respect, we have first to consider the relation between and . The total magnetic moment associated with the free layer is given by

where has been introduced above and coincides with the area of the surface of the film. The magnetization is obtained by dividing the total magnetic moment by the volume of the film V:

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

where is the Landè factor for electrons,

where

which can be further normalized by dividing both sides by

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

where

where

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

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

By measuring the time in units of , and introducing the following definitions,

equation (2.91) can be written in the compact form

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

where

Equation (2.95) is formally identical to LLG when there are no current-driven torque term. With the definition of the generalized effective field

The basic difference between the ordinary effective field and the generalized effective field is that the first one can be derived by the gradient of a free energy, while the second one cannot.