Basic Equations for Magnetization Dynamics

2.1 Landau–Lifshitz equation

The Landau–Lifshitz equation for magnetization dynamics in ferromagnets can be construed as a dynamic constitutive relation that is compatible with micromagnetic constraints. To better understand the origin and nature of this equation, it is appropriate to start with a brief discussion of the micromagnetic description of ferromagnets subject to classical electromagnetic fields [10,79].

Micromagnetics is a continuum theory, which is highly nonlinear in nature and includes effects on rather different spatial scales such as short-range exchange forces and long-range magnetostatic effects. In micromagnetics, the state of the ferromagnet is described by the differentiable vector field (r,t) representing the local magnetization at every point inside the ferromagnet. When the temperature is well below the Curie temperature of the ferromagnet, the strong exchange interaction prevails over all other forces at the smallest spatial scale compatible with the continuum hypothesis. This fact is taken into account by imposing the following fundamental constraint:

M(r,t)|=Ms,(2.1)

which means that the magnitude of the local magnetization vector at each point inside the ferromagnet is equal to the spontaneous magnetization s at the given temperature . The direction of (r,t) is in general nonuniform, i.e., it varies from point to point. At equilibria, the spatial distribution of (r,t) results in extrema of an appropriate Gibbs–Landau free energy L(M(.);Ha). This free energy depends on the applied magnetic field a and the temperature . We omit the dependence of L and s on , since in the subsequent discussion the temperature will always be assumed to be uniform in space and constant in time.

The micromagnetic free energy L for a ferromagnet occupying the region is expressed as the following volume integral:

L(M(.);Ha)=∫Ω[AMs2((∇Mx)2+(∇My)2+(∇Mz)2)+fAN(M)−μ02M⋅HM−μ0M⋅Ha]dV.(2.2)

The first term inside the integral represents the exchange energy, which penalizes nonuniformities in the magnetization orientation. The constant is the so-called exchange stiffness constant; its value in ferromagnets is usually of the order of 10−11 J m−1. The second term AN(M) describes crystal anisotropy effects, while the two last terms represent magnetostatic energy and energy of interaction with the external magnetic field. The magnetostatic contribution is governed by the field M. This field is determined by solving the following magnetostatic Maxwell equations:

×HM=0,∇⋅HM=−∇⋅M,(2.3)

subject to the appropriate interface conditions at the ferromagnet surface. The applied field a is produced by external sources and, in subsequent discussion, it will be considered as a given vector function of space and time. The micromagnetic free energy may contain additional terms describing other energy contributions, for example magnetoelastic effects. These additional terms are beyond the scope of our discussion.

To find equilibrium magnetization states under given applied field a, the free energy variation GL with respect to arbitrary variations of the vector field (r) subject to the constraint (2.1) must first be determined. By using standard variational calculus, one obtains that GL corresponding to magnetization variation M(r) is given by the expression:

GL=−μ0[∫ΩHeff⋅δMdV−2Aμ0Ms2∮Σ∂M∂n⋅δMdS],(2.4)

where the second integral is over the surface of the ferromagnet, while /∂n represents the derivative with respect to the outward normal to . The effective field eff is defined as:

eff=Ha+HM+HAN+HEX,(2.5)

where AN and EX are the anisotropy field and the exchange field, respectively:

AN=−1μ0∂fAN∂M,HEX=2Aμ0Ms2∇2M.(2.6)

At equilibrium, GL=0 for any arbitrary variation M consistent with the constraint (2.1). Such a variation of will be of the form:

M=M×δv,(2.7)

where v is a small but otherwise arbitrary space-dependent vector. By substituting Eq. (2.7) into Eq. (2.4) and by taking into account that GL=0 for any arbitrary space-dependent variation v, one finds that at each point in the following equation is valid:

×Heff=0,(2.8)

whereas at each point on :

×∂M∂n=0i.e.∂M∂n=0.(2.9)

The above two forms of the boundary condition are equivalent because M/∂n is perpendicular to as a consequence of (2.1). Equation (2.8) is known as Brown’s equation; it expresses the fact that the local torque exerted on the magnetization by the effective field must be zero at equilibrium [133,134]. The boundary condition given by Eq. (2.9) is valid when no surface anisotropy is present. Surface anisotropy may give rise to pinning effects that substantially alter the response of the ferromagnet to external magnetic fields. In particular, spatially nonuniform magnetization modes may appear under spatially uniform driving fields in ellipsoidal ferromagnetic particles.

It is important to stress that Brown’s equation determines all possible magnetization equilibria regardless of their stability. However, according to the thermodynamic principle of free energy minimization, only L minima will correspond to stable equilibria and, thus, will be in principle physically observable. The information on the nature of equilibria can be obtained by computing the second variation of L and determining if it is positive under arbitrary variations of the vector field (r), subject to the constraint (2.1).

When ×Heff≠0, the system is not at equilibrium and will evolve in time according to some appropriate dynamic equation. The equation originally proposed by Landau and Lifshitz [429] is mostly used for the description of magnetization dynamics. This equation is based on the idea that in a ferromagnetic body the effective field eff will induce a precession of local magnetization (r,t) of the form:

M∂t=−γM×Heff,(2.10)

where >0 determines the precession rate. In the following, for simplicity we shall identify with the gyromagnetic ratio associated with the electron spin, which yields =2.2⋅105A−1ms−1. The dynamics described by Eq. (2.10) is such that the magnitude (length) of magnetization M| is conserved. Indeed, ⋅∂M/∂t=0 is an immediate consequence of Eq. (2.10). Thus, Eq. (2.10) is consistent with the fundamental...