Adiabatic domain wall motion and Landau-Lifshitz damping

Abstract

Recent theory and measurements of the velocity of current-driven domain walls in magnetic nanowires have re-opened the unresolved question of whether Landau-Lifshitz damping or Gilbert damping provides the more natural description of dissipative magnetization dynamics. In this paper, we argue that (as in the past) experiment cannot distinguish the two, but that LandauLifshitz damping nevertheless provides the most physically sensible interpretation of the equation of motion. From this perspective, (i) adiabatic spin-transfer torque dominates the dynamics with small corrections from non-adiabatic effects; (ii) the damping always decreases the magnetic free energy, and (iii) microscopic calculations of damping become consistent with general statistical and thermodynamic considerations. I. BACKGROUNDExperiments designed to study the effect of electric current on domain wall motion in magnetic nanowires show that domain walls move over large distances with a velocity proportional to the applied current. 1,2,3,4,5,6,7,8,9,10 Most theories ascribe this behavior to the interplay between spin-transfer (the quantum mechanical transfer of spin angular momentum between conduction electrons and the sample magnetization) and magnetization damping of the Gilbert type. 11 Contrary to the second point, we argue in this paper that LandauLifshitz damping 12 provides the most natural description of the dynamics. This conclusion is based on the premises that damping should always reduce magnetic free energy and that microscopic calculations must be consistent with statistical and thermodynamic considerations.Theoretical studies of current-induced domain wall motion typically focus on one-dimensional models where current flows in the x-direction through a magnetization M(x) = MM(x). When M is constant, the equation of motion isṀThe precession torque −γM × H depends on the gyromagnetic ratio γ and an effective field µ 0 H = −δF/δM which accounts for external fields, anisotropies, and any other effects that can be modelled by a free energy F [M] (µ 0 is the magnetic constant). The spintransfer torque N ST is not derivable from a potential, but its form is fixed by symmetry arguments and model calculations. 13,14,15,16,17,18,19,20,21 A local approximation 22 (for current in the x-direction) isThe first term in (2) occurs when the spin current follows the domain wall magnetization adiabatically, i.e., when the electron spins remain largely aligned (or antialigned) with the magnetization as they propagate through the wall. The constant υ is a velocity. If P is the spin polarization of the current, j is the current density, and µ B is the Bohr magneton,The second term in (2) arises from non-adiabatic effects.The constant β is model dependent. The damping torque D in (1) accounts for dissipative processes, see 23 for a review. Two phenomenological forms for D are employed commonly: the LandauLifshitz form 12 with damping constant λ,and the Gilbert form 11 with damping constant α,The difference between the two is usually very small and almo...

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“…(ii) As alluded to earlier, the validity of the Gilbert damping term in the presence of spin torque is being debated [27,19]. Because this perspective is primarily concerned with sustained precession in the nano-pillar and point-contact geometries, the following simple argument shows that a …”

Electrical rectification effect in single domain magnetic microstrips: A micromagnetics-based analysis

“…However, when the current flows in a magnetic medium with a continuously varying magnetization, the situation is more complex. As a result, several forms for this so-called spin transfer torque (STT) have been proposed [5,6,7,8], and the appropriate equation for magnetization dynamics has even been questionned [9,10].…”

Respective influence of in-plane and out-of-plane spin-transfer torques in magnetization switching of perpendicular magnetic tunnel junctions

“…(2), where the [αa ⊥m × (m ×p)] component allows the system to change its energy even if a = 0. That turns us to the still open discussion [45][46][47][48][49][50][51][52] of physical validity of Gilbert damping and Landau damping formulation in the magnetization dynamics equation. Although it is generally claimed that LL and LLG equations are mathematically equivalent, we can see a significant difference when the STT terms are added: the field-like STT term written in the LL equation is fully conservative, and it cannot change the system energy if Eq.…”