Irreversible thermodynamics of non-uniform insulating ferromagnets

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Abstract

Two types of damping are commonly applied to describe ferromagnets at the phenomenological level: one by Landau and Lifshitz, and the other by Gilbert. This work successively applies the methods of irreversible thermodynamics to insulators, insulating paramagnets, and uniform insulating ferromagnets, in the last case uniquely obtaining Landau–Lifshitz damping. These methods are then applied to non-uniform insulating ferromagnets, for which new, non-current-related, spin fluxes and spin torques appear. These may be relevant to the dynamics of magnets in confined geometries, where boundary conditions impose complex non-trivial textures. Study of systems with very large damping might help distinguish between the two proposed forms of damping.

Introduction

The present work is directed to two topics in the damping of insulating ferromagnets, both studied with the methods of irreversible thermodynamics: (1) damping in uniform ferromagnets, where two forms of phenomenological damping are commonly employed, one by Landau and Lifshitz (LL), and the other by Gilbert; (2) damping in non-uniform insulating ferromagnets, which becomes relevant for non-monodomain nanomagnets. Although damping of non-uniform conducting ferromagnets is timely, it is of value to clearly establish the effects of non-uniformity before considering the effects of conduction.

The theory of damping in a ferromagnet dates to the 1935 work of Landau and Lifshitz, who wrote down an equation of motion that included precession and damping, the latter intended to be phenomenological [1]. Damping in paramagnetic systems was developed in the 1950s by a number of others, notably Bloch and co-workers. A number of phenomenological and microscopic theories were developed at that time, the one by Gilbert becoming the most popular, and currently in wide use [2]. One aspect of the theory of Gilbert (but not of LL) is that it supports the possibility of critical damping [3]. Nevertheless, the equations of motion of Landau and Lifshitz, and of Gilbert, are mathematically equivalent in the sense that, by redefining parameters for one theory one can get exactly the same equations as for the other theory. However, as we argue, they are not physically equivalent.

Many experimentalists, and many theorists working on microscopic mechanisms, often choose to interpret their results for damping within the framework of Gilbert. Nevertheless, there are strong physical reasons to prefer the framework of Landau and Lifshitz. First, it is a consequence of irreversible thermodynamics [4], [5]. Second, it has been derived both by projector operator techniques [6], [7], [8] and by Langevin equation methods [9]. Moreover, Landau–Lifshitz damping does not require that one renormalize the effective g-factor because of damping processes.

Landau and Lifshitz merely wrote down their form for the damping. At the time of their work, the theory of irreversible thermodynamics was still in its infancy. This was despite the early work of Einstein, Langevin, and others on near-equilibrium damping of particles in a fluid, and despite the work of Onsager on the symmetry of the kinetic coefficients that appear in phenomenological theories of damping for near-equilibrium macroscopic systems [10]. Landau had not yet developed his theory of superfluidity of 4He [11], which later Khalatnikov systematized, implicitly using the methods of irreversible thermodynamics [12]. The concept of the order parameter (M for a ferromagnet) was simultaneously being developed by Landau, but the concept of broken symmetry (e.g., the direction of M for a ferromagnet), a product of the early 1960s, was far in the future. The theory of irreversible thermodynamics, which applies to systems near equilibrium and subject to motions that vary slowly in space and time, relative to appropriate mean-free paths and collision times, was not developed until the late 1940s and early 1950s [13], [14].

In the late 1960s Halperin and Hohenberg applied the methods of irreversible thermodynamics to a number of broken symmetry systems, including isotropic ferromagnets and antiferromagnets [15]. If the resulting equations lead to excitations for which the frequency ω0 when the wavevector k0, as occurs for broken symmetry systems, then the resulting theory is said to be “hydrodynamic”. In the early 1970s a number of works used broken symmetry to develop the irreversible thermodynamics of liquid crystals [16], [17] and superfluid 3He [18], [19], [20]. Shortly thereafter a text applying irreversible thermodynamics to broken symmetry systems appeared [21]; and the topic is treated in at least two recent texts [22], [23]. Irreversible thermodynamics has been applied to numerous other systems, among them supersolids [24], [25], [26], ferrofluids [27], spin glasses [28], [29], and semiconductors (this last not being a broken symmetry system) [30].

Despite some late 1960's predictions of the demise of magnetism, research on this subject has, if anything, become more important, active, and widespread. Given that real ferromagnetism (with anisotropy and external magnetic fields, so there is no broken symmetry) is so relevant to real-world magnetic memory, the study and/or use of ferromagnets extends far beyond the borders of pure research. Nevertheless, the theory of damping in ferromagnets and paramagnets, using the principles of irreversible thermodynamics (which leads to Landau–Lifshitz damping) is not well known, perhaps because of terseness of presentation [4], [5]. This may be a contributing factor to the continued use of the Gilbert form of damping.

As indicated above, the present work derives LL damping for an insulator using the methods of irreversible thermodynamics, both for its own sake and because it should precede studies of the electrical conduction aspect of non-uniform conducting ferromagnets. An ultimate motivation is study of the phenomenon of current-induced spin torque, whereby spin is transferred, by spin-polarized current, from one part of a magnetic texture to another [31], [32], [33]. The formal equivalence of LL and Gilbert damping is retained even when, for a conductor, one includes current-induced spin torque [9]. In particular, the present work emphasizes that LL damping is unavoidable within irreversible thermodynamics; Gilbert damping simply cannot occur within this framework. As noted above, this removes a difficulty associated with Gilbert damping, whereby the g-factor, a measure of the Zeeman interaction, is renormalized by damping effects. Moreover, as we show, the methods of irreversible thermodynamics readily yield expressions for damping in non-uniform ferromagnets, whereas there is no obvious means to do this using the approach of Gilbert.

An outline of this work is as follows. Section 2 discusses the basics of irreversible thermodynamics, and discusses the intrinsic time-reversal signature of extensive and intensive thermodynamic variables, and of their sources and fluxes, results which are applicable to all of the systems to be studied. Section 3 then considers insulators, insulating paramagnets, and uniform insulating ferromagnets, obtaining their equations of motion using the methods of irreversible thermodynamics. Section 4 discusses LL damping and Gilbert damping in more detail, and suggests that experimental study of systems with large damping might settle the issue of LL vs. Gilbert damping. Section 5 considers insulating ferromagnets with non-uniform magnetic textures (e.g. domain walls). To our knowledge, the methods of irreversible thermodynamics have not been applied previously to such systems. Even without an electric current (properly, without a gradient of the electrochemical potential) the theory predicts a number of new terms in the spin flux and in the spin torque, which should have an especially large effect on the dynamics of small (but not monodomain) magnets, where boundary conditions impose rapid spatial variation of textures. Eq. (35) gives an approximate equation for dM/dt that has four new dissipative terms associated with a non-uniform magnetic texture, and which disappear for a uniform magnet. Section 6 presents a summary.

We note that this work serves as a precursor to the study of both conduction and magnetism [34].

Section snippets

Basics of irreversible thermodynamics

Basic terminology employed in irreversible thermodynamics are thermodynamic force, thermodynamic flux, thermodynamic source, intrinsic time-reversal signature, time-reversal signature, dissipative, and reactive. The essential idea behind irreversible thermodynamics is that the fluxes and sources are proportional to the forces, and whether the fluxes and sources are dissipative or reactive (non-dissipative), or a combination of the two, depends on the intrinsic time-reversal signature of the

Three thermodynamic systems

We begin by considering the insulator, a simple but commonly encountered system for which the only thermodynamic force is the temperature gradient, and there is only one thermodynamic flux and one thermodynamic source. We then successively add additional degrees of freedom, which leads to new forces, fluxes, and sources, and off-diagonal Onsager coefficients connecting fluxes and/or sources associated with a given intensive thermodynamic quantity with another thermodynamic force.

Landau–Lifshitz versus Gilbert damping

In the absence of spin flux Qi, the equation of motion for a uniform ferromagnet, (12), takes the formtM=-γμ0M×H*+N.For fixed |M|, we have A=0, so N=-λμ0M^×(M×H*), which yields the Landau–Lifshitz form of damping:tM=-γμ0M×H*-λμ0M^×(M×H*).

On the other hand, Gilbert argues, by analogy to damping of a particle using the Rayleigh damping term in a modified Hamiltonian formulation of mechanics, that the damping form should go as M^×tM [2]. However the driving term tM: (1) is not

Analogy between magnets and fluids in porous media

It is not uncommon to read of an analogy between magnets and fluids, with reference to magnetic hysteresis as evidence for magnetic “viscosity”. Although magneticians certainly know what they mean by this, the analogy to fluids is a bit more complex than one might imagine, and we take the opportunity here to clarify this point.

In fluids, viscosity is a measure of diffusion of linear momentum. The proper magnetic analog is the magnetic diffusion constant. What, then, is the fluid analog of

Non-uniform insulating ferromagnet

This system's thermodynamics is described by (10), its constitutive relation by (19) and its dynamical equations are (5), (3), (12). However, now the structure of the equilibrium system is characterized by both M and iM, which permits additional terms in the fluxes and sources.

The energy flux is still given by (14), but now the entropy flux has three additional terms. Explicitly,jis=-κTiT-LsQμ0M·iH*-LsM1μ0(iM)·H*-LsM2μ0(M^·iM)(M^·H*)-LsMμ0(M^×iM)·H*.Thus, if there is a torque

Damping and normal modes

Damping can be treated as a perturbation when applied to the normal modes, so the effect of the primed transport coefficients (reactive) can be neglected [44]. Moreover, because simulations consider only the transverse degrees of freedom (so M^·iM=0) and constant temperature, the relevant part of the transverse part of the equation of motion for M is all that we consider now. In addition to the Larmor term and the Landau–Lifshitz term, we must include the C, C, LQN1, LQN3, LNQ1, and LNQ3

Summary

We have considered the implications of irreversible thermodynamics for magnetic insulators, including paramagnets, uniform ferromagnets, and nonuniform ferromagnets. Two apparently new terms appear in the off-diagonal transport properties associated with temperature gradients and field gradients. As advertised, we show that irreversible thermodynamics uniquely predicts Landau–Lifshitz damping for a uniform ferromagnet. In addition, the irreversible thermodynamics of non-uniform ferromagnets

Acknowledgments

We would like to acknowledge valuable conversations with Carl Patton, Mark Stiles, and Andy Zangwill. The support of the DOE, through grant DE-FG02-06ER46278 is gratefully acknowledged.

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