The Righi-Leduc effect refers to the thermal analogue of the Hall effect, for which the electric current is replaced by the heat current and the electric field by the temperature gradient. In both cases, the magnetic field generates a transverse forc
e that deviates the carriers (electron, phonon, magnon) in the direction perpendicular to the current. In a ferromagnet, the magnetization plays the role of the magnetic field, and the corresponding effect is called anomalous Hall effect. Furthermore, a second transverse contribution due to the anisotropy, the planar Hall effect, is superimposed to the anomalous Hall effect. We report experimental evidence of the thermal counterpart of the Hall effects in ferromagnets, namely the magnon Hall effect (or equivalently the anomalous Righi-Leduc effect) and the planar Righi-Leduc effect, measured on ferromagnets that are either electrical conductor (NiFe) or insulator (YIG). The study shows the universal character of these new thermokinetic effects, related to the intrinsic chirality of the anisotropic ferromagnetic degrees of freedom.
The angular dependence of the thermal transport in insulating or conducting ferromagnets is derived on the basis of the Onsager reciprocity relations applied to a magnetic system. It is shown that the angular dependence of the temperature gradient ta
kes the same form as that of the anisotropic magnetoresistance, including anomalous and planar Hall contributions. The measured thermocouple generated between the extremities of the non-magnetic electrode in thermal contact to the ferromagnet follows this same angular dependence. The sign and amplitude of the magneto-voltaic signal is controlled by the difference of the Seebeck coefficients of the thermocouple.
The two spin-channel model is generalized to the case of transport of ferromagnetic excitations in electric conductors and insulators. The two channels are defined by reducing the ferromagnetic degrees of freedom to a bivaluated variable, i.e. to an
effective spin one-half. The reduction is performed after defining the local magnetic configuration space by a sphere $Sigma_x$, and integrating the relevant physical quantities over the two hemispheres $Sigma_x^{uparrow}$ and $Sigma_x^{downarrow}$. The configuration space is then extended to the $x$ direction for non-uniform magnetization excitations. The transport equations for both magnetic moments and magnetic energy are deduced, including the relaxation from one channel to the other. The heat transport equations for ferromagnets is deduced.
The interpretation of some recent measurements of spin-dependent voltage for which the electric conduction does not play a role rises some new fundamental questions about the effects of spin-dependent heat currents. A two spin-channel model is propos
ed in order to describe the effect of out-of-equilibrium spin-dependent heat carriers in electric conductors and insulators. It is shown that thermal spin-accumulation can be generated by the heat currents only over an arbitrarily long distance for both electric conductors or electric insulators. The diffusion equations for thermal spin-accumulation are derived in both cases, and the principle of its detection based on Spin-Nernst effect is described.
The gyromagnetic relation - i.e. the proportionality between the angular momentum $vec L$ (defined by an inertial tensor) and the magnetization $vec M$ - is evidence of the intimate connections between the magnetic properties and the inertial propert
ies of ferromagnetic bodies. However, inertia is absent from the dynamics of a magnetic dipole (the Landau-Lifshitz equation, the Gilbert equation and the Bloch equation contain only the first derivative of the magnetization with respect to time). In order to investigate this paradoxical situation, the lagrangian approach (proposed originally by T. H. Gilbert) is revisited keeping an arbitrary nonzero inertial tensor. A dynamic equation generalized to the inertial regime is obtained. It is shown how both the usual gyromagnetic relation and the well-known Landau-Lifshitz-Gilbert equation are recovered at the kinetic limit, i.e. for time scales above the relaxation time $tau$ of the angular momentum.
A general definition of the Spintronics concept of spin-pumping is proposed as generalized forces conjugated to the spin degrees of freedom in the framework of the theory of mesoscopic non-equilibrium thermodynamics. It is shown that at least three d
ifferent kinds of spin-pumping forces and associated spin-currents can be defined in the most simple spintronics system (the Ferromagnetic/Non-Ferromagnetic metal interface). Furthermore, the generalized force associated to the ferromagnetic collective variable is also introduced in an equal footing, in order to describe the coexistence of the spin of the conduction electrons (paramagnetic spins attached to $s$-band electrons) and the ferromagnetic-order parameter. The dynamical coupling between these two kinds of magnetic degrees of freedom is presented, and interpreted in terms of spin-transfer effects.
The dynamical equation of the magnetization has been reconsidered with enlarging the phase space of the ferromagnetic degrees of freedom to the angular momentum. The generalized Landau-Lifshitz-Gilbert equation that includes inertial terms, and the c
orresponding Fokker-Planck equation, are then derived in the framework of mesoscopic non-equilibrium thermodynamics theory. A typical relaxation time $tau$ is introduced describing the relaxation of the magnetization acceleration from the inertial regime towards the precession regime defined by a constant Larmor frequency. For time scales larger than $tau$, the usual Gilbert equation is recovered. For time scales below $tau$, nutation and related inertial effects are predicted. The inertial regime offers new opportunities for the implementation of ultrafast magnetization switching in magnetic devices.
A mesoscopic description of spin-transfer effect is proposed, based on the spin-injection mechanism occurring at the junction with a ferromagnet. The effect of spin-injection is to modify locally, in the ferromagnetic configuration space, the density
of magnetic moments. The corresponding gradient leads to a current-dependent diffusion process of the magnetization. In order to describe this effect, the dynamics of the magnetization of a ferromagnetic single domain is reconsidered in the framework of the thermokinetic theory of mesoscopic systems. Assuming an Onsager cross-coefficient that couples the currents, it is shown that spin-dependent electric transport leads to a correction of the Landau-Lifshitz-Gilbert equation of the ferromagnetic order parameter with supplementary diffusion terms. The consequence of spin-injection in terms of activation process of the ferromagnet is deduced, and the expressions of the effective energy barrier and of the critical current are derived. Magnetic fluctuations are calculated: the correction to the fluctuations is similar to that predicted for the activation. These predictions are consistent with the measurements of spin-transfer obtained in the activation regime and for ferromagnetic resonance under spin-injection.
