Variational principle for magnetisation dynamics in a temperature gradient

By applying a variational principle on a magnetic system within the framework of extended irreversible thermodynamics, we find that the presence of a temperature gradient in a ferromagnet leads to a generalisation of the Landau-Lifshitz equation with an additional magnetic induction field proportional to the temperature gradient. This field modulates the damping of the magnetic excitation. It can increase or decrease the damping, depending on the orientation of the magnetisation wave vector with respect to the temperature gradient. This variational approach confirms the existence of the magnetic Seebeck effect which was derived from thermodynamics and provides a quantitative estimate of the strength of this effect. Copyright c © EPLA, 2015 Introduction. – The effect of a thermal spin torque on the magnetisation dynamics has attracted a lot of attention recently [1–5]. In a conductor, the spin dependence of the transport properties implies that a heat current induces a spin current, and consequently, a torque on the magnetisation [6,7]. In an insulator, this transport model does not apply. In this publication, we show that extended irreversible thermodynamics leads to a variational principle for the magnetisation which predicts the existence of an additional magnetic induction field proportional to the temperature gradient in the Landau-Lifshitz equation. In our previous work [8], we called this effect the “magnetic Seebeck effect” since the Seebeck effect refers to the presence of an electric field induced by a temperature gradient. This effect should not be confused with the transport phenomenon known as the spin Seebeck effect [9–11]. Classical irreversible thermodynamics (CIT) [12–14] requires the system of interest to be at local equilibrium. Transport phenomena are then described by phenomenological relationships between current densities and generalised forces so as to fulfill the second law of thermodynamics. When a system does not satisfy the condition of local equilibrium, it can be described within the framework of extended irreversible thermodynamics (EIT) where the current densities are considered as additional state variables [15]. In this article, we show that this approach provides an expression for the magnetic Seebeck effect in terms of the thermal properties of the magnetisation. Variation of the internal energy. – In the absence of a magnetic excitation field, the magnetisation M is collinear to the magnetic induction field obtained by performing the variation of the internal energy with respect to the magnetisation δu/δM , as pointed out by Gurevich [16]. In the presence of a magnetic excitation field b, the Landau-Lifshitz equation describes the precession of the magnetisation M about this magnetic induction field. Since the magnetisation is locally out of equilibrium, we use the framework of extended irreversible thermodynamics. According to this framework, the internal energy density u (M ,∇×M) is a function of the magnetisation M and of the magnetisation current jM = ∇×M that are, in turn, functions of the position r. According to the variational equation (A.9) established explicitly in the appendix, the variational derivative of the internal energy in the bulk of the system reads δu δM = ∂u ∂M +∇× ( ∂u ∂ (∇×M) ) . (1) The variational principle used by Gurevich et al. [16] and Bose et al. [17] assumes that the internal energy density is a function of the magnetisation M and the gradient of