Role of Entropy in Domain Wall Motion in Thermal Gradients

Abstract

Thermally driven domain wall (DW) motion caused solely by magnonic spin currents was forecast theoretically and has been measured recently in a magnetic insulator using magneto optical Kerr effect microscopy. We present an analytical calculation of the DW velocity as well as the Walker breakdown within the framework of the Landau Lifshitz Bloch equation of motion. The temperature gradient leads to a torque term acting on the magnetization where the DW is mainly driven by the temperature dependence of the exchange stiffness, or in a more general picture by the maximization of entropy. The existence of this entropic torque term does not rest on the angular momentum transfer from the magnonic spin current. Hence, even DWs in antiferromagnets or compensated ferrimagnets should move accordingly. We further argue that the entropic torque exceeds that of the magnonic spin current.Spin caloritronics is a new field of research focused on the combined transport of spin, charge, entropy, and energy in magnetic systems [1]. A good example for a spin caloritronic phenomenon is the spin Seebeck effect, the occurrence of spin currents or spin accumulation due to temperature gradients, which was discovered first in metallic ferromagnets [2]. Later on, this effect was also found in dilute magnetic semiconductors [3] and even in insulators [4]. In the latter case the existence of the spin Seebeck effect can only be explained by pure magnonic spin currents not resting on any charge transport.Spin caloritronic phenomena call for an exploitation in magnetic devices. In Ref.[5] the existence of thermally driven domain wall (DW) motion in a temperature gradient was demonstrated by computer simulations based either on the stochastic Landau-Lifshitz-Gilbert equation of motion for an atomistic spin model or-in a more micromagnetic picture-on the Landau-Lifshitz-Bloch (LLB) equation of motion for a thermally averaged spin polarization. The microscopic explanation for DW motion in temperature gradients rests on the diffusive motion of magnons from the hot end of a nanowire towards the colder end [6] and the transfer of angular momentum pushing the wall in the direction opposite to the magnonic spin current. Theoretical investigations based on these arguments give estimates for the DW velocity which can be achieved [7][8][9][10].In this work we will focus on a more thermodynamic point of view. At finite temperature T a DW is a thermodynamic object where the free energy ΔFðTÞ is the thermodynamic potential that is minimized. The free energy of the DW can be expressed as the difference between a magnetic system with DW minus the free energy of the same system without DW [11]. It is connected to the internal energy ΔU of the wall and its entropy ΔS via ΔF ¼ ΔU − TΔS. The temperature dependence of the different thermodynamic potentials are illustrated in Fig. 1 for a DW in a high anisotropic material. The calculations are from Ref. [11] for an atomistic spin model for FePt. However, the general features of a DW free energy do not depend on the ...

Inertia-Free Thermally Driven Domain-Wall Motion in Antiferromagnets

“…For a DW at finite temperature, the free energy is ΔFðTÞ ¼ ΔU − TΔS, where ΔU is the internal energy and ΔS the entropy of the DW. It is a monotonically decreasing function of temperature [9][10][11]. This rather general argument explains a DW motion towards the hotter parts of the sample where the free energy is lower [11][12][13] and it can be expected to hold for other magnetic textures as well.…”

“…The exchange stiffness is weaker for higher temperature and therefore, an effective torque on the DW is created driving it towards the hotter region [11].…”

Thermally induced magnonic spin current, thermomagnonic torques, and domain-wall dynamics in the presence of Dzyaloshinskii-Moriya interaction

“…Naturally, the magnonic spin current does not exert a magnonic pressure on the DW's surface, while the angular momentum is still transferred. In this case the DW moves to the hot edge and this process is predetermined by the free energy of the DW [14]. When the width of the DWs exceeds a critical value, the magnonic spin current is totaly reflected by DW.…”

“…After considering Eq. (12) and averaging over the fast oscillations in time, the expression of the exchange thermomagnonic torque, −l A M × (∇ 2 M), splits into two different parts: the magnonic adiabatic spin transfer torque, −u∂ x m 0 [16][17][18], and the exchange entropic torque, l A /2m 0 × (∂ x ρ)(∂ x m 0 ) [14,18].…”

“…The thermally induced DW dynamics has been studied intensively in recent years [11][12][13][14][15]. The effect of applied thermal bias is twofold: (i) it generates the adiabatic thermomagnonic spin transfer torque driving DWs from the hot to the cold edge [16][17][18], and (ii) it forms the nonadiabatic thermomagnonic * levan.chotorlishvili@gmail.com torque, also known as entropic torque or dissipative torque.…”