Microscopic calculation of thermally induced spin-transfer torques

Abstract

Spin-transfer torques, both reactive and dissipative, induced by temperature gradients in conducting ferromagnets are calculated microscopically for smooth magnetization textures. Temperature gradients are treated a la Luttinger by introducing a fictitious gravitational field that couples to the energy density. The thermal torque coefficients obtained by the Kubo formula contain unphysical terms that diverge towards zero temperature. Such terms are caused by equilibrium components and should be subtracted before applying the Einstein-Luttinger relation. Only by following this procedure a familiar Mott-like formula is obtained for the dissipative spin-transfer torque. The result indicates that a fictitious field that couples to the entropy rather than energy would solve the issue from the outset.

Geometrical contributions to the exchange constants: Free electrons with spin-orbit interaction

“…For example, the Berry phase theory of DMI allows us to relate DMI to the spin-orbit torque [4], to ground-state spin-currents [7], and to ground-state energy currents which need to be subtracted in order to extract the inverse thermal spin-orbit torque [6]. Similarly, torques due to the exchange interaction need to be considered in the theory of thermally induced spintransfer torques [17], and a Green's function expression of exchange is well suited for this purpose.…”

Microscopic theory of magnon-drag electron flow in ferromagnetic metals

“…Here, we introduced the temperature gradient ∂ i T through the combination, −∂ i ψ −∂ i T /T . This is justified for operators A of which the average vanishes naturally in the equilibrium state, where ∂ i T /T + ∂ i ψ = 0 holds [16,17]. Therefore, the response to (−∂ i T /T ) is obtained as the response to (−∂ i ψ) [16].…”

“…where u i = 2Jq i is the magnon velocity, ω λ is the Matsubara frequency of the external perturbation ψ, and we have set Q = 0 for simplicity. The terms linear in D q are "corrections" arising from the δ-function in the relation [17],…”

Theory of Cross-correlated Electron–Magnon Transport Phenomena: Case of Magnetic Topological Insulator

“…To calculate the response of a physical quantity Âi to a temperature gradient, we follow Luttinger 16,17) and use the formula, Âi = lim ω→0 (iω…”

“…( 21)], has the same T -dependence as that from a purely electronic process, (κ EQ el,el ) xy [Eq. (17)], but the magnitude is much larger. In fact, with 13) |M z | = 0.1 ∼ 0.01eV, r 0 = 0.5nm, and |M z |/J ex ∼ 1, where J ex ≡ J/r 2 0 , the ratio…”