Gilbert damping in metallic ferromagnets from Schwinger-Keldysh field theory: Intrinsically nonlocal, nonuniform, and made anisotropic by spin-orbit coupling

Felipe Reyes-Osorio and Branislav K. Nikolić

Phys. Rev. B 109, 024413 – Published 12 January 2024

Abstract

Understanding the origin of damping mechanisms in the magnetization dynamics of metallic ferromagnets is a fundamental problem for nonequilibrium many-body physics of systems in which quantum conduction electrons interact with localized spins assumed to be governed by the classical Landau-Lifshitz-Gilbert (LLG) equation. It is also of critical importance for applications because damping affects energy consumption and the speed of spintronic and magnonic devices. Since the 1970s, a variety of linear-response and scattering theory approaches have been developed to produce widely used formulas for computation of the spatially independent Gilbert scalar parameter as the magnitude of the Gilbert damping term in the LLG equation. The Schwinger-Keldysh field theory (SKFT), largely unexploited for this purpose, offers additional possibilities, such as to rigorously derive an extended LLG equation by integrating quantum electrons out. Here we derive such an equation whose Gilbert damping for metallic ferromagnets is nonlocal, i.e., dependent on all localized spins at a given time, and nonuniform, even if all localized spins are collinear and spin-orbit coupling (SOC) is absent. This is in sharp contrast to standard lore, in which nonlocal damping is considered to emerge only if localized spins are noncollinear—for such situations, direct comparison using the example of a magnetic domain wall shows that SKFT-derived nonlocal damping is an order of magnitude larger than the previously considered one. Switching on SOC makes such nonlocal damping anisotropic, in contrast to standard lore, in which SOC is usually necessary to obtain a nonzero Gilbert damping scalar parameter. Our analytical formulas, with their nonlocality being more prominent in low spatial dimensions, are fully corroborated by numerically exact quantum-classical simulations.

Received 20 September 2023
Revised 8 December 2023
Accepted 11 December 2023

DOI:https://doi.org/10.1103/PhysRevB.109.024413

Physics Subject Headings (PhySH)

Dissipative dynamics Magnetization dynamics Quantum field theory Spin pumping

Landau-Lifschitz-Gilbert equation Nonequilibrium Green's function

Condensed Matter, Materials & Applied PhysicsParticles & FieldsStatistical Physics & Thermodynamics

Authors & Affiliations

Felipe Reyes-Osorio and Branislav K. Nikolić ^*

Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA

^*bnikolic@udel.edu

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Vol. 109, Iss. 2 — 1 January 2024

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Images

Figure 1
Schematic view of (a) classical localized spins, modeled by unit vectors $M_{n}$ (red arrows), within an infinite metallic ferromagnet defined on a cubic lattice in one to three dimensions (one dimension is used in this illustration) and (b) a finite-size metallic ferromagnet (central region) attached to semi-infinite NM leads terminating in macroscopic reservoirs, whose difference in electrochemical potentials injects charge current, as commonly done in spintronics. The localized spins interact with conduction electron spin $〈 \hat{s} 〉$ (green arrow) via $s d$ exchange of strength $J_{s d}$ , while both subsystems can experience external magnetic field $B$ (blue arrow). (c) Nonlocal damping $λ_{n n^{'}}^{D}$ [Eq. (10)] obtained from SKFT vs the distance $| r_{n} - r_{n^{'}} |$ between two sites $n$ and $n^{'}$ of the lattice for different dimensionalities $D$ of space.
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Figure 2
(a) Time evolution of two localized spins $M_{n}$ , located at sites $n = 1$ and $n^{'} = 3$ within a chain of 19 sites in the setup of Fig. 1, computed numerically using the TDNEGF+LLG scheme [40, 43, 56, 57]. The two spins are collinear at $t = 0$ and point along the $x$ axis, while magnetic field is applied along the $z$ axis. (b) The same information as in (a), but for two noncollinear spins with angle $\in {0, 45, 90, 135, 180}$ between them. (c) and (d) Effective damping extracted from TDNEGF+LLG simulations (red dashed line) vs the one from SKFT [the black solid line plots the 1D case in Eq. (10)] as a function of the site $n^{'}$ of the second spin. The two spins are initially parallel in (c) and antiparallel in (d). The Fermi wave vector of conduction electrons is chosen to be $k_{F} = π / 2 a$ , where $a$ is the lattice spacing.
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Figure 3
(a) Comparison of trajectories of localized spins $M_{n}^{z} (t)$ in the setup in Fig. 1, whose central region is a 1D metallic ferromagnet composed of five sites, using LLG equation (9) with SKFT-derived nonlocal damping (solid red lines) vs the LLG equation with conventional spatially independent $α_{G} = 0.016$ (black dashed line). This value of $α_{G}$ is obtained by averaging nonlocal damping over the whole ferromagnet. The dynamics of $M_{n} (t)$ is initiated by an external magnetic field along the $z$ axis, while all five localized spins point along the $x$ axis at $t = 0$ . (b) Comparison of spin current $I_{R}^{S^{z}} (t)$ pumped [56, 57, 61] by the dynamics of $M_{n} (t)$ for the two cases [i.e., nonuniform $M_{n} (t)$ for nonlocal damping vs uniform $M_{n} (t)$ for conventional damping] from (a). The Fermi wave vector of conduction electrons is chosen to be $k_{F} = π / 2 a$ .
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Figure 4
(a) Comparison of magnetic DW velocity $v_{DW}$ vs DW width $w$ extracted from numerical simulations using the extended LLG equation (9) with SKFT-derived nonlocal damping [Eq. (10); red line], the extended LLG equation (1) with SMF-derived nonlocal damping from Ref. [13] [Eq. (2); blue line] or SMF-derived nonlocal damping from Ref. [19] [with an additional term compared to Ref. [13]; green line; see Eq. (14)], and the conventional LLG equation (1) with local Gilbert damping [i.e., $η = 0$ in Eq. (2); black line]. (b) Spatial profile of the DW within a quasi-1D ferromagnetic wire at time $t = 410 ℏ / J$ , where $J$ is the exchange coupling between $M_{n}$ at NN sites, as obtained from the SKFT-derived extended LLG equation (9) with nonlocal damping $λ_{n n^{'}}^{2 D}$ [Eq. (10)]. (c) and (d) The corresponding spatial profiles of nonlocal damping across the DW in (b) using the SKFT-derived expression [Eqs. (9) and (10)] and the SMF-derived [13] expression [second term on the right-hand side of Eq. (2)], respectively.
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