Ellipticity effects on diffusive magnon spin and heat transport in easy-plane ferromagnets

Technical Summary: Ellipticity Effects on Diffusive Magnon Spin and Heat Transport in Easy-Plane Ferromagnets

Problem Statement
In magnetic insulators, spin excitations (magnons) can be circularly or elliptically polarized. While circular polarization occurs in isotropic systems, elliptical polarization arises naturally in the presence of magnetic anisotropies transverse to the equilibrium magnetization. Although previous studies have investigated magnon transport in various geometries and the role of soft magnons near critical points, the specific contribution of magnon ellipticity to diffusive spin and heat transport in easy-plane ferromagnets remains unresolved. Specifically, it is unclear how the ellipticity of the spin wave, mediated by the dispersion relation and the angular momentum per magnon, modifies the magnon-spin and magnon-thermal conductivities in both three-dimensional (3D) and two-dimensional (2D) systems.

Methodology
The authors model the system using the Landau-Lifshitz-Gilbert (LLG) equation with a perpendicular magnetic anisotropy term. They derive the magnon dispersion relation ($\omega_K$) and the spin per magnon ($s_m$) analytically, showing that finite anisotropy leads to elliptical orbits and modifies the spin from the standard $\hbar$ value.

To quantify transport, the study employs the Boltzmann transport equation within the relaxation time approximation. The authors assume a quasi-equilibrium Bose-Einstein distribution driven by gradients in magnon chemical potential ($\nabla \mu_m$) and magnon temperature ($\nabla T_m$). They calculate the magnon-spin current ($j_{ms}$) and magnon-heat current ($j_{mQ}$) for both 3D and 2D systems. The analysis utilizes a perturbative expansion in terms of the anisotropy parameter $G$ (related to the ellipticity) to isolate the ellipticity-induced corrections to the transport coefficients. The calculations involve introducing ultraviolet ($K_\infty$) and infrared ($K_0$) momentum cutoffs to regularize the integrals, interpreting these as physical limits of the continuum transport theory and the validity of the diffusive regime, respectively.

Key Contributions and Results
The paper provides explicit analytical expressions for the magnon-spin conductivity ($\sigma_m$) and magnon thermal conductivity ($\kappa_m$), separating the contributions from circular ($G=0$) and elliptical ($G \neq 0$) cases.

1. Magnon-Spin Transport:

   • In easy-plane systems ($G > 0$), increasing the anisotropy (and thus ellipticity) leads to a reduction in the magnon-spin conductivity.
   • In hard-axis systems ($G < 0$), ellipticity results in an enhancement of spin transport.
   • The correction is linear in the anisotropy constant $G$.
   • In 2D systems, the reduction in spin conductivity is more pronounced (approximately 29% for typical YIG parameters) compared to 3D systems (approximately 0.4%), due to the enhanced infrared sensitivity of 2D transport integrals.

2. Magnon-Heat Transport:

   • Unlike spin transport, the magnon thermal conductivity is always enhanced by ellipticity, regardless of whether the system has an easy or hard magnetic axis.
   • The correction to thermal conductivity enters quadratically with the anisotropy constant $G$.
   • Consequently, the enhancement is generally small in the diffusive regime for realistic parameters (e.g., $\sim 10^{-9}\%$ in 3D and $\sim 10^{-8}\%$ in 2D for YIG at 300 K).

3. Spin Seebeck/Peltier Coefficients:

   • The coefficients $S_{12}$ (Spin Seebeck) and $S_{21}$ (Spin Peltier) are affected by ellipticity. $S_{12}$ follows the same trend as spin conductivity (reduced for easy-plane, enhanced for hard-axis). $S_{21}$ is reduced for easy-plane systems but the magnitude of the correction is small.

Significance and Claims
The authors claim that their study clarifies the role of magnon ellipticity in diffusive transport, distinguishing it from other mechanisms that enhance magnon transport in thin films.

• Clarification of Dimensional Crossover: The results suggest that the observed orders-of-magnitude enhancement of magnon-spin conductivity in very thin YIG films (dimensional crossover) does not originate from ellipticity. Instead, the authors attribute such enhancements to the suppression of scattering channels (reduction of the relaxation time $\tau$) in the quasi-2D limit.
• Distinct Mechanisms for Spin vs. Heat: The paper establishes that ellipticity has opposing or distinct effects on spin and heat transport: it generally suppresses spin transport in easy-plane magnets while universally enhancing heat transport (though the latter effect is quantitatively small in the diffusive regime).
• Theoretical Framework: The work provides a minimal framework for studying magnon transport properties as a function of ellipticity, treating ellipticity not as an independent variable but as a consequence of magnetic anisotropy that simultaneously modifies the dispersion and the spin per magnon.

The authors conclude that while the quantitative impact of ellipticity on thermal conductivity is negligible in the diffusive regime, its effect on spin transport in 2D systems is significant enough to be considered in the design and interpretation of magnonic devices, particularly those involving thin films or specific anisotropy configurations.