Hidden Magnon Berry Curvature drives Vertical Magnon Transport

TL;DR

This work identifies a hidden, in-plane component of magnon Berry curvature (HMBC) in quasi-2D magnets and shows that it can drive vertical magnon transport (VMT) when in-plane gradients of magnetic field or temperature are applied. Using a semiclassical wave-packet picture combined with Boltzmann transport and a bosonic BdG framework, the authors derive HMBC from a pseudo-$Z$ operator and show that the vertical current is tied to in-plane BC components and their dipoles; linear and nonlinear VMT coefficients are associated with HMBC and hidden magnon BCD extensions. They predict measurable VMT in realistic systems, including buckled honeycomb lattices and bilayer CrI$_3$ CrX$_3$, with distinct behavior across AFM/FM order and stacking, governed by layer symmetry and TRS. The findings provide a new route to probe magnon geometry in layered magnets and suggest potential spintronic applications for sensing vertical field gradients and engineering geometry-driven magnon transport in van der Waals materials.

Abstract

We predict an in-plane, or hidden, Berry curvature (BC) for magnons in electrically insulating quasi-2D magnets and demonstrate that the hidden magnon Berry curvature (HMBC) gives rise to a previously unrecognized form of vertical, out-of-plane, magnon transport. Combining a semiclassical framework with Boltzmann transport theory, we show that the vertical magnon transport (VMT) currents respond both linearly and nonlinearly to the in-plane gradients of magnetic field and temperature. The linear transport coefficients are tied to the total hidden magnon BC, while the nonlinear (second-order) coefficients for the magnetic field and temperature gradients are determined by the hidden magnon BC dipole and the hidden extended magnon BC dipole, respectively. Using linear spin-wave theory, we find that the hidden magnon BC over the Brillouin zone is given by the expectation value of a pseudo-${\cal Z}$ operator, representing vertical displacements, evaluated in the space of paraunitary matrices that diagonalize the magnon Hamiltonian. We estimate VMT in spin models of realistic magnets with ferro- and antiferromagnetic order, including the buckled honeycomb (BHC) lattice and bilayer Chromium trihalide (CrX$_3$; X = Cl, Br, I) systems. In BHC, both linear and nonlinear VMT arise when time-reversal symmetry is broken by Dzyaloshinskii-Moriya interactions. In CrX$_3$ systems, the nonlinear coefficients dominate, while the linear responses vanish due to time-reversal symmetry. Both systems exhibit distinctive features across a broad range of temperatures and parameters. Therefore, our prediction of VMT and its characteristic signatures is directly testable in present-day magnonic experiments, especially in atomically thin, few-layered van der Waals magnets.

Figures (6)

• Figure 1: Vertical (out-of-plane) magnon current in quasi-2D systems: A quasi-2D magnetic insulator consisting of a few atomically thin layers is placed on the $xy$-plane. The system is subjected, either separately or simultaneously, to a spatially varying magnetic field ${{\textbf{B}}}=B(x)\hat{\textbf{z}}$ with an in-plane gradient and a thermal gradient along the $\hat{\textbf{x}}$ direction arising from a temperature differential $T_B-T_A$. The hidden magnon Berry curvatures of the quasi-2D magnet give rise to a vertical (out-of-plane) magnon current.
• Figure 2: Buckled honeycomb lattice (a) Top view of the lattice showing Heisenberg interactions, with coupling strength ${\rm J}_1$ and ${\rm J}_2$, connecting sublattice sites A with B, and Dzyaloshinskii Moriya (DM) interactions (directions shown by arrows) coupling sites of the same sublattice with strength $D_A$, $D_B$ for $A$, $B$ sublattices, respectively. Panel (b) shows the side view of the lattice, $\xi$ is the relative vertical shift between sublattices A and B. Panel (c) shows the spin orientations, typical magnon-bands and the two hidden (in-plane) components of magnon Berry-curvature (BC), $\Omega^{(x)}$ and $\Omega^{(y)}$, for the lowest energy magnon-mode in the AFM ordered phase. Panel (f) shows the same plots as panel (c) for the FM order. The magnon-BC components within the first BZ (hexagon) are plotted in units of $\xi$ with yellow (blue) regions indicating positive (negative) values. The VMT responses arising from temperature gradient ($\nabla T$) -- $\sigma^z_{x},\sigma^z_{y},D^z_{xx},D^z_{xy}$, $D^z_{yy}$, and those arising from magnetic field gradient ($\nabla {\textbf{B}}$) -- $\sigma^{zz}_{x},\sigma^{zz}_{y},D^{zzz}_{xx},D^{zzz}_{xy}$, $D^{zzz}_{yy}$, for the AFM case are plotted in panels (d) and (e), respectively, using parameters ${\rm J}_1=1.0,{\rm J}_2=0.7, D_A=-D_B=0.35,\kappa_A=\kappa_B=0.01$. The corresponding VMT responses for the FM case are shown in (g) and (h), respectively, with ${\rm J}_1=1.0,{\rm J}_2=0.7, D_A=D_B=0.1, \kappa_A=0.1,\kappa_B=0.01$.
• Figure 3: AB and AB$'$ stacked bilayer CrI$_3$: (a) and (b) show AB- and AB$'$-stacking geometries for bilayer CrI$_3$ with FM and AFM orders, respectively. Bottom layer (described by parameters ${\rm J}_1,\lambda_1,\kappa_1$) and top layer (described by parameters ${\rm J}_2,\lambda_2,\kappa_2$) are vertically separated by $\xi$ and coupled with exchange couplings ${\rm J}_\perp$. Panel (c) shows magnetic ordering, (d) magnon-bands and (e) shows the hidden (in-plane) components of magnon Berry-curvature, $\Omega^{(x)}$ and $\Omega^{(y)}$, for the AB-stacked FM for a typical set of parameter values. Panels (i--k) show similar plots as (c--d) for the AB$'$-stacked AFM. The magnon-BC within the first BZ (hexagon) are presented in units of $\xi$ with yellow (blue) regions indicating positive (negative) values. The linear VMT responses, $\sigma^z_{x,y}$ and $\sigma^{zz}_{x,y}$, vanish due to time-reversal symmetry holding for both stacking geometries. (f) The temperature ($T$) variation of the non-linear VMT responses, $D^{z}_{xx},D^{z}_{yy},D^{zzz}_{xx}$, $D^z_{yy}$ for the AB-stacked ferromagnet with parameters ${\rm J}_{\perp}=0.6$, $\delta={\rm J}_2/{\rm J}_1=\lambda_2/\lambda_1=\kappa_2/\kappa_1=0.7$. Responses appearing in (f) plotted by varying inter-layer coupling strength ${\rm J}_\perp$ in panel (g) for parameters $T=0.1$, $\delta=0.7$, and plotted in (h) by varying the layer-asymmetry parameter $\delta$ with $T=0.1$, ${\rm J}_\perp=0.6$. Panels (l, m) show the same quantities as panels (f, g) plotted for the AB$'$-stacked AFM phase, using the parameters ${\rm J}_{\perp}=0.04$, $\delta=0.7$ in (l), and $T=0.3$, $\delta=0.7$ in (m). A magnified view of the comparatively small but finite responses $D^z_{xy}$, $D^{zzz}_{xy}$ appearing in (l) shown in panel (n). A C$_3$ rotation symmetry in the AB-stacked phase causes $D^{z}_{xy}, D^{zzz}_{xy}$ to vanish and forces $D^{z}_{xx}=D^{z}_{yy}$ and $D^{zzz}_{xx}=D^{zzz}_{yy}$; no such symmetry exists for the AB$'$-stacked phase. The remaining parameters are set to ${\rm J}_{1}=2.2,\lambda_1=0.09,\kappa_{1}=0.01$.
• Figure 4: Panel (a) shows the VMT responses as functions of ${\rm J}_2$ (vertical-bond coupling) for the AFM state, using parameters ${\rm J}_1=1.0$, $D_A=-D_B=0.35$, $\kappa_A=\kappa_B=0.01$, and $T=0.4$. Panel (b) presents the corresponding responses as functions of $D$ (with $D=D_A=-D_B$) for ${\rm J}_1=1.0$, ${\rm J}_2=0.7$, $\kappa_A=\kappa_B=0.01$, and $T=0.4$. Panel (c) displays $D^{zz}_{xx}$, $D^{zz}_{xy}$, and $D^{zz}_{yy}$ for the AFM state as functions of temperature, using ${\rm J}_1=1.0$, ${\rm J}_2=0.7$, $D_A=-D_B=0.35$, and $\kappa_A-\kappa_B=0.2$. Panel (d) shows the VMT responses for the FM state as functions of ${\rm J}_2$, with parameters ${\rm J}_1=1.0$, $D_A=D_B=0.1$, $\kappa_A=0.1$, $\kappa_B=0.01$, and $T=0.4$. Panel (e) plots the same responses versus $D$ (where $D=D_A=D_B$) for ${\rm J}_1=1.0$, ${\rm J}_2=0.7$, $\kappa_A=0.1$, $\kappa_B=0.01$, and $T=0.4$. Panel (f) shows $D^{zz}_{xx}$, $D^{zz}_{xy}$, and $D^{zz}_{yy}$ for the FM state as functions of temperature for ${\rm J}_1=1.0$, ${\rm J}_2=0.7$, $D_A=D_B=0.1$, $\kappa_A=0.1$, and $\kappa_B=0.01$.
• Figure 5: The bilinear VMT responses generated by simultaneous application of temperature gradient $(\nabla T)$ and magnetic field gradient-- namely $(\nabla {\textbf{B}})$--$D^{zz}_{xx},D^{zz}_{xy}$ and $D^{zz}_{yy}$--are shown for the AB-stacked phase of CrI$_3$ as functions of temperature in panel (a) using parameters ${\rm J}_{\perp}=0.6$, $\delta=0.7$, and as functions of interlayer coupling ${\rm J}_\perp$ in panel (b), using $T=0.1$, $\delta=0.7$. Corresponding responses for the AB$'$-stacked phase are as functions of temperature in panel (c), using ${\rm J}_{\perp}=0.04$, $\delta=0.7$ and as functions of ${\rm J}_\perp$ in panel (d) using $T=0.3$, $\delta=0.7$. All ther parameters are fixed at ${\rm J}_1=2.2,\lambda_1=0.09$ and $\kappa_1=0.01$.