Stacking theory for bilayer two-dimensional magnets

Abstract

Two-dimensional unconventional magnetism has recently attracted growing interest due to its intriguing physical properties and promising applications in spintronics. However, existing studies on stacking-induced unconventional magnetism mainly focus on specific materials and stacking configurations. Here, we develop a general symmetry-based stacking theory for two-dimensional magnets. We first introduce spin layer groups as the fundamental symmetry framework, providing the essential magnetic symmetry information for the stacking theory. Based on this framework, we construct the complete set of 448 collinear spin layer groups for describing two-dimensional collinear magnets. Subsequently, we develop a general magnetic stacking theory applicable to arbitrary magnetic systems and derive its general solutions. Using CrF$_3$ as an illustrative example, we show how this theory enables designs of two-dimensional unconventional magnetism, as validated by first-principles calculations. We realize two-dimensional fully compensated ferrimagnetism through our stacking theory. Our work provides a general symmetry-guided platform for discovering and designing stacking-induced unconventional magnetism.

Figures (2)

• Figure 1: Classification of collinear spin layer groups and schematic illustration of the bilayer stacking in magnetic materials. (a) The 448 collinear spin layer groups (cSLGs) are classified into five categories according to the magnetic states they describe, including 80 ferromagnetic or ferrimagnetic (FM/FIM), 122 $\mathcal{T}\boldsymbol{\tau}$ antiferromagnetic ($\mathcal{T}\boldsymbol{\tau}$ AFM), 88 $\mathcal{P}\mathcal{T}$ antiferromagnetic ($\mathcal{P}\mathcal{T}$ AFM), 92 altermagnetic (AM) and 66 type-IV 2D collinear magnetic (Type-IV) cSLGs. (b) Starting from a monolayer $S$, a stacking operation $\{E \,\|\, \hat{\tau}_z\} \{c \,\|\, \hat{O}\}$ is applied to generate $S'$, which is then stacked onto $S$ to form a bilayer $B$.
• Figure 2: The crystal structures of monolayer CrF$_3$ and twisted bilayer CrF$_3$ with a twist angle of $11.43^\circ$, together with the band structure of bilayer CrF$_3$. (a) The crystal structures of monolayer CrF$_3$. A monolayer CrF$_3$ is stacked into a bilayer via stacking operations$\{c\,\|\, C^{11.43}_z \mid \mathbf{0}\}$, where $C^{11.43}_z$ denotes a rotation of $11.43^\circ$ about the $z$ axis. For $c = E$, the resulting bilayer CrF$_3$ (b) hosts fully compensated ferrimagnetism, whereas for $c = T$, the bilayer CrF$_3$ (c) exhibits altermagnetism. (d) Band structure of the twisted bilayer CrF$_3$ shown in (c), with the high-symmetry points in reciprocal space shown in (e).