Spin Orientation Driven Polarization in Collinear Magnets

Abstract

In a collinear magnet, the predominant magnetic moments are collectively aligned along a specific spatial orientation, and this alignment may yield intriguing phenomena such as spin orientation driven polarization. It is well known that spin orientation driven polarization is a relativistic effect that widely occurs in various type-II multiferroics. However, a universal theory that describes such a phenomenon and directs the corresponding materials discovery is lacking. Here, we revisit the magnetic structures of collinear magnets and explore the spin-orientation-dependent phenomena therein. Based on symmetry principles, we analyze the spin point groups (SPGs) that are associated with collinear magnets in the non-relativistic regime, demonstrate how relativistic spin-orbit interaction reduces each SPG to various magnetic point groups that are associated with different magnetic alignments, and classify the SPGs with respect to spin orientation driven polarization. We employ our theory to elucidate the mechanisms of spin orientation driven polarization in a variety of type-II multiferroics. Combined with first-principles simulations, we further show that polarization may be driven in nonpolar collinear antiferromagnets (e.g., CuFeS$_2$) by reorienting their magnetic alignments. Our theory provides guidelines for designing and discovering materials with spin orientation driven polarization, which will benefit the development of spintronics based on type-II multiferroics and related materials.

Figures (2)

• Figure 1: Examples of scalar and vector magnetic structures. Panel (a) sketches four different scalar magnetic structures. The $+m$, $+m^\prime$, $-m$ and $-m^\prime$ scalar magnetic moments are represented by red, pink, blue and cyan spheres, respectively. Panel (b) sketches the $\mathbf{e}_\alpha$ orientation vector that describes the magnetic alignment along the $\alpha$ direction ($\mathbf{e}_\alpha$ being a unit vector). Panel (c) shows four vector magnetic structures resulting from the direct products between the scalar magnetic structures in panel (a) and the $\mathbf{e}_\alpha$ orientation vector in panel (b). Panel (d) is a scalar magnetic structure for Ba$_2$CoGe$_2$O$_7$. Panel (e) sketches $\mathbf{e}_x$, $\mathbf{e}_{xy}$ and $\mathbf{e}_z$ orientation vectors. Panel (f) shows three vector magnetic structures resulting from the direct products between the scalar magnetic structure in panel (d) and the orientation vectors in panel (e). In panels (d)--(f), Ba, Ge, and O ions are not shown.
• Figure 2: Spin reorientation driven polarization in CuFeS$_2$. Panel (a) sketches the ground state magnetic structure associated with a $\mathbf{e}_z$ orientation vector. The Fe ions are denoted by gray spheres, while the Cu and S ions are not displayed. The red and blue arrows represent the vector magnetic moments carried by Fe ions. Panel (b) shows the polarization of CuFeS$_2$ as a function of the $\theta$ orientation angle. As defined in the inset of panel (b), $\theta$ is the angle between $\mathbf{e}_x$ and $\mathbf{e}_\theta$, where $\mathbf{e}_\theta$ lies in the plane perpendicular to $\mathbf{e}_z$. The first-principles results are marked by unfilled circles, and the solid line is obtained by a sinusoidal function fitting (coefficient of determination being about 99.98%).