Electric polarization driven by non-collinear spin alignment investigated by first principles calculations

TL;DR

This work addresses the microscopic origin of spin-induced electric polarization in type-II multiferroics and extends beyond simple spin-current pictures by presenting a fully first-principles framework based on the relativistic KKR-GF method. Polarization is parameterized by three-site, spin-antisymmetric parameters ${\cal P}_{ij,k}^{\alpha\beta,\mu}$ derived from Green's function perturbation theory, enabling site-resolved predictions of spin-orbit–driven polarization. Applied to Cr$_2$O$_3$, MnI$_2$, MnO$_2$, CuCrO$_2$, and AgCrO$_2$, the method yields finite polarization for non-collinear spin orders and reveals material-specific distributions across metal and oxygen sites, consistent with symmetry constraints and, in several cases, experimental trends. This ab initio decomposition clarifies the relative roles of three-site SOC-driven contributions and complements lattice-displacement (inverse-DMI) effects, offering a framework to guide design of type-II multiferroics.

Abstract

We present an approach for first principles investigations on the spin driven electric polarization in type II multiferroics. We propose a parametrization of the polarization with the parameters calculated using the Korringa-Kohn-Rostoker Green function (KKR-GF) formalism. Within this approach the induced electric polarization of a unit cell is represented in terms of three-site parameters. Those antisymmetric with respect to spin permutation are seen as an ab-initio based counter-part to the phenomenological parameters used within the inverse-Dzyaloshinskii-Moriya-interaction (DMI) model. Due to their relativistic origin, these parameters are responsible for the electric polarization induced in the presence of a non-collinear spin alignment in materials with a centrosymmetric crystal structure. Beyond to this, our approach gives direct access to the element- or site-resolved electric polarization. To demonstrate the capability of the approach, we consider several examples of the so-called type II multiferroics, for which the magneto-electric effect is observed either as a consequence of an applied magnetic field (we use Cr$_2$O$_3$ as a prototype), or as a result of a phase transition to a spin-spiral magnetic state, as for instance in MnI$_2$, CuCrO$_2$ and AgCrO$_2$.

Figures (9)

• Figure 1: Element-resolved induced electric polarization in Cr$_2$O$_3$ for the spin configurations $'udud'$ (upper panel) and $'duud'$ (lower panel) (see text). Panels (b) and (d) show the $P^x$ and $P^y$ components of the induced dipole moments on Cr atoms represented as a function of the tilting angle $\theta$. Panels (a) and (c) show schematically the orientation of the polarization vectors in the unit cell (blue arrows).
• Figure 2: Results for Cr$_2$O$_3$ with the ground-state spin configuration $'udud'$: arrows show the dipole moments induced on the Cr sites due to a spin rotation for the Cr1 and Cr2 atoms (top left) (i.e. the spin moments $\vec{m}_1$ and $\vec{m}_2$ shown in Fig. \ref{['fig_1:Cr2O3']}(a)) away from collinear antiferromagnetic alignment, and due to a spin rotation for the Cr3 and Cr4 atoms, i.e. $\vec{m}_3$ and $\vec{m}_4$ (see Fig. \ref{['fig_1:Cr2O3']}(a)). The arrows on the bottom left and right panels show the corresponding induced dipole moments of the O atoms.
• Figure 3: The Cr-Cr exchange interactions in Cr$_2$O$_3$ as a function of the Cr-Cr distance $R_{0j}^{Cr_a-Cr_b}$.
• Figure 4: Element-resolved electric polarization in MnI$_2$ due to proper screw magnetic structures characterized by propagation vectors $\vec{q} || \langle 1 \bar{1} 0 \rangle$ (a) and $\vec{q} || \langle 1 1 0 \rangle$ (b), represented as a function of the angle $\Theta = (\vec{q}\cdot \vec{R}_{01})$ where $\vec{R}_{01}$ corresponds to the position of a nearest-neighbor Mn atom at $\vec{R}_{01} = (0.866,0.5,0)a$. The three plots in panels (c) and (d) represent the three components of the antisymmetric (DMI-like) electric polarization parameters ${\cal P}^{x \nu,{\rm a}}_{ij,j}$ (c) giving access to the electric polarization due to spin rotations within the $yz$ plane, and the parameters ${\cal P}^{y \nu,{\rm a}}_{ij,j}$ (d) representing electric polarization due to spin rotations within the $xz$ plane. The left, middle and right plots corresponding to $\nu = x, y, z$, respectively. The absolute values of the parameters in panel (c) are: (left, $\nu = x$) $|\vec{P}_2| = |\vec{P}_3| = 1.61\; \mu{\rm C/m^2}$, $|\vec{P}_1| = |\vec{P}_4| = 1.14\; \mu{\rm C/m^2}$; (middle $\nu = y$) $|\vec{P}_2| = |\vec{P}_3| = 0.38\; \mu{\rm C/m^2}$, $|\vec{P}_1| = |\vec{P}_4| = 0.00\; \mu{\rm C/m^2}$; (right $\nu = z$) $|\vec{P}_2| = |\vec{P}_3| = 0.12\; \mu{\rm C/m^2}$, $|\vec{P}_1| = |\vec{P}_4| = 0.33\; \mu{\rm C/m^2}$. The absolute values of the parameters in panel (d) are: (left, $\nu = x$) $|\vec{P}_2| = |\vec{P}_3| = 0.18\; \mu{\rm C/m^2}$, $|\vec{P}_1| = |\vec{P}_4| = 0.00\; \mu{\rm C/m^2}$; (middle $\nu = y$) $|\vec{P}_2| = |\vec{P}_3| = 1.18\; \mu{\rm C/m^2}$, $|\vec{P}_1| = |\vec{P}_4| = 1.99\; \mu{\rm C/m^2}$; (right $\nu = z$) $|\vec{P}_2| = |\vec{P}_3| = 0.27\; \mu{\rm C/m^2}$, $|\vec{P}_1| = |\vec{P}_4| = 0.00\; \mu{\rm C/m^2}$.
• Figure 5: The element resolved electric polarization for MnO$_2$ for a proper screw (left panel) and cycloidal (right panel) spin modulation with the propagation vector $\vec{q}\, ||\, \hat{z}$ (a) and $\vec{q}\, ||\, \hat{y}$ (b). In the case of a cycloidal spin modulation, the spin moments are rotated within the $xz$ plane for $\vec{q}\, ||\, \hat{z}$ and within the $yz$ plane for $\vec{q}\, ||\, \hat{y}$. The results are presented as a function of the angle $\Theta = \vec{q}\cdot\vec{R}_{01}$ between the nearest neighbor spin moments connected by the radius-vector $\vec{R}_{01}$, with the nearest neighbor atoms located at $\vec{R}_{01} = (\pm 0.5, \pm 0.5, \pm 0.327)a$.