Electric polarization, toroidal moment, spin canting, and chirality induced by Dzialoshinsky–Moriya interactions in aV3cluster analog of multiferroics

Abstract

The electric polarization and toroidal moments in the spin-frustrated ${\mathrm{V}}_{3}$ single-molecular magnet (${\mathrm{V}}_{3}$) with the out-of-plane $({D}_{z})$ and in-plane $({D}_{x},{D}_{y})$ Dzialoshinsky--Moriya (DM) interaction are studied. It is shown that the spin-current mechanism of polarization proposed by Katsura, Nagaosa, and Balatsky (KNB) [Phys. Rev. Lett. 95, 057205 (2005)] is the driving force of the polarization ${P}_{z}$ induced by the ${D}_{x}$ coupling and directed perpendicular to the ${\mathrm{V}}_{3}$ plane, ${P}_{z}||Z$, in the field $\mathbf{B}||Z.$ The KNB mechanism leads to polarization ${P}_{z}^{}$ which increases nonlinearly with increasing field, reaches a maximum at the avoided level-crossing field ${B}_{\mathrm{A}1}$ and then gradually decreases in ${\mathrm{V}}_{3}^{\mathrm{R}}$ with the ground state of the right vector chirality. Spin dynamics (field-dependent spin canting) in the ground state of this system has the form of the opening of the $({\mathbf{S}}_{1},{\mathbf{S}}_{2},{\mathbf{S}}_{3})$ umbrella by the increasing field up to the maximum canting angle $\ensuremath{\alpha}=\ensuremath{\pi}/4$ and, then, closing it in high fields. In the excited state of the avoided level-crossing structure, the spin dynamics is opposite. ${\mathrm{V}}_{3}^{\mathrm{L}}$ with the ground state of the left chirality demonstrates polarization ${P}_{z}^{}$ in the spin-collinear state and a lack of polarization in the spin-frustrated state. The ${D}_{y}$ coupling results in the field-dependent toroidal magnetic moment ${T}_{z}$. The ${D}_{x},{D}_{y}$ coupling leads to the in-plane $\ensuremath{\phi}$ canting of the spins and coexistence of ${P}_{z}^{}$ and ${T}_{z}$. The coupling between the right vector chirality ${\ensuremath{\kappa}}_{\mathrm{R}}$ and magnetic field ${B}_{z}$ is nonlinear, the negative vector chirality ${\ensuremath{\kappa}}_{\mathrm{L}}$ changes the sign at the level crossing. The polarization ${P}_{z}$ and toroidal moment ${T}_{z}$ are proportional to the radial and tangential in-plane spin fluctuations, respectively, and depend nonlinearly on the value ${\ensuremath{\kappa}}_{\mathrm{R}}$ of the vector chirality, while magnetic moment ${\ensuremath{\mu}}_{z}$ is linear proportional to ${\ensuremath{\kappa}}_{\mathrm{R}}.$ The DM ${\mathrm{V}}_{3}^{}$ nanomagnets are the cluster analogs of multiferroics.