Dynamical multiferroicity

An appealing mechanism for inducing multiferroicity in materials is the generation of electric polarization by a spatially varying magnetization that is coupled to the lattice through the spin-orbit interaction. Here we describe the reciprocal effect, in which a time-dependent electric polarization induces magnetization even in materials with no existing spin structure. We develop a formalism for this dynamical multiferroic effect in the case for which the polarization derives from optical phonons, and compute the strength of the phonon Zeeman effect, which is the solid-state equivalent of the well-established vibrational Zeeman effect in molecules, using density functional theory. We further show that a recently observed behavior—the resonant excitation of a magnon by optically driven phonons—is described by the formalism. Finally, we discuss examples of scenarios that are not driven by lattice dynamics and interpret the excitation of Dzyaloshinskii-Moriya-type electromagnons and the inverse Faraday effect from the viewpoint of dynamical multiferroicity.

Figure 1

Duality of magnetization and polarization. (a) A spatially varying magnetization induces a polarization, as is well known for example in multiferroic ${\mathrm{TbMnO}}_3$ [2, 3, 8, 10]. (b) A temporally varying polarization induces a magnetization as shown in this work.

Figure 2

Magnetic moments from ionic loops. Schematic motion of ions in a diatomic ${\mathrm{A}}^{+}{\mathrm{B}}^{-}$ material driven by perpendicular optical phonons. The circular motions of the ions create local magnetic moments, ${\mathbf{m}}_{\text{A}}$ and ${\mathbf{m}}_{\text{B}}$. The area covered by the ionic loop of the lighter ion (here ${\mathrm{B}}^{-}$) is larger, and therefore its local magnetic moment overcompensates that of the heavier ion. This leads to a macroscopic magnetic moment, $\mathbf{M}$.

Figure 3

Zeeman splitting of degenerate phonon modes. Sketch of the phonon Zeeman splitting as a function of the external magnetic field $\mathbf{B}$. The phonons with their magnetic moment $\mathbf{M}$ aligned parallel to the external magnetic field $\mathbf{B}$ has a lower energy than the phonons with their moment aligned antiparallel to the field.

Figure 4

Calculated time-varying magnetic moment from optically driven phonons in ${\mathrm{ErFeO}}_3$. (a) Time evolution of the phonon magnetic moment $\mathbf{M}$ in units of the nuclear magneton ${\mu }_{\text{N}}$ in ${\mathrm{ErFeO}}_3$ after excitation with an ultrashort terahertz pulse. The phonon magnetic moment oscillates with a large amplitude with the difference frequency, ${\omega }_{-}={\omega }_1-{\omega }_2$, and with a small amplitude with the sum frequency, ${\omega }_{+}={\omega }_1+{\omega }_2$. The schematic pump pulse has a duration of $130\text{fs}$. (b) Time evolution of the normal mode coordinates $Q_{\alpha }$ of the excited high-frequency IR modes ($\alpha ={\text{B}}_{3\text{u}},{\text{B}}_{2\text{u}}$) in units of $\sqrt{\mu }\text{Å}$, where $\mu$ is the atomic mass unit.

Figure 5

Dzyaloshinskii-Moriya-type electromagnon symmetry in ${\mathrm{TbMnO}}_3$. Spin cycloid oriented in the $a\text{−}b$ plane with ferroelectric polarization ${\mathbf{P}}_{\text{FE}}$ along $a$. To excite a Dzyaloshinskii-Moriya electromagnon, the ac electric field of the subterahertz radiation E has to be aligned along $c$ and the induced magnetization m is along $b$.