Dynamical magnetoelectric effects induced by the Dzyaloshinskii- Moriya interaction in multiferroics

We study the dynamical interplay between ferroelectricity and magnetism in a multiferroic with a helical magnetic order. We show that the dynamical exchange-striction induces a biquadratic interaction between the spins and transverse phonons resulting in quantum fluctuations of the spontaneous ferroelectric polarization P in the ferroelectric phase. The hybridization between the spin wave and the fluctuation of the electric polarization leads to low-lying transverse phonon modes. Those are perpendicular to P and to the helical spins at small wave vector but then turn parallel to P at a wave vector close to the magnetic modulation vector. For helical magnetic structure, the spin chirality which determines the direction of P, also possesses a long-range order. Due to the dynamical Dzyaloshiskii-Moriya interaction, the spin-chirality is strongly coupled to the spin fluctuation which implies an on-site inversion of the spin-chirality in the ordered spin-1/2 system and results in a finite scattering intensity of polarized neutrons from a cycloidal helimagnet. Introduction. – Multiferroic compounds in which the electric and the magnetic order coexist are in the focus of current research. Of a particular interest is the possibility of controlling the direction of the spontaneous electric polarization by a magnetic field [1]. Beside this technological relevance of such a strong interplay between the magnetic and electric order parameters it is also fundamentally interesting to understand how such a coupling comes about and what is the microscopic mechanism behind the magnetoelectric (ME) coupling in multiferroics. Among the family of multiferroics, two different type of manganites, RMnO3 (R= Gd, Tb, Dy) [2] and RMn2O5 (R= rare earth, Tb,Y, Bi) [3] play a special role as they exhibit different microscopic ME coupling mechanisms: In the perovskite multiferroic RMnO3, it’s shown experimentally that the onset of helical magnetic order induces spontaneous ferroelectric (FE) polarization, which can be well described by the so-called spin-current model [4]. In these compounds, the spin-orbit coupling within the d(p)orbitals of magnetic(oxygen) ions produces an electric polarization of the form [5], P ∼ Si × Sj . Non-collinearity of the spins Sj (at sites j) is strictly required in the spincurrent model. In the main FE phase of RMn2O5 the electric dipole moments are directed along the b-axis, the spins however are almost collinear in the ab plane which indicates that a spin-orbit-driven mechanism can not be the primary source for FE in RMn2O5. As an alternative explanation, the (super)exchange-striction [6, 7] is believed to be the origin of ferroelectricity in RMn2O5, P ∼ Si · Sj . On the other hand, inspecting carefully the dynamical properties of the multiferroics, we find both, the antisymmetric Dzyaloshiskii-Moriya (DM) interaction and the symmetric magnetostriction play an essential role and need to be taken into account. Based on the spin-current model, the dynamical properties of DM interaction were studied in Ref. [8–10]. A collective ME excited mode, so-called electromagnon, was theoretically predicted. Moreover, it was found [8] that this new low-lying mode is perpendicular to both the spontaneous polarization and the helical wave vector. It corresponds to a rotation of the spin plane with respect to the axis of the helical wave vector, the rotation frequency is √ SJD, where S is the spin value, J is the exchange coupling, and D is the magnetic anisotropy.