Second harmonic generation on incommensurate structures: The case of multiferroic ${\text{MnWO}}_4$

A comprehensive analysis of optical second harmonic generation on an incommensurate (IC) magnetically ordered state is presented using multiferroic ${\text{MnWO}}_4$ as model compound. Two fundamentally different SHG contributions coupling to the primary IC magnetic order or to secondary commensurate projections of the IC state, respectively, are distinguished. Whereas the latter can be described within the formalism of the 122 commensurate magnetic point groups the former involves a breakdown of the conventional macroscopic symmetry analysis because of its sensitivity to the lower symmetry of the local environment in a crystal lattice. Our analysis thus foreshadows the fusion of the hitherto disjunct fields of nonlinear optics and IC order in condensed-matter systems.

Figure 1

Overview of the magnetic phases (Refs. 35, 36), the symmetry-breaking order parameters (Refs. 33, 34), and the corresponding commensurate point groups for multiferroic ${\text{MnWO}}_4$.

Figure 2

Optical transitions and paramagnetic SHG contributions in ${\text{MnWO}}_4$. (a) Linear absorption spectrum of ${\text{MnWO}}_4$ obtained with $x$-polarized light propagating along the $y$ axis of the crystal. At low temperature two intra-atomic and one charge-transfer transitions are identified, and labeled accordingly. (b) SHG spectra obtained in the paramagnetic phase with light incident along the $x$ axis. Only crystallographic SHG from ${\chi }_{zyz}^{\text{MD},\text{EQ}}$ is observed.

Figure 3

Commensurate SHG in the multiferroic phase of ${\text{MnWO}}_4$. [(a) and (b)] C-SHG spectra obtained with $k\parallel x\left({\chi }_{yzz}^{\text{ED}},{\chi }_{yyy}^{\text{ED}}\right)$ and $k\parallel z\left({\chi }_{yxx}^{\text{ED}}\right)$. The exceptionally large signal yield at 1.95 eV is due to multiferroic phase matching. (c) Temperature dependence of the SHG intensity from ${\chi }_{yzz}^{\text{ED}}$, ${\chi }_{yxx}^{\text{ED}}$, and ${\chi }_{yyy}^{\text{ED}}$. Since the latter component is quite weak it was measured at a photon energy where SHG yield from the other SHG tensor components have a comparable magnitude so that admixtures to the ${\chi }_{yyy}^{\text{ED}}$ are small enough to be neglected. All the signals are associated to ED-SHG contributions that are present in the multiferroic AF2 phase only. Note that the same arbitrary scale is used for the SHG intensity in Figs. 3, 4.

Figure 4

Incommensurate SHG in the magnetically ordered phases of ${\text{MnWO}}_4$. (a) IC-SHG spectra obtained with $k\parallel y$ for which C-SHG contributions of any multipole order are not allowed. The temperature dependence in the inset reveals that IC-SHG is present in all three magnetic phases (and in the paramagnetic state, see text). [(b) and (c)] Polarization dependence of the IC-SHG signal at 8 K and 2.22 eV for light incident parallel to the $y$ axis. $x^{\prime }$ and $z^{\prime }$ denote the axes of the rotated coordinate system defined by the polarization dependence of the IC-SHG signal. P and A denote the polarization of the incident light and of the detected SHG component, respectively. In (b) P and A are rotated simultaneously while in (c) P is rotated while A remains constant.

Figure 5

Model explaining the presence of “forbidden” SHG contributions in crystals with IC order. (a) A commensurate modulation of the crystal structure retains the twofold axis of the structure. (b) An IC modulation violates the twofold symmetry locally. Additional SHG contributions may therefore emerge.