Optically probed symmetry breaking in the chiral magnet ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$

We report on the linear optical properties of the chiral magnet ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$, specifically associated with the absence of inversion symmetry, the chiral crystallographic structure, and magnetic order. Through spectroscopic ellipsometry, we observe local crystal-field excitations below the charge-transfer gap. These crystal-field excitations are optically allowed due to the lack of inversion symmetry at the Cu sites. Optical polarization rotation measurements were used to study the structural chirality and magnetic order. The temperature dependence of the natural optical rotation, originating in the chiral crystal structure, provides evidence for a finite magnetoelectric effect in the helimagnetic phase. We find a large magneto-optical susceptibility on the order of $\mathcal{V}\left(540\mathrm{nm}\right)\sim {10}^4\mathrm{rad}/\mathrm{T}\mathrm{m}$ in the helimagnetic phase and a maximum Faraday rotation of $\sim 170\mathrm{deg}/\mathrm{mm}$ in the ferrimagnetic phase. The large value of $\mathcal{V}$ can be explained by considering spin cluster formation and the relative ease of domain reorientation in this metamagnetic material. The magneto-optical activity allows us to map the magnetic phase diagram, including the skyrmion lattice phase. In addition to this, we probe and discuss the nature of the various magnetic phase transitions in ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$.

Figure 1

The $3d$ crystal-field splittings for the Cu-I (trigonal bipyramid) and Cu-II (square pyramid) ions. The dipole-active local crystal-field (CF) excitations are indicated with red arrows, and the charge-transfer excitations with blue arrows. Intersite excitations between the Cu ions, illustrated with a gray arrow, lie outside the experimentally accessible energy range of this work.

Figure 2

Optical conductivity at 15 and $300\mathrm{K}$ as obtained by ellipsometry. The onset of charge-transfer excitations is observed at about $2.5\mathrm{eV}$ at $15\mathrm{K}$. Between 1.0 and $2.0\mathrm{eV}$ local crystal-field excitations are observed with conductivity values not surpassing $50{\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$. At $15\mathrm{K}$, pronounced peaks are located at 1.2, 1.4, and $1.5\mathrm{eV}$. The features broaden with increasing temperature, obscuring the $1.4\mathrm{eV}$ peak at $300\mathrm{K}$, as seen in the inset.

Figure 3

(a) Temperature-dependent absorption spectra in the range of the transmission window as measured on the $1-\mathrm{mm}$-thick sample. (b) Temperature dependence of the onset $E_g$ of excitations across the gap, determined at $\alpha =70{\mathrm{cm}}^{-1}$. The fit (red line) is based on the empirical Varshni equation. (c) ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ is transparent for green light. The (111) sample, with a thickness $d\approx 221\pm 3\mu \mathrm{m}$, is glued on a copper sample holder with a 2.0-mm-diameter hole.

Figure 4

The natural optical activity ${\theta }_{\mathrm{NOA}}$ per sample thickness $d$ in deg/mm measured along the crystallographic (111) axis with probe photon energy $E_{\mathrm{probe}}=2.30\pm 0.03\mathrm{eV}$. The crystal is levorotatory. Above the Curie temperature $T_C\approx 58\mathrm{K}$, the temperature dependence is weak. Below $T_C$ an enhancement of ${\theta }_{\mathrm{NOA}}$ is observed, which is attributed to the finite magnetoelectric coupling in the helimagnetically ordered state.

Figure 5

The Faraday rotation per sample thickness, ${\theta }_{\mathrm{F}}/d$, as a function of field for different temperatures. With increasing field, ${\theta }_{\mathrm{F}}$ increases until a plateau is reached, marking the phase transition between conical and ferrimagnetic order. At $15\mathrm{K}$ this phase transition is induced around an applied field of $B_a\approx 195\mathrm{mT}$. At lower fields, in the $15\mathrm{K}$ curve around $B_a\approx$ 60 mT, the helical-conical phase transition becomes apparent as a kink in the field dependence of ${\theta }_{\mathrm{F}}/d$.

Figure 6

The magnetic phase diagram of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ as obtained from the Faraday rotation. $B_a$ is the applied magnetic field along the crystallographic (111) direction. The left panel shows the magnetic phase diagram for a large $(B_a,T)$ range, whereas the right panel gives a zoom-in around the skyrmion lattice phase (SkL). The triangles indicate the conical-ferrimagnetic phase boundary. Diamonds give the helical-conical phase boundary. The SkL phase boundary is indicated by the circles. The phase boundary between ordered phases and the paramagnetic phase is indicated by the squares. The color mapping indicates the second derivative of the Faraday rotation. For a quantitative comparison of the critical fields with results obtained by other techniques, one needs to take into account the demagnetization factor (see main text).

Figure 7

Faraday rotation across the paramagnetic-helimagnetic phase transition, normalized by applied field $B_a$. The curves are offset by $(\pi /1.8)\times {10}^4\mathrm{rad}/\mathrm{T}\mathrm{m}$ with respect to each other. Within the helical phase the scaled rotation attains a value of ${\theta }_{\mathrm{F}}/\left(B_{a}d\right)\approx 1.7\times {10}^4\mathrm{rad}/\mathrm{T}\mathrm{m}$. The weak first-order transition into the helical phase is marked by black circles. The fluctuation-disorder regime is located between the black dot and the red diamond.

Figure 8

The magneto-optical susceptibility ${\chi }_{\mathrm{MO}}$ as a function of temperature at different applied field strengths. The phase transition from the paramagnetic phase to one of the ordered phases shows a rescaling from first order (for the para to helical/conical/SkL phases) to second order (para-ferri) with increasing field. The changes in ${\chi }_{\mathrm{MO}}$ below $T_C\approx 58\mathrm{K}$ show the temperature-induced conical-helical and ferrimagnetic-conical phase transitions.

Figure 9

The field-even rotation ${\theta }_{\mathrm{even}}$, shown here per sample thickness $d$ for some selected temperatures. The curves are offset by $-1\mathrm{deg}/\mathrm{mm}$ for clarity. The phase transition from conical to ferrimagnetic order again becomes apparent. The jagged feature at the helical-conical phase transition is due to hysteresis in ${\theta }_{\mathrm{F}}$.

Figure 10

Optical conductivity in the region of the crystal-field excitations for different temperatures. The inset depicts the spectral weight (integral of the optical conductivity between 1.0 and $2.0\mathrm{eV}$) as a function of temperature.