Chiral order and multiferroic domain relaxation in $\mathrm{Na}\mathrm{Fe}{\mathrm{Ge}}_2{\mathrm{O}}_6$

The magnetic structure and the multiferroic relaxation dynamics of ${\mathrm{NaFeGe}}_2{\mathrm{O}}_6$ were studied by neutron scattering on single crystals partially utilizing polarization analysis. In addition to the previously reported transitions, the incommensurate spiral ordering of ${\mathrm{Fe}}^{3+}$ moments in the $ac$ plane develops an additional component along the crystallographic $b$ direction below $T\approx 5\mathrm{K}$, which coincides with a lock-in of the incommensurate modulation. The quasistatic control of the spin-spiral handedness, respectively of the vector chirality, by external electric fields proves the invertibility of multiferroic domains down to the lowest temperature. Time-resolved measurements of the multiferroic domain inversion in ${\mathrm{NaFeGe}}_2{\mathrm{O}}_6$ reveal a simple temperature and electric-field dependence of the multiferroic relaxation that is well described by a combined Arrhenius-Merz relation, as it has been observed for ${\mathrm{TbMnO}}_3$. The maximum speed of domain wall motion is comparable to the spin-wave velocity deduced from an analysis of the magnon dispersion.

Figure 1

The nuclear structure (with data from Ref. [35]) is shown in (a) by utilizing the crystallographic visualization software vesta3 [38]. Both (b) and (c) sketch the magnetic structure of the intermediate SDW phase and of the low-temperature spin-spiral phase, respectively.

Figure 2

(a), (b) Display exemplary $Q_H$ and $Q_L$ scans for different SF channels. The temperature-dependent components $k_h$ and $k_l$ of the propagation vector $\mathit{k}=\left(k_{h}0k_l\right)$ are, respectively, shown in (c) and (d). The blue and red data correspond to cooling and heating sequences, respectively. In (e) and (f) the peak intensity of reflection $\mathit{Q}=(0.6780-1.075)$ is plotted for all SF channels as function of temperature. All intensities were corrected for the finite flipping ratio and all measurements were performed on sample SI.

Figure 3

Polarized neutron scattering in (1 0 0)/(0 1 0) geometry. The temperature dependence of SF channels measured at $\mathit{Q}=\left(0.3230.08\right)$ is shown in (a) and (b). The $Q$-space position was adjusted with the determined temperature-dependent values of the incommensurate propagation vector [see Figs. 2 and 2]. In both (c) and (d) the chiral ratio is plotted as a function of temperature and for different electric-field amplitudes applied along $a^{★}$.

Figure 4

All recorded hysteresis loops of the chiral ratio shown in (a)–(d) were obtained by driving the applied external electric field (along $a^{★}$) slowly (with a period duration of about $45\min$) between the two polarities. Both slopes of each hysteresis loop were fitted by a hyperbolic tangent in order to estimate the values of respective coercive fields, which are plotted in (e).

Figure 5

All displayed switching curves in (a)–(f) were recorded by utilizing the stroboscopic method at IN12 and by measuring on sample SII. The blue data points refer to the recorded chiral ratio and the red line corresponds to the fit. The time dependence of the applied electric field follows a rectangular shape with inversion at the beginning and in the middle of each panel.

Figure 6

At 4F1 the relaxation processes of the chiral ratio, which is plotted in (a) and (b), were recorded by counting both SF channels $I_{x\overline{x}}$ and $I_{\overline{x}x}$ as a function of time. The electric field ($\left|E\right|=0.12{\mathrm{kV}\mathrm{mm}}^{-1}$) was manually switched after the saturation of the chiral ratio for the respective handedness was reached. For both curves, sample SII was measured.

Figure 7

(a), (b) Display the temperature- and electric-field-dependent relaxation times, respectively. Up and down triangles refer to ${\tau }_a$ and ${\tau }_b$ from IN12 data, whereas crosses correspond to ${\tau }_a$ from the recorded 4F1 data. The whole IN12 data set was fitted simultaneously by the combined Arrhenius-Merz law [see Eq. (7)] and the plotted colored solid lines refer to the fit results shown for fixed electric fields in (a) or fixed temperatures in (b). The 4F1 data were not included to the fit.

Figure 8

Additional set of measurements of the relaxation times from a second experiment are displayed in (a) and (b). The combined Arrhenius-Merz law is valid even at very low temperatures. A slightly different activation constant and critical relaxation time with respect to results shown in Fig. 7 arise from a renewed sample setup (see text).

Figure 9

The different scaling relations for the relaxation times shown in (a) and (b) clearly determine the combined Arrhenius-Merz law to be the correct description of the multiferroic domain inversion. For the comparison only IN12 data from the first experiment (see Fig. 7) were used.

Figure 10

The plots in a) and b) display exemplary two recorded constant-energy scans at $T=3.9\mathrm{K}$ around the incommensurate zone center $\mathit{Q}=\left(0.67700.9252\right)$ along $H$ and $L$, respectively. Both panels c) and d) show the complete measured data sets, whereby red data points refer to the fitted peak maxima and the red lines correspond to fits of the respective dispersion.

Figure 11

Both panels show the calculated dispersion that was obatined by using SpinW code [60]. Hereby, the red data points correspond to the measured spin wave energies from Fig. 10.