Flop of Electric Polarization Driven by the Flop of the Mn Spin Cycloid in Multiferroic ${\mathrm{TbMnO}}_3$

Using in-field single-crystal neutron diffraction, we have determined the magnetic structure of ${\mathrm{TbMnO}}_3$ in the high field $\mathbf{P}\parallel a$ phase. We unambiguously establish that the ferroelectric polarization arises from a cycloidal Mn spin ordering, with spins rotating in the $ab$ plane. Our results demonstrate directly that the flop of the ferroelectric polarization in ${\mathrm{TbMnO}}_3$ with applied magnetic field is caused from the flop of the Mn cycloidal plane.

Figure 1

(a, b) Illustrations of the two cycloidal Mn magnetic structures proposed to cause ferroelectricity in ${\mathrm{TbMnO}}_3$. In both cases the magnetic propagation vector $\tau$ is parallel to the $b$ axis. In zero field [panel (a)] cycloidal order, Mn spins rotate around the $a$ axis ($S_i\times S_{i+1}$) and rotate wholly within the $bc$ plane. The ferroelectric polarization via the antisymmetric DM interaction is produced along the $c$ axis. The high field magnetic structure determined in this work is shown in panel (b); it yields a $\mathbf{P}\parallel a$ ferroelectric polarization that arises from an $ab$ cycloid where Mn spins rotate around the $c$ axis. In both panels we also show the magnetic ordering of Tb spins. In the zero field case (a), the magnetic propagation vectors of Tb and Mn spins are clamped, and Tb spins point along the $a$ axis forming a spin-density wave [1]. The canted antiferromagnetic ordering of Tb spins for $H\parallel b=5\mathrm{T}$ determined in this work is shown in panel (b). (c) Here we depict an anharmonic $ab$ spiral where the phase difference between spins 1 and 2 is $\omega =\pi /2$. In such a case the angle between spins 1 and 2 is different from that between 2 and $1+b$ yielding an alternating scalar product along the $b$ axis. Amplitudes of Tb and Mn spins are not to scale.

Figure 2

Single-crystal neutron diffraction measurements from the E4 diffractometer. Here we show scans along (0, $k$, 1) in reciprocal space as a function of temperature measured in $H\parallel b=5\mathrm{T}$. The intensity of these scans is plotted in color coding with corresponding scales above each panel. (a) Portion of the data showing the temperature dependence of the A mode reflection (0, $ϵ$, 1). Here the wave number $ϵ$ varies from 0.282 at $T_N$ to 0.266 at 15 K before it locks discontinuously to the commensurate value of $\frac{1}{4}$ below 11 K. The weak third harmonic reflection was also observed in these scans to disappear at this transition. These data are shown in an enhanced color scale on the same panel and in the same location in the $T\mathrm{\text{−}}Q$ map. (b) Temperature evolution of the $Pbnm$ forbidden reflection (0,0,1).

Figure 3

Results of refinements of the magnetic structure at 8.5 K and $H\parallel b=5\mathrm{T}$. In panels (a) and (b) we show the variation of the $M_x$ and $M_y$ components of the Mn spins for the ions located at $z=0$ (black squares) and $z=\frac{1}{2}$ [gray (red) squares] along the magnetic propagation vector. Note that the components along $a$ and $b$ are out of phase so as to yield a cycloidal ordering shown in Fig. 1. In panels (c) and (d) we show the comparison between observed and calculated magnetic structure factors ($F^2$) for the analysis of the Mn $\tau =\frac{1}{4}$ reflections and nuclear plus Tb magnetic reflections, respectively. Since the Tb magnetic propagation vector is $\tau =0$, the nuclear and magnetic reflections are not separated.