Figure 1

(a) Magnetic phase diagram of perovskite ${\mathrm{Gd}}_{1-x}{\mathrm{Tb}}_x{\mathrm{MnO}}_3$ in $x\mathrm{\text{−}}T$ plane (upper panel) and $x\mathrm{\text{−}}H(||b)$ plane at 15 K (lower panel). The $x=0.3$ compound, for example, undergoes the transitions, at first to the collinear antiferromagnetic (collinear AFM) and paraelectric (PE) phase at the Néel temperature $T_N$, then to the ferroelectric (FE) and cycloidal spin order phase at the Curie temperature $T_C$, and finally to the $A$-type antiferromagnetic (A-AFM) phase at $T_A$. (b) Cycloidal spin structure of the ferroelectric $P||a$ phase of ${\mathrm{Gd}}_{0.7}{\mathrm{Tb}}_{0.3}{\mathrm{MnO}}_3$ ($x=0.3$) revealed in this study; Mn spins rotate mostly within the $ab$ plane with the commensurate magnetic wave number $q_m=1/4$. (c) Cycloidal magnetic structure whose spins rotate within the $bc$ plane with the incommensurate magnetic wave number $q_m\sim 0.27$ in the ferroelectric $P||c$ phase of ${\mathrm{TbMnO}}_3$. Magnetic moments of rare-earth ions (Tb and Gd) are omitted in these figures.Reuse & Permissions

Figure 2

Temperature dependence of (a) magnetization along the $c$ axis ($M||c$), (b) electric polarization along the $a$ axis ($P||a$) measured in a warming run after an electric field cooling procedure ($E=\pm 100\mathrm{kV}/\mathrm{m}$), and (c)  the magnetic wave number $q_m$ in the neutron magnetic reflection (0 $q_m$ 3) as well as derived from the lattice modulation wave number $q_l(=2q_m)$ in the (0 $q_l$ 0) synchrotron x-ray diffraction [24]. ${}^{160}\mathrm{Gd}_{0.7}{\mathrm{Tb}}_{0.3}{\mathrm{MnO}}_3$ and ${}^N\mathrm{Gd}_{0.7}{\mathrm{Tb}}_{0.3}{\mathrm{MnO}}_3$ denote the samples using ${}^{160}\mathrm{Gd}$ and the naturally abundant Gd (${}^N\mathrm{Gd}$), respectively.Reuse & Permissions

Figure 3

(a) Comparison between observed ($F_{\mathrm{obs}}$) and calculated ($F_{\mathrm{cal}}$) structural factors of magnetic reflections of ${}^{160}\mathrm{Gd}_{0.7}{\mathrm{Tb}}_{0.3}{\mathrm{MnO}}_3$ in the $P||a$ phase at 20 K. (b) Alignment of Mn-spin moments as a function of $y$ (along the $b$ axis) in the $a\times 4b\times c$ magnetic unit cell. Note that there are two basal Mn-O planes in a unit cell with $z=0$ and 0.5 (along the $c$ axis). The antiferromagnetic coupling in the $a$ and $b$ components and the ferromagnetic coupling in the $c$ components between these two planes (i.e., mirror symmetry perpendicular to the $c$ axis) are assumed.Reuse & Permissions

Figure 4

(a) Profiles of polarized neutron magnetic satellite reflection (0 $q_m$ 3) in the $P||a$ state of ${}^{160}\mathrm{Gd}_{0.7}{\mathrm{Tb}}_{0.3}{\mathrm{Mn}}_3$ at 20 K; (a) $P_a>0$ and (b) $P_a<0$. Insets show the relation between the spin helicity and the direction of electric polarization. (c) The temperature dependence of $\mathit{Q}$-integrated intensities of the magnetic reflections around $\mathit{Q}=(0q_{m}3)$ and (0 0 3). The flipper off and on correspond to the incident neutron spin $S_n$ parallel ($I_{\uparrow }$) and antiparallel ($I_{\downarrow }$), respectively, to the scattering vector ${\mathit{Q}}_s$.Reuse & Permissions