$E$-type noncollinear magnetic ordering in multiferroic $o-{\mathrm{LuMnO}}_3$

Multiferroic orthorhombic $o-{\mathrm{LuMnO}}_3$ exhibits large ferroelectric polarization induced by an $E$-type magnetic order. Recently, the $E$-type magnetic phase in ${\mathrm{LuMnO}}_3$ was proposed to feature magnetic moments tilted away from the collinear ordering. We employed neutron diffraction to determine the symmetry of the magnetic order in $o-{\mathrm{LuMnO}}_3$. We observed that below $T_N=39$ K, the ${\mathrm{Mn}}^{3+}$ spins order into an incommensurate amplitude-modulated phase that obeys the Pbnm crystal symmetry and is paraelectric. The incommensurate phase locks into a commensurate phase at $T_C=35.5$ K described by a fully antiferromagnetic and noncollinear $E$-type order. This noncollinear $E$-type ordering breaks the spatial inversion symmetry and induces a spontaneous polarization at $T_C$. At $T=2$ K, an appreciably large electric polarization was observed similar to that of other orthorhombic manganites featuring $E$-type magnetic order. We also present a Pbnm symmetry-allowed Dzyaloshinskii-Moriya interaction that explains the noncollinear $E$-type order in the commensurate phase. These results are in qualitative agreement with the type of distortions from collinear $E$-type antiferromagnetic order found using Monte Carlo simulation for rare-earth manganites [M. Mochizuki et al., Phys. Rev. B 84, 144409 (2011)].

Figure 1

Powder neutron scattering patterns from ${\mathrm{LuMnO}}_3$ measured using HRPT as a function of scattering angle $2\mathrm{\Theta }$ at (a) $T=50$ K showing only nuclear scattering, and (b) $T=2$ K showing additional magnetic peaks such as (0, 1/2, 0) and (0, 1/2, 1). (c,d) Bragg peak powder patterns measured using DMC at $T=38$ and 5 K. (a)–(d) The vertical bars indicate magnetic and nuclear (upper row) Bragg peaks. The bottom solid line indicates the difference between the experiment and the model.

Figure 2

Schematic illustrations of the magnetic structure (a), (b) at $T_C=35.5<T<T_N=39$ K and (c),(d) at $T<T_C$ projected onto the $bc$ plane and the $ab$ plane, respectively. In the amplitude-modulated incommensurate phase (a),(b) the ${\mathrm{Mn}}^{3+}$ spins are oriented along the $\mathbf{b}$ axis and tilted toward the $\mathbf{a}$ axis. In the commensurate phase (c),(d) the spins ordering with the noncollinear $E_1$ type are shown. The spins in the noncollinear $E_1$ type are shown tilting away from the $\mathbf{b}$ axis toward the $\mathbf{c}$ axis ($\theta \sim \pm 3.9^{\circ }$). It violates the inversion symmetry along the $\mathbf{a}$ axis inducing a ferroelectric polarization below $T_C$ along the $\mathbf{a}$ axis, (d) +$P{}_a$($E_1$) (yellow arrows). Here, the tilting and magnitude of moments are exaggerated as a guide to eye.

Figure 3

Temperature dependence of the ${\mathrm{Mn}}^{3+}$ moment (a) $|m^a|$, (b) $|m^b|$, (c) $|m^c|$, and (d) the magnitude of the total moment with varying temperature. In the incommensurate phase, the magnitude is scaled by $\sqrt{2}$. The vertical dotted lines indicate the $T_C=35.5$ K and $T_N=39$ K.

Figure 4

(a) Temperature dependence of the magnetic Bragg peak intensity at Q = ($0,1/2,1$) in the commensurate phase and the intensities at the Q = ($0,\mathit{q},1$) and Q = ($0,1-\mathit{q},1$) Bragg peak positions for $0.47<\mathit{q}\le 0.5$. (b) Comparison of the temperature dependence of magnetic reflections ($0,\mathit{q},1$) and ($1,\mathit{q},1$). (c) Temperature dependence of the magnetic wave vector, $q$, along the $\mathbf{b}$ axis. (d) Temperature dependence of the magnetic correlation length. The vertical dotted lines indicate $T_C=35.5$ K and $T_N=39$ K.

Figure 5

(a), (b) Temperature variations of the real and imaginary parts of dielectric constants, ${\varepsilon }_r^{\prime }$ and ${\varepsilon }_r^{\prime \prime }$, respectively, measured at a frequency $f=100$ kHz on cooling (solid circle) and warming (open circle). (c) $T$ variation of the spontaneous electric polarization ($P$) measured after electric field cooling of $E_p=3000$ kV/m. The dotted lines mark $T_N$ and $T_C$. Inset: Switching of the ferroelectric polarization with a poling electric field of $\left|E\right|=833$ kV/m is shown. (d) $T$ variations of $P$ after cooling under different applied $E$ fields. (e) $T$ variations of the $P$ normalized to the value at $T=2$ K. (f) Poling $E$-field dependence of the electric polarization magnitude at $T=2$ K for ${\mathrm{LuMnO}}_3$ and ${\mathrm{TmMnO}}_3$ [9]. The lines are guides to the eye. The vertical dotted lines indicate $T_C$ and $T_N$.

Figure 6

Temperature dependence of the lattice constants for $T<50$ K. The vertical dotted lines at $T_N=39$ K and $T_C=35.5$ K indicate the onset temperature for incommensurate and commensurate Mn spin ordering, respectively.

Figure 7

Schematic of the spin structure in the $E_1$ domain projected on the $bc$ plane. Blue arrows here resemble the collinear $\mathbf{c}$-axis spin component. The DMI coupling parameter ${\alpha }_c$ switches sign along the $\mathbf{c}$ axis.

Figure 8

$E$ and ${\alpha }_c$ dependence of $|S^a|/|S^b|$ (left panels) and $|S^c|/|S^b|$ (right panels) in the ground-state spin configuration of the model described in Ref. [14], obtained as described in the text. Here, $E$ and $D$ represent the single-ion anisotropy in the $ab$ plane and along the $\mathbf{c}$ axis, respectively, whereas ($\alpha ,\beta ,\gamma$) are DM interactions in the $ab$ plane and the $\mathbf{c}$ axis. The upper, middle, and lower panels show the results for different values of the relevant interactions with spin-orbit origin. The black star represents the canting size corresponding to the coupling parameters in Ref. [14]. The dark areas in (e),(f) represent the coupling strengths, which correspond to the experimentally observed canting of the magnetic moments. All the coupling strengths are presented in meV.

Figure 9

(a) The refined magnetic structure of ${\mathrm{LuMnO}}_3$ in the commensurate phase (FPstudio) [22]. Here, we show the spin structure of the noncollinear $E_1$ domain. (b) Schematic illustration of the spin structure projected on the $bc$ plane defined by sublattice magnetization along the $\mathbf{b}$ axis ($E_1^b$) and (c) along the $\mathbf{c}$ axis (${\mathrm{O}}_1^c$). Lu (red), Mn (green), and O (blue) ions are represented by solid spheres. Black arrows here resemble magnetization components.