Field-induced linear magnetoelastic coupling in multiferroic ${\mathrm{TbMnO}}_3$

We have used in-field neutron and x-ray single-crystal diffraction to measure the incommensurability $\delta$ of the crystal and magnetic structures of multiferroic ${\mathrm{TbMnO}}_3$. We show that the flop in the electric polarization at the critical field $H_C$, for field $H$ along the $a$ and $b$ axes, coincides with a first-order transition to a commensurate phase with propagation vector $\kappa =(0,\frac{1}{4},0)$. In-field x-ray diffraction measurements show that the quadratic magnetoelastic coupling breaks down with applied field as shown by the observation of the first harmonic lattice reflections above and below $H_C$. This indicates that magnetic field induces a linear magnetoelastic coupling.

Figure 1

(Color online) Crystal and magnetic structures of ${\mathrm{TbMnO}}_3$ (Pbnm, $a=5.302$, $b=5.857$, $c=7.402\mathrm{\AA }$) in the $ab$ plane. Long (darker) and short (lighter) Mn-O bond ordering arises from staggered $3d_{3x^2-r^2}∕3d_{3y^2-r^2}$ orbital ordering found above $T_N$. (a) A model of the IC magnetic structure of ${\mathrm{TbMnO}}_3$ below $T_N$ (Ref. 7). The arrows through Mn atoms indicate the direction of Mn spins. (b) A model of the Mn spin structure for ${\mathrm{TbMnO}}_3$ for $H>H_C$. Here arrows from O atoms indicate the direction of displacement from the average structure as to increase (decrease) the angle $\varphi$ on the basis of FM (AFM) coupling between adjacent Mn ions. The large arrows on ferroelectrically active octahedra indicate the direction of the predicted polarization from these displacements.

Figure 2

(Color online) (a,d) Electric polarization $P\Vert a$ and $P\Vert c$ of ${\mathrm{TbMnO}}_3$ for $H\Vert a$ and $H\Vert b$. These data were compiled from field-cooled measurements reported by Kimura et al. (Ref. 11). (b) Variation of $\delta$ as a function of $H\Vert a$ for reflections $(0,{\delta }_m^{\mathrm{Mn}},1)$, $(0,{\delta }_m^{\mathrm{Tb}},1)$ measured using neutron diffraction (filled blue and red circles, respectively) and $(0,3-\delta ,8)$ measured with x-ray diffraction (filled green squares). For $H>H_C^a$ the C reflections $(0,\frac{1}{4},1)$ and $(0,\frac{1}{2},1)$ (open blue circles) were measured using neutron diffraction while the $(0,2\frac{3}{4},8)$ reflection with x rays (open green squares). (c) Variation of the integrated intensity as a function of $H\Vert a$ for the same reflections as in panel (b). For the $(0,{\delta }_m^{\mathrm{Tb}},1)$ reflection the logarithmic scale on the right axis is used. The field dependence of the intensity of the FM (002) reflection is also shown (filled red squares). (e) Variation of $\delta$ as a function $H\Vert b$, for the $(0,{\delta }_m^{\mathrm{Tb}},1)$ reflection and the C reflection $(0,\frac{1}{3},1)$ due to Tb spin ordering (filled and unfilled red circles, respectively). On the same panel we show the results of x-ray diffraction measurements of the $(0,2{\delta }^{\mathrm{Mn}},3)$ reflection (filled blue circles) and the C reflections $(0,\frac{1}{2},3)$ and $(0,\frac{1}{4},3)$ (open blue circles). (f). Variation of the integrated intensity as a function of $H\Vert b$ for the reflections shown in panel (e). For the $(0,{\delta }_m^{\mathrm{Tb}},1)$ reflection the logarithmic scale on the right axis is used. The field dependence of the intensity of the FM (022) reflection is also shown (filled red squares). Integrated intensity in panels (c) and (d) are shown in arbitrary units. All measurements are taken at 2 K.

Figure 3

(Color online) (a) X-ray diffraction scans of the first harmonic $(0,3-\mathrm{K},8)$ reflection as a function of field with $H\Vert a$ at $T=10\mathrm{K}$, measured on X21. (b) Neutron-diffraction scans through the first harmonic $(0,{\delta }_m^{\mathrm{Mn}},1)$ reflection as a function of $H\Vert a$.

Figure 4

(Color online) (a) Scans of the second harmonic IC lattice reflection at $(0,2{\delta }^{\mathrm{Mn}},3)$ with increasing field at 2.6 K measured on BW5. The C reflection at $(0,\frac{1}{2},3)$ is shown for $H>H_C^b$. (b) Similarly, scans of the first harmonic $(0,\frac{1}{4},4)$ reflection of the C phase with increasing field at 2.6 K for $H>H_C^b$ are shown.