Magnetic-field-induced transitions in multiferroic ${\text{TbMnO}}_3$ probed by resonant and nonresonant x-ray diffraction

Multiferroic ${\text{TbMnO}}_3$ is investigated using x-ray diffraction in high magnetic fields. Measurements on first- and second-harmonic structural reflections due to modulations induced by the Mn and Tb magnetic order are presented as function of temperature and field oriented along the $a$ and $b$ directions of the crystal. The relation to changes in ordering of the rare earth moments in applied field is discussed. Observations below $T_N^{\text{Tb}}$ without and with applied magnetic field point to a strong interaction of the rare earth order, the Mn moments, and the lattice. Also, the incommensurate to commensurate transition of the wave vector at the critical fields is discussed with respect to the Tb and Mn magnetic order, and a phase diagram on basis of these observations for magnetic fields $H\parallel a$ and $H\parallel b$ is presented. The observations point to a complicated and delicate magnetoelastic interaction as function of temperature and field.

Figure 1

(Color online) Temperature dependence of the specific heat $\left(C_p\right)$ and magnetic susceptibility $\chi$ of a small single crystal from the ${\text{TbMnO}}_3$ boule. For the magnetization measurements a field of 0.01 T was applied along the $c$ axis. The $C_p\left(T\right)$ and $\chi \left(T\right)$ data indicate three successive phase transitions at 7, 27, and 41 K corresponding to $T_N^{\text{Tb}}$, $T_s$, $T_N$, respectively.

Figure 2

(Color online) Temperature dependence at zero field and 9.5 keV photon energy of the (a) intensity and (b) $k$ component of the wave vector of the $A$-mode $\left(0,2+2{\delta }^{\text{Mn}},9\right)$ and $F$-mode $\left(0,2+2{\delta }^{\text{Mn}},8\right)$ reflections. The respective temperatures $T_N$, $T_s$, and $T_N^{\text{Tb}}$ are indicated by dashed lines. In addition data for the $A$-mode $\left(0,2{\delta }^{\text{Mn}},3\right)$ reflection measured at 100 keV are plotted. Due to a systematic offset the absolute wave numbers are not accurate.

Figure 3

(Color online) Tb-superlattice reflection as function of temperature. (a) shows the intensity of $\left(0,2+2{\delta }^{\text{Tb}},9\right)$ and $\left(0,{\delta }^{\text{Tb}},3\right)$. (b) shows the variation of the propagation vector of the $\left(0,2+2{\delta }^{\text{Tb}},9\right)$ superlattice reflection. In the inset, a scan over the first- and second-harmonic peak along the $b^{\ast }$ direction is shown.

Figure 4

(Color online) Temperature dependence of (a) intensities and [(b)–(e)] wave number of the first- and second-harmonic structural Mn- and Tb-superlattice reflections in the Tb-ordered phase $T<T_N^{\text{Tb}}$ at zero field.

Figure 5

(Color online) Magnetic-field dependence of (a) wave vector and (b) intensity of the second-harmonic $\left(0,2+2{\delta }^{\text{Mn}},9\right)$ (circles) and first-harmonic $\left(0,3-{\delta }^{\text{Mn}},9\right)$ (triangles) superlattice reflection at a sample temperature of $T=27\text{K}$ for $H\parallel a$. Magnetic-field dependence of (c) wave vector and (d) intensities of the CM and IC superlattice reflection for magnetic field $H\parallel a$ at a sample temperature $T=15\text{K}$. At ${\mu }_0{H}_C^a=9\text{T}$ the reflections lock in at CM positions. The horizontal dashed lines show the half and quarter integer positions of the CM first- and second-harmonic reflections above $H_C^a$.

Figure 6

(Color online) Field dependence at $T=4\text{K}$ of the terbium and manganese superlattice reflections with magnetic field $H\parallel a$. (a) peak positions (closed symbols) and (b) the corresponding intensities (open symbols) are shown. Open rectangles show the (0, 0, 4) main Bragg intensity divided by a factor 100. The inset in (b) shows the peak profiles of the $\left(0,3-{\delta }^{\text{Tb}},9\right)$ and $\left(0,2+2{\delta }^{\text{Tb}},9\right)$ reflections for different fields.

Figure 7

(Color online) $K$ scans of the $\left(0,{\delta }^{\text{Tb}},5\right)$ and $\left(0,1-{\delta }^{\text{Tb}},5\right)$ reflection as function of field $H\parallel a$ at a sample temperature $T=4\text{K}$ without polarization analysis. The inset shows an energy scan over the $\text{Tb}{L}_2$ absorption edge with and without polarization analysis in the $\pi -{\sigma }^{\prime }$ channel. The total signal only shows the resonant intensity. The fluorescence background is subtracted using an energy scan performed off the reflection position.

Figure 8

(Color online) Incommensurability $\delta$ of the superlattice reflections $\left(0,2{\delta }^{\text{Mn}},4\right)$ and $\left(0,1-2{\delta }^{\text{Mn}},4\right)$ below $H_C^b$ (closed circles) and (0, 0.25, 4), (0, 0.5, 4), and (0, 0.75, 4) above $H_C^b$ (open circles), measured at (a) $T=2\text{K}$, (b) $T=10\text{K}$, and (c) $T=20\text{K}$ as function of magnetic field $H\parallel b$. At $T=2\text{K}$ in addition $\left(0,{\delta }^{\text{Tb}},5\right)$ and $\left(0,2{\delta }^{\text{Tb}},5\right)$ reflections are shown below and above $H_C^{\text{Tb}}$ (open triangles).

Figure 9

(Color online) (a) Intensities as function of field of the IC $\left(0,{\delta }^{\text{Mn}},3\right)$ and $\left(0,2{\delta }^{\text{Mn}},4\right)$ reflections below $H_C^b$ and the CM (0, 0.25, 4), (0, 0.5, 4), and (0, 0.75, 4) above $H_C^b$. (b) The respective wave vectors (0 $k$ 4) for the intensities in (a) are shown.

Figure 10

(Color online) Hysteresis of the superlattice reflections measured at $T=10\text{K}$. Solid circles show the data measured with increasing field, open circles the data measured with decreasing field. The inset shows the superlattice reflections at the phase transition for increasing field.

Figure 11

(Color online) Intensities of the (0 0.55 3), (0 0.55 4), and (0 0.66 5) reflections below $H_C^b$ and (0 0.25 4) and (0 0.5 4) above $H_C^b$ as function of magnetic field $H\parallel b$ measured at a sample temperature of $T=4\text{K}$.

Figure 12

(Color online) (a) Temperature dependence of the intensities of the Mn and Tb first- and second-harmonic superlattice reflections measured at magnetic field ${\mu }_{0}H=3\text{T}$. (b) Shows the corresponding $k$ components of the wave vector. The measurement is performed by going from the intermediate Tb-ordered phase through $T_N^{\text{Tb}}$ into the purely Mn ordered phase.

Figure 13

(Color online) $k$ scans over (a) $\left(0,\delta ,5\right)$ and (b) $\left(0,2\delta ,5\right)$ positions as function of magnetic field $H\parallel b$ and temperature below $T_N^{\text{Tb}}$.

Figure 14

(Color online) (a) Intensity behavior of CM and IC superlattice reflections measured at $H=6\text{T}$ below (CM) and above (IC) the critical temperature. (b) Corresponding wave number ${\delta }^{\text{Mn}}$ as function of temperature for the first-harmonic $F$-mode reflection. (c) $k$ scans in the region of coexistence of the CM and IC phase at the critical temperature.

Figure 15

(Color online) Phase diagram of ${\text{TbMnO}}_3$ for fields (a) $H\parallel a$ and (b) $H\parallel b$. In (a) the light-shaded area represents the intermediate phase region and the dark-shaded area, the Tb AF ordered state found for field $H\parallel a$. In (b) the green line represents $T_N^{\text{Tb}}$ as found for magnetic field $H\parallel b$. The green shaded area shows the intermediate CM Tb-ordered phase for $H\parallel b$. The IC to CM transition for the manganese order is shown for $H\parallel a$ and $H\parallel b$. The vertical dashed line defines the onset of the first-harmonic reflections.