X-ray scattering study of the order parameters in multiferroic $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$

We report on an extensive investigation of the multiferroic compound $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$ using x-ray scattering techniques. Nonresonant x-ray magnetic scattering (NRXMS) was used to characterize the domain population of the single crystal used in our experiments. This revealed that the dominant domain is overwhelmingly $A$ type. The temperature dependence of the intensity and wave vector associated with the incommensurate magnetic order was found to be in good agreement with neutron scattering data. X-ray resonant scattering experiments were performed in the vicinity of the Mn $K$ and Tb $L_3$ edges in the high-temperature collinear phase, the intermediate temperature cycloidal and ferroelectric phase, and the low-temperature phase. In the collinear phase, where according to neutron diffraction only the Mn sublattice is ordered, resonant $E1\text{−}E1$ satellites were found at the Mn $K$ edge associated with $A$-type but also $F$-type peaks. Detailed measurements of the azimuthal dependence of the $F$-type satellites (and their absence in the NRXMS experiments) leads us to conclude that they are most likely nonmagnetic in origin. We suggest instead that they may be associated with an induced charge multipole. At the Tb $L_3$ edge, resonant $A$- and $F$-type satellites were observed in the collinear phase again associated with $E1\text{−}E1$ events. We attribute these to a polarization of the $\mathrm{Tb}5d$ states by the ordering of the Mn sublattice. On cooling into the cycloidal and ferroelectric phase, a new set of resonant satellites appear corresponding to $C$-type order. These appear at the Tb $L_3$ edge only. In addition to a dominant $E1\text{−}E1$ component in the $\sigma \text{−}{\pi }^{\prime }$ channel, a weaker component is found in the preedge with $\sigma \text{−}{\sigma }^{\prime }$ polarization and displaced by $-7\mathrm{eV}$ with respect to the $E1\text{−}E1$ component. Comprehensive calculations of the x-ray scattering cross section were performed using the FDMNES code. These calculations show that the unrotated $\sigma \text{−}{\sigma }^{\prime }$ component of the Tb $L_3$ $C$-type peaks appearing in the ferroelectric phase contains a contribution from a multipole that is odd with respect to both space and time, known in various contexts as the anapole. Our experiments thus provide tentative evidence for the existence of a type of anapolar order parameter in the rare-earth manganite class of mulitferroic compounds.

Figure 1

(Color online) The crystal structure of $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$ in its high-temperature (Pbnm) phase shown in projection down the $\mathbf{c}$ axis.

Figure 2

(Color online) Definitions of the nomenclature used to label the polarization state of the photon beam and the coordinate system used to resolve the components of the magnetic moments relative to the incident and scattered beams.

Figure 3

(Color online) Summary of azimuthal scans of various diffraction peaks recorded at different temperatures and photon energies. The data were recorded in the $\sigma \text{−}{\pi }^{\prime }$ channel and have been normalized to unity. The lines are guides to the eyes.

Figure 4

(Color online) Summary of the nonresonant x-ray magnetic scattering observed on ID20 in scans of wave-vector transfer parallel to ${\mathbf{b}}^{*}$ through various positions in reciprocal space associated with the $A$-, $F$-, or $C$-type magnetic structures. The data were recorded in the ferroelectric and cycloidal phase at $T=14\mathrm{K}$, at an azimuth of $\psi =0$, and in the $\sigma \text{−}{\pi }^{\prime }$ channel.

Figure 5

(Color online) Summary of the nonresonant x-ray magnetic scattering observed on ID20 in scans of wave-vector transfer parallel to ${\mathbf{b}}^{*}$ through various positions in reciprocal space associated with the $A$-, $F$-, or $C$-type magnetic structures. The data were recorded in the ferroelectric and cycloidal phase at $T=14\mathrm{K}$, at an azimuth of $\psi =0$, and in the $\sigma \text{−}{\sigma }^{\prime }$ channel.

Figure 6

(Color online) Temperature dependence of the intensity and modulation wave vector of various satellites. In (a) and (b), we compare the Mn $K$ edge and nonresonant data, all recorded in $\sigma \text{−}{\pi }^{\prime }$. For ease of comparison, the intensities have been normalized to unity at low temperature. The data were taken at ID20.

Figure 7

(Color online) (a) Fluorescence spectrum in the vicinity of the Mn $K$ edge. [(b) and (c)] Representative energy dependence of the x-ray resonant scattering near the Mn $K$ edge at various satellite positions and for various polarization geometries. The data were taken at ID20.

Figure 8

(Color online) Representative scans of the wave-vector transfer with the photon energy tuned to the peak of the resonance at the Mn $K$ edge. The scans shown are through the positions of $F$-type [(a) and (b)] and $A$-type [(c) and (d)] reflections and were recorded in the $\sigma \text{−}{\pi }^{\prime }$ channel at ID20.

Figure 9

(Color online) Representative scans of the wave-vector transfer with the photon energy tuned to the peak of the resonance at the Tb $L_3$ edge. The scans shown are through the positions of $F$-type [(a) and (b)] and $A$-type [(c) and (d)] reflections and were recorded in the $\sigma \text{−}{\pi }^{\prime }$ channel at ID20.

Figure 10

(Color online) (a) Fluorescence spectrum in the vicinity of the Tb $L_3$ edges. [(b) and (c)] Representative energy dependence of the x-ray resonant scattering near the Tb $L_3$ edges at various satellite positions and for various polarization geometries. The data were taken on XMaS.

Figure 11

(Color online) Summary of azimuthal scans of various diffraction peaks recorded at different temperatures and photon energies. The data were recorded in the $\sigma \text{−}{\pi }^{\prime }$ channel and have been normalized to unity. The lines are guides to the eyes. The azimuth $\psi =0$ corresponds to the $\mathbf{c}$ axis in the scattering plane.

Figure 12

(Color online) Temperature dependence of the intensity and modulation wave vector at the Tb $L_3$ edge recorded in $\sigma \text{−}{\pi }^{\prime }$. For ease of comparison, the intensities have been normalized to unity at low temperature.

Figure 13

(Color online) Energy dependence of XRS at the $(4,-q_{\mathrm{Mn}},0)$ satellite in the vicinity of the Mn $K$ edge collected using a normal beam geometry with $\pi$ incident photons on ID20.

Figure 14

(Color online) Energy dependence of XRS at the $(0,3-q_{\mathrm{Mn}},0)$ satellite in the vicinity of the Tb $L_3$ edge recorded on ID20. The data were taken at a temperature of $T=14\mathrm{K}$, and the azimuth was set to be $\psi =20°$, i.e., close to the minimum in the intensity of the response in the $\sigma \text{−}{\pi }^{\prime }$ channel [Fig. 11b].

Figure 15

(Color online) (a) Energy scan taken at the Tb ordering wave vector $q_{\mathrm{Tb}}=0.42$ at $T=2\mathrm{K}$ in the $\sigma \text{−}{\pi }^{\prime }$ channel, where only a weak XRS is observed. (b) Energy scan taken at the Mn ordering wave vector $q_{\mathrm{Mn}}=0.28$ at $T=2\mathrm{K}$, where no significant XRS is observed.

Figure 16

(Color online) (a) $k$ scan taken at the Tb ordering wave vector $q_{\mathrm{Tb}}=0.42$ at $T=2\mathrm{K}$ in the $\sigma \text{−}{\pi }^{\prime }$ channel. The broad width of the scattering indicates that the underlying magnetic order is only short ranged ordered at this temperature. (b) $k$ scan taken at the Mn ordering wave vector $q_{\mathrm{Mn}}=0.28$ at $T=2\mathrm{K}$, indicating that the induced polarization on the Tb ions is long ranged ordered at this position.

Figure 17

(Color online) Summary of scans of the wave-vector transfer along $[0,k,0]$ [(a) and (b)] XRS data around the primary satellite positions $(0,k\pm q_{\mathrm{Mn}},0)$ at the Tb $L_3$ edge for azimuthal angles of $\psi =0$ and 90° in the $\sigma \text{−}{\pi }^{\prime }$ channel. (c) Nonresonant, charge satellites at $(0,k\pm 2q_{\mathrm{Mn}},0)$ in the $\sigma \text{−}{\sigma }^{\prime }$ channel.

Figure 18

(Color online) Representative results of FDMNES calculations for $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$ in the vicinity of the Mn $K$ edge for the [(a)–(c)] collinear and [(d)–(f)] cycloidal phases. The calculated x-ray absorption near edge structure (XANES) is given by the dot-dashed line in (a). The figure allows for a comparison of the calculated intensities of [(a) and (d)] $A$ type, [(b) and (e)] $F$-type, and [(c) and (f)] $C$-type studied in our experiments. The $\sigma \text{−}{\pi }^{\prime }$ and $\sigma \text{−}{\sigma }^{\prime }$ channels are represented by the solid red and dashed blue lines, respectively.

Figure 19

(Color online) Representative results of FDMNES calculations for $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$ in the vicinity of the Tb $L_3$ edge for the [(a)–(d)] collinear and [(e)–(h)] cycloidal phases. The calculated XANES is given by the dot-dashed line in (a). The figure allows for a comparison of the calculated intensities of [(a) and (e)] $A$ type, [(b) and (f)] $F$ type, and [(c), (d), (g), and (h)] $C$ type studied in our experiments. The $\sigma \text{−}{\pi }^{\prime }$ and $\sigma \text{−}{\sigma }^{\prime }$ channels are represented by the solid red and dashed blue lines, respectively.

Figure 20

(Color online) Azimuthal dependence of XRS at the $(0,3-q_{\mathrm{Mn}},0)$ satellite in the vicinity of the Tb $L_3$ edge recorded on ID20. The $\sigma \text{−}\pi$ data were taken at an energy corresponding to an $E1\text{−}E1$ event around $7.520\mathrm{keV}$, while the $\sigma \text{−}\sigma$ data were taken $7\mathrm{eV}$ below this energy. The lines have been calculated using the FDMNES package. In the unrotated channel, the main contribution is calculated to be from $E1\text{−}E1$ XRMS associated with a splitting of the $\mathrm{Tb}5d$ bands induced by the cycloidal order on the Mn sublattice. For the unrotated channel, the weak preedge peak is calculated to be $E1\text{−}E2$, arising from an anapole, i.e., a multipole which is odd with respect to both time and parity. The data in the unrotated channel have been multiplied by a factor of 5.