Interplay between spin dynamics and crystal field in the multiferroic compound ${\mathrm{HoMnO}}_3$

In the multiferroic hexagonal manganite ${\mathrm{HoMnO}}_3$, inelastic neutron scattering and synchrotron-based THz spectroscopy have been used to investigate the spin waves associated with the Mn order together with Ho crystal-field excitations. While the Mn order sets in first below 80 K, a spin reorientation occurs below 37 K, a rare feature in the rare-earth manganites. We show that several Ho crystal-field excitations are present in the same energy range as the magnons, and that they are all affected by the spin reorientation. Moreover, several anomalous features are observed in the excitations at low temperature. Our analysis and calculations for the Mn spin waves and Ho crystal-field excitations suggest Mn-Ho coupling mechanisms as well as electric charge redistribution affecting the dynamics.

Figure 1

Sketch of the ${R\text{MnO}}_3$ crystallographic and magnetic structures. (a) Projection of the different atomic positions onto the $ab$-plane, omitting the oxygens for clarity. The Mn ions on the $6c$ Wyckoff site have an $x$ coordinate within the $ab$-plane close to 1/3 and a $z$ coordinate along the $c$-axis equal to 0 (red spheres) or 1/2 (black spheres). The rare-earths occupy two different Wyckoff sites $(2a$ and $4b)$. The main intraplane $\left(J\right)$ and interlayer $\left(K_{1,2}\right)$ exchange interactions between the Mn magnetic moments are indicated. (b) Magnetic arrangement of the Mn ions with zero propagation vector corresponding to the four one-dimensional irreducible representations ${\mathrm{\Gamma }}_i$ of the $P{6}_{3}cm$ space group. The homometric pairs, indistinguishable through neutron scattering if $x_{\text{Mn}}=1/3$, are indicated by the boxes. $x$ is along the $a$-axis, $y$ is perpendicular to it in the $ab$-plane, and $z$ is along the $c$-axis. ${\varphi }_p$ and $\mathrm{\Delta }\varphi$ are, respectively, the angle of the magnetic moment in the $p$ plane and the dephasing of this spin with respect to the corresponding one in the adjacent plane.

Figure 2

Inelastic neutron-scattering color maps in ${\mathrm{HoMnO}}_3$ obtained gathering several constant-$Q$ energy scans corrected for the detailed balance factor. Vertical bars give the neutron counts in arbitrary units. The dots added on the color maps were obtained by Gaussian fits of the individual scans. (a) Maps for the $(1,0,\ell )$ reciprocal space direction at different temperatures, ranging from 65 K down to 1.5 K. The spin reorientation occurs at $T_{\text{SR}}=37$ K. The upper spin-wave branch (labeled $M)$ associated with Mn order is observed around $45{\mathrm{cm}}^{-1}$. The lower branch $\left(M^{\prime }\right)$ can be observed at 65, 45, and 27 K. The dashed white lines are guide to the eyes. The spectral weight is typical of the ${\mathrm{\Gamma }}_4$ configuration at 65 and 45 K, but it becomes typical of ${\mathrm{\Gamma }}_3$ at 27 K. In addition, flat modes (0–4), likely Ho crystal-field excitations, are also observed. For the temperatures below 27 K, the spin waves gradually fade away into the CF excitations, and only nondispersive excitations remain at 1.5 K (below $T_{\text{Ho}})$. (b) Maps for the (1,0,0) and (1,0,1) reciprocal space positions as a function of temperature. The dashed vertical lines indicate the transition temperatures.

Figure 3

THz absorption spectra of ${\mathrm{HoMnO}}_3$ for the three different orientations of the electromagnetic wave e, h fields with respect to the crystal $c$-axis, using two different samples. Upper panels: absolute absorptions. Lower panels: relative absorptions with a reference above the magnetic ordering temperature at 98 or 130 K.

Figure 4

Details of the THz spectra around bands 1 and 2. Left: relative absorption measured for two different orientations of e and h on the same sample. Below $T_{\text{SR}}$, two new excitations appear abruptly: band 1 for $\mathbf{h}\perp \mathbf{c}$ and band 2 for $\mathbf{h}\parallel \mathbf{c}$. Middle: results of the Gaussian fits for the THz absorption peak position as a function of temperature for mode 1 in red and mode 2 in black (closed and open symbols refer to $\mathbf{h}\perp \mathbf{c}$ and $\mathbf{h}\parallel \mathbf{c}$, respectively). The results from neutron measurements are also reported as open/closed star symbols for the peak positions at the $(1,0,0)/(1,0,1)$ reciprocal space position. The gray boxes correspond to the THz measurements cutoff. Right: results of Gaussian fits for the THz absorption peaks with the area and width as a function of temperature.

Figure 5

Crystal-field scheme and selection rules of the parameter set 1 for the two different holmium sites with a tentative matching of the calculated CF transitions with the observed THz bands: Two transitions are expected for site $4b$ corresponding to band 2 and 3 and three for site $2a$ corresponding to bands 4, 5, and 6. The selection rules and the symmetry of the levels are indicated below.

Figure 6

(a)–(d) Calculated CF THz absorption spectra for both orientations of the THz magnetic field h and for Ho sites $4b$ (blue) and $2a$ (yellow), using parameter set 1 at 40 K, and zero magnetic field (a) at 6 K and (b) at 6 K with a 2 T magnetic field parallel to the $c$-axis. The dashed line and the two gray arrows for $\mathbf{h}\parallel \mathbf{c}$ indicate the splitting of the lowest energy Ho $4b$ CF excitation under a magnetic field. Panel (d) corresponds to the calculations performed using the parameter set 2 at 6 K and under a magnetic field of 2 T along the $c$-axis with a tentative matching of the calculations with the measured bands of excitations. (e) Measured THz spectra at 40 and 6 K for comparison.

Figure 7

(a) Mn spin-wave calculations along $(h,0,0)$ for spin fluctuations within the $ab$-plane (upper panel) and along the $c$-axis (lower panel). The calculated CF levels are added for the Ho $4b$ site (red and blue lines) as well as the Ho $2a$ site (yellow and green lines). The coupling of these CF excitations with the Mn magnons is always allowed by symmetry for all but the dashed line.

Figure 8

Spin dynamics in ${\mathrm{HoMnO}}_3$. (a) ${\mathrm{Mn}}^{3+}$ spin waves and ${\mathrm{Ho}}^{3+}$ CF excitations measured by INS along the $(h,0,1)$ reciprocal space direction. These data have been taken on IN8 at 2 K with a medium energy resolution of about 0.8 meV. $M$ and $M^{\prime }$ denote the Mn dispersive spin waves that can be seen at the zone center. Labels 1–6 refer to nondispersive Ho CF lines. (b) Comparison with spin-wave calculations using the model described in the text for the ${\mathrm{\Gamma }}_{1,3}$ configurations of the Mn magnetic structure. The following values of the parameters have been used: $J=2.5$ meV, $D_z=0.65$ meV, $K_1=0.005$ meV, $K_2=0.008$ meV, and $D_x=D_y=0.0025$ meV. (c),(d) Spin-wave calculations along $(1,0,\ell )$ for the four different possible ${\mathrm{\Gamma }}_i$ (among which are two homometric pairs with $x_{\text{Mn}}=1/3)$. The calculations were performed with the same parameters for the Hamiltonian as for panel (b) except for the hierarchy of $K_1$ and $K_2$ on the one hand and $D_x$ and $D_y$ on the other hand according to Table 2. (e,f) Spin-wave calculations along $(h,0,0)$ in the ${\mathrm{\Gamma }}_{1,3}$ configurations for spin fluctuations, respectively, along the $c$-axis (e) and within the basal hexagonal plane (f). The color vertical bars give the neutron counts in arbitrary units for all the panels. All the spin-wave calculations have been performed using the SpinWave software [45]. Note that in all the calculations of panels (b)–(f), modest but finite values of $D_x$ and $D_y$ were taken into account to ensure the stability of the various ${\mathrm{\Gamma }}_i$; see Table 2. This is responsible for the rise of the small gaps at the zone centers.

Figure 9

Temperature dependence of the ordered magnetic moments Ho $4b$ extracted from the refinement of the neutron-diffraction data (green dots) compared to the calculated magnetic moment (blue line) induced by a constant magnetic field of 2 T due to the coupling with the Mn magnetism. The red dots show the Mn ordered moment also extracted from the neutron-diffraction data.

Figure 10

Details of the THz spectra around bands 3 and 4. Left: relative absorption measured for two different orientations of e and h on the same sample. Right: results of Gaussian fits for the absorption peaks with the position, area, and width as a function of temperature. A maximum of three peaks was used. Closed and open symbols refer to $\mathbf{h}\perp \mathbf{c}$ and $\mathbf{h}\parallel \mathbf{c}$, respectively. Note the complexity of the spectra with several components that are all affected by $T_{\text{SR}}$.

Figure 11

Details of the THz spectra around the magnon excited for a THz h field perpendicular to the $c$-axis. Left: Absorption measured at different temperatures for two different directions of the THz e field on two different samples. Right: results of the Gaussian fits of the absorption peak with the position, area, and width as a function of temperature obtained for $\mathbf{e}\parallel \mathbf{c}$ (with similar results for $\mathbf{e}\perp \mathbf{c})$. Note the emergence of the magnon $M$ below $T_N$ and how it is affected by $T_{\text{SR}}$.

Figure 12

Details of the THz spectra around band 5 when $\mathbf{h}\parallel \mathbf{c}$. Left: absolute and relative absorption measured at different temperatures. Right: results of Gaussian fits for the absorption peaks with the position, area, and width as a function of temperature. Two peaks are present. The lower-energy peak appears smoothly when lowering the temperature and can be fitted below 50 K. The higher-energy one, the stronger, is affected at $T_{\text{SR}}$ with a small shift in energy and increased width and area.

Figure 13

Details of the THz spectra around band 6 when $\mathbf{h}\perp \mathbf{c}$. Left: absolute and relative absorption measured at different temperatures. Right: results of Gaussian fits for the absorption peaks with the position, area, and width as a function of temperature. Two peaks are present that are both affected at $T_{\text{SR}}$.