Antiferromagnetic resonance in TmFeO3 at high temperatures

Temperature dependence of the antiferromagnetic resonance spectrum in a thulium orthoferrite (TmFeO3) poly-crystalline sample was studied by transmission spectroscopy in the frequency range of 0.1–0.75 THz and temperature range of 300–670 K, up to its Néel temperature. We observed quasi-ferromagnetic resonance and the quasi-antiferromagnetic resonance modes. The temperature dependence of the resonance frequencies, linewidths, and amplitudes were extracted. The resonance spectrum of TmFeO3 near its Néel temperature was directly observed and a drastic drop in resonance frequency for both modes was observed.

Thulium ferrite crystallizes in orthorhombic Pbnm space group. Below the Néel temperature T N = 635 K [36] Fe 3+ ions orient in a Gtype antiferromagnetic order. Superexchange Dzialoshinskii-Moriya interaction leads to canting of neighbouring antiferromagnetically oriented spins, that gives a net magnetisation m, making this material a weak ferromagnet. Above the spin-reorientation transition at about 90 K, the symmetry of the magnetic system is that of the Γ 4 phase, with antiferromagnetic vector G laying along a axis and m in c axis [1]. Due to the spin canting, the antiferromagnetic resonance has two modes, the quasi-antiferromagnetic resonance (qAFMR) mode at higher frequency, exited when a dynamical magnetic field h is parallel to the magnetization (h‖m), and the quasi-ferromagnetic resonance (qFMR) mode at lower frequency, excited when h⊥m.

Samples and experiment
Thanks to the development of frequency extenders to vector network analyzers (VNA), continuous-wave spectroscopic measurements up to 1.5 THz can be rapidly conducted with a high frequency resolution and with a very high dynamic range [37][38][39][40]. We investigated a pelletized powder sample of TmFeO 3 of d = 1.28 mm in thickness and 15 mm in lateral diameter. TmFeO 3 was produced from Tm 2 O 3 and Fe 2 O 3 . The two powders were mixed in stoichiometric amount and pressed into a pellet. The pellet was annealed for a week in air at 1200 • C with intermediate grinding. X-ray diffraction experiment confirmed that our sample is thulium ferrite [41] with a precision on the chemical composition of about 1%. During the THz transmission experiment, the pellet was placed between two oversized cylindrical metallic waveguides of 11 mm in inner diameter. These were fixed inside a cylindrical ceramic furnace under PID-controlled temperature measured with a Ktype thermocouple positioned close to the sample. The terahertz radiation generated by the source frequency-extender passed the sample and was detected by the detector frequency-extender, as illustrated in Fig. 2. After stabilizing temperature T of the sample, we obtained a transmission spectrum as a function of radiation frequency f. Recorded spectra were obtained by averaging 20 single spectra measured with an intermediate frequency bandwidth of 100 Hz. The procedure was repeated separately in frequency bands of 100-170 GHz, 200-350 GHz, 330-500 GHz, 480-750 GHz using different sets of frequency-extenders. We recorded the relative power of transmitted radiation in units of dB and the phase of transmitted electric field in units of degrees. Thus, a complex transmission data matrix S 21 (f, T) was obtained.

Results and analysis
In each frequency band, we changed temperature starting from the highest temperature with a step of ΔT = − 1 K. The data displayed in Fig. 3(a,c) are temperature derivative spectra obtained by subtracting amplitude of two neighbouring spectra and for the phase Our result shows two strong resonances softening with rising temperature when approaching T N . The signal from the lower-frequency mode (qFMR) disappears near room temperature. Actually, the qFMR resonance mode is still present, it does not show up in temperaturedifferential spectra because it is almost temperature-independent in the temperature range of 300-400 K. Periodical horizontal modes are Fabry-Pérot type cavity modes of the sample related to its thickness. Features of alternating amplitude at 0.56 THz are related to changes in resonant absorption by water vapor. In Fig. 3(b,d), we show a fit of temperature-differential spectra using an electrodynamics-based model [40]: where t is transmittance of a plane electromagnetic wave normalincident on a parallel-plane slab of infinite lateral dimensions and thickness d:   Table 1.
with an isotropic slab material described by permeability μ(f, T) and permittivity ∊(f, T).
The assumptions that were used to obtain the fit in Fig. 3(b,d) are explained in following paragraphs. Because the sample is composed of randomly oriented crystal grains, both the modes can be excited at any direction of incident wave polarization. Also, because the crystal grains fill only p = 0.64 of the sample volume [42], we assume effective dielectric function [43,44] ̅̅̅ where ∊ c and μ c are mean crystalline permittivity and permeability, respectively. In order to carry out a quantitative analysis of the observed magnetic resonances, we fitted the amplitude of temperature-differential spectra with Eq. (3), assuming heuristic material properties of permittivity and permeability. In our model, ∊ is described by a single oscillator, that is a valid approximation because our frequency range is far from optical phonons [45]. We assumed that the oscillator strength in ∊ is weakly dependent on temperature: where where j = 1 stands for the qFMR and j = 2 stands for the qAFMR, Δμ j is a response to a static magnetic field, f j are modes frequencies and g j modes widths.
In order to better describe the temperature dependence of the antiferromagnetic resonance, we first fitted the resonance frequency, amplitude, and linewidth of the modes in temperature intervals, 10 K in width. Within each such interval, we put amplitudes and widths constant with temperature and we assumed that frequencies of the resonances are described by a second order polynomial. Results of these temperature interval fittings are shown as points in Fig. 4(a,b). We show in Fig. 4(a) middle frequencies of modes obtained in each temperature interval. Theses results allow us to assume some simple functional dependences that are valid in the entire experimental temperature range. First, we assumed that the amplitudes of the resonances Δμ j are constant with temperature. Second, we assumed a modified power law [46]: where f (0) j has frequency units, T N is a parameter close to the Néel temperature and a power factor expected to be β j ≈ 1/3. We modified the classical power law by putting κ j ∕ = 0. Finally, we assumed that temperature dependence of widths of modes is described by a sum of a linear and an exponential functions where the first therm is dominant at lower temperatures and the latter dominates close to T N . We comment on these assumptions in the next section.

Discussion
The global fitting result, using the above assumptions, are displayed in Fig. 3(b) and (d). The fit parameter values are given in Table 1. Functions used to describe modes frequencies, amplitudes and widths are drawn as solid lines in Fig. 4. Parameters in Table 1 give good estimations of properties of magnetic resonance only in the experimental ranges of 300-650 K and 0.1-0.7 THz. As displayed in Fig. 4(a), the frequencies of both modes decrease with temperature in the temperature range from 300 K to T N . The frequency of the qFMR mode is almost constant near room temperature, which can be clearly observed in the differential spectra as displayed in Fig. 3, in which the resonance can be barely seen. Indeed, fit parameters of κ 1 and κ 2 show that the modification of power law behaviour is only Fig. 4. The temperature dependence of the resonance in frequency (a), amplitude (b) and width (c). Points come from 10 K intervals fits. The solid lines are calculated using Eq. (7)-(8) with parameters from Table 1. The qFMR is marked purple and the qAFMR is marked green. In (a), orange points come from Ref. [1]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) necessary in the case of the qFMR. This is related to a temperature dependence of the magnetic anisotropy field in TmFeO 3 that governs the frequency of this mode. This antiferromagnet undergoes spinreorientation transition at about 90 K, where the qFMR frequency drops nearly to zero. The actual values of β j (Table 1) depart from the expected critical exponent of 1/3, most likely because the fits are obtained for a relatively large temperature range below T N . Fit values of T N ≈ 639 K for both modes agree with each other within estimated uncertainties. The obtained value is slightly higher than in the literature (T N = 635 K) [36]. This discrepancy might come from a precision of the temperature measurement in our system. As seen in Fig. 4(a), our points are at higher frequencies and higher temperatures than those of Ref. [1]. It is doubtful that this could be a result of an error in frequency measurement, since both experiments used monochromatic radiation sources of very precise frequency. An error in temperature measurement would have to be as large as about 20 K at around T = 500 K, that would lead to a giant discrepancy in the estimated T N . Thus, the difference between our results and those of Ref. [1] must come from an actual difference between measured samples. From Fig. 4(c), we see that the width of the qFMR mode (purple) experiences a slight linear decrease with temperature, while the width for the qAFMR mode (green) first experiences a linear increase. Our results are 2-3 times larger than values reported in the literature for a single crystal sample [1], that is possibly because of the poly-crystalline nature of our sample. Exponential growth of the linewidth close to the T N is probably due to sample temperature inhomogenity or inhomogenity of T N in the sample. We can estimate the distribution of temperatures that could cause such a broadening by solving Fig. 4(b), the amplitudes of both modes are constant. The amplitude of the qFMR is about twice of that of the qAFMR. This is expected, since, in our poly-crystalline sample, scalar μ c is an average of the permeability tensor, where the qFMR has almost equal amplitudes in μ xx and μ yy , while the qAFMR is present only in μ zz . Off-diagonal components of the permeability tensor are antisymmetric and they cancel each other in the average. Obtained amplitudes are close to those in Ref. [1], taking into account the poly-crystalline nature of our sample, in so that we observe about 1/3 of the qAFMR amplitude (9.0⋅10 − 4 ) and 2/ 3 of the qFMR amplitude (8.5⋅10 − 4 ).
In Fig. 5, we show measured spectra at three different temperatures, T = 377 K, T = 558 K, T = 573 K, as example of our analysis. We can clearly see at different temperatures, resonance frequencies and lineshapes experience changes. We observed also effects of interaction of magnetic resonance with electromagnetic standing waves in the sample slab [14,40], that are readily taken into account by electrodynamics calculations. Light-matter interaction in our case is in a weak regime, because of low quality factors of cavity modes in the sample-slab. Nevertheless, the electromagnetic cavity modes have a dramatic and nontrivial effect on observed lineshapes, as shown in Fig. 5b and c.
In Fig. 6 we show magnetic anisotropy factors K = μ 0 H A M 0 , where H A is anisotropy field and M 0 is saturation magnetisation of a sublattice.
Taking that the static magnetic susceptibility χ ⊥ = M 0 /2H E , where H E is exchange field, and that χ ⊥ = 3Δμ 2 /ρ, where ρ = 8⋅10 3 kg/m 3 is TmFeO 3 density and the factor 3 comes from the fact the we measured about 1/3 of the qAFMR amplitude in our poly-crystalline sample. Then, solving the simplified Kittel's equation we find anisotropy energy density for each mode. The magnetic phase of TmFeO 3 is Γ 4 , thus, for the qFMR mode K 1 is an anisotropy factor in the ac plane and for the case of the qAFMR mode the calculated K 2 anisotropy factor is for the ab plane [1]. We see that K 2 has almost a linear dependence on temperature, whereas K 1 has a more complex behaviour due to the spin-reorientation transition at about 90 K. These values agree in order of magnitude with the values from Ref. [1]. Jumps of K 1 and K 2 at T = 460 K are artifacts caused by the qAFMR passing through the water vapor transmission line at 0.56 THz that was changing its amplitude during the measurement. In Fig. 7, we show experimental data in an oversaturated scale that shows a very narrow feature, the frequency of which is growing with temperature from 0.2 THz at about 320 K to about 0.48 THz at 650 K. We identified this feature as the qAFMR of hematite (α-Fe 2 O 3 ) [47]. From its amplitude (≈ 0.04 dB/K), compared to the amplitude measured in a 0.5mm-thick single crystal hematite (≈ 30 dB/K), we can estimate that the amount of hematite impurity in our sample was about 0.2%. That is lower than the accuracy of the X-ray diffraction experiment (Fig. 1).  Fig. 4(a,b) and lines where obtained from fits presented in Fig. 4(a,b) using Eqs. (9) and (7) with parameters from Table 1.

Summary
In summary, we measured the temperature dependence of the antiferromagnetic resonance in TmFeO 3 poly-crystalline sample at temperatures up to 650 K, that is over its Néel temperature (T N ). Two antiferromagnetic resonance modes, the quasi-ferromagnetic and the quasi-antiferromagnetic were observed. When temperature is approaching the T N , both resonance modes experience sharp drop in frequency. The resonance frequency, resonance width and amplitudes were extracted and fitted with an electrodynamic model. The qFMR mode resonance is nearly temperature-independent near room temperature, which may be an advantage for its use in information processing. The amplitudes of both resonance modes stay constant. The widths of both modes change linearly with temperature, at temperatures far below T N , and both increase sharply near T N , possibly because of inhomogeneous broadening. Using our fits we have obtained the estimation of magnetic anisotropy factors in TmFeO 3 . Our quantitative analysis shows that light-matter coupling plays crucial role in observed lineshape. We show that our technique can detect contamination with hematite that are smaller than those detected by X-ray spectroscopy.
The data that support the findings of this study are available from the corresponding author upon reasonable request. Support by the Sino-Swiss Science and Technology Cooperation (SSSTC) Grant No. EG-CN_02_032019 is gratefully acknowledged. The VNA and frequency extenders were funded by EPFL and the SNF R'Equip under Grant No. 206021_144983.