Antiferromagnetic resonance in $\alpha$-Fe$_2$O$_3$ up to its N\'eel temperature

Hematite ($\alpha$-Fe$_2$O$_3$) is an antiferromagnetic material with a very low spin damping and high N\'eel temperature. The temperature dependence of the antiferromagnetic resonance in a bulk single crystal of hematite was characterized from room temperature up to the N\'eel temperature in the frequency range of 0.19-0.5 THz. From these data, the N\'eel temperature was estimated as 966 K.

Hematite (α-Fe 2 O 3 ) is an antiferromagnetic material with a very low spin damping and high Néel temperature. The temperature dependence of the antiferromagnetic resonance in a bulk single crystal of hematite was characterized from room temperature up to the Néel temperature in the frequency range of 0.19-0.5 THz. From these data, the Néel temperature was estimated as 966 K.
Hematite crystallizes in an approximately hexagonal structure with space group R3c. Precise measurements show that the actual symmetry is monoclinic C2/c 15 . In the weak ferromagnetic state the ferromagnetic moment m is along a axis 16,17 (magnetic symmetry C2'/c' 15 ) or b axis 2,16 (magnetic symmetry C2/c) 1,15 ). Owing to the spin canting, the antiferromagnetic resonance has two modes: at a higher frequency, the quasiantiferromagnetic resonance (qAFMR) mode which is excited when the dynamical magnetic field h is parallel to the magnetization (h m), and at low frequencies, the quasiferromagnetic resonance (qFMR) mode, excited by a dynamical magnetic field perpendicular to the magnetization (h ⊥ m). In this work, we characterize the qAFMR mode up to the Néel temperature, which was observed previously up to about 600 K 2 and below room temperature 14 .
The qFMR mode has non-zero frequency only with external field applied 18,19 and falls below our experimental range. To our knowledge, the qFMR mode was investigated only at room temperature and below [20][21][22] .
Development of frequency extenders for vector network analyzers (VNA) allows continuous-wave spectroscopic measurements up to 1.5 THz to be conducted with a high frequency resolution and with a very high dynamic range 12,23-26 . Our sample was a natural single crystal of α-Fe 2 O 3 of d = 0.5 mm in thickness and 10×10 mm 2 in lateral dimensions. The normal to the sample surface was (11)(12)(13)(14)(15)(16)(17)(18)(19)(20).  Figure 1 shows a schematic of our THz setup, where terahertz radiation generated by the source extender was guided to the sample using oversized metallic waveguides. Transmitted radiation was guided from the sample to the detector using identical oversized metallic waveguides. Detector measured incoming power and phase of the electric field. Temperature scans started from the highest temperature with a step of ∆T = −0.25 K. We chose ∆T so that change of the resonance frequency with temperature is smaller than the line width of the resonance, which is particularly narrow at lower temperatures. After stabilizing sample temperature T with a PID controller, we measured transmission as a function of radiation frequency f . The recorded spectra were obtained by averaging 20 frequency sweeps measured with 100 Hz bandwidth of extenders intermediate frequency bandwidth. This procedure was applied at two frequency bands of 0.19-0.35 THz and 0.33-0.5 THz using different sets of frequency-extenders. We recorded relative transmitted radiation power in dB and the phase of the transmitted electric field in deg units. This way, a complex transmission data matrix S 21 (f, T ) was obtained.
The source extender emits linearly polarized beam of THz radiation and the detector extender detects only radiation of a linear polarization that matches its waveguide. Rotation of the polarization plane about the optical axis is possible by rotating the source and detector about the optical axis. The data presented here, were obtained with parallel source and detector polarizations.
Raw spectra are dominated by interference occurring  between components of the experimental setup. Since this pattern very weakly depends on temperature, we eliminated it by calculating temperature-derivative spectra, that is, by subtracting successive spectra from one another. We calculated magnitude temperaturederivative spectra and phase derivative: (2) Our main results are presented in Fig. 2a (magnitude) and Fig. 2c (phase). They were obtained with the THz magnetic field acting in the basal plane of the crystal, as determined from polarization-dependent results in Fig.  3. At a perpendicular polarization, with THz magnetic field along c axis, the resonance is almost not excited. We can notice the the resonance peak shows a tiny splitting of about 1 GHz. Result of angular dependence obtained with a c-cut sample shows a more complex dependence on polarization angle that is different for a higher and a lower modes of the qAFMR (green circles and blue rectangles in Fig. 3b). This angular dependence suggests that the observed splitting is related to different magnetic domains in our samples.
We observed a strong and narrow resonance ( Fig. 2b  and c), the frequency of which is rising with temperature at above room temperature (Fig. 2a), reaching a maximum of about 484 GHz at about 685 K and then dropping sharply as T N is approached. The periodic, almost horizontal pattern is related to the Fabry-Pérot type cavity modes inside the sample. We can account for these temperature-differential spectra, as shown in Fig. 2 (b,  d), by using an electrodynamics-based model 26 : where t is the transmittance of a plane electromagnetic wave at normal-incident on a parallel-plane slab of infinite lateral dimensions and thickness d Here, r = ( We assume an isotropic slab material characterized by its permeability µ(f, T ) and permittivity (f, T ). Our crystal is not isotropic, but in the case of incidence at a normal angle to the crystal surface, the crystal being cut in one of its principal axis, we approximate it as isotropic, that is, we assume that there is no rotation of polarization in the crystal. At other angles of incident beam polarization, we get different values of . However, as long as we were interested in the behaviour of the antiferromagnetic resonance, we determined permittivity at the polarization angle where the resonance is excited the strongest. Since this occures when the dynamical magnetic field is in the basal plane, the that we determined in this communication describes response of hematite when exposed to dynamical electric field acting in the c axis. The assumptions made for both µ and are the following. We found that our data are sensitive to a temperature dependence of the dielectric response function. The simplest model that accounts for our data is a real second order polynomial of temperature. This approximation is justified because our frequency range is far from optical phonons (the lowest at about 7.5 THz) [27][28][29] and dielectric absorption is negligible. Thus, we assumed where T 0 = 900 K, 900 = 23.405, a = 1.139 · 10 −2 K −1 , b = 6.434 · 10 −6 K −2 . To account for the magnetic resonance, we assumed that the permittivity has the form of two Lorentzian functions to account for the observed splitting of the qAFMR mode Here, ∆µ 1 + ∆µ 2 = ∆µ gives the total static magnetic susceptibility, f r is the lower qAFMR frequency, g 1 and g 2 are widths of the two split resonances and ∆f is the splitting of about 1 GHz. In the next paragraphs we describe these parameters. In order to describe the temperature dependence of the antiferromagnetic resonance, we first fitted the resonance frequency, amplitude, and line widths in temperature intervals of 2 K around some temperature T (l) 0 , where l is the interval number. Within each such interval, we put amplitudes and widths independent of temperature and we fit the resonance frequency with a first order poly- T (T − T 0). Results of these temperature interval fittings are shown as green crosses in Fig. 2(a,b,c). We show in Fig. 4

(a) middle frequency f (l)
T 0 obtained in given temperature interval. These results allowed us to assume some simple functional dependences that are approximately valid in the entire experimental temperature range. The global fitting result, using the above assumptions, are displayed in Fig.2(b) and (d). The fit parameter values are given in Tab. I. The functions used to describe modes frequencies, amplitudes and widths are drawn as solid yellow lines in Fig. 4. To describe the observed temperature dependence of the resonant frequency (Fig. 4a), we took the dependence of the resonant frequency to follow a modified power law 30 where T N is a parameter close to the Néel temperature and the power factor β is expected to be about 1/2. We modified the classical power law adding a 5-th order polynomial that lets us heuristically describe its nontrivial dependence on temperature. This non-monotonic dependence of resonance frequency on temperature is caused by the strong temperature dependence of the magnetic anisotropy field 31 that governs the frequency of this mode. It is related to the spin-reorientation transition at about 260 K 2 , with one of its effects being that the qAFMR mode frequency has a minimum at this temperature. The fit value of T N ≈ 966 K is within the spread of literature values of T N = (946 − 972) K 1,3-5 . These inconsistencies in the literature might stem from errors in temperature measurement in different experiments. However, it may be also due to different models used to determine T N , when it is determined by extrapolation.
We assumed that the observed splitting (Fig. 4b) can be described in the following way which express our expectation that the splitting is caused by presence of two types of domains characterized by slightly different anisotropy fields (factor δ) and slightly different critical temperatures (factor δT N ). The second term dominates at high temperatures. Inhomogeneous nature of this splitting is justified by the measurements for 0001 cut sample (Fig. 3b) that show that upper and lower modes have different dependence on polarization angle. The observed temperature dependence of the splitting can be well fitted (Fig. 4b) with the Eq. 8 using reasonable values of δ and δT N (Tab. I).
Taking into account the scatter in Fig. 4d, we took amplitudes of the resonance ∆µ 1 = ∆µ 2 = 1 2 ∆µ to be independent of temperature. This scatter is probably due to imperfect interval fits, especially pronounced at around 700 K and close to T N , owing to low signal obtained by temperature-differential transmission in these temperature ranges.
We assumed that the temperature dependence of the observed width (Fig. 4c) of both modes is described by the same function consisting of a sum of a constant and an exponential, g(T ) = g 400 + g e e gs(T −T N ) , where the latter dominates close to T N . It could be due to either: 1) temperature inhomogenity, 2) inhomogen- The maximum T N + − T N − that is necessary to account for the observed broadening is about 6 K, which is much larger than estimation of δT N = 1.2 K obtained from the splitting of the resonance. Therefore, we conclude that the observed broadening with temperature is mainly due to rising spin damping.
We notice that the linewidth drops strongly around 620 K and 740 K (two arrows in Fig. 4c). Since at these two temperatures, the frequency of the resonance is about 475 GHz, we can guess that this effect is due to interaction of the magnetic resonance with either some electromagnetic mode in the setup or some electric transition in the crystal that has a temperature-independent frequency.
There are features in Fig. 2 that are possibly caused by interaction with cavity modes than cannot be taken into account within our one-dimensional model. The effects of some of such modes are shown with blue arrows in the inset of Fig. 2a and a particular example of a lineshape distortion is shown in Fig. 5 at T = 346 and 354 K. We can clearly see that, at different temperatures, the lineshape changes and some of these features are not reproduced by our global one-dimensional model. Blue arrows in Fig. 2a show that these are indeed related to modes of cavities that cannot be taken into account by the global model, which show-up as almost temperatureindepedent horizontal lines distorting the resonance in the inset of Fig. 2a that are not reproduced by the fit in inset of Fig. 2b. These distortions cause apparent fluctuations of width and amplitude of the resonance line in the obtained temperature-interval fit values.
We observed weak interaction of magnetic resonance with electromagnetic standing waves, as found previously 26 . These effects are readily taken into account by a classical electrodynamic model. Light-matter interaction when the sample is its own cavity is in a weak regime, because the quality factor of modes of dielectric cavity is low. Nevertheless, these electromagnetic cavity modes already have a dramatic and nontrivial effect on the observed lineshapes, as shown in Fig. 4b and Fig. 4c.
In Fig. 4e we also included the magnetic anisotropy factor, which we deduced from equation K = µ 0 H A M 0 , where H A is anisotropy field and M 0 is a saturation magnetisation of a sublattice. To do so, we took into account that the static magnetic susceptibility is given by where H E is exchange field, and that χ ⊥ = ∆µ 2 /ρ m , where ρ m = 5.27 · 10 3 kg/m 3 is the hematite mass density 32 . Then, using the Kittel's equation we found the mass density of the magnetic anisotropy energy. We see in Fig. 4e that K has a complex behaviour, which is known to be caused by the spin-reorientation transition that occurs at about 260 K. Hematite single crystals present a very narrow resonance around room temperature, about 1 GHz. The resonance frequency increases sharply above room temperature and reaches a maximum of 484 GHz at about 685 K and then, the mode softens steeply when approaching the Néel temperature T N . Above 690 K, the observed width of the resonance rises exponentially with temperature. Weak light-matter coupling was observed between the qAFMR mode and the standing waves inside the slab itself. The qAFMR mode presents a splitting that was ascribed to domains in the sample. 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.
The authors have no conflicts to disclose. The data that support the findings of this study are available from the corresponding author upon reasonable request.