Spin-wave coupling to electromagnetic cavity fields in dysposium ferrite

Coupling of spin-waves with electromagnetic cavity field is demonstrated in an antiferromagnet, dysprosium ferrite (DyFeO3). By measuring transmission at 0.2-0.35 THz and sweeping sample temperature, magnon-photon coupling signatures were found at crossings of spin-wave resonances with Fabry-Perot cavity modes formed in samples. The obtained spectra are explained in terms of classical electrodynamics and a microscopic model.


INTRODUCTION
Light-matter coupling [1] is a topic of great interest in solid state physics research because of their hybrid quantum nature [2]. In the THz range, phonon-polaritons are a well-know example of a light-matter coupling. Recently, polaritons in the THz region were shown in twodimensional electron gases with intersubband transitions [3] and with plasmons [4]. In these systems, a strongcoupling regime can be achieved when losses are smaller than the exchange rate between light and matter [5], giving rise to the vacuum Rabi splitting. Polaritons are composite particles, which are studied in basic research on quantum optics [2], and can be considered for use in quantum computing and quantum memories [6][7][8][9]. The coupling of electromagnetic cavity-modes to magnons has been investigated in ferromagnets at GHz frequencies [6,10,11] as well as coupling of magnons with superconducting qubits [12,13]. The Purcell enhancement and the vacuum Rabi splitting were demonstrated in ferromagnetic materials [14][15][16][17].
It is interesting to investigate magnon-photon coupling in antiferromagnetic materials [1,18] in view of their high frequency dynamics. However, we are only aware of one experimental report [19] on bulk magnon-polaritons in thulium ferrite (TmFeO 3 ). In this letter, we show an interaction of antiferromagnetic magnons with electromagnetic cavity modes of a sample shaped as a slab. By sweeping temperature, we change the magnetic resonance frequency, while frequency of electromagnetic cavity mode is almost constant. As a consequence, we observe avoided crossings of spin-wave modes and cavity modes, a behavior which is characteristic of polaritons. SAMPLES We report on the magnon-phonon coupling in dysprosium ferrite DyFeO 3 (DFO), an orthogonally-distorted perovskite. It is an antiferromagnet with a Néel temperature of about 640 K [20,21], and with a strong Dzyaloshinskii-Moriya interaction leading to a weak fer-romagnetism. Symmetry breaking due to the spin canting allows two antiferromagnetic resonance modes to be excited: the quasi-ferromagnetic and the quasiantiferromagnetic [20][21][22][23], which both excited by the magnetic component of radiation [24].
At around 60 K the material passes the spin-reorientation transition, which causes a strong softening of the quasiantiferrromagnetic mode [21,22,25]. Single crystals of DFO show anitferromagnetic ordering of Dy atoms below 4 K, which allows a magnetoelectric coupling [26]. Our measurements were performed above room temperature, where the quasi-ferromagnetic resonance has lower frequency than the quasi-antiferromagnetic resonance and both resonances monotonously soften with rising temperature.
We report on investigations of polycrystaline samples of two different thicknesses, thus having different cavitymode spectra. Both samples where first produced in larger pressed powder pellets. To obtain a specific thickness we used a diamond wire saw. Finally, samples were cut into disks of 8 mm diameter with a Nd:Yag 1064 nm laser. We estimated thicknesses with a confocal microscope by measuring mean heights of samples positioned on a glass substrate. These heights and their standard deviations are (1037±27) µm and (638±18) µm. Since both surfaces show some roughness, as indicated above, we assumed that thicknesses of samples are measured heights minus twice their standard deviations. This assumption allowed us to obtain a good agreement between our dielectric-constant fits for different samples and the literature values [23]. EXPERIMENT A sample was placed in an enlongated-cylinder ceramic furnace, which allowed to control the sample temperature up to 500 • C. Temperature was measured with two thermocouples placed close to the sample, on either sides and on its opposite corners. Thus, we can estimate a maximum systematic error of our temperature measurements to be 10 K at our highest temperature. The furnace power supply was PID-controlled. The THz radiation was guided to the sample using two oversized metallic waveguides of 8 mm diameter, which were placed inside the furnace. The sample was located in a magnetic field generated by water-cooled split-coils. The strength of the magnetic induction field was controlled by a computercontrolled power supply (Fig. 1).
Our THz spectrometer uses frequency extenders to a vector network analyzer (VNA) [27]. The sourceextender produces, with help of a corrugated hornantenna, an approximately Gaussian beam of THz radiation. The detector-extender measures the amplitude and the phase of the THz electric field coupled to its waveguide. This complex signal S 21 (f, T, H) is a function of frequency f , temperature T and magnetic field H. We report its amplitude in dB units and its phase in degrees. This transmitted signal, in the case of our samples, was mostly governed by absorption. The incoming radiation was coupled to the detector using a corrugated horn-antenna, identical with that at the source.
In order to extract a signal related to a magnetic resonance, we used two methods. We measured transmission through a sample, while either sweeping temperature T or modulating magnetic-field H. The first method allows us to obtain temperature-differential spectra by subtracting averaged spectra measured at subsequent temperatures, with ∆T = 1 K. This technique was used also to measure spin-wave resonances in bismuth ferrite at high [28] and low temperatures [29]. The second method allows us to obtain magnetic-field-differential spectra, Magnetic-field dependence of antiferromagnetic resonances is rather weak, therefore the signal is much weaker. Thus, here we used a modulation technique, were the field was modulated on-off with am amplitude ∆H = 1.38 k0e and a period of the order of 20 sec. During one period, two spectra were measured, one with field and the other without. Some fraction of a second was used to wait for the magnetic field to stabilize after the current was changed. In order to extract the magneticfield-dependent component of the S 21 , we wrote a software calculating fast Fourier transforms (FFT) of signals measured at each THz frequency and comparing these with an FFT of a record of the field modulation. This data treatment may be thought of as resembling the standard lock-in technique, with the biggest difference being that with our software we obtain signals in hundreds of channels simultaneously, were each channel be.
Physically, the most important difference between the two above-described methods is that in temperaturedifferential spectra, an interference pattern in a sample slab is visible because broadband refractive index changes with temperature. In magnetic-field-differential spectra the interference pattern is absent. Furthermore, if a resonance does not change with applied magnetic field, it is absent in magnetic-field-differential spectra and, if a resonance is locally independent of temperature, it is absent in temperature-differential spectra.

MODEL
If a material has a resonance, the refractive index around this resonance causes a change of light dispersion in this medium. This leads to the creation of two polariton states and a gap between them which represents the coupling of the resonance with the electromagnetic radiation. We consider transmission through a slab of thickness d that has infinite lateral dimensions.
We model the magnetic susceptibility µ(f, T, H) of the slab material using a sum of lorentzian distributions where, for the m-th magnetic resonance, f m is its frequency, γ m its width and ∆µ m is its input to the zerofrequency magnetic susceptibility. In the case of DFO, we have M = 2 with m = 1 corresponding to the quasiferromagnetic and m = 2 to the quasi-antiferromagnetic resonance.
The slab material has a dielectric function (f, T ), which in our frequency range is governed by the lowfrequency tails of phonon resonances: where, for the s-th phonon, f s is its frequency, γ s is its width and ∆ s is its input to the zero-frequency dielectric constant. In our experimental frequency range, i.e. far from phonons resonant frequencies, we can use a linear approximation of the above formula around a convenient point where a and b are real. This parametrization of (f, T ) was necessary, because our literature search did not indi-cate any experimental data on phonons central frequencies, widths or amplitudes for dysprosium ferrite. The complex refractive index of a material is n(f, T, H) = p (f, T )µ(f, T, H) + (1 − p), where factor p = 0.64 is related to a mass fraction of material to air in our pelletized samples [29]. This value is obtained from density measurements and is close to the maximum density of random-packed hard spheres [30]. When using this assumption, we obtain a good agreement with the literature values on dielectric function of DFO [23]. The complex wave vector is k 1 = 2πf n/c, where c is the speed of light in vacuum. Transmission of electric field t(f, T, H) in the case of a slab of a thickness d is given by a well-known formula [31]: where ρ 2 = (n − 1) 2 /(n + 1) 2 is the square of the reflection coefficient at the vacuum-material interface. For frequencies far away from resonances, Eq. 6 shows an interference pattern, related to subsequent cavity modes of a slab, called Fabry-Perot modes. However, when a material undergoes a resonance at a frequency close to a mode of a cavity, both the resonant and the cavity lines are altered by the matter-photon interaction. We calculated the function to fit amplitude of temperature-differential spectra, and to analyze amplitude of magnetic-field-differential spectra, where ∆T = 1 K is the temperature step and ∆H is the amplitude of H-field modulation. We calculated phase of temperature-differential spectra using the following equation:

RESULTS
The interference pattern visible in our experimental results reflects the presence of Fabry-Perot cavity modes in the sample slab. At 300 GHz, the wavelength in air is about 1 mm, so that, in our polycrystaline DFO samples with n ≈ 3.3, the half-wavelength is about 150 µm. With rising temperature, the refractive index increases, which shortens the period of the interference pattern. The growth of the refractive index with temperature is due to the softening and broadening of the phonon modes, with center at above 2 THz. As a consequence, the dielectric function depends on temperature and therefore, the interference pattern is also temperature-dependent. This gives rise to a series of minima and maxima in the transmission which dependent weakly on temperature. The magnetic susceptibility takes into account the effect of the magnetic resonances on the transmission. The magnetic resonance frequencies have a much stronger temperature-dependence than the that of the interference pattern. As a result, resonances cross a number of interference pattern maxima as temperature rises.
The experimental temperature-differential signal amplitude for the 1.0-mm-thick DFO sample is shown in Fig.  2a. These data show clearly that the resonant lines are distorted when they cross the sequence of sample-cavity modes. The amplitude and the widths of the resonances are altered because of their interaction with the electromagnetic standing waves. This is accounted for by Eq. 7 as shown in Fig. 2c. Thus, we find that this model reproduces most of the important features of Fig. 2a. The fitting parameters are parameters of the simplified dielectric function (Eq. 5) and of the magnetic susceptibilty (Eq. 3). We assumed that resonance frequencies have a dependence on temperature described by a power law, applicable when approaching the Néel temperature T N [32]: where f * m has a unit of frequency and β m ≈ 1 3 . We also assumed that the magnetic resonances have a linear dependence of widths and amplitudes on temperature, so that for H = 0: ∆µ m (T ) = ∆µ m (1 + u m (T − T We found good agreement between our values of the real part of the dielectric function and the published values [23]. However, our results on the imaginary part are about three times larger than in Ref. 23. Since, we measured polycrystaline sample, this discrepancy might be caused by scattering, which, in our model, is accounted for by the imaginary part of (f, T ). The Néel temperature is close to values obtained using the Mössbauer effect [32] and the Raman spectroscopy [33]. The widths of resonances are 2-3 times larger than those in single crystal samples, while the amplitudes of the magnetic resonances are 2-3 time smaller [23]. This is likely to result from the polycrystaline nature of our samples.
The phase of the obtained fit (Fig. 2d) reveals clearly the interactions between electromagnetic waves and magnetization dynamics. We used this phase prediction to estimate the cavity mode-magnetic resonance coupling strength. Thus, we superimposed on the phase plot  2d the prediction of the harmonic coupling model [1,34] where f ± indicates lower and upper polariton frequencies, f (l) is the l-th cavity mode frequency, f m with m = 1, 2 are the resonances frequencies and κ is the coupling strength. We calculated modes undergoing four in-teractions with the same coupling strength κ. From the low-temperature side, we took the lower polariton between the l-th cavity mode and the quasiferromagnetic resonance (m = 1), then the upper polariton with the (l − 1)-th cavity mode, then the lower polariton with the quasiantiferromagnetic resonance (m = 2) and, finally, the upper polariton mode with the (l − 2)-th cavity modes.The phase maximum of ∂S 21 /∂T is observed under c(l − 1/2) = 2n(f, T )f (l) d, where n(f, T ) is the refractive index as obtained from the fit, which allowed us to calculate f (l) . The lowest visible mode at ≈ 230 GHz has l = 6. Resonant modes frequencies were obtained from the same fit to the experimental ∂S 21 /∂T magnitude. The temperature dependence of both coupled and uncoupled modes is presented in Fig. 2d. We found κ = 0.25 as the best match for our transmission model fit.
Precise measurement of the signal phase is more difficult than measurement of its magnitude since it is more prone to noise and instabilities. Since the 1.0-mm-thick sample gives stronger signal, we were able to measure the temperature-differential phase of transmitted electric field, as presented in Fig. 2b. This result was obtained by averaging 32 spectra. We subtracted a medium phase for each temperature, in order to remove random offsets from our results. Despite high noise, the basic features predicted in the phase plot (Fig. 2d) are observed, including a strong light-matter interaction with avoided crossings, which can be distinguished in the center of the figure at T = 540 K, f = 270 GHz (Fig. 2b). For the 0.6-mm-thick sample, the temperaturedifferential results are presented in Fig. 3a. Because of the smaller thickness, the period of the interference is larger. We observe distortions of the magnetic resonance lines at the crossings with the interference pattern. The fit to these data is presented in Fig. 3c. With this sample, we show an example of spectrum obtained by magnetic field modulation (Fig. 3b). It is clear that the resonant lines oscillate while crossing subsequent cavity modes, even though the interference pattern is hard to distinguish in the background. To explain these magneticfield-differential spectra we assumed H-field linear corrections to existing fit to temperature-differential data. This is justified because for an antiferromagnetic material, a field of ∆H = 1.38 kOe can be considered small. Therefore, γ m (T, H) = γ m (T )(1+g H H) with m = 1 for the quasiferromagnetic resonance and m = 2 for the quasiantiferromagnetic resonance. The resulting plot is presented in Fig. 3d for the 0.6-mm-thick sample, while fit parameters for both samples are listed in Tab. II. We found that, in our polycrystaline DFO samples, the magnetic-field-differential spectra are mostly governed by changes of widths of resonances (g  (Fig. 3c). SUMMARY We have shown magnon-photon coupling in hightemperature antiferromagnets. We observe evidence of the coupling of the THz-frequency magnetic resonance to the standing modes of the optical resonator formed by the sample itself. As in other polaritonic systems, we can observe avoided crossings depending on the difference between the cavity mode and the magnetic resonance frequency. Our research shows new perspectives for spin-cavitronics based on the high-frequency dynamics of antiferrromagnets . Experimental spectra for the 0.6 µm-thick sample: (a) amplitude of the temperature-differential signal (b) amplitude of magnetic-field-differential signal. Fits of the model to the data: (c) temperature-differential (d) magnetic-field differential.