Spin-wave coupling to electromagnetic cavity fields in dysposium ferrite

TL;DR

This paper demonstrates the coupling of spin-waves with electromagnetic cavity fields in dysprosium ferrite, revealing magnon-photon interactions through transmission measurements and theoretical modeling.

Contribution

It presents the first experimental observation of spin-wave and electromagnetic cavity field coupling in an antiferromagnet, supported by classical and microscopic models.

Findings

Magnon-photon coupling signatures observed at resonance crossings

Spectra explained by classical electrodynamics and microscopic theory

Coupling demonstrated in the 0.2-0.35 THz frequency range

Abstract

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.

Equations

$$\frac{\partial S_{21}}{\partial T}=\Delta T^{-1}(S_{21}(f,T+\Delta T,0)-S_{21}(f,T,0)),$$

$${\mu }_c(f,T,H)=1+\sum _{m=1}^M\frac{\Delta {\mu }_m(T,H)f_m^2(T,H)}{f_m^2(T,H)-f^2-if{\gamma }_m(T,H)},$$

$${ϵ}_c(f,T)={ϵ}_{00}+a(f-f_0)+b(T-T_0).$$

$$t(f,T,H)=\frac{(1-r^2)e^{ikd}}{1-r^2{e}^{i2kd}},$$

$$\Vert \frac{\partial S_{21}}{\partial T}\Vert =\Delta T^{-1}20lo{g}_{10}\Vert \frac{t(f,T+\Delta T,0)}{t(f,T,0)}\Vert$$

$$\arg (\partial S_{21}/\partial T)=\arg (t(f,T+\Delta T,0)-t(f,T,0)).$$

$$f_{\pm }=\frac{1}{2}\left(f_{(l)}+f_m\pm \sqrt{(f_{(l)}-f_m{)}^2+4{\kappa }^2{f}_{(l)}}\right),$$

$$\kappa =\frac{g_s{\mu }_B}{2h}\sqrt{\frac{{\mu }_{0}h}{2}p\rho },$$

Full text

antiferromagnetism, antiferromagnetic resonance, magnons, electromagnons, spin waves, optical cavities, cavity modes, light-matter coupling, THz, THz spectroscopy, cavitronics

Coupling of matter and electromagnetic radiation Mills and Burstein (1974) is a topic of great interest in solid state physics research because of their hybrid quantum nature. Khitrova et al. (2006) In the THz range, phonon-polaritons are a well-know example of a light-matter coupling. Recently, polaritons in the THz region were shown with intersubband transitions,Anappara et al. (2009); Todorov et al. (2010) cyclotron resonance Bayer et al. (2017) and plasmons Yu et al. (2018) in two-dimensional electron gases, as well as with intermolecular transitions in organic materials. Damari et al. (2019) In these systems, a strong-coupling regime can be achieved when losses are smaller than the exchange rate between light and matter, Paravicini-Bagliani et al. (2017) giving rise to the vacuum Rabi splitting. Polaritons are composite particles, which are studied in basic research on quantum optics,Khitrova et al. (2006) and can be considered for use in quantum computing and quantum memories.Wesenberg et al. (2009); Schuster et al. (2010); Kubo et al. (2010); Huebl et al. (2013); Frisk Kockum et al. (2019) The coupling of electromagnetic cavity-modes to magnons was researched intensively in ferromagnets at GHz frequencies, Huebl et al. (2013); Yao et al. (2015); Castel et al. (2017) meeting with expectations of energy-efficient spintronic devices. Matsukura et al. (2015) Coupling of magnons with superconducting qubits was also investigated. Tabuchi et al. (2015); Lachance-Quirion et al. (2017) The Purcell enhancement and the vacuum Rabi splitting were demonstrated in ferromagnetic materials.Zhang et al. (2014); Tabuchi et al. (2014); Bai et al. (2015); Maier-Flaig et al. (2016)

It is interesting to investigate magnon-photon coupling in antiferromagnetic materials Mills and Burstein (1974) in view of their high-frequency spin dynamics,Kampfrath et al. (2011); Jungwirth et al. (2016); Baierl et al. (2016); Nemec et al. (2018); Schlauderer et al. (2019) comparing to that of ferromagnets. As this phenomenon is readily taken into account by classical electrodynamics, it was accounted for in the analysis of optical investigations of antiferromagnetic materials, for instance in Ref. Balbashov et al. (1985); Kozlov et al. (1993); Mikhaylovskiy et al. (2014, 2015a, 2015b). Some experimental reports focused on characterization of interaction of antiferromagnetic magnons with photons in FeF2 Sanders et al. (1978); Jensen et al. (1997), NiO Li et al. (2011), TmFeO3 Grishunin et al. (2018) and in ErFeO3. Li et al. (2018); Sivarajah et al. (2019) Most of these reports use classical electrodynamics. A microscopic picture was developed in Ref. Li et al., 2018. Here, we confront the classical electrodynamic model with predictions of a microscopic model by estimating, quantitatively from our data, the strength of the interaction of antiferromagnetic spin-waves with electromagnetic cavity modes in dysprosium ferrite DyFeO3 (DFO).

Dysprosium ferrite is an orthogonally-distorted perovskite. It shows antiferromagnetic ordering below a Néel temperature of about 640 K White et al. (1982). Dzyaloshinskii-Moriya leads to a weak ferromagnetismIn DFO, the spin canting allows two antiferromagnetic resonance modes to be excited: the quasi-ferromagnetic (qFMR) and the quasi-antiferromagnetic (qAFMR),Zvezdin and Matveev (1979); White et al. (1982); Balbashov et al. (1985); Koshizuka and Hayashi (1988); Kozlov et al. (1993) which are excited by the magnetic component of radiation.Reid et al. (2015)

Our measurements were performed above 400 K, where both resonances, monotonously soften with rising temperature. We used polycrystaline samples for their isotropic properties, as large anisotropic properties in crystals might hinder the coupling. Białek et al. (2019) We have created disk-shaped samples of 1.0 and 0.6 mm thicknesses, thus having different cavity-mode spectra. A sample was placed in a furnace, which allowed to control temperature up to 700 K. By using quasioptical methods, we measured transmission with our continuous-wave THz spectrometer based on frequency extenders to a vector network analyzer (VNA). Caspers et al. (2016) This complex signal $S_{21}(f,T,H)$ is a function of frequency $f$, temperature $T$ and magnetic field $H$. We report its power amplitude in dB units and its phase in degrees. In order to extract a signal related to magnetic resonances, we measured transmission through a sample at different temperatures $T$. We obtained temperature-differential spectra by subtracting averaged spectra measured at subsequent temperatures:

[TABLE]

with $\Delta T=1$ K. We used this technique to measure spin-wave resonances in bismuth ferrite at high Białek et al. (2018) and low temperatures.Białek et al. (2019)

In classical electrodynamics, the dispersion of electromagnetic radiation in a material with a resonance is modified. This leads to creation of two polariton states split by a frequency representing coupling of the resonance with the electromagnetic radiation.Mills and Burstein (1974) In DFO, we model magnetic susceptibility $\mu_{c}(f,T,H)$ using lorentzian distributions

[TABLE]

where, for the $m$-th magnetic resonance, $f_{m}$ is its frequency, $\gamma_{m}$ its width and $\Delta\mu_{m}$ is its input to the zero-frequency magnetic susceptibility. In the case of DFO, $M=2$, with $m=1$ corresponding to the qFMR and $m=2$ to the qAFMR.

Ferrites have dielectric functions $\epsilon_{c}(f,T)$, which, in our experimental frequency range, i.e. far from phonons resonant frequencies, can be approximated as linear around a convenient point ($T_{0}=400$ K, $f_{0}=0.3$ THz)

[TABLE]

For our polycrystalline samples, we assume an effectiveBirchak et al. (1974); Heller (1945) magnetic susceptibility as $\sqrt{\mu}=p\sqrt{\mu_{c}}+(1-p)$ and effective dielectric function as $\sqrt{\epsilon}=p\sqrt{\epsilon_{c}}+(1-p)$, where the factor $p=0.64$ is a volume fraction of material to air in our pelletized samples.Białek et al. (2019) This value is the maximum density of random-packed hard spheres.Song et al. (2008) This assumption allows us to obtain values of $\epsilon_{c}$ and $\mu_{c}$ that are comparable with literature values for single crystal samples.Kozlov et al. (1993); Lobo et al. (2007) The complex wave vector is $k=2\pi f\sqrt{\epsilon\mu}/c$, where $c$ is the speed of light in vacuum. Transmission of electric field $t(f,T,H)$ though a slab of a thickness $d$ and infinite lateral dimensions is: Born and Wolf (1980)

[TABLE]

where $r^{2}=(\sqrt{\epsilon}-\sqrt{\mu})^{2}/(\sqrt{\epsilon}+\sqrt{\mu})^{2}$ is the square of the reflection coefficient at the vacuum-material interface. For frequencies far away from resonances, Eq. 4 implies an interference pattern, related to subsequent cavity modes of a slab, called Fabry-Pérot modes. At 300 GHz, in our polycrystalline DFO samples with the refractive index $\sqrt{\epsilon\mu}\approx 3.6$, the wavelength is about 300 $\mu$m. With rising temperature, $\sqrt{\epsilon\mu}$ increases, which shortens the period of the interference pattern. The magnetic susceptibility takes into account the effect of the magnetic resonance. When its frequency is close to a mode of a cavity, both the magnon and the cavity lines are altered by the matter-photon interaction. Since, the magnetic resonance frequencies have a much stronger temperature dependence than that of the interference pattern, they cross several interference pattern minima as temperature rises. We calculated

[TABLE]

to fit amplitude of temperature-differential spectra, where $\Delta T=1$ K is the temperature step. We calculated phase of temperature-differential spectra using

[TABLE]

The experimental temperature-differential signal amplitude for the 1.0-mm-thick DFO sample is shown in Fig. 1a.

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. 5 as shown in Fig. 1c. Thus, we find that this model reproduces most of the important features of Fig. 1a. The fitting parameters are parameters of the simplified dielectric function (Eq. 3) and of the magnetic susceptibility (Eq. 2). We assumed that resonance frequencies have a temperature dependence described by a power law $f_{m}^{*}(1-T/T_{N})^{\beta_{m}}$, applicable when approaching the Néel temperature $T_{N}$,Eibschütz et al. (1966) where $f_{m}^{*}$ has a unit of frequency and $\beta_{m}\approx\frac{1}{3}$ in the case of DFO. To improve the quality of our fits, we assumed that magnetic resonances have linear dependences of their widths and amplitudes on temperature. These formulas and values of fit parameters are given in the supplementary materials.

We measured the temperature-differential phase of transmitted electric field, as presented in Fig. 1b. Despite high noise, features predicted in Fig. 1d are observed in the experiment. The phase reveals clearly interactions between electromagnetic waves and magnetization dynamics. In Fig. 2a, a spectrum obtained at a temperature between crossings points shows that the qAFMR can be accounted for with a harmonic model. In Fig. 2b, the spectrum obtained at the temperature of the crossing with the $l=6$ cavity mode shows a structure that cannot be explained using a single oscillator. It is due to two polariton states which are not well-separated, i.e. they are in a weak coupling regime.Khitrova et al. (2006)

We used this phase prediction to estimate the cavity mode-magnetic resonance coupling strength. Thus, we superimposed on the Fig. 1d predictions of the harmonic coupling model Mills and Burstein (1974); Huebl et al. (2013); Harder et al. (2016)

[TABLE]

where $f_{\pm}$ indicates upper and lower polariton frequencies, $f_{(l)}$ is the $l$-th cavity mode frequency, $f_{m}$ with $m=1,2$ are the resonances frequencies. The coupling strength is given by Huebl et al. (2013)

[TABLE]

where $g_{s}=2$, $\rho$ is density of resonators and $p=0.64$ is the mass filling factor of our polycrystalline sample. Song et al. (2008); Białek et al. (2019) Equation 8 assumes that a magnon is coupled to a single electromagnetic cavity mode with a coupling strengthSoykal and Flatté (2010) $g_{s}\mu_{B}B_{0}/2h$, where $B_{0}=\sqrt{\mu_{0}hf_{(l)}/2V_{(l)}}$ is the magnetic component of vacuum fluctuations.Niemczyk et al. (2009) Amplitude of these is only $\approx 5\cdot 10^{-11}$ T for an electromagnetic mode at 0.3 THz taking the volume $V_{(l)}\approx 16\pi 10^{-9}$ m3 of the 1-mm-thick sample. This is about 3 orders of magnitude smaller than the amplitude of the THz field in our experiment. However, in an ensemble of $N$ resonators that collectively interact with a cavity mode, coupling is increased by a factor $\sqrt{N}$. Raizen et al. (1989); Soykal and Flatté (2010); Huebl et al. (2013); Harder et al. (2016); Li et al. (2018) Thus, the collective coupling strength $\kappa$ depends only on oscillators density $\rho=N/V_{c}$ and the ratio $p$ of a crystalline volume $V_{c}$ to a cavity volume $V_{(l)}$ Huebl et al. (2013); Li et al. (2018). We take $\rho=\frac{1}{3}\rho_{Fe}$, where $\rho_{Fe}=1.76\times 10^{28}$ m*-3* is density of iron atoms in DFO.Eibschütz (1965) This factor reflects the fact that in a polycrystalline material, on average, only $1/3$ of magnetic dipoles are excited by linear polarized electromagnetic wave. Under this assumption $\kappa=1.78\times 10^{4}$ Hz1/2, which results in a splitting $2\kappa\sqrt{f_{(l)}}\approx 19.5$ GHz with a cavity mode of $f_{(l)}=0.3$ THz.

Using Eq. 7, we calculated modes undergoing subsequent interactions with the same coupling strength $\kappa$. We determined $f_{(l)}$ from the condition $c(l-1/2)=2\Re{(\sqrt{\epsilon\mu})}f_{(l)}d$ for minimum of $\arg(\partial S_{21}/\partial T)$. In the case of 1-mm-thick sample, the lowest visible mode at $\approx 230$ GHz has $l=6$. Resonant modes frequencies were obtained from the same fit to the experimental $\partial S_{21}/\partial T$ magnitude. For the 1.0-mm-thick sample, the temperature dependence of both coupled and uncoupled modes is presented in Fig. 1d.

In summary, we have observed coupling of spin-waves with electromagnetic fields in high-temperature antiferromagnet DyFeO3. We observed the coupling of THz-frequency magnetic resonances to modes of cavities formed by the samples themselves. Our research demonstrates a possibility for a spin-cavitronic system based on antiferrromagnetic resonances. To account for our results, we applied both classical electrodynamics and a microscopic magnon-photon coupling model,Huebl et al. (2013) taking into account the polycrystalline nature of our samples.