Strong coupling between magnons in a chiral magnetic insulator Cu{}_{2}OSeO{}_{3} and microwave cavity photons

Strong coupling between magnons in a chiral magnetic insulator CuOSeO and microwave cavity photons

L.V. Abdurakhimov l.abdurakhimov@ucl.ac.uk London Centre for Nanotechnology, University College London, London WC1H 0AH, United Kingdom    S. Khan London Centre for Nanotechnology, University College London, London WC1H 0AH, United Kingdom    N.A. Panjwani London Centre for Nanotechnology, University College London, London WC1H 0AH, United Kingdom    J.D. Breeze Department of Materials, Imperial College London, Exhibition Road, London SW7 2AZ, United Kingdom London Centre for Nanotechnology, Imperial College London, Exhibition Road, London SW7 2AZ, United Kingdom    S. Seki RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan    Y. Tokura RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan Department of Applied Physics and Quantum Phase Electronics Center (QPEC), University of Tokyo, Tokyo 113-8656, Japan    J.J.L. Morton London Centre for Nanotechnology, University College London, London WC1H 0AH, United Kingdom    H. Kurebayashi h.kurebayashi@ucl.ac.uk London Centre for Nanotechnology, University College London, London WC1H 0AH, United Kingdom
July 14, 2019
Abstract

Anticrossing behavior between magnons in a multiferroic chiral magnet CuOSeO and a two-mode dielectric resonator cavity was studied in the temperature range 5–100 K. We observed a strong coupling regime between ferrimagnetic magnons and two microwave cavity modes with a cooperativity reaching 3600. We measure a coupling strength which scales as the square root of the net magnetization, , and demonstrate that the magnetic phase diagram of CuOSeO can be reconstructed from the measurements of the resonance frequency of a dispersively-coupled cavity mode. An additional avoided crossing between a helimagnon and a cavity mode was observed at low temperatures. Our results reveal a new class of magnetic systems where strong coupling of microwave photons to non-trivial spin textures can be observed.

Introduction.

Strong coupling between microwave photons and particle ensembles is a general phenomenon in light-matter interactions that has been observed in a broad range of condensed-matter systems, including ensembles of magnetically ordered spins Huebl et al. (2013); Tabuchi et al. (2014); Zhang et al. (2014); Goryachev et al. (2014); Abdurakhimov et al. (2015); Goryachev et al. (2017); Morris et al. (2017), paramagnetic spins Chiorescu et al. (2010); Schuster et al. (2010); Kubo et al. (2010); Breeze et al. (2017), and two-dimensional electron systems Muravev et al. (2011); Scalari et al. (2012); Abdurakhimov et al. (2016). A common feature of ensemble coupling is that the coupling strength between a photon and particles scales with the square root of , , in accordance with the Dicke model Agarwal (1984); Imamoğlu (2009); Garraway (2011). Studies on strong coupling in spin systems are particularly interesting due to possible applications of hybrid spin-ensemble-photon systems for quantum information processing as quantum memories Schoelkopf and Girvin (2008); Kurizki et al. (2015) and quantum transducers Blum et al. (2015). The spin-ensemble coupling strength can be extremely large in magnetically ordered systems due to their high spin densities, and extensive studies of strong coupling to magnons – the quanta of spin wave excitations in magnetically ordered systems – have been performed recently in experiments on ferrimagnetic insulators. In particular, new magnon-cavity-coupling phenomena have been observed in yttrium iron garnet (YIG), such as coherent coupling between a magnon and a superconducting qubit Tabuchi et al. (2015), microwave-to-optic-light conversion Hisatomi et al. (2016); Haigh et al. (2016), cavity-mediated coherent coupling between multiple ferromagnets Zhang et al. (2015); Lambert et al. (2016), spin pumping in a coupled magnon-photon system Bai et al. (2015), and cooperative cavity magnon-polariton dynamics in feedback-coupled cavities Yao et al. (2017).

So far, most studies of strong coupling in magnetic materials have focused on ferrimagnetic materials with the -like Heisenberg exchange interaction between neighbor spins and . In chiral magnets, the spin-spin exchange interaction consists of two terms; besides the symmetric Heisenberg interaction which favors collinear spin structures, there is an additional antisymmetric -like Dzyaloshinskii-Moriya (DM) interaction which tends to twist neighbor spins. As a result of the interplay between Heisenberg and DM exchange interactions, various non-collinear spin textures can be formed in chiral magnets, such as helical, conical, and Skyrmion spin structures Not (). Studies of the coupling between microwave photons and non-collinear spin textures is a potentially rich and largely unexplored area.

A chiral magnetic insulator copper-oxoselenite CuOSeO crystallizes in a non-centrosymmetric cubic structure with 16 copper ions Cu per unit cell (space group , lattice constant Å Larrañaga et al. (2009)). The basic magnetic builing block of CuOSeO is a tetrahedral cluster formed by four Cu spins in a 3-up-1-down spin configuration, which behaves as a spin triplet with the total spin Janson et al. (2014). This magnetic structure has been visualized elsewhere Seki et al. (2012a); Janson et al. (2014). Due to a combination of Heisenberg and DM exchange interactions, the system of clusters forms helical, conical, ferromagnetic and Skyrmion magnetic phases in an applied external magnetic field below the Curie temperature  K, as shown in Fig. 1(a). Above , the system is paramagnetic. Besides being a chiral magnet, CuOSeO is also of particular interest due to its multiferroic and magnetoelectric properties Seki et al. (2012a, b); Okamura et al. (2013); Mochizuki and Seki (2015); Ruff et al. (2015).

In this Letter, we report a study of anticrossing behavior between magnonic modes in CuOSeO and microwave modes of a dielectric resonator cavity. In particular, we observed a strong coupling regime between magnons in the field-induced ferrimagnetic phase and photons in a two-mode microwave cavity. The coupling strength temperature dependence was found to follow that of the square root of the net magnetization, and we demonstrate that the magnetic phase diagram can be determined by the measurement of the frequency of a cavity mode. In the helical phase, normal-mode splitting was detected between a helimagnon mode and a cavity mode.

Figure 1: (color online) (a) Magnetic phase diagram of CuOSeO. (b) Experimental setup for reflection microwave spectroscopy. (c-d) Strong coupling between magnons in CuSeO and multiple microwave cavity modes: (c) experimental data of microwave reflection as a function of the applied external field and the microwave probe frequency (temperature  K, microwave input power  mW), (d) microwave reflection calculated using Eq.(1) and parameters described in the text.

Experimental details.

We performed microwave X-band spectroscopy studies of CuOSeO in the temperature range 5–100 K using a helium-flow cryostat. The experimental setup is schematically shown in Fig. 1(b). A sample of single crystal CuOSeO was inserted into a commercial Bruker MD5 microwave cavity consisting of a sapphire dielectric ring resonator mounted inside a metallized plastic enclosure. The sample mass was  mg, corresponding to a total effective spin number . The shape of the sample was close to semi-ellipsoidal, with the lengths of semi-axes being 1.5 mm, 1.5 mm, and 2 mm, and a flat plane being oriented along the long ellipsoid axis. The orientation of crystallographic axes of the sample relative to the cavity axis was chosen arbitrarily. The cavity supported two microwave modes SM (); Mongia and Bhartia (1994): the primary mode TE with the resonance frequency of about  9.74 GHz and the hybrid mode HE with the resonance frequency of about  9.24 GHz. By adjusting the position of a coupling loop antenna, it was possible to tune the quality factor of the primary-mode resonance. In our measurements, we used a slightly under-coupled cavity with . The quality factor of the hybrid-mode resonance did not depend on the position of the coupling antenna, and was about . In our experiments, the microwave reflection S-parameter was measured as a function of external magnetic field and microwave probe frequency.

Figure 2: (color online) (a) Microwave response at 64 K. Dashed lines correspond to theoretical estimates calculated by using Eq.(1) and  MHz. (b) Microwave response at 70 K. Dashed lines correspond to theoretical estimates calculated by using Eq.(1) and  MHz. (c) Temperature dependence of the coupling strength in ferromagnetic (FM) and paramagnetic (PM) phases: experimental data (squares) and theoretical estimations for for  Oe (dashed black line) and  Oe (solid red line).

Results.

Figure 1(c) shows typical data from microwave reflection measurements at temperature  K, obtained from raw experimental data by background-correction processing SM (). The input microwave power was  mW. Two avoided crossings are visible at the degeneracy points where two cavity modes would otherwise intersect a magnon mode. The magnon mode corresponds to a uniform spin precession (Kittel mode) with frequency , where is the applied magnetic field, is the demagnetizing field, and  GHz/T is the electron gyromagnetic ratio. Here, we assume that anisotropy fields are small and can be neglected.

The interaction between two cavity modes and a magnon mode can be described by the following Hamiltonian in the rotating-wave approximation (RWA):

where () is the creation (annihilation) operator for microwave photons at frequency , () is the creation (annihilation) operator for microwave photons at frequency , () is the creation (annihilation) operator for magnons at frequency , and () is the coupling strength between the magnon mode and the first (second) cavity mode.

In order to extract numerical values of coupling strengths and and other parameters from the experimental data, we used the following equation obtained from input-output formalism theory SM ():

(1)

where

and () is the damping rate of the first (second) cavity mode, () is the coupling rate between the first (second) cavity mode and the output transmission line, and is the damping rate of the magnonic mode. Damping rates represent linewidths (FWHM) of the corresponding modes.

We reproduce the data shown in Fig. 1(c) by using Eq. (1) with the following parameters:  MHz,  MHz,  MHz,  MHz,  MHz,  MHz,  Oe and  MHz (see Fig. 1(d)). Thus, the coupling strengths are much greater than the damping rates of both cavity and magnon modes, and , and strong coupling regimes are realized for both avoided crossings. From the obtained value of the ferromagnetic resonance linewidth , we estimate the Gilbert damping parameter at 5 K which is consistent with the literature Stasinopoulos et al. (2017). Cooperativity parameters are much higher than unity: and . Moreover, ratios and are close to the condition of the ultra-strong coupling regime (), where the coupling strength is comparable with the frequency of the degeneracy point of an avoided crossing, and new physics beyond RWA can be explored Zhang et al. (2014); Goryachev et al. (2014).

The obtained values of coupling strengths and between magnons and cavity modes are in relatively good agreement with the theoretical estimates

(2)

where is the vacuum permeability,  mm ( mm) is the mode volume of the primary (hybrid) cavity resonance, and the coefficient () describes the spatial overlap between the primary (hybrid) cavity mode and the magnon mode SM (). Substituting other known parameters into the equations, we obtain  MHz and  MHz. Slight discrepancies between theoretical and experimental values of coupling strengths can be caused by the excitation of additional spin-wave modes in the sample which are visible as additional faint narrow lines in the experimental data, and the resulting reduction of the effective number of spins involved in the uniform Kittel mode precession Boventer et al. (2018).

Figure 3: (color online) Magnon-photon coupling at low magnetic fields. (a) Microwave reflection at 5 K. The magnetic field sweep was performed from low to high field values (“field-up”). Dashed line corresponds to a helimagnon mode, and an avoided crossing between the helimagnon mode and the cavity mode is clearly visible. and are critical magnetic fields of helical-to-conical and conical-to-ferromagnetic phase transitions, respectively. (b) Microwave reflection at 5 K. The magnetic field sweep was performed from high to low values (“field-down”). The transition from ferromagnetic to conical phase demonstrated hysteretic behavior with the abrupt bistability-like shift of the cavity mode frequency at the field . (c) The magnetic phase of CuOSeO sample reconstructed from the “field-up” measurements of and .

Experimental values of the coupling strengths depended strongly on temperature. In particular, the temperature dependence of the coupling strength is shown in Fig. 2. Assuming that the effective total number of spins is proportional to the net magnetization , Maier-Flaig et al. (2017), we find that the experimental data can be well described by the function normalized to the value of coupling strength at low temperatures (see Fig. 2(c)), where is calculated using the Weiss model of ferromagnetism SM (). In our experiments, anticrossing behavior was independent of microwave probe power, consistent with observations of strong coupling in other systems Chiorescu et al. (2010); Huebl et al. (2013); Abdurakhimov et al. (2015).

In order to understand strong coupling dynamics in helical and conical magnetic phases, we investigated coupling phenomena at low magnetic fields (see Fig. 3). At temperatures  K, we detected an additional avoided crossing in the helical magnetic phase. According to our analysis SM (), the observed anticrossing behavior is caused by a normal-mode splitting between a cavity mode and a magnon mode in the helical phase — a so-called helimagnon Kataoka (1987); Onose et al. (2012); Schwarze et al. (2015); Weiler et al. (2017) — characterized by the wave vector , where is the wave vector of the helical spin spiral. The coupling strength is  MHz, and linewidths of the helimagnon and cavity modes are  MHz and  MHz, respectively SM (). Therefore, the observed normal-mode splitting can be attributed to Purcell effect (, Zhang et al. (2014)), where the decay of microwave cavity photons is enhanced due to their interaction with lossy magnons. We could not detect at temperatures  K which is consistent with the literature Weiler et al. (2017).

Another interesting feature of the low-field response is the magnetic-field dependence of the cavity resonance frequency. We identify two singularities at magnetic field values and which can be attributed to magnetic phase transitions in CuOSeO (see Fig. 3(a,c) and Fig.1(a)). In the helical phase , the net magnetization — and, hence, the magnon-photon coupling to the uniform-precession Kittel mode — is negligible, and only the coupling to helimagnons can be observed. In the conical phase , the net magnetization , and the cavity mode is dispersively coupled to the Kittel mode which results in the shift of the cavity resonance frequency. It should be noted that the observed transition from helical to conical phases is relatively smooth, which can be related to the fact that in the helical phase the spin system forms a multidomain structure of flat helices Seki et al. (2012a), and, since the external DC magnetic field was not aligned along the high symmetry directions of the crystal structure in our measurements, domains gradually reoriented themselves with the increase of the magnetic field Schwarze et al. (2015). In the ferromagnetic phase , the net magnetization is close to its maximum value (saturation magnetization), the coupling strength to the Kittel mode is high, and the shift of the cavity resonance with the increase of the magnetic field is large.

The magnetic field value of the conical-to-ferromagnetic transition was found to depend on the direction of magnetic field sweep (Fig. 3(a-b)). Besides this hysteretic behavior, a jump in the cavity resonance frequency was observed at which can be a manifestation of a bistability present in the system. A power-dependent bistability in the coupled magnon-photon system has been discussed recently in the literature Wang et al. (2017). However, that bistability mechanism is not directly applicable to our data, since in our experiments the bistability did not demonstrate any dependence on the microwave power SM ().

Conclusions.

We performed a study of magnon-photon coupling in helical, conical and ferrimagnetic phases of CuOSeO. In particular, we achieved a strong coupling regime between a ferromagnetic magnon mode and multiple microwave cavity modes. The strong coupling regime extends to low magnetic fields which allows us to use a dispersively-coupled microwave cavity mode as a probe for the sensing of magnetic phase transitions in CuOSeO. We also detected a normal-mode splitting between microwave photons and helimagnons in the helical phase in the Purcell-effect regime. These findings establish a new area of studies of strong coupling phenomena in multiferroic chiral magnetic systems, paving the way for new hybrid systems consisting of non-trivial spin textures coupled to a single microwave photon via magnetic and magnetoelectric interactions.

Acknowledgements.
This work was partly supported by the Grants-In-Aid for Scientific Research (Grants No. 17H05186 and No. 16K13842) from JSPS, and by the European Union’s Horizon 2020 programme (Grant Agreement Nos. 688539 (MOS-QUITO) and 279781 (ASCENT)); as well as the Engineering and Physical Science Research Council UK through UNDEDD project (EP/K025945/1).

References

Supplemental Material for “Strong coupling between magnons in a chiral magnetic insulator CuOSeO and microwave cavity photons”

Appendix A Correction of the experimental data background

An example of raw experimental data is shown in Figure S1(a). We can clearly see two avoided crossings and a background pattern consisting of horizontal stripes. The background is caused by the presence of a standing wave in coaxial cables between the microwave resonator and the vector network analyzer due to impedance mismatch at the input of the microwave resonator. To eliminate that background, we reconstructed the standing-wave profile by a piecewise-defined function with different pieces being taken at different values of magnetic field, where the standing-wave background was not affected by the avoided crossings. Background-corrected data was obtained by subtracting the standing-wave profile from the raw experimental data, and adding an offset value to keep the minimum and maximum levels of at the same level. An example of background-corrected data is shown in Figure S1(b).

Figure S1: (color online) The standing-wave background correction. (a) An example of raw experimental data ( K). (b) The same data after the correction.

Appendix B Microwave TE and HE modes of a dielectric resonator

We performed numerical simulations of the sapphire dielectric ring resonator cavity by using CST Microwave Studio software (see Fig. S2). The sample was modeled as a dielectric sphere with the radius  mm and the relative electrical permittivity Ruff et al. (2015). We find that the cavity supports two microwave modes: a primary TE mode with the resonance frequency of about  9.8 GHz (Fig. S2(a-b)) and a hybrid HE mode with the resonance frequency of about  9.56 GHz (Fig. S2(c-d)). The microwave magnetic field of the primary mode is parallel to the axis of the dielectric resonator, and magnetic energy density is concentrated inside the cavity bore (Fig.S2(a)). The effective magnetic mode volume for the primary cavity resonance can be estimated as

(S1)

where is the vacuum permittivity, is the relative electrical permittivity at a given point, is the microwave electric field, is the vacuum permeability, is the relative permeability, is the microwave magnetic field, is the maximum value of the microwave magnetic field, and are microwave electrical and magnetic energy densities, and the integration is taken over the total volume of the system. From numerical simulations, we obtain  mm.

Next, we introduce an overlap coefficient to take into account a slight non-uniformity of the distribution of the microwave magnetic field across the sample which affects our calculations of coupling strengths described in the main text:

(S2)

where is the volume of the sample used in the numerical model, and the integration is taken over the sample volume. Numerically, we find for the primary mode.

The hybrid HE mode is a double-degenerate axially-asymmetric mode. At the site of the sample, the magnetic component lies in the plane perpendicular to the axis of the dielectric resonator. The HE mode is degenerate, and two mutually perpendicular orientations of microwave magnetic field — two polarizations — are possible. Typical distributions of microwave magnetic and electric energy densities for one of two possible polarizations are shown in Fig. S2(c-d)). In calculations of the coupling strength described in the main text, we take into account only one polarization of the HE mode for which the microwave magnetic field at the site of the sample is perpendicular to the external DC magnetic field (a hybrid mode with another polarization does not interact with a ferromagnetic mode). By using equations similar to Eq. (S1S2), we estimate that the effective magnetic mode volume is  mm, and the overlap coefficient is for the magnetic component of the hybrid mode.

Figure S2: (color online) Results of numerical simulations of the dielectric resonator cavity. Distributions of magnetic () and electric () energy densities in the center horizontal plane, which is perpendicular to the axis of the dielectric ring resonator, are shown for the fundamental cavity mode (a-b) and the hybrid cavity mode (c-d). Dashed white circles represent the contour of the dielectric ring resonator.

Appendix C Input-output formalism: strong coupling between a magnon mode and a two-mode cavity

Following input-output formalism theory Gardiner and Zoller (2000); Walls and Milburn (2008), we can write the following Heisenberg-–Langevin equations:

where and are an external input and output fields, and are coupling constants between cavity fields and the external fields. The Hamiltonian describes the cavity modes and the magnon mode, where the Hamiltonian is defined above, and describes the dissipation inside the cavity due to the interaction with a heat bath.

Figure S3: (color online) A schematic representation of the two cavity fields, the magnon mode, and the input and output fields for a single-sided cavity.

By assuming that solutions should be in the form of , , etc., we can obtain the following equations:

where , , and are resonant frequencies of the cavity modes and the magnon mode, respectively, and are dissipation rates of the cavity modes, and is the dissipation rate of the magnon mode.

By solving those equations, we find the following expression for :

where the following notations are used:

Appendix D Temperature dependence of magnetization

Figure S4: (color online) Temperature dependence of magnetization at different values of the external magnetic field: 0 T (blue curve) and 0.3 T (red curve) according to numerical calculations.

We will use the Weiss model of ferromagnetism Blundell (2001) to calculate the net magnetization in the ferromagnetic phase as a function of the temperature and the external magnetic field , . The ferromagnetic interaction is modeled by introducing the molecular field

where is a constant parameter. For a ferromagnet, .

The magnetization can be found by solving self-consistently the following non-linear equations:

(S3)
(S4)

where the saturation magnetization is

is the Brillouin function given by

and is a g-factor, is the Bohr magneton, is a spin number, is the Boltzmann constant, is the temperature, and is a spin density.

It can be shown that the parameter is related to the Curie temperature as Blundell (2001)

For CuOSeO, , , (lattice constant Å), and, by assuming that  K, equations (S3-S4) can be solved numerically. Results of numerical calculations for magnetic field values 0 T and 0.3 T are shown in Fig. S4.

Appendix E Coupling between a photon and a helimagnon

Figure S5: (color online) A normal-mode splitting between a cavity mode and a helimagnon mode. (a) Microwave reflection at 5 K. (b) Results of the numerical calculation of by using equations and parameters described in the text.

The helical phase of CuOSeO is characterized by a multidomain structure of flat helices, where the propagation vectors of the different helices are pinned along the preferred axes of the system. The accurate description of helimagnons in the helical phase requires taking into account the cubic anisotropies of CuOSeOSchwarze et al. (2015), which is outside the scope of this paper. Instead, we will use the analyitcal equation used for the description of magnons in the conical phase Weiler et al. (2017):

(S5)

where is the electron gyromagnetic ratio, is the demagnetization factor along the direction of the vector, is the internal conical susceptibiliy ( for CuSeOSchwarze et al. (2015)), is the applied external field, and is the critical field for the transition between conical and ferromagnetic phases ( T in our experiments). The mode number describes the relation between the helimagnon wave vector and the wave vector of the helical spiral , . By adjusting the parameters and , we identify the observed helimagnon mode as degenerate mode (the demagnetization factor ), which is shown by the dashed line in Fig. S5.

In order to characterize the observed avoided crossing quantitatively, we will use the following equation for the microwave reflection Zhang et al. (2014):

(S6)

where and are the frequency and the linewidth of the cavity mode, respectively, is the coupling rate between the cavity mode and an antenna, and are the frequency and the linewidth of the helimagnon mode, respectively, and is the coupling strength between the cavity mode and the helimagnon mode. We found that experimental data, shown in Fig. S5(a), can be reproduced by using the following parameters:  MHz,  MHz,  MHz,  MHz, and  MHz (see Fig. S5(b)).

Appendix F Additional experimental data on the bistability behavior

The factors influencing the bistability formation are not fully understood and require further investigations. For the same experimental conditions, the bistability feature was occasionally observed at different magnetic fields (see Fig. S6(a-b)). In the majority of the measurements, the bistability occured at magnetic fields slightly above 600 Oe, and did not demonstrate any consistent dependence on microwave probe power (compare Fig. S6(a) and Fig. S6(c)).

Figure S6: (color online) Microwave response at temperature 5 K. The magnetic field sweep was performed from high to low field values (“field-down” sweep). (a) Microwave probe power 0.5 mW. (b) Microwave probe power 0.5 mW (another measurement). (c) Microwave probe power 5 mW.

References

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