Observation of room temperature exchange cavity magnon‐polaritons in metallic thin films

Cavity magnonics has become an intriguing ﬁeld due to its potential to enable next-generation technologies centered around controlling information exchange in hybrid resonant systems. Investigating the tunability of magnon-photon coupling is key to advancing the ﬁeld. Here, the observation of coupling between the ﬁrst order magnon mode in a metallic thin ﬁlm with a cavity photon mode is reported. An electromagnetic perturbation theory that takes account of perpendicular standing spin waves and their respective dissipation is utilized to estimate the coupling strength. The metallic thin ﬁlm exhibits notably lower dissipation for the higher-order magnon mode, which is not observed in a thin ﬁlm magnetic insulator. As such, and given that metallic Kittel magnons typically exhibit lower coherence times than their insulator counterparts, the excitation and coupling to speciﬁc higher order modes could lengthen these times compared to previous observations, which may be useful for future integration into quantum devices.


Introduction
The field of cavity magnonics has expanded rapidly in the past decade, tracing its roots to experimental studies of microwave resonator coupling to the magnetic modes of yttrium iron garnet (YIG), in thin films coupled to planar resonator structures, [1][2][3] and small spheres inside 3D cavities, [4][5][6] both showing that strong coupling between many-body spin systems and photons DOI: 10.1002/qute.202300420 is achievable, giving rise to a hybrid quasiparticle [7] of a new class, the cavity magnon-polariton (CMP).
Control over the coupling [8][9][10] between these light-matter excitations could yet be an integral part of one of the most potentially transformative applications of CMPs in the realization of optical communication between distant quantum computers. [11]In the last decade, it has been demonstrated that superconducting qubits can indirectly couple to magnons in YIG spheres through cavity photons. [12,13]Concurrently, the subfield of cavity opto-magnonics [14] has further enlightened the resonant interaction of optical light with magnons, and microwaveto-optical information conversion using magnons as an intermediary [15][16][17][18] is becoming established, with on-chip integration of this functionality [19] giving promise to the fusion of superconducting, magnonic and photonic technologies.These advances have made CMPs an important candidate for engineering the quantum information transduction [20,21] essential to future quantum networking.Crucial to pursuits in this area on the microwave-magnonic side, is the enhanced coupling strength between photons and spins that results from using a large ensemble of spins. [22,23]In ferromagnets these ensembles are bound together by Heisenberg exchange.Because of the relative strength of the exchange field, spins constitute an effective ordered medium, which at nanoscale and above can be described as a continuous magnetization field. [24]Disorder in these media comes about through a variety of competing internal energies, [25] and external fields can trigger excitations which propagate through it in a number of ways. [26]hese are known as spin waves, or magnons, which are essentially disruptions in the strict ordering leading to a difference in phase of neighboring spin precession, if this disruption is to the exchange ordering then we have exchange spin waves.
[29][30] Exchange spin waves can be excited as quantized standing modes across a film's thickness and when the exciting field is similarly confined to standing modes in cavity resonators, the continued magnetic-dipole interaction leads to coupled-oscillator behavior.This exchange mode coupling has previously been studied in thin films of YIG, [31][32][33] and coupling to magnetostatic (MS) volume waves has also been observed. [34]Despite the current dominance of YIG in cavity magnonics, and magnonics in general due to low damping rates, [35] YIG thin films are currently very difficult to integrate into millikelvin nanoscale circuits.The primary substrate for growth is gadolinium gallium garnet (GGG), which promotes higher losses in YIG at millikelvin temperatures, [36] and makes it difficult to directly integrate on-chip, [37] creating a need for new growth and fabrication techniques, [38] or alternative materials.The metallic magnet permalloy (Py), has also been previously studied in this context, [39][40][41][42] and notably, can be fabricated directly onto superconducting resonators, [39,40] making it a viable candidate for hybrid systems.However, there has been no reported observation of coupling to higher order magnon modes in Py films.Here, we report on such a coupling to a cavity mode in a 3D resonator at room temperature.We utilize an electromagnetic perturbation theory extended to account for perpendicular standing spin waves to predict the coupling and account for different magnon dissipation rates.Discussion of these different rates has been rare in this context, so we offer an extension to the case of metal versus insulator and a comparison with coupling to similar modes in a YIG film.In contrast to the magnetic insulator YIG, we observe that metallic exchange modes of Py have markedly lower dissipation than the Kittel mode, an effect, which has previously been observed in ferromagnetic resonance studies using a range of metallic magnets. [43]Thus, we expect that photon coupling to specific higher order magnon modes could lengthen coherence times compared to previous observations, offering applicability, in for instance, quantum integration beyond insulator YIG.

Results
Before investigating magnon-photon hybridization, we find it useful to characterize the exchange magnon modes-both experimentally and theoretically-to identify their resonant frequency range (that we will later match with that of microwave resonators) as well as dissipation rates.
We start by modelling the spin wave modes using the standard Landau-Lifshitz (LL) equation in the form dM∕dt = −|| 0 (M × H eff ) with a view to including damping at resonance phenomeno-logically further in the analysis.We define magnetization as M = ẑM s + m x,y exp j(k ⋅ r + t) with constant ẑ component of saturation magnetization M s and transverse components m x,y that vary harmonically in time and space across the film thickness and with frequency .The wavevector term k ⋅ r is evaluated in the transverse field directions with k = n d for the thin film, [29,44] standing mode order n, and thickness d.The constant ẑ component is a reasonable assumption given that the transverse components are of significantly smaller magnitude in the low-power, linear regime.We consider this magnetization as defined for each individual spin wave mode leading to individual resonance equations and dynamic susceptibilities.
We define an effective field containing a static component ẑH 0 in which we assume the effect of anisotropy is included, demagnetization, and exchange terms [44] as where the demagnetizing term For a very thin film with no out-of-plane magnetization (along ŷ), so that it can be assumed that D y = 1, D x , and D z = 0, the LL equation yields the following solution This describes the resonant frequencies of perpendicular standing spin wave modes with dispersion k mediated by the exchange interaction, which takes effect in the continuum approximation through the effective exchange stiffness.Spins at the surfaces of ferromagnetic thin films experience a pinning effect due to surface anisotropy, which, even if small, allows standing modes to form. [29]e then performed Vector Network Analyzer (VNA) FMR measurements on Py thin film samples and obtain the map of the magnon modes shown in Figure 1.The darker areas correspond to higher microwave absorption in which we can see the Kittel ferromagnetic mode and first order exchange mode, these are overlaid with the resonance curves from the Equation (1) shown as dashed lines, where FMR (n = 0) is in light blue and exchange (n = 1) in black.The dashed lines use  typical parameters for 100 nm thick Py films [45] of  0 M s = 0.9 T and ∕2 = 31.4GHzT −1 , while fitting for A = 8.1 pJm −1 which is slightly lower than reported in ref. [45], this value is the result of fitting to the spin wave mode peaks in the data presented in Figure 1b, further detail can be found in the Supporting Information.We note that in the theoretical analysis, modes up to n = 2 can be seen within the same frequency range as the experimental spectrum, but they are not present in the experimental data.This is expected for a uniform RF field, qualitatively, as only odd order modes will have a component of the magnetization that does not cancel with transverse components of opposite phase. [29]We make further note of this in the Supporting Information.
Knowing the frequency range at which the modes with profile shown in Figure 1a are observable, we can then design a microwave resonator that meet the coupling condition.That is, not only a microwave photon resonant frequency that match that of the magnetic modes, but also an RF magnetic field and static field profile that meets the FMR condition, as shown in the coordinate system also in Figure 1a.For this, we designed an aluminium 3D microwave resonator of dimensions 54×24×7 mm 3 with TE 410 mode resonant at frequency  c ∕2 ≈ 12.8 GHz.The 6×6mm 2 , 100nm thick Py sample is placed at the bottom centre of the cavity where to good approximation there is no electric field and the RF magnetic field oscillates only in the transverse field direction x (as shown in Figure 2b).The static field is applied along ẑ.
To test the coupling of the Py magnons to cavity photons we take S 11 reflection measurements with the resulting spectrum shown in Figure 2c.We note that in the cavity used here, the cavity itself absorbs far more of the input power than the 100 nm Py film.As such the coupling cannot be discerned in the raw spectrum data and it is necessary to perform a background subtraction to unveil the coupled modes, more details can be found in the Supporting Information.Upon background subtraction, we can observe a coupling signature -a central branch branch of coupled eigenmodes which is typically a result of level splitting.This splitting occurs as the magnon mode approaches resonance with the cavity mode where the two-level, cavity-magnon system hybridises and level repulsion of the cavity magnon-polariton is observed.Because of the presence of two magnon modes in the thin film, this process occurs twice.
In order to study this behavior further, we can use materials' magnetic susceptibility, (), in combination with electromagnetic perturbation theory to predict the magnon-photon coupling strength. [46]In order to obtain (), we again solve LL, but now accounting for a a time-varying RF field, h x,y , transverse to the applied field.Here, this is taken to be along x, in line with the geometries shown in Figure 2a,b.This can be included in H eff as h x,y = he jt x.Then, one can obtain coupled linear equations for the magnetization components in terms of the RF field components.These equations can be expressed in matrix form, providing a second-order tensor for ().Since in our experiment we only consider the x-component of the RF field, the relevant component of () is (2) We can then include this into a perturbation theory, where we have a single cavity perturbation but distinct susceptibilities corresponding to each spin wave mode.These susceptibilities  n () are summed to account for the total contribution of the film to the cavity perturbation when the cavity is in resonance with both magnetic modes.Following closely from previous derivation, [46] details of which can be found in the Supporting Information, we arrive at an eigenvalue equation of the form where W p denotes the energy of the microwave magnetic field in the sample region, and W c is the total energy stored in the cavity.This ratio is analogous to the field overlap often used in other studies of magnon-photon coupling.Equation ( 4) can be solved for  using numerical root finding techniques.The positive real solutions correspond to the resonant eigenmodes of the system, we note that for high mode numbers the accuracy of these solutions will be limited by numerical precision.The solutions of Equation ( 4), using the same parameters as those used in Figure 1, are shown as white dashed lines in Figure 2c over the experimental S 11 spectra-both in excellent agreement with one another.These solutions are obtained using the field energy ratio W p ∕W c ≈ 5.94 × 10 −7 (see Supporting Information for details).
Such excellent agreement however, requires that the effect of the cavity and magnon dissipation are accurately captured.These are related to the cavity and magnon resonance linewidths, Δ c and Δ r,n , respectively, which can be included into Equation ( 4) by simply adding imaginary terms to the respective resonant frequencies as follows:  c →  c + jΔ c and  r,n →  r,n + jΔ r.n .We obtain these linewidths for each system separately where the cavity dissipation is determined from the mode linewidth of the bare cavity and the magnon dissipation is determined from the broadband FMR measurements.We use the half-width at half-maximum (HWHM) of the bare cavity frequency spectrum around the resonant mode, and the field HWHM of the magnon modes from FMR at the cavity frequency.
To estimate magnon dissipation we fit a double Lorentzian function to the S 21 data at the frequencies corresponding the the cavity resonances used in the main text for the Py and YIG.This fitting is detailed further in the Supporting Information.
To find the cavity mode dissipation we use a single Lorentzian, where the variables now correspond to the cavity S 11 peaks, and the HWHM is readily in the frequency domain, these fits can be seen in Figure 3.The bare resonator has a frequency of ∕2 ≈ 12.73 GHz at the TE 41 mode, which changes to ∕2 ≈ 12.80 GHz when the Py sample is placed at the bottom of the cavity, the cavity dissipation rate also changes from Δ c ∕2 ≈ 45.0 MHz to Δ c ∕2 ≈ 36.0MHz.These shifted values are used in the calculation of coupling strength.
Using these estimates we numerically determine the coupling strengths using perturbation theory, by analyzing the gap between eigenmodes calculated from Equation (4), as g r,0 ∕2 = 13.44 ± 1.40 MHz and g r,1 ∕2 = 16.37 ± 1.90 MHz for the Kittel and first order mode, respectively.Because of the relatively large cavity and magnon dissipations here, these polaritons are in the weak coupling regime. [4]We have also tested additional Py films of different thickness to see if the same exchange mode coupling is observed, this can be found in the Supporting Information.
It is worth noting that our measurements show a lower dissipation rate for the first-order spin wave mode, as can be seen in Figure 4b.This has previously been observed by Li et al. [43] in ferromagnetic resonance studies using films of similar thickness to ours and we note that it is counter to what has been observed in the higher order modes of YIG spheres. [47]As such, it is also useful to compare metal versus insulator exchange modes.
In Figure 5, we look at the hybridization between magnon modes in a good quality 3 × 3 mm 2 , 0.735 micron thick YIG film with the TE 110 mode of the same cavity, which is resonant at  c ∕2 ≈ 6.872 GHz with a dissipation of Δ c ∕2 ≈ 1.1 MHz.The resonance characteristics here change negligibly upon introduction of the sample.In this mode configuration the field energy ratio W p ∕W c ≈ 2.35 × 10 −7 .We use typical parameters for YIG [44] of  0 M s = 0.178 T, ∕2 = 28 GHz/T,  ex = 3 × 10 −16 m 2 , along with estimated magnon dissipation Δ r,0 ∕2 ≈ 1.78 ± 0.5 MHz and Δ r,1 ∕2 ≈ 1.87 ± 0.9 MHz for the Kittel and first order mode, respectively.In this data we can clearly see once again a double mode splitting associated with the presence of exchange mode magnon-photon coupling.These polaritons seem to be on the boundary between the magnetically induced transparency and strong coupling regime. [4]We estimate the coupling strengths as g r,0 ∕2 = 2.361 ± 0.082 MHz and g r,1 ∕2 = 2.355 ± 0.157 MHz for the Kittel and first order modes, respectively.Because of the close separation of the magnon modes in this YIG sample compared to the Py, different behavior is seen in the central coupled eigenmode branch, it seems the case that the cavity mode no longer steps down before the next magnon mode is brought into resonance, this may indicate that there is some coupling between the cavity mode and both magnon modes simultaneously in this central region.In the Supporting Information we offer comparison of this spectrum to theoretical prediction from microwave scattering theory.

Discussion
We reported on the observation of exchange cavity magnonpolaritons in metallic thin films at room temperature and presented an extension of electromagnetic perturbation theory to account for the coupling and dissipation of higher order magnon modes.With the ability to predict coupling strength for different magnon modes, the perturbation theory can be a valuable tool in the design of multi-mode cavity magnonic systems.Accurate prediction of coupling strength at cryogenic temperatures, [46] is critical for interfacing magnonic and superconducting technology.
In previous study of coupling to higher order magnon modes, few have captured the different rates of dissipation, [19,47,48] and only Zhang et al offer much discussion, focusing on the effect of temperature on damping and subsequent coupling extinction in YIG spheres, [47] The mode-order dependent damping offers another route to magnon dissipation tunability in the context of cavity magnonics, where as discussed by Li et al, this can be be varied depending on the thickness of the film. [43]iven that different regimes of coupling [4,50] can be achieved through dissipation and coupling strength tuning, something that has recently been studied with both electrical current [51] and temperature, [49] utilising significant differences in magnon dissipation between modes in Py could lead to a new avenue for magnetic control of microwave signals.With films of appropriate thickness, one could simply change the magnetic field strength to steer between different coupling regimes.Thus, this form of dissipation tunability may offer uses YIG cannot.Other relative advantages of Py are the lower bias fields needed to achieve GHz resonance, the large magnetization which naturally increases coupling strength, and fabrication advantages leading to closer proximity with the photon resonator. [39,52]The trade-off for quantum coherent applications is the relatively large magnon dissipation when compared to YIG, but as we have shown, this is somewhat mitigated by using the first-order mode with little difference in coupling strength.Additionally, we note that it may be of further interest to explore higher order modes in low damping metallic thin films such as CoFe [53] for coupling and on-chip integration with superconducting resonators in the context of cavity magnonics.

Sample Preparation
Samples were fabricated by DC magnetron sputtering using elemental nickel and iron targets and a base pressure of 6 × 10 −9 Torr.Depositions were performed on substrates of silicon with a 200 nm thermal oxide at room temperature, deposition rate of 1.2 Ås −1 , and in the presence of 0.8m Torr argon.The sample structure was Ta(5 nm)/Py(100 nm)/Ta(5 nm), where tantalum was used as a seed and capping layer, the latter of which was employed to avoid sample oxidization.

Measurement
VNA-FMR was performed using a microstrip PCB fabricated inhouse by milling Rogers Corporation TMM10i, 50 Ω SMA connectors were then soldered on to the microstrip line, and ported to a Rohde and Schwarz ZVA 40, where S 21 measurements where performed by taking a frequency sweep at each current step.The magnetic field was measured using a hall probe.For the cavity measurements a high conductivity aluminium cavity with one port was used and S 11 reflection measurements were performed.
The microstrip S 21 measurement presented of the Py film was taken at -10dBm input power, while the cavity measurement was taken at 15dBm.Cavity measurement of the YIG was at -15dBm power.

Figure 1 .
Figure 1.a) Diagram depicting the Kittel and first-order exchange spin wave mode for schematic 1D spin chains corresponding to standing thickness modes in a thin film.The static magnetic field H 0 applied in the ẑ direction sets the direction of precessional motion, the oscillating magnetic field supplied by the cavity field h rf in the x direction provides a alternating torque that continuously re-excites this precession.b) FMR Spectrum for 100nm Py film obtained using a microstrip line.Light blue dashed lines fit to the Kittel mode (n = 0), and black to the exchange mode (n = 1).The experimental absorption is shown more clearly inset.

Figure 2 .
Figure 2. a) Schematic of microwave resonator setup used in the experiment.The rectangular resonator (silver), with dimensions i = 27 mm, ii = 24 mm, iii = 7 mm, is excited through an SMA port (gold) connected to a coaxial cable.In (b) we see the field profile of the TE 410 mode (simulated with COMSOL 6.1) that resonates at about 12.8 GHz.Through the FMR spectrum of Figure1bwe can predict the magnetic field strength needed to bring the magnon modes into resonance with this cavity mode.In (c) we see a signature of level-splitting typically associated with cavity magnonpolaritons, resulting from bringing each magnon mode into resonance with the cavity mode by tuning the magnetic field strength.Both the resonances from Equation (1), and the coupled eigenmodes obtained numerically from Equation (4) agree well with the observed spectrum here.

Figure 3 .
Figure 3. a) Bare resonator mode and resonator with YIG film sample, at the TE110 mode frequency.Showing a very small change in the resonance characteristics when the magnetic sample is present (bias magnetic field is set far from resonance for this measurement).b) Bare resonator mode and resonator with Py film sample, at the TE410 mode frequency.The presence of the metallic Py sample significantly alters the resonance characteristics, causing a shift up in frequency and a decrease in linewidth.This is possibly due to the metallic sample altering the impedance of the resonator.

Figure 4 .
Figure 4. Double Lorentzian to Kittel and first order spin wave mode to estimate the corresponding field linewidths Δ 0 H 1∕2 for YIG (a) and Py (b).

Figure 5 .
Figure 5. CMP modes for the YIG thin film.White dashed lines are from the solutions to Equation (4) while black and cyan dashed lines are solutions to Equation (1).