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Abstract
We report on the results of finite difference time domain (FDTD) simulations of the terahertz response of a split ring resonator (SRR) metamaterial coupled to a hypothetical antiferromagnetic material (AFM) characterized by a magnon resonance. We find a hybridization of the SRR’s local magnetic field and the magnon, which manifests as an avoided crossing in the far-field transmission spectrum. We show that the strong light-matter coupling can be modelled via a two coupled oscillator model. We further evaluate the SRR-AFM coupling strength by varying the physical separation with a dielectric spacer between them. We find strong coupling for spacers thinner than 3 μm, suggesting far-field transmission measurements of metamaterial near-fields to be a versatile platform to investigate magnetic excitations of quantum materials.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Strong coupling between light and resonant normal modes in normal materials can produce hybrid light–matter modes such as polaritons [1,2], magnon-polaritons [3], and exciton-polariton hybrid modes [4–6]. Such modes have applications in laser engineering and quantum computing [4,7–9]. Recent efforts have focused on engineering strong coupling (and thus hybridization) between light and the resonant cavity modes [10–12]. Strong coupling between light and vibrational modes [12], excitons [13], spin excitations [14], and even control over electronic phase transitions [15–17] have been demonstrated.
Other structures beyond cavities also exhibit strong light–matter coupling. Metamaterials, artificial structures composed of periodically repeating metal and dielectric meta-atoms, offer another avenue for engineering light–matter hybridization [18–21]. Metamaterials behave like smooth, continuous media when the meta-atom dimensions are much smaller than incident electromagnetic radiation’s wavelength. Driven metamaterials undergo an inductive-capacitive (LC) resonance from the geometric inductance and capacitance of metallic meta-atoms. Carefully designing the meta-atom geometry allows one to engineer electrodynamic properties not observed in naturally occurring materials, including negative index of refraction, perfect lensing, and giant optical activity [19,20,22–24].
Metamaterials’ strong resonant modes, reminiscent of strong cavity coupling, suggest that they could also hybridize with low energy modes of naturally occurring materials. Studies have shown [25] coupling between a split ring resonator (SRR) metamaterial and a phonon in a thin film of SiO$_2$. These experiments show that designing a metamaterial resonance frequency to match an infrared active phonon’s resonant frequency causes the metamaterial’s near-fields and phonon mode to hybridize and manifest as a Rabi splitting. Additional studies have probed strong coupling between metamaterials and 2D electron gasses, superconductors, and electronic phase transitions [26–31].
Using metamaterials to excite magnetic responses has also been reported [32–34]. Antiferromagnetic materials, whose ultrafast optical control has been studied for applications in spintronics and information processing [35–37], is often characterized with far-field terahertz (THz) measurements. Far-field THz measurements at normal or near-normal incidence, however, often have small or moderate magnetic fields (0.25 A/m) [38] and they constrain the incident THz electric and magnetic fields to be parallel to sample surfaces, leaving out-of-plane features unexplored. Mukai et. al. [32] showed that the local, magnetic near-field from a THz pulse could be amplified 20-fold by incorporating a single SRR. The increased magnetic fields allowed for an efficient in-plane interrogation of the antiferromagnetic (AFM) resonance in HoFeO$_3$, as probed by near-field Faraday rotation measurement. A question still remains: can strong coupling between a metamaterial with an AFM simply manifest in far-field THz transmission experiments? If so, one could study strong light-matter coupling by performing far-field transmission experiments on metamaterials deposited on magnetically active substrates, such as AFMs or more exotic spin liquid candidates [39].
To answer this, we numerically evaluate the coupling of SRRs on an AFM by performing finite difference time domain (FDTD) calculations to simulate the far-field THz time domain spectroscopy (THz TDS). We show that the metamaterial’s near-fields hybridize with the AFM’s low energy magnons and produce an avoided crossing. We derive a two coupled oscillator model, show that it accurately describes the simulation, and use it to quantify the avoided crossing. We examine the role of the magnon’s magnetic dipole oscillator strength, quantify the spatial depth at which hybrid modes exist, and conclude metamaterial-mode coupling to be a viable platform for studying magnetic materials.
2. Methods
We applied FDTD numerical techniques using MIT’s Electromagnetic Equation Propagation (MEEP) software [40] to the geometry depicted in Fig. 1(A). Model time domain THz pulses, Gaussians with full width, half maximum of 0.493 ps, impinge on the material stack with a wavevector antiparallel to the Y-axis. The THz electric (magnetic) field $E_\omega$ ($H_\omega$) points along the X- (Z-) axis. An infinitesimally thin layer of perfect electric conductor metallization was selected to reduce computational complexity. Periodic boundary conditions were defined on the computational cell’s walls along the Z- and X- axes to simulate an infinite lattice.
Fig. 1. SRR - AFM coupling geometry A. A single unit cell, where L (30 $\mu$m), S (5 $\mu$m), G (5 $\mu$m), T$_1$ (between 0 $\mu$m and 3 $\mu$m), T$_2$ (500 $\mu$m), and P (35 $\mu$m) represent the metamaterial’s side length, strip width, gap length, spacer thickness, AFM thickness, and the periodicity, respectively. An infinitesimally thin layer of perfect electric conductor metallization was used. The polarization of the electric field is along the X-axis, which is parallel to the SRR gap. B. Near magnetic field enhancement factor in the dielectric spacer 1 $\mu$m below the SRR, computed in CW analysis. The SRR significantly enhances the incoming field amplitude. C. The simulated AFM crystal with the spins parallel to the incident THz H-field to prevent excitation from only the incident THz pulse.
The simulated material stack consists of an SRR, a dielectric spacer of variable thickness to evaluate the coupling depth, and the AFM. Metamaterials are modelled with a lumped circuit representation [19,20,22]. The “ring" has an inductance; the “split" in the ring has a capacitance. The SRR metamaterial geometry was selected because it exhibits strong confinement of enhanced magnetic near-fields near resonance [32,34]. The metamaterial dimensions L, S, G, and P were engineered to place the SRR resonant modes in the bandwidth accessible by typical THz time domain spectrometers (0.2 - 2.0 THz) [38,41]. When exposed to the incident THz electromagnetic wave, the THz electric field capacitively couples to the SRR gap and causes a current to flow through the ring. This current produces a magnetic field pointing along the Y-axis, whose magnitude is significantly larger than the magnetic field of the incident THz pulse. We compute the magnetic field enhancement by sourcing continuous wave THz radiation at the SRR on a non-magnetic dielectric spacer. We compute the magnetic field enhancement as the ratio of the Y-axis magnetic field in the plane 1 $\mu$m below the SRR to the Z-axis magnetic field of an incident continuous THz wave sourced at the SRR's resonant frequency with a T$_1$ and T$_2$ of 500 $\mu$m and 0 $\mu$m, respectively (shown in Fig. 1(B)). The magnetic field enhancement 1 $\mu$m below the SRR’s center is 15.2, comparable to those measured [32] and simulated in related metamaterials [42].
Our model AFM emulates a hypothetical two-sublattice easy-axis antiferromagnet with a 4-fold symmetry along the Z-axis. The easy-axis points in the Z-direction and thus the spins are consequently aligned and anti-aligned in the Z-axis (Fig. 1(C)). Such an AFM would exhibit resonant modes in the form of magnons that can be excited by a uniform THz magnetic field. We assume few meV scale exchange and anisotropy energy scales, placing the magnon resonance frequency in accessible terahertz bandwidths [43]. We have chosen to set the AFM’s relative permittivity to be 6, a typical number for AFMs such as CaFe$_2$O$_4$ [44–46], in order to keep the SRR’s LC-resonance of the SRR within the THz frequency bands.
The AFM was modeled with a magnetic permeability whose frequency dependence is defined by a Lorentz oscillator model:
In order to explore a wide range of quantum magnets, $\omega _{AFM}$ was varied from 0.1 THz to 2 THz in 50 GHz increments. This also allows us to simulate different hypothetical magnon modes, as well as the potential temperature dependence of the magnon frequency. As the temperature is lowered beyond the Néel temperature, magnons can blueshift by a few hundreds of GHz [48,49], which is within the range of our simulated parameters.
Free space THz TDS experiments also require measurement of a reference signal, wherein THz electromagnetic waves are detected without a sample present. We imitate such a reference by simulating an additional wave that is transmitted through a non-magnetic material with the same dielectric permittivity as our AFM. This ensures that minimal phase difference between the sample and reference signals. We then computed the ratio of the Fourier transforms of the sample and reference signals, the transmission amplitude, from which the SRR - AFM coupling was derived.
To characterize this coupling, we performed two different sets of simulations. First, we observed the effects of oscillator strength on the coupling strength by tuning the constant $\alpha$. This was varied from 1 to 50 at increments of 5. Second, we measured the spatial extent of the coupling by introducing a spacer in between the two materials in the form of a non-magnetic dielectric of thickness T$_1$ between the SRRs and AFM. The permittivity of the spacer was matched to that of the AFM in order to avoid internal reflections within the material stack. The coupling constant for all oscillator strengths and spacer thicknesses were then calculated.
3. Results
To establish features arising from SRR / AFM magnon hybridization, we simulate the following geometries: when the SRRs are absent and only the AFM is present (Fig. 2(A), red), when the SRRs are present and the AFM is absent (Fig. 2(A), blue), and when both the SRRs and AFM are present (Fig. 2(B)). When the SRRs are absent, the AFM does not respond to the applied THz pulse. This is expected, as the exciting THz H-field H$_\omega$ interacts with only the frequency-independent component of $\stackrel {\leftrightarrow }{\mu }_{r}(\omega )$, which is has unit permeability. When the SRR is added but the AFM is removed, the sample responds via an 886 GHz LC resonance that doesn’t depend on AFM magnon resonant frequency, as expected.
Fig. 2. The far-field THz transmission of A. only the AFM (red) and only the SRRs with a non-dispersive material stack (blue). The magnon mode is present in the AFM only, but it is optically inactive. The SRR only transmission shows a dip in transmission at the SRR LC resonance. B. Transmission through SRRs on model AFM material, showing SRR LC and magnon hybridization manifesting as an avoided crossing.
When both the SRRs and the AFM are present (Fig. 2(B)), however, we see features arising from the SRR near-field - AFM interaction. For the situation where the magnon frequency is substantially lower or higher than the SRR LC resonance frequency, the transmission is dominated by the SRR-only features. In the case that the magnon frequency matches the LC resonance, an avoided crossing manifests. At this avoided crossing, the transmission decreases due to the excitation of a hybrid SRR - magnon mode.
The transmission also shows weak features at large magnon frequencies, even when the magnon frequency is a few times larger than the SRR resonance frequency. Since this feature is not excited by incident electric and magnetic fields in Fig. 2(A), this decreased transmission must also be caused by an SRR-magnon hybrid mode. The SRR is being driven off its primary resonance, and thus the near-fields are weaker, leading to weaker hybridization.
4. Discussion
We use a model of two coupled oscillators to understand the results of the numerical simulations. Here, the SRR can be modelled as a lumped element resonator with a parallel inductance $L$, capacitance $C$, and radiative / conductive loss $R$ [50]. The AFM substrate can be pictured as the SRR inductor’s magnetic core that makes the effective inductance $L_{eff}=\mu _r(f) L$ and capacitance $C_{eff}$. From the lumped circuit model, the LC circuit has current $I(t)$ that must satisfy:
Equation (3) is solved by a Fourier transform, namely monochromatic plane waves of the form $I(t) = I_0\exp {(-i \omega t)}$. Substituting Eq. (1) for $\mu _r(f)$: produces
We can simplify by defining $\omega _{SRR} = 1/\sqrt {\varepsilon LC}$ and $\gamma _{SRR} = R/L$. Assume that the avoided crossing gap is small, namely that $\omega _{AFM} - \omega _{SRR} \equiv \beta$, where $\beta$ is small compared to $\omega _{AFM}$ and $\omega _{SRR}$. Substituting this definition into Eq. (4), simplifying, and ignoring terms of order $\beta ^2$ yields:
Equation (5)’s form is similar to that of a two coupled oscillator model, which relates the behavior of the two distinct modes into a single hybridized mode via a coupling constant $V$ [51–53]. Because of this similarity, the right hand side of Eq. (5) is analogous to the square of the coupling constant $V$.
In the strong coupling regime $2V>\gamma _{SRR}-\gamma _{AFM}$ (which occurs when $\Omega > 0$), the coupling constant can be found by [51]:
To compute coupling constant $V$, we must know Rabi splitting $\Omega$, $\gamma _{SRR}$, and $\gamma _{AFM}$. $\gamma _{AFM}$ is specified in the AFM definition. $\gamma _{SRR}$ (and $\omega _{SRR})$ are characteristics of the SRR dimensions and are extracted from simulation with no AFM (shown in Fig. 3(A)), yielding $\gamma _{SRR}$ of 114 GHz (and $\omega _{SRR}/2\pi = 886$ GHz). The Rabi splitting frequency $\Omega$ is quantified as the frequency separation between the left and right local minima in transmission magnitude when the SRR resonant frequency matches the AFM magnon frequency (see Fig. 3(A)). The center frequency and Rabi splitting are well-matched to the capabilities of THz TDS techniques. The largest simulated oscillator strength, $\alpha = 50$ ($\alpha \sigma _0$ = 0.113), produces the largest extracted coupling constant $V$, which was found to be 0.45 meV $\sim$ 109 GHz.
Fig. 3. Analysis of hybrid mode formation with $\alpha = 50$ and $\alpha \sigma _0$ = 0.113 A. SRR LC resonance with a frequency-independent permeability (blue) and when the AFM magnon resonance matches that of the SRR (red). The result is an avoided crossing with Rabi frequency $\Omega$. B. Validity of coupled oscillator model. The horizontal and diagonal dashed lines indicate the non-hybridized SRR and magnon resonances, circles (squares) show the low (high) energy modes, and solid lines show two coupled oscillator model solutions, which are in good agreement with simulation.
The system eigenfrequencies $\omega$ were computed by solving eqn. (7) [51]:
The solutions to Eq. (7) (solid black lines) and the uncoupled SRR/AFM modes (dashed red lines) are shown in Fig. 3(B). When compared to the hybrid SRR - AFM system’s eigenmodes derived from the simulation (red squares and blue dots) the two oscillator model captures well the avoided crossing of the SRR/AFM system.
The nature of the hybrid mode can be further explored by adjusting the oscillator strength and varying the spacer thickness to examine the depth at which such modes exist. Tuning only the magnon oscillator strength $\alpha$ (Fig. 4(A)) strongly impacts the hybrid mode formation. At low oscillator strength, only the pure SRR resonance manifests. Increasing $\alpha$ broadens the resonance, which separates into two distinct modes at high oscillator strength. Since the coupling equation derived for the circuit model (Eq. (5)) indicates a linear dependence to $\sqrt {\alpha }$, the results in the inset of Fig. 4(B) adds validity to this model. The coupling strength $V$ at each oscillator strength was extracted and fit to a linear equation (valid only when the Rabi splitting is greater than the linewidth):
Fig. 4. Tuning of hybrid mode formation. A. Effects of tuning oscillator strength on transmission, showing no Rabi splitting at low oscillator strength and (inset) coupling that goes as $\sqrt {\alpha }$, per the two coupled oscillator model. B. SRRs to AFM spacer thickness dependence on transmission (with $\alpha = 50$). A Rabi splitting is observed at zero spacer thickness. Rabi splitting vanishes when the spacer thickness reaches 3.0 $\mu$m. The two coupled oscillator coupling $V$ versus spacer thickness is show in the inset.
From this fit shown in the inset of Fig. 4, $y$ and $A$ were determined to be 0.065 meV $\sim$16 GHz and 0.054 meV $\sim$ 13.1 GHz respectively. Though readily measurable with ideal THz TDS systems, far-field THz transmissions of strong SRR-AFM coupling will be more clear if one uses an AFM whose peak imaginary permeability is large. One such material is YFeO$_3$, which has a peak imaginary permeability of 0.27, oscillator strength of 8.1 $\times 10^{-4}$, and a $\gamma _{AFM}$ of 0.9 GHz, and has been prepared in 0.5 mm-thick single crystalline samples [54,55].
To evaluate the coupling’s spatial extent, the thickness of the dielectric spacer between the SRRs and AFM was adjusted (Fig. 4(B)). The modes have clear separation with zero spacer layer thickness. As the dielectric spacer thickness increases, the Rabi frequency decreases monotonically until the mode separation vanishes when the system enters the weak coupling regime with $\sim$3 $\mu$m spacer thickness. We quantify this behavior by deriving the coupling constants at each separation (Fig. 4(B), inset). The coupling is strongest at the AFM surface, decreases monotonically in spacer thickness on a few-micron scale. Experimentally engineered coupling will be most visible in experiment by patterning SRRs at most a few microns from a high-oscillator strength quantum magnetic material, which is readily achievable with current nanofabrication technologies.
5. Summary
The coupling between a THz metamaterial and AFM was numerically evaluated to understand the interaction between the SRR’s LC resonance and a magnon in an AFM material. The metamaterial was used to enhance incident THz-frequency magnetic fields along an axis not accessible by conventional far-field THz TDS. When the magnon resonance and SRR resonance are frequency-matched, the SRR near-fields and magnon form a hybrid modes, which manifest as an avoided crossing. This phenomenon can be understood with a coupled two oscillator model. The coupling constant, on the order of 100 GHz, shows strong dependence on the magnon oscillator strength. Materials with small oscillator strength (i.e. peak imaginary permeability on the order of 0.01) will likely not show a measurable frequency splitting. The strong coupling regime was found to occur when the AFM was within 3 $\mu$m of the SRR. These simulations motivate the experimental exploration of the magnon-photon coupling with far-field THz transmission experiments on magnetically ordered materials with metamaterials. Thick films of single-crystalline YFeO$_3$, for example, lie within the parameter space simulated here and are excellent candidates for measurement. Future work should also take advantage of optimized metamaterial designs [42] to engineer ultrastrong metamaterial-AFM coupling. The convenient geometry, relative ease of far-field experiments, and large H-field enhancement by the SRR allows for strong magnetic fields, whose coupling to more exotic magnetic orders could prove fruitful.
Funding
Institute for Materials Research, Ohio State University (EMRG-00030, IMR-FG0168); National Science Foundation (DMR-2011876).
Acknowledgments
The authors thank Thuc Mai, Yufei Li and Chase Lyon for comments on the manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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