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. 2026 Oct 12. Online ahead of print. doi: 10.1039/d6na00610h

Exceptionally high Verdet constant in gold nanodisc arrays

Gajendra Mulay a,b, Shraddha Choudhary a,c, Ashwin A Tulapurkar d, Daria O Ignatyeva e, Vladimir I Belotelov e, Shriganesh S Prabhu a, Venu Gopal Achanta a,f,✉
PMCID: PMC13717621  PMID: 42959816

Abstract

Magneto-optical effects in non-magnetic noble metals can be invoked by leveraging the in-plane Lorentz force at engineered plasmonic resonances. We demonstrate a 2-D array of gold nanodiscs designed to host a hybrid resonance of localized plasmon and surface lattice modes. The structure exhibits a Verdet constant of 1.98 × 106°·T−1 m−1, corresponding to a Faraday rotation of −0.15° at a 1 T magnetic field. This Verdet constant represents a 15-fold enhancement over a continuous, unpatterned gold film of the same thickness measured under identical conditions and is highly competitive with many plasmon-enhanced diamagnetic nanostructures. Numerical simulations show that the magnetic field independent rotation is due to structure imperfections and thus with better tolerance in nanofabrication, one could get much higher absolute Faraday rotation at these resonances. These findings offer new opportunities for harnessing strong magneto-plasmonic effects in optoelectronic devices by patterning non-magnetic metal contacts.

Coupling localized plasmons to surface lattice modes in gold nanodisc arrays strongly enhances Faraday rotation in non-magnetic noble metal.graphic file with name d6na00610h-ga.webp

Magneto-optical (MO) effects arising from the breaking of time-reversal symmetry in materials are manifested through the off-diagonal elements of the dielectric constant tensor.1 The strength of the Faraday effect is quantified by the Verdet constant, which typically ranges from 103–104° T−1 m−1 for inorganics to 104–106° T−1 m−1 for alkali gases, materials such as Terbium Gallium Garnet (TGG)2 or organic polymers.3 While the enhancement of MO effects in transparent ferromagnetic materials using plasmons, Mie resonances, or metasurfaces4–15 is well-established, the weaker response in non-magnetic noble metals is also a subject of significant investigation.16–19 Inducing a strong MO effect in materials such as gold, which are standard for electrical contacts in optoelectronics, remains a key opportunity for the design of novel active devices. The key challenge is in enhancing the weak off-diagonal polarizability (αxy) of the metal which is related to the off-diagonal elements of the dielectric tensor. Off-diagonal polarizability is proportional to the applied magnetic field, which limits the possibilities to enhance MO effects, as the magnetic fields needed to see large MO effects are prohibitively high, exceeding 10 s of Tesla. An alternative is to enhance the αxy at a designed resonance in a structured metal nanostructure.

Hybrid resonances, formed by coupling Localized Surface Plasmon Resonances (LSPR) with Surface Lattice Resonances (SLR), are known to produce collective optical responses that are substantially narrower than the localized resonance of an isolated particle and have been studied to demonstrate novel functionalities such as low-threshold lasing, refractive-index sensing, and enhanced light–matter interaction.20–24 In this work, we propose and demonstrate a 2-D gold nanodisc array that supports a hybridized LSPR–SLR mode to significantly enhance the Faraday effect, achieving an exceptionally large Faraday rotation and Verdet constant in a purely non-magnetic noble metal. This enhancement is driven by the coupling of individual LSPR modes into a collective SLR, which creates a strong, narrow resonance in the Faraday spectrum. Additionally, the nanostructuring itself modifies the gyrotropic component of the permittivity of gold nanodiscs, thereby amplifying the magneto-optical response. At this resonance, we measured a Verdet constant that is three orders of magnitude higher than in other non-magnetic metal nanostructures. This gives a total Faraday rotation that is 15 times greater than that measured on continuous, unpatterned gold film of the same nominal thickness deposited on the same substrate. We compare these with literature values for diamagnetic materials and nanostructures. Both the Verdet constant and the Faraday rotation angle show a giant enhancement compared to those reported in nanostructures with non-magnetic metals, demonstrating the potential of resonant structures designed for specific applications.

Structure supporting hybridized LSPR and SLR

We study nanostructures consisting of gold nanodiscs with a diameter D = 300 nm and height h = 80 nm arranged in a 2-D array with period, P = 500 nm, 550 nm, and 600 nm on a quartz substrate (Fig. 1a). Gold nanodisc arrays were fabricated by a multi-step process involving sputtering of gold thin films, electron beam lithography for nanopatterning, and reactive ion etching for transferring the pattern to the metal film (SI S1).

Fig. 1. Surface lattice mode (a) Schematic of an array of gold nanodiscs and the geometry of the applied optical and external magnetic fields. (b) The extinction cross-section and Faraday rotation spectrum of a single Au nanodisc were calculated using modified long-wavelength approximation (MLWA).21 The inset shows the electric field distribution at 860 nm resonant wavelength calculated at the centre of the disc using FDTD. (c) SEM image of P = 550 nm array. (d) Experimental and (e) simulated transmittance spectra of the nanodisc arrays. The blue-shaded region shows the extinction cross-section (Qext) for comparison. (f) Simulated electric field profile for 550 nm period gold nanodisc array showing the lattice mode excited by light polarized along the x-axis.

Fig. 1

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The periods were chosen to excite the coupled LSPR and SLR modes. In metal nanodiscs, excitation of LSPR significantly modifies the effective polarizability tensor,25 so the extinction cross-section Inline graphic has a maximum at a wavelength depending on the nanodisc diameter D and height h (blue curve in Fig. 1b). This maximum is associated with excitation of an electric dipole in the nanodisc (inset of Fig. 1b). Moreover, in the vicinity of this resonance condition, an electrical dipole, transversal to that excited by the induced light is also generated under the impact of the external magnetic field directed along the axis of the nanodisc, which leads to the appearance of the off-diagonal αxy polarizability component. This results in the substantial broadband increase of the Faraday rotation Inline graphic in the vicinity of LSPR (orange line in Fig. 1b). A coupled-dipole description of this response, and of its modification by the lattice, is given in SI S2.

While LSPR in nanoparticles enhances the local field over sub-wavelength dimensions, their arrangement in 2-D plasmonic crystal lattice results in the interaction between the neighboring nanodiscs and enables the coupling of diffraction modes with the plasmon resonances.20,26 This coupling becomes essential if a diffracted wave propagates along the surface of the array,26–28i.e. under the conditions close to the Rayleigh anomalies (RA) determined by:

graphic file with name d6na00610h-t3.webp 1

where λ is the wavelength of the incident light, θ is the angle of incidence, P is the period of the structure, nβ represents the refractive index of the environment (either nair or nquartz), and ax and by signify distinct diffraction orders of the grating. It follows from eqn (1) that for some angle of incidence and lattice period, Rayleigh anomaly (RA) may coincide with LSPR, which leads to the diffractive coupling of individual LSPR of nanodiscs and formation of the hybrid SLR.

The proposed structure consists of an air superstrate with n = 1 and a fused silica substrate, without an index-matching layer. The substrate refractive index nquartz = 1.46 is used for the substrate side diffraction channel in eqn (1) and in all simulations. At normal incidence, the first-order substrate side Rayleigh anomaly occurs at λRA = nquartz.P, giving 730, 803, and 876 nm for lattice periods of 500, 550, and 600 nm, respectively. For the P = 550 nm array shown in the SEM image in Fig. 1c, the Rayleigh anomaly is located at 803 nm, whereas the hybridized LSPR and SLR response observed in the measured and simulated transmittance occurs near 860 nm, corresponding to a red shift of approximately 57 nm from the Rayleigh anomaly (SI S2). A square lattice was deliberately selected for two reasons. First, at normal incidence, its lowest order reciprocal lattice vectors form two degenerate pairs, (±1, 0) and (0, ±1). This gives a single, well defined Rayleigh anomaly condition, λRA = nP, unlike the hexagonal or honeycomb lattices studied previously.29–32 Second, the geometry is particularly well suitable for Faraday measurements because the fourfold in-plane symmetry of a square array of circular nanodiscs does not introduce intrinsic linear birefringence that could contribute to polarization rotation independently of the applied magnetic field.

To study in more detail the interaction between the LSPR in individual nanodiscs and the SLR appearing in the lattice, finite difference time domain simulations (SI S3) and angle-resolved white light transmission measurements are carried out, see Fig. 2 and SI S4. The dashed lines in transmission spectra are the analytically calculated RAs for each array using eqn (1). While keeping the diameter of the nanodisc constant, when the period of the array is varied, the spectral position of the RA changes and, therefore, the hybridized LSPR–SLR mode for different periods appears at different wavelengths. At normal incidence, the Rayleigh anomaly occurs at 730 nm, 803 nm, and 876 nm for P = 500 nm, 550 nm, and 600 nm, respectively. The Faraday rotation extrema do not need to coincide with these analytical RA wavelengths because the measured magneto-optical response is a Fano-like hybrid response resulting from interference between the broader LSPR and the narrower lattice contribution.

Fig. 2. Simulated (a) and measured (b) transmittance spectra for P = 550 nm array for different incident angles. Dashed lines are the dispersion calculated using eqn (1).

Fig. 2

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Faraday rotation and Verdet constant of gold nanodisc arrays

The resonant interaction of light with gold nanodiscs is accompanied by resonances in the Faraday rotation (Fig. 3(a and b)). The details on the experimental setup, including information about subtraction of substrate and field-independent contributions arising from slight structural imperfections33 is given in SI S5 and S6.

Fig. 3. Faraday rotation measured and simulated Faraday rotation for nanodisc arrays (a) with different periods and B = 1 T; (b) for different external magnetic fields and P = 550 nm (see the legends).

Fig. 3

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The Faraday rotation of the Au nanodisc arrays exhibits a prominent Fano-shaped resonance near the SLR. This asymmetric profile arises from the hybrid coupling between the narrow SLR and the broad LSPR modes. For instance, a Faraday angle peak of 0.15° is observed at λ = 790 nm, and a dip of −0.15° is observed at λ = 880 nm for a lattice period P = 500 nm at B = 1 T. Such a large Faraday rotation is caused by the interplay of two different factors. First, SLR in nanodisc arrays produces narrower resonances than LSPR of individual discs (compare Fig. 1b, d and e),34 which means stronger light–matter interaction and thus enhances the Faraday rotation. The second important mechanism for the enhancement of the Faraday effect is related to the nanostructuring of Gold. In diamagnetic metals, off-diagonal elements of the permittivity tensor responsible for the Faraday rotation are determined by the Lorentz force acting on free electrons. In the case of nanostructures, the surface charge carriers play an important role in their magnetism, whose properties obviously differ from those of bulk electrons. Recent studies showed a giant enhancement in the magnetic susceptibility of nanostructured gold due to the surface effects35,36 and, therefore, gyrotropic terms of the permittivity tensor are enhanced.35,36 Both reasons lead to a significant enhancement of the Faraday effect in the nanostructured gold, as we observed here.

The Faraday rotation measured for the continuous gold film is only 0.01°, so the two-dimensional nanodisc array provides a 15-fold enhancement relative to a continuous, unpatterned gold film of the same nominal thickness, deposited on the same substrate. The enhancement arises from the resonant response of the patterned structure (SI S6). While the array geometry plays a role in the resonance response, we compared the square lattice with the continuous film, since the Verdet constant is defined per unit propagation length according to θF(λ) = v(λ)Bh, making a film with identical thickness and deposition history the appropriate reference for keeping the material and optical path length fixed while isolating the effect of patterning.37 Accounting for the reduction in the amount of gold introduced by patterning, which is approximately 0.28, 0.23 and 0.20 for P = 500, 550 and 600 nm, respectively, gives an enhancement per unit gold of approximately 53-, 64-, and 76-fold. A strongly detuned control array, for which no Rayleigh anomaly falls within the LSPR spectral range, can further separate the contribution of the collective lattice mode from that of the individual disc LSPR. The agreement between measured and simulated Faraday rotation is noticeably better for P = 550 nm and 600 nm than for P = 500 nm (Fig. 3a), which can be understood from the detuning between the Rayleigh anomaly and the isolated disc LSPR near 860 nm. For P = 500 nm, the Rayleigh anomaly is approximately 130 nm blue shifted from the LSPR, whereas for P = 600 nm the two are nearly coincident. Under strong detuning, the response is more strongly dominated by the LSPR, and the resulting Fano asymmetry becomes particularly sensitive to the exact LSPR position, thereby becoming sensitive to variations in disc diameter and height that are introduced during fabrication. Additionally, the enhancement of Re(εxy) associated with nanostructuring used in the simulations was calibrated against the smooth film rotation at 850 nm and is treated as dispersion independent. So, the resulting uncertainty is expected to be more significant for the P = 500 nm array, whose Faraday peak occurs near 790 nm.

As expected, the Faraday rotation shows a linear dependence on the applied magnetic field, exhibiting pronounced resonant behavior near the SLR wavelength (Fig. 3b and SI S6). Thus, the Verdet constant v is calculated using the relation θF(λ) = v(λ)Bh (Fig. 4). Among the various structures, the one with a lattice period of 500 nm exhibited the highest Verdet constant of 1.98 × 106° T−1 m−1 at 880 nm. As shown in Table 1, this value represents a significant enhancement for a plasmonic system and is orders of magnitude higher than that of many other non-magnetic materials.

Fig. 4. Verdet constant for different lattice periods as a function of wavelength.

Fig. 4

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Table 1. The Verdet constant (v) reported in different non-magnetic materials and structures with linear MO response.

Material description Origin of MO v(λ) (×104 deg T−1 m−1) Ref.
Chiral polyfluorene π-conj./B term −38 @410 nm 38
Phthalocyanine A term −33 @800 nm 39
Au NPs/BK7 LSPR 28.98 @532 nm 40
Au NPs solution SPR 0.24 @ 575 nm 41
Au NP 2D-array LSPR + SLR 186 @790 nm This study
P = 500 nm −198 @880 nm
Au NP 2D-array LSPR + SLR 173 @850 nm This study
P = 550 nm −177 @900 nm
Au NP 2D-array LSPR + SLR 144 @890 nm This study
P = 600 nm −169 @940 nm
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The enhancement of the Faraday rotation is often accompanied by a significant decrease in transmittance in resonant plasmonic structures. Thus, an important point from a practical point of view is the value of the figure of merit42,43 FOM = |θF|T0.5 at specific wavelengths (SI S7). Due to the Fano shape, the maximum magnitudes of the Faraday rotation (Fig. 3a and b) are spectrally shifted from the SLR resonance center where transmittance is at a minimum (Fig. 1d and e). The structure with P = 500 nm yields a FOM of 0.07° with a simultaneous transmittance of 0.23. This value is close to what is usually obtained not only in non-magnetic,40 but also in magnetic plasmonic structures.6,44–52

Conclusion

In this work, we have demonstrated that a carefully designed plasmonic nanostructure composed of a non-magnetic metal can generate a prominent magneto-optical (MO) response. By fabricating a gold nanodisc array that supports a hybridized LSPR–SLR mode, we achieved a significant enhancement in Faraday rotation. Our central finding is a field-dependent Verdet constant of 1.98 × 106° T−1 m−1 corresponding to a field-dependent Faraday rotation of −0.15 deg at 1 T. Furthermore, the structure exhibits a promising figure of merit (FOM) of 0.07° based on the simultaneous high transmittance of 0.23, indicating its potential for device applications. These values are not only 15 times larger than those of a continuous, unpatterned gold film of the same thickness, and approximately 53 times larger when normalized to the amount of gold remaining after patterning, but are also comparable with many plasmon-enhanced magnetic nanoparticle systems. In summary, these results open new possibilities for leveraging strong magneto-plasmonic effects in common, non-magnetic metals, enabling the design of novel active and non-reciprocal components for integrated optoelectronic devices.

Conflicts of interest

The authors declare no conflicts of interest.

Supplementary Material

NA-OLF-D6NA00610H-s001

Acknowledgments

DST/INT/RUS/RSF/P-76/2023(G) project by Department of Science and Technology, Government of India, and Tata Institute of Fundamental Research, Mumbai. DOI and VIB acknowledge the Russian Science Foundation project No. 24-42-02008. All authors at institutions affected by sanctions have contributed to this scientific work in a personal capacity. DOI and VIB acknowledge the Russian Quantum Centre.

Data availability

Data will be provided on reasonable request.

Supplementary information: SI 1. See DOI: https://doi.org/10.1039/d6na00610h.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

NA-OLF-D6NA00610H-s001
NA-OLF-D6NA00610H-s001.pdf (2MB, pdf)

Data Availability Statement

Data will be provided on reasonable request.

Supplementary information: SI 1. See DOI: https://doi.org/10.1039/d6na00610h.

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