Enhanced Faraday rotation and magneto-optical figure of merit in gold grating/graphene/silicon hybrid magneto-plasmonic devices Open Access

Single layer graphene (SLG), an atomic layer thick material bonded with carbon atoms, has been widely studied for photonic device applications from the visible to Terahertz (THz) frequency range due to its unique band structure and optical properties.^1–5 In the THz frequency range, graphene serves as a candidate material for optical modulators and Faraday rotators. Giant magneto-optical (MO) Faraday rotation has been demonstrated theoretically and experimentally in magnetically biased graphene.⁶ However, due to the low cyclotron frequency of graphene materials, large Faraday rotation is only observable at several hundred GHz frequencies with experimentally achievable magnetic fields, limiting their application frequency range.^6,7 Due to the diamagnetic nature of graphene, a high magnetic field of several Tesla is also required to yield several degrees of Faraday rotation,⁷ which calls for device designs to enhance the magneto-optical properties. Several device structures have been proposed to tackle the above challenges, such as, utilizing THz magnetoplasmonic resonance in graphene nanostructures,^4,7–11 and combining graphene with Fabry–Pérot cavities^8,12–14 or photonic crystals^15–17 to enhance the light-matter interaction. However, due to the single atomic layer thickness and low carrier density in graphene, the oscillator strength of graphene magnetoplasmonic resonance is low, leading to reduced Faraday rotation and higher scattering loss at high THz frequency range, whereas photonic crystal structures usually require multilayer thin films and relatively complex device structures. Meanwhile, for nonreciprocal photonic device applications, simultaneous enhancement of the optical transmission and the Faraday rotation angle is required. To evaluate the performance of a MO device, the magneto-optical figure of merit (FOM) can be defined as $θ*T$⁠, where θ is the Faraday rotation angle and T is the transmission.¹⁸ Experimentally, single layer graphene can show a FOM of 5.2 at 2.4 THz.⁶ It is even harder to achieve high FOM at higher frequencies. Therefore it is highly desired to design novel and simple device structures to enhance the magneto-optical effect, figure of merit, and extending the application frequency range of graphene nonreciprocal photonic devices.

Extraordinary optical transmission (EOT) effect, where excitation and tunneling of surface plasmonic modes in subwavelength metallic structures leading to higher transmission than predicted value, has been widely observed theoretically and experimentally.^19–22 Recently, the EOT effect has been applied to simultaneously enhance the Faraday rotation and optical transmission in gold grating and magnetic oxide thin film heterostructures in the visible and near infrared frequency range.^18,23 The coupling between the TE waveguide (WG) mode and the TM hybrid WG-plasmon mode leads to strong TE-TM mode conversion via the magneto-optical oxides, whereas the transmission is also enhanced due to the EOT effect, leading to a high device FOM. On the other hand, Au grating coupled to graphene nanoribbon structures can also induce the EOT effect in the mid-infrared, causing a stronger light-mater interaction and higher optical modulation efficiency.¹⁰ In THz, gold approaches a perfect electronic conductor. The excitation of spoof surface plasmons (SSPs) also induces EOT like effect.^24,25 However there has been no report on magneto-optical devices using the SSP and EOT effect at the THz frequency range.

In this paper, we theoretically demonstrate that a heterostructure of Au grating/SLG/Si can simultaneously yield high Faraday rotation angle and optical transmission in THz. Thanks to the enhanced light-mater interaction using Au SSP, we demonstrate high Faraday rotation angle and FOM in a wide frequency range of 0.43–24 THz. The Faraday rotation and FOM of our devices can also be higher than SLG films at their cyclotron frequencies, which is not achievable in graphene magnetoplasmonic devices. Such devices demonstrate a promising potential for THz nonreciprocal photonic device applications.

Figure 1(a) shows the schematic of the device structure. The device consists of single layer graphene sandwiched between a gold grating and a THz transparent high resistivity silicon substrate. A static magnetic field B is applied perpendicular to the graphene film surface. TM polarized THz frequency light (polarization direction perpendicular to the grating direction) is incident onto the grating, which excites a hybrid TM polarized SSP mode and a waveguide mode in the device. The transmitted light shows polarization rotation due to the magneto-optical effect of graphene. For a single layer graphene material, the conductivity tensor can be well described by the semiclassical Drude model as follows:^6,7

where E_F is the Fermi level, τ is the scattering time, and $ωc=eBVF/EF$ is the cyclotron frequency. V_F represents the Fermi velocity of the Dirac fermions in graphene (V_F = 9.5 × 10⁵ m/s).⁴ For free-standing monolayer graphene or graphene on a substrate, the Faraday rotation angle is proportional to σ_xy, whereas the transmission loss is proportional to σ_xx.⁶ In our device, the TM WG-SSP mode enhances the field intensity on graphene and the Faraday rotation angle. Meanwhile, proper thicknesses of Si and Au grating period and width also allow the EOT effect at the same frequency, therefore enhancing the device transparency and magneto-optical figure of merit at the same time. We use the finite element method with commercial software COMSOL MULTIPHYSICS to simulate the optical transmission and Faraday rotation spectra.²⁶ Considering the grating is one-dimensionally periodic, a two-dimensional (2D) model is applied, as shown in Fig. 1(b). Scattering boundary and periodical boundary conditions are applied for the top/down and left/right boundaries, respectively. We treat graphene as a frequency dependent surface current boundary condition, which is defined by the Maxwell constitutive equation $J(ω)=σ̃(ω)⋅E(ω)$⁠, where ω is the frequency of the incident light, J(ω) is the surface current, $σ̃(ω)$ is the conductivity tensor of graphene, and E(ω) is the incident electric field. We take the Fermi energy of graphene as 0.2 eV or 0.1 eV for optimized FOM in different structures. For the scattering time, we use the experiment available value τ = 0.1 ps (corresponding to a carrier mobility of 4500 cm² V⁻¹ s⁻¹). For the state of the art, higher quality graphene can be experimentally fabricated by CVD with mobility of around 25 000 cm² V⁻¹ s⁻¹,²⁷ which will lead to even better device performances. The Au material is defined by the Drude model as $εm=ε∞−ωp2/(ω2+iγω)$⁠, with ε_∞ = 1, ω_p = 1.37 × 10¹⁶ rad/s, and γ = 2.72 × 10¹³ rad/s in THz.²⁸ A dispersionless permittivity value of ε = 11.66 is applied for silicon for the frequency range studied in this paper.⁸ For different operation frequencies, we varied the Au grating period, filling factor, and the silicon thicknesses to allow the best device performance, as discussed in detail below. The applied magnetic field is fixed at 7 T for all simulations.

Figure 2(a) shows the optical transmission spectrum of the Au grating/SLG/Si device at a frequency range of 0.42 THz to 0.44 THz. The period and width of the Au gratings are 600 μm and 465 μm, respectively. The thicknesses of Au and Si are 400 nm and 510 μm, respectively. A clear transmission peak with up to 64% maximum transmittance is observed at this frequency range due to the EOT phenomenon.¹⁸ A small transmission dip is observed at around 0.431 THz in the transmission spectrum, which is attributed to the absorption of graphene when the SSP mode is excited.²⁸ Figure 2(b) shows the Faraday rotation angle and Faraday ellipticity spectrum of the device in the same frequency as Fig. 2(a), compared with free-standing SLG at the same frequency range. A significantly enhanced Faraday rotation angle approaching 20° with a sharp peak is observed at the EOT frequency in Fig. 2(a), which is 3 times higher compared to SLG at the same frequency as indicated by dashed lines in Fig. 2(b). Simultaneous enhancement of the Faraday ellipticity (blue solid line) is also observed. The Faraday ellipticity is zero at the frequency of the maximum Faraday rotation angle, indicating linear polarization of the device at this frequency. The FOM achieves 14.86 compared to 6.68 in SLG, which also shows 2.2-fold improvement. The operation frequency range can be extended to other THz frequency range by properly designing the device structure. In Figs. 2(c) and 2(d), we demonstrate the Faraday rotation angle and FOM of six optimized structures with resonance frequency ranging from 0.43 to 24 THz. For device operating at 0.431 THz, 0.645 THz, 1.99 THz, 7.97 THz, 13.1 THz, and 23.5 THz, the corresponding Au grating periods/Au grating widths/silicon thicknesses of the six devices are 600 μm/465 μm/510 μm, 400 μm/342 μm/310 μm, 80 μm/43 μm/66.3 μm, 30 μm/5.6 μm/22.7 μm, 20 μm/3 μm/15.5 μm, and 10 μm/1.9 μm/6.8 μm, respectively, as shown in Table I. Compared to SLG, an enhancement of the Faraday rotation angle by a factor of 2-7 folds is observed in all structures at the same frequency range. The maximum enhancement is around 7-fold at the frequency of 13.1 THz. Moreover, the device Faraday rotation angle exceeds the maximum Faraday rotation value that is achievable in SLG at its cyclotron resonance frequency. This result is observed in a large frequency band from 0.4 to 13.1 THz [shaded region in Fig. 2(c)], which is not observed in graphene magnetoplasmonic structures, where the Faraday rotation angle is always lower than SLG at its cyclotron resonance frequency.^7,29 The reason for the much higher Faraday rotation angle enhancement of our devices is due to the stronger1b field localization by the excited SSP mode in Au gratings compared to graphene magnetoplasmonic structures.²⁸ The FOM also increases due to the EOT of the devices, except for the enhancement of Faraday rotation angles, as shown in Fig. 2(d). 1.5- to 6-fold enhancement of the FOM compared to SLG can be observed in different devices across the whole THz frequency range that has been studied.

Figure 3 shows the optical transmission, Faraday rotation, Faraday ellipticity, and FOM of the device operating at 0.431 THz as a function of the graphene Fermi level. We first notice that the maximum Faraday rotation, Faraday ellipticity, and optical transmission are always at the same frequency range independent of E_F. This is drastically different from SLG or graphene plasmonic structures, where the maximum transparency and Faraday rotation is related to the Fermi level and the cyclotron frequency.^6,7 This difference is due to the fixed SSP frequency determined by the Au grating and silicon substrate dimensions. Figure 3(a) shows the optical transmission as a function of graphene Fermi levels. When the Fermi level is equal to 0.1 eV, the transmission of the device can reach 80% at the resonant frequency. With increasing the Fermi level up to 0.6 eV, the device gradually becomes opaque due to the interband transition characteristics of the THz absorption in graphene materials, where the density of states accessible to the absorption process is proportional to the Fermi level $D(EF)=2EFπℏ2VF2$⁠. Figure 3(b) shows the Faraday rotation spectrum as a function of graphene Fermi level. With increasing E_F from 0.1 eV to 0.6 eV, the maximum Faraday rotation angle increases from 10° to 26° for E_F = 0.3 eV, then gradually decreases to 20° for E_F = 0.6 eV, whereas the spectrum gradually broadens with increasing E_F due to the higher absorption of the graphene material. The highest Faraday rotation angle is observed at E_F = 0.3 eV, which corresponds to a matching condition between the cyclotron resonance frequency and the device EOT frequency. The same trend also applies to the Faraday ellipticity of the device, as shown in Fig. 3(c). For the device FOM shown in Fig. 3(d), the maximum FOM reaches 14 at E_F = 0.2 eV, rather than E_F = 0.3 eV where the highest Faraday rotation is observed, indicating a trade-off between the material absorption and Faraday rotation. Compared to SLG,⁶ graphene plasmonic,⁷ or graphene-based photonic crystal¹⁵ devices, the Au/SLG/Si device allows large tunability of the magneto-optical properties by electronically doping graphene at a pre-designed frequency, which is potentially applicable in THz reconfigurable optical isolators and Faraday modulators.

To further understand the device operation mechanism, we simulated the optical transmission spectrum of the Au/SLG/Si device as a function of the grating period and incident wavelength (corresponding to a frequency range of 0.417 THz–0.448 THz), as shown in Fig. 4. In THz, the dispersion relation of SSP for a 1D grating is given by²⁸ 

where β, k₀, a, p, h, and ε_d are the mode propagation constant, the free space wave vector, the interval between two gratings, the grating period, the thickness of the grating material, and the permittivity of the surrounding dielectric material, respectively. For the TE polarized incident light, only the TE WG mode can be excited in the silicon layer, and the SSP mode cannot be excited according to the Maxwell’s equations. Thus, there is no EOT phenomenon for the s-polarization case for a free-standing gold grating. But when a dielectric layer, such as silicon is placed close to the grating, a surface wave is excited, which allows for EOT for the s-polarization incident light due to the coupling between the waveguide mode in the silicon layer to the free space via the gold grating,³⁰ as shown in Fig. 4(a). By increasing the grating period, the transmission peak is red shifted to longer wavelengths. For TM polarized incident light, both the WG mode and SSP mode can be excited, as shown in Fig. 4(b). When the WG mode has the same propagating constant with the SSP mode, these two modes hybridize to a WG-SSP mode. An anticrossing spectrum shape with up and down brunches of the transmission spectrum is observed, as shown in Fig. 4(b). We indicate the TM WG-SSP mode by a black dashed line and overlay the TE WG mode in Fig. 4(a) as a red dashed line in Fig. 4(b). The two modes overlap, with two crossing points at a grating period of p = 600 and 623 μm, respectively. For such device geometries, the TE-TM mode conversion is strongly enhanced via the graphene magneto-optical effect, whereas a high optical transmission is simultaneously achieved due to the EOT effect as discussed above. Figures 4(c) and 4(d) show the E_x and E_z field (parallel to the grating) distribution when the incident light is TM polarized, with the corresponding device structure and incident wavelength indicated by a green star symbol shown in Fig. 4(b). As shown in Fig. 4(c), the E_x field is strongly localized at the two corners of the gold grating which strongly interacts with graphene at the resonance wavelength of the SSP mode. Meanwhile, the E_z field (parallel to the grating) is induced by graphene’s magneto-optical Faraday effect as shown in Fig. 4(d). This field distribution matches with the 3rd order TE waveguide mode in the silicon substrate. Thus, a TM to TE mode conversion is achieved, leading to a significantly enhanced Faraday rotation.

In summary, we demonstrate enhanced Faraday rotation and magneto-optical figure of merit in Au/SLG/Si devices at the THz frequency range. The strong hybridization between the TM WG-SSP mode and the TE waveguide mode in such devices can significantly boost the Faraday rotation and magneto-optical FOM of the graphene material in a frequency range from 0.43 to 24 THz. Such devices are advantageous for its simpleness and wide operation frequency range. The strategy of combining a noble metal SSP mode with graphene opens a new way for THz MO device applications.

This work was supported by National Natural Science Foundation of China (Nos. 61475031 and 51522204), the Fundamental Research Funds for the Central Universities (ZYGX2014Z001), the Science Foundation for Youths of Sichuan Province (2015JQO014), and the Open Foundation for Key Laboratory of Multi-spectral Absorbing Materials and Structure of Ministry of Education (ZYGX2016K009-2).