Enhanced magnetic circular dichroism by subradiant plasmonic mode in symmetric graphene oligomers at low static magnetic fields

J. Q. Liu; S. Wu; P. Wang; Q. K. Wang; Y. B. Xie; G. H. Sun; Y. X. Zhou

doi:10.1364/OE.27.000567

1. Introduction

Nanoheterostructures with an enhanced magneto-optical (MO) activity are interesting from both fundamental and technological points of view, which make many applications in fhe field of nanophotonic devices possible [1,2]. Under external magnetic field, graphene shows different absorption for right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) light, which is called magnetic circular dichroism (MCD) [3]. In recent years, based on surface plasmon resonances, MO response such as MCD [4,5] and Faraday rotation (corresponding to the polarization rotation) have attracted much attention in metallic and graphene nanostructures [6,7]. For example, many new methods are reported to improve Faraday rotation angle in nanostructure when the magnetic field is parallel to the propagation of the incident light exciting the plasmon resonances [8,9]. Large MO response comes from an increase of the magnetic Lorentz force induced by the large collective movement of the conduction electrons in the nanostructures is observed when the plasmon resonance is excited [10]. An enhanced MCD may open the way for the development of a new generation of highly sensitive assays based on the detection of their MCD signals [11]. However, most of the reported MCD in metallic or graphene microstructures are based on the bright electric dipole resonance which couples with light directly, needing remarkably large external magnetic field [12,13]. It is due to the low MO activity in nonmagnetic noble metals and large radiation loss of the bright plasmonic resonances. These features limit further applications of MCD signal.

Subradiant plasmonic modes [14–16] (dark mode) are modes with vanishing dipole moments, which can confine electromagnetic energy more efficiently than bright plasmonic modes due to an inhibition of radiative losses. This character results in narrower spectral line width, making them interacting for enhanced light-matter interactions [17] and for developing enhanced biological and chemical sensors [18,19], detectors, etc. In composite metallic or graphene microstructures, many experimental and theoretical results are reported that Fano resonance [20–25] occurs in the extinction, reflection or transmission spectra as the subradiant plasmonic mode couples with bright plasmonic mode. However, to the best of our knowledge, subradiant modes and Fano resonance in plasmonic microstructure are mainly investigated in the case of linearly polarized light illumination. For magneto-optical materials or microstructures, how MCD associated with LCP and RCP depends on the Fano resonance is not clear.

In the present work, we show that subradiant plasmonic mode and Fano resonance can be realized in a symmetric oligomers consisting of six graphene disks in each unit cell due to the near field coupling. We numerically verify that a dipole moment at the location of one particular graphene disk (nanoparticle) is moved through every location on the ring by symmetric rotations. Particularly, under external magnetic field, we demonstrate that the subradiant plasmonic mode has different excitation efficiencies for LCP and RCP incident waves, respectively. Based on this distinctive feature, remarkable MCD effect is revealed with relatively low external static magnetic field. We show that 24.25% absorption difference is attainable under 0.4 T magnetic field with our designed graphene oligomer-metal substrate array microstructure, which is 3 times larger than MCD based on electric dipole plasmon resonance approach in simple periodic array. The results open the way for many potentially interesting magneto-plasmonic systems and active magneto-optical metamaterials.

2. Model and method

Figure 1(a) schematically shows a periodic array of graphene oligomers consisting of six disks with radii of 60 nm in each unit cell. The periodic array is formed in a square lattice of a period p_x = p_y = 600 nm on a metal substrate (silver) spaced with a insulator (such as SiO₂) layer (see the inset of Fig. 1(b)). In a perpendicular external static magnetic field, the permittivity of graphene can be expressed as the following tensor [26]

ε_{g} = 1 + \frac{i}{ω t ε_{0}} (\begin{matrix} σ_{x x} & σ_{x y} & 0 \\ σ_{y x} & σ_{y y} & 0 \\ 0 & 0 & σ_{d} \end{matrix})

where t = 0.5 nm denotes the effective thickness of graphene disk in z direction, ε₀ is the permittivity of vacuum, and σ_d does not depend on external magnetic field being σ_d = σ_xx (B = 0). The conductivity tensor of graphene can be represented as [27]

σ_{x x} = σ_{y y} = \frac{e^{2} | E_{f} |}{π ℏ^{2}} \frac{i (ω + i τ)}{{(ω + i / τ)}^{2} - ω_{c}^{2}}

σ_{x y} = - σ_{y x} = \frac{e^{2} | E_{f} |}{π ℏ^{2}} \frac{ω_{c}}{{(ω + i / τ)}^{2} - ω_{c}^{2}}

where E_f is the Fermi energy,

τ = μ E_{f} / e V_{f}^{2}

is the electron relaxation time with μ = 10⁴ cm²/(V.s) being the carrier mobility [28,29], ω, ω_c = eBV_f²/E_f and

ℏ

denote the angular frequency of incident light, the cyclotron frequency of electrons and Plank constant, respectively. V_f = 10⁶ m/s is the Fermi velocity in graphene. The Fermi energy is fixed to E_f = 0.5 eV unless otherwise stated. The designed graphene oligomers arrays can be fabricated by electron beam lithography and the reflection spectrum can be measured with Fourier transform infrared. In the simulation, the finite element method [30] is used to calculate the reflection and absorption spectra. The three-dimensional simulations were performed for a single unit cell and periodic boundary conditions with plane wave normally incident in z direction. The perfectly matched layer absorbing boundary conditions are applied at both end of the computing space in z direction. The thickness of graphene disks is assumed to be 0.5 nm and the corresponding meshes are set to 10 nm in the x, y plane, whereas they are set as 0.1 nm in the z direction. The permittivity of graphene layers without and in a static magnetic field was treated as described by Eq. (1). In the calculation, we assume the spacer layer is SiO₂ with a permittivity of

ε_{d}

= 3.9 [31–33], where this assumed value gives an approximation of the glass permittivity in our simulated frequency range. The permittivity of metal silver is taken from [34].

Fig. 1 Schematic of graphene oligomers array and simulated reflection, absorption spectra. (a) Periodic array of symmetric ring of graphene oligomers consisting of 6 graphene disks with radii of 60 nm in each unit cell. (b) The simulated absorption spectra under x-polarized plane incident wave illumination with (black dashed curve with circles) and without (red dashed curve) external magnetic field. The reflection spectrum for external magnetic field B = 0 T is displayed with blue curve. The gap between graphene disk is 40 nm (R = 160 nm). The inset shows the side view of oligomers array under incident plane wave, where the orange short lines describe the graphene oligomers. The character peaks are labeled with ①,② in the absorption spectrum.

Download Full Size | PDF

3. Results and discussion

To clearly understand the absorption behaviors in our designed graphene oligomers under LCP and RCP illumination, we first demonstrate the response of this structure under x-polarized plane incident wave. Figure 1(b) displays the calculated reflection (absorption) spectrum with two distinct dips (peaks) and an asymmetric Fano line shape in the reflection (absorption) spectrum with (black curve) and without external magnetic field (blue and red curves). It is clear that the absorption peak around 12.7 $μ m$ (labeled as ②) is broader and higher than the absorption peak (labeled as ①) at 12 $μ m$ . This remarkable feature will be explained in the following paragraph according to the plotted electric field E_z distributions at the wavelengths of peaks ① and ② displayed in Fig. 1(b). For comparison, we also illustrate the absorption spectrum under external magnetic field B = 0.5 T (black curve with circles), which almost overlaps with the absorption spectrum under B = 0 T.

In Fig. 2(a), we find that the intensities and directions (labeled as red arrows) of the electric dipoles in the graphene disks depend on the position of each disk in the arranged ring composite microstructure. Under x-polarization incident light, in Fig. 2(a), the electric dipoles in the upper and lower of the unit cell oscillate out-phase with respect to the middle two dipoles in the x direction (see the rede arrows in Fig. 2(a)), while the component of electric dipoles cancels with each other in the y direction. For the electric field distribution shown in Fig. 2(b), we can find that all the graphene disks almost show in-phase resonance, acting as electric dipoles arranged in the same direction (x direction). Thus, the resonant absorption peak exhibits narrower line shape for peak ① than that of peak ② around wavelength of 12.7 $μ m$ . According to the well established plasmon hybridization theory [35,36], we can conclude that the absorption peak ① corresponds to a subradiant mode while the peak ② is a superradiant (bright) plasmonic mode [36]. Both of the subradiant and superradiant modes are nondegenerate eigenmodes in the simulated symmetric ring-type oligomer consisting of 6 graphene disks [37]. They interference with each other, leading to a Fano line shape in the reflection/absorption spectrum [37]. In our designed graphene oligomer with metal (silver) mirror structure, the silver film acts as a perfect mirror, which enhances the intrinsic Fano-resonance intensity [25]. At the same time, the Fabry–Perot resonance introduced by the spacer layer and silver substrate improves the absorption of the resonant peak in the absorption spectrum. In Figs. 2(c) and 2(d), the distributions of electric field component E_z plotted at resonant peaks ① and ② in Fig. 1(b) under external magnetic field B = 0.5 T are presented, where the dashed arrow denote the directions of electric dipoles in Figs. 2(a) and (b) without external magnetic field. Although the absorption spectrum with external magnetic field B = 0.5 T in Fig. 1(b) presents little difference compared with that of the case B = 0 T, some of the excited electric dipoles show slight rotation relatively to the corresponding electric dipoles illustrated in Figs. 2(a)-2(b), especially in the subradian mode shown in Fig. 2(c). These rotations are due to the cyclotron motion of the free carriers in graphene under the external magnetic field.

Fig. 2 Distributions of electric field component E_z at wavelengths corresponding to the resonant peaks of ①,② shown in Fig. 1(c) without ((a)-(b)) and with ((c)-(d)) external magnetic field B. In panels (c) and (d), the dashed arrows denote the directions of electric dipole shown in panel (a) and (b), respectively. The red arrows denote the directions of electric dipole. Panel (e) displays the absorption spectra for different ring radii.

Download Full Size | PDF

In Fig. 2(e), the dependence of resonant modes on the gap between graphene disks ispresented with R = 140, 160, 180 nm (the corresponding graphene disk gaps are 20, 40, 60 nm), respectively. With the increase of graphene disk gap, the superradiant mode with wider line shape at longer wavelength exhibits a blue-shift, while the subradiant mode reveals a red-shift. Particularly, the intensity of subradiant peak decreases with the increase of gap between graphene disks, indicating that it is a collective coupling behavior of the ring shape oligomer microstructures.

Based on the above discussion of optical response under x-polarized incident, we now turn to the case of circularly polarized incident wave under external magnetic field. The side view of considered oligomers is illustrated in the right inset of Fig. 3(a), where external static magnetic field is perpendicular to the surface of graphene oligomers array. In the calculation, we define MCD = $A_{l} - A_{r}$ , where A_l, A_r denote the absorption of LCP and RCP, respectively. According to Eq. (1), the permittivity of graphene has off-diagonal components under external magnetic field, so we can find that the absorption spectrum shows different response to the LCP and RCP incident wave. Around the wavelength of 12.19 $μ m$ for sharp absorption of LCP, a dramatically increase of MCD (up to 24.25%) is observed, which is two times larger in intensity than the MCD at superradiant mode absorption peak around 12.54 $μ m$ . To compare the increased MCD signal with the reported MCD based on single plasmonic dipole resonant, we also calculated the absorption and MCD spectra for simple periodic array consisting of single graphene disk in each unit cell with identical filling factor, as shown in Fig. 3(b). One can find that the MCD peak is only 8% in Fig. 3(b), which is three times smaller than MCD peak of the subradiant mode displayed in Fig. 3(a). Interestingly, we also could find that the line shape of MCD peak due to superradiant plasmonic resonant in Fig. 3(a) is similar to the MCD line shape displayed in Fig. 3(b). This is well understood from the mechanism that both of superridant mode in Fig. 3(a) and the dipole mode in Fig. 3(b) are bright plasmonic mode, which couples directly with incident light.

Fig. 3 (a) The absorption (dashed and dot-dashed curves) and MCD (curve with triangular) spectra for LCP and RCP incident wave. The inset illustrates the side view of graphene oligomers array illuminated with circular polarized light under external magnetic field B. The radii of ring oligomers is 160 nm and other parameters are identical to those describe in section 2. (b) The absorption and MCD spectra for a simple periodic array with one graphene disk in each unit cell. The filling factor is identical to the case of panel (a) with oligomer composite structures. Other parameters are the same as panel (a).

Download Full Size | PDF

In order to explain the enhanced MCD in Fig. 3(a), the corresponding electric field E_z and its amplitude distributions for the MCD peak wavelength are illustrated in Figs. 4(a)-4(d). As seen in Fig. 4(a), for LCP incidence, both electric dipoles in the x and y directions are excited, where the electric dipoles in the x direction display out-phase distributions but in the y direction they oscillate in phase, indicating a subradiant resonant mode. However, in Fig. 4(b), one can find that the electric dipoles present out-phase distributions in both the x and y directions if they are projected on these two axes, respectively. The remarkable difference of plasmonic modes excited by LCP and RCP can be clearly seen by comparing the red solid and dashed arrows in Fig. 4(b). This indicates that the subradiant mode for RCP excitation is a darker subradiant mode compared with the mode of LCP shown in Fig. 4(a). In other words, the subradiant mode corresponding to LCP could be regarded as a quasi-dark plasmonic mode, since its coupling with y polarized component of incident light is more efficient than that of RCP subradiant mode illustrated in Fig. 4(b). Due to this fact, we can also verify this point from their amplitude distributions displayed in Figs. 4(c)-4(d), where the amplitude in Fig. 4(c) is remarkably stronger than that of Fig. 4(d). Thus, around the subradiant mode and Fano resonance wavelength, the absorption for LCP is larger than that of RCP, which is the origin of the enhanced MCD peak in Fig. 3(a). On the other hand, the line width of RCP absorption peak shown in Fig. 3(a) is narrower than that of LCP at the peak wavelength of subradiant resonant mode around 12 um, further indicating the latter is a quasi-subradiant mode. For the different coupling efficiencies of the subradiant mode with LCP and RCP incident waves, it could be understood that graphene exhibits different permittivity for LCP and RCP under external magnetic field [12]. The cyclotron motion of the free carriers due to external magnetic field induces the electric dipoles on grapheme disks to rotate in the x-y plane, leading to the redistribution of the electric dipoles and the above mentioned quasi-dark and darker subradiant plasmonic modes.

Fig. 4 Panels (a) and (b) show the distributions of the electric field component E_z for LCP (a) and RCP (b) at the MCD peak wavelength shown in Fig. 3(a). The dashed arrows in panels (b) and (f) illustrate the corresponding directions of oscillating electric dipoles excited by LCP in panels (a) and (e). Panels (c) and (d) represent the electric field amplitudes corresponding to panel (a) and (b). (e) and (f) describe the simulated electric field component E_z at the resonant wavelength of superadiant mode around 12.54 $μ m$ displayed in Fig. 3(a).

Download Full Size | PDF

For the case of superradiant plasmonic modes, as can be seen in Figs. 4(e)-4(f), the electric dipoles in the x and y directions are excited and their projections on the x, y directions (i.e. the directions of incident field) are almost in phase resonant, except one electric dipole on the left graphene disk in Fig. 4(e). Therefore, the absorption spectrum illustrates wider line shape and the intensity of MCD mainly depends on the different loss denoted by the imaginary part of permittivity, which can also be used to explain why the MCD shows similar line shape around the superradiant modes as the MCD spectrum for single graphene disk periodic array displayed in Fig. 4(b).

We further elucidate the dependence of MCD on the intensity of external magnetic field in Fig. 5. It is found that the subradiant mode at the wavelength of 12.17 um is less sensitive to the variation of external magnetic field, which increases from 20% to 30% as the external magnetic field B increases from 0.3 to 0.5 T. However, the MCD intensity for the superradiant plasmonic resonance around 12.7 $μ m$ shows less sensitivity to the increase of external magnetic field, increasing only 10% as the external magnetic field B varies from 0.3 T to 0.5 T. This phenomenon highlights the superiority of our proposed mechanism to enhance to MCD signal based on subradiant plasmonic mode.

Fig. 5 MCD spectra for different external magnetic fields. The geometry parameters are the same as Fig. 3(a).

Download Full Size | PDF

Finally, we discuss the tunability of MCD in our studied mechanism. The resonant wavelength of graphene disk (without external magnetic field) can be approximately predicted by Eq. (3) in [38], where the dielectric constant of graphene and its surrounding media, the diameter of the graphene disks determine the resonant wavelength. In addition, the coupling of the adjacent graphene disks also plays a crucial role in the resonant MCD peak. The frequency of enhanced MCD could be tuned actively through the variation of Fermi level with a back gate, which can be realized by designing an appropriate substrate [39] or changing the designed oligomers structure as the corresponding antidot lattice array as demonstrated experimentally in [40].

4. Conclusions

In summary, we have demonstrated an effective mechanism to enhance MCD through the subradiant plasmonic mode in a designed symmetric graphene oligomers. Our numerical simulation illustrates that the excitation efficiencies and orientation of each graphene disk depend on its position in the ring-shape oligomers, which induces subradiant plasmonic mode and Fano resonant line shape. In the mid-infrared region, we show MCD signal is dramatically enhanced to 24.25% under a relatively low external magnetic field of 0.4 T, which is three times larger than previously reported mechanism based on the resonance of electric dipole plasmonic mode. This giant MCD is attributed to the remarkably different excitation efficiency of subdradiant plasmonic mode due to the collective coupling under circular polarization incidence and external magnetic field.

Funding

National Natural Science Foundation of China (11664020); the Project for Distinguished Young Scholars of Jiangxi Province (20171BCB23098); the Natural Science Foundation of Jiangxi Province (20161BAB201002, 20151BAB207056).

References

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]

2. H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296(4), 56–62 (2007). [CrossRef] [PubMed]

3. P. J. Stephens, “Magnetic circular dichroism,” Annu. Rev. Phys. Chem. 25(1), 201–232 (1974). [CrossRef]

4. M. A. Zaitoun, W. R. Mason, and C. T. Lin, “Magnetic circular dichroism spectra for colloidal gold nanoparticles in xerogels at 5.5 K,” J. Phys. Chem. B 105(29), 6780–6784 (2001). [CrossRef]

5. areF. Pineider, G. Campo, V. Bonanni, C. de Julián Fernández, G. Mattei, A. Caneschi, D. Gatteschi, and C. Sangregorio, “Circular magnetoplasmonic modes in gold nanoparticles,” Nano Lett. 13(10), 4785–4789 (2013). [CrossRef] [PubMed]

6. I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7(1), 48–51 (2011). [CrossRef]

7. M. Tymchenko, A. Yu. Nikitin, and L. Martín-Moreno, “Faraday rotation due to excitation of magnetoplasmons in graphene microribbons,” ACS Nano 7(11), 9780–9787 (2013). [CrossRef] [PubMed]

8. J. Q. Liu, S. Wu, Y. X. Zhou, M. D. He, and A. V. Zayats, “Giant Faraday rotation in graphene metamolecules due to plasmonic coupling,” J. Lightwave Technol. 36(13), 2606–2610 (2018). [CrossRef]

9. M. Tamagnone, T. M. Slipchenko, C. Moldovan, P. Q. Liu, A. Centeno, H. Hasani, A. Zurutuza, A. M. Ionescu, L. M. Moreno, J. Faist, J. R. Mosig, A. B. Kuzmenko, and J. M. Poumirol, “Magnetoplasmonic enhancement of Faraday rotation in patterned graphene metasurfaces,” Phys. Rev. B 97(24), 241410 (2018).

10. B. Sepúlveda, J. B. González-Díaz, A. García-Martín, L. M. Lechuga, and G. Armelles, “Plasmon-induced magneto-optical activity in nanosized gold disks,” Phys. Rev. Lett. 104(14), 147401 (2010). [CrossRef] [PubMed]

11. B. Han, X. Q. Gao, J. W. Lv, and Z. Y. Tang, “Magnetic circular dichroism in nanomaterials: new opportunity in understanding and modulation of excitonic and plasmonic pesonances,” Adv. Mater. 1, 1801491 (2018). [CrossRef]

12. M. Wang, Y. Q. Wang, M. B. Pu, C. G. Hu, X. Y. Wu, Z. Zhao, and X. Luo, “Circular dichroism of graphene-based absorber in static magnetic field,” J. Appl. Phys. 115(15), 154312 (2014). [CrossRef]

13. K. Yang, M. Wang, M. Pu, X. Wu, H. Gao, C. Hu, and X. Luo, “Circular polarization sensitive absorbers based on graphene,” Sci. Rep. 6(1), 23897 (2016). [CrossRef] [PubMed]

14. F. Hao, E. M. Larsson, T. A. Ali, D. S. Sutherland, and P. Nordlander, “Shedding light on dark plasmons in gold nanorings,” Chem. Phys. Lett. 458(4), 262–266 (2008). [CrossRef]

15. D. E. Gómez, Z. Q. Teo, M. Altissimo, T. J. Davis, S. Earl, and A. Roberts, “The dark side of plasmonics,” Nano Lett. 13(8), 3722–3728 (2013). [CrossRef] [PubMed]

16. D. R. Chowdhury, X. Su, Y. Zeng, X. Chen, A. J. Taylor, and A. Azad, “Excitation of dark plasmonic modes in symmetry broken terahertz metamaterials,” Opt. Express 22(16), 19401–19410 (2014). [CrossRef] [PubMed]

17. W. Cao, R. Singh, I. A. Al-Naib, M. He, A. J. Taylor, and W. Zhang, “Low-loss ultra-high-Q dark mode plasmonic Fano metamaterials,” Opt. Lett. 37(16), 3366–3368 (2012). [CrossRef] [PubMed]

18. Z. Liu, M. Yu, S. Huang, X. Liu, Y. Wang, M. Liu, P. Pan, and G. Liu, “Enhancing refractive index sensing capability with hybrid plasmonic–photonic absorbers,” J. Mater. Chem. C Mater. Opt. Electron. Devices 3(17), 4222–4226 (2015). [CrossRef]

19. Z. Liu, G. Liu, S. Huang, X. Liu, P. Pan, Y. Wang, and G. Gu, “Multispectral spatial and frequency selective sensing with ultra-compact cross-shaped antenna plasmonic crystals,” Sens. Actuators B Chem. 215(12), 480–488 (2015). [CrossRef]

20. J. Chen, T. Q. Zha, T. Zhang, C. J. Tang, Y. Yu, Y. J. Liu, and L. B. Zhang, “Enhanced magnetic fields at optical frequency by diffraction coupling of magnetic resonances in lifted metamaterials,” J. Lightwave Technol. 35(1), 71–74 (2017). [CrossRef]

21. J. Chen, C. J. Tang, P. Mao, C. Peng, D. P. Gao, Y. Yu, Q. G. Wang, and L. B. Zhang, “Surface-plasmon-polaritons-assisted enhanced magnetic response at optical frequencies in metamaterials,” IEEE Photonics J. 8(1), 1 (2016). [CrossRef]

22. S. Wu, J. Q. Liu, L. Zhou, Q. J. Wang, Y. Zhang, G. D. Wang, and Y. Y. Zhu, “Electric quadrupole excitation in surface plasmon resonance of metallic composite nanohole arrays,” Appl. Phys. Lett. 99(14), 141104 (2011). [CrossRef]

23. Z. Fang, J. Cai, Z. Yan, P. Nordlander, N. J. Halas, and X. Zhu, “Removing a wedge from a metallic nanodisk reveals a fano resonance,” Nano Lett. 11(10), 4475–4479 (2011). [CrossRef] [PubMed]

24. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010). [CrossRef] [PubMed]

25. X. Zhu, H. Shi, S. Zhang, Q. Liu, and H. Duan, “Constructive-interference-enhanced Fano resonance of silver plasmonic heptamers with a substrate mirror: a numerical study,” Opt. Express 25(9), 9938–9946 (2017). [CrossRef] [PubMed]

26. W. H. Wang, S. P. Apell, and J. M. Kinaret, “Edge magnetoplasmons and the optical excitations in graphene disks,” Phys. Rev. B Condens. Matter Mater. Phys. 86(12), 125450 (2012). [CrossRef]

27. H. X. Da and C. W. Qiu, “Graphene-based photonic crystal to steer giant Faraday rotation,” Appl. Phys. Lett. 100(24), 241106 (2012). [CrossRef]

28. W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012). [CrossRef] [PubMed]

29. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004). [CrossRef] [PubMed]

30. COMSOL http://www.comsol.com/

31. J. Li, J. Tao, Z. H. Chen, and X. G. Huang, “All-optical controlling based on nonlinear graphene plasmonic waveguides,” Opt. Express 24(19), 22169–22176 (2016). [CrossRef] [PubMed]

32. Yu. V. Bludov, N. M. R. Peres, and M. I. Vasilevskiy, “Graphene-based polaritonic crystal,” Phys. Rev. B Condens. Matter Mater. Phys. 85(24), 245409 (2012). [CrossRef]

33. V. Semenenko, S. Schuler, A. Centeno, A. Zurutuza, T. Mueller, and V. Perebeinos, “Plasmon−plasmon interactions and radiative damping of graphene plasmons,” ACS Photonics 5(9), 3459–3465 (2018). [CrossRef]

34. S. Wu, Z. Zhang, Y. Zhang, K. Zhang, L. Zhou, X. Zhang, and Y. Zhu, “Enhanced rotation of the polarization of a light beam transmitted through a silver film with an array of perforated S-shaped holes,” Phys. Rev. Lett. 110(20), 207401 (2013). [CrossRef] [PubMed]

35. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A Hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003). [CrossRef] [PubMed]

36. Y. Sonnefraud, N. Verellen, H. Sobhani, G. A. E. Vandenbosch, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Experimental realization of subradiant, superradiant, and fano resonances in ring/disk plasmonic nanocavities,” ACS Nano 4(3), 1664–1670 (2010). [CrossRef] [PubMed]

37. B. Hopkins, A. N. Poddubny, A. E. Miroshnichenko, and Y. S. Kivshar, “Revisiting the physics of Fano resonances for nanoparticle oligomers,” Phys. Rev. A 88(5), 053819 (2013). [CrossRef]

38. H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012). [CrossRef] [PubMed]

39. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011). [CrossRef] [PubMed]

40. M. Tamagnone, T. M. Slipchenko, C. Moldovan, P. Q. Liu, A. Centeno, H. Hasani, A. Zurutuza, A. M. Ionescu, L. Martin-Moreno, J. Faist, J. R. Mosig, A. B. Kuzmenko, and J. M. Poumirol, “Magnetoplasmonic enhancement of Faraday rotation in patterned graphene metasurfaces,” Phys. Rev. B 97(24), 241410 (2018). [CrossRef]

Enhanced magnetic circular dichroism by subradiant plasmonic mode in symmetric graphene oligomers at low static magnetic fields

Abstract

1. Introduction

2. Model and method

3. Results and discussion

4. Conclusions

Funding

References

Cited By

Figures (5)

Equations (3)

Optics Express