In this paper, dispersion properties and field distributions of surface magneto plasmons (SMPs) in double-layer graphene structures at room temperature are studied. It is found that, the dispersion curves of both symmetric and antisymmetric SMPs modes split into several branches/bands when a magnetic field is applied perpendicularly to the graphene surface. Surprisingly, the lowest energy SMP band has anomalous dependence on the applied magnetic field, different to the other higher bands. In addition, the symmetric and antisymmetric modes can be decoupled if the two graphene layers possess different properties, such as different Fermi energies. Furthermore, electric components of the surface modes which are parallel to the graphene surfaces but perpendicular to the propagation direction (i.e. the transverse-electric mode) are no longer zero caused by the Lorentz force on the free electrons.
© 2014 Optical Society of America
Plasmonics, a new branch of photonics based on surface plasmons (SPs), has been enormously developed since the discovery of its extraordinary optical transmission property through subwavelength hole arrays [1, 2]. SPs are essentially electromagnetic waves that are confined on an interface between a dielectric and a conductor (usually a metal), caused by the interaction of electromagnetic fields and free electrons . Due to the high confinement, SPs are widely applied into nano optical communication systems, sensing, imaging, photolithography fabrication, spectroscopy, and among others [4–7].
Graphene, which is a monolayer of carbon atoms tightly packed into a two-dimensional (2D) honeycomb lattice, has been used as a plasmonic material in recent years [8–17]. Because graphene charge carriers are massless Dirac fermions [18, 19], graphene plasmonics have two intriguing properties compared to metal-based plasmonics: (1) SPs on graphene can be tuned by gating or doping [8–11]. (2) Graphene supports not only transverse magnetic (TM) modes like metals, but also transverse electric (TE) modes . Besides, SPs wavelength of graphene is much smaller than that of metals due to the large wave vector of SPs on graphene [10, 20–23]. Therefore, graphene has a great potential for realizing tunable and highly-integrated plasmonic devices.
On the other hand, it is known that when an external magnetic field is applied on a metal or a semiconductor, SPs can be modulated (often called surface magneto plasmons (SMPs)) [24–34], and sometimes they have completely different properties, such as the nonreciprocal effect [35–37]. In the presence of a magnetic field perpendicular to a 2D graphene sheet, the massless free carriers in graphene result in non-equidistant Landau levels (LLs) and specific electron-electron excitations [38–41]. For a single layer graphene (SLG), it is found that due to the applied magnetic field, the dispersion curve of the SMP mode splits into a series of branches. In addition, not only TM-polarized SMPs, but also quasi-transverse-electric (QTE) modes can be generated and supported  in SLG.
In this paper, we study the SMPs modes in a graphene-dielectric sandwich (GDS) structure [43–45] in which a dielectric layer is placed between two SLGs. Unlike in a bilayer structure, where the two SLGs are directly stacked on each other , the complicated interlayer hopping effects in the double-layer structure can be neglected. Compared with SLGs, we find that, the SMPs have one symmetric (SM) mode and one anti-symmetric (AM) mode in a GDS structure, which is similar to the SPs in a double-layer graphene  in the absence of the magnetic field. However, when a magnetic field is applied, these two modes of SMPs split into several bands due to the separation of LLs. In addition, the lowest energy SMP band arising from the partially occupied LLs at nonzero temperatures  has an anomalous dependence on the external magnetic field intensity, different to the other higher energy bands. Furthermore, due to the Lorenz force applied on free carriers, the electric component perpendicular to the propagating direction and parallel to the graphene surface is no longer zero. This leads to a TE-polarized component. Due to the small value of the Hall conductivity σyz, the external magnetic field has little effect on the coupling between the SMs and AMs. However, if the two SLGs possess different Fermi-energies, e.g. achieved by doping or gating, decoupling between the two modes can be well achieved. Our results suggest that this structure can be applied for mode controlling and magnetic sensors. The remainder of this paper is organized as follows. In Sec. II, we give the GDS structure and discuss the conductivity of graphene in an external magnetic field. In Sec. III, the dispersion of the SMPs in GDS is derived. In Sec. IV, we present the dispersion and field distribution of SMPs in symmetric GDS structures, i.e. σ1 = σ2, ε1 = ε2 = ε3, and μ1 = μ2 = μ3. The dispersion of SMPs in asymmetric GDS structures is discussed in Sec. V. Finally, we give the conclusion in Sec. VI.
2. Magneto-optical conductivities of graphene
We consider the GDS structure lied on the y-z plane, as depicted in Fig. 1(a). The two graphene layers are separated by a distance d. The conductivities of the two layers are denoted by σ1 and σ2 respectively. The relative permittivities and permeabilities of the materials between/above/below the two layers are denoted by ε1/2/3 and μ1/2/3, respectively. The SMPs wave propagates along the z-axis. We first obtain the conductivity of graphene under an external magnetic field [42, 47]. When an external magnetic field B is applied along the –x-axis, LLs in SLG are given by, where vF ≈106m/s is the Fermi velocity in graphene, and n is the LL index. –e (smaller than zero) is the electron charge. In the following part, for convenience, we use E1 as a measure of the applied magnetic fields. According to the random phase approximation, the conductivity of a SLG becomes a tensor, which is written as42],48, 49], it is much smaller comparing with the frequency, and has little influence on our following results. Therefore, in order to obtain a relatively simple expression of σyy and σyz, we consider it as a constant in the calculations as .Δintra,n andΔinter,n are calculated by Δintra,n = En+1-En andΔinter,n = En+1 + En, respectively. ћ and kB are the Planck and Boltzmann constant, respectively. The two terms in the bracket in Eqs. (2a) and (2b) correspond to the intra- and interband transitions, respectively.
Because the imaginary part of σyy and the real part of σyz give the main contribution to the dispersion of graphene SMPs , we plot the conductivities of Im(σyy) and Re(σyz) as a function of angular frequency ω with and without external magnetic fields, shown in Fig. 1(b). The parameters are chosen as EF = 0.05eV, T = 300K, Γ = 0.03EF. Then the spectrum in the calculation [0.01EF/ħ, 3EF/ħ] covers a frequency range from the terahertz to the near infrared. It is found that when E1 = 0, the imaginary part of σyy is always positive. This means TM-polarized SMP modes are always supported. When a magnetic field with an intensity of E1 = EF is applied (the corresponding applied magnetic field is 1.85Tesla), both Im(σyy) and Re(σyz) show several peaks in the spectrum. It is known that when T = 0K, graphene has only one intraband conductivity resonance and infinite interband conductivity resonances . However, at a nonzero temperature, Fermi function is no longer a step function and some LLs are partially occupied, resulting in an additional resonance at a low frequency . In Fig. 1(b), because nF (E0) = 1, nF (E1) = 0.5 and nF (Ei≥2) ~0, the intraband conductivity σyy can be simplified from Eq. (2a) as:Eq. (4) that σintra,yy1 and σintra,yy2 contribute to the resonances corresponding to ћω = E1-E0 and ћω = E2-E1, separately, as shown in Fig. 1(b). The third peak is caused by interband transitions: ћω = E2-E-1. The three peaks indicate the splitting of SMP modes into branches when a magnetic field is applied, because they only exist for Im(σyy) > 0 [20, 42]. There are also QTE modes existing under the applied magnetic field. For the reason that the QTE modes are weakly confined, in the following, we only consider the highly confined SMPs modes in a GDS structure.
3. Dispersion relation of SMPs in a GDS structure
The external magnetic field leads to different dispersion relations of SMP modes in a SLG structure . It is expected that new phenomena will also be observed in a GDS structure. In the following, we present the derivations of the dispersion relation of SMPs in a GDS structure. The electric field components of a SMPs mode propagating along the + z-axis can be written as , where β is the propagation constant, ω is the angular frequency of incident wave. In order to ensure the electromagnetic fields attenuate at both sides of a graphene layer, Ex Ey and Ez in the GDS structure are expressed byEq. (6) becomes the dispersion relations of the SP modes in a GDS structure ; when d→∞, Eq. (6) becomes the dispersion relations of the SMPs on a SLG sheet  under external magnetic field, respectively.
4. Symmetric structures
In this section, we will focus on a symmetric GDS structure, i.e. σ1 = σ2, the materials above/ between/below the two graphene sheets are the same. Without loss of generality, we will set ε1 = ε2 = ε3 = 1, μ1 = μ2 = μ3 = 1. In this situation, the two modes can be obtained from Eq. (6),Eqs. (7a) and (7b) into Eqs. (8a) and (8b), we haveEqs. (9a) and (9b) correspond to the SM and AM in GDS in , respectively.
In Fig. 2, we plot the dispersion relations of the SMPs modes with and without a magnetic field (E1 = EF) calculated by Eq. (9). The spacing of the two graphene layers are d = 0.003 λ. The propagation constant is normalized by the Fermi wave vector kF = EF/(ħvF). The other parameters are the same as those in Fig. 1(b). Without the magnetic field, two continuous SP modes are supported, corresponding to the SM (black solid line) and AM (black dashed line) in Fig. 2. As β increases, the two modes become closer to each other, and converge to the surface mode of SLG at β→∞ (not shown in the figure). When the magnetic field is applied, due to the existence of the discrete LLs, the SMPs dispersion is divided into three bands: [0.47, 0.75]EF/ħ, [1.05, 1.85]EF/ħ, and [2.47, 2.72]EF/ħ. Each band has two modes, SM (red solid) and AM (blue dashed lines). Similar to the SP modes without an applied magnetic field, the AMs have a larger β than the SMs for SMPs, and these two modes become closer when β increases. The much larger β of the SMPs modes than that in free space (green dashed line) indicates that the SMPs modes have strong confinements.
In order to further study the propagation of SMPs in the GDS structure, we plot the field distributions of the electric field components in Fig. 3. Because Ex and Ey have a phase shift of π/2 with respect to Ez, we use the real part of Ez and imaginary parts of Ex and Ey in the simulations. Without loss of generality, the frequency is chosen as ħω = 1.5EF, which is located in band 2. Field distributions in band 1 and band 3 are similar to the results for band 2, shown in Fig. 2. It is noted that, for SM, both Ez and Ey are symmetric with respect to the x = 0 plane. However, the electric component Ex perpendicular to the surface is antisymmetric. The similar phenomenon can be observed for the AM. The field volumes of both Ez and Ex are very small, in the range of 6 × 10−3λ above and below the two graphene layers. When a magnetic field is added, the confinement is enhanced (see the blue and red curves in Figs. 3(a) and 3(b)), caused by the increased propagation constant for ħω = 1.5EF, as shown in Fig. 2. An interesting phenomenon is that Ey is no longer zero when B is applied. It has a phase shift of π with respect to Ez as shown in Fig. 3(c). From the classical point of view, it attributes to the Lorentz force on the oscillating electrons in the y-direction, although this effect is very weak since Ey has a magnitude of ~10−5 of that of Ez and Ex.
We then investigate how the magnetic field affects the SMP modes in a GDS structure. The dispersions of the 3 bands of SMPs in Fig. 2 under magnetic fields of E1 = 0.9EF, E1 = 1.0EF, and E1 = 1.1EF are plotted in Figs. 4(a)–4(c), respectively. From Figs. 4(a) and 4(b), it is found that with the applied magnetic field intensity increases, the dispersion curves of the band 2 and band 3 moves towards higher frequencies. This phenomenon can be understood by the shift of the LLs. Since the excitations of SMPs only appear at frequencies corresponding to Im(σyy)>0, the SMPs bands should move with frequency bands of Im(σyy)>0. From Fig. 1(b), one can find that these frequency bands always locate at frequencies higher than the frequency difference between two LLs (Δintra,1 = E2-E1, Δintra,0 = E1-E0, and Δinter,1 = E2-E-1, as shown in Fig. 1(b)), which are proportional to the square root of the external magnetic fields by the definition of En, i.e. , . Therefore, when B increases, the SMPs bands should move towards higher frequencies for all the three bands. Surprisingly, this change is only found in band 2 and band 3 as shown in Fig. 4(d). An anomalous shift is found in band 1: the dispersion curve moves to lower frequencies when the magnetic field increases (as marked in the red shadow in Fig. 4(d)). In order to explain this anomalous effect in details, we turn to the dispersion equations of Eq. (9). When the magnetic field is in the order of E1~EF, the last term in Eqs. (9a) and (9b) is negligible (in the order of ~10−4). Then substituting Eq. (4b) into Eq. (9), we haveEq. (10) that as Im(σyy) decreases, when β increases. While with the increase of B, the increase of in band 1 is much smaller than those of Δintra,0 and Δinter,1 in band 2 and band 3, respectively. Thus the overall effect is that the dispersion curve is moved downward in frequencies as shown in Fig. 4(c) (see also Fig. 4(d)).
It is noted that although the external magnetic field has an obvious effect on the shift of the dispersion curves, it is difficult to find its effect on the coupling between the SMs and AMs. The reason is that the main difference between the two modes caused by the magnetic field relies on the last term . However, σ1,yz is always much smaller compared with η1, even when the magnetic field is very strong.
We also study the effect of the spacing d (from 0.003 λ to 0.01 λ) on the SMP modes in the GDS structure. The dispersions of SMPs of the three bands, compared with those of SLG, are plotted in Figs. 5(a)–5(c), respectively. The magnetic field intensity is E1 = EF. The other parameters are the same as those in Fig. 2. It is found that, in all the three bands, the separation between the SMs and AMs becomes narrower with the increase of the spacing d. Because these two modes are caused by the coupling of the SLG SMP of the two graphene sheets, they reduce to SLG SMP modes when d becomes large. In addition, when d increases, the field confinement is enhanced for the SMs, while reduced for the AMs, as shown in Figs. 5(d) and 5(e), respectively. This is caused by the opposite moving directions of the SMs and AMs with d.
5. Asymmetric structures
When the symmetry of the structure is broken, we shall use Eq. (6) instead of Eqs. (8a) and (8b) to calculate the dispersions and field distributions of GDS SMPs. In this situation, we use modified symmetric (MS) and antisymmetric (MA) modes to represent them. For asymmetric structures, we first consider the situation when the Fermi levels of the two graphene sheets in the GDS structure are different, e.g. one Fermi level keeps as 0.05eV while the other changes to EF = 0.06eV. The dispersion relation is depicted in Fig. 6(a). For simplicity, we only plot the second band of SMP. The other two bands have similar results. A special feature of the dispersion curve compared with that in Fig. 2 is that the two SMP modes are decoupled as β→∞. By comparing the dispersions of a SLG (shown as the dashed lines in Fig. 6(a)), we find that these two modes approach SLG SMP of EF = 0.05eV and EF = 0.06eV, respectively. This indicates that as β increases, the SMP modes of each graphene layer are more and more confined on each surface, thus decreasing the coupling of the surface modes of the two layers. The field distribution plotted in Fig. 6(b) also proves the decoupling. Most energy of Ez is confined on the upper graphene layer for the MS mode, while the lower layer for the MA mode. However, due to the weak coupling, the MS and MA modes have similar phase distributions as those for symmetric and antisymmetric modes, respectively.
We also consider asymmetric materials, consisting of 3 dielectric layers: Al2O3, SiO2, and air, i.e. ε1 = 6, ε2 = 3.8, and ε3 = 1, with a magnetic field of E1 = EF. The two SMP modes are plotted in Fig. 7(a). The SMP modes on the interfaces of Al2O3(ε1 = 6)/Air(ε2 = 1) and Al2O3(ε1 = 6)/SiO2 (ε2 = 3.8) in a SLG structure are also plotted (dashed lines), respectively, for comparison. It is seen that the MS mode and MA mode are decoupled due to the asymmetric structure. From the dispersion curves in Fig. 7(b), we can infer that the MS mode corresponds to the lower graphene layer (the interface of Al2O3 / Air), and the MA mode relates to the upper layer mode (the interface of Al2O3 / SiO2).
In conclusion, in this paper we study the SMPs in GDS structures above zero temperature. The dispersions and field distributions of the SMPs are calculated. It is found that due to the LLs in graphene, the two modes of SMPs split into braches/bands when an external magnetic field is applied. In addition, Ey is no longer zero due to the applied Lorentz force. For T≠0, an additional intraband SMP band with an anomalous dependence on the external magnetic field appears. Although the magnetic field has little effect on the coupling of the symmetric and antisymmetric modes, the decoupling of these two modes can be achieved by varying the doping levels of the two graphene layers. The study of the SMPs in GDS may open a new avenue to realize novel ultra-confined graphene-based plasmonic and photonic devices.
This work is supported from Nanyang Technological University (NTU), Singapore by the start-up grant (grant number M58040017), partially by the Ministry of Education, Singapore (MOE2011-T2-2-147 and MOE2011-T3-1-005), and the CNRS International – NTU - Thales Research Alliance (CINTRA) Laboratory, UMI 3288, Singapore 637553. The support from NCET, China, and Project-sponsored by SRF for ROCS, SEM, China (To Dr. Bin Hu), as well as the National Basic Research Program of China (973 Program Grant no. 2013CBA01702) are also acknowledged.
References and links
1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]
2. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58(11), 6779–6782 (1998). [CrossRef]
5. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70(1), 1–87 (2007). [CrossRef]
6. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photon. 4(2), 83–91 (2010). [CrossRef]
7. T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008). [CrossRef]
8. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photon. 4(9), 611–622 (2010). [CrossRef]
10. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012). [CrossRef] [PubMed]
12. A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011). [CrossRef]
13. B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012). [CrossRef] [PubMed]
14. A. Yu. Nikitin, F. Guinea, F. Garcia-Vidal, and L. Martin-Moreno, “Slow-light dark solitons in insulator–insulator–metal plasmonic waveguides,” Phys. Rev. B 85, 081405 (2012). [CrossRef]
15. P. Huidobro, A. Nikitin, C. González-Ballestero, L. Martín-Moreno, and F. García-Vidal, “Superradiance mediated by graphene surface plasmons,” Phys. Rev. B 85(15), 155438 (2012). [CrossRef]
17. L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011). [CrossRef] [PubMed]
19. A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009). [CrossRef]
21. Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the grapheme–SiO₂ interface,” Nano Lett. 11(11), 4701–4705 (2011). [CrossRef] [PubMed]
22. J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012). [PubMed]
23. Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012). [PubMed]
24. G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: Combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013). [CrossRef]
25. J. C. Banthí, D. Meneses-Rodríguez, F. García, M. U. González, A. García-Martín, A. Cebollada, and G. Armelles, “High magneto-optical activity and low optical losses in metal-dielectric Au/Co/Au-SiO(2) magnetoplasmonic nanodisks,” Adv. Mater. 24(10), OP36–OP41 (2012). [PubMed]
26. V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011). [CrossRef] [PubMed]
27. V. Bonanni, S. Bonetti, T. Pakizeh, Z. Pirzadeh, J. Chen, J. Nogués, P. Vavassori, R. Hillenbrand, J. Åkerman, and A. Dmitriev, “Designer magnetoplasmonics with nickel nanoferromagnets,” Nano Lett. 11(12), 5333–5338 (2011). [CrossRef] [PubMed]
28. B. Hu, Q. J. Wang, S. W. Kok, and Y. Zhang, “Active focal length control of terahertz slitted plane lenses by magnetoplasmons,” Plasmonics 7(2), 191–199 (2011). [CrossRef]
29. B. Hu, Q. J. Wang, and Y. Zhang, “Slowing down terahertz waves with tunable group velocities in a broad frequency range by surface magneto plasmons,” Opt. Express 20(9), 10071–10076 (2012). [CrossRef] [PubMed]
32. E. P. Fitrakis, T. Kamalakis, and T. Sphicopoulos, “Slow-light dark solitons in insulator–insulator–metal plasmonic waveguides,” J. Opt. Soc. Am. B 27, 1701–1706 (2010). [CrossRef]
34. M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. 41(1-8), 1–416 (2001). [CrossRef]
35. R. E. Camley, “Nonreciprocal surface waves,” Surf. Sci. Rep. 7(3-4), 103–187 (1987). [CrossRef]
36. J. J. Brion, R. F. Wallis, A. Hartstein, and E. Burstein, “Theory of surface magnetoplasmons in semiconductors,” Phys. Rev. Lett. 28(22), 1455–1458 (1972). [CrossRef]
38. Yu. A. Bychkov and G. Martinez, “Magnetoplasmon excitations in graphene for filling factors ν≤6,” Phys. Rev. B 77(12), 125417 (2008). [CrossRef]
39. O. Berman, G. Gumbs, and Y. Lozovik, “Magnetoplasmons in layered graphene structures,” Phys. Rev. B 78(8), 085401 (2008). [CrossRef]
40. 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 (2010). [CrossRef]
41. W. Wang, J. Kinaret, and S. Apell, “Excitation of edge magnetoplasmons in semi-infinite graphene sheets: Temperature effects,” Phys. Rev. B 85(23), 235444 (2012). [CrossRef]
42. A. Ferreira, N. M. R. Peres, and A. H. Castro Neto, “Confined magneto-optical waves in graphene,” Phys. Rev. B 85(20), 205426 (2012). [CrossRef]
43. E. Hwang and S. D. Sarma, “Plasmon modes of spatially separated double-layer graphene,” Phys. Rev. B 80(20), 205405 (2009). [CrossRef]
44. T. Stauber and G. Gómez-Santos, “Plasmons and near-field amplification in double-layer graphene,” Phys. Rev. B 85(7), 075410 (2012). [CrossRef]
45. C. H. Gan, H. S. Chu, and E. P. Li, “Synthesis of highly confined surface plasmon modes with doped graphene sheets in the midinfrared and terahertz frequencies,” Phys. Rev. B 85(12), 125431 (2012). [CrossRef]
47. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007). [CrossRef]
48. Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4(7), 532–535 (2008). [CrossRef]
49. E. Hwang, B. Y.-K. Hu, and S. Das Sarma, “Inelastic carrier lifetime in graphene,” Phys. Rev. B 76(11), 115434 (2007). [CrossRef]