Magneto-plasmonics in graphene-dielectric sandwich

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 OCIS codes: (240.6680) Surface plasmons; (230.3810) Magneto-optic systems; (230.7370) Waveguides. 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). 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). 3. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824– 830 (2003). 4. E. 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Introduction
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 [3].Due to the high confinement, SPs are widely applied into nano optical communication systems, sensing, imaging, photolithography fabrication, spectroscopy, and among others [4][5][6][7].
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][25][26][27][28][29][30][31][32][33][34], and sometimes they have completely different properties, such as the nonreciprocal effect [35][36][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][39][40][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 [42] in SLG.
In this paper, we study the SMPs modes in a graphene-dielectric sandwich (GDS) structure [43][44][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 [46], 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 [45] 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 [41] 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 σ yz , 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. σ 1 = σ 2 , ε 1 = ε 2 = ε 3 , and μ 1 = μ 2 = μ 3 .The dispersion of SMPs in asymmetric GDS structures is discussed in Sec.V. Finally, we give the conclusion in Sec.VI.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 σ 1 and σ 2 respectively.The relative permittivities and permeabilities of the materials between/above/below the two layers are denoted by ε 1/2/3 and μ 1/2/3 , 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 2 ( ) 2

Magneto-optical conductivities of graphene
n F E sign n n eB v =  , where v F ≈10 6 m/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 E 1 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 as where σ yy and σ yz are the longitudinal and Hall conductivities, respectively.If the temperature is T and the Fermi energy is E F , the elements in the tensor are given by, respectively [42], where , and E n is the LLs as defined above.ω is the incident angular frequency.Γ is the scattering rate.Although Γ depends on the frequency, temperature, and LL index [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 σ yy and σ yz , we consider it as a constant in the calculations as [47].Δ intra,n andΔ inter,n are calculated by Δ intra,n = E n+1 -E n andΔ inter,n = E n+1 + E n , respectively.ћ and k B 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 σ yy and the real part of σ yz give the main contribution to the dispersion of graphene SMPs [42], we plot the conductivities of Im(σ yy ) and Re(σ yz ) as a function of angular frequency ω with and without external magnetic fields, shown in Fig. 1(b).The parameters are chosen as E F = 0.05eV, T = 300K, Γ = 0.03E F .Then the spectrum in the calculation [0.01E F /ħ, 3E F /ħ] covers a frequency range from the terahertz to the near infrared.It is found that when E 1 = 0, the imaginary part of σ yy is always positive.This means TM-polarized SMP modes are always supported.When a magnetic field with an intensity of E 1 = E F is applied (the corresponding applied magnetic field is 1.85Tesla), both Im(σ yy ) and Re(σ yz ) 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 [42].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 [41].In Fig. 1(b), because n F (E 0 ) = 1, n F (E 1 ) = 0.5 and n F (E i≥2 ) ~0, the intraband conductivity σ yy can be simplified from Eq. (2a) as: intra, intra, 1 intra, 2 , yy yy yy where It can be inferred from Eq. ( 4) that σ intra,yy1 and σ intra,yy2 contribute to the resonances corresponding to ћω = E 1 -E 0 and ћω = E 2 -E 1 , separately, as shown in Fig. 1(b).The third peak is caused by interband transitions: ћω = E 2 -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(σ yy ) > 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.

Dispersion relation of SMPs in a GDS structure
The external magnetic field leads to different dispersion relations of SMP modes in a SLG structure [42].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, E x E y and E z in the GDS structure are expressed by where A x/y/z , B x/y/z , C x/y/z and D x/y/z are amplitudes, , k 0 is the wave vector in vacuum.From the Maxwell equations and the boundary conditions of the doublelayer graphene structure, after some derivations, we can have the dispersion of SMPs in a GDS structures as ) coth( ) ε 0 and μ 0 are the permittivity and permeability in vacuum, respectively.η 1 is the impedance of the material between the graphene sheets.When B = 0, Eq. ( 6) becomes the dispersion relations of the SP modes in a GDS structure [45]; when d→∞, Eq. ( 6) becomes the dispersion relations of the SMPs on a SLG sheet [42] under external magnetic field, respectively.

Symmetric structures
In this section, we will focus on a symmetric GDS structure, i.e. σ 1 = σ 2 , the materials above/ between/below the two graphene sheets are the same.Without loss of generality, we will set In this situation, the two modes can be obtained from Eq. ( 6), Substitute Eqs.(7a) and (7b) into Eqs.(8a) and (8b), we have When B = 0, it is easily found that Eqs.(9a) and (9b) correspond to the SM and AM in GDS in [45], respectively.In Fig. 2, we plot the dispersion relations of the SMPs modes with and without a magnetic field (E 1 = E F ) 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 k F = E F /(ħv F ).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]E F /ħ, [1.05, 1.85]E F /ħ, and [2.47, 2.72]E F /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 E x and E y have a phase shift of π/2 with respect to E z , we use the real part of E z and imaginary parts of E x and E y 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 E z and E y are symmetric with respect to the x = 0 plane.However, the electric component E x perpendicular to the surface is antisymmetric.The similar phenomenon can be observed for the AM.The field volumes of both E z and E x are very small, in the range of 6 × 10 −3 λ 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 E y is no longer zero when B is applied.It has a phase shift of π with respect to E z 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 E y has a magnitude of ~10 −5 of that of E z and E x .
Re(E z ) x/λ 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 E 1 = 0.9E F , E 1 = 1.0EF , and E 1 = 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(σ yy )>0, the SMPs bands should move with frequency bands of Im(σ yy )>0.From Fig. 1(b), one can find that these frequency bands always locate at frequencies higher than the frequency difference between two LLs (Δ intra,1 = E 2 -E 1 , Δ intra,0 = E 1 -E 0 , and Δ inter,1 = E 2 -E -1 , as shown in Fig. 1(b)), which are proportional to the square root of the external magnetic fields by the definition of E n , i.e. i ntra,0 . 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 E 1 ~EF , the last term are in the order of ~1, it can be inferred from Eq. ( 10) that as Im(σ yy ) decreases, when β increases.While with the increase of B, the increase of i ntra,1 (2  2) B Δ ∝ − in band 1 is much smaller than those of Δ intra,0 and Δ inter,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

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 E F = 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 E F = 0.05eV and E F = 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 E z 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: Al 2 O 3 , SiO 2 , and air, i.e. ε 1 = 6, ε 2 = 3.8, and ε 3 = 1, with a magnetic field of E 1 = E F .The two SMP modes are plotted in Fig. 7

Conclusion
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, E y 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.

Fig. 1 .
Fig. 1.(a) Schematic of the SMPs propagating in a double-layer graphene structure when a magnetic field is applied along the -x axis.(b) Graphene conductivity elements Im(σ yy ) and Re(σ yz ) as a function of frequency without the magnetic field (black dotted line), and with a magnetic field of E 1 = E F (red solid and blue dashed lines).The solid black lines indicate the electron-electron transitions between LLs of intraband (E 1 -E 0 , E 2 -E 1 ) and interband (E 2 -E-1 ).The parameters are chosen as E F = 0.05eV, d = T = 300K, Γ = 0.03E F .

Fig. 2 .
Fig. 2. Dispersion relations of SMPs without (black lines) and with (red and blue lines) external magnetic fields.The symmetric modes (SMs) and antisymmetric modes (AMs) are denoted by solid and dashed lines, respectively.The green line is the dispersion of light in free space.The spacing between the two graphene layer is d = 0.003λ.The intensity of the magnetic field is E 1 = E F .The other parameters are the same as Fig. 1.

Fig. 3 .
Fig. 3. Electric field distributions of ħω = 1.5EF with and without the external magnetic fields.(a) Re(E z ); (b) Im(E x ); (c) Im(E y ).The parameters are the same as used in Fig. 2. The black solid and dashed lines denote the symmetric and antisymmetric modes of B = 0, respectively.The red solid and blue dashed lines denote the symmetric and antisymmetric modes of E 1 = E F , respectively.
Eqs. (9a) and (9b) is negligible (in the order of ~10−4 σ 1,yz is always much smaller compared with η 1 , even when the magnetic field is very strong.

Fig. 4 .Fig. 5 .
Fig. 4. Effects of the magnetic field on the SMP modes of the GDS structure.The symmetric and antisymmetric modes are denoted by solid and dashed lines, respectively.3 magnetic fields with intensities of E 1 = 0.9E F (green lines), E 1 = 1.0EF (blue lines), and E 1 = 1.1EF (blue lines) are compared.(a) Band 3; (b) Band 2; (c) Band 1; (d) Comparison of Im(σ yy ) under the 3 magnetic fields.The other parameters are the same as those in Fig. 2.

Fig. 6 .
Fig. 6.(a) SMPs modes of band 2 of a GDS when the Fermi energies of the two graphene sheets are different (the red and blue solid lines).The upper graphene sheet is E F1 = 0.05eV while the lower is E F1 = 0.06eV.For comparison, SMPs on SLG with magnetic fields of E F = 0.05eV and E F = 0.06eV are also plotted (the dashed lines).(b) Field distribution of Re(E z ) of the two modes at ħω = 1.3EF .The other parameters are the same as Fig. 2.
(a).The SMP modes on the interfaces of Al 2 O 3 (ε 1 = 6)/Air(ε 2 = 1) and Al 2 O 3 (ε 1 = 6)/SiO 2 (ε 2 = 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 Al 2 O 3 / Air), and the MA mode relates to the upper layer mode (the interface of Al 2 O 3 / SiO 2 ).