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Magneto-plasmonics in graphene-dielectric sandwich

Bin Hu,1,2 Jin Tao,2 Ying Zhang,3 and Qi Jie Wang2,4,* 1School of Optoelectronics, Beijing Institute of Technology, Beijing 100081, China

2OPTIMUS, Photonics Centre of Excellence, School of Electrical & Electronic Engineering, Nanyang Technological University, 50 Nanyang Ave., 639798 Singapore

3Singapore Institute of Manufacturing Technology, 71 Nanyang Drive, 638075 Singapore 4CDPT, Centre for Disruptive Photonic Technologies, Nanyang Technological University, 637371 Singapore

*[email protected]

Abstract: 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

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#217493 - $15.00 USD Received 21 Jul 2014; revised 15 Aug 2014; accepted 20 Aug 2014; published 2 Sep 2014(C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021727 | OPTICS EXPRESS 21727

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1. 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 [4–7].

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 [20]. 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 [42] 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 [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 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

x zy

B

d

ε2, μ2

ε1, μ1

ε3, μ3

σ1

σ2SMPs

(a)

ħω/EF

Con

duct

ivit

y (e

2 /h)

Im(σyy), E1 = 0

Im(σyy), E1 = EF

Re(σyz), E1 = EF

(b)

E2-E1 E1-E0 E2-E-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(σyy) and Re(σyz) as a function of frequency without the magnetic field (black dotted line), and with a magnetic field of E1 = EF (red solid and blue dashed lines). The solid black lines indicate the electron-electron transitions between LLs of intraband (E1-E0, E2-E1) and interband (E2-E-1). The parameters are chosen as EF = 0.05eV, d = T = 300K, Γ = 0.03EF.

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 2( ) 2n FE sign n n eB v= , 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 as

0 0 0

0 ,

0yy yz

y yy

g

z

σ σ σσ σ

= −

(1)

where σyy and σyz are the longitudinal and Hall conductivities, respectively. If the temperature is T and the Fermi energy is EF, the elements in the tensor are given by, respectively [42],


( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )

21 12

22intra, intra,

1 1

22inter, inte

0

r,

,

22

2

F n F n F n F nF

n n

F n F n F n

y

F n

n n

yn

n E n E n E n Ee

i

n E n E n E n E

i eB vi

i

σ ωπ ω

ω

+ +

+

∞

=

+

− + − − −= + Γ+ Γ Δ − Δ

− − + − − + Δ − Δ + Γ

(2a)

( ) ( ) ( ) ( )

( ) ( )

22

1 1

2 22 2intra, in e ,

0

t r2 2

1 1,

F F n F n F ny

n

zn

F n

n

en E neB v

i i

E n E n Eσπ

ω ω

+ +

∞

=

= −

+ Γ +

− − −

Γ

+ −

× + Δ − Δ −

(2b)

where nF(En) = 1/{exp[(En-EF)/kBT] + 1} is the Fermi Dirac distribution, and En 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 σyy and σyz, we consider it as a constant in the calculations as [47].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 [42], 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 [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 nF (E0) = 1, nF (E1) = 0.5 and nF (Ei≥2) ~0, the intraband conductivity σyy can be simplified from Eq. (2a) as:

intra, intra, 1 intra, 2 ,yy yy yyσ σ σ= + (3)

where

( ) ( ) ( )( ) ( ) ( )

02

12

2intra, 1

0

2

1 1 0

22

,yF

yF

Fi eBn E n Ee

i E E E Ev

iσ ω

π ω−

= − − −

+ Γ+ Γ

(4a)

( ) ( ) ( )( ) ( ) ( )

12

22

2intra, 2

1

2

2 2 1

22

.yF

yF

Fi eBn E n Ee

i E E E Ev

iσ ω

π ω−

= − − −

+ Γ+ Γ

(4b)

It can be inferred from 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 [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 ( ) ( ), , z tx y z

iE E E e β ω−=E , 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 by

1 1

2

3

/ / / /( /2

/ /

)/ / / /

( /2)

/ 2 / 2

/ 2 ,

/ 2x y z

x xx y z x y z

x dx y z x y z

x d

A e B e d x d

E C e x d

D e x d

κ κ

κ

κ

−

− −

+

+ − ≤ ≤= ≥ ≤ −

(5)

where Ax/y/z, Bx/y/z, Cx/y/z and Dx/y/z are amplitudes, 1/2/2 2

1/2 3 1/ //3 02 3kκ β ε μ= − , k0 is the wave

vector in vacuum. From the Maxwell equations and the boundary conditions of the double-layer graphene structure, after some derivations, we can have the dispersion of SMPs in a GDS structures as

[ ]

[ ] [ ]

[ ]{ }

2

2

2 22 2 2 2 22 1 2 1

2

3

33 1 3 2

3

3 1

22 22

1 1

2 1

1

1

sin( )

sin( ) sin( )

1 sin( )

1

0

1 yz yz

yz yz

d

d d

d

η σ ηϕ ϕϕ ϕ

ψ ψ κ

κ κψ ψ

σϕ

σ σκ

ϕ

ηϕ ϕ

+

+

⋅ + + +

+ =−

(6)

in which

10 12(3)

2(1(2

2(

)

3)1

1 13)coth( ) yyi d iϕ

μωμ μσμ

κκ

κ κ+= + (7a)

111

2

2(3)2(3)

1 0 11(2)

(3)

coth( ) zzdi iε κ

ψ σ κκε εωε κ

+= − (7b)

ε0 and μ0 are the permittivity and permeability in vacuum, respectively. η1 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.

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),

[ ] [ ]2 22 2 2 20 1 0 12 2 2

0 11 1 1

21 1

0 1 2

1 sin h( ) sin h( ) s1 0in h( )r rzy zy

r r

d d dκ ψ κ κε ε ψ

ϕϕ ϕε

μ μ μσ σε

μ − − + =

− −

(8a)


[ ] [ ]2 22 2 2 20 1 0 12 2 2

0 11 1 1

21 1

0 1 2

1 sin h( ) sin h( ) s1 0in h( )r rzy zy

r r

d d dκ ψ κ κε ε ψ

ϕϕ ϕε

μ μ μσ σε

μ − + + =

− −

(8b)

Substitute Eqs. (7a) and (7b) into Eqs. (8a) and (8b), we have

12 201

11 1

0

1 1 1

1

tan 1 tan 1 02

h h2 zz zz y

rz

d di i

κ κ ωμσ ηε κ

σ σεωκ +

+ + − =

−

(9a)

2 201

1 1 1

11 1

101coth 1 oth

21 0

2czz zz yz

d di i

κ κ ωμσ ηε κ

σ σεωκ +

+ + =

− +

(9b)

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 (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.

β/kF

ħω/E

F

Band 1

Band 2

Band 3

SM, E1=0AM, E1=0

SM, E1=EF

AM, E1=EF

Vacuum

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 E1 = EF. The other parameters are the same as Fig. 1.

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.

Re(Ez)

x/λ

Im(Ex) Im(Ey)

(b)(a) (c)

SM, E1=0

AM, E1=0SM, E1=EF

AM, E1=EF

Fig. 3. Electric field distributions of ħω = 1.5EF with and without the external magnetic fields. (a) Re(Ez); (b) Im(Ex); (c) Im(Ey). 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 E1 = EF, respectively.

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. i ntra,0 2 BΔ ∝ , i nter,1 (2 2) BΔ ∝ + . 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 1

2 21 yzη σ in Eqs. (9a) and (9b) is negligible (in the

order of ~10−4). Then substituting Eq. (4b) into Eq. (9), we have


2 2

02 21

1 00h Itan 1 ) 0

2m( zz

r

kdk

βε

β σωε

− − −+

= (10a)

2 2

0

1

2 2

010 Icoth ) 0

2m(1 zz

r

kdk

ββ σ

ωε ε +

−− =

− (10b)

As both 2 20tan h( / 2)k dβ − and 2 2

0coth( / 2)k dβ − are in the order of ~1, it can be

inferred from Eq. (10) that as Im(σyy) decreases, when β increases. While with the increase of

B, the increase of i ntra,1 (2 2) BΔ ∝ − 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 1

2 21 yzση± . However, σ1,yz is always much smaller compared with η1,

even when the magnetic field is very strong.

ħω/EF

Im(σ

yy)/

(e2 /h

)

(d)

Band 1 Band 2 Band 3

ħω/E

F

β/kF

Band 3

Band 2

Band 1

(a)

(b)

(c)

B ↑

B ↑

B ↑

E1=0.9EF

E1=1.0EF

E1=1.1EF

E1=1.1EFE1=1.0EF

E1=0.9EF

E1=1.1EF

E1=1.0EF

E1=0.9EF

B ↑

B ↑ B ↑

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 E1 = 0.9EF (green lines), E1 = 1.0EF (blue lines), and E1 = 1.1EF (blue lines) are compared. (a) Band 3; (b) Band 2; (c) Band 1; (d) Comparison of Im(σyy) under the 3 magnetic fields. The other parameters are the same as those in Fig. 2.

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.

ħω/E

F

β/kF

Band 3

Band 2

Band 1

(a)

(b)

(c)

(x-d/2)/λ

|Ez|2

|Ez|2

(x-d/2)/λ

(d) (e)Symmetric

modeAntisymmetric

mode

SM, d ↑ AM, d ↑

SM, d ↑AM, d ↑

SM, d ↑ AM, d ↑

d ↑d ↑

Fig. 5. Effects of the spacing d on the symmetric (red lines) and antisymmetric (blue lines) SMP modes of the GDS structure. (a)-(c) Dispersions of SMPs with various graphene separations of band 3, band2, and band1, respectively. The separations are chosen as d = 0.003λ (solid lines), d = 0.01λ (dashed lines) and d = 0 (dotted lines). (d)-(e), Ez intensity distributions above the GDS structure of the symmetric and antisymmetric modes, respectively. d is increased form 0.003λ (solid lines) to 0.006λ (dashed lines) and 0.01λ (dotted lines).

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.

β/kF1

ħω/E

F1

Band 2

MS mode

MA mode

EF=0.34eV, SLGEF=0.4eV, SLG

ħω=1.3EF

x/λ

Re(Ez)

(a)

(b)

EF1=0.34eV

MS modeMA mode

EF2=0.4eV

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 EF1 = 0.05eV while the lower is EF1 = 0.06eV. For comparison, SMPs on SLG with magnetic fields of EF = 0.05eV and EF = 0.06eV are also plotted (the dashed lines). (b) Field distribution of Re(Ez) of the two modes at ħω = 1.3EF. The other parameters are the same as Fig. 2.

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).


β/kF

ħω/E

F Band 2

ħω=1.1EF

x/λ

Re(Ez)

(a)

(b)

ε3=1MS modeMA mode

ε1=6ε2=3.8

MS mode

MA mode

ε1(6)/ε2(1), SLGε1(6)/ε2(3.8), SLG

Fig. 7. (a) SMPs modes (solid lines) of band 2 of a GDS structure with ε1 = 6, ε2 = 3.8, ε3 = 1. Compared with SMP modes on SLG interfaces of Al2O3 / Air (red dashed line) and Al2O3 / SiO2 (blue dashed line). (b) Field distribution of Re(Ez) of the MS and MA modes at ħω = 1.1EF withε1 = 6, ε2 = 3.8, ε3 = 1. The other parameters are the same as Fig. 2.

6. 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, 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.

Acknowledgments

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.


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