Propagation of surface plasmon polaritons in monolayer graphene surrounded bynonlinear dielectric mediaS. Baher, and Z. Lorestaniweiss

Citation: Journal of Applied Physics 124, 073103 (2018); doi: 10.1063/1.5031191View online: https://doi.org/10.1063/1.5031191View Table of Contents: http://aip.scitation.org/toc/jap/124/7Published by the American Institute of Physics

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Propagation of surface plasmon polaritons in monolayer graphenesurrounded by nonlinear dielectric media

S. Baher and Z. Lorestaniweissa)

Physics Department, Lorestan University, Khorramabad, Iran

(Received 29 March 2018; accepted 29 July 2018; published online 17 August 2018)

The propagation of s-polarized surface plasmon polaritons (SPPs) was investigated in a monolayer

graphene sheet surrounded by two dielectric media on each side, so that one or both sides of the

media were linear or nonlinear with Kerr-type nonlinearity. The plasmonic properties including the

wave propagation index neff , the penetration depth, the time-averaged power flow and the spatialprofile of electric and magnetic fields were calculated for the following structures: Linear medium/

Graphene/Linear medium (L/G/L), Nonlinear medium/G/L (NL/G/L) and NL/G/NL. The analysis

of the nonlinear coefficient effect on the SPP properties showed that increasing the nonlinearity in

NL/G/L enhanced neff . However, for a smaller difference between the nonlinearity of layers, neffdecreased in NL/G/NL. By comparing between the proposed structures, it was found that while

large values of neff can be obtained from L/G/L, its frequency confinement is smaller than that ofNL/G/L and NL/G/NL. Furthermore, NL/G/L and NL/G/NL were able to support localized nonlin-

ear modes, leading to enhanced frequency confinement of transverse electric (TE) waves in

the presence of nonlinearity. Increasing the nonlinearity in NL/G/L confined the spatial profile

of the electric field near the graphene interface, indicating the existence of surface plasmon

solitons. The influence of the graphene chemical potential l on the plasmonic properties of thestructures was also investigated. In this case, it was found that the plasmonic properties can be con-

trolled by l. Our calculations may solve the difficulties in TE surface plasmons for application inoptics and plasmonics. Published by AIP Publishing. https://doi.org/10.1063/1.5031191

I. INTRODUCTION

In recent years, a great deal of attention has been given

to the field of plasmonics, in particular, the nonlinear plas-

monics. This is because that it can support the emergence of

significant and novel phenomena including negative refrac-

tion, cloaking and super-sensing, providing various applica-

tions in plasmonic devices such as plasmonic sensors,1,2

optoelectronics3 and metamaterial cloaking.4 In this way, the

plasmonic field is dealing with the generation, manipulation

and detection of surface plasmon polaritons (SPPs). The

SPPs are quasi-particles originating from the coupling of

light and surface plasmons (SPs).5 In turn, SPs are electro-

magnetic (EM) waves propagating along the boundary sur-

face between a metal (or a semiconductor) and a dielectric.5

In fact, these are transverse magnetic (TM) modes accompa-

nied by collective oscillations of surface charge confined

near the interface, decaying exponentially in the transverse

direction.

Nonlinear plasmonics is a fast-growing research field

encompassing a wide range of applications. Notably, the

nonlinear response of plasmonic systems has been observed

in metal films and metallic nanostructures. In the field of

nanophotonics, controlling light at scales considerably

smaller than its wavelength together with some other inter-

esting phenomena occurring in the implementation of most

metamaterials is related to plasmonics.

The plasmonic materials (mostly metals) have signifi-

cant losses in a frequency range of interest.6 To overcome

this limitation on the metal-based plasmonic geometries,

physicists were motivated to investigate plasmons and their

losses in new materials with outstanding properties.7 For

example, graphene plasmons can overcome the difficulties

arising from the s-polarized transverse electric (TE) modes.

Due to the lack of high field confinement in systems with a

parabolic band structure, the TE modes are not common to

metal-based plasmonic structures.8,9 As a gapless material,

graphene can be doped to high values of electron and hole

concentrations by applying voltage externally.10 In addition,

the collective excitation of plasmons in graphene holds great

potential for technological applications.11 Another unique

feature of graphene is its capability to tune optical conduc-

tivity via electric and magnetic fields, gate potential12,13 and

chemical doping.14

The dynamic frequency-dependent conductivity of gra-

phene, r xð Þ, can be determined in the framework of linearresponse theory (LRT) and random phase approximation

(RAP), leading to the Kubo formula.15 In turn, this comprises

intraband and interband contributions: r xð Þ ¼ rinter xð Þþrintra xð Þ ¼ r0 xð Þ þ ir00 xð Þ. The imaginary part of gra-phene conductivity, r00 xð Þ; may be positive or negative.9When r00 xð Þ is negative, the TE mode can exist in monolayergraphene.9 However, the resulting mode is bounded in a very

narrow frequency interval defined as: 1:67 < �hxl < 2,9 where

�h and l are the Planck constant and the graphene chemicalpotential, respectively. To overcome the difficulty in identify-

ing the s-polarized TE SPPs from the incident EM wave ina)Author to whom correspondence should be addressed: lorestani.zi@fs.lu.ac.ir

0021-8979/2018/124(7)/073103/11/$30.00 Published by AIP Publishing.124, 073103-1

JOURNAL OF APPLIED PHYSICS 124, 073103 (2018)

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plasmonic devices, the use of multilayer graphene sheets16

and strained graphene flakes17 while also considering the spa-

tial dispersion of graphene18 has been suggested. Moreover,

merging graphene with a waveguide structure19 and the use

of a nonlinear substrate on graphene20 have improved the TE

mode.

The TE mode is very sensitive to the difference between

two dielectric media embedded in the graphene sheet,8 exhibit-

ing a very low propagation loss.9,21 The presence of the nonlin-

ear dielectric media together with graphene may overcome the

difficulties in using s-polarized modes for nano-optic applica-

tions. The effect of a Kerr-type nonlinear substrate on the dis-

persion and propagation distance of graphene plasmons has

been theoretically studied in the literature.20

As investigated by Bludov et al.,22 in the case of coatinga nonlinear substrate with a graphene layer, nonlinear TE

plasmons were strongly localized in the vicinity of the gra-

phene interface, in contrast to linear TE SPPs in graphene

with a weakly localized field. Also, interestingly, they found

that the TE modes existed even if the imaginary part of con-

ductivity was positive.22

In this study, a theoretical model of s-polarized TE modes

supported by a three-layer structure consisting of a graphene

sheet bounded symmetrically on each side by another semi-

infinite dielectric material is presented. The real and homoge-

neous dielectric functions of the media are considered to be

linear or nonlinear. Moreover, the dielectric functions, ej,entering into the proposed structure are assumed to have the

following general form: ej ¼ ejL þ aj Ejj j2, where ejL is the lin-ear part of the dielectric constant, aj is the self-focusing coeffi-cient (aj > 0) and Ejj j is the amplitude of the electric field.Additionally, aj is assumed to be independent of Ejj j.

The analytical expression for the dispersion relation and

plasmonic properties such as the wave propagation index,

neff , the penetration depth, d, the time-averaged power flowper unit length, Ptot, and the spatial profile of electric andmagnetic fields is calculated for the following structures:

Linear medium/Graphene/Linear medium (L/G/L), Nonlinear

medium/G/L (NL/G/L) and NL/G/NL. The effect of the non-

linear coefficient on the SPP properties is analyzed, showing

that neff is enhanced when increasing the nonlinearity in NL/G/L. However, neff decreases in NL/G/NL for a smaller dif-ference between the nonlinearity of layers. It is shown that

NL/G/L and NL/G/NL can support localized nonlinear modes

and the presence of nonlinearity leads to an enhancement in

the frequency confinement of TE waves. Finally, the influ-

ence of l on the plasmonic properties is investigated.This paper is outlined as follows: In Sec. II, the theoreti-

cal model is described and a first integral of the linear wave

equations is obtained for related integral constants. The

scheme for solving these equations is examined for three

structures in Subsections II A–II C. The results, some discus-

sions and possible extensions are given in Sec. III.

II. THE THEORETICAL MODEL

Initially, a monolayer graphene interface between two

homogeneous semi-infinite linear dielectrics e1 and e2 istaken into consideration, so that the graphene is placed at

z ¼ 0, according to the 3D schematic representation shownin Fig. 1. In this way, the interface is in the x� y plane andthe surrounding layers are labeled by the integer index j (¼1and 2), while also choosing the wave propagation along the xdirection parallel to the interface. The starting point for the

general analysis of the surface wave is as follows:

c2r2~E � ej@2~E

@t2¼ 0; (1)

where c is the speed of light and x is the angular frequencyof the wave. For s-polarized TE modes, the field components

of EM waves can be expressed as given as follows:

~E x; zð Þ ¼ 0; Ey; 0ð Þ eibx�ixt; (2a)

~H x; zð Þ ¼ Hx; 0; Hzð Þeibx�ixt: (2b)

Here, bð¼ kxÞ is the in-plane wave vector along the x direc-tion. From Eqs. (1) and (2), one obtains

@2Ej zð Þ@z2

þ x2

c2ej � b2

� �Ej zð Þ ¼ 0; (3)

where Ej denotes the y component of the electric field EjðzÞfor the layer j. By integrating Eq. (3) with respect to z, onegets the following equation:

@Ej@z

� �2þ gj E2j

� �¼ Cj; (4)

where Cj is the constant of integration for the layer j and thecoordinate z is measured perpendicular to it. The generalform of gjðE2j Þ; as a quadratic function of the electric fieldamplitude, is given by

gj E2j

� �¼ 2

ðE0

x2

c2ej � b2

� �E0dE0

¼ x2

c2ejL � b2

� �E2j þ

x2

c21

2ajE

4j (5)

and the integration constants Cj and Cjþ1 are then related toeach other as follows:23,24

FIG. 1. The 3D schematic representation of graphene-based plasmonic struc-

tures (L/G/L, NL/G/L, and NL/G/NL) with dielectric constants e1 and e2.

073103-2 S. Baher and Z. Lorestaniweiss J. Appl. Phys. 124, 073103 (2018)

Cjþ1 ¼ Cj þx2

c2ejLþ1 � ejL þ

1

2aj E

2j

� �E2j : (6)

In Subsections II A–II C, the theoretical approach is

applied to three geometrical structures in which the dielectric

functions of the media surrounding the graphene sheet are

either linear or nonlinear.

A. L/G/L

As schematically presented in Fig. 1, L/G/L is a simple

structure consisting of a graphene layer surrounded by two

linear dielectric media in which the dielectric functions are

independent of the electric field amplitude. A necessary con-

dition for guided or surface waves is to consider EyðzÞ and@Ey@z to decay exponentially as zj j ! 1. This is attained whenboth arbitrary constants C1 and C2 are zero [see Eq. (4)].Hence, the required form of Eq. (5) in L/G/L is given by the

following expression:

gj E2j

� �¼ x

2

c2ej � b2

� �E2j : j ¼ 1 and 2: (7)

With the given function gj E2j

� �, the solution of Eq. (4)

can be written as

E1 zð Þ ¼ E01 e�q1z; z > 0 8að ÞE2 zð Þ ¼ E02 eq2z; z < 0; 8bð Þ

(

where E01 ¼ E1 z ¼ 0ð Þ ¼ E0 and E02 ¼ E2 z ¼ 0ð Þ ¼ E0.From the following condition: q2j ¼ b2 � x

2

c2 ej, where b >xcffiffiffiffi

ejp

(for a surface wave), the wave vector q2j should be realand positive. Moreover, the parameter b is the same for bothlayers due to the invariance in the y direction.

The boundary conditions at z ¼ 0 for EM waves yield

E1 z ¼ 0ð Þ ¼ E2 z ¼ 0ð Þ; 9að ÞH1;x z ¼ 0ð Þ � H2;x z ¼ 0ð Þ ¼ cl0r xð ÞE1 z ¼ 0ð Þ; 9bð Þ

(

where Hi;x ¼ c�ix @~E@z and r xð Þ is the complex surface conduc-

tivity of monolayer graphene, which can be expressed as

follows:9

r xð Þ ¼ rinter xð Þ þ rintra xð Þ; (10)

where the intraband and interband contributions at kBTl ! 0are separately defined by

rintra xð Þ ¼ ie2

p�h�hx

l

; (11)

rinter xð Þ ¼ e2

4�hh

�hxl� 2

� �� i

pln

2þ �hxl

2� �hxl�

266664

377775; (12)

where e denotes the electron charge, �h is the reduced Planckconstant, x is the frequency, l is the chemical potential, Trepresents the temperature in K and kB is the Boltzmann con-stant. By applying the boundary conditions shown in Eqs.

(9a) and (9b) for z ¼ 0, one obtains the following dispersionrelation:

q1 þ q2 ¼ ixl0r xð Þ: (13)

The left-hand side of the above equation is real and positive.

Therefore, the conditions Re r xð Þ½ � ¼ 0 and Im r xð Þ½ � < 0are required.

B. NL/G/L

To investigate the propagation of SPP modes in the

structure NL/G/L, the approach developed in Sec. II A, is

used here. Accordingly, Eq. (5) changes as follows:

gj E2j

� �¼

x2

c2ej � b2

� �E2j ; j ¼ 2; 14að Þ

x2

c2ejL � b2

� �E2j þ

x2

c21

2ajE

4j ; j ¼ 1: 14bð Þ

8>>><>>>:

By inserting the above equations in Eq. (4), one may

obtain components of the EM wave as given below:

E1 zð Þ ¼ffiffiffiffiffi2

a1

rcq1x

1

cosh q1 zþ z1ð Þ½ �

H1x ¼ �iffiffiffiffiffi2

a1

rc2q21x2

sin h q1 zþ z1ð Þ½ �cos h q1 zþ z1ð Þ½ �ð Þ2

;

z > 0; 15að Þ

E2 zð Þ ¼ E0eþq2z

H2x ¼ þic

xq2E0e

þq2z ; z � 0: 15bð Þ

8>>>>>>>>>>><>>>>>>>>>>>:

Here, the parameter z1 is the second integration constantdetermining the position of the soliton peak in the nonlinear

medium and is calculated as follows:

z1 ¼ �1

q1cosh�1

ffiffiffiffiffiffiffiffiffiffi2

a1E20

scq1x

24

35; (16)

Applying the boundary conditions [i.e., Eq. (9)] for z ¼ 0leads to the following dispersion relation:

q1 tanh q1z1ð Þ þ q2 ¼ il0xr xð Þ: (17)

This expression has been obtained previously by Bludov

et al.,22 demonstrating the validity of the approach used inthe present study. In the absence of nonlinearity (i.e., a1 ! 0and z1 !1), Eq. (17) is reduced to Eq. (13). It is worthmentioning that since the function tan hu never exceeds

unity, a necessary condition for the solution is:il0xr�q2

q1

< 1. Using the relation 1

cosh q1 z1ð Þ½ � ¼ffiffiffiffia12

px

cq1E0 together with

the known function tan hu ¼ 6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

cos hu

�2q, Eq. (17)

changes to the following form:

q21 ¼e1L � e2ð Þ þ

1

2a1E

20 þ c2l20r2

�2l0cr

0@

1A

2

xc

� �2: (18)

073103-3 S. Baher and Z. Lorestaniweiss J. Appl. Phys. 124, 073103 (2018)

This is a simple equation with explicit dependence on the

electric field intensity at the interface.

Next, the role of Ptot in the graphene-based plasmonicstructure is studied. In this case, Ptot is calculated in the xdirection, given by25

Ptot ¼X2j¼1

Pjx ¼1

2

ðSxj�

dz; (19)

where j denotes the index of each layer, the 12

factor is due to

the average time, and Sxj�

is the time-independent x compo-nent of the Poynting vector which can be expressed as follows:

Sxj� ¼ Re Eyj Hzjn o: (20)

By using Eq. 17ð Þ; the above quantity is expressed as

Ptot ¼c2

x3b

al0

q1 þ q2ð Þ2

2q2þ c

2

xbl0a

1

2q2rð Þ2; (21)

which is a function of x, l and the nonlinearity.

C. NL/G/NL

For the general case in which both surrounding media

have Kerr-type self-focusing nonlinearity, the solution

approach is a direct extension of what was used in Sec. II B.

To determine the dispersion relation, one needs to integrate

across the interface, incorporating the boundary conditions.

Therefore, using the same approach presented in Sec. II B,

for the electric and magnetic field components of TE modes,

one obtains

E1 zð Þ¼ffiffiffiffiffi2

a1

rcq1x

1

cosh q1 zþ z1ð Þ½ �

H1x¼�iffiffiffiffiffi2

a1

rc2q21x2

sinh q1 zþ z1ð Þ½ �cosh q1 zþ z1ð Þ½ �ð Þ2

;

a1 > 0; z> 0 22að Þ

E2 zð Þ¼ffiffiffiffiffi2

a2

rcq2x

1

cosh q2 z� z2ð Þ½ �

H2x¼�iffiffiffiffiffi2

a2

rc2q22x2

sinh q2 z� z2ð Þ½ �cosh q2 z� z2ð Þ½ �

�2 ;

a2> 0; z< 0: 22bð Þ

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:The above solutions are known as soliton solutions. The soli-

ton peak position zj can be calculated as follows:

zj ¼ �1

qjcosh�1

ffiffiffiffiffiffiffiffiffi2

ajE20

scqjx

24

35; j ¼ 1 and 2: (23)

Using the boundary conditions expressed in Eq. (9), one

may get

ffiffiffiffi2

aj

s� cqj

x1

cosh qjzj½ �¼ E0; 24að Þ

ffiffiffiffiffi2

a1

r� c

2q21x2

sin q1 zþ z1ð Þ½ �cos h q1 zþ z1ð Þ½ �ð Þ2

�ffiffiffiffiffi2

a2

r� c

2q22x2

sin q2 z� z2ð Þ½ �cos h q2 z� z2ð Þ½ �

�2 ¼ icl0rE0; 24bð Þ

8>>>>><>>>>>:

in which the required form of the dispersion relation is determined after some algebraic computations, as given below

q21 ¼þ2 e1L � e2Lð Þa1E20 þ il0rcð Þ2 �

1

2a2E

20 þ a1E20

�þ e1 � e2Lð Þ

� �2� a2E20a1E20

4 il0crð Þ2

264

375: (25)

Furthermore, using Eqs. (19) and (25), Ptot in NL/G/NLcan be expressed as follows:

Ptot ¼bl0

c2

x3q2a2

1� tan h q2z2½ �

�þ q1

a1tan h q1z1½ � � 1

�� �

:

(26)

III. RESULTS AND DISCUSSION

As stated previously, the propagation of SPPs in geo-

metrical structures consisting of a graphene layer surrounded

by a semi-infinite dielectric layer is investigated in this

study. The influence of NL and l on the SPP propertiesincluding neff ¼ cbx , the behavior of electric and magnetic

fields, d ¼ 1ReðbÞ and Ptot are also taken into consideration. It

is assumed that the Kerr-type nonlinearity is in the following

form: ej ¼ ejL þ aj Ejj j2. The frequency is chosen to be in therange of 1:67 < �hxl < 2. However, due to the absence of TEmodes in some frequencies, the x range becomes narrowerfor the linear case, even if r00 xð Þ < 0.

The graphene conductivity parameters are chosen as fol-

lows: l ¼ 0:2 eV and the charged particle scattering, C ¼ 1s,where s is the relaxation time, is set to zero. The linearmedium permittivity is also considered to be the same as that

of the linear part of the nonlinear medium i.e., e1 ¼ e1L¼ 2:89 and e2 ¼ e2L ¼ 2:80. For all structures, the electricfield intensity, E0, is identical and equal to E0 ¼ 106 Vm. Forthe structures NL/G/L and NL/G/NL, a1E20 ¼ 0:01, unlessotherwise stated.

073103-4 S. Baher and Z. Lorestaniweiss J. Appl. Phys. 124, 073103 (2018)

Since q21 is required to be real and positive for the exis-tence of SPPs [see Eqs. (18) and (25)], the solutions are

obtained in the case of surface waves with neff > e1L. Hence,the real part of conductivity is neglected, i.e., Re r xð Þ½ � ¼ 0.

In Figs. 2, 3, 5, and 6, the first column (panels a and b),

the second column (panels c and d) and the third column

(panels e and f) correspond to the structures L/G/L, NL/G/L,

and NL/G/NL, respectively. Figure 2 presents the variation

of neff and d as a function of x. As is seen, neff decreaseswith increasing x and L/G/L has a higher neff than that ofthe other structures. Note that due to the absence of the non-

linearity coefficient (i.e., aj ¼ 0) in L/G/L, there is only onecurve with higher neff in the smaller x range, compared toNL/G/L and NL/G/NL. The strength of nonlinearity is deter-

mined by the parameter a1E20 in NL/G/L. As is evident inFigs. 2(b) and 2(c), neff (d) of NL/G/L increases (decreases)with increasing a1E20. In other words, one can improve neffand d in NL/G/L by tuning the a1 value.

To investigate the effect of the nonlinear Kerr coeffi-

cient, a dimensionless parameter is defined for NL/G/NL as

follows: t ¼ a2E20

a1E20¼ a2a1. As is inferred, neff and d depend on

the nonlinear Kerr coefficient. While increasing t ða2 > a1)enhances d in NL/G/NL, it leads to a decrease in the corre-sponding neff [see Figs. 2(e) and 2(f)]. It should be noted thatsince the difference between a2 and a1 is small ( 0:4 eV, neff varies slowly as a function of l; therefore,large neff values can be achieved in NL/G/L and NL/G/NLfor l � 0:4 eV. From Figs. 3(d) and 3(f), while increasing lcontinuously enhances d, the variation of d in NL/G/NL ismore pronounced than that in NL/G/L, leading to higher

localization [Note that Figs. 3(d) and 3(f) are in good agree-

ment with the electric field in Figs. 6(c) and 6(e)]. The varia-

tion of neff and d versus the nonlinearity coefficients ofthe nonlinear structures is plotted in Fig. 4. In the case of

NL/G/L, Figs. 4(a) and 4(b) show the dependence of neff andd on changes in a1E20. In other words, neff ðdÞ increases(decreases) as a function of a1E20. However, as can be seen inFigs. 4(c) and 4(d), neff ðdÞ of NL/G/NL decreases (increases)with increasing t. For example, enhancing a1E20 of NL/G/Lfrom 0.4902 to 0.7396 increases (decreases) neff ðdÞ from94.1536 (0.0060 lm) to 134.6142 (0.0042 lm) at x¼ 0:35 eV and l¼ 0.2 eV. This may find applicability fornonlinear devices based on graphene materials, improving

their SP properties.26 In fact, integrating graphene with plas-

monic devices may also provide an opportunity to develop

FIG. 2. The variation of ne ff and d as a function of x in: (a) and (b) L/G/L, (c) and (d) NL/G/L, and (e) and (f) NL/G/NL. In panels (c) and (d), the solid(blue), dashed (red), and dashed-dotted (black) curves refer to a1E20 ¼ 0:01, 0:050, and 0:09, respectively, whereas they refer to t ¼ 0:1, 0:5, and 0:9 in panels(e) and (f). Other parameters include e1L ¼ 2:89, e2L ¼ 2:80 and l ¼ 0:2 eV.

073103-5 S. Baher and Z. Lorestaniweiss J. Appl. Phys. 124, 073103 (2018)

planar optoelectronic devices.27 In this direction, Xiao et al.28

recently introduced graphene (grown by a chemical vapor

deposition technique) in an Au plasmonic nanoresonator

array as an encapsulating medium. Using graphene with

l¼ 0.26 eV for simulation calculations enabled them to bestreproduce the shifts observed in the experimental extinction

spectra of a concentric ring/disc cavity, thereby determining

the graphene dielectric properties.28

To study the evolution of plasmon-soliton and plasmon-

polariton waves in the proposed structures, variations of the

electric field spatial profile are depicted in Fig. 5, where the

corresponding characteristics are shown as contour plots in the

second row. Noticeably, the mode evolutions are more obvious

in the contour plots than in the spatial profiles. In this direction,

the dark blue regions in the corresponding contour plots indi-

cate that the electric field amplitude is quenched and the bright

regions show the evolution of the localized soliton peaks.

Figure 6 presents the spatial profile of electric and magnetic

fields for two frequencies (A and B) depicted in the inset show-

ing neff as a function of x. Increasing x broadens the electricfield amplitude width in L/G/L [Fig. 6(a)], while also locating

its peak position at the graphene interface. It is also found that

while HxðzÞ is different in each side of the graphene layer andits magnitude decreases with increasing x at the grapheneinterface, HxðzÞ can possess negative values in L/G/L [see Fig.6(b)]. Alternatively, for NL/G/L, the electric field curves show

one peak at z1Aj j ¼ 0:2130 and z1Bj j ¼ 0:3231 located insidethe nonlinear medium. In this case, increasing x increases thewidth of the spatial soliton, keeping its peak position out of the

interface for larger z1j j. It is notable that the same results werereported elsewhere [see Fig. 3(c) in Ref. 22].

Furthermore, it is evident from Fig. 6 that the nonlinear

medium leads to enhanced field intensity. Based on the insets

of Fig. 6, the TE modes with neff ;A > neff ;B show higher peaksand lower widths in the corresponding curve A compared to

curve B. Since the real part of neff corresponds to the SPwavelength kSP ¼ 2pReðbÞ, the electric field with higher neffresults in lower kSP; indicating more stable SPP modes.Therefore, decreasing the x (thus increasing neff ) confinesthe electric field near the graphene interface, while also sta-

bilizing the SPP mode. The reason behind the two soliton

peaks observed in NL/G/NL [Fig. 6(c)] is the presence of

two nonlinear media, arising from the nonlinear dielectric

term [Eqs. (22a) and (22b)]. Thus, the asymmetry obtained

in the peak heights of Fig. 6(c) emerges due to the different

nonlinear term (a2 > a1). On the contrary, for an identicalnonlinear term (i.e., t ¼ 1), one would obtain symmetricpeaks.

The comparison between the variation of HxðzÞ in L/G/Land NL/G/L shows that the magnetic field behavior of the

nonlinear structure is different from that of the linear one. In

fact, increasing x shifts the spatial profiles with a positive(negative) maximum (minimum) Hx value away from thegraphene interface, which leads to broader and smaller

curves according to Figs. 6(c) and 6(d). In NL/G/NL, there

are two soliton peaks with different amplitudes in both sides

of graphene for each x so that the spatial soliton peak ampli-tudes are reduced with increasing x. Meanwhile, both peaks

FIG. 3. The variation of ne ff and d as a function of l for three different x in: (a) and (b) L/G/L, (c) and (d) NL/G/L, and (e) and (f) NL/G/NL. In panels (a) and(b), the solid (blue), dashed (red) and dashed-dotted (black) curves refer to x ¼ 0:342; 0:346; and 0:350 eV, respectively, whereas they refer to x ¼ 0:35,0:37, and 0:40 eV in panels (c)–(f). Other parameters include: e1L ¼ 2:89, e2L ¼ 2:80, a1E20 ¼ 0:01; and t ¼ 0:1.

073103-6 S. Baher and Z. Lorestaniweiss J. Appl. Phys. 124, 073103 (2018)

broaden and shift from the interface simultaneously [see

Figs. 6(e) and 6(f)].

To study the effect of nonlinearity on the spatial profile

of the electric field in NL/G/L, the variation of EyðzÞ wasinvestigated for different nonlinearities, and the results

obtained are demonstrated in Fig. 7(a). As is seen, the width

of the spatial solitons decreases with increase in the nonlinear

term a1E20, and their peak positions shift toward the interface.As a result, one can achieve the TE surface wave with stron-

ger localization and higher intensity on changing the nonlin-

ear term.

It is also found that the peak positions shift toward the

graphene interface when decreasing t, thereby reaching thehigh localization [see Fig. 7(b)]. On the other hand, the vari-

ation of EyðzÞ is plotted in Fig. 8 using different values of l.Based on Fig. 8(a), although the amplitude heights remain

the same, increasing l broadens the profile curves of L/G/Land detaches them from the graphene interface. In Fig. 8(b),

the corresponding curves of NL/G/L shift toward the gra-

phene interface when decreasing l, leading to stronger andsharper electric field amplitudes. In Fig. 8(c), increasing l ofNL/G/NL broadens and weakens both electric fields while

also detaching them from the interface.

With regard to the localization of the electric field in

L/G/L, based on Eq. (8), the electric field maximum posi-

tion is always located at z¼ 0 [Fig. 8(a)]. For NL/G/L andNL/G/NL, the electric field position is obtained using Eqs.

(16) and (23). In turn, this is a function of x, a1E20 and q.Accordingly, the electric field position changes as a func-

tion of the variables at each z. In other words, apartfrom controlling the nonlinear modes by l, it is possible tocontrol them through a1E20. If z¼ 0 for NL/G/L, one cansee that the propagation characteristics of the surface

polaritons are similar to those of L/G/L. This means that

the maximum electric field is located in the interface

between the linear and nonlinear media. In this case, the

FIG. 4. The variation of ne ff and d as a function of the nonlinear term a1E20 for three different x in: (a) and (b) NL/G/L, and (c) and (d) NL/G/NL. The solid(blue), dashed (red) and dashed-dotted (black) curves refer to x ¼ 0:35, 0:37, and 0:39 eV, respectively. Other parameters include: e1L ¼ 2:89, e2L ¼ 2:80,and l ¼ 0:2 eV.

073103-7 S. Baher and Z. Lorestaniweiss J. Appl. Phys. 124, 073103 (2018)

localization of the electric fields is negligible. If z 6¼ 0,localization of the modes is much stronger with respect to

the linear dielectric, due to the nonlinearity of the graphene

substrate. For NL/G/NL, there are two nonlinear dielectrics

so that the localization of the modes is stronger than that in

L/G/L and NL/G/L. Therefore, it can be seen that the non-

linear medium significantly affects the localization of the

surface modes [Figs. 8(b) and 8(c)].

FIG. 6. The spatial profile of electric and magnetic fields along the z direction in: (a) and (b) L/G/L, (c) and (d) NL/G/L, and (e) and (f) NL/G/NL. The curvesA and B correspond to the respective points in the inset image (ne ff as a function of x) so that they correspond to x ¼ 0:342 and 0:346 eV [panels (a) and (b)],and x ¼ 0:35 and 0:36 eV [panels (c)–(f)], respectively. The white and pink regions correspond to two different media separated by graphene.

FIG. 5. The 3D evolution of the spatial profile of surface plasmon polaritons and the corresponding contour plots in: (a) and (b) L/G/L, (c) and (d) NL/G/L,

and (e) and (f) NL/G/NL. The parameters involved are as follows: l ¼ 0:2 eV, x ¼ 0:342 eV [panel (a)] x ¼ 0:35 eV and a1E20 ¼ 0:01 [panel (b)], and,x ¼ 0:35 eV and t ¼ 0:1 [panel (c)].

073103-8 S. Baher and Z. Lorestaniweiss J. Appl. Phys. 124, 073103 (2018)

Finally, based on Eqs. (21) and (26), Fig. 9 illustrates the

variation of Ptot as a function of l and the nonlinear termsa1E20 and t for three different x. From Figs. 9(a) and 9(b),increasing l and a1E20 monotonically reduces Ptot in NL/G/L.Taking into account the nonlinear medium with a large value

of a, the energy carried by the SPP waves is reduced and con-fined near the graphene interface. These behaviors of Ptot are in

good agreement with the amplitude of the electric field investi-

gated in Fig. 7(a). In addition to decreasing with increasing l,it is seen that Ptot decreases as t increases in NL/G/NL [see

Figs. 9(c) and 9(d)]. This means that, for a smaller difference

between the two nonlinearities, the power flow can reach

higher values. On the other hand, since increasing l enhancesits optical conductivity, perfect conductor-like behavior may be

achieved in the graphene layer, extending the field of surface

plasmon modes into the dielectric medium.

IV. CONCLUSIONS

In summary, the dispersion relations were obtained for

the s-polarized TE surface plasmons in the following

graphene-based structures with linearity or Kerr-type nonlin-

earity: L/G/L, NL/G/L, and NL/G/NL. By calculating the

plasmonic properties, large values of neff were obtained fromL/G/L with the smaller frequency confinement than that of

NL/G/L and NL/G/NL. The presence of nonlinearity in NL/

G/L and NL/G/NL led to enhanced frequency confinement

of TE waves, supporting the localized nonlinear modes. The

spatial profile of SPPs in NL/G/NL was found to be different

from that in L/G/L at z 6¼ 0, while being similar to NL/G/L atz ¼ 0. The presence of two bounded nonlinear media led tostronger localization, revealing two peaks in the spatial pro-

file of SPPs. It was also found that the plasmonic properties

including neff , d and Ptot were highly dependent on l.

FIG. 7. The spatial profile of the electric field in: (a) NL/G/L with different a1E20 values and (b) NL/G/NL with different t values. The white and pink regionscorrespond to two different media separated by graphene. In panel (a), the solid (blue), dashed (red), and dashed-dotted (black) curves refer to

a1E20 ¼ 0:01; 0:05, and 0:09, respectively, whereas they refer to t ¼ 0:1; 0:5 and 0:9 in panel (b). Other parameters involved are the same as those of Fig. 4.

FIG. 8. The spatial profile of the electric field for different values of l in: (a) L/G/L, (b) NL/G/L, and (c) NL/G/NL. In all panels, the solid (blue), dashed (red),and dashed-dotted (black) curves refer to l ¼ 0:3; 0:7; and 1:2 eV, respectively. Other parameters involved are the same as those in Fig. 3.

073103-9 S. Baher and Z. Lorestaniweiss J. Appl. Phys. 124, 073103 (2018)

Furthermore, the spatial profile of the electric field was con-

fined near the graphene interface when increasing the nonlin-

earity in NL/G/L. This, in turn, indicated the existence of

surface plasmon solitons. Our study provides evidence that

the plasmonic properties can be controlled by adjusting both

l of graphene and the nonlinearity of media.

1S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).2J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance

sensors,” Sens. Actuators B: Chem. 54(1–2), 3–15 (1999).3N. Sharma et al., “Fuchs Sondheimer–Drude Lorentz model and Drudemodel in the study of SPR based optical sensors: A theoretical study,”

Opt. Commun. 357, 120–126 (2015).4S. Unser, I. Bruzas, J. He, and L. Sagle, “Localized surface plasmon reso-

nance biosensing: Current challenges and approaches,” Sensors 15(7),15684–15716 (2015).

5G. Grosso and G. Parravicin, Solid State Physics, 2nd ed. (AcademicPress, Elsevier, 2014).

6I. Avrutsky, R. Soref, and W. Buchwald, “Sub-wavelength plasmonic

modes in a conductor-gap-dielectric system with a nanoscale gap,” Opt.

Express 18(1), 348–363 (2010).

7F. H. L. MinovKoppensich, D. E. Chang, S. Thongrattanasiri, F. J. G.

Garcia, and d Abajo, “Graphene plasmonics: A platform for strong light-

matter interactions,” Opt. Photonics News 22(12), 36–36 (2011).8X. Y. He, J. Tao, and B. Meng, “Analysis of graphene TE surface plas-

mons in the terahertz regime,” Nanotechnology 24(34), 345203 (2013).9S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,”

Phys. Rev. Lett. 99(1), 016803 (2007).10F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. Ron

Shen, “Gate-variable optical transitions in graphene,” Science 320(5873),206–209 (2008).

11A. Croy, D. Midtvedt, A. Isacsson, and J. M. Kinaret, “Nonlinear damping

in graphene resonators,” Phys. Rev. B 86(23), 235435 (2012).12C. Zhang, L. Chen, and Z. Ma, “Orientation dependence of the optical

spectra in graphene at high frequencies,” Phys. Rev. B 77(24), 241402(2008).

13Y. Jiang et al., “A planar electromagnetic “black hole” based on graphe-ne,” Phys. Lett. A 376(17), 1468–1471 (2012).

14M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared

frequencies,” Phys. Rev. B 80(24), 245435 (2009).15V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical con-

ductivity in graphene,” J. Phys.: Condens. Matter 19(2), 026222 (2006).16B. Wang et al., “Strong coupling of surface plasmon polaritons in mono-

layer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).

FIG. 9. The variation of Ptot as a function l and the nonlinear terms a1E20 and t for three different x in: (a) and (b) NL/G/L, and (c) and (d) NL/G/NL. The solid(blue), dashed (red), and dashed-dotted (black) curves refer to x ¼ 0:35, 0:37; and 0:39 eV, respectively. Other parameters involved are the same as those inFig. 4.

073103-10 S. Baher and Z. Lorestaniweiss J. Appl. Phys. 124, 073103 (2018)

https://doi.org/10.1016/S0925-4005(98)00321-9https://doi.org/10.1016/j.optcom.2015.08.092https://doi.org/10.3390/s150715684https://doi.org/10.1364/OE.18.000348https://doi.org/10.1364/OE.18.000348https://doi.org/10.1364/OPN.22.12.000036https://doi.org/10.1088/0957-4484/24/34/345203https://doi.org/10.1103/PhysRevLett.99.016803https://doi.org/10.1126/science.1152793https://doi.org/10.1103/PhysRevB.86.235435https://doi.org/10.1103/PhysRevB.77.241402https://doi.org/10.1016/j.physleta.2012.03.018https://doi.org/10.1103/PhysRevB.80.245435https://doi.org/10.1088/0953-8984/19/2/026222https://doi.org/10.1103/PhysRevLett.109.073901

17F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, “Linear response

correlation functions in strained graphene,” Phys. Rev. B 84(19), 195407(2011).

18J. S. Gomez-Diaz, J. R. Mosig, and J. Perruisseau-Carrier, “Effect of spa-

tial dispersion on surface waves propagating along graphene sheets,” IEEE

Trans. Antennas Propag. 61(7), 3589–3596 (2013).19M. Hajati and Y. Hajati, “Investigation of plasmonic properties of gra-

phene multilayer nano-ribbon waveguides,” Appl. Opt. 55(8), 1878–1884(2016).

20L. Wang et al., “Surface plasmons at the interface between graphene andKerr-type nonlinear media,” Opt. Lett. 37(13), 2730–2732 (2012).

21A. V. Gorbach, “Nonlinear graphene plasmonics: Amplitude equation for

surface plasmons,” Phys. Rev. A 87(1), 013830 (2013).22Y. V. Bludov et al., “Nonlinear TE-polarized surface polaritons on graphe-

ne,” Phys. Rev. B 89(3), 035406 (2014).

23S. Baher and M. G. Cottam, “Theory of nonlinear guided and surface plas-

mon–polaritons in dielectric films,” Surf. Rev. Lett. 10(01), 13–22 (2003).24O. N. Vassiliev and M. G. Cottam, “Optically nonlinear s-polarized elec-

tromagnetic waves in multilayered symmetric dielectrics,” Surf. Rev. Lett.

7(01n02), 89–102 (2000).25J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley and Sons,

New York, 1999).26J. Lee, A. Lee, and H. K. Yu, “Graphene protected Ag nanowires: Blocking

of surface migration for thermally stable and wide-range-wavelength trans-

parent flexible electrodes,” RSC Adv. 6(88), 84985–84989 (2016).27Z. Fang et al., “Graphene-antenna sandwich photodetector,” Nano Lett.

12(7), 3808–3813 (2012).28Y. Xiao et al., “Probing the dielectric response of graphene via dual-band

plasmonic nanoresonators,” Phys. Chem. Chem. Phys. 15(15), 5395–5399(2013).

073103-11 S. Baher and Z. Lorestaniweiss J. Appl. Phys. 124, 073103 (2018)

https://doi.org/10.1103/PhysRevB.84.195407https://doi.org/10.1109/TAP.2013.2254443https://doi.org/10.1109/TAP.2013.2254443https://doi.org/10.1364/AO.55.001878https://doi.org/10.1364/OL.37.002730https://doi.org/10.1103/PhysRevA.87.013830https://doi.org/10.1103/PhysRevB.89.035406https://doi.org/10.1142/S0218625X03004573https://doi.org/10.1142/S0218625X00000129https://doi.org/10.1039/C6RA17173Ghttps://doi.org/10.1021/nl301774ehttps://doi.org/10.1039/c3cp43896a

s1lcor1s2d1d2ad2bd3d4d5d6f1s2Ad7d8d9d10d11d12d13s2Bd14d15d16d17d18d19d20d21d22d23s2Cd25d26s3f2f3f4f6f5s4f7f8c1c2c3c4c5c6c7c8c9c10c11c12c13c14c15c16f9c17c18c19c20c21c22c23c24c25c26c27c28