Plasmons in spatially separated double-layer graphene nanoribbonsMehran Bagheri and Mousa Bahrami

Citation: Journal of Applied Physics 115, 174301 (2014); doi: 10.1063/1.4873639 View online: http://dx.doi.org/10.1063/1.4873639 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Control of graphene nanoribbon vacancies by Fe and N dopants: Implications for catalysis Appl. Phys. Lett. 101, 064102 (2012); 10.1063/1.4742890 Ab-initio study of co-doped zigzag graphene nanoribbons AIP Conf. Proc. 1447, 805 (2012); 10.1063/1.4710247 Electronic transport properties on transition-metal terminated zigzag graphene nanoribbons J. Appl. Phys. 111, 113708 (2012); 10.1063/1.4723832 Plasmons in electrostatically doped graphene Appl. Phys. Lett. 100, 201105 (2012); 10.1063/1.4714688 Control of the plasmon in a single layer graphene by charge doping Appl. Phys. Lett. 99, 082110 (2011); 10.1063/1.3630230

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Plasmons in spatially separated double-layer graphene nanoribbons

Mehran Bagheri1,a) and Mousa Bahrami21Laser and Plasma Research Institute, Shahid Beheshti University, G. C., Evin, Tehran 19835-63113, Iran2ICFO-Institut de Ciencies Fotoniques, Parc Mediterrani de la Tecnologia, 08860 Castelldefels (Barcelona),Spain

(Received 2 January 2014; accepted 16 April 2014; published online 1 May 2014)

Motivated by innovative progresses in designing multi-layer graphene nanostructured materials in

the laboratory, we theoretically investigate the Dirac plasmon modes of a spatially separated

double-layer graphene nanoribbon system, made up of a vertically offset armchair and metallic

graphene nanoribbon pair. We find striking features of the collective excitations in this novel

Coulomb correlated system, where both nanoribbons are supposed to be either intrinsic

(undoped/ungated) or extrinsic (doped/gated). In the former, it is shown the low-energy acoustical

and the high-energy optical plasmon modes are tunable only by the inter-ribbon charge separation.

In the later, the aforementioned plasmon branches are modified by the added doping factor. As a

result, our model could be useful to examine the existence of a linear Landau-undamped

low-energy acoustical plasmon mode tuned via the inter-ribbon charge separation as well as

doping. This study might also be utilized for devising novel quantum optical waveguides based on

the Coulomb coupled graphene nanoribbons. VC 2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4873639]

I. INTRODUCTION

Owing to their distinctive and appealing properties like

the extreme confinement of carriers,1 the large conductivity

and low loss, the crystalinity, the tunability in doping

level,2,3 and the exotic many-body phases,4–8 mostly origi-

nated from the linear band structure near the Fermi points,

the graphene-based nano-systems of different shapes and

sizes have been recently spurred enormous interest. These

chiral electronic systems propose abundant potentials for

both new physics and technological applications in manufac-

turing graphene devices for photodetectors,9,10 quantum

information,11 and many other fields in quantum photonics

and plasmonics.12–26

Furthermore, the problem of screening effects and con-

fined plasmons in various types of nanostructured graphene

sheets, ranging from monolayer, bilayer, double-layer, multi-

layer, and the array of nano-ribbons have been scrutinized in

systematic ways in a fairly number of papers.25–46 In a large

variety of the aforementioned attempts, the low-lying excita-

tions are treated by the 2D Dirac-Weyl Hamiltonian, which

considers carriers in graphene as massless fermions.47 It is

turned out that plasmons in graphene-based nanosystems,

including 2D,32–41,45 1D,25–31,46 and nano-disk,42 show unique

behaviors like low damping rates, tunability, and confining

and controlling the electromagnetic energy at subwavelength

scales appropriate for many applications in quantum informa-

tion, optoelectronics, and sensors.12–14,20–23,25–27,31,44,48–51

More importantly would be plasmons in various geometries

of spatially separated multi-layer nanostructured graphene-

based materials. Recently, the authors of Ref. 45 have scruti-

nized acoustical and optical collective excitations of spatially

separated and Coulomb correlated layered graphenes. Authors

of Ref. 25, using macroscopic treatments, have examined gra-

phene nanoribbon-based plasmonic waveguides for a pair of

nanoribbons configured either horizontally or vertically and

for ribbons wider than 250 A. Also, a detail calculation on

plasmons in specially separated and coplanar interacting gra-

phene nanoribbons has been performed in Ref. 46. The signif-

icance of confined plasmons in nanoribbon systems should be

manifested in their capabilities for creating and designing

nanometer-sized and hybridized plasmonic waveguides and

plasmon circuits through tuning the nano-ribbon widths, the

light wave-length, the doping level, the dielectric background,

and the inter-ribbon charge separation.25,27,46

Motivated by the aforementioned potential applications

of the Coulomb interacting graphene nanosystems25,26,45,46

and also other nanoscale counterparts,52,53 we theoretically

investigate the behavior of plasmon modes for a spatially

separated double-layer graphene nanoribbon (SSDLGNR)

system, characterized by a vertically offset Coulomb inter-

acting graphene nanoribbon pair. Our analysis is within the

frame of the random phase approximation (RPA) and based

on the k � p model.29,39,54,55 Each isolated graphene nanorib-

bon in the SSDLGNR system could be engineered to exhibit

a rich variety of behavior ranging from metallic to insulating

phases. For simplicity, we assume that two nanoribbons

forming the SSDLGNR system are armchair and metallic

with the same width. We show when both nanoribbons are

assumed to be either intrinsic (I) or extrinsic (E), for non-

zero values of the inter-ribbon charge separations our model

predicts two plasmon branches, a low-energy acoustical and

a high-energy optical mode. In the intrinsic case, the collec-

tive excitations are tuned only by the inter-ribbon charge

separation between two Coulomb interacting nanoribbons. In

the extrinsic case, an extra factor to control these plasmon

modes is supplied by doping. As a result, we exhibit that

the SSDLGNR system supports long wavelength limita)Electronic address: mh-bagheri@sbu.ac.ir.

0021-8979/2014/115(17)/174301/8/$30.00 VC 2014 AIP Publishing LLC115, 174301-1

JOURNAL OF APPLIED PHYSICS 115, 174301 (2014)

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Landau-undamped acoustical plasmons for the small enough

inter-ribbon charge separations which is hardly manifested

in other 3D, 2D, and 1D nano-structured systems.54 This

present study might provide an understanding to design

graphene plasmonics and quantum optical devices with

potential applications to quantum controlled systems

established by the Coulomb interacting graphene

nanoribbons.12,20,25,26,56

The paper is organized as follows: In Sec. II, we present

main equations of our model. Plasmon modes are discussed

in Sec. III. Section IV contains our conclusions and the per-

spective of potential applications.

II. FORMALISM

Figure 1 schematically represents a double layer gra-

phene nanoribbon, characterized by a pair of vertically offset

infinite, mono-layer, armchair, and metallic graphene nano-

ribbons. We suppose the armchair edges of the nanoribbon

are located at x¼ 0 and x¼W, so the SSDLGNR system is

translationally invariant in the y–direction has, with length

L!1 and width W. Also, we assume no inter-layer tunnel-

ing, only the inter-layer Coulomb interaction is important,

the interaction-induced gap predicted in bilayer graphene

systems is therefore ignored.57–59 To calculate the matrix

elements of the Coulomb potential required for computing

the plasmon branches, one would acquire the single-particle

four components wavefunction for an isolated, metallic, and

armchair graphene nanoribbon near the Fermi points K and

K0 within the k � p approximation as follows:4,29

wkkx;kyðx; yÞ ¼ 1ffiffiffiffiffiffiffiffiffiffi

4WLp

ke�ihðkx;kyÞ

1

!eikxx

�ke�ihðkx;kyÞ

1

!e�ikxx

0BBBBB@

1CCCCCAeikyy; (1)

where the corresponding energy eigenvalue is expressed by

Ekðkx; kyÞ ¼ k�hvF

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

x þ k2y

q, with the chiral index k denoting

the conduction (k¼þ1) and valence (k¼�1) subbands,

respectively. vF � 106 ms�1 is the 2D Fermi velocity for gra-

phene. hðkx; kyÞ ¼ tan�1ðk=kxÞ determines the direction of

the pseudospin in the nanoribbon. The longitudinal and

transverse wavevectors are expressed by ky � k and kx

� kðnÞ ¼ np=W � 4p=3ffiffiffi3p

acc, respectively, which is

obtained through the edge boundary conditions with n inte-

ger pointing to the subband index. The width of the nanorib-

bon is expressed by W ¼ ðN þ 1Þffiffiffi3p

acc=2, where N and

acc � 1:42 A are the number of dimer lines and the carbon-

carbon bond length, respectively. When N¼ 3m – 1 in which

m is an integer, the armchair is metallic with no gap between

the conduction and valence subbands. Therefore, l¼ h¼ 2 mcorrespond to the lowest lying linear conduction subband

(l,þ) and the highest lying linear valence subband (h,–),

respectively, in which Ek¼61ðkðlÞ ¼ kðhÞ ¼ 0; k ¼ 0Þ ¼ 0.4

Using Eq. (1), the matrix elements of the intra-

layer(l ¼ �) and inter-layer(l 6¼ �) Coulomb potentials

could be written in the following form:

Vl�n1; n2; n3; n4

k1; k2; k3; k4

ðq; k; k0;Wl;W�; zl; z�Þ

¼ Fln1; n4

k1; k4

ðk; k þ qÞF�n2; n3

k2; k3

ðk0; k0 � qÞvl�n1;n2;n3;n4

� ðq;Wl;W�; zl; z�Þ; l; � ¼ 1; 2; (2)

where n1, n2, n3, and n4 denote the subband indices and k1,

k2, k3, and k4 are served as chiral indices. zl(Wl) and z�ðW�Þdepict the positions of the nanoribbons in the z–direction

(Fig. 1). Generally, two nanoribbons forming the SSDLGNR

system might have different widths. One would calculate the

form factor representing the overlap of states given by

Fjn; n0

k; k0ðk; k0Þ ¼ 1þ kk0ei hjðkðnÞ;kÞ�hjðkðn0Þ;k0Þ½ �

2; j ¼ l; �: (3)

With exploiting the 4-component spinorial wavefunction

expressed by Eq. (1) into Eq. (2), one would obtain the one-

dimensional Fourier transformed Coulomb interaction as

follow:

vl�n1;n2;n3;n4

ðq;Wl;W�; zl; z�Þ

¼ 2e2

�b

ð1

0

du

ð1

0

du0cos½ðn1 � n4Þpu�cos½ðn2 � n3Þpu0�

�K0 jqj½ðuWl � u0W�Þ2 þ ðzl � z�Þ2�12

� �; (4)

where �b stands for the background dielectric function and

K0 is the zeroth order modified Bessel function of the second

kind. Due to the spatial symmetry properties of the Coulomb

potential and according to the parity of the single-particle

states, it is easy to verify that the nonzero components of the

expression vl�n1;n2;n3;n4

meet the conditions jn1 � n2jþjn3 � n4j ¼ 2s, with s is an integer.29 For metallic armchair

nanoribbons within a two-subband model, only the subband

indices n1¼ n2¼ n3¼ n4¼ l¼ h, where k(l)¼ k(h)¼ 0, are

involved. Hereafter, for the sake of simplicity, we define

l ¼ h � 0. To quantify our arguments, we begin with com-

puting the Fourier components of the electron-electron

Coulomb interaction expressed by Eq. (4) as a function of

the wave vector for different inter-ribbon charge separations.

In the numeric, we choose the background dielectric to be

�b ¼ ð�SiO2þ 1Þ=2 � 2:5 as the averaged dielectric constant

FIG. 1. Schematic of a spatially separated double-layer graphene nanoribbon

system consisting of two infinite mono-layer metallic armchair graphene

nanoribbons embedded in silica with edges located at x¼ 0 and x¼W. Two

layers with the same width W are taken away by d ¼ jz2 � z1j in the z–direc-

tion and coupled only through the inter-layer Coulomb interaction.

174301-2 M. Bagheri and M. Bahrami J. Appl. Phys. 115, 174301 (2014)

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of the SSDLGNR system embedded in the silica and

introduce the dimensionless energy and wavevector as

C ¼ c=aB � 12:42 eV and �q ¼ qaB, where c ¼ �hvF

� 6:58 eVA and aB� 0.53 A is the Bohr radius, respec-

tively.7 We choose two nanoribbons of the same width

W1¼W2� 14.75 A, corresponding to the number of dimer

lines N¼ 11, which are located at z1¼ 0 and z2¼ d. It worth

noting that for the case N¼ 11 and within the two-subband

approximation, the subband index of the positive and nega-

tive energies contributed to the Coulomb matrix elements is

actually l¼ h¼ 8.4 As shown in Fig. 2, the dimensionless

Fourier components of the electron-electron Coulomb inter-

action as a function of the dimensionless wave vector

for five different inter-ribbon charge separations d/w¼ 0

(red-solid line), d/w¼ 0.1 (olive-dashed line), d/w¼ 0.5

(blue-dotted line), d/w¼ 1.5 (violet-dashed-dotted line), and

d/w¼ 4.5 (magenta-dashed-double-dotted line) are plotted.

The potential diverges as q ! 0, appropriate to the long

wavelength limit calculations, and speedily falls down with

increasing the inter-ribbon charge separation.

The collective charge density excitations of the system

can be obtained through finding the zeros of the determinant

of the generalized dielectric matrix. Here, the RPA dynami-

cal dielectric function is a 2� 2 matrix whose elements are

obtained via the general expression

�l�n1n3n2n4

k1k2k3k4

ðq;xÞ ¼ dl�dn1n2dn3n4

dk1k2dk3k4

� vl�n1n3

n2n4

ðq;Wl;W�; zl; z�Þ

�Pð0Þ��n2n4

k1k3

ðq;xÞ; l; � ¼ 1; 2; (5)

where dij is the Kronecker delta function and Pð0Þ��nn0

kk0ðq;xÞ

stands for the noninteracting mono-layer graphene nanorib-

bon polarizability function expressed by54,55

Pð0Þ��nn0

kk0

ðq;xÞ

¼ gs

2L

Xk

nF½EknðkÞ� � nF½Ek0

n0 ðkþ qÞ�Ek

nðkÞ�Ek0

n0 ðkþ qÞþ �hðxþ igÞjF�nn0

kk0

ðk;kþqÞj2

¼ð

dk

2pnF½Ek

nðkÞ�Ek

nðkÞ �Ek0

n0 ðkþ qÞþ �hðxþ igÞjF�nn0

kk0

ðk;kþ qÞj2

�ð

dk

2pnF½Ek0

n0 ðkÞ�Ek

nðk� qÞ�Ek0

n0 ðkÞþ �hðxþ igÞjF�nn0

kk0

ðk�q;kÞj2;

(6)

where gs¼ 2 being the spin degeneracy, nF is the Fermi

distribution function, and g! 0þ. In the zero-temperature

limit, nF ¼ #ðeF � eÞ, where # and eF are the step function

and Fermi energy, respectively. In the intrinsic (undoped/

ungated) case, the Fermi wavevector kundopedF ¼ 0; while in

the extrinsic(doped/gated) case, the linear density n is related

to the Fermi wave vector through kdopedF ¼ np=2.55 One would

choose the linear density to be ndoped ’ 106=cm, which yields

the Fermi wavevector KdopedF ¼ EF=C ’ 0:015=A with the

Fermi energy EdopedF ’ 0:1 eV. From Eq. (6), we can realize

that the polarization function is only non-zero over those inter-

vals in the k–space where both the Fermi distribution function

nF as well as the form factor F are at the same time non-zero.

In the Appendix, we present detail expressions concerning the

real and imaginary parts of both intrinsic and extrinsic polar-

ization functions to be fed into their corresponding dielectric

functions. In Sec. III, we will present the explicit form of the

dielectric function for the problem at hand.

III. PLASMON MODES

The plasmon modes are obtained once the real and

imaginary parts of the dielectric function become simultane-

ously zero. Using Eqs. (4) and (6) for a two-subband model,

where contributions to the dielectric matrix come only from

ðl;þÞ � 0 and ðh;�Þ � 0 subbands, the dielectric function

of the SSDLGNR system expressed by Eq. (5) simplifies to

�sðq;xÞ ¼ ½1�Pð0Þ11s ðq;xÞv11

0000ðq;W1; z1Þ�� ½1�Pð0Þ22

s ðq;xÞv220000ðq;W2; z2Þ�

�Pð0Þ11s ðq;xÞPð0Þ22

s ðq;xÞ

� v120000ðq;W1;W2; z1; z2Þ

� �2; s ¼ I;E; (7)

where Pð0Þs denotes the polarization function for the

intrinsic(I) and extrinsic(E) cases. In Figs. 3(a) and 3(b), the

real and imaginary parts of the intrinsic �intrinsicðq;xÞ and ex-

trinsic �extrinsicðq;xÞ dielectric expressions as functions of

the energy for qaB � 0:001 and qaB � 0:01ð> kdopedF aB

� 0:0083Þ, respectively, with the inter-ribbon charge separa-

tion d=w� 0.5, are depicted. As shown, the overall behavior

of the dielectric function, either intrinsic or extrinsic, is dif-

ferent from that of the usual massive one-dimensional fer-

mionic systems. However, one would find that the structure

of the real parts of the dielectric functions are regular over

their corresponding single-particle excitations and diverge at

FIG. 2. The Fourier components of the Coulomb potential as functions of

the dimensionless wave vector for different inter-ribbon charge separations

d/w¼ 0 (red-solid line), d/w¼ 0.1 (olive-dashed line), d/w¼ 0.5 (blue-dotted

line), d/w¼ 1.5 (violet-dash-dotted line), and d/w¼ 4.5(magenta-dashed-

double-dotted line) obtained from Eq. (4). The number of dimer lines is

N¼ 11 corresponding to the nano-ribbon width W1¼W2� 14.75 A.

174301-3 M. Bagheri and M. Bahrami J. Appl. Phys. 115, 174301 (2014)

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the boundaries. Plasmon modes concerned with the intrinsic

and extrinsic transitions are obtained once both real and

imaginary part of the dielectric function simultaneously

become zero, as zoomed in to clearly show two cross points.

Therefore, a plasma consisting of two spatially separated and

distinct components, depending sensitively on the spatial

charge separation between two components forming the

plasma, could show two longitudinal collective oscillations,

a low-energy acoustical (out-of-phase) and a high-energy

optical(in-phase) collective mode, assuming the carriers

forming the plasma have the same charge.54

In Fig. 4, for the nano-ribbon width W1¼W2� 14.75 A,

corresponding to N¼ 11, and for five various inter-ribbon

charge separations d/w¼ 0 (dark-yellow-solid line),

d/w¼ 0.01 (green-diamond line), d/w¼ 0.1 (blue-left-tri line),

d/w¼ 0.5 (cyan-right-tri line), and d/w¼ 1.5 (magenta-circle

line), we calculate the intrinsic acoustical plasmon (IPad=w¼?)

and the intrinsic optical plasmon (IPod=w¼?), denoted by arrows,

in the SSDLGNR system. The Landau-damped line, represent-

ing the intrinsic single-particle excitations (ISPE) with edges

obtained from the poles of the intrinsic dielectric function, is

shown by red-dashed line. As shown, when d/w¼ 0, two

aforementioned low-energy and high-energy plasmon modes

are degenerate. For a finite value of the inter-ribbon charge

separation, this degenerate plasmon branch is split into a linear

acoustical branch IPad=w with a distinct slop just above the

ISPE line and a non-linear optical branch IPod=w with a gap in

between. Upon further increasing the inter-ribbon charge sepa-

ration, the gap between the low-energy and the high-energy

branches is decreased. More importantly, for the large enough

inter-ribbon charge separations, where d=w> 1, the

low-energy acoustical branch is no longer a linear line and

asymptotically approaches the high-energy optical branch.

This is because when v120000ðqÞ ! 0, occurred for larger

values of the ratio d/w, two interacting graphene nano-ribbons

become uncoupled and the plasmon modes of the

SSDLGNR system move towards the plasmon modes of a

single-layer graphene nano-ribbon. As a significant result, the

Landau-undamped linear acoustical plasmon branch could

then be achievable based on the intrinsic SSDLGNR system,

where the inter-ribbon charge separation d/w is small enough.

One would numerically find the condition for the existence of

the linear Landau-undamped acoustical branch xap � va

gq, with

vag to be the intrinsic acoustical plasmon group velocity in

such a way that the critical value dc of the inter-ribbon charge

separation d is found under the condition vag ¼ vF.53

FIG. 3. The real part(dashed-blue line) and imaginary part(solid-red line) of

the intrinsic (a) and extrinsic (b) dynamical dielectric function of the

SSDLGNR system for the finite transferred momentum qaB� 0.001 and

qaB� 0.01, respectively. We choose the inter-ribbon charge separation

d/w¼ 0.5, the number of dimer lines N¼ 11, and the ribbon width

W1¼W2� 14.75 A.

FIG. 4. The low-energy intrinsic acoustical plasmon IPad=w and the high-

energy intrinsic optical plasmon IPod=w, exhibited by arrows, in the

SSDLGNR system for five various inter-ribbon charge separations d/w¼ 0

(dark-yellow-solid line), d/w¼ 0.01 (green-diamond line), d/w¼ 0.1 (blue-

left-tri line), d/w¼ 0.5 (cyan-right-tri line), and d/w¼ 1.5 (magenta-circle

line). The red-dashed line, depicting the ISPE, shows intrinsic Landau

damped line. For d/w¼ 0, where two nano-ribbons are unified, we obtain a

degenerate plasmon branch. For a finite value of the inter-ribbon charge sep-

aration, this degenerate plasmon mode is split into two plasmon branches,

one intrinsic acoustical IPad=w¼0:01;0:1;0:5;1:5 and another intrinsic optical

IPod=w¼0:01;0:1;0:5;1:5. For small enough d/w, the low-energy plasmon line

remains linear. However, for large enough inter-ribbon charge separations,

this plasmon modes behaves in a non-linear way and approaches the high-

energy optical mode. We choose the ribbon width W1¼W2� 14.75 A, corre-

sponding to N¼ 11.

174301-4 M. Bagheri and M. Bahrami J. Appl. Phys. 115, 174301 (2014)

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In Fig. 5, for the equal density ndoped1 ¼ ndoped

2 ’ 106=cm

and for five various inter-ribbon charge separations d/w¼ 0

(dark-yellow-solid line), d/w¼ 0.01 (green-diamond line),

d/w¼ 0.1 (blue-left-tri line), d/w¼ 0.5 (cyan-right-tri line),

and d/w¼ 1.5 (magenta-circle line), the low-energy extrinsic

acoustical plasmon EPad=w and the high-energy extrinsic opti-

cal plasmon EPod=w, exhibited by arrows, are plotted. The col-

ored area show the extrinsic Landau-damped (orange) and

Landau-undamped (cream) regions of the plasmon modes in

the x–q plane, which their corresponding edges are obtained

from the poles of the extrinsic dielectric function. The overall

behavior of the extrinsic plasmon modes is similar to that of

the intrinsic ones, except for a cusp occurred at q¼ kF. We at-

tribute this cusp to the bipartite feature of the extrinsic polar-

ization function (expressed by Eq. (A10) in the Appendix).

Comparing the extrinsic and intrinsic plasmon branches

reveals that they are truly matched for q< kF; while for

q> kF, they depart from the Landau-damped line ISPE with

different slopes. This theme stems from the fact that the

intrinsics and extrinsic polarizabilities are actually the same

over q< kF, whereas they are different when q> kF.

Therefore, in the extrinsic case, the behavior of the low-

energy and high-energy plasmon branches are governed by

doping factor as well as the inter-ribbon charge separation at

the same time. It is worth mentioning that for a very small

inter-ribbon charge separation, i.e., d=w� 1, this cusp is

nearly melted away and the low-energy branch is completely

linear over all energies.

To show the influence of doping and consequently the

position of the Fermi cusp on the extrinsic plasmon modes,

we show the plasmon branches for unequal densities

ndoped1 ’ 105=cm (red-right-tri line) and ndoped

2 ’ 106=cm

(blue-left-tri line) at the fixed layer separation d/w¼ 0.5 and

for the ribbon width W1¼W2� 14.75 A in Fig. 6. As a

result, when the density is decreased, the extrinsic

low-energy and high-energy plasmon modes shift to the

lower energies and their gap becomes narrower. As shown,

the extrinsic plasmon modes are crucially sensitive to the

inter-ribbon charge separation as well as doping. Here, one

would also be able to numerically find a critical value of the

inter-ribbon charge-separation for which the extrinsic acous-

tical plasmon group velocity equals to the Fermi velocity for

a fixed value of either doping or the inter-ribbon charge

separation.

IV. CONCLUSION

In conclusion, we investigate the collective excitations

of a spatially separated double-layer graphene nanoribbon

system, made of two vertically offset metallic and armchair

graphene nanoribbons. For the intrinsic and extrinsic cases,

we obtain both the low-energy acoustical and the high-

energy optical plasmon modes for different inter-ribbon

charge separations. We find that plasmon branches in the

intrinsic case are engineered only via the ribbon separation;

moreover, the additional handle of doping is supplied in the

extrinsic one. This study anticipates an either intrinsic or ex-

trinsic Landau-undamped acoustical plasmon mode, which

might be engineered based on either the inter-ribbon charge

separation or doping. Our model could be useful to create

quantum controlled devices based on the interacting gra-

phene nanoribbons. Also, the existence of a linear Landau-

undamped acoustical plasmon mode may provide with us an

understanding of the plasmon-based superconductivity based

on the interacting massless fermions.

FIG. 5. The low-energy extrinsic acoustical plasmon EPad=w and the high-

energy extrinsic optical plasmon EPod=w, exhibited by arrows, in the

SSDLGNR system for five various inter-ribbon charge separations d/w¼ 0

(dark-yellow-solid line), d/w¼ 0.01 (green-diamond line), d/w¼ 0.1 (blue-

left-tri line), d/w¼ 0.5 (cyan-right-tri line), and d/w¼ 1.5 (magenta-circle

line). The shaded area show the extrinsic Landau-damped (orange) and

Landau-undamped (cream) regions. For d/w¼ 0, where two nano-ribbons

are the same, two plasmon branches EPad=w¼0 and EPo

d=w¼0 are degenerate.

For a finite value of the inter-ribbon charge separation, this degenerate plas-

mon mode is split into two plasmon branches IPad=w¼0:01;0:1;0:5;1:5 and

IPod=w¼0:01;0:1;0:5;1:5. The cusp-like behavior of the plasmon branches is

directly related to the bipartite feature of the extrinsic polarizability. For

small enough d/w, the low-energy plasmon line remains linear. However, for

large enough inter-ribbon charge separations, this plasmon modes behaves

in a non-linear way and approaches the high-energy optical mode. We

choose the ribbon width W1¼W2� 14.75 A, corresponding to N¼ 11.

FIG. 6. Extrinsic plasmon modes of the SSDLGNR system for the fixed

inter-ribbon charge separation d/w¼ 0.5 and for two different densities

ndoped1 ’ 106=cm(red-right-tri line) and ndoped

2 ’ 105=cm(blue-left-tri line).

The lower edge of the extrinsic single-particle excitations is called the

Landau-damped line (olive-dashed line). We choose the ribbon width

W1¼W2� 14.75 A, corresponding to N¼ 11.

174301-5 M. Bagheri and M. Bahrami J. Appl. Phys. 115, 174301 (2014)

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ACKNOWLEDGMENTS

The authors acknowledge the financial support received

from Shahid Beheshti University through Project No.

600/1478.

APPENDIX: THE POLARIZATION FUNCTION

This appendix contains the explicit expressions associ-

ated with the real and imaginary parts of the polarization

function for the intrinsic and extrinsic cases.54,55 The form

factor expressed by Eq. (3) is nonzero only for the not back-

scattered right-forward and left-forward transitions between

two states jki and jk0i, with the angle between their corre-

sponding spinors given by Dhjn;n0 ðk; k0Þ ¼ hjðkðnÞ; kÞ

�hjðkðn0Þ; k0Þ ¼ tan�1ðk=kðnÞÞ � tan�1ðk0=kðn0ÞÞ. As men-

tioned, for the two-subband model with linear band structure

around the Fermi energy, we have kðn ¼ ðl;þÞÞ ¼ kðn0¼ ðh;�ÞÞ ¼ 0. It yields Dhðk; k0Þ ¼ pðsgnðkÞ � sgnðk0ÞÞ=2,

where sgnðxÞ ¼ x=jxj is the usual sign function.

1. Intrinsic: Inter-subband transitions

In the intrinsic (undoped/ungated) case, at the zero-

temperature T¼ 0, where eF ¼ 0 and kF¼ 0, only inter-

subband transitions from the highest valence subband (h,–)

to the lowest conduction subband (l,þ) are contributed to the

nano-ribbon polarizability and vice versa. The form factor

expressed by Eq. (3) reads jFj2 ¼ 1� cosðDhðk; k0ÞÞ, where

Dhðk; k0Þ ¼ p or equivalently sgnðkÞ � sgnðk0Þ ¼ 2.

Therefore, the whole intrinsic (I) polarization function is

Pð0Þ��I ¼ Pð0Þ��lhþ�þPð0Þ��hl

�þin which l ¼ h � 0 with the layer

index � ¼ 1; 2, dropped hereafter.29,30 Starting with the

k¼þ and k0 ¼ � term given by

Pð0Þlhþ�ðq;xÞ ¼ � 1

4p

ðdk

1� cosp2ðsgnðk � qÞ � sgnðkÞÞ

� �ðjk � qj þ jkjÞ�hvF þ �hðxþ igÞ ;

(A1)

where the solutions of the expression sgn(k – q) – sgn(k)¼ 2

define the interval of the integration to be 0 k q. The cor-

responding terms of the real and imaginary parts are then

evaluated to yield

<ePð0Þlhþ�ðq;xÞ ¼ � q

2p1

�hvFqþ �hx; (A2)

and

=mPð0Þlhþ�ðq;xÞ ¼ q

2dð�hvFqþ �hxÞ: (A3)

Similarly, moving to the k¼� and k0 ¼ þ term given by

Pð0Þhl�þðq;xÞ ¼ 1

4p

ðdk

1� cosp2ðsgnðkÞ � sgnðk þ qÞÞ

� �ð�jkj � jk þ qjÞ�hvF þ �hðxþ igÞ ;

(A4)

where the solutions of the expression sgn(k) –

sgn(kþ q)¼�2 define the interval of the integration to be

�q k 0. The associated terms of the real and imaginary

parts are thus taken the following forms, respectively,

RePð0Þhl�þðq;xÞ¼� q

2p1

�hvFq� �hx¼ <ePð0Þlh

þ�ðq;�xÞ; (A5)

and

JmPð0Þhl�þðq;xÞ ¼ � q

2dð�hvFq� �hxÞ ¼ �=mPð0Þlh

þ�ðq;�xÞ:

(A6)

The real and imaginary parts of the entire intrinsic polariza-

tion function have the following forms, respectively:

<ePð0ÞI ðq;xÞ ¼ �1

p�hvFq2

ð�hvFqÞ2 � ð�hxÞ2; (A7)

and

=mPð0ÞI ðq;xÞ ¼q

2½dð�hvFqþ �hxÞ � dð�hvFq� �hxÞ�: (A8)

It is worth noting that the real part of the static polarizability

becomes <ePð0ÞI ðq; 0Þ ¼ �1=p�hvF in which one would

define the density of states (per spin, per valley, and per unit

length) of the metallic armchair graphene nano-ribbon at the

Fermi surface via DOSðeF ¼ 0Þ=ðgrgvÞ ¼ 1=p�hvF, with the

spin degeneracy gr¼ 2 and the valley degeneracy gv¼ 2.60

In spite of having presented a closed-form expression for the

polarization function of the nano-ribbon, there exist obvious

differences with the main features of the Lindhard function

in the one-dimensional jellium model.55 We should empha-

size that our results for the intrinsic case are completely dif-

ferent with those obtained by authors in Ref. 46. The most

important differences between their results and ours are

these: (1) they have found a logarithmic (step) function for

the real (imaginary) part of the polarizability for undoped

metallic graphene nanoribbons, whereas we obtain a rational

(delta) function for the real (imaginary) quota of the polar-

ization function;29,30 (2) the real part of the static polarizabil-

ity that they find is normalized to zero, which is not true for

metallic nano-ribbons with a finite density of states at the

Fermi point, while our outcome meets the normalization

condition of the polarizability through �<ePð0ÞI ðq; 0Þ=DOS¼ 1; (3) they have not expressed their results in terms of

Pð0Þlhþ�

and Pð0Þhl�þ

,30 yet we clarify how the whole polarizability

stems from these terms and how the real and imaginary parts

are symmetrized in the x–space. It could be useful to men-

tion that no logarithmic behavior for the real part of the

polarization function for the undoped metallic armchair gra-

phene nanoribbons is addressed in Refs. 29 and 30 as well.

2. Extrinsic: Intra-subband transitions

We will now evaluate the graphene nanoribbon polariz-

ability for the extrinsic (doped/gated) case. For the sake of

simplicity, we assume two isolated n–doped GNR with

174301-6 M. Bagheri and M. Bahrami J. Appl. Phys. 115, 174301 (2014)

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eF > 0, where their Fermi levels are located in the lowest

lying linear conduction subband (l,þ). The associated form

factor expressed by Eq. (3) takes the form jFj2 ¼ 1

þcosðDhðk; k0ÞÞ, where Dhðk; k0Þ ¼ 0 or equivalently sgnðkÞ�sgnðk0Þ ¼ 0. The whole polarization function is thus

Pð0Þ��E ¼ Pð0Þ��llþþ

, with l � 0 and the layer index � ¼ 1; 2,

dropped hereafter. We proceed by exploiting the k¼þ and

k0 ¼ þ term given by

Pð0Þll

þþ

ðq;xÞ

¼ 1

4p

ðdk

nF½Eþl ðkÞ� 1þ cosp2ðsgnðkÞ� sgnðk þ qÞÞ

� � ðjkj � jk þ qjÞ�hvF þ �hðxþ igÞ

� 1

4p

ðdk

nF½Eþl ðkÞ� 1þ cosp2ðsgnðk�qÞ� sgnðkÞÞ

� � ðjk � qj � jkjÞ�hvF þ �hðxþ igÞ :

(A9)

In the first term of Eq. (A9), the solutions of the expression

sgn(k) – sgn(kþ q)¼ 0 accompanied by nFð¼ 1Þ 2 ½�kF; kF�define the integral in the k–space to be

Ðdk ¼

�q�kF

dk

þÐ kF

0dk for q < kF and

Ðdk ¼

Ð kF

0dk for q > kF. Likewise,

in the second term of Eq. (A9), one would obtainÐdk ¼

Ð 0

�kFdk þ

Ð kF

q dk for q < kF andÐ

dk ¼Ð 0

�kFdk for

q > kF. Therefore, the real and imaginary parts of the entire

extrinsic polarization function of the nanoribbon are found to

be, respectively,

<ePð0ÞE ðq;xÞ ¼� 1

p�hvFq2

ð�hvFqÞ2 � �hxÞ2q kF;

� 1

p�hvFqkF

ð�hvFqÞ2 � �hxÞ2q > kF;

8>>>><>>>>:

(A10)

and

=mPð0ÞE ðq;xÞ

¼

q

2dð�hvFqþ �hxÞ � dð�hvFq� �hxÞ½ � q kF;

kF

2dð�hvFqþ �hxÞ � dð�hvFq� �hxÞ½ � q > kF:

8>><>>: (A11)

Calculating the real part of the static extrinsic polarization

function, we have

� <ePð0ÞE ðq; 0ÞDOSðeF 6¼ 0Þ ¼

1 q kF;

kF

qq > kF;

8><>: (A12)

which becomes constant over q kF and falls off for

q > kF, with a cusp at q¼ kF. This bipartite feature of the ex-

trinsic polarizability could also be a different behavior of

this massless and chiral quasi-one-dimensional fermionic

system from a one-dimensional jellium system.55 Here,

because only the first subband of the positive energy is taken

into account, we find DOSðeF 6¼ 0Þ ¼ DOSðeF ¼ 0Þ before

the first Van Hove singularity in the DOS spectrum of the

armchair metallic nanoribbon. Moreover, there are some-

thing different about our analytical findings with those

obtained by authors in Ref. 46 for the polarizability of the

extrinsic metallic and armchair graphene nanoribbons.

Clearly, the polarization function derived here is a bipartite

function, while they find a single-part one, which requires

the static extrinsic polarization function to be singular at

q¼ 0 and it is decayed with each increment of q. This is in

contrast with having a constant finite value for the density of

state of the metallic armchair nano-ribbon before the first

Van Hove singularity in the DOS spectrum. Finally, our cal-

culations show that once kF ! 0, the extrinsic polarization

converts into the intrinsic one.

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