Superradiance mediated by Graphene Surface Plasmons

8.9 MB·English

by  Paloma A. Huidobro

Superradiance mediated by Graphene Surface Plasmons
P. A. Huidobro,1,∗ A. Y. Nikitin,2,3 C. Gonz´alez-Ballestero,1 L. Mart´ın-Moreno,2 and F.J. Garc´ıa-Vidal1
1Departamento de F´ısica Teo´rica de la Materia Condensada,
Universidad Auto´noma de Madrid, E-28049 Madrid, Spain
2Instituto de Ciencia de Materiales de Arago´n and Departamento de F´ısica de la Materia Condensada,
CSIC–Universidad de Zaragoza, E-50009 Zaragoza, Spain
3A. Ya. Usikov Institute for Radiophysics and Electronics,
NAS of Ukraine, 12 Academician Proskura Street, 61085 Kharkov, Ukraine
We demonstrate that the interaction between two emitters can be controlled by means of the
efficient excitation of surface plasmon modes in graphene. We consider graphene surface plasmons
2
supportedbyeithertwo-dimensionalgraphenesheetsorone-dimensionalgrapheneribbons,showing
1
in both cases that the coupling between the emitters can be strongly enhanced or suppressed. The
0
super- and subradiant regimes are investigated in the reflection and transmission configurations.
2
Importantly,thelengthscaleofthecouplingbetweenemitters,whichinvacuumisfixedbythefree

n space wavelength, is now determined by the wavelength of the graphene surface plasmons that can
a be extremely short and be tuned at will via a gate voltage.
J

1
3 I. INTRODUCTION radiation properties of several emitters are enhanced or
suppressed. Weintroducethetotalnormalizeddecayfac-
]
l Inthepastfewyears,plasmonicshasemergedasaway tor, γ, which characterizes the superradiant regime, and
l
a to control light at the subwavelength scale [1–3]. Its po- discussitstunability. Weanalyzetheevolutionofγ with
h the vertical distance between the emitters and the sheet
tentiality is based on the properties of surface plasmons
-
s (SPs), which are surface electromagnetic (EM) waves in the transmission configuration, and with the in-plane
e distancebetweentheemittersinthereflectionconfigura-
coupled to charge carriers. These surface waves propa-
m
tion. InSec. IVweconsiderthecouplingoftwoemitters
gate along a metal-dielectric interface and are character-
. by GSP in one-dimensional (1D) graphene ribbons [21].
t ized by their subwavelength light confinement and long
a We show that confining in 1D leads to longer interac-
propagation lengths. Recently, it has been shown that
m
tion ranges between the emitters. Moreover, we study
graphene [4], which has remarkable optical [5] and opto-
- how the coupling between the emitters is affected by the
d electronic [6] properties, also supports the propagation
n of SPs. These EM modes bounded to a graphene sheet presence of the ribbon’s edges. Finally, our main results
o have been studied theoretically [7–12] and very recently are summarized in Sec. V.
c
observed in experiments [13, 14].
[
ThepropertiesofSPsingraphenehaveattractedgreat
1
attention as they appear as an alternative for many of II. COUPLING BETWEEN AN EMITTER AND
v
the functionalities provided by noble-metal SPs with the GRAPHENE SURFACE PLASMONS
2
9 advantage of being tunable by means of a gate poten-
4 tial. For instance, by designing spatially inhomogeneous First, let us characterize the coupling to GSPs for an
6 conductivity patters in a graphene sheet one can have emitter at frequency ω decaying in the vicinity of a free-
.
1 a platform for transformation optics and THz metama- standing 2D graphene sheet. A sketch of the system
0 terials [15]. Additionally, strong light-matter interaction under study can be seen in Fig. 1 (a). The graphene
2 betweenSPsandquantumemittersingraphenehasbeen sheet is placed in the x−y plane and has a conductiv-
1
proposed based on the high decay rates of emitters close ity σ(ω), obtained in the random-phase approximation
:
v to graphene sheets [16]. The field patterns excited by [22, 23]. This quantity depends on the chemical poten-
i
X a nanoemitter in graphene were analyzed in Ref. 17, tial of the graphene sheet, µ, the temperature, which we
demonstrating high field enhancements and long prop- consider to be T = 300 K, and the carriers’ scattering
r
a agation lengths for the SPs. Furthermore, fluorescence time τ, for which we use a value taken from the theo-
quenching in graphene has been proposed as a probe of retical predictions [24] such that E = h/τ = 0.1 meV.
τ
the evidence of plasmons [18, 19]. Theemitter,modeledinthepointdipoleapproximation,
The paper is organized as follows: in Sec. II we in- is placed at a distance z from the sheet and has a dipole
troduce the theoretical formalism and characterize the moment p(cid:126).
couplingbetweenanemitterandtheGSPssupportedby Whentheemitterisclosetothegraphenesheet,itcan
a 2D graphene sheet. In Sec. III we study the inter- decay through three different mechanisms: radiation to
action between two emitters mediated by the 2D GSPs. free-space, excitation of GSPs or coupling to absorption
The presence of GSPs leads to interesting phenomena losses in the graphene sheet. The emitter’s total decay
such as the super- and subradiant regimes, where the rate is proportional to the imaginary part of the Green’s
2
tensorofthesystem,Gˆ((cid:126)r,(cid:126)r,ω),thedipolemomentp(cid:126)and Purcell factor can be obtained for very thin metal films
the free-space momentum k =ω/c: [25] when the thickness is much smaller than the skin
0
depth, which is challenging from the fabrication point of
2k2|p(cid:126)|2 (cid:110) (cid:104) (cid:105) (cid:111)
Γ= 0 (cid:126)u Im Gˆ((cid:126)r,(cid:126)r,ω) (cid:126)u (1) view, as opposed to graphene. For higher frequencies or
¯h(cid:15) p p
0 smaller µ, larger Purcell factors in the plasmonic region
where (cid:126)up is the unitary vector in the direction of the can be obtained in graphene: for instance, Γ/Γ0 ≈ 103
dipole moment, For the sake of simplicity, we take a at λ0 =64µm for the same chemical potential.
dipolemomentperpendiculartothegraphenesheet(p(cid:126)=
p(cid:126)u ), and then the relevant component of the tensor is (a) 1.0
z 103
Gzz [17]. This leads to the following expression for the µ=0.2eV
Purcell factor, which is the total decay rate normalized 0.8 102 ΓGSP/Γ
0
to the free-space decay rate (Γ0 =k03|p(cid:126)|2/(3π¯h(cid:15)0)): 10 Γ/Γ
0
Γ = 3Re(cid:20)(cid:90) ∞dqq3 (cid:0)1−r (q)e2ik0qzz(cid:1)(cid:21) (2) β0.6 1 Γrad/Γ0
Γ 2 q p
0 0 z 0.4
10-1
whereweintegrateoverthenormalizedparallelwavevec- 10-4 10-3 10-2 10-1 1
tor q =k(cid:107)/k0, qz =(cid:112)1−q2 is the momentum in the di- 0.2 z/λ0 p1
rection perpendicular to the sheet, with Im(q )≥0, and z σ
z
r (q)=−αq /(αq +1)isthereflectioncoefficientofthe
p z z
graphene layer for p-polarization, with α = 2πσ/c be- (b)2.0
µ=0.2eV
ing the normalized conductivity. The pole of r (q) gives
p µ=0.1eV
the dispersion relation of the GSPs propagating in the
1.5 µ=0.05eV
graphene sheet, which appear when Im(σ) > 0, i.e., be- γ
vac
low a critical frequency h¯ω ≈ 2µ. The contribution of
0
GSPs to the total decay rate can be calculated from the
1.0
γ
pole in rp(q): p
1
ΓGSP = 3πRe(cid:34)iqp2e2ik0qzpz(cid:35) (3) 0.5 z yx zz σ
Γ 2 α
0
p
√ 0.0 2
whereq = 1−α−2 andqp =−α−1 arethenormalized
p z 0.0 0.1 0.2 0.3 0.4 0.5
momentum components of the GSP. z/λ
0
The inset panel in Fig. 1 (a) shows the Purcell fac-
tor (solid red line) at 2.4 THz and for µ = 0.2 eV as a FIG. 1: (a) β factor for an emitter at 2.4THz as a func-
function of the emitter-graphene distance z normalized tion of the distance to the graphene sheet, z, normalized
to the free-space wavelength, λ0 = 124µm. The physi- to the free-space wavelength, λ0, for different values of
cal parameters µ and λ were chosen, as shown in Ref. the chemical potential (µ = 0.2, 0.1 and 0.05 eV). Inset
0
panel: total decay rate (Γ/Γ ) and decay rates through
17, to provide a good compromise in the trade-off be- 0
the plasmonic (ΓGSP/Γ ) and radiative(Γrad/Γ ) channels.
tweenconfinementandpropagationlengthfortheGSPs. 0 0
(b) Super/subradiance between two emitters mediated by a
Threedifferentregionscanbeidentifiedintheinsetpanel
graphenesheetwhenthetwodipolesinteractintransmission
in Fig. 1 (a) according to the decay mechanisms: (i) a through it. γ is plotted as a function of the vertical distance
radiativeregionatlargedistances(z ≥λ0/10forthecho- of the dipoles to the graphene sheet for the three values of
senparameters),wheretheemitterisfarenoughfromthe the chemical potential µ. The red line shows the vacuum
graphene sheet and the total decay rate follows Γrad/Γ0 interaction γvac.
(dotted blue line), which corresponds to the integration
of the radiative modes in Eq. 2 (0<q <1); (ii) a region The parameter that accounts for the efficiency of the
(λ /10 ≥ z ≥ λ /100) where the dominant decay chan- coupling to GSP, the β factor, is defined as the ratio of
0 0
nel is the coupling to GSPs and the total decay equals the emitter’s decay rate through GSP to its total decay
ΓGSP/Γ (green dashed line); and (iii) a lossy region rate, β = ΓGSP/Γ. Fig. 1 (a) studies the possibility of
0
when the emitter is very close to the sheet (z ≤λ /100). tuning β with the chemical potential. Three values of µ
0
Importantly, and as the figure shows, the decay rate of are considered: µ=0.2, 0.1 and 0.05 eV. For each value
the emitter can be enhanced by several orders of mag- of µ there is a range of z’s where β is close to 1, which
nitude. Here we are interested in the plasmonic region, corresponds to the region where the decay rate is dom-
wheretheGSP-contributiontothePurcellfactorreaches inated by the plasmonic channel (see inset panel). The
values larger than 100 for the parameters we have cho- regionofhighβcanbedynamicallytunedwiththechem-
sen. It is interesting to note that similar values of the ical potential, which is in turn controlled by means of an
3
electrostatic gating or a chemical doping. In particular, (a)2.0
2 4 6 8
whenthechemicalpotentialisdecreasedto0.1eV(green r/λ
SP
line)and0.05eV(blueline),theGSPappearsatlargerq- ν = 2.4 THz
1.5
vectors, theGSPismoreconfinedtothegraphenesheet,
and the range of distances where β is high is narrower.
The capability of tuning plasmonic properties by means
γ 1.0
of a gate potential is the most important advantage of
graphene compared to thin metal layers.
0.5 p p z=2µm
1 2 1
z r z 1+βJ(k r)e−r/Lp z=15µm
σ 0 SP 2
III. SUPERRADIANCE IN TWO-DIMENSIONAL γ z=50µm
0.0 vac 3
GRAPHENE SHEETS
0.0 0.5 1 1.5 2.0 2.5
r/λ
(b) 0
Theefficientandtunablecouplingofanemittertothe 2.0
5 10 15 20 25
SP modes propagating in a graphene sheet can be used r/λ
SP
tomodifytheinteractionbetweentwoemittersplacedin ν = 7.4 THz
1.5
thevicinityofthe2Dsheet,similarlytoSPsinmetalsur-
faces [20]. In order to study the GSP-mediated coupling
between the two emitters, we introduce the normalized γ1.0
decay factor, γ, defined as the ratio of the total decay
rateofasystemwherethetwoemittersinteractthrough γ (z=0.2µm)
graphene to the decay rate of two uncoupled emitters in 0.5 p1 p2 1+β π2r e−r/Lp
thepresenceofthegraphenesheet. Thus, itaccountsfor z r z σ γ
vac
the modification of the collective decay rate due to the 0.0
presence of the second emitter and reads: 0.0 0.5 1 1.5 2.0 2.5
r/λ
0
Γ +Γ +Γ +Γ
γ = 11 12 21 22 (4)
Γ +Γ
11 22 FIG.2: Tuningsuper-andsubradiance: γfactorasafunction
ofthein-planedistancerbetweentwodipolesinthereflection
The decay rate Γ is the contribution to the decay
ij configuration for two different frequencies. (a) At ν = 2.4
rate of a dipole p(cid:126) placed at (cid:126)r due to the presence
i i THz, when the dipoles are placed at z = 2µm (β = 0.98,
1
of a dipole p(cid:126)j placed at (cid:126)rj and can be written in Γ/Γ0 = 105), z2 = 15µm (β = 0.55, Γ/Γ0 = 2) and z3 =
terms of the Green’s function that connects (cid:126)r to (cid:126)r : 50µm (β = 0.01, Γ/Γ = 0.92). The free-space wavelength
i j 0
(cid:110) (cid:104) (cid:105) (cid:111)
Γ = 2k2|p(cid:126)|2/(¯h(cid:15) ) p(cid:126) Im Gˆ((cid:126)r ,(cid:126)r ,ω) p(cid:126) . Note that is λ0 = 124µm, the plasmon wavelength λp = λ0/3.5 and
ij 0 0 i i j j the propagation length L = 14.9λ . The dashed black line
p p
for i=j we obtain the decay rate in Eq. 1, i.e., Γ=Γii. correspondstothefree-spaceinteractionandthedashedgray
Thevalueofγ characterizestworegimes: whenγ >1the linetoEq. 7forhighβ. (b)Atν =7.4THz,whenthedipoles
interaction is enhanced due to the presence of graphene are at z = 0.2µm, with β = 0.98 and Γ11/Γ0 = 2250. The
and the system is superradiant. Correspondingly, when free-spacewavelengthisλ0 =41µm,theplasmonwavelength
is λ =λ /10 and the propagation length is L =20λ . The
γ < 1, there is an inhibition of the dipole-dipole in- p 0 p p
dashedblacklineshowsthefree-spaceγ andthegraylinethe
teraction and the system is subradiant. Interestingly, a
decay of the interaction.
graphene sheet allows for two different configurations of
the emitters: interaction in reflection (both emitters at
the same side of the sheet) or in transmission (emitters
where the − sign comes from the fact that the dipoles
placed at opposite sides).
areanti-parallelandthetransmissioncoefficientist(q)=
Let us first consider two emitters interacting through
1/(αq +1). The γ factor as a function of z/λ is plot-
the graphene sheet placed at opposite sides of the sheet, z 0
ted in Fig. 1 (b) for the same parameters used in panel
i.e., in the transmission configuration [see the sketch in
(a). The red line shows γ when the emitters are placed
Fig. 1 (b)]. In order to study the behavior with the
in free-space and interact only through radiative modes:
distance to the sheet, we locate the emitters at the same
z (|z | = |z |) and at (cid:126)r = 0. We assume the dipole γvac = 0 at z << λ0 because the opposite phase of
1 2 (cid:107) the dipole moments inhibits the radiation and when z
momentstobeofthesamemodulebutanti-parallel,p(cid:126) =
1
−p(cid:126) = p(cid:126)u . The decay rate related to the interaction increases γvac oscillates around 1 with λ0. When the
2 z
graphene sheet is present, the interaction between the
between the emitters, Γ , is needed to determine γ and
12
emitters is strongly modified at the subwavelength scale.
is related to the transmission coefficient:
In the limit of large z, in correspondence with the dis-
ΓT 3 (cid:20)(cid:90) ∞ q3 (cid:21) tances where β ≈0 in panel (a), the emitters couple via
12 =− Re dq t(q)e2ik0qzz (5)
Γ 2 q radiative modes and γ approaches γvac. On the other
0 0 z
4
hand, in the range of z where the plasmonic coupling isconfinedtothe2Dgraphenesheet. Whenthedistance
between the emitters starts to dominate, β (cid:54)= 0, γ devi- to the graphene sheet is increased, β decreases and γ de-
atesfromγvac,and,asthedistancebetweentheemitters viates from the analytical expression. For z2 = 15µm
and the sheet decreases, the system turns from subradi- the β factor is 0.55 and the shape of γ reflects the fact
ant to superradiant. For each value of µ (0.2, 0.1 and thattheemitterdecaysbothtoGSPandradiatively. Fi-
0.05 eV), the value of z/λ where β starts to grow from nally, when the distance to the sheet is large enough to
0
0to1,istheonsetoftheseparationbetweenγ andγvac. haveβ ≈0, suchasz =50µm, thevacuuminteractionis
Thus, the superradiant regime, controlled by high cou- recovered. Therefore,ourresultsshowthatalargerinter-
pling to GSP, can be tuned by means of µ. In the limit action length scale and a modification of the super- and
z << λ , where β = 0 again and losses dominate, the subradiant regimes can be achieved in a subwavelength
0
interaction reaches γ = 2, in contrast to the free-space scale for the appropriate choice of parameters.
value in this limit, γvac =0. The reason for this is that, Thetunabilityofgrapheneenablesustoreacharegime
in this limit, the integrals in Eq. 2 and Eq. 5 are dom- where the interaction between the two emitters can be
inated by the contributions coming from large q, where controlled at very deep subwavelength scales. Although
q =i|q | and Γ is controlled by −Im[r(q)] and ΓT by the tuning can also be done via µ, here we show a situa-
z z 11 12
−Im[t(q)]. Since the imaginary part of both coefficients tion where the tuning parameter is the frequency. When
is the same, Γ = ΓT and thus γ = 2. It is interesting the two emitters interact in reflection [see Fig. 2 (b)] at
11 12
to note that this sign change comes from the continuity 7.4THzandµ=0.2eV,γ(redline)isverydifferentfrom
conditions of the electromagnetic fields. the one corresponding to the free-space situation (black
Let us now study how the interaction evolves with the dashed line). Increasing the frequency while maintain-
in-plane distance between the emitters, r = |(cid:126)r −(cid:126)r |, ingthechemicalpotential,resultsinalargermomentum
1(cid:107) 2(cid:107)
where (cid:126)r = (x,y). We take two dipoles interacting for the GSP, q , which leads to a tighter confinement as
(cid:107) p
through the graphene sheet in the reflection configura- well as a reduction in the propagation length. Thus, the
tion,assketchedinFig. 2(a). Inthiscaseweplaceboth interaction varies in a λ = λ /10 scale, as opposed to
p 0
of them at the same distance z to the sheet, separated the free-space interaction, dominated by λ [Fig. 2 (b)
0
by an in-place distance r, and with dipole moments of shows both scales r/λ and r/λ ]. The decay of the γ
0 p
thesamemagnitude,parallelandpointinginthevertical factor(grayline),isgivenbythesquare-rootdecaychar-
direction, p(cid:126) = p(cid:126) = p(cid:126)u . In order to determine the γ acteristic of 2D interactions for distances shorter than
1 2 z
factor we need the interaction decay rate in reflection: the propagation length, that in this case is L = 1.85λ
p 0
(19.2λ ). For larger distances, losses start to dominate
ΓR12 = 3Re(cid:20)(cid:90) ∞dqq3J (k qr)(cid:0)1−r(q)e2ik0qzz(cid:1)(cid:21) (6) and thpe interaction decays exponentially, according to
Γ0 2 0 qz 0 0 Eq. 7.
where the in-plane dependence is given by the zeroth-
order Bessel function, J . The γ factor as a function of
0 IV. SUPERRADIANCE IN ONE-DIMENSIONAL
r/λ is plotted in Fig. 2 (a) at 2.4THz and µ = 0.2
0 GRAPHENE RIBBONS
eV when the dipoles are at three different separations
to the sheet, z , z and z . First, for z = 2µm, we
1 2 3 1
For completeness, we have also considered the possi-
know from Fig. 1 (a) that the coupling to GSPs is very
bility of confining GSPs in 1D graphene ribbons, which
efficient,β =0.98,andΓ/Γ =105. Inthissituation,the
0
could provide a platform for long distance entanglement
interactionbetweenthetwoemittersismediatedbyGSP,
betweentwoemitters,asproposedinRef. 27. Compared
and, consequently, the length scale of the interaction is
to the GSPs propagating in a 2D graphene sheet at the
controlled by λ = λ /3.5 as opposed to the free-space
p 0
samefrequency, theribbon-GSPshaveahigherq vector,
interaction,dominatedbyλ (dashedline). Whenβ ≈1,
0
thus it is more tightly confined to the graphene layer.
ananalyticalexpressionforγ canbeobtainedinthepole
On the other hand, while the coupling of two emitters
approximation:
mediated by 2D-GSPs is dominated by the dimensional-
√
γ =1+βJ (k q r)e−r/Lp (7) ity factor, e−r/Lp/ r (see Fig. 2), it is expected that in
0 0 p
the case of 1D-GSPs this coupling will decay as e−x/Lp,
where L is the propagation length of the GSP, given by enablinglongrangeinteractionbetweentheemitterspro-
p
L = λ /[2πIm(q )], and equal to 14.9λ in this case. vided L is long enough. Let us consider a free-standing
p 0 p p p
Asitcanbeseenintheplot, theanalytical(graydashed graphene ribbon of width ∆ at |y| < ∆/2, placed at
line)andexact(redline)calculationsofγ coincide. Since z = 0 with its axis along the x direction (see Fig. 3).
the propagation length is large enough, the decay length We take ν = 2.4 THz and the fundamental mode of a
of the interaction is then given by the Bessel function, ribbon of width ∆ = 5µm, that originates from the hy-
(cid:112)
that decays as 2/(πr). This is a dimensionality factor, bridization of two edge modes and has even parity of
comingfromthefactthatthe3Dinteractioninfree-space E with respect to the ribbon axis [21]. The field pro-
z
5
tialdecaygivenbyapropagationlengthL =20λ . Our
Re(E) (a) p p
z
resultsdemonstratethatgrapheneribbonscouldbeused
tocontrolthelengthscaleoftheinteractionbetweentwo
∆ graphene ribbon emitters thanks to the efficient excitation of GSPs.
λ
e 0
ol
p
di (b)
y
x
graphene ribbon
Whenanalyzinggrapheneribbonsitisworthstudying
the transition across the the ribbon, i.e., how the cou-
(c)
2 pling between two emitters is affected by the presence of
1.8 ∆=5µm=λ0/25 γγribbon edges (see Fig. 4). For this reason, we consider the evo-
z=3.12µm=λ/40 vac lution of the interaction in the reflection configuration,
0
1.6 1+β e-x/Lp the same as in Fig. 3 (c), and also plotting the evolu-
tion of γ(x) as a function of the lateral distance from
1.4
the center of the ribbon (see Fig. 4 (a)). As sketched
1.2 in the inset panel, we displace both emitters perpendic-
1 ularly to the ribbon axis (along y), from the ribbon’s
γ
center y = 0 and passing through the ribbon’s edge at
0.8
y =∆/2,uptoy =2∆. Asthefigureshows,γ(x)evolves
0.6 from reflecting a high coupling to GSP at y = 0, to fol-
p lowing the free-space interaction at y = 2∆, i.e., β goes
0.4 p 2
1 x z from ≈ 1 to ≈ 0 in a length scale of the order of 2∆.
z σ
0.2 ∆ Additionally, we also study the evolution of the interac-
tionbetweentwoemitterswithoppositedipolemoments

0 0.5 1 1.5 2 2.5 3 3.5 4
in the transmission configuration for two situations and
x/λ
0 plot γ at x = 0 as a function of y in panel (b). First,
we displace the emitters simultaneously such that both
FIG. 3: Interaction mediated by graphene ribbons. (a,b)
of them are at the same y [see sketch (1)]. At y =0, the
Electricfieldprofileforadipoledecayingtotheribbon-GSP.
The dipole is placed at x=0, y =0 and z =λ /40 in panel emitterscoupletotheribbon-GSPandthesystemisina
0
(a) and at z = λ /10 in panel (b) [the same would be ob- superradiant state (red line), as opposed to the situation
0
tained for z > λ /10]. (c) Super-radiance mediated by the in free space, that is subradiant (inset panel, red line).
0
ribbon-GSP mode shown in panel (a). The green line shows When the emitters are displaced from the ribbon’s cen-
the exponential decay of the interaction.
ter, but are still on top of the ribbon, i.e. |y| < ∆/2, γ
is only slightly modified. Once the dipoles pass the rib-
bon’s edge, the coupling to the 1D GSP is reduced, and
file of the fundamental mode, obtained by means of the thereforeγdecreases. Subradianceisquicklyreached,ap-
Finite Element Method (COMSOL software), is shown proaching the free-space value, γ = 0, which is achieved
for two situations, where the emitter is placed at dis- when the emitters are placed at a distance 2∆ from the
tances, z =3.12µm and z =12.4µm, as shown in Fig. 3 ribbon’s center. In the second situation, one emitter is
(a,b). Foradistancetotheribbonof3.12µm,i.e. λ /40, kept at the ribbon’s center and the other is displaced
0
β ≈ 1, the GSP mode is excited with a very high effi- perpendicularly to the ribbon’s axis [see sketch (2)]. In
ciency and its field structure is clearly seen in panel (a). this case, the system is always superradiant for the dis-
On the other hand, when the emitter is not sufficiently tances considered: γ starts at 2 and approaches 1, while
close to the ribbon, β << 1 and it couples mostly to in free space γ is of the order of 0.05 at y =∆ (blue line
radiation, as can be seen in panel (b) for an emitter at in the inset panel). The reason for this lies on the fact
z = λ /10 where the field snapshot virtually coincides that the emitter that is fixed always couples to the GSP.
0
with an spherical wave. For the case with β ≈ 1, Fig. 3 Remarkably, with only one emitter efficiently coupled to
(c) shows γ (blue line) for two emitters interacting in re- theribbon-GSP,theinteractionbetweenbothemittersis
flectionthroughtheribbon-GSP.Similartothe2DGSP, very different to the vacuum case. Our results for both
a subwavelength modification of the interaction can be configurations (reflection and transmission) demonstrate
achieved. However, since the ribbon-GSP is much more thatthecouplingbetweenemittersmediatedby1D-GSP
confined(λ =λ /6.1),thismodificationcanbeachieved is very insensitive to the lateral displacement and that
p 0
in a shorter length scale. Moreover, propagation in 1D the effective lateral extension of these 1D-GSP is of the
allows for a longer interaction range, with an exponen- order of ∆/2 measured from the edge’s ribbon.
6
(a) (b) 2
y= 2∆
γ γ(1)
r 1.8 t
γ γ(2)
vac (1) t
y= 1.6∆ 1.6
p
1
x 1.4 p (2)
∆ 1 ∆
y = 1.2∆ p 1.2 y y
2
γ γ 1 p2 p2
y= 0.8∆

p 0.8 y/λ
1 e 0
g 0 0.02 0.04 0.06 0.08
y = 0.4∆ ∆ y 0.6 s ed 0.06 γvacuum
0.4 n’ 0.04
o (1)
b
y= 0 0.2 rib 0.02
∆/2 (2)
0
0.0
0 1 2 3 4 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
x/λ y/λ
0 0
FIG. 4: Dependence of the dipole-dipole coupling in graphene ribbons on the lateral separation from the center of the ribbon,
y = 0. (a) γ factor between two dipoles in the reflection configuration as a function of x for several lateral separations from
the center of the ribbon (blue lines). The scale in each sub-panel is between 0 and 2. The red line shows the corresponding
interaction in vacuum. The inset panel shows a top view of the structure. (b) γ factor as a function of the lateral separation
y in the transmission configuration for two cases. First (red line), γ factor when the two dipoles are displaced simultaneously,
as shown in sketch (1). The dots correspond to numerical simulations and the continuous line is a guide-to-the-eye. Second
(blueline),γ factor(fromsimulations)whentheupperdipoleisfixedatthecenteroftheribbonandthelowerisdisplaced,as
shown in sketch (2). The inset panel shows the corresponding interaction when the dipoles are placed in vacuum.
V. CONCLUSION used as efficient platforms to modify the interaction be-
tweentwoemitterswhentheyareplacedintheirvicinity.
In conclusion, we have studied the tailoring of the
interaction between two emitters mediated by surface
plasmon modes in a graphene sheet. We have shown ACKNOWLEDGEMENT
that within a certain range of distances to the graphene
sheet, the decay rate of one emitter can be fully dom- ThisworkhasbeensponsoredbytheSpanishMinistry
inated by the graphene surface plasmons. Due to this ofScienceandInnovationunderContractNo. MAT2008-
efficient coupling, the enhancement of the decay rate 06609-C02 and Consolider Project Nanolight.es. P.A.H.
of the emitter, or Purcell factor, can be enhanced by acknowledgesfundingfromtheSpanishMinistryofEdu-
several orders of magnitude. The interaction between cationthroughgrantAP2008-00021andA.Y.N.acknowl-
twoemittersmediatedbythegrapheneplasmonsintwo- edges the Juan de la Cierva Grant No. JCI-2008-3123.
dimensional graphene sheets can thus be controlled at
a subwavelength scale and can be tuned by means of
external parameters. We have studied the appearance
of the super- and subradiant regimes, both in the reflec-
∗ Electronic address: [email protected]
tionandtransmissionconfigurations. Additionally,when
[1] W. L. Barnes, A. Dereux, T. W. Ebbesen, Nature 424,
the interaction is confined to one-dimension in graphene
824 (2003).
ribbons, a longer interaction range between the emitters
[2] S. A. Maier, H. A. Atwater, J. Appl. Phys. 98, 011101
andaverydeepsubwavelengthcontroloftheinteraction (2003).
can be achieved. Here, the lateral confinement leads to [3] S.I.Bozhevolnyi,C.Genet,T.W.Ebbesen,Phys.Today
much higher enhancement factors, very deep subwave- (May) 44 (2008).
length length scales for the coupling and longer interac- [4] A. H. Castro-Neto, F. Guinea, N. M. R. Peres, K. S.
Novoselov,A.K.Geim,Rev.Mod.Phys.81,109(2009).
tion ranges. By considering the lateral displacement of
[5] FBonaccorso,Z.Sun,T.Hasan,A.C.Ferrari,Nat.Pho-
the emitters from the ribbon’s axis, we have also shown
tonics 4, 611 (2010).
that the coupling to the graphene surface plasmons sup-
[6] M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L.
ported by the ribbon is very robust. Our results show Ju, F. Wang, X. Zhang, Nature 474, 64 (2011).
that both graphene sheets and graphene ribbons can be [7] Kenneth W. -K. Shung, Phys. Rev. B 34, 979 (1986).
7
[8] G.Campagnoli,E.Tosatti,inProgressonElectronProp- [18] K. A. Velizhanin, A. Efimov, Phys. Rev. B 84, 085401
erties of Metals, edited by R. Girlanda et al. (Kluwer, (2011).
Dordrecht, 1989), p. 337. [19] Go´mez-Santos, G. and Stauber, T., Phys. Rev. B 84,
[9] O. Vafek, Phys. Rev. Lett. 97, 266406 (2006). 165438 (2011).
[10] G. W. Hanson, J. Appl. Phys. 103, 064302 (2008). [20] D. Mart´ın-Cano, L. Mart´ın-Moreno, F. J. Garc´ıa-Vidal,
[11] M. Jablan, H. Buljan, M. Soljaci´c, Phys. Rev. B 80, E. Moreno, Nano Lett. 10, 3129 (2010).
245435 (2009). [21] A. Y. Nikitin, F. Guinea, F. J. Garc´ıa-Vidal, L. Mart´ın-
[12] A. A. Dubinov, V. Y. Aleshkin, V. Mitin, T. Otsuji, V. Moreno, Phys. Rev. B 84, 161497(R) (2011).
Ryzhii, J. Phys. Condens. Matter 23, 145302 (2011). [22] B.Wunsch,T.Stauber,F.Sols,F.Guinea,NewJ.Phys
[13] L. Ju, B. Geneg, J. Horng, C. Girit, M. Martin, Z. Hao, 8, 318 (2006).
H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, F. Wang, [23] E. H. Hwang, S. Das Sarma, Phys. Rev. B 75, 205418
Nature Nanotech. 6, 630 (2011). (2007).
[14] Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. [24] E. H. Hwang, S. Adam, S. Das Sarma, Phys. Rev. Lett.
S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. 98, 98, 186806 (2007).
Dom´ınguez, M. Thiemens, M. M. Fogler, M. J. Tauber, [25] C. A. Marocico, J. Knoester, Phys. Rev. A 84, 053824
A.H.Castro-Neto,C.N.Lau,F.Keilmann,D.N.Basov, (2011).
Nano Lett. 11, 4701-4705 (2011). [26] D. Dzsotjan, J. Kastel, M. Fleischhauer, Phys. Rev. B
[15] A. Vakil, N. Engheta, Science 332, 1291-1294 (2011). 84, 075419 (2011).
[16] F. H. L. Koppens, D. E. Chang, F. J. Garc´ıa de Abajo, [27] A. Gonza´lez-Tudela, D. Mart´ın-Cano, E. Moreno, L.
Nano Lett. 11, 11, 3370-3377, (2011). Mart´ın-Moreno, C. Tejedor, F. J. Garc´ıa-Vidal, Phys.
[17] A. Y. Nikitin, F. Guinea, F. J. Garc´ıa-Vidal, L. Mart´ın- Rev. Lett. 106, 020501 (2011).
Moreno, Phys. Rev. B 84, 195446 (2011).