arXiv:1001.0011v2  [cond-mat.mes-hall]  16 Apr 2010Guided plasmons in graphene p-njunctions
E. G. Mishchenko,1A. V. Shytov∗,1and P. G. Silvestrov2
1Department of Physics and Astronomy, University of Utah, Sa lt Lake City, Utah 84112, USA
2Theoretische Physik III, Ruhr-Universit¨ at Bochum, 44780 Bochum, Germany
Spatial separation of electrons and holes in graphene gives rise to existence of plasmon waves
conﬁned to the boundary region. Theory of such guided plasmo n modes within hydrodynamics of
electron-hole liquid is developed. For plasmon wavelength s smaller than the size of charged domains
plasmon dispersion is found to be ω∝q1/4. Frequency, velocity and direction of propagation of
guided plasmon modes can be easily controlled by external el ectric ﬁeld. In the presence of magnetic
ﬁeld spectrum of additional gapless magnetoplasmon excita tions is obtained. Our ﬁndings indicate
that graphene is a promising material for nanoplasmonics.
PACS numbers: 73.23.-b, 72.30.+q
Introduction . Breakthrough progress in synthesis and
characterization has made graphene [2] a promising ob-
ject for nanoelectronics. Operation of graphene-based
transistors [3] and other components would rely on the
propertiesofits single-particle excitations–electronsand
holes. However, one can also envisage a completely dif-
ferent set of applications which employ collective excita-
tions, such as plasmons. Currently, plasmon excitations
in metallic structures are a subject of nanoplasmonics, a
new ﬁeld which has emerged at the conﬂuence of optics
and condensed matter physics with one of the aims be-
ing the developing of plasmon-enhanced high resolution
near-ﬁeld imaging methods [4, 5]. Another objective is
possible utilization of plasmons in integrated optical cir-
cuits. However, perspectives of graphene for nanoplas-
monics are largely unexplored since plasmon modes of
graphene ﬂakes have not been addressed so far. As our
results indicate a great amount of control over graphene
plasmon properties makes it a very promising material
for applications.
Fundamentally, the spectrum of collective chargeoscil-
lations reﬂects the long-rangenature of Coulomb interac-
tion. In conventional two dimensional systems, such as
those created in semiconducting heterostructures, plas-
mons are gapless, ω2(q) = 2πe2nq/m∗, withnandm∗
being electron density and eﬀective mass, respectively
[6]. Such oscillations can be treated hydrodynamically.
In clean graphene at zero temperature the plasmon fre-
quency,ω2∝ |EF|, vanishes with decreasing the doping
levelEF. It has been argued [7] that the interaction be-
tweenelectronsandholesintheﬁnalstatecanmodifythe
response functions of Dirac fermions and open up a pos-
sibility for the propagation of charge oscillations at low
frequencies ω < qv, wherevis electron velocity. Still, hy-
drodynamic( ω > qv)analogofconventionalplasmonsre-
mains absent unless either temperature is non-zero [8] or
graphene is driven away from the charge neutrality point
by doping or gating [9]. Expectedly, in both cases plas-
mon spectrum has the conventional form, ω(q)∝q1/2.
In the present paper we investigate spectra of hydro-
dynamic plasmons in spatially inhomogeneous grapheneﬂakes. Realistic graphene samples are typically subject
to disorder potential and mechanical strain [10] that lead
totheformationofchargedelectronandholepuddles[11]
with boundaries between nandpregions being the lines
ofzerochemicalpotential. Moreover,controlled p-njunc-
tions can be made with the help of metallic gates [12].
Alsop-njunctions can be created by applying electric
ﬁeld within the plane of a graphene ﬂake, see Fig. 1a.
The ﬁeld separates electrons and holes spatially in a way
that allows control of both the amount of induced charge
(and thus plasmon frequency) and spatial orientation of
the junction (the direction of plasmon propagation).
b)2d 2d
Ea)
0n n
p p
FIG.1: Twotypesofgraphene p-njunctions: a)ﬁeld-induced,
b) gate-induced. Dot-dashed line indicates boundary betwe en
electron and hole regions and, correspondingly, the direct ion
of plasmon propagation. In case of ﬁeld-induced junction it
is controlled by the direction of external electric ﬁeld E0.
Below, we demonstrate that such p-njunctions can
guide plasmons. We show the existence of charge oscil-
lations which are localized at the junction and have the
amplitude decaying with the distance to the junction.
For wavelengths shorter than the width of the charged
domains, we ﬁnd the plasmon spectrum of the form,
ω2
n(q) =αne2v
¯h/radicalbigg
q|ρ′
0|
e, (1)
whereρ′
0is the gradient of equilibrium charge density
at the junction, vis electron velocity, and n= 0,1,2,...2
enumerates the solutions. The lowest mode has α0=
4√
2πΓ(3/4)/Γ(1/4)≈3.39.
Below we derive this result and discuss plasmon prop-
erties for the two types of p-njunctions: electric ﬁeld
controlled and gate controlled, as shown in Fig. 1.
Hydrodynamics of charge density oscillations. We uti-
lize the hydrodynamic approach to describe the motion
of charged Dirac fermions. The rate of change of electric
current density Jdue to dynamic electric ﬁeld Efollows
from the usual intra-band Drude conductivity with the
corresponding density of states [13],
˙J(r,t) =e2
π¯h2|µ(r)|E(r,t), (2)
determined by the local value of chemical potential µ(r)
as measured from the Dirac point (positive for electrons
and negative for holes). Electric current is related to the
variation of charge density δρby means of the continuity
equation,
δ˙ρ(r,t)+∇·J(r,t) = 0. (3)
Finally, the variation of charge density produces electric
ﬁeld according to the Coulomb law [14],
E(r,t) =−∇/integraldisplay
d2r′δρ(r′,t)
|r−r′|. (4)
Equations (2)-(4) give a closed system for plasmon exci-
tations in graphene ﬂakes. We apply it to a p-njunction
created in a strip inﬁnite along the y-axis (direction of
plasmon propagation). Using the Fourier representation,
δρ(r,t) =δρ(x)exp(iqy−iωt), and eliminating Eand
Jwe arrive at the equation for the oscillating part of
electron density,
ω2δρ(x)+2e2v√π¯h/braceleftBigg
d
dx/radicalbigg
|ρ0(x)|
ed
dx−q2/radicalbigg
|ρ0(x)|
e/bracerightBigg
×/integraldisplayd
−ddx′δρ(x′)K0(|q||x−x′|) = 0,(5)
HereK0is the modiﬁed Bessel function and 2 dis
the width of graphene ﬂake. Within the Thomas-
Fermi approximation equilibrium charge density ρ0(x)
is related to the chemical potential via ρ0(x) =
−sgn(µ)eµ2(x)/π¯h2v2(electron charge is taken to be
−e). This follows from the condition that the electro-
chemical potential µ(x)−eφ(x) is constant throughout
the system. The solutions of Eq. (5) will now be consid-
ered for large and small plasmon momenta separately.
Short wavelength, q≫1/d. In this case the decay
of plasmon density δρ(x) occurs over a distance much
smaller than the width of the system and the limits
of integration in Eq. (5) can be extended to inﬁnity.
Assuming (cf. Eq. (11) below) the linear dependence,
ρ0(x) =ρ′
0x, we observe that the integro-diﬀerentialequation (5) acquires obvious scaling property. Intro-
ducing the variable ξ=qxwe arrive at the plasmon
spectrum in the form (1), with dimensionless constants
αndetermined from the eigenvalue problem:
−2√π/parenleftbiggd
dξ/radicalbig
|ξ|d
dξ−/radicalbig
|ξ|/parenrightbigg
×/integraldisplay∞
−∞dξ′δρ(n)(ξ′)K0(|ξ−ξ′|) =αnδρ(n)(ξ).(6)
Interestingly, this integro-diﬀerential equation allows a
complete analytic solution, though the detailed analysis
is beyond the scope of this paper. Our main ﬁndings
are as follows. Solutions are enumerated by n= 0,1,2...
with even/odd numbers corresponding to even/odd den-
sity proﬁle, δρ(n)(−ξ) = (−1)nδρ(n)(ξ). Surprisingly,
eigenvalues are doubly-degenerate and given by
α2n=α02n+1
4n+1·3·7··(4n−1)
1·5··(4n−3), α2n+1=α2n.(7)
At large distances all modes have exponential depen-
dence,δρ(n)(ξ)∼e−|ξ|, while at |ξ| ≪1 even and
odd solutions exhibit diﬀerent behavior, δρ(even)∼1−
const/radicalbig
|ξ|andδρ(odd)∼sign(ξ)//radicalbig
|ξ|. The ﬁrst pair
of solutions (belonging to the lowest eigenvalue α0) in
the Fourier representation δρ(n)(k) =/integraltext
dξδρ(n)(ξ)eikξ
acquires a simple form:
δρ(0)(k)∝1
(1+k2)3/4, δρ(1)(k)∝k
(1+k2)3/4.(8)
Long wavelength, q≪1/d. In contrast to the above
result (1) plasmon spectrum at small qis sensitive to a
speciﬁc realization of the p-njunction. We address the
long-wavelength behavior of plasmons in ﬁeld controlled
junctions. We expect this case to be of more interest,
in addition it allows a more complete description. Be-
fore analyzing plasmons in this structure, we discuss the
equilibrium density proﬁle. As shown in Fig. 1a the ﬂake
of width 2 dis placed in external electric ﬁeld E0applied
along the x-direction. The equilibrium density distribu-
tionρ(x) is found from,
E0x+sgn(x)¯hv
e/radicalbiggπ
e|ρ0(x)|+2/integraldisplayd
0dx′ρ0(x′)lnx+x′
|x−x′|= 0,
(9)
where it is used that ρ0(x) =−ρ0(−x). Prior to solv-
ing Eq. (9) it is instructive to analyze validity of the
semiclassical approach. The ﬁrst condition implies that
the change of the electron wavelength is smooth on the
scale of itself, d/dx(¯hv/µ)≪1. Estimating µ(x)∼eE0x
we obtain that the distance to the p-njunction line
(x= 0) should exceed the characteristic electric ﬁeld
lengthlE=/radicalbig
e/E0≪x. The second condition requires
that the electron wavelength is small compared with the
width of the system, d≫¯hv/µ. Noting that in graphene3
¯hv∼e2we can rewrite this second condition simply as
lE≪d. Thus, the Thomas-Fermi equation (9) for the
equilibrium charge density and the hydrodynamic equa-
tion (5) for its variation are applicable as long as
lE≪d, q≪1/lE. (10)
However, the ratio of qand 1/dcan be arbitrary. For a
moderate external electric ﬁeld ∼104V/m the value of
electric length lE∼0.4µm and the ﬁrst of the conditions
(10) is satisﬁed easily for micron-sized samples.
AnalyticsolutionofEq.(9)ispossiblewhenthe second
term is small, in which case the charge density is [15]
ρ0(x) =E0x√
d2−x2. (11)
Substituting this expression back into Eq. (9) we ob-
serve that the second term is indeed negligible as long
asx≫l2
E/d. This is assured whenever the condi-
tions (10) are satisﬁed. It is also worth pointing out
that Eq. (11) justiﬁes the linear approximation for the
charge density used in deriving Eq. (1) for q≫1/d, with
ρ′
0/e= 1/(l2
Ed).
We now turn to the analysis of plasma oscillations
propagating on top of the density distribution, Eq. (11).
For small plasmon momenta, q≪1/d, electric ﬁeld ex-
tends beyond the width of the ﬂake and the equation (5)
needs to be supplemented with the boundary condition,
which ensures that electric ﬁeld (and thus the current)
vanishes at the edges, x=±d:
P/integraldisplayd
−ddxδρ(x)
x±d= 0. (12)
The spectrum of the lowest symmetric mode can be most
easily found by integrating Eq. (5) across the width of
the ﬂake. The ﬁrst term in the brackets will then van-
ish exactly due to the boundary condition (12). The
remaining integral can now be calculated to the log-
arithmic accuracy with the help of the approximation
K0(q|x−x′|) =−lnq|x−x′|:
/integraldisplayd
−ddx/radicalbigg
|ρ0(x)|
eln(q|x−x′|)≈2dΓ2(3/4)
lE√πln(qd).
(13)
Eqs. (5) and (13) combine to give the equation, [ ω2−
ω2
0(q)]/integraltextd
−ddxδρ(x) = 0, that yields the dispersion of the
gapless symmetric plasmon,
ω2
0(q) = Γ2(3/4)4e2vd
π¯hlEq2ln(1/qd),(14)
reminiscent of the plasmon spectrum in quasi-one-
dimensional wires, The remaining modes, n≥1, are
gapped. For these modes/integraltextd
−ddxδρ(x) = 0 and simple
procedure of integrating Eq. (5) over the width of theﬂake is not useful. Instead, the equation for the n-th fre-
quency gap can be obtained by setting q= 0 in Eq. (5).
We observe that
ω2
n(0) =βne2v
¯hlEd, (15)
whereβnare the eigenvalues of the equation,
2√πd
dξ/radicalbig
|ξ|
(1−ξ2)1/4/integraldisplay1
−1dξ′δρ(n)(ξ′)
ξ−ξ′=βnδρ(n)(ξ).(16)
The zeroth mode β0= 0, see Eq. (14), is found ana-
lytically: δρ(0)∝1//radicalbig
1−ξ2. It describes charge dis-
tribution in the strip in response to a (uniform along
xdirection and smooth along y-direction) change of its
chemical potential [16]. Other solutions of Eq. (16) are
found numerically:
β1= 1.41, β2= 6.49, β3= 6.75,... (17)
With increasing nthe eigenmodes of integro-diﬀerential
equation (16) oscillate faster, but in generaldo not follow
the oscillation theorem familiar from quantum mechan-
ics. In particular, the solutions with n= 0 andn= 3 are
even while n= 1,n= 2 are odd [17].
Finally, we mention the case of a gate-controlled p-n
junction, Fig.1b. Theequilibriumdensityproﬁleislinear
nearx= 0 and saturates for large |x|[18]. Eq. (1) is still
applicable for q >1/d. In the limit q <1/done should
take into account the screening of long-range Coulomb
interaction by metallic gates. In this case the logarithm
in the spectrum of the gapless plasmon disappears, and
the lowest mode Eq. (14) becomes sound-like.
Magnetoplasmons. If external magnetic ﬁeld Bis ap-
plied perpendicularly to the plane of graphene the plas-
mon spectra acquire new modes. The equation of motion
(2) should now be modiﬁed to include the Lorentz force,
˙J(r,t) =e2
π¯h2|µ(x)|E(r,t)−ev2
cµ(x)J×B.(18)
The relative coeﬃcient between electric and magnetic
terms in this equation follows from the expression for
the Lorentz force acting on a single particle. The last
term has opposite sign for electrons and holes. Note that
the frequency of cyclotron motion ωB(x) =ev2B/cµ(x)
in graphene p-njunctions is position-dependent. The
remaining equations (3)-(4) are intact in the presence of
magnetic ﬁeld. The boundary condition requires now the
vanishing of the normal component of electric current at
the boundary, rather than simply vanishing of the elec-
tric ﬁeld, as in Eq. (12). Eliminating JandEwe arrive
at the generalization of equation (5),
δρ(x)+2e2
π/braceleftbigg
q2Z −q
ω(ωBZ)′−d
dxZd
dx/bracerightbigg
×/integraldisplayd
−ddx′δρ(x′)K0(|q||x−x′|) = 0,(19)4
whereZ(x) =|µ(x)|/(ω2
B(x)−ω2).
The most interesting eﬀect described by Eq. (19) is
the appearance of a set of new modes, chiral magne-
toplasmons, similar to those considered in Ref. [19] for
conventional 2D electron systems with smooth bound-
aries. To ﬁnd their dispersion in strong magnetic ﬁelds,
whenω≪ωB(x) (the exact condition is given below),
one should retain only the second term in Eq. (19).
Noticing that ( ωBZ)′=πl2
Bρ′
0(x)/e=πl2
B/(l2
Ed), where
lB=/radicalbig
¯hc/eBis the magnetic length, we arrive at the
integral equation
−2c
Bq
ωdρ0(x)
dx/integraldisplayd
−ddx′δρ(x′)K0(|q||x−x′|) =δρ(x).(20)
SinceK0is positive, propagation of magnetoplasmons
withq >0is quenched, indicative oftheir chiral property
[20]. As seen from Eq. (20), the plasmon density δρ(x) is
concentratedwhere ρ′
0(x) isthestrongest. Thederivative
of the charge density in ﬁeld-induced junctions (11) fea-
tures strong singularitynearthe edges of the ﬂake. Thus,
low-frequency magnetoplasmon spectrum is strongly de-
pendent on the microscopic regularization of this singu-
lar behavior and is, therefore, beyond the scope of the
Thomas-Fermi approximation used throughout this pa-
per.
Thegate-induced junctions, however, allow a rather
simple analytical description of these modes if we ap-
proximate that ρ′
0(x) =e/l2
Edfor|x| ≤dandρ′
0(x) = 0
for|x|> d. The oscillating density δρ(x) then vanishes
for|x|> d. The solution inside the strip, |x| ≤d, can
be easily found for q≫1/d, where one can assume the
range of integration in Eq. (20) to be inﬁnite. The eigen-
functions of Eq. (20) are simply given by sin[ q⊥(x+d)],
with the values of q⊥=πn/2ddetermined from the con-
dition,δρq(±d) = 0. The spectrum of magnetoplasmons
is then found to be,
ωn(q) =−2πe2l2
B
¯hl2
Edq/radicalbig
q2+π2n2/4d2, n= 1,2...(21)
The magnetoplasmon spectrum (21) is derived under
the assumption that magnetic ﬁeld is strong, ωB(d)≫ω,
which implies that lB≪lE. In order to neglect the ﬁrst
and third terms in the brackets in Eq. (19) one has to
ensure that q≪(lE/lB)4/d. This condition might turn
out to be more orless restrictivethan the hydrodynamics
condition q≪1/lE, depending on the particular value of
the ratio lB/lE. Note that the smallness of this ratio is
not in contradiction to the non-quantized description of
electron motion in magnetic ﬁled. The latter is valid as
long as the ﬁlling factor is large, eEd≫ωB(d), which
means that lB≫l2
E/d. For magnetic ﬁeld ∼1T, and
lB∼25nm, using the estimate below Eq. (10) that lE∼
400nm we conclude that the width of the ﬂake should
exceedd >10µm. The magnetoplasmon modes (21) are
∼(lB/lE)2slowerthan electrons. Note that these modesare undamped since single-particle excitations cannot be
induced at frequencies below cyclotron frequency ωB.
Conclusions . Graphene p-njunctions are among the
most simple and promising applications of this material.
Single-electron properties of p-njunctions have been ex-
tensively studied. In the present paper we investigated
their collective excitations both with and without mag-
netic ﬁeld. We anticipate that plasmon modes will be
crucial for the optical response of graphene nanostruc-
tures and realistic samples containing electron-hole pud-
dles. High degree of experimental control should make
them of special interest to nanoplasmonics and electron-
ics. Among the most promising applications of plasmons
inp-njunctions we envisage a possibility of a “plasmon
transistor” [4]. In particular, by simply switching the
direction of electric ﬁeld from across the ﬂake to along
it (and back) the propagation of plasmons can be facil-
itated (or prevented). In addition, as follows from the
above Eqs. (1), (11), the plasmon velocity can be con-
trolled with simple change in the magnitude of electric
ﬁeld. This is in a sharp contrast to plasmons in metal-
lic nanostructures, whose spectra are typically ﬁxed once
devices are fabricated.
Acknowledgments. Useful discussions with M. Raikh
and O. Starykh are gratefully acknowledged. This
work was supported by DOE, Grant No. DE-FG02-
06ER46313. P.G.S. was supported by the SFB TR 12.
[*] Present address: School of Physics, University of Exete r,
EX4 4QL, U.K.
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