Elementary Electronic Excitations in Graphene Nanoribbons

0.25 MB·English

by  L. Brey

Elementary Electronic Excitations in Graphene Nanoribbons.
L. Brey1 and H. A. Fertig2,3
1. Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049 Madrid, Spain
2. Department of Physics, Indiana University, Bloomington, IN 47405
3. Department of Physics, Technion, Haifa 32000, Israel
7 (Dated: February 4, 2008)
0
0 We analyze the collective mode spectrum of graphene nanoribbons within the random phase
2 approximation. In the undoped case, only metallic armchair nanoribbons support a propagating
plasmon mode. Landau damping of this mode is shown to be suppressed through the chirality of
n
the single particle wavefunctions. We argue that undoped zigzag nanoribbons should not support
a
plasmonexcitationsbecauseofabroadcontinuumofparticle-holeexcitationsassociatedwithsurface
J
states,intowhichcollectivemodesmaydecay. Dopednanoribbonshavepropertiessimilartothoseof
1
semiconductornanowires,includingaplasmonmodedispersingasq√ lnqW andastaticdielectric
3 −
response that is divergent at q=2kF.

]
l PACSnumbers:
l
a
h
- I. INTRODUCTION doped graphene nanoribbon. We calculate the low en-
s ergy charged collective excitations, and analyze the dif-
e
ferences between conventional semiconductor-based one
m Recent improvements in the processing of graphite
dimensional systems and graphene nanoribbons.
. havemadepossibletheisolationofsinglegraphenelayers
t We now summarize our results. Within the RPA, we
a [1]. These carbon sheets are very promising for micro-
find that for most undoped armchair nanoribbons there
m electronic applications because they support the electric
are no plasmon modes associated with interband transi-
field effect: by applying a gate voltage a high mobility
-
d twodimensionalelectronorholegascanbecreated[2,3]. tions. The onlyexceptionismetallicnanoribbons,which
n Thegraphenesheetscouldbefurtherprocessedintomul- support a plasmon that disperses as q√ lnq, with q
−
o the momentum of the excitation along the nanoribbon.
titerminal devices with graphene nanoribbons acting as
c Unlike plasmons in semiconductor based 1D systems, in
[ wires connecting different nanodevices. This possibility
has motivated much recent work on carbon-based nano- metallicnanoribbonstheenergyoftheplasmonsdoesnot

1 depend on the level of doping of the system. The case
electronics.
v ofundoped zigzagnanoribbonsturns outto be consider-
The electronic properties of graphene nanoribbons de-
7 ably more complicated, and we do not carry through a
8 pend strongly on their size and geometry [4, 5]. This
full RPA analysis as in the armchairchase. However,we
7 dependence occurs in part because edges of the ribbons
arguethat a broadparticle-holecontinuum in the zigzag
1 parallel to different symmetry directions require differ-
case implies that any poles in the inversedielectric func-
0 ent boundary conditions. For nanoribbons with zigzag
7 tionwillbedamped,sothattherecannotbepropagating
edges (“zigzag ribbons”) the boundary conditions lead
0 plasmon modes. Finally, we show that doped semicon-
to a particle-like and a hole-like band with evanescent
/ ductor nanoribbons support low energy plasmons that
t wavefunctions confined to the surfaces [5], which con-
a disperse as q√ lnq and, in the armchair case, depend
m tinuously evolve into the well-known zero energy surface −
onthedensityofcarriersintheribboninastandardway.
states [6, 7] as the nanoribbon width gets larger. In the
- This paper is organized as follows. In Section II we
d case of armchair edge nanoribbons (“armchairribbons”)
introduce the Dirac Hamiltonian used for describing the
n thebandstructureismetallicwhenthewidthofthesam-
electronic properties of graphene. In subsections II-A
o ple in lattice constants units has the form 3M+1, with
c and II-B we review the non-interacting electronic struc-
M an integer, and insulating otherwise.
: ture of armchair and zigzag nanoribbons. Section III
v With respect to the lowest energy states, graphene
is dedicated to defining a generalized dielectric function
i
X nanoribbonsareonedimensional(1D)systems. However, for graphene nanoribbons, and to presenting results for
theirbandstructureisratherdifferentthanthatofmore
r collective charge excitations in intrinsic (undoped) and
a conventional 1D systems. In armchair metallic nanorib-
dopedgraphenenanoribbonswhichmaybe derivedfrom
bons the dispersions of the electron and hole bands are
it. We conclude in Section IV with a summary of the
linear in momentum, whereas in armchair semiconduc-
results.
tor nanoribbons the low energy dispersion is quadratic.
On the other hand, zigzag nanoribbons have a peak in
the density of states at the Fermi energy. This affects
II. MODEL HAMILTONIAN.
the low energy physics and the screening properties of
graphene nanoribbons. In this work we compute, within
the Random Phase Approximation (RPA) [8], the dy- Graphene is a honeycomb structure of covalently
namical polarizability and the dielectric constant of a bonded carbon atoms, which should be treated as a tri-
2
angular lattice with two basis atoms, denoted by A and the momentumk ,andthe couplingbetweenthemotion
y
B, per unit cell. In graphene the low energy dispersions in the x and y directions only enters in the direction of
of the electron and hole bands are linear near the points the isospin of the state, θ .
kn,ky
K = 2π(1, 1 ) and K = 2π( 1, 1 ) of the Brillouin
a0 3 √3 ′ a0 −3 √3
zone. Here a is the triangular lattice parameter of the
0
graphenestructure. Inthek papproximation[9,10]the B. Zigzag nanoribbon.
·
wavefunctions are expressed in terms of envelope func-
tions[ψA(r),ψB(r)]and[ψA′ (r),ψB′ (r)]forstatesnearthe The boundary conditions for zigzag nanoribbons are
K and K′ points, respectively. The envelope wavefunc- vanishing of the wavefunctiononthe A sublattice at one
tions may be combined into a four vector which satisfies edge, x=0, and on the B sublattice at the other, x=W.
the Dirac equation HΨ=εΨ, with Thegeometryofthezigzagnanoribbondoesnotmixthe
valleys, and solutions near the K valley with wavevector
0 k ik 0 0
x− y ky are degenerate with solutions near the K′ valley with
k +ik 0 0 0
H =γa0  x 0 y 0 0 k ik  , wavevector-ky.
 0 0 k +ik − x0− y  For the K valley and a given value of ky the nanorib-
 − x y  bons wavefunctions take the form
(1)
whereγ =√3t/2,withtthe hoppingamplitudebetween i ssinh(z x) eikyy
Ψ = n (4)
nearestneighbor carbonatoms. Note that k denotes the n,s,ky (cid:18)sinh((W x)zn) (cid:19) C
−
separationinreciprocalspaceofthewavevectorfromthe
K (K) point in the upper left (lower right) block of the where C is the appropriate normalization constant. The
′
Hamiltonian. corresponding eigenvalues are ε = sγa k2 z2 with
The bulk solutions of Hamiltonian (1) have en- 0q y − n
s= 1. Theboundaryconditionsleadtoatranscendental
einrgitehse εv=asllγeay0|kin|dewx.ith Tsh=e±1wavaenfduncatrioensdetgaekneertahtee equ±ation for the allowed values of zn,
form [e iθk,s,0,0]eikr/C for the K valley, and
− k z
[0,0,eiθk,s]eikr/C for the K′ valley. Here θk = ky +−zn =e−2Wzn . (5)
arctank /k , and C = 2L L is a normalization con- y n
x y x y
stant, with Lx and Ly tphe sample dimensions. For k > 1/W, equation (5) has solutions with real val-
y
ues of z which correspond to states localized at the sur-
face. There are two branches of such surface states, one
A. Armchair nanoribbon.
with positive energies, and the other with negative en-
ergies. These energies approach zero as k increases. In
y
In an armchair nanoribbon the termination at the undoped graphene the Fermi energy is located between
edges consists of a line of A-B dimers, so that the wave- thesesurfacestatebranches. Fork <0,therearenoso-
y
function amplitude should vanish on both sublattices at lutions with real z and surface states are absent. The
the extremes, x=0 and x=W +a0/2, of the nanoribbon. solutions to Eq. 5 have purely imaginary z and rep-
Tosatisfythisboundaryconditionwemustadmixvalleys resent confined states. For values of k in the range
y
[5], and the confined wavefunctions have the form, 0 < k < 1/W, surface states from the two edges are
y
strongly mixed and are indistinguishable from confined
e−iθkn,ky eiknx states. In zigzag nanoribbons the allowed values of z
eikyy  seiknx  n
Ψ = (2) and the corresponding wavefunctions in the transverse
n,s 2 W +a0/2 Ly e−iθkn,ky e−iknx  direction x depend strongly on the value of the longitu-
p p  −se−iknx  dinal momentum of the state ky. This is a consequence
ofthechiralityimplicitintheDiracHamiltonian,Eq.(1).
with energies ε (k )=sa γ k2 +k2. The allowed val-
n,s y 0 n y
q
ues of k satisfy the condition [5]
n
III. DIELECTRIC CONSTANT.
2πn 2π
k = + . (3)
n
2W +a0 3a0 Innanoribbonstheelectronsareconfinedtomovejust
in the y direction, and the momentum in this direction,
ForawidthoftheformW=(3M+1)a ,the allowedval-
0 k , is a good quantum number. The motion in the x di-
y
uesofkn,kn = 32aπ0 (cid:16)2M2M++1+1n(cid:17),createdoublydegenerate rection is confined and it is described by a band index n
states for n = 2M 1, and allow a zero energy state and the conduction or valence character s. The form of
6 − −
when k 0. Nanoribbons of widths that are not of the dielectric constant depends strongly on whether the
y
→
this form have nondegenerate states and do not include motion in the longitudinal and transverse directions of
a zero energy mode. Thus these nanoribbons are band thewirearecoupled,asinthecaseofthezigzagnanorib-
insulators. The allowed values of k are independent of bon, or decoupled, as in the armchair nanoribbon. In
n
3
the latter case the dielectric constant can be described
as a tensor in the subband indices whereas in the for- 6
mer case it also depends on the momentum k of the
y
electronic states. For undoped ribbons, we will analyze (cid:12)
(cid:19)
the armchair case in detail, whereas for the zigzag case (cid:72)/ 4 (0,0)
we will argue that propagating plasmons are unlikely to 2e (1,1)
exist. The results for doped graphene nanoribbons are ) ( ((23,,23))
qualitatively similar to those of standard semiconductor W 2 (0,2)
nanowires for both cases. q (1,3)
(
k
vi,
0
A. Armchair nanoribbon.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
For armchair nanoribbons the motion of the carri-
qW
ers on the wire is coupled with the transverse motion
onlythroughtherelativephasesofthewavefunctions[cf.
Eq.(2)]. In this situation the generalized dielectric func- FIG. 1: (Color online) Different matrix elements of the
tion [11, 12] for the nanoribbon is given by Coulomb interaction as function of qW for a armchair ter-
minated nanoribbon, Equation 9.
ε (q,ω)=δ δ v (q)Π (q,ω) (6)
ijmn im jn ijmn m,n
−
wherethe functionΠ (q,ω)is ageneralizedirreducible
mn
intervalley scattering nor changes the sublattice index,
polarizability function. In this expression i,j,m,n are
as should be appropriate in the long wavelength limit.
composite indices specifying the subband index and
Note that the interaction is strictly zero for arbitrary q
the valence/conduction band character s = 1. The
± if i j + m n is an odd integer.
wavevectorq is along the direction in which the electron
| − | | − |
motion is free, and δ is a Kronecker delta function. In Fig. 1 we plot different matrix elements of the
im
The explicit expression for Π takes the form Coulomb interaction as a function of the momentum q.
Clearlyv ismuchlargerthantheothers,anditdiverges
0,0
gs f(ǫn′(ky +q)) f(ǫn(ky)) as ln(qW) when q 0. Thus, in graphene nanorib-
Πn,n′(q,ω)= Ly Xky ǫn′(ky+q)−ǫ−n(ky)−~ω × breolne−vsa,nbtottrhanisnittiohnesuant→dloopngedwaanvedlednogpthedsincavsoelvs,etshuebboannldy
Fn,n′(ky,ky+q) , (7) indices which satisfy the condition i j=m n=0. For
− −
the low energy excitations in the long wavelength limit
where g is the spin degeneracy, f(ǫ) is the Fermi distri-
s
we need only include the Coulomb matrix element v ,
bution function, ǫ (k ) is the subband dispersion, and 0,0
n y
and can to a good approximation ignore the others.
Fn,n′(ky,ky + q) is the square of the overlap between
wavefunctions, The charge density excitations are obtained by find-
ing the zeros of the real part of the determinant of the
1 dielectric matrix Eq.(6). These excitations are well de-
Fns,n′s′(ky,ky +q)= (1+ss′cosθ), (8)
2 finedprovidedthey arenotdampedby the continuumof
electron-hole pair excitations, defined by a finite imagi-
with θ the angle between the wavevectors (k ,k ) and
n y nary part of the determinant of the dielectric matrix.
(kn′,ky+q). The Coulombmatrixelements forthe arm-
In undoped graphene nanoribbons at zero tempera-
chair nanoribbon take the form
ture, only transitions from the valence subbands, s=-1
1 1
to conduction subbands , s=1, contribute to the polar-
v (q)=v (qW)= du du
ijmn |i−j|,|m−n| Z0 Z0 ′ izability. Moreover, if we include only v0,0, the lowest
energy transitions are from the valence (s= 1) to con-
cos[π(i j)u]cos[π(m n)u]v(qW u u ), (9)
′ ′ −
− − | − | duction (s = 1) bands which have the same subband
where the one-dimensional Fourier transform of the the index n. The dielectric tensor has the form
Coulomb interaction has the form [13]
2e2 ε(n1,s),(n1,−s),(n2,s′),(n2,−s′)(q,ω)=
vq(x−x′)=v(q|x−x′|)= ǫ0 K0(q|x−x′)|) . (10) δss′δn1,n2 −v0,0(q)Π(n2,s′),(n2,−s′)(q,ω) (11)
Here ǫ is the background dielectric constant and K (x) Thezerosofthe determinantofthis matrixarethe zeros
0 0
isthezeroth-ordermodifiedBesselfunctionofthesecond of the function
kind, which diverges as -ln(x) when x goes to zero.
Inorderto obtainthelastexpressioninEq. 9wehave 1 v (Π +Π ) 1 v χ , (12)
0,0 n ,n+ n+, n 0,0 n
assumed that the Coulomb interaction neither produces − − − − ≡ −
Xn Xn
4
3
)
n 2
(cid:70)
(cid:54)
00 1 F=0
-v F(cid:143)0
al(1 0 qa0=0.005
e -1 qa0=0.01
R
-2
0.00
X Data
)
n-0.02 Right Left
(cid:70)(cid:54) Col 2 vs Col 3 movers. movers.
00-0.04 WCol= 6 v2s C5ola 7
v 0
1--0.06
(
m
-0.08
I FIG.3: (Color online)Schematicrepresentationoftheband
structure and low energy electron-hole pair excitations in
0.00 0.05 0.10 0.15 0.20 0.25 0.30
intrinsic (a) metallic armchair nanoribbons and (b)zigzag
(cid:74)a /t
0 nanoribbons. In metallic armchair nanoribbons the chirality
of the wavefunctions allows only transitions between carriers
movingin thesamedirection toenterthepolarizability; con-
FIG. 2: (Color online)Real and imaginary part of the di- tributionsfrombackscatteringvanish. Foragivenwavevector
electric constant, Eq. (12), versus ω for an undoped metallic q, because of thelinear dispersion, all theelectron hole pairs
armchair nanoribbon of width W=25a0. Dotted and dashed enteringthepolarizability havethesameenergyγa0q. Inthe
lines correspond to wavevectors qa0=0.005 and qa0=0.01 re- caseofzigzagnanoribbonstheexistenceofsurfacestatesleads
spectively. toacontinuuminenergiesforelectron-holepairswithagiven
q.
with
peak visible. In the limit δ 0 this peak becomes a
χ (q,ω)= gs 2∆n(ky,q) delta function.) →
n L (~ω+iδ)2 (∆ (k ,q))2 ×
y Xky − n y The collapse of the particle-hole continuum into a sin-
glelineintheωvs. qplaneisverycommoninonedimen-
F (k ,k +q) (13)
n ,n+ y y
− sionalmetallicsystems. However,itspresencehereturns
where ∆ (k ,q)=ǫ (k +q)+ǫ (k ). out to depend in a fundamental way on the chirality of
n y n y n y
In Fig. 2 we sh|ow the th|e r|eal an|d imaginary parts the Dirac fermions. This enters through the overlap F
of the dielectric constant as function of frequency for an that appears in the polarizability, and guarantees that
armchairnanoribbonofwidthW=25a . Thelowestsub- for a given q only particle-hole excitations of a single
0
banddisperseslinearlyfromzerosothatthe nanoribbon frequency contribute to Eq. 13; i.e., ∆n=0 is indepen-
is metallic. Above this, the next lowest energy subband dent of ky when F = 0. The overlap F = (1+cosθ)/2,
6
pocacrutrosfftohrekd1ie≈lec0t.r1i3cac−0o1n.stIanntFiisg.2clowsee tsoeeutnhiatyt tahnedrtehael fweirtehntθtthhaenaznegroleobneltywfeoernθ(0=,kπy). aFnodr (e0x,ckitya+tioqn)s, aiscrdoisfs-
imaginarypart has a non-vanishing value for frequencies the Dirac point, this corresponds to forward scattering,
largerthantwicetheconfinementenergyofthefirstsub- with a single energy for a given q. Although there is a
band. The frequency ω˜ = γa ( k2+q2 +k ) defines continuum of energies for a backscatteredparticle, these
0 1 1
the edge for the continuum of nponinteracting intersub- do not contribute because the wavefunction overlap F
band electron-hole excitations. Note that the dielectric strictly vanishes. This is illustrated in Fig. 3(a).
constant does not cross zero in the vicinity of these in- From the top panel in Fig. 2 one may see that the
tersubband excitation energies. The absence of such ze- real part of the dielectric function vanishes at a fre-
rosindicatestherearenocollectiveexcitationsassociated quency close to but just above the particle-hole excita-
withinterbandtransitions,whichinanycaseforenergies tionfrequency. Since theimaginarypartofthedielectric
greater than ~ω˜ would have a finite lifetime due to Lan- function vanishes at this frequency, one obtains a single,
dau damping. sharplydefinedplasmonmode. This is theonlycollective
For much lower frequencies, the imaginary part of the excitation that occurs in undoped graphene nanoribbons.
dielectric function remains zero down to an energy InFig.4 we plot the intrabandplasmonfrequency as a
∼
γa q, where a peak reveals the existence of electron-hole function of qW. It is clear in the figure that in the long-
0
excitations at a single frequency. This correspondsto an wavelength limit, qW < 0.001, the Bessel function can
intraband excitation in the metallic subband. (We note be approximated by -ln(qW) and the plasmon disperses
thatinthelowerpanelofFig. 2thisisslightlybroadened asq ln(qW). Keepingonlyintrabandtransitions,the
−
due to the small imaginary iδ added to ω to make the effecptive polarizability entering in Eq. 12 for metallic
5
4
W=25a (a)
6 0 s) W=24a0
t
(cid:12) uni 3 k a =0.0108
(cid:16)(cid:22) . F 0
(cid:19) 4 b k a =0.0155
(cid:20) ar 2 F 0
(cid:74)(cid:11) ) ( kFa0=0.0216
(cid:18) (cid:19)
(cid:90) 2 (cid:32)
(cid:90)
(cid:3) 1
,
q
(
(cid:70)
0 0
0.00 0.01 0.02 0.03 0.04 (b)
qW 0.25
0.20
mFIoGn.d4i:sp(Ceroslioornoinnlainme)eTtahlleiccognratipnhueonuesnlianneoirnidbibcoantecsatlhcuelpaltaesd- /(cid:90)(cid:74)0.15 kFa0=0.0216
within the RPA. The dashed line shows the local longwave-
0.10 k a =0.0155
lengthplasmondispersion,Eq.16. Thedottedlineisthesingle F 0
particle excitation. 0.05
k a =0.0108
F 0
0.00
armchair nanoribbons can be obtained analytically for 0.0 0.5 1.0 1.5 2.0 2.5 3.0
finite temperatures and dopings, q/k
F
g γa q
s 0
χ (q,ω,β,µ)= f (q,β,µ) (14)
0 −π (~ω)2 (γa q)2 1
0
− FIG. 5: (Color online)(a) The static polarizability as func-
with tion of the wavevector of a doped semiconductor armchair
nanoribbon for different densities. (b)Intrasubband plasmon
1 1+e−βµ dispersion of thesame system.
f (q,β,µ)= βγa q+2ln ,
1 βγa0 (cid:20)− 0 1+e−β(γa0q+µ)(cid:21)
(15)
with β = 1/k T and µ the chemical potential. This suchthermo-plasmonexcitations. The temperature only
B
result applies both for doped and undoped nanoribbons. slightly alters the plasmon dispersion. Finally, we stress
In the longwavelength (q 0) limit we can easily show the strong dependence of the plasmon dispersion on the
that the plasmon frequenc→y is given by, nanoribbon width. This indicates that plasmon spec-
troscopy can be used as a characterization of the width
2g e2 1/2 of the nanoribbon.
~ω s γa q2 ln(qW) f (q,β,µ) .
p 0 1
≃(cid:18) πǫ (cid:19) −
0 p p
(16)
B. Doped semiconductor armchair nanoribbon.
Some comments on this result are in order. First, the
plasmon has exactly the same dispersion as the normal
one-dimensional plasmon of a metallic nanowire. How- We next analyze the low energy excitations of an
ever, at zero temperature, in graphene nanoribbons the electron-doped armchair nanoribbon. We confine our-
plasmonfrequencyisindependentofthechemicalpoten- selves to the case that the Fermi energy lies in the low-
tial and therefore of the density. A classical one dimen- est energy subband. In this situation the plasmon cor-
sionalplasmonhasadependence√n ,wheren isthe responds to excitations within the same subband. The
1D 1D
one-dimensional carrier density. A similar anomaly oc- polarizability can be approximated by
curs in two dimensional doped graphene where the plas-
2 kF ∆(k ,q)
m(no2nD)f1r/e2qucelanscsyicablehdaepveesndaesn(cne2[D1)41,/145,,c1o6m].paTrheidswanitohmtahlye Π0,0 = π Z−kdFkyF0+,0+(ky,ky+q)(~ω)2−∆y(ky,q()127)
inthedensitydependenceoftheplasmonsisadirectcon-
with∆(k ,q)=ǫ (k +q)-ǫ (k ), andthe collectiveex-
sequence of the quantum relativistic nature of graphene y 1 y 1 y
| | | |
itations are obtained from the equation
[16]. Second, in undoped two dimensional graphene a
finite temperature produces thermo-plasmon excitations
1 v (q)Π (q,ω)=0 (18)
0,0 0,0
associated with transitions between thermally activated −
electrons and holes [17, 18]. In metallic nanoribbons The Fermi wavevector appearing in Eq.17 is related to
the absence of backscattering precludes the existence of the one dimensional density of extra carriers in the
6
nanoribbonbytheexpressionk =πn /2. InFig. 5(b) unity only for transitions from valence states, s=-1 to
F 1D
weplottheplasmonenergyforthiscaseandfordifferent the conduction states s=1. Unlike the armchair case, in
densities. As expected in one dimensional systems, the zigzag nanoribbons there is a broad particle-hole contin-
plasmon behaves as q√ lnqW. However,in contrast to uumofexcitations,fromzeroenergyupto γ,forsmall
− ∼
the metallic case, the low energy subband dispersion is q. This occurs because of the existence of surface states
parabolic, and for small q the overlap between electron which are essentially at zero energy over a broad range
andholewavefunctions,F (k ,k +q)=(1+cosθ)/2, of k , allowing arbitrarily low energy particle-hole exci-
0+,0+ y y y
is close to unity. Thus these plasmons have the usual tations for any q smaller than an appreciable fraction of
√n dependence on the electron density. The possibil- the Brillouinzone size, as illustratedin Fig. 3(b). These
1D
ity of backscattering is also reflected in the static polar- necessarilydampanyplasmonpolesthatmightotherwise
izability. In Fig. 5(a) we plot the static polarizability appear in the inverse dielectric function, and we expect
as a function of the momentum. This quantity is loga- no propagating plasmon modes for the undoped zigzag
rithmically divergent at q=2k as in the standard one nanoribbon.
F
dimensional electron gas. This is a consequence of the The doped case is essentially the same as that of the
perfectnestingandthepossibilityofbackscatteringfrom doped armchair nanoribbon and standard semiconduct-
-k to k . It suggests that at sufficiently low tempera- ingnanowires: onemaylinearizethesingleparticlespec-
F F
ture, the system may be unstable to the formation of a trum around the Fermi energy, leading to a narrow line
charge density wave when coupling to lattice phonons is of particle-hole excitations, above which a plasmon pole
included. should appear with dispersion q√ lnqW.
−
C. Zigzag nanoribbon.
IV. SUMMARY
Thecalculationofplasmonpolesinthezigzagnanorib-
bon case is complicated by the fact that the parameter In this paper we have analyzed the collective mode
zn in the wavefunctions in Eq. 4 is a function of ky. In spectrumofgraphenenanoribbons. Intheundopedcase,
this case the Coulomb matrix element cannot be writ- we find only metallic armchair nanoribbons support a
ten only as a function of the momentum transfer q. The propagatingplasmonmode. This mode mode may prop-
generalized dielectric function within the RPA obeys agate undamped because the chirality of the wavefunc-
tions prevents it from decaying into particle-hole pairs.
ǫijmn(ky,ky′;q,ω)=δimδjnδky,ky′ −vijmn(ky,ky′;q)× Wearguedthatundopedzigzagnanoribbonswillnotsup-
f(ǫ (k +q)) f(ǫ (k )) port plasmon excitations because this supports a broad
m y n y
gsgvǫ (k +q) ǫ (k−) ~(ω+iδ) (.19) continuumofparticle-hole excitationswith low wavevec-
m y n y
− − tors, and there is nothing to protect a plasmon from de-
Here g =2 is the valley degeneracy. The matrix element caying into this. Doped nanoribbons turn out to have
v
of the Coulomb interaction has the form properties similar to those of semiconductor nanowires,
including aplasmonmodedispersingasq√ lnqW with
W W a coefficientthatvanishes whenthe doping−vanishes(ex-
vijmn(ky,ky′;q)=Z dxZ dx′Ψ∗i,ky(x)·Ψj,ky+q(x)× ceptinthemetallicarmchaircase),andastaticdielectric
0 0
response that is divergent at q =2k .
vq(x−x′)Ψ∗m,ky′+q(x′)·Ψn,ky′(x′)(20) F
where Ψ Ψ denotes a dot product of the vectors ap- Acknowledgements. The authors thanks P. de
∗
·
pearing in Eq. 4. The collective excitation spectrum is Andr´es, F.Guinea and P.L´opez-Sancho for helpful dis-
obtained by the condition that the determinant of the cussions. This work was supported by MAT2006-03741
dielectric matrix, Eq. (19), vanishes. In the case of un- (Spain) (LB) and by the NSF throughGrant No. DMR-
doped graphene the dielectric function is different from 0454699(HAF).
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