El El grafenografeno
como sistema electrónico en como sistema electrónico en DD
= 2= 2
El grafeno
ha despertado recientemente gran interés:
por sus posibles aplicaciones, al ser un material conductor, flexible, casi transparente y muy resistente
desde un punto de vista fundamental,por representar el comportamiento de campos relativistas en dos dimensiones
La coordinación del carbono en compuestos bidimensionales La coordinación del carbono en compuestos bidimensionales ha dado lugar a una secuencia de descubrimientos de nuevos ha dado lugar a una secuencia de descubrimientos de nuevos materiales en los últimos tiemposmateriales en los últimos tiempos
1985 1991 2004
El primer paso para entender las propiedades de estos materialesEl primer paso para entender las propiedades de estos materiales consiste en entender el comportamiento de los planos de consiste en entender el comportamiento de los planos de grafenografeno
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−=
∑∑
⋅−
⋅
0
0
a
via
vi
a
a
e
etH p
p
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=
00
yx
yxF ipp
ippvH
The observed properties were actually consistent with the The observed properties were actually consistent with the dispersion expected for electrons in a honeycomb latticedispersion expected for electrons in a honeycomb lattice
ELECTRONIC PROPERTIES OF GRAPHENEELECTRONIC PROPERTIES OF GRAPHENE
Expanding around each corner of the Expanding around each corner of the BrillouinBrillouinzone, we obtain the zone, we obtain the hamiltonianhamiltonian
for a for a twotwo‐‐component component fermionfermion
((DiracDirac
hamiltonianhamiltonian))
We have to introduce a We have to introduce a DiracDirac
fermionfermion
for each independent Fermi point, at whichfor each independent Fermi point, at which
pppσ FF vvH ±=⋅= )( , ε
∑′
+ ′−=rr
tb tH,
)( )( rr ψψ
)2/3cos()2/cos(4)2/(cos41 2xyy apapaptE ++±=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
0 0
φ
φ
i
i
F ee
vHk
k
ELECTRONIC PROPERTIES OF GRAPHENEELECTRONIC PROPERTIES OF GRAPHENE
In the absence of In the absence of scatterersscatterers
that may induce a large momentumthat may induce a large momentum‐‐transfer, backscattering is thentransfer, backscattering is thensuppressed suppressed (H. (H. SuzuuraSuzuura
and T. Ando, Phys. Rev. and T. Ando, Phys. Rev. LettLett. 89,266603 (2002)). . 89,266603 (2002)).
The scattering by impurities is quite unconventional in The scattering by impurities is quite unconventional in graphenegraphene, due to the , due to the chiralitychirality
of electrons.of electrons.When a When a quasiparticlequasiparticle
encircles a closed path in momentum space, it picks up a Berry encircles a closed path in momentum space, it picks up a Berry phase of phase of ππ
ψψ σπ )2/(2 zie→
⎟⎟⎠
⎞⎜⎜⎝
⎛±
=−
2/
2/
21
φ
φ
ψi
i
ee
⊃
⊂
AA
][||||||~ **222⊃⊂⊃⊂⊃⊂⊃⊂ +++=+ AAAAAAAAw
0|||| 22)2/(2* <−== ⊂⊂
−
⊃⊂ AAeAA zi σπ
ELECTRONIC PROPERTIES OF GRAPHENEELECTRONIC PROPERTIES OF GRAPHENE
Another way of explaining the suppression of backscattering Another way of explaining the suppression of backscattering is by considering that, for the is by considering that, for the masslessmassless
DiracDirac
fermions, the fermions, the pseudospinpseudospin
gives rise to the conserved quantitygives rise to the conserved quantity
This also explains the peculiar properties of electrons when tunThis also explains the peculiar properties of electrons when tunneling across potential barriers:neling across potential barriers:the transmission probability is equal to 1 at normal incidence, the transmission probability is equal to 1 at normal incidence, and 0 for backscatteringand 0 for backscattering
M. I. M. I. KatsnelsonKatsnelson, K. S. , K. S. NovoselovNovoselov, and , and A. K. A. K. GeimGeim, Nature Physics , Nature Physics 2, 620 (2006) , 620 (2006)
ppσ ⋅
that changes sign upon the inversion of the momentum.that changes sign upon the inversion of the momentum.
MANYMANY‐‐BODY EFFECTS IN GRAPHENEBODY EFFECTS IN GRAPHENE
The singleThe single‐‐particle properties are significantlyparticle properties are significantlyrenormalized due to the strong Coulomb interaction:renormalized due to the strong Coulomb interaction:
(J. G., F. Guinea and (J. G., F. Guinea and M. A. H. M. A. H. VozmedianoVozmediano,,Phys. Rev. B 59, R2474 (1999))Phys. Rev. B 59, R2474 (1999))
)/log( )( )/log( )(
1 1
0
kFkk
Fk
gvgv
GG
ωβωγωω
Λ⋅−Λ−⋅−≈
Σ−=
kσkσ
Fveg 16/ with 2≡
222
2
08
),(ω
ω−
−=Πqqq
Fv
GrapheneGraphene
is a system with remarkable manyis a system with remarkable many‐‐body properties, starting with the behavior of itsbody properties, starting with the behavior of itselectronelectron‐‐hole excitations. The polarization ishole excitations. The polarization is
In the In the undopedundoped
system, there are no electronsystem, there are no electron‐‐hole hole excitations nor excitations nor plasmonsplasmons
into which the electrons into which the electrons can decay can decay (J. G., F. Guinea and M.A.H. (J. G., F. Guinea and M.A.H. VozmedianoVozmediano, , NuclNucl
. Phys. B424, 595 (1994)). Phys. B424, 595 (1994))
En el En el grafenografeno
la interacción de la interacción de CoulombCoulomb
no está no está apantallada, y conduce a propiedades electrónicas exóticasapantallada, y conduce a propiedades electrónicas exóticas
)/log(1)( 02 ωω EgZ −≈
RenormalizaciónRenormalización de la función de la función de ondas del electrónde ondas del electrón
Razón de desintegración de Razón de desintegración de las las cuasipartículascuasipartículas
ωω 2~)( gΓ
J. G., F. Guinea and
M.A.H. Vozmediano,Phys. Rev. B 59, R2474 (1999)
)(),(
ωωω ψ
Γ+⋅−=
iZvZ
kGvF kσ
CURVATURE IN GRAPHENECURVATURE IN GRAPHENE
Can Can wewe
learnlearn
somethingsomething
fromfrom
thethe
effecteffectofof
curvaturecurvature
in in thethe
carboncarbon
layerlayer??
AgainAgain, , thethe
inducedinduced
changechange
ofof
topologytopology
requiresrequirestopologicaltopological
defectsdefects, , andand
thesethese
may may givegive
riserise
totointerestinginteresting
featuresfeatures
in in thethe
electronicelectronic
structurestructure. .
TOPOLOGY OF GRAPHENE SHEETSTOPOLOGY OF GRAPHENE SHEETS
the contribution of a pentagon as the contribution of a pentagon as
and the contribution of a heptagon asand the contribution of a heptagon as
We can compute the contribution of a hexagon to We can compute the contribution of a hexagon to χχ
asas
, 61 1
25
35 =+−=Δχ
, 0 1 3 2 =+−=Δχ
TheThe
allowedallowed
geometriesgeometries
ofof
thethe
carboncarbon
latticeslattices
are are constrainedconstrained
by by thethe
valuesvalues
ofof
thethe
EulerEulercharacteristiccharacteristic
χχ
, , whichwhich
isis
expressedexpressed
in in termsterms
ofof
thethe
numbernumber
ofof
handleshandles
hh
ofof
a a givengiven
topologytopologyas as χχ
= 2 = 2 ––
2 2 h . h .
χχ
can be can be alsoalso
expressedexpressed
forfor
a a givengiven
latticelattice
as as faces# edges# vertices# +−=χ
61 1
27
37 −=+−=Δχ
ThisThis
isis
whywhy
12 12 pentagonspentagons
are are onlyonly
neededneeded
toto
closeclose
thethe
carboncarbon
latticelattice
intointo
a a sphericalspherical
shapeshape,,providedprovided
thatthat
theirtheir
effecteffect
isis
notnot
counterbalancedcounterbalanced
by by thethe
negativenegative
curvaturecurvature
ofof
thethe
heptagonsheptagons..
En el En el casocaso
de de loslos
fullerenosfullerenos, , loslos
anillosanillos
pentagonalespentagonales
induceninducenla la curvaturacurvatura
de la red, de la red, peropero
tambiéntambién
introducenintroducen
frustraciónfrustración
entreentre
laslas
dos dos subredessubredes..
TOPOLOGICAL DEFECTS IN GRAPHENE TOPOLOGICAL DEFECTS IN GRAPHENE
⎟⎟⎠
⎞⎜⎜⎝
⎛ΨΨ
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛ΨΨ
´´ 0110
´´
K
K
K
K
⎟⎟⎠
⎞⎜⎜⎝
⎛ −Φ=
00
2
i
iA
πφ
TheThe
pentagonal pentagonal carboncarbon
ringsrings
can be can be formedformed
by a by a cutcut
andand
pastepasteoperationoperation
in in thethe
planeplane. . ThisThis
induces induces anan
effectiveeffective
rotationrotation
ofof
pp/3/3atat
thethe
junctionjunction, , whichwhich
impliesimplies
in in turnturn
thethe
exchangeexchange
ofof
thethe
twotwoDiracDirac
valleysvalleys
TheThe
exchangeexchange
ofof
thethe
twotwo
DiracDirac
valleysvalleys
isis
onlyonly
feltfelt
whenwhen
makingmaking
a complete a complete turnturn
aroundaroundthethe
topologicaltopological
defectdefect. . ThereforeTherefore, , thethe
effecteffect
can be can be mimickedmimicked
by a by a lineline
ofof
effectiveeffective
flux flux ΦΦthreadingthreading
thethe
pentagonal pentagonal ringring, , actingacting
onon
thethe
((K K , , KK´́) ) spacespace
( ) 2
0110
exp πφ φ =Φ⇔⎟⎟⎠
⎞⎜⎜⎝
⎛−
=∫ Adi
J. G., F. Guinea and M.A.H. J. G., F. Guinea and M.A.H. VozmedianoVozmediano, , NuclNucl. Phys. B 406, 771 (1993). Phys. B 406, 771 (1993)
TOPOLOGY OF GRAPHENE SHEETSTOPOLOGY OF GRAPHENE SHEETS
While the number of defects needed to change the topology of theWhile the number of defects needed to change the topology of the
carbon lattice is small, carbon lattice is small, they induce a strong effect in the electronic properties of the they induce a strong effect in the electronic properties of the material material
J. G., F. Guinea and M.A.H. J. G., F. Guinea and M.A.H. VozmedianoVozmedianoNuclNucl. Phys. B 406, 771 (1993). Phys. B 406, 771 (1993)
nnnii Ψ=Ψ−∇⋅ ) ( εAγ
22
22 21 gjRj −⎟⎠⎞
⎜⎝⎛ +=ε
In the fullerenes, the combined effect of the In the fullerenes, the combined effect of the 12 pentagonal rings is consistent with the 12 pentagonal rings is consistent with the field of a monopole, whose charge is dictated field of a monopole, whose charge is dictated by the total fluxby the total flux
23
2
41
12
1 i== ∑
=
ππ
g
ByBy
approximatingapproximating
thethe
effectiveeffective
fieldfield
by by ananisotropicisotropic
flux flux atat
thethe
sphericalspherical
surfacesurface
ofof
thethefullerenefullerene, , thethe
DiracDirac
equationequation
forfor
thethe
curvedcurvedlatticelattice
becomesbecomes
We can also investigate the effects of negative curvature in We can also investigate the effects of negative curvature in graphenegraphene. The simplest instance is . The simplest instance is a carbon a carbon nanotubenanotube‐‐graphenegraphene
junction junction
CARBON NANOTUBECARBON NANOTUBE‐‐GRAPHENE JUNCTIONSGRAPHENE JUNCTIONS
The The nanotubenanotube‐‐graphenegraphene
junction requires an amount of negative curvature correspondingjunction requires an amount of negative curvature corresponding
toto6 heptagons. This is consistent with the fact that, in any conti6 heptagons. This is consistent with the fact that, in any continuum geometry matching the nuum geometry matching the graphenegraphene
plane with the plane with the nanotubenanotube, we find the Euler characteristic, we find the Euler characteristic
The above procedure describes the construction of junctions withThe above procedure describes the construction of junctions with
zigzig‐‐zagzag
nanotubesnanotubes
of type of type
((6n6n,0) ,0) . . When the heptagons are regularly distributed, these are the onlyWhen the heptagons are regularly distributed, these are the only
possible geometries,possible geometries,together with the junctions made of armchair (together with the junctions made of armchair (6n6n,,6n6n) ) nanotubesnanotubes..
1 2 −== ∫ Rgxdχ
There is a general, compact way of describing the There is a general, compact way of describing the nanotubenanotube‐‐graphenegraphene
junctions, when junctions, when the topological defects (heptagons) are regularly distributed. Wthe topological defects (heptagons) are regularly distributed. We can think of all possible e can think of all possible geometries as assemblies of triangular blocks of honeycomb lattigeometries as assemblies of triangular blocks of honeycomb lattice ce
CARBON NANOTUBECARBON NANOTUBE‐‐GRAPHENE JUNCTIONSGRAPHENE JUNCTIONS
This shows again that the number of heptagonal carbon rings is aThis shows again that the number of heptagonal carbon rings is always the same (6). It also lways the same (6). It also becomes clear that junctions with armchair becomes clear that junctions with armchair nanotubesnanotubes
are possible, with geometries (are possible, with geometries (6n6n,,6n6n) .) .
Within
each
class, , allall
thethe
DOS DOS looklook
veryvery
similar, similar, eveneven
forfor
differentdifferent
geometriesgeometries
ofof
thethe
nanotubenanotube, , withwith
thethe
positionposition
ofof
thethe
mainmain
featuresfeatures
scaledscaled
in in inverseinverse
proportionproportion
toto
thethe
radiusradius
RR
ofof
thethe
tubetube. .
ThisThis
leadsleads
toto
thinkthink
thatthat
therethere
may may existexist
aa
unifiedunified
descriptiondescription
in in termsterms
ofof
thethe
DiracDirac
equationequation
in in thethe
curvedcurved
spacespace
CARBON NANOTUBECARBON NANOTUBE‐‐GRAPHENE JUNCTIONSGRAPHENE JUNCTIONS
)12,12(
)0,18(
)0,48(
)0,54(
( ) ±± Ψ=Ψ∇⋅ εAσ ieivF m
LOCALIZED STATES LOCALIZED STATES
What is then responsible for the What is then responsible for the peaks within the depleted DOS peaks within the depleted DOS at very low energies?at very low energies?
ItIt
isis
thenthen
possiblepossible
toto
havehave
localizedlocalized
statesstates. . TakingTaking
thethe
maximummaximum
flux flux ΦΦ
= 3= 3pp
, , wewe
havehave
forfor
instanceinstance
We may look for bound states of the We may look for bound states of the DiracDirac
equation, that can only take place at equation, that can only take place at εε
= 0 = 0
0 0
21
21
21
210
=⎟⎟⎠
⎞⎜⎜⎝
⎛
ΨΨ
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+Φ
±∂+∂
+Φ
∂−∂±
±
B
A
r
r
rrir
rriri
π
π
θ
θ m
We find a state with We find a state with nn
= 1 which has an amplitude decaying in both the plane and the = 1 which has an amplitude decaying in both the plane and the nanotubenanotube..Similarly, we have another localized state with Similarly, we have another localized state with nn
= = ‐‐11
in the other in the other sublatticesublattice
of the of the graphenegraphenelayer. These localized states are then consistent with the abovelayer. These localized states are then consistent with the above
lowlow‐‐energy peak in the DOS.energy peak in the DOS.
3/
)12,12(πieq ±= 3/
)0,18(πieq ±=
0 0 , ~
0 , ~ )/(
02
0 <=ΨΨ
>=ΨΨ++
+−+
zee
Rrer
BinRzn
A
Binn
A
θ
θ
GRAPHENE WORMHOLESGRAPHENE WORMHOLES
⎟⎟⎠
⎞⎜⎜⎝
⎛
ΨΨ
=⎟⎟⎠
⎞⎜⎜⎝
⎛
ΨΨ
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+Φ
±∂+∂
+Φ
∂−∂±<
±<
±<
±<
,
,
,
, 0
21
21
21
210
B
A
B
A
r
r
rrir
rriri ε
π
π
θ
θ m
In the lower branch of this geometry, we have the same In the lower branch of this geometry, we have the same DiracDirac
spinorsspinors
that we had before that we had before
We can also speculate about the possibility of We can also speculate about the possibility of forming forming graphenegraphene
wormholes connecting two wormholes connecting two graphenegraphene
layers. These can be considered as the layers. These can be considered as the addition of two addition of two nanotubenanotube‐‐graphenegraphene
junctions, junctions, doubling the number of heptagonal defects. doubling the number of heptagonal defects.
But, in order to match them with the But, in order to match them with the spinorsspinors
in the upper branch, we have to invert therein the upper branch, we have to invert therethe direction of the the direction of the azimuthalazimuthal
angle angle
⎟⎟⎠
⎞⎜⎜⎝
⎛
ΨΨ
=⎟⎟⎠
⎞⎜⎜⎝
⎛
ΨΨ
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+Φ
±∂−∂
+Φ
∂+∂±>
±>
±>
±>
,
,
,
, 0
21
21
21
210
B
A
B
A
r
r
rrir
rriri ε
π
π
θ
θ m
GRAPHENE WORMHOLESGRAPHENE WORMHOLES
We can analyze the local DOS at the junctions, for different angWe can analyze the local DOS at the junctions, for different angular ular momentamomenta. In the case . In the case of a short (54,0) of a short (54,0) nanotubenanotube
bridge between two bridge between two graphenegraphene
layers:layers:
The peaks around the Fermi level can be explained in terms of loThe peaks around the Fermi level can be explained in terms of localized zero modes:calized zero modes:
1 =q 3/2 πieq ±= 1 -q =3/ πieq ±=
0 , ~ 0 21,
2,, =ΨΨ⇒=Ψ⎟
⎠⎞
⎜⎝⎛ +∂+∂ +
<−+
<+< B
innAAr er
rir
θθ
0 , ~ 0 21,
2,, =ΨΨ⇒=Ψ⎟
⎠⎞
⎜⎝⎛ +∂−∂ +
>−−+
>+> B
innAAr er
rir
θθ
ThusThus
wewe
seesee
thatthat
thethe
onlyonly
localizedlocalized
statesstates
correspondcorrespond
toto
nn
= 0, +1, = 0, +1, ‐‐1. 1. TheThe
statestate
withwith
nn
= 0 = 0 turnsturnsout out toto
be a be a normalizablenormalizable
zerozero
modemode, , whilewhile
thosethose
withwith
nn
= = ±1 are quasi±1 are quasi‐‐bound states, with a bound states, with a norm that diverges logarithmically with the size of the layer.norm that diverges logarithmically with the size of the layer.
To summarize, To summarize,
GrapheneGraphene
seems a quite exciting material from the experimental as well aseems a quite exciting material from the experimental as well as from the s from the theoretical point of view, with many aspects largely unexploredtheoretical point of view, with many aspects largely unexplored
graphenegraphene
has a natural tendency to develop has a natural tendency to develop
ripples, pointing at an intrinsic instability of ripples, pointing at an intrinsic instability of the flat surface that comes possibly from the the flat surface that comes possibly from the
interaction of the electrons with the fluctuations interaction of the electrons with the fluctuations
in the curvature of the carbon sheetin the curvature of the carbon sheet
regarding fundamental physics, regarding fundamental physics, graphenegraphene
provides the possibility of studying the provides the possibility of studying the interaction of strong curvature of the space interaction of strong curvature of the space with relativistic fields, allowing to study with relativistic fields, allowing to study effects that would be the analogue of the effects that would be the analogue of the
interaction with event horizons (creation of interaction with event horizons (creation of particles, Hawking radiation)particles, Hawking radiation)