electronic properties of graphene · electronic properties of graphene ( 1/2) 4. 2. n = + h e σ....

ELECTRONIC PROPERTIES OF GRAPHENEELECTRONIC PROPERTIES OF GRAPHENE

J. GonzJ. GonzáálezlezInstituto de Estructura de la Materia, CSIC, Instituto de Estructura de la Materia, CSIC, SpainSpain

1985 1991 2004


GrapheneGraphene

has opened the way to understandhas opened the way to understandthe behavior of electrons in D = 2the behavior of electrons in D = 2

remarkable properties are observed from the remarkable properties are observed from the theoretical point of viewtheoretical point of viewit has sparked great expectations of reaching it has sparked great expectations of reaching very large very large mobilitiesmobilities

But But …… some challenges have to be faced:some challenges have to be faced:

samples have significant corrugationsamples have significant corrugationthe interaction with the substrate and the interaction with the substrate and boundary conditions modify significantlyboundary conditions modify significantlythe transport properties the transport properties

From E. From E. StolyarovaStolyarova

et al.et al., ,

Proc. Natl. Acad. Proc. Natl. Acad. SciSci. 104, 9209 (2007). 104, 9209 (2007)

From E. From E. StolyarovaStolyarova

et al.et al., ,

Proc. Natl. Acad. Proc. Natl. Acad. SciSci. 104, 9209 (2007). 104, 9209 (2007)


)2/1( 4 2

+= Nhe

xyσ

The first experimental observations and measurement of unusual The first experimental observations and measurement of unusual transport properties pointed at the existence of a conical dispetransport properties pointed at the existence of a conical dispersion rsion of of quasiparticlesquasiparticles

in in graphenegraphene

the electric field effect shows that a substantial the electric field effect shows that a substantial concentration of electrons (holes) can be inducedconcentration of electrons (holes) can be inducedby changes in the gate voltageby changes in the gate voltage

the response to a magnetic field is also the response to a magnetic field is also unusual, as observed in particular in the unusual, as observed in particular in the quantum Hall effectquantum Hall effect

gVn ∝From K. S. From K. S. NovoselovNovoselov

et al.et al.,,Nature 438, 197 (2005)Nature 438, 197 (2005)

From K. S. From K. S. NovoselovNovoselov

et al.et al.,,Nature 438, 197 (2005)Nature 438, 197 (2005)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

∑∑

⋅−

⋅

0

0

a

via

vi

a

a

e

etH k

k

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=

00

yx

yxF ikk

ikkvH

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


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

εεε )( , )( ∝±= nvF kk

∑′

+ ′−=rr

tb tH,

)( )( rr ψψ

)2/3cos()2/cos(4)2/(cos41 2xyy akakaktE ++±=

A direct evidence of the conical dispersion has been obtained wiA direct evidence of the conical dispersion has been obtained with angleth angle‐‐resolvedresolvedphotoemission spectroscopyphotoemission spectroscopy


These experiments are also useful to provide a measure of the inThese experiments are also useful to provide a measure of the interaction effects in teraction effects in graphenegraphene

A. A. BostwickBostwick

et al.et al., Nature Phys. 3, 36 (2007), Nature Phys. 3, 36 (2007)

A. A. BostwickBostwick

et al.et al., Nature Phys. 3, 36 (2007), Nature Phys. 3, 36 (2007)

) ( Akσkσ ce

FF vvH −⋅→⋅=

∑ ∫′

′+ ⋅′−=rr

r

rtb ceitH,

)( ))/(exp( )( rdlAr ψψ

The peculiar quantization of the Hall conductivity can be explaiThe peculiar quantization of the Hall conductivity can be explainednedsatisfactorily by the coupling of the satisfactorily by the coupling of the DiracDirac

fermions to gauge fields:fermions to gauge fields:

This leads to Landau levels quantized This leads to Landau levels quantized according to the expressionaccording to the expression

which corresponds to the gauge prescriptionwhich corresponds to the gauge prescription

BNevNE FN22 )sgn( =

QUANTUM HALL EFFECT IN GRAPHENE QUANTUM HALL EFFECT IN GRAPHENE

TheThe

quantum quantum HallHall

effecteffect

isis

actuallyactually

quite quite robustrobust

andand

shouldshould

persistpersist

eveneven

in in thethe

presencepresence

ofof

curvaturecurvature

ofof

thethe

samplessamples. In . In thethe

case case ofof

thethe

shellsshells

ofof

MWNTsMWNTs

withwith

radiusradius

R R = 20 = 20 nmnm, , forfor

instanceinstance, ,

wewe

can can predictpredict

thethe

sequencesequence

ofof

bandband

structuresstructures

forfor

magneticmagnetic

fieldfield

strengthstrength

BB

= 0, 5, 10, 20 T := 0, 5, 10, 20 T :

S. S. BellucciBellucci, J. G., F. Guinea, P. , J. G., F. Guinea, P. OnoratoOnorato

and E. and E. PerfettoPerfetto,,

J. Phys.: J. Phys.: CondensCondens. Matter . Matter 1919, 395017 (2007), 395017 (2007)

QUANTUM HALL EFFECT IN GRAPHENE QUANTUM HALL EFFECT IN GRAPHENE

ELECTRONIC TRANSPORT IN GRAPHENEELECTRONIC TRANSPORT IN GRAPHENE

However, there are open questions in the case of the conductivitHowever, there are open questions in the case of the conductivity. This is usuallyy. This is usuallycomputed in computed in BoltzmannBoltzmann

transport theory in terms of the density of states transport theory in terms of the density of states D D as as

s

1 )(

1 )(

1σμεεσ

+=lFF ne

)( )( )( 22FFFF Dve ετεεσ =

This argument seems to be also in agreementThis argument seems to be also in agreementwith recent experiments where the influence of with recent experiments where the influence of the background dielectric constant is shown.the background dielectric constant is shown.The best fit to the conductivity is given by The best fit to the conductivity is given by

The linear dependence observed experimentally on gate voltage imThe linear dependence observed experimentally on gate voltage implies that plies that σσ should beshould beproportional to the electron density. proportional to the electron density.

in the case of shortin the case of short‐‐range range scatterersscatterers, , 1/1/k k , implies that , implies that σσ = const.= const.in the case of charged longin the case of charged long‐‐range range scatterersscatterers, , k k , implies that , implies that σσ εε ²²

From C. Jang From C. Jang et al.et al., Phys. Rev. , Phys. Rev. LettLett. 101, 146805 (2008) . 101, 146805 (2008)

JOSEPHSON EFFECT IN GRAPHENEJOSEPHSON EFFECT IN GRAPHENEThere have been also observations of There have been also observations of supercurrentssupercurrents, when , when graphenegraphene

is contacted is contacted with superconducting electrodeswith superconducting electrodes

FromFrom

H. B. H. B. HeerscheHeersche

et al.et al., ,

NatureNature

446446, 56 (2007), 56 (2007)

The reason why The reason why supercurrentssupercurrents

may exist at the may exist at the DiracDirac

point is that Cooper pairs have apoint is that Cooper pairs have anonvanishingnonvanishing

propagation even at vanishing charge density propagation even at vanishing charge density

22220

)0(

81),( ωω −−=

=kk F

FT

vv

D From J. G. and E. From J. G. and E. PerfettoPerfetto,,Phys. Rev. B 76, 155404 (2007) Phys. Rev. B 76, 155404 (2007)

⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

0 0

φ

φ

i

i

F ee

vHk

k


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 The scattering by impurities is quite unconventinalunconventinal

in 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


Another way of explaining the Another way of explaining the suppresionsuppresion

of backscattering 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))

)(),(

ωωω

Γ+⋅−=

iZvZ

GvF kσ

k

)/log(1)( 2 ωω Λ−≈ gZ

MANYMANY‐‐BODY EFFECTS IN GRAPHENEBODY EFFECTS IN GRAPHENEThe electron propagator is strongly The electron propagator is strongly renormalized at low energies:renormalized at low energies:

The effective coupling flows to zerThe effective coupling flows to zero at low o at low energies due to the renormalization of the Fermi velocity. energies due to the renormalization of the Fermi velocity. The The quasiparticlequasiparticle

weight weight

ZZ

is renormalized asis renormalized as

Fveg 16/ 2≡

but stays finite at low energies due to the vanishing of but stays finite at low energies due to the vanishing of g .g .

Is there room for new phases in Is there room for new phases in graphenegraphene? ?

A Hubbard interaction may be added A Hubbard interaction may be added on top of the longon top of the long‐‐range interaction: range interaction:

though it gives rise to irrelevant couplings though it gives rise to irrelevant couplings

)/1( 4 2 NOgggxx

x +−−=Λ∂

∂Λ

New critical points may arise then at strongNew critical points may arise then at strongcoupling coupling

I. F. I. F. HerbutHerbut, Phys. Rev. , Phys. Rev. LettLett. . 9797, 146401 (2006), 146401 (2006)


In the doped system, the decay of In the doped system, the decay of quasiparticlesquasiparticles

is possible due to is possible due to intrabandintraband

electronelectron‐‐hole hole excitations:excitations:

The imaginary part of the selfThe imaginary part of the self‐‐energy is energy is )(log

),( Im 2

22

ωωωω ∝⎟⎟

⎠

⎞⎜⎜⎝

⎛∝Σ

Fvek

(J. G., F. Guinea and (J. G., F. Guinea and M. A. H. M. A. H. VozmedianoVozmediano, , Phys. Rev. Phys. Rev. LettLett. . 7777, 3589 (1996)), 3589 (1996))

However, the dependence of the QP decay is subtle, as reflected However, the dependence of the QP decay is subtle, as reflected in the singular behavior in the singular behavior

kkk FF vv ),( Im ∝+Σ ε

0 ),( Im =−Σ εkk Fv

The QP decay rate is now: The QP decay rate is now:

( )FF kkkk −−∝− log )( 21τ

E. H. Hwang, Ben Yu‐Kuang

Hu, and S. Das Sarma, Phys. Rev. B Phys. Rev. B 7676, 115434 (2007), 115434 (2007)

LOWLOW‐‐ENERGY ELECTRONIC PROPERTIESENERGY ELECTRONIC PROPERTIES

For comparison, we may derive the For comparison, we may derive the quasiparticlequasiparticle

decay rate in the case of a screeneddecay rate in the case of a screenedCoulomb interaction:Coulomb interaction:

In the limit , we have a finite In the limit , we have a finite quasiparticlequasiparticle

decay rate, with two different decay rate, with two different regimes regimes

0→δ

222

2

220

4

22

)(22

1

1

),( )()(

)( Im

),( Im

qq

qq

qqk

qkγ

kk

qq

q

q q

k

F

qqk

aFqk

q

F

vddqe

Vi

vdqdie

v

−Ω+Ω∂∂

Ω∝

−−+−−−⋅+−

∝

Σ−=

−

+

−

∫∫

∫∫

l

φ

ωεωω

ωωω

τ

δ

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

<⎟⎟⎠

⎞⎜⎜⎝

⎛

>⎟⎟⎠

⎞⎜⎜⎝

⎛

∝

−

−

−

13222

122

1

ll

l

kk

kk

FF

FF

vve

vve

τ

(J. G., F. Guinea and M. A. H. (J. G., F. Guinea and M. A. H. VozmedianoVozmediano, , Phys. Rev. Phys. Rev. LettLett. . 7777, 3589 (1996), also consistent, 3589 (1996), also consistentwith E. H. Hwang, B. Y.with E. H. Hwang, B. Y.‐‐K. K. HuHu, and S. , and S. DasDas

SarmaSarma, Phys. Rev. B , Phys. Rev. B 7676, 115434 (2007)) , 115434 (2007))


We now turn to phonons as the relevant source of scattering at We now turn to phonons as the relevant source of scattering at low carrier densities. At sufficiently large energy/temperature low carrier densities. At sufficiently large energy/temperature we we have the contribution of optical phonons. The QP decay rate is have the contribution of optical phonons. The QP decay rate is

),( )()(

)( Im

),( Im

2222

1

qqk

Fqkq

F

Di

vdqdig

v

ωεωω

ωωω

τ

qqkqkσ

kk

−−+−−−⋅+−

∝

Σ−=

∫∫

−

This gives rise to a decay rate linearly proportional to the QP This gives rise to a decay rate linearly proportional to the QP energy, above the phonon energy. energy, above the phonon energy.

Using Using BoltzmannBoltzmann

transport theory, we obtain a transport theory, we obtain a resistivityresistivity

that does not depend on carrier density that does not depend on carrier density and is and is lineralylineraly

proportional to temperatureproportional to temperature

)( )( )( 22FFFF Dve ετεεσ =

J. G., E. J. G., E. PerfettoPerfetto, Phys. Rev. , Phys. Rev. LettLett. . 101101, 176802 (2008), 176802 (2008)

E. H. Hwang and S. Das

Sarma,Phys. Rev. B 77, 115449 (2008)


There is also interesting physics below the scale of the outThere is also interesting physics below the scale of the out‐‐ofof‐‐plane phonons. These couple to theplane phonons. These couple to theelectron charge and have therefore a strong hybridization with eelectron charge and have therefore a strong hybridization with electronlectron‐‐hole pairs. In the RPA,hole pairs. In the RPA,

We observe the appearance of very soft phonon modes We observe the appearance of very soft phonon modes near the K point of near the K point of graphenegraphene, which are right below the , which are right below the particleparticle‐‐hole continuumhole continuum

222

222

022

02

0

8/2/2

2 ),(

ωωωεωω

ωω

−

−−+−

≈

qq

q

F

F

vvgi

D

The hybrid states give rise to a cubic dependence of the The hybrid states give rise to a cubic dependence of the QP decay rate on energy, that competes with the lowerQP decay rate on energy, that competes with the lowerbound given at very low energies by the decay into bound given at very low energies by the decay into acoustic phononsacoustic phonons

J. G., E. J. G., E. PerfettoPerfetto, Phys. Rev. , Phys. Rev. LettLett. . 101101, 176802 (2008), 176802 (2008)

LOWLOW‐‐ENERGY ELECTRONIC PROPERTIESENERGY ELECTRONIC PROPERTIES

The existence of soft phonon modes changes significantly the lowThe existence of soft phonon modes changes significantly the low‐‐energy electronicenergy electronicproperties. We have for instance the properties. We have for instance the quasiparticlequasiparticle

decay ratedecay rate

We have now quite different behaviors depending on the energy raWe have now quite different behaviors depending on the energy range: nge:

8

/2/2

),(222

2222

0

20

2

ωω

ωωωω

−

−−

−=

qqq

F

F

vvgQ( )),(

),( )()(

)( Im

),( Im

00

2

22

)(22

1

qq

q

q

kqq

qqk

qkγ

kk

ΩΩ∂∂

Ω∝

−−+−−−⋅+−

∝

Σ−=

∫∫

∫∫

−

Qddqg

Di

vdqdig

v

qqk

aFqk

q

F

δφ

ωεωω

ωωω

τ

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

<⎟⎟⎠

⎞⎜⎜⎝

⎛

>⎟⎟⎠

⎞⎜⎜⎝

⎛

∝−

020

332

2

2

02

2

1

ωω

ω

τ

kk

kk

FF

F

FFF

vv

vg

vvvg

, where, where

(consistent with C.(consistent with C.‐‐H. Park H. Park et al.et al., , Phys. Rev. Phys. Rev. LettLett. . 9999, 086804 (2007) ), 086804 (2007) )

J. G. and E. J. G. and E. PerfettoPerfetto, , Phys. Rev. Phys. Rev. LettLett. . 101101, 176802 (2008), 176802 (2008)

RESISTIVITY AND MOBILITY IN GRAPHENERESISTIVITY AND MOBILITY IN GRAPHENEThe theoretical results have to matched with the experimental meThe theoretical results have to matched with the experimental measures of the asures of the resistivityresistivity..

It is assumed that the It is assumed that the resistivityresistivity

has a has a TT‐‐independentindependentcontribution from impurities, another from acousticcontribution from impurities, another from acousticphonon scattering, and some extra contribution givingphonon scattering, and some extra contribution givingthe nonlinear behaviorthe nonlinear behavior

),( )( )( ),( nlacimp TVTVTV ggg ρρρρ ++=

The hope is to be able to remove the contribution The hope is to be able to remove the contribution from impurities to remain with the intrinsic source from impurities to remain with the intrinsic source of of resistivityresistivity

(phonons), in which case the mobility (phonons), in which case the mobility would diverge at low carrier density aswould diverge at low carrier density as

J.‐H. Chen et al., Nature Nanotech. 3, 206 (2008)

J.‐H. Chen et al., Nature Nanotech. 3, 206 (2008)

ρμ

ne1 =

To conclude, To conclude,

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 viewtheoretical point of view

from the theoretical point of view, the question of the minimum from the theoretical point of view, the question of the minimum conductivity is still conductivity is still a matter of debatea matter of debate

from a practical point of view, one has to be able to tailor thefrom a practical point of view, one has to be able to tailor the

graphenegraphene

structure atstructure atthe the nanoscalenanoscale, as well as to suppress the extrinsic scattering mechanisms tha, as well as to suppress the extrinsic scattering mechanisms that are thet are themain source of resistivity at presentmain source of resistivity at present

electronic properties of graphene · electronic properties of graphene ( 1/2) 4. 2. n = + h e σ....

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