Drude weight, plasmon dispersion, and a.c. conductivity in doped graphene sheets

2.4 MB·English

by  Saeed H. Abedinpour

Drude weight, plasmon dispersion, and a.c. conductivity in doped graphene sheets
Saeed H. Abedinpour and G. Vignale
∗
Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
A. Principi and Marco Polini
†
NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy
Wang-Kong Tse and A.H. MacDonald
Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA
1 We demonstrate that the plasmon frequency and Drude weight of the electron liquid in a
1 doped graphene sheet are strongly renormalized by electron-electron interactions even in the long-
0 wavelength limit. This effect is not captured by the Random Phase Approximation (RPA), com-
2 monly used to describe electron fluids and is due to coupling between the center of mass motion
andthepseudospindegreeoffreedomofthegraphene’smasslessDiracfermions. Makinguseofdia-
n
grammaticperturbationtheorytofirstorderintheelectron-electroninteraction,weshowthatthis
a
J couplingenhancesboththeplasmonfrequencyandtheDrudeweightrelativetotheRPAvalue. We
alsoshowthatinteractionsareresponsibleforasignificantenhancementoftheopticalconductivity
2
at frequencies just above the absorption threshold. Our predictions can be checked by far-infrared
2
spectroscopy or inelastic light scattering.

]
l
l
a I. INTRODUCTION of an individual electron. The exact plasmon frequency
h expression is correctly captured by the RPA2,3,6, but
- also by rigorous arguments7 in which the selection of
s
The first theory of classical collective electron density
e a particular center-of-mass position breaks the system’s
m oscillations in ionized gases by Tonks and Langmuir1 in Galileaninvarianceandplasmonexcitationsplaytherole
the 1920’s helped initiate the field of plasma physics.
. of Goldstone bosons. In two-dimensional (2D) systems
t The theory of collective electron density oscillations in (cid:112)
a V =2πe2/q so that ω (q →0)= 2πne2q/m, where e
m metals, quantum in this case because of higher electron q pl
is the magnitude of the electron charge.
densities, was developed by Bohm and Pines2,3 in the
-
d 1950’s and stands as a similarly pioneering contribution Electrons in a solid, unlike electrons in a plasma or
n to many-electron physics. Bohm and Pines coined the electrons with a jellium model6 background, experience
o term plasmon to describe quantized density oscillations.
a periodic external potential created by the ions which
c Today plasmonics is a very active subfield of optoelec-
[ breaks translational invariance and hence also Galilean
tronics4,5, whose aim is to exploit plasmon properties in
invariance. Solid state effects can lead in general to a
1 order to compress infrared electromagnetic waves to the
renormalization of the plasmon frequency, or even to the
v nanometer scale of modern electronic devices. This wide
absence of sharp plasmonic excitations. In semiconduc-
1
importanceofplasmonsacrossdifferentfieldsofbasicand
9 tors and semimetals, however, electron waves can be de-
2 appliedphysicsfollowsfromtheubiquityofchargedpar- scribed at super-atomic length scales using k·p theory8,
ticlesandfromthestrengthoftheirlong-rangeCoulomb
4 which is based on an expansion of the crystal’s Bloch
. interactions. Hamiltonianaroundbandextrema. Inthesimplestcase,
1
0 The physical origin of plasmons is very simple. When for example for the conduction band of common cubic
1 electrons in free space move to screen a charge inho- semiconductors,thisleadsusbacktoaGalilean-invariant
1 mogeneity, they tend to overshoot the mark. They are parabolic band continuum model with isolated electron
v: thenpulledbacktowardthechargedisturbanceandover- energy Ec(p) = p2/(2mb). The crystal background for
i shootagain,settingupaweaklydampedoscillation. The electron waves appears only via the replacement of the
X
restoring force responsible for the oscillation is the av- bareelectronmassbyaneffectivebandmassm . Inthis
b
r erage self-consistent field created by all the electrons. typeofk·pGalilean-invariantinteractingelectronmodel,
a
Because of the long-range nature of the Coulomb in- validformanysemiconductorandsemiconductorhetero-
teraction, the frequency of oscillations ω (q) tends to junction systems, the plasmon dispersion is accurately
pl
be high and is given in the long wavelength limit by givenbytherandomphaseapproximation(RPA)2,3,6and
ω2(q → 0) = nq2V /m where n is the electron den- itisgivenbytheclassicalformulaquotedabovewiththe
pl q
sity, m is the bare electron mass in vacuum, and V is replacements m → m and e2 → e2/(cid:15), (cid:15) being the high-
q b
the Fourier transform of the Coulomb interaction. This frequencydielectricconstantofthesemiconductormate-
simple explicit plasmon energy expression is exact be- rial. The absence of electron-electron interaction correc-
cause long-wavelength plasmons involve rigid motion of tions to plasmon frequencies at very long wavelengths in
the entire plasma, which does not involve the complex thesesystems, hasbeendemonstratedexperimentallyby
exchange and correlation effects that dress6 the motion means of inelastic light scattering9,10.
2
The situation turns out to be quite different in
graphene – a monolayer of carbon atoms tightly packed
in a 2D honeycomb lattice11–14. When k ·p theory is
applied to graphene it leads to a new type of electron
fluid model, one with separate Dirac-Weyl Hamiltonians
for electron waves centered in momentum space on one
of two honeycomb lattice Brillouin-zone corners K and
K :
(cid:48)
(cid:88)
HˆD =(cid:126)v ψˆk†,α(σαβ ·k)ψˆk,β . (1)
k,α,β
Here v is the bare electron velocity, k is the k · p
momentum, α,β are sublattice pseudospin labels, and
σ = (σx ,σy ) is a vector of Pauli matrices which
αβ αβ αβ
act on the sublattice pseudospin degree-of-freedom. It
follows that the energy eigenstates for a given p have
pseudospins oriented either parallel (upper band) or an-
tiparallel (lower band) to p. Physically, the orientation
of the pseudospin determines the relative amplitude and
the relative phase of electron waves on the two distinct FIG.1: BreakdownofGalileaninvarianceingraphene. Panel
graphene sublattices. 1a) shows the occupied electronic states in the upper band
of graphene in the ground state. Notice that every state is
Electron-electron interactions in graphene are de-
characterized by a value of momentum (the origin of the ar-
scribed by the usual non-relativistic Coulomb Hamilto-
row) and a pseudospin orientation (the direction of the ar-
nian15
row). Panel 1b) shows the occupied states after a Galilean
1 (cid:88) boost. Anobserverridingalongwiththeboostwouldclearly
HˆC = Vq ρˆqρˆ q , (2) see that the orientation of the pseudospins has changed. It
2S q=0 − lookslikethepseudospinsaresubjectedtoa“pseudomagnetic
(cid:54) field”thatcausesthemtotilttowardsthe+xˆ direction. The
where S is the sample area, V = 2πe2/((cid:15)q) is the appearance of this pseudomagnetic field is the signature of
q
broken Galilean invariance. In contrast, in a Galilean invari-
2D Fourier transform of the Coulomb interaction ((cid:15) be-
ant system [Panels 1c) and 1d)] the energy eigenstates are
ing an effective average dielectric constant), and ρˆ =
q characterized by momentum only: an observer riding along
(cid:80)k,αψˆk† q,αψˆk,α is the usual density operator. Electron with the boost would not see any change in the character of
carriers −with density n can be induced in graphene by the occupied states.
purely electrostatic means, creating a circular 2D Fermi
surface in the conduction band with a Fermi radius k ,
√ F
which is proportional to n16. The model described by with an oscillatory motion of the pseudospins. Since ex-
Hˆ =HˆD+HˆC requires an ultraviolet wavevector cutoff, change interactions depend on the relative orientation of
kmax, which should be assigned a value corresponding to pseudospins they contribute to plasmon kinetic energy
thewavevectorrangeoverwhichHˆDdescribesgraphene’s and renormalize the plasmon frequency even at leading
π bands. This corresponds to taking kmax ∼1/a0 where order in q.
a0 ∼ 1.42 ˚A is the carbon-carbon distance. This model In this article we present a many-body theory of this
is useful when kmax is much larger than kF. subtle pseudospin coupling effect and discuss the main
The feature of graphene that is ultimately responsible implications of our findings for theories of charge trans-
forthelargemany-bodyeffectsontheplasmondispersion portandcollectiveexcitationsindopedgraphenesheets.
andtheDrudeweightisbroken Galilean invariance. The Our manuscript is organized as follows. In Sect. II we
lattice reference frame remains present in the continuum introduce the most important definitions and the basic
modelthroughthecouplingbetweenmomentaandpseu- linear-response functions that control the plasmon dis-
dospins. The oriented pseudospins provide an “ether” persion, the Drude weight, and the a.c. conductivity. In
against which a global boost of the momenta becomes Sect. III we present the approach we have used to cal-
detectable. This is explained in detail in the caption of culate the Drude weight and the a.c. conductivity, i.e.
Fig. 1. diagrammatic perturbation theory, and the main ana-
Fig.1explainswhytheplasmonfrequencyingraphene lytical results. Identical results can be obtained using a
is so strongly affected by exchange and correlation. In a kineticequationapproachwhichwillbedetailedinasep-
plasmon mode the region of occupied states (Fermi cir- aratepublication17 thatisfocusedonadifferentapplica-
cle) oscillates back and forth in momentum space un- tion. Ourmainnumericalresultsbasedonthisapproach
der the action of the self-induced electrostatic field. In are illustrated in Sect. IV. In Sect. V we emphasize how
graphene, this oscillatory motion is inevitably coupled density-density and current-current response functions
3
donotleadtothesameresultsfortheDrudeweightand We thus see from Eq. (6) that the Drude weight com-
a.c. conductivity due to the presence of a rigid cutoff in pletely controls the plasmon dispersion at long wave-
momentumspace. Finally,inSect.VIwesummarizeour lengths. When electron-electron interactions are ne-
findings and draw our main conclusions. Three Appen- glected D tends to the RPA Drude weight D = 4ε σ ,
0 F 0
dices(A-C)highlightsomeimportanttechnicalaspectsof where σ = e2/(4(cid:126)) is the so-called universal19–23
0
the diagrammatic calculation, while Appendix D reports frequency-independent inter-band conductivity of a neu-
on the generalized form of the continuity equation that tral graphene sheet. In the same limit m →(cid:126)k /v and
pl F
applies in the presence of a rigid momentum cutoff, and
ε2 gα q
Appendix E reports numerical results for Thomas-Fermi ω2(q →0)= F ee , (8)
screened interactions. pl (cid:126)2 2 kF
where g = g g = 4 is a spin-valley degeneracy factor
s v
and we have introduced the dimensionless fine-structure
II. FORMULATION coupling constant α = e2/((cid:15)(cid:126)v), the ratio between the
ee
Coulombenergyscalee2k /(cid:15)andthekineticenergyscale
F
The collective (plasmon) modes of the system de- (cid:126)vk . (The fine-structure constant can be tuned exper-
F
scribed by Hamiltonian Hˆ can be found by solving the imentally by changing the dielectric environment sur-
following equation6, rounding the graphene flake24,25.) Eq. (8) is the well-
knownRPA26–29resultfortheplasmondispersionatlong
1−Vqχ(cid:101)ρρ(q,ω)=0 , (3) wavelengths.
In the following Section we calculate D exactly to
where χ(cid:101)ρρ(q,ω) is the so-called proper18 density-density first order in the fine-structure constant αee by means
response function. In the q →0 limit of interest here we of diagrammatic perturbation theory, demonstrating in
can neglect the distinction between the proper and the the process that its value is substantially enhanced by
full causal response function electron-electroninteractions. Ourresultsdependonthe
electron density via the ultraviolet cutoff Λ = k /k .
χ (q,ω) max F
χ (q,ω)= (cid:101)ρρ . (4) Thus, the momentum sums that appear in the evalua-
ρρ 1−V χ (q,ω)
q(cid:101)ρρ tionofthediagramswillberestrictedinsuchawaythat
only single-particle states with wave vectors k ≤ k
We show below that max
are involved. The value of Λ varies from ∼20 for a very
D q2 high-density graphene system with n ∼ 1013 cm−2 to
ωlim0qlim0(cid:60)e χρρ(q,ω)= πe2 ω2 , (5) ∼100 for a density n∼5×1011 cm−2 just large enough
→ → to screen out the unintended30 inhomogeneities present
where D is a as yet unidentified density- and coupling- insamplesonsubstrates. Wewillseethatourresultsare
constant-dependent quantity. Note the order of limits in only weakly dependent on Λ. If this were not true the
Eq.(5): thelimitω →0istakeninthedynamicalsense, Dirac model for electron-electron interactions in doped
i.e. vq (cid:28) ω (cid:28) 2εF. Here εF = vkF is the Fermi energy graphene would not be useful and it would be neces-
and 2εF is the threshold for vertical inter-band electron- sarytocorrectlyaccountforinteractioneffectsatenergy
hole excitations. Using Eq. (5) in Eq. (3) and solving for scales beyond those for which the model is valid. The
ω we find that, to leading order in q, cutoff appears only in the well-known electron-electron
interaction enhancement of the quasiparticle velocity.
(cid:115)
2πe2n
ω (q →0)= q , (6)
pl
(cid:15)m
pl
III. DIAGRAMMATIC PERTURBATION
THEORY
where we have introduced the “plasmon mass”, m =
pl
πe2n/D.
In the dynamical limit, the imaginary-part of the a.c. In Fig. 2 we show the diagrams that contribute to
conductivity, σ(ω) = ie2ωlimq 0χρρ(q,ω)/q2, has the the density-density response function χρρ(q,ω) up to
form → first order in the coupling constant; the bare bubble
diagram, two self-energy-correction diagrams, and one
D
(cid:61)m σ(ω)→ . (7) vertex-correction diagram. As usual, partial cancella-
πω tionsbetweentheself-energyandvertexcorrectionsplay
an essential role. In Fig. 2 solid lines are noninteracting
ItthenfollowsfromastandardKramers-Kr¨oniganalysis
Green’s functions31,
that the real-part of the conductivity has a δ-function
peak at ω = 0: (cid:60)e σ(ω) = Dδ(ω). Thus the quantity G(k,ω)= 1 (cid:88) 11σ+µσk , (9)
D introduced in Eq. (5) is the Drude weight. In the 2 ω−ξ +iη
k,µ k,µ
µ= 1
presenceofdisordertheδ-functionpeakisbroadenedinto ±
aDrudepeak,buttheDrudeweightispreservedforweak where 11 is the identity matrix in pseudospin space,
σ
disorder. σ =σ·k/k, ξ =µvk−ε , and η =η sgn(−ξ )
k k,µ F k,µ k,µ
4
and
a) b)
π q2
ρˆq ρˆ−q ρˆq ρˆ−q (cid:61)m χ(ρVρ)(q,ω)=−16αeeν(εF)ω
×[Θ(ω−1)J (ω,Λ)+{ω →−ω}] ,
V
c) d) (13)
with
ρˆq ρˆ q ρˆq ρˆ q
− −
V (ω,1)−V (ω,1)
2 0
J (ω,Λ) =
V
2
ω (cid:90) Λ F(ω,k)
+ P dk (14)
FIG.2: Feynmandiagramsfortheproperdensity-densityre- 2 1 k2−ω2
sponse function χ (q,ω) up to first order in the electron-
ρρ
electron interaction. Panel a) The bare bubble diagram. and F(ω,k) = 2kV (ω,k)+ω[V (ω,k)+V (ω,k)]. Here
1 0 2
Panel b) Vertex correction. Panels c) and d) Self-energy dia- V (k,k )aredimensionlessCoulombpseudopotentials32,
m (cid:48)
grams.
(cid:90) 2π dθ exp(−imθ)
V (k,k )= ,
m (cid:48) (cid:113)
(with η = 0+). Dashed lines are electron-electron inter- 0 2πq + k2+k 2−2kk cos(θ)
TF (cid:48) (cid:48)
actions.
(15)
All wave vectors, q, k, and k , which appear below
(cid:48) where q = 4α is the Thomas-Fermi screening wave
TF ee
are measured in units of k , while frequencies and en-
F vector (in units of k ). Making use of these formulas it
ergies are in units of 2ε . Below we set (cid:126) → 1. We F
F is easy to check that the vertex correction to the Drude
also introduce the density-of-states at the Fermi energy,
weight [i.e. Eq. (12)] is negative and that the integral in
ν(ε )=2ε /(πv2).
F F thesecondlineofEq.(14)convergesinthelimitΛ→∞.
The diagrams in Fig. 2 are first evaluated in the limit
The last two diagrams are “self-energy” corrections,
of small q (i.e. to order q2) and finite ω. Then, for the
(SE)
denoted by χ (q,ω), which physically describe the
realpartofχ wewillretainonlythetermsthatscaleas ρρ
ρρ
q2/ω2forω →0andthuscontributetotheDrudeweight, modification of the response function due to exchange
energy corrections to the quasiparticle dispersion. We
while for the imaginary part of χ , which controls the
ρρ
find that (see Appendix C for more details)
real part of the a.c. conductivity, we calculate the full
frequency-dependent function.
Proceedinginthismannerweseethattheemptybub- (cid:60)e χ(SE)(q,ω) = 1 α ν(ε )q2[V (1,1)+2V (1,1)
ble in Fig. 2 (i.e. the noninteracting diagram), de- ρρ 32 ee F ω2 0 1
notedbyχ(0)(q,ω),reproducesthenoninteractingDrude 1 q2
ρρ + V (1,1)]+ α ν(ε )
weight D , as expected (see Appendix A): 2 32 ee F ω2
0
(cid:90) Λ
(cid:60)e χ(0)(q,ω)= D0 q2 = 1ν(ε )q2 (10) × dk [V0(1,k)−V2(1,k)] (16)
ρρ 4πv2e2 ω2 8 F ω2 1
and and
(cid:61)m χ(ρ0ρ)(q,ω)=−1π6ν(εF)qω2 [Θ(ω−1)+{ω →−ω}] , (cid:61)m χ(ρSρE)(q,ω)=−1π6αeeν(εF)qω2
(11)
×[Θ(ω−1)J (ω,Λ)+{ω →−ω}] ,
where Θ(x) is the usual Heaviside step function and the SE
(17)
notation “{ω →−ω}” means that we have to add to the
first term in square brackets an identical term in which
ω is interchanged with −ω. where
The next diagram is the so-called “vertex correction”,
 1
denoted by χ(ρVρ)(q,ω), which physically represents the JSE(ω,Λ)= ωΣ(ω,Λ)−∂ωΣ(ω,Λ)
dressing of the external driving field by the internally . (18)
generated exchange field. We find that, up to order q2 Σ(ω,Λ)= 1(cid:90) Λdk kV1(ω,k)
and for ω →0 (see Appendix B for details), 2
1
(cid:60)e χ(V)(q,ω) = − 1 α ν(ε )q2[V (1,1)+2V (1,1) InwritingEq.(17)wehaveexcludedatermproportional
ρρ 32 ee F ω2 0 1 toδ(ω−1)whichisanartifactofperturbationtheoryas
+ V (1,1)] (12) we explain below.
2
5
IV. DRUDE WEIGHT RENORMALIZATION electron-electron interactions reduce the pseudospin sus-
AND A.C. CONDUCTIVITY ceptibility36 and therefore increase the pseudospin stiff-
ness,i.e. theenergythatisrequiredtoalignpseudospins
We now combine the terms calculated in the previous alongagivendirection. Accordingtothediscussiongiven
Section. Extensive cancellations occur between vertex intheIntroduction,thelargerpseudospinstiffnessresults
corrections and self-energy contributions. For example in higher energy of plasma oscillations. This observation
the first term on the right hand side of Eq. (16) cancels is consistent with the fact that in a Landau Fermi liquid
the vertex contribution (12). The final result for the description36 the suppression of the pseudospin suscep-
Drude weight to first order in α is tibility would be driven by the many-body enhancement
ee
of the density of states factor v(cid:63)/v, while interactions
D α (cid:90) Λ between the quasiparticles produce the opposite effect.
=1+ ee dk [V (1,k)−V (1,k)] . (19)
D 4 0 2 Once again we must conclude that the many-body en-
0 1
hancement of the plasmon frequency is intimately con-
The real part of the a.c. conductivity is given, to the nectedtothemany-bodyenhancementofthequasiparti-
same order, by cle velocity v(cid:63)/v. Any physical mechanism that reduces
v(cid:63)/v without affecting the interaction between quasipar-
(cid:60)e σ(ω) ticles could in principle result in a reduction of the plas-
=Θ(ω−1){1+α [J (ω,Λ)+J (ω,Λ)]} .
ee V SE
σ mon frequency and Drude weight.
0
(20) The Drude weight D and the real part of the a.c. con-
Eqs. (19)-(20) are the most important results of this ar- ductivityσ(ω)areplottedasfunctionsofdopingnandω
ticle. respectivelyinFigs.3and4. WeobservethattheDrude
weight (and hence the coefficient of q1/2 in the plasmon
dispersion relation) is substantially enhanced above the
A. Long-range interactions noninteracting value. As we showed above in Eqs. (21)-
(22),theenhancementgrowsslowly(logarithmically)asa
For unscreened Coulomb interactions [q = 0 in functionofΛ[itgrowslinearlywithΛforshort-rangein-
TF
Eq. (15)] V (1,k) decays as 1/k at large k (see, for ex- teractions – see Eq. (24) in Sect. IVB]. The a.c. conduc-
0
ample, Ref. 32): we thus find that tivity is likewise enhanced above the threshold ω =2εF.
As we have mentioned above, an unphysical term pro-
D v(cid:63) portional to δ(ω −1) has been omitted from Eq. (20).
= +βα , (21)
D v ee Thissingularcontributionisduetotheshiftoftheinter-
0
bandabsorptionthresholdfromthebarevalue2ε tothe
F
where dressed value 2ε(cid:63). In first-order perturbation theory the
F
v(cid:63) α absorptionshiftappearsasaδ(ω−1)contributiontothe
ee
=1+ ln(Λ) (22) integrand for frequency integrals.
v 4
Fig. 4 shows that for ω (cid:29) 2ε but ω (cid:28) 2Λε the
F F
is the well-known logarithmic velocity enhancement33 a.c. conductivity approaches the high-frequency univer-
(see also Refs. 32,34,35) and salvalueσ0. However,asω becomescomparableto2ΛεF
the conductivity decreases and, in fact, has an unphys-
1 (cid:90) Λ (cid:20) 1 (cid:21) ical logarithmic divergence at ω = 2ΛεF. This makes
β ≡ lim dk V (1,k)− −V (1,k)
0 2 perfect sense, since our model is only valid for energies
4Λ→∞ 1(cid:18) (cid:19) k that are much smaller that the cutoff energy 2ΛεF. As
1 1 2 ln4
= − + −4C + (cid:39)−0.017 , (23) long as this condition is met, the calculated spectrum
8 4π 3 4 is essentially independent of the cutoff. This is a very
satisfactory feature of the present calculation.
C (cid:39) 0.916 being Catalan’s constant. Notice that in
Eq. (23) we have taken the limit Λ → ∞ since the inte-
grand decays like 1/k2 for k → ∞ [β reaches its Λ = ∞
B. Short-range interactions
asymptotic value reported in Eq. (23) already at values
of Λ as small as ≈ 10]. For all dopings of experimental
It is instructive to examine how the results presented
relevance Λ (cid:29) 1: in this regime the doping dependence
in the main body of this Section change if the electron-
ofD islogarithmicandstemsfromthevelocityenhance-
electroninteractionisassumedtobeofultra-shortrange
ment factor (22). The enhancement or suppression of D
in space, i.e. V = v = const and all other moments V
with respect to D depends on the relative strength of 0 m
0 with m ≥ 1 are zero. The calculations can be carried
the two terms in Eq. (21), which have opposite sign. In
out in a completely analytical fashion with the following
the low-density Λ (cid:29) 1 regime the velocity enhancement
results:
completely dominates the “β-term” and D/D >1. The
0 (cid:12)
enhancement of D, or, equivalently of the plasmon fre- D (cid:12)(cid:12) =1+ αeev¯(Λ−1) , (24)
quency, can be understood qualitatively by noting that D (cid:12) 4
0 sr
6
where v¯=(cid:15)k v/(2πe2), and
F
(cid:12)
(cid:60)e σ(ω)(cid:12)
(cid:12) =Θ(ω−1)[1+αeeJsr(ω,Λ)] , (25)
σ (cid:12)
0 sr
where
(cid:26) (cid:20) (cid:21)(cid:27)
v¯ ω (ω+1)(Λ−ω)
J (ω,Λ)= −1+ ln . (26)
sr 2 2 (ω−1)(Λ+ω)
Note that J (ω,Λ) becomes independent of Λ for Λ →
sr
∞. Itiseasytoseethat,inthislimit,(cid:60)eσ(ω)approaches
the universal value for large ω, since J (ω,∞) goes to
sr
zero like v¯/(6ω2) for ω → ∞. If, on the other hand,
ω is allowed to tend to infinity before Λ then unphysi-
cal cutoff-related features appear, such as a logarithmic
divergence at ω =Λ.
Let us make a brief comment on the limit of zero
FIG.3: (Coloronline)TheratioD/D0 betweenthe interact- doping of our theory. In this limit we predict that the
ing value of the Drude weight D, calculated from Eq. (19), strength of the Drude peak vanishes as the absorption
and the RPA value D = 4ε σ is plotted as a function of
0 F 0 threshold at 2ε moves toward zero. Then for finite but
electrondensityn(inunitsof1012 cm−2)fordifferentvalues F
arbitrarily small ω the optical conductivity approaches
ofgraphene’sfine-structureconstantα . Thevalueα =0.9
ee ee the universal value σ , yielding a result that is indepen-
isbelievedtobeappropriateforgraphenedepositedonSiO , 0
2
dent of α .
the other side exposed to air. Note that D/D >1 and that ee
0
it depends weakly on carrier density.
V. DENSITY RESPONSE VERSUS CURRENT
RESPONSE
In many models of electronic systems, gauge invari-
ance and the continuity equation allow us to express the
a)
density-densityresponsefunctionintermsofthecurrent-
current response function. In the present model the re-
lation would take the form
b)
v2q2 vq
χ (q,ω)= χ (q,ω)+ (cid:104)[σˆ ,ρˆ ](cid:105) , (27)
ρρ ω2 σσ ω2 q −q
where σˆ is the longitudinal component (parallel to q)
q
of the pseudospin-density fluctuation and [σˆ ,ρˆ ] is an
q q
anomalous commutator37, reminiscent of the co−mmuta-
torofFourier-component-resolveddensityfluctuationsin
a 1D Luttinger liquid38. Because the current in the
Dirac model is proportional to the pseudospin density,
χ (q,ω)inEq.(27)isthecurrent-currentresponsefunc-
σσ
tion. Eq.(27)worksperfectlyatthenoninteractinglevel,
but fails when electron-electron interactions are taken
FIG. 4: (Color online) Panel a) Deviation of the real part of into account. Due to the cutoff in momentum space, it
the a.c. conductivity (cid:60)e σ(ω) from the noninteracting uni- turns out that the continuity equation,
versalvalueσ =e2/(4(cid:126)),asafunctionoffrequencyω/(2ε ),
0 F
calculatedfromEq.(20)forseveralvaluesofthefinestructure i∂ ρˆ =q·jˆ , (28)
t q q
constant α and for n = 1.5×1012 cm−2 (Λ = 50). Panel
ee
b) Same as in the main panel but for a fixed value of the with the current-density operator jˆ given by
finestructureconstant(α =0.9)andtwodifferentvaluesof q
ee
doping (corresponding to Λ = 50 and 100). Notice that the (cid:88)
dependenceoftheconductivityonthevalueofthecutoffΛis jˆq =v ψˆk† q,ασαβψˆk,β , (29)
almost invisible at low frequencies and becomes visible only k,α,β −
at frequencies several times ε .
F
is no longer satisfied in the interacting system (see Ap-
pendix D).
7
Similar conclusions have been reached by turbation theory, this effect is captured by the vertex
Mishchenko39 in the context of calculations of the correction. In graphene on the other hand, the velocity
a.c. conductivity of undoped graphene sheets. This topic enhancement occurs over a broad range of energies cen-
has indeed attracted considerable interest39–43 and has tered on the Dirac point, not the Fermi surface. Density
been at the center of a dispute.39,41 Mishchenko39, in response is influenced by enhanced velocities which in-
particular, was the first to clarify that the calculation creasetheenergycostofchangingtheelectrondensity. In
of σ(ω) based on the density-density response function ordinary two-dimensional electron gases the logarithmic
predicts a very small correction over the noninteracting velocity enhancement vanishes in any event once screen-
conductivity, which is also in agreement with the ing is accounted for. In graphene, on the other hand
experimental findings19,20, while methods based on the theenhancementcomesfrominteractionsatwavevectors
current-current response function41 or on the kinetic much larger than the Fermi energy at which only inter-
equation39 predict very different results. We have also band screening, which does not change the long range
found44 that a naive application of Eq. (27) to the inter- 1/r behavior, is relevant. Neither screening nor vertex
actingsystemproducesstronglycutoff-dependentresults corrections fully counter the enhanced Drude weight due
for both the Drude weight and the a.c. conductivity of to graphene’s Dirac point velocity enhancement.
the doped system. Moreover, it yields a qualitatively We note that a recent experiment45 has clearly estab-
different behavior of the Drude weight with respect to lished the presence of a strong Drude peak which devel-
thatfoundinthepresentarticle[seeEq.(19)andFig.3]. opsingrapheneasthecarrierdensityisincreased. These
The Drude weight calculated from the current-current authors conclude that the Drude weight is reduced com-
response is suppressed44 rather than enhanced. On the pared to the noninteracting electron theory, instead of
other hand, if Eq. (27) is corrected to take into account being increased as predicted by this theoretical analysis.
the modification of the continuity equation due to e-e It will be interesting to see whether or not this exper-
interactions, then the results of the present analysis are imental conclusion changes as sample quality improves
recovered. The conclusion is that it is generally safer and it becomes possible to more cleanly separate inter-
in an effective low-energy theory with a rigid cutoff band and Drude conductivity contributions over a wider
to work with the density-density response function regime of carrier density.
χ (q,ω) rather than with the current-current response
ρρ Before concluding we would like to mention that a
function, since the calculation of the former places less
new theoretical paper reporting a study of the effect
weightonhigh-energyintermediatestates,whicharenot
of electron-electron interactions on the conductivity of
properly described. This observation is consistent with dopedgraphenesteets46hasappearedrecently. (Werefer
the findings of other authors39. An alternative solution
the reader to Ref. 47 for studies of band-structure, dis-
is to use a modified Coulomb interaction39,43 such that
order, phonon, and strain effects.) A quantitative com-
the continuity equation (27) remains valid even in the
parison between our findings and theirs is not possible
interacting system.
since their study takes disorder into account while our
study is for clean graphene. However, we would like to
stress that the authors of this paper have treated inter-
VI. DISCUSSION AND CONCLUSIONS actions only at the vertex level neglecting self-energy ef-
fects. Eventhough itiswell knownthat this approxima-
Large Fermi velocity enhancements due to exchange tion is not “conserving”48 (e.g. breaks gauge-invariance)
interactions, like the velocity enhancement that occurs eveninastandardparabolic-bandelectronliquid,wecan
in graphene, are common in the theory of solids. Often adopt the same strategy in our calculations and artifi-
theroleofvelocityenhancementsarefullycancelledinre- cially “switch-off” self-energy contributions to (cid:60)e σ(ω)
sponse functions by vertex corrections. For example the [diagrams in panels c) and d) in Fig. 2]. If this proce-
Fermi velocity of an ordinary two-dimensional electron dure is followed, we find the results shown in Fig. 5: it is
gas diverges when screening is neglected, but the corre- evidentfromthispointthattheneglectofself-energyin-
sponding reduction in density-density response is absent sertionsisresponsibleforthereductionof(cid:60)eσ(ω)below
when vertex corrections like those which appear in this σ0 at large frequencies found in Ref. 46.
paper are included. From this point of view the main Finally, let us comment on the broader implications of
finding of this paper, supported by an explicit first order our results. Effects similar to those described in this ar-
perturbationtheorycalculation,isthatnosuchcancella- ticle occur in graphene bilayers49 and are also expected
tion occurs for graphene’s well-known velocity enhance- in other few-layer systems. The lack of Galilean invari-
ment. Thedifferenceiseasytounderstand. Inthecaseof ancealsoaffectsthecyclotronresonancefrequencywhen
anordinarytwo-dimensionalelectrongasthevelocityen- the 2D sheet of graphene is placed in a perpendicular
hancementisduetotherapidvariationoftheunscreened magnetic field50–55 since Kohn’s theorem56, which as-
exchange self-energy as one goes from occupied states to serts the absence of many-body effects in cyclotron res-
emptystatesacrosstheFermisurface. Whenthedensity onance, is not applicable in this case. (The impact of
changes, the energy at which the velocity peak occurs broken Galilean invariance on the collective cyclotron
also changes, negating its influence on response. In per- motion in graphene has been studied in Ref. 57 in the
8
Appendix A: Calculation of the noninteracting
contribution to the dynamical density-density
response function
The noninteracting response function χ(0)(q,ω), i.e.
ρρ
the empty bubble diagram in Fig. 2a), reads
χ(0)(q,ω) = 1 (cid:88)(cid:90) +∞ d(cid:15) Tr[G(k ,(cid:15))G(k ,(cid:15)+ω)]
ρρ S 2πi − +
k −∞
= 2(cid:90) d2k (cid:88) nk−,µ−nk+,ν
(2π)2 ω+ξ −ξ +iη
µ,ν k−,µ k+,ν
× F (k ,k ) . (A1)
µν +
−
Herek =k±q/2andTr=gTr ,whereTr isthetrace
σ σ
overps±eudospindegrees-of-freedom,andF (k,k)=1+
µν (cid:48)
FIG. 5: (Color online) The real part of the a.c. conductivity µνcos(φk−φk(cid:48)), φk being the angle between k and the
xˆ axis.
(inunitsofσ )asafunctionofω/(2ε )calculatedbyinclud-
0 F
ing only the vertex correction [i.e. by neglecting self-energy We are interested in the long-wavelength q → 0 limit
diagrams in panels c) and d) in Fig. 2] for α =0.9 and dif- ofthedensity-densityresponsefunction: wethusexpand
ee
ferentvaluesofdopingn(correspondingtoΛ=100,200,and Eq. (A1) to second order in q:
1000). Thesedatahavebeencalculatedusinglong-range(un-
screened) interactions, i.e. qTF = 0 in Eq. (15). We remind q2sin2(φ )
tohrdeerreapderetrutrhbaatt,ioinntthheeorayb,s(cid:60)eneceσ(oωf)d=iso0rdfeorr aωn<d w2εith.inNfiotriscte- Fµν(k−,k+)=(1+µν)−µν 2k2 k , (A2)
F
that(cid:60)eσ(V)(ω)convergesrapidlyinthelimitΛ→∞[since,
where we have assumed that q is along the xˆ (i.e. that
as already stated in the main text, the integral in the second
φ = 0). When µ = ν = −1, the “Lindhard ratio” in
lineofEq.(14)convergesinthesamelimit]andthatitdrops q
Eq. (A1) vanishes. For µ = −ν, Eq. (A2) is already of
well below the universal value σ at large ω.
0 orderq2andonecanjustputq =0intheLindhardratio.
Forµ=ν =+1weneedinsteadtoexpandtheLindhard
ratio to second order in q with the result
high-temperature hydrodynamic regime in which Lan- nk−,+−nk+,+ = qcos(φk) δ(kF−k)
dau levels are not well resolved.) Undoubtedly much ω+ξk−,+−ξk+,++iη ω
interesting physics, potentially useful for applications in vq2cos2(φ )δ(k −k)
k F
opto-electronics, hasstilltobelearnedfromthestudyof + ω2 .
graphene and other58 non-Galilean invariant systems.
(A3)
Here we have used n (cid:39)n ∓qcos(φ )δ(k −k)/2,
k±,+ k,+ k F
and we have dropped iη in the denominator since the
expansion on the right hand side is always real. The
first term in Eq. (A3) gives zero contribution to the re-
sponsefunctionafterangularintegration. Now,replacing
Eqs. (A2) and (A3) in Eq. (A1), we find
Acknowledgments
4vq2 (cid:90) d2k
χ(0)(q,ω) = cos2(φ )δ(k −k)
ρρ ω2 (2π)2 k F
S.H.A.andG.V.acknowledgesupportfromNSFGrant + q2(cid:88)(cid:90) d2k sin2(φk) nk,µ−nk,µ¯ ,
No. 0705460. M.P. acknowledges financial support by (2π)2 k2 ω+2µvk+iη
the 2009/2010 CNR-CSIC scientific cooperation project. µ
Work in Austin was supported by the Welch Founda- (A4)
tion and by SWAN. S.H.A. acknowledges support and
the kind hospitality of the IPM in Tehran, Iran, during where µ¯ = −µ. Performing the integral over k and in-
the final stages of this work. We thank Igor Herbut for troducing dimensionless variables, one finds Eqs. (10)
useful comments and clarifications about the theory of and (11) for the real and imaginary parts of the non-
the undoped system. interacting response function, respectively.
9
Appendix B: Calculation of the vertex correction
The vertex correction [i.e. see Fig. 2b)] contribution
to the density-density response function reads
χ(ρVρ)(q,ω) = −S12 (cid:88)Vk−k(cid:48)(cid:90) +∞ 2dπ(cid:15)i(cid:90) +∞ 2dπ(cid:15)(cid:48)iTr(cid:2)G(k−,(cid:15))G(k+,(cid:15)+ω)G(k+(cid:48) ,(cid:15)(cid:48)+ω)G(k−(cid:48) ,(cid:15)(cid:48))(cid:3)
k,k(cid:48) −∞ −∞
= −116(cid:90) (d2π2k)2 (cid:90) (d22πk)(cid:48)2 µ,ν(cid:88),µ(cid:48),ν(cid:48)Vk−k(cid:48)ω+nξkk−−,,µµ−−ξnkk++,ν,ν+iηω+nξkk−(cid:48)−(cid:48),,µµ(cid:48)(cid:48)−−ξnkk+(cid:48)+(cid:48),ν,ν(cid:48)(cid:48)+iη
× Tr[(11σ+µσk−)(11σ+νσk+)(11σ+ν(cid:48)σk+(cid:48) )(11σ+µ(cid:48)σk−(cid:48) )] . (B1)
The trace in the previous equation can be expanded in complete expression of w is quite cumbersome and will
2
powers of q: not be reported here. The only terms that contribute to
χ(V)(q,ω) up to order q2 are given by
ρρ
Tr[(11σ+µσk−)(11σ+νσk+)(11σ+ν(cid:48)σk+(cid:48) )(11σ+µ(cid:48)σk−(cid:48) )]
(cid:88)∞
=16n=0wn(k,k(cid:48),φq;µ,ν,µ(cid:48),ν(cid:48))qn . w2|ν=µ¯,ν(cid:48)=µ¯(cid:48) = sin(φk2)kskin(φk(cid:48))Fµµ(cid:48)(k,k(cid:48)) . (B5)
(cid:48)
(B2)
If we choose q =qxˆ, it is straightforward to show that Collecting all terms up to O(q2) we can write χ(V)(q,ω)
ρρ
in the following form:
(1+µν)(1+µν )
w0 = (cid:48) (cid:48) Fµµ(cid:48)(k,k(cid:48)) , (B3)
2
2
and χ(V)(q,ω)= (cid:88)χ(V n)(q,ω) , (B6)
ρρ ρρ−
w =(µ−ν)(µ +ν )sin(φk)sin(φk(cid:48) −φk)+Perm, (B4) n=0
1 (cid:48) (cid:48)
4k
(V n)
where“Perm”isobtainedfromthefirstterminEq.(B4) where χρρ− (q,ω) denotes terms proportional to wn.
byinterchangingprimedwithnon-primedvariables. The We find that
χ(ρVρ−0)(q,ω) = −2(cid:90) (d2π2k)2 (cid:90) (d22πk)(cid:48)2 Vk−k(cid:48)[1+cos(φk(cid:48) −φk)]ω+nvk|k−,+|−−vn|kk++,+|+iηω+nvk|k−(cid:48)(cid:48),+|−−vn|kk+(cid:48)+(cid:48),+|+iη
2q2 (cid:90) d2k (cid:90) d2k − −
= −ω2 (2π)2 (2π)(cid:48)2vk−k(cid:48)δ(kF−k)δ(kF−k(cid:48))cos(φk)cos(φk(cid:48))[1+cos(φk(cid:48) −φk)] . (B7)
Introducing the Coulomb pseudopotentials V , performing the integrations over k and k, and using dimensionless
m (cid:48)
variable we find immediately Eq. (12).
For n=1 we find
χ(ρVρ−1)(q,ω) = −2ωq2 (cid:88)(cid:90) (d2π2k)2 (cid:90) (d22πk)(cid:48)2Vkk−k(cid:48)δ(kF−k(cid:48))sin(φk)cos(φk(cid:48))sin(φk(cid:48) −φk)µω(+nk2,µµv−kn+k,iµ¯η)
µ
= −q2kF (cid:88)P(cid:90) ∞dk V0(k,kF)−V2(k,kF) . (B8)
8π2ω ω+2µvk+iη
µ kF
Since this contribution vanishes for ω →0 it does not contribute to the renormalization of the Drude weight. Never-
theless, its imaginary part for finite positive frequency gives precisely the term in the first line of Eq. (14).
10
Finally, for n=2, we find that
χ(ρVρ−2)(q,ω) = −q22 (cid:88)µµ(cid:48)(cid:90) (d2π2k)2 (cid:90) (d22πk)(cid:48)2Vkk−kk(cid:48) sin(φk)sin(φk(cid:48))(ω+2(µ1v−k+nki,η+))((ω1−+n2µk(cid:48),v+k) +iη)Fµµ(cid:48)(k,k(cid:48))
µ,µ(cid:48) (cid:48) (cid:48) (cid:48)
= − q2 (cid:88)P(cid:90) ∞dk (cid:90) ∞dk V0(k,k(cid:48))+V2(k,k(cid:48))+2µµ(cid:48)V1(k,k(cid:48)) . (B9)
25π2 (cid:48) (ω+2µvk+iη)(ω+2µvk +iη)
µ,µ(cid:48) kF kF (cid:48) (cid:48)
It is possible to show that this expression does not scale Appendix C: Calculation of the self-energy
like ω 2 for ω → 0: thus it does not contribute to the insertions
−
renormalizationoftheDrudeweight. Theimaginarypart
ofEq.(B9)atfinitefrequencygivespreciselythetermin
the second line of Eq. (14). Summing the contributions
to (cid:61)m χ(V)(q,ω) coming from the two terms with n=1 Wenowturntocalculatethetwofirst-orderself-energy
ρρ
and n=2 we find Eq. (13). diagrams in Figs. 2c) and d). The first diagram reads
χ(ρSρE−c)(q,ω) = −(cid:90) (d2π2k)2 (cid:90) (d22πk)(cid:48)2Vk−k(cid:48)(cid:90) +∞ 2dπ(cid:15)i(cid:90) +∞ 2dπ(cid:15)(cid:48)iTr(cid:2)G(k−,(cid:15))G(k+,(cid:15)+ω)G(k+(cid:48) ,(cid:15)(cid:48)+ω)G(k+,(cid:15)+ω)(cid:3)
= −116 (cid:88) (cid:90) (d2π2k)2 (cid:90) (d22πk)−(cid:48)2∞Vk−k(cid:48)nk+(cid:48)−,γ∞
µ,ν,λ,γ
(cid:104) (cid:105)
× (cid:90) +∞ d(cid:15) Tr (11σ+µσk−)(11σ+νσk+)(11σ+γσk+(cid:48) )(11σ+λσk+) , (C1)
2πi((cid:15)−ξ +iη )((cid:15)+ω−ξ +iη )((cid:15)+ω−ξ +iη )
k−,µ k−,µ k+,ν k+,ν k+,λ k+,λ
−∞
It is easy to show that for λ=ν the trace becomes
(cid:104) (cid:105)(cid:12)
Tr (11σ+µσk−)(11σ+νσk+)(11σ+γσk+(cid:48) )(11σ+λσk+) (cid:12)(cid:12)λ=ν =16 Fµν(k+,k−) Fγν(k+,k+(cid:48) ) , (C2)
and that the contribution arising from λ = −ν averages Inserting Eq. (C2) in Eq. (C1) we find that
to zero after performing the integrations over k and k.
(cid:48)
χ(ρSρE−c)(q,ω) = −µ(cid:88),ν,γ(cid:90) (d2π2k)2 (cid:90) (d22πk)(cid:48)2Vk−k(cid:48)nk+(cid:48) ,γ(cid:90)−+∞∞ 2dπ(cid:15)i((cid:15)−ξk−,µF+µνiη(kk−+,,µk)(−(cid:15))+Fγων(−k+ξk,+k,+(cid:48)ν)+iηk+,ν)2
= ∂ωµ(cid:88),ν,γ(cid:90) (d2π2k)2 (cid:90) (d22πk)(cid:48)2Vk−k(cid:48)nk+(cid:48) ,γFµν(k+,k−)Fγν(k+,k+(cid:48) )ω+nξkk−−,,µµ−−ξnkk++,ν,ν+iη . (C3)
For the diagram in Fig. 2d) we find that
χ(ρSρE−d)(q,ω) = −∂ωµ(cid:88),ν,γ(cid:90) (d2π2k)2 (cid:90) (d22πk)(cid:48)2Vk−k(cid:48)nk−(cid:48) ,γFµν(k−,k+)Fγν(k−,k−(cid:48) )ω+nξkk−−,,νν−−ξnkk++,µ,µ+iη . (C4)
We now sum Eqs. (C3) and (C4) together and separate the inter-band channel F (k ,k ) is already of order
µν +
inter-band (ν = −µ) from intra-band (ν = µ) terms. In −