Effects of optical and surface polar phonons on the optical conductivity of doped graphene

0.36 MB·English

by  Benedikt Scharf

Effects of optical and surface polar phonons on the optical conductivity of doped
graphene
Benedikt Scharf,1 Vasili Perebeinos,2,∗ Jaroslav Fabian,1 and Phaedon Avouris2
1Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany
2IBM Research Division T. J. Watson Research Center, Yorktown Heights, New York 10598, USA
(Dated: January 22, 2013)
Using theKubo linear response formalism, we study theeffects of intrinsic graphene optical and
3 surfacepolarphonons(SPPs)ontheopticalconductivityofdopedgraphene. Wefindthatinelastic
1 electron-phononscatteringcontributessignificantlytothephonon-assistedabsorptionintheoptical
0 gap. At room temperature, this midgap absorption can be as large as 20-25% of the universal ac
2 conductivityfor graphene on polar substrates (such as Al2O3 or HfO2) dueto strong electron-SPP
n coupling. The midgap absorption, moreover, strongly depends on the substrates and doping levels
a used. With increasing temperature, the midgap absorption increases, while the Drude peak, on
J the other hand, becomes broader as inelastic electron-phonon scattering becomes more probable.
1 Consequently,theDrudeweight decreases with increasing temperature.
2
PACSnumbers: 81.05.ue,72.10.Di,72.80.Vp
] Keywords: graphene,optical conductivity, phonons, substrate
l
l
a
h I. INTRODUCTION sorption in heavily doped samples. Moreover, the op-
- tical conductivity in the presence of a magnetic field,
s
e Sinceitwasfirstisolatedin2004,1graphene,amaterial theso-calledmagneto-opticalconductivity,hasalsobeen
m investigated theoretically,32,33 with Ref. 33 taking into
composedofasinglelayerofcarbonatomsarrangedina
. two-dimensional (2D) honeycomb lattice, has attracted account the coupling between electrons and Einstein
t
a immense interest2–4 due to its excellent transport and phonons.
m optical properties,2,4–9 which make it an attractive can-
- didate for possible applications in nanoscale electronics
d
and optoelectronics.10–12 One particular field which has Besides the aforementioned studies on the optical
n
receivedconsiderableattention,bothexperimentally13–16 conductivity, the role played by different phonons has
o
c aswellastheoretically,17–25 istheoptical(orac)conduc- also been studied in the context of heat dissipa-
[ tivity in graphene,that is,the frequency-dependentcon- tion mechanisms34–36 and current/velocity saturation in
ductivity. The main feature that can be observed in the graphene,37 which plays an important role in electronic
2
optical conductivity is that for frequencies larger than RF applications11 and also for transport38,39 in the sim-
v
9 twice the absolute value of the chemical potential µ, the ilar system of carbon nanotubes. Inelastic scattering ei-
2 optical conductivity is roughly given by σ0 = e2/(4~), ther by intrinsic graphene optical phonons40 or surface
3 the so-called universal ac conductivity.13,14 For frequen- polar phonons37,41–45 (SPPs) is thought to give rise to
3 cies below 2µ, the optical conductivity is greatly re- the saturation of the current in graphene and affect the
1. duced, which| c|an be explained within a single-particle low field carrier mobility.46,47 However, from transport
1 model where transitions induced by photons with ener- experimentsaloneitisdifficulttoidentifytheroleplayed
2 gies ~ω < 2µ are forbidden due to Pauli’s exclusion bySPPsfromthepolarsubstratesbecauseofthecompli-
1 principle. Th|e|ability to tune the optical properties of cationsarisingfromchargetrapswhichcanbepopulated
v: graphene has been explored for use in broadband light thermally48 or by the high electrical fields.49
i modulators.26–28 One figure of merit for this application
X
is the modulation depth of the optical absorption. In
r
a experiments, however, one does not observe the optical Here,weshowthatthetemperaturedependenceofthe
conductivity to vanish completely, as one would expect midgap absorption can be significantly stronger in the
fromthe simple single-particleargumentgivenabove. In presence ofSPPs as comparedto suspended grapheneor
addition, a substantial Drude weight loss has been re- graphene on a non-polar substrate such as diamond-like
ported in graphene on SiO .16 carbon. Ourmaingoalinthismanuscriptistostudythe
2
To describe this behavior, mechanisms involving dis- optical conductivity in the presence of phonons. While
order and/or phonons, both of which can account for the impact of optical phonons has been studied in sev-
a finite absorption below 2µ, have been studied the- eral earlier works,20–22 the effect of SPPs on the optical
oretically in both monolaye|r2|0–22,25,29 and bilayer30,31 conductivity in graphene has yet to be analyzed. In this
graphene. In addition to these single-electron effects, paper,weuselinearresponsetheorytoderiveaKubofor-
excitonic effects23 as well as effects arising from the mulafortheopticalconductivity,whichisthenevaluated
Coulomb interaction24 have also been considered, but for suspended graphene as well as graphene on different
were found to have a negligible effect on the midgap ab- polar substrates, where SPPs are present.
2
II. MODEL meV.67 Furthermore, we need to know the products
M˜ Mλλ˜ (k q,q)Mλ˜λ′ (k, q) of the electron-
Λ ≡ vv˜,Λ − v˜v′,Λ −
To describe the electronic (single-particle)bandstruc- v˜
phononPcouplingmatrixelementsforeachphononbranch
ture of graphene, we use the Dirac-cone approximation,
in order to calculate the electron self-energy. The cou-
where the Hamiltonian can be written as
pling of both Γ phonons is given by51,52,54
Hˆe = λǫkcˆ†λksvcˆλksv, (1) ~D2
k,Xs,v,λ M˜ΓLO +M˜ΓTO = 2NMΓω (1+λλ′)δvv′, (5)
c Γ
with ǫ = ~v k. Here, k, s, and v denote the momen-
k F whereN isthenumberofunitcells,M thecarbonmass,
tum, spin, and valley quantum numbers, λ the conduc- c
andD 11.2eV/˚Athestrengthoftheelectron-phonon
tcˆion (λthe=co+r1re)sapnodndvinalgencrceea(tλion=an−d1)anbnainhdilsa,ticˆo†λnksovpearnad- couplinΓg≈.42 ThecouplingoftheKTO phononmodetothe
λksv electronsingrapheneistwiceaslargeasthatofphonons
tors,andvF ≈108 cm/sthe Fermivelocityingraphene.4 at the Γ-point51–53 and is described by68
Since the goal of this work is to study and compare
the effects of several different phonons on the optical ~D2
conductivity of graphene, we need to take into account M˜KTO = 2NMΓω 1+λλ′−λ˜ λe−iθ+λ′eiθ δvv′,
the interaction with these phonons. A general phononic c K h (cid:0) (cid:1)i (6)
Hamiltonian reads as whereθ θ θ andθ =arctan(k /k ). Ourchoice
≡ k− k−q k x y
ofthe opticalphonondeformationpotential lies between
Hˆ = ~ω (q)pˆ† pˆ , (2)
ph Λ qΛ qΛ the LDA results of Refs. 51 and 52 and the GW results
Xq,Λ fromRef.55. Asmallerelectron-intrinsicopticalphonon
couplingassuggestedinRef.56wouldfurtherreducethe
where different phonon branches are labeled as Λ, the
absorption below 2µ in suspended graphene.
phononmomentumasq,andthecorrespondingfrequen- | |
Here, we include SPPs in our model as follows: There
cies and creation (annihilation) operators as ω (q) and
Λ are two surface optical(SO) phonons in polar substrates
pˆ† (pˆ ). Whereas Eqs. (1) and (2) describe isolated
qΛ qΛ that interact with the electrons in graphene and whose
systems of electrons and phonons, respectively, the cou-
dispersion can again be approximated by substrate-
pling between those systems is given by
specific, constant frequencies ω and ω , and their
SO1 SO2
electron-phonon coupling matrix elements read as43,47
Hˆ = Mλλ′ (k,q) pˆ† +pˆ
e−ph vv′,Λ −qΛ qΛ
λXksvλ′Xqv′Λ (cid:16) (cid:17) (3) M˜ = π2e2FΛ2(q)e−2qz0 1+λλ′+λ˜ λe−iθ+λ′eiθ δ ′,
×cˆ†λ(k+q)svcˆλ′ksv′, Λ NAq h (cid:0) (cid:1)i (v7v)
where Mvλvλ′′,Λ(k,q) is the electron-phonon coupling ma- Awh=er3e√e3=a2|/e|2itshtehaeraebasoofluttheevgarlaupehoefntehueneilteccterllo,nacha1r.g4e2,
trixelement.50Hence,thetotalHamiltonianofourmodel ˚Athedistancebetweentwocarbonatoms,z 3.5≈˚Athe
reads as 0 ≈
van der Waals distance between the graphene sheet and
Hˆ =Hˆe+Hˆph+Hˆe−ph. (4) tthhee smuabgstnriatuted,eanofdtthheepFor¨olahrliizcahticoonupfileinldg.5F7Λ2T(qh)edFers¨ocrhilbicehs
coupling is given by34,58
In this work, we investigate two different types of
phononsthatcoupletotheelectronsingraphene: intrin- ~ω 1 1
sic graphene optical phonons and SPPs, that is, surface F2 (q)= SO1 (8)
SO1 2π ε +ε(q) − ε +ε(q)
phononsofpolarsubstrateswhichinteractwiththe elec- (cid:20) i 0 (cid:21)
trons in graphene via the electric fields those phonons
and
generate. Intrinsic graphene acoustic phonons are not
included in our model because their effect on the optical ~ω 1 1
F2 (q)= SO2 (9)
conductivity is negligible as has been shown in Ref. 20. SO2 2π ε +ε(q) − ε +ε(q)
The dominant electron-optical phonon coupling51–54 is (cid:20) ∞ i (cid:21)
due to longitudinal-optical (LO) and transverse-optical with the optical, intermediate, and static permittivi-
(TO) phonons at the Γ-point and TO phonon at the ties ε , ε, and ε of the substrate as well as the
∞ i 0
K-point. In the vicinity of the Γ point, the disper- static, low temperature dielectric function ε(q) = 1 +
sion of both, LO and TO phonons (denoted by Γ 2πe2Π (q,ω =0)/(κq), where κ is the background di-
LO g
and Γ ), can be approximated by the constant energy electric constant and Π (q,ω) the polarization function
TO g
~ω 197 meV. Near the K and K′ points, on the of graphene as calculated in Refs. 59,60. The dielectric
Γ ≈
other hand, only the TO phonon (denoted by K ) con- function ε(q) accounts for the screening of the Coulomb
TO
tributes to the electron self-energy and its dispersion in interaction in the graphene sheet above the polar sub-
this region can again be assumed as flat, ~ω 157 strate. If the effect of screening is to be disregarded, we
K ≈
3
Al2O3a h-BNb HfO2c SiCd SiO2e have the form
ε0 12.53 5.09 22.0 9.7 3.90 Im[ΣΛ(ω+i0+)]=
εi 7.27 4.575 6.58 - 3.36 [n +n (~ω ~ω)] g (~ω+µ ~ω ) (12)
ε∞ 3.20 4.10 5.03 6.5 2.40 +Λ[n +FDn (~Λω−+~ω)]Λg (~ω+−µ+~Λω )
~ω [meV] 56.1 101.7 21.6 116.0 58.9 Λ FD Λ Λ Λ
SO1
~ω [meV] 110.1 195.7 54.2 - 156.4 with the functions g (ǫ)= AD2 ǫ/(2M ~ω v2).
F2SO2[meV] 0.420 0.258 0.304 0.735 0.237 Similarly, the effeΛct of −the tΓw|o| SPPc mΛodFes (Λ =
SO1
F2 [meV] 2.053 0.520 0.293 - 1.612 SO1,SO2) is described by
SO2
Im[ΣΛ(k,ω+i0+)]=
λ
TABLE I: Optical, intermediate, and static permittivities as [n +n (~ω ~ω)] hΛ(k,~ω+µ ~ω ) (13)
well as frequencies and bare Fr¨ohlich couplings for the SPP +Λ[n +FDn (~Λω−+~ω)]λhΛ(k,~ω+−µ+~Λω ),
scatteringonthesubstratesAl2O3,hexagonalBN,HfO2 SiC, Λ FD Λ λ Λ
and SiO2.
where
a Refs. 58,61
b Refs. 42,61,62 2π
c Refs. 42,58,61 πe2 F2(q)e−2qz0
hΛ(k,ǫ)= dθ Λ (ǫ +λǫcosθ)
d Refs. 42,63 λ −2(~v )2 q | |
e Ref. 42, which uses averages of valuesfrom Refs. 43,47,58. F Z0
(14)
with q ǫ2+ǫ2 2ǫǫ cosθ/(~v ). In Eqs. (12)
≡ k− k F
set ε(q)=1 in Eqs. (8) and (9), for which we obtain the and (13), we have introduced the Fermi-Dirac and Bose-
p
bare Fr¨ohlich couplings presented in Table I. Einsteindistributionfunctions,nFD/BE(ǫ)=1/[exp(βǫ)±
1],whereβ =1/(k T)(withT andk beingthetemper-
Employing standard diagrammatic perturbation B B
theory50,64,65 and inserting the specific expressions for ature and the Boltzmann constant, respectively), nΛ =
the matrix elements Mvλvλ′′,Λ(k,q) (for more details, we nthBeE(f~oωlloΛw),inagn,dwteheignchoerme itchael epffoetcetntoiaflpµola=ronµi(cTs)h.iftIsn,
refer to the Appendix A), we find that, up to first
that is, the real parts of the self-energies and set
non-vanishing order, the electronic spectral function is
Σ (k,ω+i0+) iIm[Σ (k,ω+i0+)]. The total (re-
diagonal in the four quantum numbers λ, k, s, and v λ ≡ λ
tarded)self-energyinourmodelisthuscompletelyimag-
and is given by
inary.
As detailed in the Appendix B, the real part of the
(k,ω)=
Aλ optical conductivity can be calculated from the spectral
1 function (10) via the formula69
2Im ,
− (cid:26)ω+i0+−[λǫk−µ+Σλ(k,ω+i0+)]/~(cid:27) ∞ ∞
(10) σ v2
where Σλ(k,iνn) denotes the imaginary-time self-energy σ(ω)=π02ωF dω′ dkkAλ(k,ω′)
attheimaginary(fermionic)frequencyiνn. Inthelowest Xλλ′−Z∞ Z0
oisrdjuers,ttthheecosuntmribouftitohnetcootnhtreibseultf-ioennesrgfryomduethteopdhiffoenroennst ×Aλ′(k,ω′+ω)[nFD(~ω′)−nFD(~ω+~ω′)],
(15)
phononsΛofourmodel. Moreover,weincludescattering
which includes the universal ac conductivity σ =
0
at Coulomb impurities in our model by adding the con- e2/(4~). In the following, we will use Eqs. (10)-(14) to
tribution ΣCo(k)= i~/[2τ(k)] to the self-energy, where
− numerically calculate the spectral function, which is in
we use the transport scattering time τ(k) as calculated
turn used to calculate the real part of the optical con-
in Ref. 66, and the total self-energy reads as
ductivity numerically via Eq. (15). Those calculations
are conducted for several different substrates (as well as
Σ (k,iν )=ΣCo(k)+ΣΓ(iν )+ΣK(iν )
λ n n n (11) suspendedgraphene)with the correspondingparameters
+ΣSλO1(k,iνn)+ΣSλO2(k,iνn). summarized in Table I.
The contribution ΣCo(k) has been added to the self-
energy to model the lineshape of the Drude absorption III. RESULTS
peak. Throughout this work we use the impurity con-
centration of n = 5 1011 cm−2, which is low enough In Fig. 1, the optical conductivities70 (for a fixed
i ×
tonotaffectthemidgapabsorptionofgrapheneonpolar chemical potential µ = 0.3 eV) at two temperatures
substrates significantly. T = 1 K (inset) and T = 300 K are shown for sus-
The imaginary parts of the contributions from the op- pended graphene as well as graphene on several differ-
tical phonons at the Γ and K points (Λ = Γ,K) to the ent substrates: Al O , hexagonal BN, HfO , SiC, and
2 3 2
retarded self-energy depend only on the frequency and SiO . For comparison, we have also included the optical
2
4
1 1
Drude peak
0.75
0.8
0.75 0.6 T = 1 K 0.5 T = 1 K
interband
0.4
absorption µ = 0.2 eV
σ0 0.2 0.25 µ = 0.3 eV
σ(ω)/ 0.5 0 pa0bh.2soonropn0t.i-4oanssis0t.6ed AHSilfC2OO s23 us bsuusbtbsrsatrttraeatete σ(ω)σ/0 01 (a) µ = 0.4 eV
0.75
0.25 SiO substrate
2
BN substrate
0.5
suspended T = 300 K
no phonons
0 0.25
0 0.2 0.4 _hω [eV]0.6 0.8 1 0 (b)
0 0.2 0.4 0.6 0.8 1 1.2
_hω [eV]
FIG. 1: (Color online) Calculated frequency dependence
of the (real part of the) optical conductivity of suspended
FIG. 2: (Color online) Calculated frequency dependence of
graphene and graphene on several different substrates for
n = 5×1011 cm−2 and µ = 0.3 eV at T = 300 K and the (real part of the) optical conductivity of graphene on a
Ti=1 K (inset). SiO2 substrate for ni = 5×1011 cm−2 and several different
chemical potentials at (a) T =1 K and (b) T =300 K.
conductivity of suspended graphene and graphene with-
pends on the particular polar substrate used and is, in
out any phonon contribution (at κ = 1). Figure 1 has
particular,determinedbytheinterplaybetweentheSPP
beencalculatedusingtheparametersgivenfromTableI,
frequencies ω and the Fr¨ohlich couplings F2 : the
the dielectric function ε(q), and κ = (1+ε0)/2 as the SOi SOi
background60,66 dielectric constant. smaller the ω or the larger F2 , the larger is the
SOi SOi
The profiles in Fig. 1 illustrate the main features that midgap absorption.
the effect electron-phonon coupling has on the optical Impurity scattering has also an influence on the
conductivity: Whereas there is a gap with a width 2µ midgap absorption: By calculating the absorption spec-
| |
intheabsorptionspectrumofthepurelyelectronicsingle- tra using just the Coulomb impurity scattering with
particlemodel,wheredirecttransitionsbetweentheelec- κ = 1 and no phonons for µ = 0.3 eV the Kubo model
tronicstatesintheconductionandvalencebandsarefor- shows that while at T = 300 K the midgap absorption
bidden for energies 0<~ω <2µ due to Pauli blocking, is 2.3% for n = 5 1011 cm−2 (see also Fig. 1), it in-
thereisafiniteabsorptioninthi|sr|egioninthepresenceof creases to 4.3i% for×n = 1012 cm−2. At T = 500 K,
i
phonons. Thisfiniteabsorptionislargelyduetophonon- these numbers are 5% and 7%, respectively. We note
assistedtransitions which give rise to distinct sidebands, that an obvious first estimate of the midgap absorp-
the onsetsofwhichcanclearlybedistinguishedfromthe tion could have been obtained by using a Drude model
Drudepeakatlowtemperaturesandlowimpuritydensi- σ(ω)/σ0 = 4µ/[π~τ(ω2 + 1/τ2)], where τ is a scat-
| |
ties(seetheinsetinFig.1). Ifthephotonenergyexceeds tering time. For n = 5 1011 cm−2 and n = 1012
2µ, direct (interband) transitions become possible, re- cm−2, the Coulombiscatter×ing mobilities are γi 13000
su|lt|ing in a steep rise of the optical conductivity. cm2/(Vs) and γ 6500 cm2/(Vs), respectively≈, corre-
≈
At higher temperatures, one can see that the Drude sponding to midgap absorptions from the Drude model
peakisbroadenedasmorephononsbecomeavailableand of σ(µ/~)/σ0 ≈4~evF2/(πγµ2)=0.7% and 1.4%, respec-
electron-phonon scattering becomes more probable. For tively. Thus,thissimpleestimateusingtheDrudemodel
HfO andAl O substrates,thephononsidebandsmerge significantlyunderestimatestheresultsobtainedfromthe
2 2 3
with the Drude peak resulting in a very broad Drude fullcalculations. Indeed,thedeviationsbetweentheesti-
peak at room temperature. Furthermore, the profiles of matefromtheDrudemodelandthefullKuboformalism
theopticalconductivityaremuchsmoothercomparedto calculationbecomeevenmorepronouncedifphononsare
those at T = 1 K and distinct onsets of phonon side- taken into account.
bands can no longer be observed as the profiles of the Figure 2 shows the optical conductivity for graphene
optical conductivity are smeared out by thermal broad- ona SiO substrate at different temperatures and chem-
2
ening. Finally, Fig. 1 shows that the so-called “midgap ical potentials. Apart from the trends in the behavior of
absorption”, that is, the absorption at ~ω = µ, is sig- the opticalconductivity discussedabove,one canclearly
nificantly enhanced for graphene on polar substrates as see different gaps in the absorption spectrum, given by
comparedto suspended grapheneorgrapheneonnonpo- 2µ for each chemical potential. Another feature that
| |
lar substrates: Whereas the midgap absorption at room can be discerned from Fig. 2 is that the maximal value
temperatureisabout5-6%ofσ forsuspendedgraphene, of the phonon-mediated absorption in the gap increases
0
it can be as high as 20-25% of σ for graphene on HfO with increasing chemical potential (doping level). More-
0 2
or Al O . Hence, the midgap absorption strongly de- over, we note that due to the electron-hole symmetry of
2 3
5
AlO substrate AlO substrate
2 3 2 3
HfO substrate HfO substrate
30 2 30 2
SiC substrate SiC substrate
SiO substrate SiO substrate
2 2
BN substrate BN substrate
%] suspended %] suspended
σ [020 σ [020
ω)/0 ω)/0
σ( σ(
10 10
0 0
0.2 0.25 0.3 0.35 0.4 0 100 200 300 400 500
µ [eV] T [K]
FIG. 3: (Color online) Calculated dependenceof the midgap FIG.5: (Coloronline)Calculatedtemperaturedependenceof
absorption (at ~ω0 = µ) on the chemical potential for sus- the midgap absorption (at ~ω0 =µ) for suspended graphene
pendedgrapheneandgrapheneonseveraldifferentsubstrates, and graphene on several different substrates, ni = 5×1011
n =5×1011 cm−2, and T =300 K. cm−2, and µ=0.3 eV.
i
1
used,isthatasecondclearlydistinguishablephononside-
1
0.8 bandpeak(due totheSPPs)cannowbeobservedinthe
0.75 0.6 T = 1 K absorption spectra even at room temperature for SiO
2
0.4 and SiC substrates. For BN substrates, one can even
σ0 0.2 find two such peaks at room temperature. We suggest,
ω)/ 0.5 0 0.2 0.4 0.6 AlO substrate therefore, that measurements of the midgap absorption
σ( Hf2O23 substrate in graphene on different substrates could help to clarify
0.25 SSiiCO s usubbstsrtaratete the still controversial34,43,44,58,61 issue concerning the ef-
2
BN substrate fect of screening on the electron-SPP coupling strength.
suspended
Wealsoinvestigatethetemperaturedependenceofthe
no phonons
0 midgap conductivity (that is, at ~ω = µ) within our
0
0 0.2 0.4 _hω [eV]0.6 0.8 1 model in Fig. 5 on several different substrates. At tem-
peratures below 100 K, the midgap absorption does not
FIG. 4: (Color online) Calculated frequency dependence depend strongly on the temperature. At about 100 K,
of the (real part of the) optical conductivity of suspended anincreaseofthe opticalconductivity at~ω0 =µ is pre-
graphene and graphene on several different substrates for dicted to take place. Also, the smaller the energy of the
ni = 5×1011 cm−2 and µ = 0.3 eV at T = 300 K and dominantphononcontributingtothegapabsorption,the
T =1 K (inset) if bare Fr¨ohlich couplings are used. stronger is the temperature dependence in Fig. 5.
Finally, we relate the midgap absorption to the spec-
tral weight of the Drude peak. Describing the graphene
theDiracHamiltonian,theprofilesoftheopticalconduc- opticalconductivityinthenon-interactingsingle-particle
tivity would look the same for p-doped graphene. The picture, the spectral weight of the bare Drude peak is
dependence of the midgap absorption on the chemical ω′
potential at room temperature is shown in Fig. 3, again I0 = dωσ(ω) = 2µσ0/~ = D0/2, where D0 is the
| |
forsuspendedgrapheneandgrapheneonseveraldifferent 0
bare DRrude weight and ω′ is some characteristic fre-
substrates. In the region studied here between µ = 0.2
quencymuchlargerthanthe scatteringrate,butsmaller
eVandµ=0.4eV,themidgapabsorptiondecreaseswith
than both the lowest energy of the optical phonon ~ω
increasingchemicalpotentialforgrapheneonsubstrates, opt
and 2µ. In the presence of phonons, the total spec-
withthedecaybeingmostpronouncedforHfO2. Thedif- tral w|eig|ht has to be conserved. The spectral weight
ferencesintheshapeofthephononmediatedgapabsorp-
contribution due to the midgap absorption can be ap-
tion reflect the momentum dependence of the electron-
2|µ|
phonon interaction in Eq. (5)-(7).
proximated at low temperatures as I = dωσ(ω)
In particular,Fig. 4 revealsa striking difference in the gap ωopt ≈
absorptionifweuseabareunscreenedFr¨ohlichcouplings. ασ (2µ ~ω )/~, where α is the averaRged value of
0 | | − opt
If screening is not accounted for, we find that the opti- σ(ω), that is, of the real part of the optical conductiv-
cal conductivity in the optical gap is greatly enhanced ity, in units of σ . If we further assume that the entire
0
compared to the situation where screening is used. The spectral weight lost at the Drude peak is transferred to
most noticeable feature if the bare Fr¨ohlich coupling is theopticalgapandthat~ω 2µ,withinthispicture
opt ≪ | |
6
%] 2105 (a) Al2O3 substrate garbaspohrpetnieononatdγirffele=ren0t. Bsuebcsatursaetessusepaecnhdepdosgsreaspshdeinffeeraenndt
σ [010 SsuiOsp2e snudbesdtrate backgrounddielectric constants andthus differenttrans-
σω()/0 05 palosrotdsicffaetrteenritn.g times, the residual values at γrel = 0 are
100
%] 90
D [080 IV. CONCLUSIONS
D/
70 (b)
20 (c) We have studied the effects of intrinsic graphene op-
V]15 tical phonons and SPPs on the optical conductivity of
e
m10 dopedgraphene. Ourfocushasbeenontheabsorptionat
τ [ frequencies ~ω < 2µ, where optical transitions are for-
_ h / 5 biddenduetoPauli|bl|ockinginacleansystem(atT =0),
0
but which can occur if phonons are present, giving rise
0 0.2 0.4 0.6 0.8 1
γ to phonon sidebands. Here, we have found that inelas-
rel
tic phonon scattering contributes significantly to the ab-
FIG. 6: (Color online) Calculated dependence of the (a) ab- sorption in the optical gap and strongly depends on the
sorption at ~ω0 = µ, (b) the Drude weight D, and (c) the substrate used: At room temperature (and µ=0.3 eV),
inverse scattering time ~/τ on the relative strength of the the midgap absorption, which is mainly due to intrinsic
electron-phonon coupling γrel for suspended graphene and opticalphonons,amountsto about5-6%ofthe universal
graphene on Al2O3 and SiO2, ni = 5×1011 cm−2, T = 300 ac conductivity for suspended graphene or graphene on
K, and µ=0.3 eV.
non-polar substrates, while the midgap absorption can
be as large as 20-25% of σ for graphene on polar sub-
0
strates (such as Al O or HfO ) due to the smaller SPP
2 3 2
the remaining Drude weight D can then be estimated as energy and strong electron-SPPcoupling. Moreover,the
D/D0 = (I0 −Igap)/I0 ≈ 1−α. Thus, from this con- midgap absorption depends on the doping level and de-
siderationwe expect that, as α increases with increasing creases with increasing µ, while the maximal value of
| |
temperatureorstrengthoftheelectron-phononcoupling, the sideband absorption at low temperatures increases.
the Drude weight is reduced. We have also investigated the temperature dependence
Figure 6 shows (a) the absorption in the optical gap of the midgap absorption which increases with increas-
as well as the fitted (b) Drude weight and (c) inverse ing temperature. The Drude peak, on the other hand,
scatteringtime forsuspended grapheneandgrapheneon becomes broader with increasing temperature as inelas-
Al O and SiO substrates with µ = 0.3 eV, T = 300 tic electron-phonon scattering becomes more important.
2 3 2
K, and n = 5 1011 cm−2 as functions of the relative Consequently, the Drude weight decreases with increas-
strength γi of×the electron-phonon coupling. Here, the ing temperature due to the stronger phonon coupling.
rel
opticalconductivitieshavebeencalculatedbyscalingthe
(products of) electron-phonon coupling matrix elements
with M˜ γ M˜ , and the Drude weight D as well Acknowledgments
Λ → rel Λ
as the inverse scattering time 1/τ have been extracted
from the optical conductivity by fitting the Drude peak We gratefully acknowledge Fengnian Xia from IBM
to a Lorentzian. As expected from the argument given for stimulating discussions and Alex Matos-Abiagueand
above, the Drude weight decreases with increasing γ , Christian Ertler from the University of Regensburg for
rel
althoughthe simple relationshipbetweenreducedDrude technicaldiscussions. B.S.andJ.F.acknowledgesupport
weightloss and the midgap absorptionas 1 α does not from DFG Grants No. SFB 689 and No. GRK 1570.
−
hold. This is because the midgap absorption does not
coincide with the averaged value of σ(ω) in the gap and
becausesomeofthespectralweightfromtheDrudepeak Appendix A: Self-energy and Green’s Function
is transferrednotonly to the opticalgap,but alsoto the
spectral region ~ω >2µ. We use standard diagrammatic perturbation theory
With increasing γ |, |electron-phonon scattering be- (with the unperturbed Hamiltonian Hˆe +Hˆph and the
rel
comes more probable and consequently the scattering perturbation Hˆe−ph) to calculate the electronic Matsub-
time decreases as can be seen in Fig. 6 (c). The corre- ara Green’s function of the system described by Eq. (4),
spondingincreaseof1/τ ismostpronouncedforgraphene
~β
on Al O and least pronouncedfor suspended graphene.
Finall2y,w3eremindthereaderthatforni =5×1011cm−2 Gvλvλ′′(k,iνn)=− dτ T cˆλksv(τ)cˆ†λ′ksv(0) eiνnτ,
impurity scattering also contributes to the absorptionin Z0 D h iE
the optical gap, which can be discerned from the finite (A1)
7
where τ and iν denote the imaginary time and Using the SPP coupling matrix elements (7), the con-
n
(fermionic) frequency, ... the thermal average, tribution from each SPP to the self-energy is again cal-
h i T
the imaginary time-ordering operator, and β = culated by introducing k′ and writing the sum as a 2D
1/(k T).50,64,65 By solving the corresponding Dyson integral. Then, the angular integration for the offdiago-
B
equation, we can express the electronic Green’s func- nal elements with respect to the band indices λ and λ′
tion via the self-energy Σλλ′,vv′(k,iνn). In writing down is of the type 2πdθ sinθf(cosθ), where the function f
Eq. (A1), we have used that, due to the conservation of
0
momentumandspin,theself-energyandthustheGreen’s depends only oRn cosθ and on the SPP considered, and
function are diagonalin k and s and do not depend ons vanishes for screened as well as unscreened SPPs. Thus,
due to spin degeneracy. the lowest order contribution from each SPP to the self-
Up to the first non-vanishing order (and omitting the energy is also diagonal in the band and valley quantum
tadpole diagram, which yields a purely real self-energy numbers.
contribution that can be absorbed in the chemical po- Combining the results discussed so far, the total con-
tential),wefindtheelectronicself-energyduetophonons tribution from all phonons is given by
for arbitrary matrix elements Mλλ′ (k,q) to be50
vv′,Λ
Σpλhλ′,vv′(k,iνn)=δvv′δλλ′Σpλh(k,iνn), (A3)
Σph (k,iν ) Σ(2) (k,iν )
λλ′,vv′ n ≈ λλ′,vv′ n where Σpλh(k,iνn) is given by
= 1 Mλλ˜ (k q,q)Mλ˜λ′ (k, q)
~Λ,Xq,λ˜,v˜n v+v˜,Λ1 −f (λ˜ǫ v˜v′),Λ − (A2) Σpλh(k,iνn)=Σ+ΓΣ(iSλνOn1)(+k,ΣiνKn)(i+νnΣ)SλO2(k,iνn), (A4)
qΛ − FD |k−q|
×"iνn−(λ˜ǫ|k−q|−µ)/~−ωΛ(q) aEnqd.(Aea2c)hasinddesivcirdibueadl acboonvterifbourtλio=n λis′ acnadlcvul=atve′d. Hfreorme,
n +f (λ˜ǫ ) wefindthat,incontrasttothecontributionsΣSO1(k,iν )
+ iνn−(λ˜qΛǫ|k−qF|D−µ)|/k~−q+| ωΛ(q)# aonpdticΣalSλOp2h(okn,oiνnns),,ΣtΓh(eiνco)ntarnibdutΣioKn(siνfr)o,mdothneλotgrdaeppheennnde
n n
on the band or momentum k.
with the Fermi-Dirac and Bose-Einstein distribution
functions, f (ǫ) = n (ǫ µ) and n = n [~ω (q)], The contribution due to Coulomb impurity scattering
FD FD − qΛ BE Λ reads as
and the chemical potential µ=µ(T) at the temperature
T. Thus, in the lowest order, the self-energy is simply
the sum of the contributions from the different phonons ΣCλλo′,vv′(k)≡δvv′δλλ′ΣCo(k), (A5)
Λ.
where
In order to calculate the self-energy, we need to
know the products of the electron-phonon coupling i~ iπn d2k′ 2πe2 2
matrix elements entering Eq. (A2), v˜ Mvλv˜λ˜,Λ(k − ΣCo(k)= 2−τ(k) =− 2 i Z (2π)2 (cid:12)(cid:12)κqε(q)(cid:12)(cid:12) δ(ǫk−ǫk′)
q,q)Mv˜λ˜vλ′′,Λ(k,−q), for the Γ, K, SO1, anPd SO2 modes. (1−co(cid:12)(cid:12)sθ)(1+(cid:12)(cid:12)cosθ) (A6)
Since n = n , it follows from Eqs. (A2) and (5)
that theqΓTcOontribuqΓtLioOn due to the optical phonons near withθ ≡θk−θk′,q=k−k′,thedielectricfunctionε(q),
andthe impurityconcentrationn.66 Since thiscontribu-
the Γ point is diagonal in the valley and band quantum i
tion is also diagonal, the total self-energy
numbers. Wecalculatetheself-energybyusingthetrans-
2foDrminatteigornalkS′/=(2kπ)−2 qdakn′kd′redpθla,cwinhgerteheSs=umNAPk′anbydθthies Σλλ′,vv′(k,iνn)=δvv′δλλ′ −2τi(~k) +Σpλh(k,iνn)
(cid:20) (cid:21)
chosen to be the angRle betwReen k and k′. ≡δvv′δλλ′Σλ(k,iνn)
Equation (6) describes the coupling of the K mode (A7)
TO
to the electrons in graphene. As above, the self-energy and, consequently, the Green’s function
contribution from the K phonons is calculated by in-
ttAerfortmdeursccpinoegnrftoakrimn′ iinangngdet±hwiθeriiintnintEegqg.rta(h6tei)onvsaunomvisehrasatnhadec2aoDnngsleienqtuθe,egnrttahllye. Gvλvλ′′(k,iνn)= iνn−[λǫk−δvµv′+δλΣλ′λ(k,iνn)]/~ (A8)
the contribution to the self-energy is diagonal with re- ≡δvv′δλλ′Gλ(k,iνn)
spect to the band quantum number. Moreover, Eqs. (6) are diagonal with respect to λ and v in our model. Fi-
and(A2)make itclearthat, eventhoughthe matrix ele- nally, the spectral function is obtained from the Green’s
mentMvλvλ′′,KTO(k,q) describesintervalleyscattering,the function via
second-ordercontributionfromtheK phonontotheself-
energy is diagonal also in the valley quantum number. (k,ω)= 2Im[ (k,ω+i0+)]. (A9)
Aλ − Gλ
8
In this work, we are interested only in the imaginary where the sums over α and β refer to the matrix γ. The
parts of the retarded self-energy. Upon replacing iν by charge current density j(r) of this state Ψ(r) can be de-
n
ω+i0+ inEq.(A2),the imaginarypartofeachcontribu- termined by a variational method:
tionΛinEq.(A2)containsaDirac-δfunction[sincethere
is no contributionfromIm(M˜Λ) asdiscussedabove]. Af- δE =E[A+δA] E[A]= d2r j(r)δA(r), (B7)
ter introducing k′ and writing the sum as a 2D integral, − −
Z
the Dirac-δ function canbe usedto calculate the k′ inte-
which yields
gral,whichthenyieldsEq.(12)fortheΓandK phonons
and Eqs. (13) and (14) for the SO and SO phonons.
1 2 ˆj(r)= ev Ψ∗(r)(γ) Ψ (r). (B8)
− F α αβ β
αβ
X
Appendix B: Kubo formula for the optical
PromotingthewavefunctionsinEq.(B8)tofieldoper-
conductivity
ators, using the eigenbasis given by Eqs. (B2)-(B4), and
taking into account the spin degeneracy, the charge cur-
1. Current density operator
rent density operator can be determined as
muOlaurfosrttahretinopgtipcoailnctonindutchteivditeyriivsatthioencuorfreantKoupbeoratfoorr-. ˆj(q)= dvλλ′(k,q)cˆ†λksvcˆλ′(k+q)sv (B9)
kλλ′sv
In the presence of an arbitrary magnetic vector poten- X
tial A(r), the (first-quantized) 2D Dirac Hamiltonian of in reciprocal space and as
graphene reads as4
1
Hˆe =vFγ.πˆ (B1) ˆj(r)= S eiq.rˆj(q), (B10)
q
X
withπˆ =pˆ+eA(r)beingthe 2Dkineticmomentumop-
in real space. Here, the dipole matrix elements read as
erator, pˆ the 2D momentum operator, and the matrices
γx = σx 1, γy = σy τz, γz = 0, where 1 is the 2 2 ev
unityma⊗trixandσ an⊗dτ arePaulimatricesreferring×to dK′ (k,q)= − F λ′ei(θk+θk+q)/2+λe−i(θk+θk+q)/2 ,
λλ,x 2
the A/B sublattices and the K/K′ points, respectively. iev h i
As discussed in Ref. 4, the 2D momentum k and the dK′ (k,q)= F λ′ei(θk+θk+q)/2 λe−i(θk+θk+q)/2 ,
λλ,y 2 −
valley K/K′ are good quantum numbers and the Hamil-
h ′ i∗
dK (k,q)= dK (k,q) .
tonian (B1) has the (valley-degenerate) eigenvalues λλ′,x/y λλ′,x/y
(B11)
(cid:2) (cid:3)
ǫ (k)=λǫ =λ~v k (B2)
λ k F
and the corresponding eigenstates 2. Kubo formula
eik.r e−iθk/2 If an external electric field
Ψλ (r)= χ (B3)
K,k √2S λeiθk/2 !⊗ K
∂A(r,t)
E(r,t)= (B12)
near the K point and − ∂t
isappliedtothe systemconsideredhere,itseffectcanbe
eik.r eiθk/2
ΨλK′,k(r)= √2S λe−iθk/2 !⊗χK′ (B4) described by
Hˆ (t)= d2rˆj(r).A(r,t) (B13)
near the K′ point, where S = NA denotes the sur- ext −
Z
face area of the graphene sample, λ = 1, θ =
± k
arctan(k /k ), and with the charge current density operator (B10). The to-
x y
talHamiltonianoftheproblemthenreadsasHˆ+Hˆ (t),
ext
1 0 where Hˆ is given by Eq. (4).
χ = , χ ′ = . (B5)
K 0 ! K 1 ! Using linear response theory [for the unperturbed
Hamiltonian Hˆ and the perturbation Hˆ (t)] and con-
ext
For an arbitrary (normalized) state Ψ(r), the energy ductingaFouriertransformationwithrespecttothetime
expectation value as a functional of the vector potential andposition,50,64,65 we find that the currentdensity due
A(r) is given by to the external field is given by
1
E[A]= d2r Ψ∗(x,y) Hˆ Ψ (x,y), (B6) δ ˆj (q,ω) = ΠR (q,ω)A (q,ω) (B14)
α e αβ β h α i −~ αβ β
Xαβ Z (cid:16) (cid:17) Xβ
9
withΠR (q,ω)beingthe(Fouriertransformed)retarded where
αβ
current-current correlation function and α and β refer-
ring to the in-plane coordinates x and y. The retarded
correlation function ΠR (q,ω) can be related to the
αβ ′ ∗
imaginary-time correlation function Dλλ (k,q) dK′ (k,q) dK′ (k,q)
αβ ≡ λλ,α λλ,β (B18)
′ ′ ∗
~β =dKλλ′,α(k,q)(cid:2)dKλλ′,β(k,q)(cid:3)
1
Παβ(q,iωn)= dτ ˆjα(q,τ)ˆjβ( q,0) eiωnτ (cid:2) (cid:3)
−S T −
Z0 D h iE
(B15)
is a real number.
by ΠR (q,ω) = Π (q,ω+i0+), that is, by replacing
αβ αβ
iω withω+i0+ inEq.(B15).50,64,65 Here,iω denotesa
n n Here, we are interested in the response to a uniform
bosonic frequency. Hence, the Kubo formula for the real
field, that is, in the case q = 0, for which Eq. (B17)
part of the conductivity reads as
becomes
Im ΠR (q,ω)
αβ
Re[σ (q,ω)]= . (B16)
αβ − h ~ω i
Re[σ (0,ω)]=δ σ(ω) (B19)
αβ αβ
If vertex corrections due to phonons in Eq. (B15) are
ignored, the phonon-dressed Green’s functions given by
Eq.(A8)expressedviatheirspectralfunctions(A9),and
the sum over k rewritten as a 2D integral, we arrive at withσ(ω)givenbyEq.(15). InordertoobtainEqs.(15)
and(B19),wehaveusedthatonlyDλλ′(k,0)dependson
2 d2k dω′ αβ
Re[σ (q,ω)]= the angleofthe k-integrationinEq.(B17)forq=0and
αβ ~ω (2π)2 2π 2π
Xλλ′ Z Z (B17) that dθ Dλλ′(k,0)=πe2v2δ .
×Dλαλλβ(′k(k,ω,q′))[nλF′D((k~ω+′)q−,ωn′F+D(ω~)ω,+~ω′)] R0 k αβ F αβ
×A A | |
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