Transport coefficients of graphene: Interplay of impurity scattering, Coulomb interaction, and optical phonons

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by  Hong-Yi Xie

Transport coefficients of graphene: Interplay of impurity scattering, Coulomb
interaction, and optical phonons
Hong-Yi Xie1,∗ and Matthew S. Foster1,2
1Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
2Rice Center for Quantum Materials, Rice University, Houston, Texas 77005, USA
(Dated: May 3, 2016)
We study the electric and thermal transport of the Dirac carriers in monolayer graphene using
the Boltzmann-equation approach. Motivated by recent thermopower measurements [F. Ghahari,
H.-Y. Xie, T. Taniguchi, K. Watanabe, M. S. Foster, and P. Kim, Phys. Rev. Lett. 116, 136802
6
(2016)], we consider the effects of quenched disorder, Coulomb interactions, and electron–optical-
1
phononscattering. ViaanunbiasednumericalsolutiontotheBoltzmannequationwecalculatethe
0
2 electrical conductivity, thermopower, and electronic component of the thermal conductivity, and
discuss the validity of Mott’s formula and of the Wiedemann-Franz law. An analytical solution for
y
the disorder-only case shows that screened Coulomb impurity scattering, although elastic, violates
a
theWiedemann-Franzlawevenatlowtemperature. Forthecombinationofcarrier-carrierCoulomb
M
andshort-rangedimpurityscattering,weobservethecrossoverfromtheinteraction-limited(hydro-
dynamic)regimetothedisorder-limited(Fermi-liquid)regime. Intheformer,thethermopowerand
2
thethermalconductivityfollowtheresultsanticipatedbytherelativistichydrodynamictheory. On

the other hand, we find that optical phonons become nonnegligible at relatively low temperatures
]
l and that the induced electron thermopower violates Mott’s formula. Combining all of these scat-
l teringmechanisms,weobtainthethermopowerthatquantitativelycoincideswiththeexperimental
a
h data.
-
s PACSnumbers: 72.80.Vp,72.15.-v,67.10.Jn,72.10.-d
e
m
. Contents B. RPA screening of Coulomb interaction,
t
a cancellation of the collinear collision
m I. Introduction 1 singularity, and plasmon enhancement in
- channel C 17
d
II. Main results 3
n
A. Quantum kinetic equation 3 References 19
o
c B. Benchmark: Impurity-only transport 4
[ C. Crossover from interaction-limited regime to

disorder-limited regime 4
2 I. INTRODUCTION
D. Optical-phonon-limited transport 6
v
2 E. All scattering mechanisms; comparison to
Electricandthermaltransportsinmonolayergraphene
6 thermopower measurements 6
are influenced by various scattering mechanisms, such
8
5 III. Boltzmann equation in the presence of as quenched impurities, interparticle interactions, and
0 impurities, Coulomb interaction, and phonons [1–25]. In weakly disordered graphene, inter-
1. optical phonons 8 action effects can become dominant at non-zero tem-
0 A. Collision integrals 8 perature; in the vicinity of zero doping the Coulomb-
6 B. Solution of linearized Boltzmann equation 11 interacting massless Dirac carriers form a relativistic
1 electron-hole plasma. In this interaction-limited regime,
1. Collision matrix 12
: hydrodynamic theory [16–18] predicts intriguing non-
v 2. Orthogonal polynomials 13
i 3. Thermodynamics 13 Fermi-liquid transport properties. First, the electron-
X
hole fluid exhibits a finite and nonvanishing dc electrical
C. Transport coefficients 14
r conductivityattheDiracpointevenintheabsenceofim-
a 1. Impurity-only transport 14
purities, due entirely to inelastic electron-hole collisions.
2. Interaction-limited transport 14
Moreover,Mott’sformula[26]andtheWiedemann-Franz
Acknowledgments 15 law [27] are violated. In a Fermi liquid, these respec-
tively determine the thermoelectric power and the elec-
A. Elliptic coordinate system for the Coulomb tronic component of the thermal conductivity from the
collisions (3.31) 15 electrical conductivity. Instead, for graphene in the hy-
drodynamic regime the thermopower at non-zero doping
approaches the thermodynamic entropy per charge, and
the thermal conductivity at the Dirac point diverges as
∗Electronicaddress: [email protected] the impurity concentration vanishes.
2
The theory predicts upper bounds for the thermoelec- addition we disregard the effects of external magnetic
tric power and electronic component of the thermal con- fields or spin-flip mechanisms.
ductivity, limited only by disorder. While violating clas- In this paper we consider the inelastic optical-phonon
sical relations between thermoelectric coefficients, the scattering and model graphene by the Hamiltonian
latterarestronglyconstrainedandinterrelatedbytherel-
ativistichydrodynamics[16,17]. Weemphasizethatthis H =H +H +V(s) +V(l) +V +V , (1.1)
0 oph imp imp int e-oph
violation of the Mott and Wiedemann-Franz relations is
different from the usual physics of narrow-gap/gapless where H describes the free Dirac fermions, H the
0 oph
semiconductors, for example, the bipolar diffusion pro- optical-phonon bath, V(s) and V(l) the quenched short-
imp imp
cess [28], where separated electron and hole currents are
rangedandlong-ranged(Coulombimpurity)disorderpo-
assumed. Strong inelastic electron-hole scattering in ul-
tentials, respectively, V the interparticle Coulomb in-
int
traclean graphene implies that a composite electron-hole
teractions, and V the electron–optical-phonon cou-
e-oph
fluid emerges [17], which cannot be decomposed into va-
pling. We assume that both the time-reversal symmetry
lence and conduction band components.
and spin SU(2) rotation symmetry are preserved in the
Three very recent experiments [29–31] have provided presence of disorder and interactions. We also presume
substantial evidence for interaction-limited transport in that the particle-hole symmetry as well as the honey-
graphene. Measurements of the electronic component comb lattice space group symmetries (translations, rota-
of the thermal conductivity near charge neutrality [29] tions, and reflections) are preserved under disorder aver-
showed large violations of the Wiedemann-Franz law age [13]. Concretely, each term in the Hamiltonian (1.1)
[32, 33]. Non-local transport in doped graphene [30] was is constructed as follows.
used to probe the viscosity of the electron fluid [34–37]. Theshort-rangedimpurityHamiltonianV(s) takesthe
imp
Finally, thermoelectric power measurements [31] showed form as introduced in Ref. 13, which incorporates five
a substantial deviation from the Mott formula. In this types of time-reversal-symmetric disorder, all assumed
work, we model the experiment in [31] using the Boltz- to be zero-mean, short-ranged, and Gaussian-correlated.
mann equation to incorporate carrier-impurity, carrier- Five independent parameters {g ,g ,g ,g ,g } char-
u A A3 m v
carrier, and carrier-optical phonon scattering mecha- acterize their statistical fluctuations. In the Boltzmann
nisms. equation these parameters appear effectively in certain
The thermopower measurements in Ref. 31 were combinations [G in Eq. (3.1)]. The term V(l) gives
0,f,b imp
performed on high-mobility graphene encapsulated by the scalar potential due to Coulomb impurities. These
hexagonal-boron-nitride. The experiment was done over are subject to the temperature and density-dependent
a large span of dopings, with charge-carrier density n ≡ static screening by the electron-hole plasma [6, 7].
ρ/(−e) ranging from zero to ±3.0×1012cm−2 [ρ is the Coulomb interactions between carriers are encoded in
charge density and e>0 is the elementary charge]. The V . Dynamical screening is treated within the ran-
int
measurementswereperformedatrelativelyhightemper- dom phase approximation [14, 40–42]. Screening is
atures (130K (cid:46) T (cid:46) 350K) in order to fulfill the non- crucial both in the low-temperature degenerate Fermi
degenerate condition kBT (cid:38) µ over much of the doping liquid phase, but also in the high-temperature, non-
span, while avoiding the electron-hole puddle regime at degenerate regime of primary interest here. Different
lowtemperaturesnearchargeneutrality[2,38,39]. Here from a single component plasma, graphene ultimately
µdenotesthechemicalpotential,determinedbythetem- screens better at higher temperatures, due to the pro-
perature and the fixed charge-carrier density n. In this liferation of thermally-excited electron-hole pairs. At in-
regime the measured thermopower is consistently larger termediate temperatures and finite charge density, the
thanthatpredictedbyMott’sformula[26],butsaturates Thomas-Fermi length reaches a maximum. The interac-
belowtheidealhydrodynamicprediction. Thisnovelfea- tion strength is encoded in the fine structure constant
ture suggests that in order to quantitatively characterize α that depends on the dielectric constants of the sub-
int
thethermoelectrictransportingraphene,oneshouldcon- strates [8]. In the kinetic theory, dynamical screening
sider additional inelastic scattering mechanisms. suppresses the “collinear” singularity of the Coulomb
We exclude acoustic phonons, since at low doping collision integral, which is due to the linear dispersion
the electron–acoustic-phonon scattering [19, 20] is quasi- of Dirac fermions and the energy conservation (see Ap-
elastic and incapable of producing large violations of pendix A). Note that for simplicity we only consider
Mott’s formula. As discussed in Ref. 20, the Bloch- two-bodycollisionprocessesthatpreservethepopulation
Gru¨neisen temperature T ≡ 2(cid:126)v k /k plays the key of electrons and holes separately. We leave the effects
BG a F B
role, where v is the acoustic phonon velocity and k of (three-body or impurity-assisted) electron-hole Auger
a F
the Fermi wavevector. Assuming the acoustic phonon imbalance relaxation processes [17] to future study.
velocity equal to 2.6×106 cm/s, one can estimate the Three types of in-plane optical-phonon modes [43] al-
√
Bloch-Gru¨neisen temperature as T ≈ 70 n K, where lowed by time-reversal symmetry [44, 45] couple to elec-
BG
thedensitynismeasuredinunitsof1012 cm−2. Theex- trons. ForsimplicityweconsideronlytheA(cid:48) modesthat
1
periment in Ref. 31 is performed in the regime T (cid:38)T correspond to the “Kekul´e” vibration of the honeycomb
BG
wheretheacoustic-phonon–scatteringisquasi-elastic. In lattice and couple the electrons between K and K(cid:48) val-
3
tering. These involve different kinematic regions of fre-
quency ω and momentum q transfer across the Coulomb
line,aschannelsAandBhave|ω|≤v q(“quasi-static”),
F
whilechannelChas|ω|≥v q (“optical”). Plasmonsap-
F
pear in channel C.
This paper is organized as follows. In Sec. II we
present the main results of our calculations and inter-
pret the experimental data in Ref. 31. In Sec. III we
transcribe the Boltzmann equation that is derived via
the Schwinger-Keldysh formalism, with the collision in-
tegrals for the impurity scattering, Coulomb interaction,
and electron–optical-phonon scattering corresponding to
theFeynmandiagramsdepictedinFig.1. Thenweintro-
duce the orthogonal-polynomial method for solving the
linearized Boltzmann equation. Results for impurity-
only and interaction-limited transport are discussed in
more detail in Sec. IIIC. The collinear singularity of
the Coulomb collision integrals and the RPA dynamical
screening are discussed in the Appendix.
FIG. 1: Diagrams representing the collision integrals in
Eq. (2.3). The arrows indicate the flow of electric charge
only and the wave vector labels correspond to incoming and
outgoing fermions on the left and right of the scattering ver-
tices, respectively. (a) Static (both short- and long-ranged) II. MAIN RESULTS
impurityscattering. (b)Coulombcollisionprocessesthatpre-
serveelectronandholenumbersseparately. Thelabelλ∈±1
In general one has the linear response relations [47]
denotes electrons (+1) or holes (−1). (c) Carrier–optical-
phonon scattering: (i) and (ii) Phonon absorption; (iii) and
J=σE +σα (−∇ T), (2.1a)
(iv) Phonon emission. ∞ r
J =Tσα E +(cid:0)κ +Tσα2 (cid:1)(−∇ T), (2.1b)
Q ∞ ∞ ∞ r
whereJisthechargecurrent,J theheatcurrent,E the
leys [44, 45]. The A(cid:48) phonons have been suggested to Q
1 electrochemicalfield,∇ T thetemperaturegradient,and
be the most relevant optical-phonon branch for influenc- r
σ, κ , and α are the electrical conductivity, thermal
ingelectricaltransportatrelativelylowtemperature[21], ∞ ∞
conductivity,andthermoelectricpower,respectively. We
possessingthelowestexcitationenergyandthestrongest
use the subscript “∞” to indicate bulk thermoelectric
coupling to electrons. We note that in the context of the
transport coefficients. In a finite (mesoscopic) sample,
BoltzmannequationthecollisionintegralforA(cid:48) phonons
1 slow imbalance relaxation (recombination-generation)
can also qualitatively describe the effect of the other
can give rise to different transport coefficients and/or
optical-phonon branches. Similar to the case of short-
a spatially inhomogeneous response [17], but we do not
rangedimpurityscattering,thecollisionintegralsfordif-
consider this possibility here. The Lorenz ratio is
ferentoptical-phononbranchesaredistinguishedonlyby
the factors (1±pˆ ·qˆ)/2 that enhance the electron for- κ
L≡ ∞, (2.1c)
ward (+) or backward (−) scattering. Furthermore, we σT
usethesingle-modeEinsteinmodel(dispersionless)H
opt
to describe the A(cid:48) phonons; the electron-phonon cou- forwhichwediscussthevalidityoftheWiedemann-Franz
plingVe-opt takest1heformintroducedinRefs.21,44,45. law L0 = π2kB2/(3e2) [27]. Solving the linearized Boltz-
Two parameters are present: The A(cid:48)-phonon frequency mann equation (3.6), inserting the distribution function
1
ω and electron-phonon coupling β (doping and tem- solution into Eq. (3.15), and comparing the result to
A(cid:48) A(cid:48)
perature dependent, see the discussion in Sec. IIE). We Eq. (2.1), we obtain the transport coefficients.
inadditionassumethatthephononsareinthermalequi-
librium,thatis,thephononkineticsarenotinvolved(no
drageffectonelectrons)sincetheoptical-phonondisper-
sion is weak [46]. A. Quantum kinetic equation
All of the scattering mechanisms that we consider are
depicted in Fig. 1. In particular, the Coulomb inter- We derive the quantum kinetic equation for electron
action mediates three scattering channels that we label (λ = +1) and hole (λ = −1) distribution functions
A, B, and C. Channel A describes intraband carrier- f (p,r,t) via the Schwinger-Keldysh technique [48],
λ
carrier scattering. Channels B and C encode interband where p is the quasiparticle wave vector, r the position,
conduction electron-valence hole (“electron-hole”) scat- andtthetime. Inthepresenceofanelectricdrivenfield,
4
the stationary Boltzmann equation takes the form hancestheLorenzconstantbyafactorL/L = 21 =4.2.
0 5
The transport coefficients due to the combination of
(cid:20) (cid:21)
λe
short-ranged disorder and screened Coulomb impurities
v ·∇ − E·∇ f (p,r)=St [{f }], (2.2)
F r (cid:126) p λ λ λ(cid:48)
areshowninFig.2(i)–(iii). Wecompareresultsobtained
by the orthogonal-polynomial algorithm to the exact re-
wherev istheFermivelocityparalleltothewavevector
F sults evaluated by Eq. (3.44). We used the dimension-
p,e>0istheelementarycharge,andEisthetotalelec-
less short-ranged impurity strength g˜ and Coulomb im-
tric field. The collision integral St [{f }] incorporates
λ λ(cid:48) puritydensityn [Eq.(3.1c)]determinedbyfittingthe
imp
thethreescatteringmechanismsintheHamiltonian(1.1)
low-temperature,density-dependentconductivitydatain
(see also Fig. 1),
Ref. 31. Thomas-Fermi screening is limited by the fine
structure constant
St [{f }]=St [f ]+St [{f }]+St [{f }],
λ λ(cid:48) imp,λ λ int,λ λ(cid:48) oph,λ λ(cid:48)
(2.3) 2e2
α = , (2.6)
where Stimp,λ[fλ] describes elastic scattering induced int (κ +κ )(cid:126)v
1 2 F
by impurities [Fig. 1(a)], St [{f }] the inelastic
int,λ λ(cid:48)
Coulomb scattering between quasiparticles [Fig. 1(b)], where κ denotes the permittivities of the media above
1,2
and Stoph,λ[{fλ(cid:48)}] the inelastic scattering of carriers by and below the graphene sheet. Here we take αint = 0.6,
optical phonons [Fig. 1(c)]. appropriateforBNencapsulation[31]. Wekeeptheorder
Assuming that the distribution functions f (p,r) of the polynomial basis up to N =16 in order to recover
λ=±1
are diagonalin valley andspin space[49], we present the the analytical result. We observe that in the presence
explicitexpressionsforthecollisionintegralsinEq.(2.3) of Coulomb impurities the Wiedemann-Franz law is in
in Sec. IIIA. In the hydrodynamic regime the response general broken. As shown in Fig. 2(iii), the Lorenz ratio
to static fields is dominated by the zero modes of the L is a function of the charge density n and temperature
inelastic carrier-carrier collision integrals, associated to T for a fixed α .
int
energy and momentum conservation [16, 17]. For this There exist two interesting limits: When T → ∞
reason we also neglect the weak off-diagonal components the effective long-ranged impurity strength vanishes
in electron-hole space, which do not directly contribute [Eq. (3.9c)], so that short-ranged impurity scattering
to the dc response [11]. dominatestransportandWiedemann-Franzlawrestores.
When T → 0 the long-ranged impurity becomes domi-
nant. At the charge neutral point n = 0, the Thomas-
B. Benchmark: Impurity-only transport Fermi wavevector QTF divided by the temperature be-
comes a constant [Eq. (B8)]. The Lorenz ratio is en-
hanced relative to the Wiedemann-Franz law by a nu-
Inthepresenceofonlyelasticscattering[seeFig.1(a)]
mericalconstantdependingonthefinestructureconstant
the linearized Boltzmann equation can be solved ex-
(L/L ≈ 2.093 for α = 0.6). The Wiedemann-Franz
actly(Sec.IIIC1). Mott’sformulaforthethermoelectric 0 int
lawisrecoveredatsufficientlyhighchargedensitiesn(cid:54)=0
power α manifestly applies [27], although the integral
∞
and/or temperatures.
form must be employed away from Fermi degeneracy.
In Fig. 2(iv) we show the Lorenz ratio L as a func-
The short-ranged-impurity only transport coefficients
tion of the fine structure constant α [appearing in
take simple expressions int
the Thomas-Fermi wavevector Eq. (B7)] at charge neu-
e2 π2k2T trality in the absence of short-ranged impurity scatter-
σ(s) =N g−1, α(s) =0, κ(s) =N B g−1.
imp h(cid:101) ∞,imp ∞,imp 3h (cid:101) ing. It is clear that for any finite αint the Wiedemann-
(2.4) Franz law is broken. Especially, for α → 0 we obtain
int
where g is the effective dimensionless short-ranged dis- L/L = 21 = 4.2 [Eq. (2.5)], which provides an upper
(cid:101) 0 5
order strength [Eq. (3.28)]. Note that the Wiedemann- bound for the Lorenz ratio induced solely by impurities.
Franzlawismanifestlysatisfied. TheparameterN isthe
numberofindependent2-componentDiracspecies,equal
to four in graphene.
C. Crossover from interaction-limited regime to
Another analytically solvable limit is the long-ranged-
disorder-limited regime
impurity-only case in the absence of screening [see
Eq.(3.45)]. Especially,atthechargeneutralpointµ=0
We combine Coulomb interactions [see Fig. 1(b)] and
the Lorenz ratio is independent of temperature,
short-rangedimpuritytoverifythepredictionsoftherel-
21π2k2 ativistichydrodynamictheory[16–18]. Thetransportco-
L(∞l),imp(QTF →0,µ=0)= 5 3e2B. (2.5) efficientsobtainedbythenumericalsolutionoftheBoltz-
mannequationareshowninFig.3. Closetochargeneu-
Here Q denotes the temperature and density- trality µ (cid:46) k T the conductivity [Fig. 3(i)] remains fi-
TF B
dependent Thomas-Fermi wavevector [Eq. (B7)]. Equa- nite. Thisreflectsthe“minimal”conductivityduetothe
tion (2.5) violates the Wiedemann-Franz law and en- electron-hole collisions.
5
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0.8
(cid:72)(cid:76) 2 (cid:72)(cid:68)(cid:76) mic renormalization effects [13]). This is very different
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0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5
n 1012cm(cid:45)2 n 1012cm(cid:45)2 impurity density, and the only other length scale is the
22ΠkLB23e1221....0055(cid:237)(cid:227)(cid:233)(cid:243)(cid:64)(cid:68)(cid:72)i(cid:237)(cid:227)(cid:233)(cid:243)i(cid:237)(cid:243)(cid:227)(cid:233)i(cid:76)(cid:237)(cid:243)(cid:227)(cid:233)(cid:237)(cid:243)(cid:227)(cid:233)(cid:237)(cid:243)(cid:227)(cid:233)22Πk(cid:237)(cid:227)(cid:233)(cid:243)LB(cid:237)(cid:227)(cid:233)(cid:243)(cid:64)23e(cid:237)(cid:227)(cid:233)(cid:243)(cid:237)(cid:243)(cid:227)(cid:233)112(cid:237)...(cid:243)(cid:227)(cid:233)1482.(cid:64)(cid:68)(cid:237)(cid:227)(cid:233)(cid:243)(cid:227)0(cid:237)(cid:243)(cid:227)(cid:233)L(cid:227)(cid:237)(cid:227)(cid:233)(cid:243)(cid:187)(cid:237)(cid:227)(cid:233)(cid:243)(cid:237)(cid:227)(cid:233)(cid:243)2(cid:227)5(cid:237)(cid:227)(cid:233)(cid:243).0T0(cid:237)0(cid:227)(cid:227)(cid:233)(cid:243)9(cid:237)(cid:227)(cid:233)(cid:243)(cid:64)3(cid:227)K(cid:237)(cid:227)(cid:233)(cid:243)(cid:68)1TTTT0(cid:237)(cid:68)(cid:227)(cid:233)(cid:243)(cid:227)(cid:61)(cid:61)(cid:61)(cid:61)0(cid:237)n(cid:227)(cid:233)(cid:243)00152(cid:227)(cid:237)(cid:227)(cid:233)(cid:243)(cid:61)000K(cid:237)(cid:227)(cid:233)(cid:243)00001(cid:227)0(cid:237)5(cid:227)(cid:233)(cid:243)KK0K(cid:237)(cid:227)(cid:233)(cid:243)(cid:227)0(cid:237)(cid:227)(cid:233)(cid:243)(cid:227) 222ΠLk3eB1223344.......5050505 (cid:72)(cid:64)(cid:144)(cid:68)iLv(cid:72)(cid:76)Αint(cid:61)g(cid:142)0(cid:61)(cid:76)0(cid:61)(cid:64),2n1(cid:61)(cid:144)50(cid:61)4L.2Αint(cid:174)(cid:68)(cid:165)(cid:61)1 trtstTehhoeuxhseeppIiienrsesetmltrreesitlimcvaimhhntiloeprteeduyoeantenlrrthwda–sBertiobrtiurtpsmhoehretgeaioicnclwtporiaceaanolorns-wtwebprgaesasaheserevsoierotennevnelfgxeoedinapdnntwbegetmetarosihirctvmpeha:deseteeρtert2cn(aea(0rtTrt1re0iul3a)[ni3re0Ksg∼e1ri–,n]3wht,gnd5hianig0iamrssohtecKpo-su(dwtbi)(cid:126)sseos.estmvewIeaFrandnpnv/tscteetktbuerdreBaierabT[ontld5ouuv)0,werit2n]aeer..;
1.0
0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6(cid:72) 0.8 (cid:76) 1.0 the latter can presumably be attributed to disorder-
n 1012cm(cid:45)2 Αint dominated transport [51].
Awayfromchargeneutralityandatintermediatetem-
FIG. 2: Impur(cid:64)ity-only t(cid:68)ransport coefficients as functions of
peratures, the thermopower shown in Fig. 3(ii) ap-
the charge-carrier density n for various temperatures. The
proaches the ideal clean hydrodynamic result
symbolsarethenumericalresultobtainedbytheorthogonal-
polynomial method and the solid curves are the analytical
α =s/en, (2.7)
result obtained by Eq. (3.44). In our calculations we use the ∞
parameters in Ref. 31: the effective short-ranged impurity
strengthg∼1.1×10−4,thelong-rangedimpurityconcentra- which is the thermodynamic entropy per charge; s de-
(cid:101)
tion n = 2.4×109cm−2, and the fine structure constant notes the entropy density. At higher densities/lower
imp
αint = 0.6. (i) Electric conductivity. The horizontal black temperatures, α∞ → 0, consistent with the Mott rela-
(diagonal red) dashed line indicates the conductivity in the tion [Eq. (2.4) for short-ranged impurity scattering]. For
absence of long-ranged impurities (in the absence of short- µ(cid:28)k T theLorenzratio[Fig.3(iv)]ismuchlargerthan
B
ranged impurities) at T =0. (ii) Thermoelectric power. (iii) that of a Fermi liquid, L/L (cid:29)1. Wiedemann-Franz re-
0
Lorenz ratio. The insert panel shows the Lorenz ratio as a covers far away from the Dirac point µ(cid:29)k T.
B
function of temperature at the charge neutral point n = 0.
As shown in the inset of Fig. 3(i), the rescaled min-
(iv) Lorenz ratio as a function of the fine structure constant
imal conductivity α2 σ is almost linear in the fine
[whichdeterminestheThomas-FermiwavevectorEq.(B7)]at int min
structure constant α for α (cid:46) 1, which reflects the
charge neutrality n=0 in the absence of short-ranged impu- int int
rities g=0. Coulomb screening effect. In the absence of screening
(cid:101)
α (cid:28) 1, we recover the results of Refs. 10 and 11. As
int
showninFig.3(iii),thenon-monotonicityofthethermal
In the hydrodynamic (interaction-dominated) regime conductivity κ∞ (or the Lorenz ratio L) as a function of
of primary interest, τin (cid:28) τel [16, 17]. Here 1/τin de- µ/kBT is simply a consequence of the ideal relativistic
notestheinelasticscatteringrateduetoelectron-electron thermodynamics [Eq. (3.54)]. In the interaction-limited
and electron-hole collisions, while 1/τ is the scattering regime, the enhancement of the Lorenz ratio diverges as
el
rate due to elastic electron-impurity and (quasi)elastic the strength of impurity scattering vanishes. The hy-
electron–acoustic-phonon collisions. (In this section we drodynamic enhancement will also dominate over that
neglect optical phonons, which are dealt with below.) attributable to Coulomb impurities, Eq. (2.5). For suffi-
Stronginelasticscatteringquicklyrelaxesfluctuationsto cientlyweakimpurityscatteringandintheabsenceofop-
local equilibrium. Intercarrier scattering is special how- ticalphonons,thehydrodynamicdescriptionshouldgen-
ever,inthatitpreservesthetotalenergyandmomentum erallyapply,regardlessofthescatteringmechanismsthat
of the Dirac fluid [16, 17]. This means that the distribu- lift the zero modes of the Coulomb collision operator. In
tion function for electrons and holes is always close to reality both short-ranged and long-ranged impurities are
Fermi-Diracinsomeco-movingreferenceframe, andthis simultaneouslypresent, andtheresultingtransportcoef-
translates into strong constraints on kinetic coefficients. ficients have similar features as shown in Fig. 3.
Atchargeneutrality,chargeflowisdecoupledfrommo- Ourresultforthethermopowerinthepresenceofboth
mentum flow, and can be relaxed by electron-hole colli- types of disorder and Coulomb carrier-carrier scatter-
sions alone. In the interaction-dominated regime, the ing, but in the absence of optical phonons, is shown
minimal conductance at the Dirac point is to a first ap- in Fig. 5(iii). There it is compared to the experimen-
proximation a function only of the dimensionless inter- tal results from [31]. Our numerical results monotoni-
6
2Σeh1011100100000145(cid:64)(cid:144)(cid:68)(cid:227)(cid:227)(cid:227)(cid:72)(cid:227)(cid:227)iΣ(cid:227)(cid:76)(cid:227)(cid:227)(cid:227)D(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)22ΑΣ(cid:227)eh(cid:227)min(cid:227)int(cid:227)(cid:227)(cid:227)(cid:227)024680(cid:227)(cid:227)(cid:227).(cid:227)0(cid:227)(cid:227)(cid:227)(cid:64)(cid:144)(cid:68)n(cid:227)(cid:227)(cid:227)(cid:227)0(cid:227)(cid:61)(cid:227)(cid:227).(cid:227)2(cid:227)(cid:227)0(cid:227)(cid:227)(cid:227)(cid:227)Α(cid:227)0(cid:227)(cid:227)(cid:227).i(cid:227)4n(cid:227)(cid:227)(cid:227)t(cid:227)(cid:227)Σ(cid:227)(cid:227)0(cid:227)(cid:227)(cid:227).(cid:227)6i(cid:72)(cid:227)(cid:227)ms(cid:227)(cid:227)(cid:227)(cid:76)(cid:227)(cid:227)0(cid:227)p(cid:227)(cid:227).8(cid:227)(cid:227)(cid:227)(cid:227)(cid:227) Αke(cid:165)B0246(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:72)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)i(cid:227)(cid:227)(cid:227)(cid:64)(cid:144)(cid:68)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)i(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:76)M(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)o(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)t(cid:227)(cid:227)(cid:227)t(cid:227)Α(cid:227)(cid:227)ke(cid:227)(cid:227)(cid:227)(cid:227)(cid:165)B(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)01234567(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:64)(cid:144)(cid:68)(cid:227)(cid:227)0(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227).(cid:227)(cid:227)(cid:227)5(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)1(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227).(cid:227)Μ(cid:227)(cid:227)(cid:227)0(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:144)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)k(cid:227)(cid:227)2(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227).(cid:227)(cid:227)B0(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)T(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)H(cid:227)(cid:227)5(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227).(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)0(cid:227)(cid:227)(cid:227)y(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)d(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)1(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)0(cid:227)(cid:227)(cid:227)r(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227).(cid:227)(cid:227)(cid:227)(cid:227)o(cid:227)(cid:227)0(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227) c[wtsFouoeinlTgtecssl.haeeeralc1cvtr(utreecDoold)astn].th.a–ielonowVtOpehntntipeahiectierenaiEglcleyp-qaFcprslti-hoe.grpsfoo.(ehnn34enoio.l.cc2ennec5tot)rsnro,acof-an(lntsi3tsmtp.he2oreai6rtinn)teod,gdcpo(tpht3eirroc.ffi3oalae9cncls-e)ispe,spinhssaoetonssrnn.tddooTun(leh3obe.no3angr2tele)hyr-,
0.0 0.5 1.0Μk1.5BT2.0 2.5 3.0 0 2 4 6ΜkB8T 10 12 14 We obtain the resistivity as a function of tempera-
2(cid:180)105 ture and charge-carrier density that qualitatively coin-
2ΚTkh(cid:165)B1251(cid:180)(cid:180)(cid:180)(cid:180)50111100000(cid:144)(cid:64)(cid:144)(cid:68)04445(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:72)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)i(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)i(cid:227)(cid:227)i(cid:227)(cid:227)(cid:76)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)22Lke(cid:227)(cid:227)0,hB(cid:227)(cid:227)(cid:227)(cid:144)(cid:227)(cid:227)12(cid:227)(cid:227)0025(cid:227)(cid:227)0000(cid:64)(cid:144)(cid:68)(cid:227)(cid:227)0(cid:227)(cid:227)(cid:227)(cid:227)2(cid:227)(cid:227)(cid:227)(cid:227)4(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)6(cid:227)(cid:227)(cid:227)(cid:227)8(cid:227)(cid:227)(cid:227)(cid:227)1(cid:227)(cid:227)0(cid:227)(cid:227)1(cid:227)(cid:227)2(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)222ΠLk3eB1015000015......000500(cid:64)(cid:144)(cid:68)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:72)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)i(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)v(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:76)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:144)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227) cdtpf(isrhiiee)edeprqaeaeCulsotmnepuowdnotlrlsiciesictsythaoia.olnbzntTepohlrthueheohsotrseernsoeeoirncsnnehtuecgesalemtiormargitrpdneeheeeerrtRc(ahaTpoteerufforr.rimp(cid:46)mee2ura1arlla.g1edlytn5geiT0iionmtanhKsucieedtt)sioye.vr,fcealasotatTinewhnsdhtedebirerv,aemittrtmoahyetblasohlisywnreseeterottiaavvhveckeimeettdliriyyr--.,
2000 ΜkBT 0.1 vated phonons is exponentially small when T (cid:28) T .
0 5 10 15 20 0 5 10 15 20 A(cid:48)
Μk T (cid:144) Μk T (ii) Crossover regime (150K (cid:46) T (cid:46) 400K). The resis-
B B
tance increases superlinearly in temperature. (iii) High-
FIG. 3: Transpo(cid:144)rt coefficients in the presence(cid:144) of Coulomb temperature regime (T (cid:38) 400K). The resistance in-
interactions and short-ranged disorder as functions of µ . creases linearly in temperature. For high enough tem-
kBT
Wetaketheshort-rangeddisorderstrength,thefinestructure peratures, the optical phonons play a similar role as im-
constant, and the order of the polynomial basis the same as purities, yet the scattering amplitude is enhanced by the
thoseinFig.2. Theblacksquaresarethenumericalresult. (i) Bose-Einstein distribution function f (T /T)∼T/T .
B A(cid:48) A(cid:48)
Conductivity. The horizontal blue dashed line indicates the The temperature dependence of the resistivity qualita-
disorder-only conductivity σi(ms)p in Eq. (2.4), while the red tively follows the Bose-Einstein distribution function of
dashed line is the “Drude” component of the hydrodynamic
the optical phonons.
conductivity σ given by the first term of Eq. (3.48a). The
D In the crossover regime, the electron–optical-phonon
insetpanelshowstheminimalconductivityatthechargeneu-
scattering is strongly inelastic. The thermopower
tralityasafunctionofthefinestructureconstant. Alinearfit
ofthenumericalresultgivesα2 σ ≈0.79+9.13α (black [Fig. 4(ii)] due to electron–optical-phonon scattering
int min int
dashedline). Thisisconsistentwiththeunscreenedresultin alone does not follow Mott’s formula [52]. Further-
[10,11]. (ii)Thermoelectricpower. Thetopreddashedcurve more, the electron-hole imbalance relaxation processes
is the ideal clean hydrodynamic result in Eq. (2.7) and the [Figs.1(c) and1(c) ]havesignificanteffectsatlowdop-
ii iv
bottom blue dashed curve is the result obtained from Mott’s ing.
formula. The insert panel is a semi-log plot for the hydrody-
namic regime. (iii) Thermal conductivity. The insert panel
shows the following “synthetic” Lorenz ratio: This is a plot
E. All scattering mechanisms; comparison to
of the thermal conductivity for a hydrodynamic relativistic
thermopower measurements
gas in the absence of impurities, normalized to the minimal
conductivity at charge neutrality, Eq. (3.54). (iv) Lorenz ra-
tioforgraphenewithCoulombinteractionsandshort-ranged Finallywecombineallscatteringmechanismstomodel
disorder only. The horizontal red dashed line indicates the the data of the experiment in Ref. 31. In order to inter-
Wiedemann-Franz law. pret the data we need first to estimate all the effective
parameters. Since the graphene sample is encapsulated
between two hexagonal-boron-nitride substrates we es-
timate the fine structure constant as α = 2e2/(κ +
int 1
κ )(cid:126)v ≈ 0.6 where κ = κ ≈ 3.8 is the dielectric
2 F 1 2
constant of boron nitride [8]. The dimensionless short-
ranged impurity strength g and the Coulomb impurity
(cid:101)
cally approach the ideal hydrodynamic limit [Eq. (2.7)] concentration n are determined by the conductivity
imp
withincreasingtemperature,exceptnearchargeneutral- data at low temperature and high doping, where inelas-
itywhereafiniteimpuritydensitysendsthethermopower tic scattering is negligible. According to this analysis
to zero as n→0 [Eq. (3.48b)]. The experimental results we have g ≈ 1.1×10−4 and n ≈ 3×109cm−2. Fi-
(cid:101) imp
instead show a saturation of the thermopower midway nally,theelectron–optical-phononcouplingisattainedby
between the Mott and hydrodynamic bounds. Below we fitting the electrical conductivity data at high tempera-
show that the additional inclusion of electron–optical- tures. Notethattoreachaquantitative agreementtothe
phonon scattering gives good agreement with the exper- experimentaldata,wehavetunedtheoptical-phononfre-
iment, Fig. 5(i). quency to T =2200K, which is a little bit higher than
A(cid:48)
7
2Ρhe0000000.......0123456(cid:72)fTT(cid:227)(cid:243)(cid:227)(cid:233)(cid:227)'(cid:64)(cid:144)(cid:68)BiA(cid:227)(cid:227)(cid:76)(cid:227)(cid:227)(cid:233)n00nn(cid:227)(cid:233)(cid:227)(cid:233)..(cid:61)00(cid:61)(cid:61)01(cid:227)(cid:233)..(cid:227)(cid:243)(cid:233)00553112(cid:227)(cid:243)(cid:233)(cid:72)(cid:144)(cid:76).(cid:227)(cid:243).(cid:233)..00(cid:227)(cid:243)(cid:233)00(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:137)(cid:137)c(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)m(cid:233)11(cid:227)(cid:243)(cid:233)004(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:45)011(cid:227)(cid:243)(cid:233)223(cid:227)(cid:243)(cid:233)0T(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)cc(cid:233)(cid:64)(cid:227)(cid:243)(cid:233)mm(cid:227)(cid:243)K(cid:233)8(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)0(cid:68)(cid:45)(cid:45)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)022(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)1(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)2(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)0(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)0(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:243)(cid:233)(cid:227)(cid:233)(cid:227)2ΑΜTVK(cid:165)00000.....02468 (cid:72)(cid:144)(cid:64)(cid:144)(cid:68)ii(cid:76) TTTTT(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)331220573700000KKKKK 2ΑΜTVK(cid:165)00000.....024680.0(cid:72)(cid:144)(cid:64)(cid:144)(cid:68)i(cid:76) 0M.5ott 1.0 1.5HTTTy(cid:61)(cid:61)(cid:61)dr3122o.5770000KKK2.5 2ΑΜTVK(cid:165)00000.....024680.0(cid:72)(cid:144)(cid:64)(cid:144)(cid:68)ii(cid:76)0.5 1.0MoHtty1d.5rTTTo(cid:61)(cid:61)(cid:61)132275.70000KKK 2.5
0 200 400 T600K 800 1000 1200 0.0 0.5 n 11.0012c1m.5(cid:45)2 2.0 2.5 n 1012cm(cid:45)2 n 1012cm(cid:45)2
1.0
iii iv
FftuhIneGc.etffio4en:cstivOoefpdttiiemcma(cid:64)elp-np(cid:68)eshiroaontnuloernses–leiamlnedcitterdcohna–trrogapentsicpdaoelrn-(cid:64)tpshitcoyon.eoffincWci(cid:68)eoenutpstlainkages 2K01..80 (cid:72)(cid:68)(cid:76) (cid:64) TTT(cid:61)(cid:61)(cid:61)(cid:68)312577000KKK 2K0.8 (cid:72)(cid:68)(cid:76) (cid:64) TTT(cid:61)(cid:61)(cid:61)(cid:68)312577000KKK
V (cid:144) V0.6 (cid:144)
so(itp)rteRinceagslt-ihspthα(cid:101)ivoo2inpttoyn=ρt1≡em[cσ.pf−.e1rEaqat.su(ra3e.f1Tu4An)(cid:48);ctα(cid:101)≡ioo2pn(cid:126)toω=fAtα(cid:48)e/o2mkptBp/e(≈r1a6tπ2u22r0)e]0faKonrd[v3ta1hr]e-. ΑΜT(cid:165)00..46 (cid:144)(cid:64) Hydro ΑΜT(cid:165)0.4 (cid:144)(cid:64) Hydro
ious charge densities. For comparison the insert panel shows 0.2 0.2
the Bose-Einstein distribution function of optical phonons Mott Mott
[Eq. (3.13)]. (ii) Thermoelectric power α∞ as a function of 0.00.0 0.5 1.0 1.5 2.0 2.5 0.00.0 0.5 1.0 1.5 2.0 2.5
density for various temperatures. The solid (dashed) curves n 1012cm(cid:45)2 n 1012cm(cid:45)2
showtheresultinthepresence(absence)oftheelectron-hole v
imbalancerelaxationprocesses[seethediagrams(c)iiand(c)iv (cid:64) 0.8 (cid:68) T(cid:61)170K (cid:64) (cid:68)
in Fig. 1]. 2 (cid:72)(cid:68)(cid:76) TT(cid:61)(cid:61)237500KK
K0.6
V (cid:144)
Μ
thevaluesreportedinRefs.21and45. Thereasonforthis T0.4 (cid:64) Hydro
(cid:165) (cid:144)
enhancement might be that A(cid:48) phonons are more rigid Α
0.2
due to substrate encapsulation or that higher-frequency
Mott
optical-phonon branches are also involved.
0.0
0.0 0.5 1.0 1.5 2.0 2.5
The fitting procedure described above gives an n 1012cm(cid:45)2
electron–optical-phonon coupling that increases with de-
creasing temperature, see [31] for details. This is pre-
FIG. 5: Thermopower as a f(cid:64)unction o(cid:68)f doping and tempera-
sumably due to a combination of ultraviolet renormal- ture including various scattering mechanisms, and compar-
ization[45,53]andthetemperature-dependentCoulomb ison to the experiment in [31]. The dotted (solid) curves
screening [21, 45]. We leave the theoretical study of the are the result of theory (experiment). The bottom red
electron–optical-phonon vertex for deeply inelastic en- and top blue dashed lines show the thermopower calculated
ergy and momentum transfers to future work. from the experimental conductivity data using Mott’s for-
We have calculated the thermopower for every combi- mula [31] and the ideal hydrodynamic result [Eq. (2.7)], re-
spectively. Weusethesameparametersandthetemperature-
nation of the scattering sources in Fig. 1 and present the
dependent optical-phonon-electron coupling strength as in
mostinformativeresultsinFig.5. AsshowninFig.5(i),
Ref.31. (i)Thermopowerincorporatingimpurities,Coulomb
our theoretical result coincides quantitatively well with
channels A and B, and all electron–optical-phonons scatter-
the experimental data if we take into account impuri-
ing processes depicted in Fig. 1(c). The “optical” electron-
ties, optical phonons, and Coulomb interactions, yet ne- hole Coulomb scattering channel C [Fig. 1(b) ], which
iii
glect the electron-hole optical scattering channel C. Fig- shows a plasmon-enhancement in the RPA, is excluded by
ure 5(ii) indicates that the result of the Mott’s relation hand. (ii) Thermopower incorporating only short- and long-
(reddashedline)merelyreflectstheimpurity-only(both ranged impurities. (iii) Thermopower incorporating impu-
short- and long-ranged) thermopower at high doping. rities and Coulomb channels A, B, and C, neglecting opti-
Fig. 5(iii) shows the results in the absence of opti- cal phonons. (iv) Thermopower incorporating all scattering
mechanisms, including the Coulomb channel C. (v) Ther-
cal phonons, but including short-ranged and Coulomb-
mopowerincorporatingdisorder,CoulombchannelsAandB,
impurityscattering,aswellascarrier-carrierchannelsA,
and optical phonons, but neglecting the optical-phonon me-
B,andC[Fig.1(b)]. Althoughgrapheneisrelativelyde-
diated electron-hole imbalance relaxation processes depicted
generate for n ≥ 1012 cm−2 [T = 1350 K], the Mott
F in Figs. 1(c) and 1(c) .
ii iv
relation is not recovered for the measured temperatures.
At these high densities, this is due to the pure intraband
electron-electron scattering in channel A. The disorder
is so weak in the experiment that we would need very dynamic in a very clean sample. We estimate that at
high densities to observe Fermi liquid behavior; in other T = 170 K, the Mott relation would be recovered only
words, it is possible to be both degenerate and hydro- at densities above n∼1013 cm−2. Comparing Fig. 5(iii)
8
(resultsintheabsenceofopticalphonons)toFig.5(i),we Comparing Fig. 5(iv) (which includes channel C) to
observe that the optical phonons significantly suppress Fig. 5(i) (which neglects it), we conclude that the as-
the thermopower at higher temperatures and drive the sociated plasmon enhancement [33, 55–57] is somehow
system further away from an ideal hydrodynamic fluid. suppressed in the experiments. We propose that this
The thermopower α is well-defined and given by suppressionmaybeduetoadditionalscreeningbymetal-
Eq. (2.7) in the absence of a mechanism for momentum lic gates that soften the plasmon dispersion, or damping
relaxation. Away from charge neutrality however, even induced by the plasmon–optical-phonon coupling [58],
within the hydrodynamic regime some such mechanism which is not accounted for in our treatment. Com-
is necessary to separately define σ and σα in Eq. (2.1a). paring Fig. 5(v) [results in the absence of electron-hole
Ingeneraltheratioαisalsosensitivetothismechanism. imbalance relaxation processes due to optical phonons,
Herethisroleisfilledbyeitherdisorderoropticalphonon Figs. 1(ii) and 1(iv)] to Fig. 5(i), we observe that these
scattering. In particular, Coulomb impurities are poorly processes also significantly affect the thermopower at
screened at low temperatures for charge-carrier densities lower charge densities.
not too large, while optical phonons become important
at higher temperatures. As discussed in Sec. IIIC2, the
optical-phonon scattering becomes nonnegligible when
thecollisionmatrixelements[Eqs.(3.28)and(3.32)]sat-
III. BOLTZMANN EQUATION IN THE
isfy (M ) (cid:38) (M ) . For a charge-carrier density
opt 00 imp 00 PRESENCE OF IMPURITIES, COULOMB
n ∼ 1012cm−2 (TF ≈ TA(cid:48) ≈ 2000K) and temperature INTERACTION, AND OPTICAL PHONONS
T < 350K, this leads to T (cid:38) T∗ ∼ T /ln(105α2 ) ∼
F (cid:101)opt
200K via a simple estimation [54], based on the param-
A. Collision integrals
eters in the experiments [31].
The plasmon pole in the dynamically-screened
Coulomb interaction can enhance the electron-hole scat- The elastic collision integral in Eq. (2.3) gives Fermi’s
tering in the Coulomb channel C [Fig. 1(b) ]. This golden rule amplitudes associated to the diagram in
iii
mechanismcouldstrengthenthehydrodynamicresponse. Fig. 1(a), and reads
St [f ]=St(s) [f ]+St(l) [f ], (3.1a)
imp,λ λ imp,λ λ imp,λ λ
where St(s) and St(l) describe the short- and long-ranged impurity scattering, respectively,
imp,λ imp,λ
(cid:90) (cid:20) (cid:18)1+pˆ·qˆ(cid:19) (cid:18)1−pˆ·qˆ(cid:19)(cid:21)
St(s) [f ]= δ((cid:15) −(cid:15) ) G +G +G [f (q,r)−f (p,r)], (3.1b)
imp,λ λ q p 0 f 2 b 2 λ λ
q
2πn (cid:90) (cid:18)1+pˆ·qˆ(cid:19)
St(l) [f ]= imp δ((cid:15) −(cid:15) ) |U (ω =0,|p−q|)|2[f (q,r)−f (p,r)]. (3.1c)
imp,λ λ (cid:126) q p 2 eff λ λ
q
In Eq. (3.1b) the effective short-ranged impurity strengths are G = (2π)2(2g +g ), G = (2π)2g , and G =
0 A A3 f u b
(2π)2(2g +g ) [13]. In Eq. (3.1c) the long-ranged impurity scattering is characterized by the Coulomb impurity
m v
number per unit area n and the static RPA Coulomb interaction |U (ω = 0,k)|2 [see Appendix B]. The Dirac
imp eff
delta function δ((cid:15) −(cid:15) ) enforces energy conservation. The terms associated to the factors (1±pˆ·qˆ)/2 describe the
q p
enhancement of forward (+) and backward (−) scattering. In Eq. (3.1), we have introduced the shorthand notation
(cid:90) (cid:90) d2q
≡ .
(2π)2
q
The Coulomb collision integral is evaluated at the RPA level associated to the three scattering processes depicted
9
in Fig. 1(b),
N (cid:90) 1+pˆ·pˆ 1+pˆ ·pˆ
St [{f }]= 2 3 4 (2π)3
int,λ λ (cid:126) 2 2
p2,p3,p4
×(cid:2)δ(3)(p+p −p −p )|U (p−p )|2(cid:8)[1−f (p,r)]f (p ,r)[1−f (p ,r)]f (p ,r)
4 2 3 eff 2 λ λ 2 λ 4 λ 3
(cid:9)
−[1−f (p ,r)]f (p,r)[1−f (p ,r)]f (p ,r) (3.2a)
λ 2 λ λ 3 λ 4
+δ(3)(p−p −p +p )|U (p−p )|2(cid:8)[1−f (p,r)]f (p ,r)f (p ,r)[1−f (p ,r)]
4 2 3 eff 2 λ λ 2 −λ 4 −λ 3
(cid:9)
−[1−f (p )]f (p)f (p )[1−f (p )] (3.2b)
λ 2 λ −λ 3 −λ 4
+δ(3)(p−p −p +p )|U (−p−p )|2(cid:8)[1−f (p,r)][1−f (p ,r)] f (p ,r)f (p ,r)
4 3 2 eff 2 λ −λ 2 λ 3 −λ 4
(cid:9)(cid:3)
−f (p,r)f (p ,r)[1−f (p ,r)][1−f (p ,r)] . (3.2c)
λ −λ 2 λ 3 −λ 4
Equations. (3.2a)–(3.2c) correspond to the Coulomb scattering channels A–C, diagrams (b)–(b) , respectively. The
i iii
quasiparticleenergyandmomentumarewritteninthethree-vectorformp≡((cid:15) ,p),andthethree-dimensionalDirac
p
delta functions δ(3)(···) describe energy and momentum conservation. Channel A, Eq. (3.2a) is electron-electron
scattering, while channels B and C, Eqs. (3.2b) and (3.2c) are electron-hole scattering processes. The RPA screened
CoulombinteractiontakestheformasshowninAppendixB. Weemphasizethatevenatchargeneutrality,dynamical
screening is crucial at finite temperature due to the thermal activation of electron-hole pairs. Interaction-mediated
“Auger” imbalance relaxation processes are suppressed because of the linear dispersion of electrons and holes [17].
Due to kinematic constraints, channels A and B act in the “quasi-static” regime |ω| ≤ v q, while channel C acts
F
in the “optical” regime |ω| ≥ v q [Fig. 7]. Here ω and q are the frequency and momentum transferred across the
F
Coulomb line.
The carrier–optical-phonon scattering is described by the diagrams in Fig. 1(c) and leads to the collision integral
(2π)2β2 s (cid:90) (cid:18)1−pˆ·qˆ(cid:19)
St [{f }]= A(cid:48) 0
oph,λ λ ω M 2
A(cid:48) q
(cid:16) (cid:8) (cid:9)
× f (ω ) δ((cid:15) −(cid:15) −ω )[1−f (p,r)]f (q,r)−δ((cid:15) −(cid:15) +ω )f (p,r)[1−f (q,r)] (3.3a)
B A(cid:48) p q A(cid:48) λ λ p q A(cid:48) λ λ
(cid:8) (cid:9)(cid:17)
+[1+f (ω )] δ((cid:15) −(cid:15) +ω )[1−f (p,r)]f (q,r)−δ((cid:15) −(cid:15) −ω )f (p,r)[1−f (q,r)] (3.3b)
B A(cid:48) p q A(cid:48) λ λ p q A(cid:48) λ λ
(2π)2β2 s (cid:90) (cid:18)1−pˆ·qˆ(cid:19) (cid:110)
+ A(cid:48) 0 δ((cid:15) +(cid:15) −ω ) f (ω ) [1−f (p,r)][1−f (q,r)] (3.3c)
ω M 2 p q A(cid:48) B A(cid:48) λ −λ
A(cid:48) q
(cid:111)
−[1+f (ω )] f (p,r)f (q,r) , (3.3d)
B A(cid:48) λ −λ
where M = 2.0 × 10−23g is the carbon atom mass where f(0)(p,r) is the local equilibrium Fermi-Dirac
λ
and s0 = 2.62˚A2 the area per carbon atom. Equa- function (β =1/kBT)
tions(3.3a,3.3b)[(3.3c,3.3d)]correspondtothediagrams
1
ipnroFciegsss.es1((cc))i,iiia[n1d(c)(ici,)iv],arreespaebcsteivnetlyf.orWaeconuostteicthpahtontohne fλ(0)(p,r)= eβ(εp−µλ)+1, εp =(cid:126)vF|p|, µλ =λµ,
ii iv
(3.5a)
scattering [20] because the acoustic-phonon velocity is
and δf (p,r) is the deviation from the local equilibrium
much smaller than the Fermi velocity. To compare to λ
and can be conveniently cast into the form
the experiment in [31], we take the A(cid:48) phonon temper-
ature TA(cid:48) ≡ (cid:126)ωA(cid:48)/kB ≈ 2200K, larger than in some 1 (cid:34) df(0)(cid:35)
pbreeevniosuusggsetsutdediesto[2b1e].strTohneglycoeunpelringyg dsterpeenngdtehntβAd(cid:48)uehatos δfλ(p,r)= β − dελp χλ(p,r). (3.5b)
renormalization and screening by the Coulomb interac-
tions [21, 45, 53]. We treat β as a fitting parameter Via the standard derivation [59], from Eq. (2.2) we ob-
A(cid:48)
when interpreting the experimental data [31]. tain the time-independent linearized Boltzmann’s equa-
tion for χ ,
λ
paWrtse,separatethedistributionfunctionfλ(p,r)intotwo fλ(cid:48)(p,z)vF·(cid:18)λeβE − p−Tλlnz∇rT(cid:19)= (cid:126)1βS(cid:102)tλ[{χλ(cid:48)}],
(3.6)
f (p,r)≡f(0)(p,r)+δf (p,r), (3.4)
λ λ λ
10
where we have introduced the electrochemical field E ≡ hand side of Eq. (3.6), the linearized collision integral
E+1∇ µ,theeffectiveFermi-Diracdistributionfunction, reads
e r
and its derivative
S(cid:102)tλ[{χλ(cid:48)}]=S(cid:102)timp,λ[χλ]+S(cid:102)tint,λ[{χλ(cid:48)}]+S(cid:102)toph,λ[{χλ(cid:48)}],
1
f (p,z)≡ , f(cid:48)(p,z)≡−∂ f (p,z), (3.7) (3.8)
λ z−λep+1 λ p λ where the impurity collision integral is
which depends on the dimensionless momentum p =
β(cid:126)v |p| and the “fugacity” z = exp(βµ). On the right
F
(s) (l)
S(cid:102)timp,λ[χλ]=S(cid:102)timp,λ[χλ]+S(cid:102)timp,λ[χλ] (3.9a)
with short- and long-ranged components
S(cid:102)t(ims)p,λ[χλ]= (cid:90) δ(p−q)(cid:20)g0+gf(cid:18)1+2pˆ·qˆ(cid:19)+gb(cid:18)1−2pˆ·qˆ(cid:19)(cid:21)(cid:2)fλ(cid:48)(q)χλ(q)−fλ(cid:48)(p)χλ(p)(cid:3), (3.9b)
q
S(cid:102)t(iml)p,λ[χλ]=γ2(cid:90) δ(p−q)(cid:18)1+2pˆ·qˆ(cid:19)(cid:12)(cid:12)(cid:12)U(cid:101)eff(ω =0,|p−q|)(cid:12)(cid:12)(cid:12)2(cid:2)fλ(cid:48)(q)χλ(q)−fλ(cid:48)(p)χλ(p)(cid:3), (3.9c)
q
the Coulomb collision integral is
(i) (ii) (iii)
S(cid:102)tint,λ[{χλ}]=S(cid:102)tint,λ[{χλ}]+S(cid:102)tint,λ[{χλ}]+S(cid:102)tint,λ[{χλ}], (3.10a)
with components corresponding to channels A–C in Fig. 1(b)
(i) (cid:90) 1+pˆ·p(cid:92)−q1+kˆ·k(cid:92)−q
S(cid:102)tint,λ[{χλ}]=2πN 2 2 δ(p−|p−q|−k+|k−q|)
k,q
×|U(cid:101)eff(p−|p−q|,q)|2Ξλp,,|λp;−λ,qλ|;k,|k−q|(cid:2)−χλ(p)+χλ(p−q)+χλ(k)−χλ(k−q)(cid:3), (3.10b)
(ii) (cid:90) 1+pˆ·p(cid:92)+q1+kˆ·k(cid:92)−q
S(cid:102)tint,λ[{χλ}]=2πN 2 2 δ(p−|p+q|+k−|k−q|)
k,q
×|U(cid:101)eff(p−|p+q|,q)|2Ξpλ,,|λp;+−qλ|,;−kλ,|k−q|(cid:2)−χλ(p)+χλ(p+q)−χ−λ(k)+χ−λ(k−q)(cid:3), (3.10c)
(iii) (cid:90) 1−pˆ·p(cid:92)−q1−kˆ·k(cid:92)−q
S(cid:102)tint,λ[{χλ}]=2πN 2 2 δ(p+|p−q|−k−|k−q|)
k,q
×|U(cid:101)eff(p+|p−q|,q)|2Ξλp,,|−pλ−;qλ|,;−kλ,|k−q|(cid:2)−χλ(p)−χ−λ(−p+q)+χλ(k)+χ−λ(−k+q)(cid:3), (3.10d)
and the carrier–optical-phonon collision integral is
(cid:90) 1−pˆ·qˆ
S(cid:102)toph,λ[{χλ(cid:48)}]=αo2ph 2
q
×(cid:104)f(cid:48)(q)(cid:8)δ(p−q−Ω )[f (Ω )+f (p)]−δ(p−q+Ω )[f (−Ω )+f (p)](cid:9)χ (q)
λ A(cid:48) B A(cid:48) λ A(cid:48) B A(cid:48) λ λ
−f(cid:48)(p)(cid:8)δ(p−q+Ω )[f (Ω )+f (q)]−δ(p−q−Ω )[f (−Ω )+f (q)](cid:9)χ (p)(cid:105) (3.11a)
λ A(cid:48) B A(cid:48) λ A(cid:48) B A(cid:48) λ λ
−α2 (cid:90) 1−pˆ·qˆ δ(p+q−Ω )(cid:110)f(cid:48)(p)(cid:2)f (Ω )+f (q)(cid:3)χ (p)+f(cid:48)(q)(cid:2)f (Ω )+f (p)(cid:3)χ (q)(cid:111).
oph 2 A(cid:48) λ B A(cid:48) −λ λ λ B A(cid:48) λ −λ
q
(3.11b)
In Eq. (3.9b) we have introduced the dimension- (G ,G ,G )/(cid:126)v2. In Eq. (3.9c), we define the
0 f b F
less short-ranged impurity strengths (g ,g ,g ) = dimensionless long-ranged impurity strength γ2 =
0 f b