Unraveling the acoustic electron-phonon interaction in graphene

1.1 MB·English

by  Kristen Kaasbjerg

Unraveling the acoustic electron-phonon interaction in graphene
∗
Kristen Kaasbjerg, Kristian S. Thygesen, and Karsten W. Jacobsen
Center for Atomic-scale Materials Design (CAMD),
Department of Physics, Technical University of Denmark
(Dated: January 24, 2012)
Usingafirst-principlesapproachwecalculatetheacousticelectron-phononcouplingsingraphene
for the transverse (TA) and longitudinal (LA) acoustic phonons. Analytic forms of the coupling
matrixelementsvalidinthelong-wavelengthlimitarefoundtogiveanalmostquantitativedescrip-
2 tion of the first-principles based matrix elements even at shorter wavelengths. Using the analytic
1 forms of the coupling matrix elements, we study the acoustic phonon-limited carrier mobility for
0 temperatures 0−200 K and high carrier densities of 1012 −1013 cm−2. We find that the intrin-
2 sic effective acoustic deformation potential of graphene is Ξeff = 6.8 eV and that the temperature
dependence of the mobility µ ∼ T−α increases beyond an α = 4 dependence even in the absence
n
of screening when the full coupling matrix elements are considered. The large disagreement be-
a
tween our calculated deformation potential and those extracted from experimental measurements
J
(18−29 eV) indicates that additional or modified acoustic phonon-scattering mechanisms are at
3
play in experimentalsituations.
2

PACSnumbers: 81.05.Hd,72.10.-d,72.20.-i,72.80.Jc
]
i
c
-s I. INTRODUCTION be identified.19 Here, kF is the Fermi wave vector,
l c the sound velocity and k the Boltzmann constant
r ph B
t (T 57 K √n for the LA phonon with n measured
m Since the experimental realization of graphene,1 its inBuGnit∼s of 1012 cm−2). Since the BG temperature cor-
electronic properties and their understanding have been
. responds to the acoustic phonon energy ~ω = ~c q
t studied extensively both experimentally and theoreti- q ph
a for full backscattering at the Fermi level, short wave-
m cally.2–4 While the intrinsic carrier mobility of graphene
length acoustic phonons are frozen out at temperatures
is predicted to be exceptionally high, the experimen-
- T < T restricting scattering processes to small scat-
d tal reality in substrate supported graphene involving BG
tering angles. The restricted phase space available for
n chargedimpurities,electron-holepuddles,surface-optical
o phonons of the substrate, and disorder typically results phonon scattering at T < TBG results in a transition
c from the linear ρ T behavior of the resistivity in the
[ in strongly reduced mobilities compared to the expected high-temperature r∼egime to a stronger ρ Tα tempera-
intrinsic value.5,6 Together with scattering on acoustic ∼
ture dependence in the BG regime where α= 4 (α =6)
1 phonons which manifests itself in a linear temperature
v in the absence (presence) of screening by the carriers
dependenceofthemobilityathighertemperatures,these
1 themselves.20,21 TheBGbehaviorinthetemperaturede-
extrinsic scattering mechanisms typically dominate the
6 pendence of the mobility (resistivity) has recently been
mobility. The linear temperature dependence character-
6 observed experimentally.7
4 istic for acoustic phonon scattering has so far been ob-
. servedinbothsupported5,7–10andsuspended11graphene Existingtheoretical20,21 andexperimental5,7–9,11 stud-
1
samples. ies of acoustic phonon-limited transport in graphene of-
0
2 With the recent improvements in sample fabrication, ten parametrize the interaction with acoustic phonons
1 the relative role of acoustic phonon scattering must in terms of a coupling to a single effective acoustic
: be expected to become increasingly important in fu- phonon. The associated deformation potential coupling
v
i ture devices. For example, samples with the commonly constant extracted from the experimentally measured
X used SiO substrate replaced by hexagonal boron ni- temperature dependence of the resistivity range from
2
r tride (h-BN) which has a lattice constant very close to 18 29eV.5,7–9,11 Ontheotherhand,theoreticalstud-
a ∼ −
that of graphene and an almost atomically flat surface ies of the acoustic electron-phonon coupling yield much
with strongly reduced disorder,12,13 have shown highly lower values on the order of 3 4.5 eV.22,23 At the same
−
improved transport characteristics with mobilities ap- time, different forms of the coupling matrix element are
proaching that of suspended graphene.8,14,15 Further- used in theoretical studies20,22 making a direct compar-
more, the high energy of the surface-optical phonons ison of the different values of the deformation potential
of h-BN results in a significant reduction of surface- difficult.
optical phonon scattering16–18 that for commonly used Eventhoughtheeffectofacousticphononscatteringon
gate oxides starts to dominate the mobility around T the carriermobility in graphene has been studied widely
150 200 K.5,9 ∼ intheliterature,20,23–27 acompletestudyconsideringthe
−
When the mobility is dominated by acoustic phonon full details of the coupling matrix element is still lack-
scattering,twotransportregimesseparatedbytheBloch- ing. The purpose of the present study is to provide a
Gru¨neisen (BG) temperature T = 2~k c /k can detailed analysis of the acoustic electron-phonon inter-
BG F ph B
2
action in graphene and to establish the intrinsic value phononenergy. Inthefollowing,thesequantitiesarecal-
of the effective deformation potential. We shall focus culated from first-principles. The phonons are assumed
on supported graphene where the flexural phonons are tobeinequilibriumandpopulatedaccordingtotheBose-
quenched and hence include only the transverse (TA) Einstein distribution function N =N(~ω ). As scat-
qλ qλ
and longitudinal (LA) acoustic phonons. We use a first- teringonboththeTAandLAphononisconsideredhere,
′
principlesmethodtocalculatetheelectron-phononinter- the total relaxation time for the K,K -valleys is given
action28,29 supported by the group-theoretical consider- by the sum of the individual phonon contributions as
ations of Ref.30. We then study the intrinsic phonon- τ−1 = τ−1. As we show in the following section, the
λ λ
limited mobility using a Boltzmann equation approach matrixelementsoftheelectron-phononcouplingdifferin
P ′
inthe temperature regime0 200Kandfor highcarrier the K and K valleys (see e.g. Fig. 1). The Boltzmann
densities n 1012 1013 cm−−2 where screening by the equation must therefore be solved explicitly in both val-
∼ −
carriers suppresses other scattering mechanisms. Using leys. In the absence of intervalley scattering which cou-
thefullcouplingmatrixelements,wefindthatatemper- ples the distribution functions in the two valleys, this
ature dependence with α>4 occurs even in the absence can be done by considering the two valleys separately.
of screening. In this case, the relaxation time entering the expression
for the mobility in Eq. (1) becomes the valley-averaged
relaxation time τ =(τ +τ )/2, where τ is the to-
K K′ K/K′
II. THEORY tal relaxation time in the individual valleys. Screening
of the electron-phonon interaction by the carriers them-
Inthefollowing,thecarriersingraphenearedescribed selves32,33 has been considered elsewhere21 and will here
by massless Dirac fermions with linear dispersion ε = be neglected.
k
~v k where v 1.0 106 m/s is the Fermi velocity. While analytic considerations have been given in
F F
Within the Boltz∼mann×equationapproach,6,25,31 the mo- Refs.20,21, we will in the present work resort to a nu-
bility in graphene in the presence of (quasi) elastic scat- merical evaluation34,35 of the relaxation time in Eqs. (3)
tering mechanisms is given by and (4). This allows us to study the acoustic phonon-
limited mobility in graphene with the full coupling ma-
µxx = σxx = evF2hτki (1) trixelementswhichhaveamorecomplexangulardepen-
ne 2 dence than most often assumed (see e.g. Ref.20). The
numericalapproachalsoallowsforaunifiedtreatmentof
where σ is the conductivity, n is the two-dimensional
xx
the high-temperature (T > T ) and Bloch-Gru¨neisen
carrier density and the density-of-states averaged relax- BG
(T < T ) regimes. We note, however, that in the low-
ation time (in units of time per energy) is defined by BG
temperature regime where ~ω k T, it is crucial that
q B
∼
1 ∂f the phonon energy is retained in the Fermi function in
τ = dε ρ(ε ) τ . (2)
k k k k Eq.(3). Inthehigh-temperatureregimethisrequirement
h i n −∂ε
Z (cid:18) k(cid:19) can be relaxed and the relaxation time can be put on a
Here, ρ(ε ) = (g g /2π~2)ε /v2 is the density of states simple analytic form.
k s v k F
of the graphene layerandg =2 and g =2 are the spin
s v
and valley degeneracies, respectively. At low tempera-
tures and high carrier densities where ε k T this III. INTERACTION WITH ACOUSTIC
F B
yields µ ev2τ /ε . ≫ PHONONS
xx ≈ F kF F
Inthe caseofacousticphononscatteringwhichcanbe
treated as a quasielastic process, the relaxation time for Inthefollowingweuseafirst-principlesDFTapproach
each of the acoustic phonons (TA and LA) is given by20 basedonafully microscopicdescriptionofboththe elec-
tronic Bloch states and the phonons to calculate the
τk1λ = k′ (1−cosθk,k′)Pkλk′11−−ffkk′, (3) Tachoeusctailccuellaetcetdrovna-lpuheosnfoonr tchoeupsoliunngds vinelogcriatipehsecnλe.f2o8r,29t,h36e
X TA and LA phonons are reported in Table I together
where θk,k′ is the scattering angle and fk = f(εk) the with other parameters used in this work. Due to their
Fermi function. The transition matrix element is given high phonon energies (> 100 meV), acoustic intervalley
by and optical phonons do not play a role in the considered
temperature range and can therefore be neglected.
2π
Pkλk′ = ~ |gqλ|2 Nqλδ(εk′ −εk−~ωqλ) ticTphheoinnotenrsacintiognrabpehtweneeencachnarbgeewcarritrtieernsainndthtehegaecnoeuras-l
q (cid:20)
X form
+ (1+Nqλ)δ(εk′ εk+~ωqλ) . (4) ~
−
(cid:21) gλ = Mλ , (5)
Here, gλ is the electron-phonon coupling, λ denotes kq s2Aρωqλ kq
kq
the acoustic phonon branches, and ~ω the acoustic where A is the area of the graphene layer, ρ is the mass
qλ
3
densityandMλ = k+qδV k isthecouplingmatrix K K’
kq h | qλ| i
elementforscatteringbetweenthetwoBlochstateskand k+q k k+q k
k+q due to a phononwith wavevector q, branchindex
λandfrequencyω =c qwherec isthesoundvelocity
qλ λ λ
for the acoustic branches. The coupling is mediated by
the change δV in the microscopiccrystalpotential due
qλ
toaunitdisplacementoftheatomsalongthemass-scaled
normal mode vector e . Due to the full microscopic
qλ
treatment of both electrons and phonons, Umklapp pro-
cesses involving reciprocallattice vectors are included in
the coupling matrix element Mλ .
kq
Figure 1 shows the absolute value of the coupling
matrix elements Mλ to the TA and LA phonons in
kq
′
the K,K valleys as a function of the two-dimensional
phonon wave vector q. The matrix elements in the two
valleys are related through time-reversal symmetry as
MKλ∗ = MK′λ . This implies that carriers in the two
kq −k,−q
valleys traveling in the same direction experience differ-
ent electron-phonon couplings. It should be emphasized
thatthecalculatedmatrixelementsincludethefullsym-
metryofboththeelectronicBlochstatesandthephonon
modes as given by the complete microscopic description.
In order to emphasize the effect of the chirality of the
carriers in graphene, the initial carrier state k is located
FIG. 1: (Color online) Electron-phonon couplings to the
on the right side of the Dirac cones 300 meV above the acoustic TA and LA phonons in the K,K′-valleys (left and
′
K,K -points as indicated by the sketch in the top of the right columns, respectively) of graphene. The contour plots
figure. As is evident from the figure, both the TA and showtheabsolutevalueofthecouplingmatrixelements|Mλ |
kq
LA phonons couple to the carriers with similar coupling for a carrier energy of 300 meV as a function of the two-
strengths. WhilebackscatteringissuppressedfortheLA dimensionalphononwavevectorq. The(white)circlescorre-
mode, the situation is reversed for the TA mode where spond to k+q vectors lying on the constant energy surfaces
forwardscattering is suppressed. In addition to suppres- of the Dirac cones given by the energy εk of the initial state
in k as sketched in the top row.
sionofforwardandbackscattering,otherdirectionswith
complete suppression of scattering also appears. This is
a consequence of the inclusion of the symmetry of both
Inthefollowing,thefirst-principlescouplingmatrixel-
phonons and electronic states.
ements areanalyzedusing the group-theoreticalanalysis
of the electron-phonon interaction presented in Ref.30.
In the long-wavelength limit, the TA and LA phonons
Parameter Symbol Value are strictly transverse and longitudinal, respectively,37
Lattice constant a 2.46 ˚A (LDA) and the electron-phonon interaction has a simple ana-
Ion mass density ρ 7.6×10−8 g/cm2 lytic representation in the two-dimensional pseudospin
Fermivelocity vF 1.0×106 m/s basis.30 Using the results of Ref.30, the coupling matrix
elements canexpressedin terms of the angles θ , θ and
Transverse sound velocity cTA 14.1×103 m/s k q
θ of the involved wave vectors. Including contribu-
Longitudinal sound velocity cLA 21.2×103 m/s tiko+nqs of order O(q), we find that the coupling matrix
Electron-phonon couplings elementsinthelong-wavelengthlimittakesthefollowing
Transverse βTA 2.8 eV form in the K-valley,
Longitudinal αLA 2.8 eV
θ +θ
Longitudinal βLA 2.5 eV |MkTqA|=qβ× sin 2θq+ k 2k+q (6)
Effectivecoupling parameters (cid:12) (cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)
Soundvelocity ceff 20.0×103 m/s and (cid:12) (cid:12)
(cid:12) (cid:12)
Deformation potential Ξeff 6.8 eV θ θ
MLA =q αcos k+q− k
| kq| × 2
(cid:12) (cid:18) (cid:19)
TABLE I: Material parameters for graphene used in the (cid:12) θ +θ
(cid:12) k k+q
present work. The phonon related parameters have been ob- (cid:12)+βcos 2θq+ , (7)
2
tained from first-principlesas described in thetext. The cal- (cid:18) (cid:19)(cid:12)
(cid:12)
culated sound velocities are in excellent agreement with the for the TA and LA phonons, respectively. (cid:12)For the
values reported in Ref.27. LA phonon, the first and second terms origina(cid:12)te from
4
the deformation potential and the gauge field cou- 40 n=1012cm(cid:2)2
pling mechanisms, respectively. The TA phonon cou-
T=0.1 T
ples only through the latter.30,38,39 Since the interaction )35 T=0.5 TBG
1 BG
is Coulombic in nature, the overall couplings given in ns(cid:1) 30 T=1.0 TBG
Eqs. (6) and (7) are here referred to as deformation po- e ( T=1.5 TBG
tential couplings. m25 T=2.0 TBG
With the coupling parameters listed in Table I, we x. ti20
findthattheanalyticexpressionsfortheelectron-phonon a
el
interaction in Eqs. (6) and (7) to a high degree repro- r15
e
duce the first-principles matrix elements for electron en- s
r10
ergiesup to 750meV. Asthe analyticcouplingmatrix ve
elements are∼based on the phonon modes in the long- In 5
wavelength limit, the agreement is slightly worsened at
0
0.0 0.5 1.0 1.5 2.0
shorterwavelengthswherethemodevectorsdeviatefrom
/
the long-wavelength modes.37 This is most pronounced (cid:0) (cid:0)F
for the TA phonon. As the LA phonon retains its long-
FIG. 2: (Color online) Inverse relaxation time (valley-
wavelength character far out in the Brillouin zone, the
averaged) for acoustic phonon scattering on the TA and LA
agreementbetweenthecouplingmatrixelementsherere- phonon in the BG regime at n = 1012 cm−2 (TBG ≈ 57 K).
mains quantitative even at shorter wavelengths. In the The full lines show the results obtained with the full matrix
BG regime where short wavelength phonons are frozen elementsgiveninEqs.(6)and(7)andcouplingconstantsex-
out, we note that it is the long-wavelength limit of the tractedfromab-initiocalculations. Thedashedlinesshowthe
coupling matrix elements that governs the scattering of result obtained with scattering on a single effective acoustic
carriers. phonon with the coupling matrix element given by Eq. (8),
Often scattering on acoustic phonons is described by a deformation potential of Ξeff = 6.8 eV and sound velocity
c=20×103 m/s.
coupling to a single effective phonon mode with a cou-
pling matrix element given by20,25
θ
Meff =Ξ qcos k,k+q (8) to establish the value of the intrinsic effective acoustic
kq eff 2 deformation potential in graphene.
(cid:18) (cid:19)
In Fig. 2 we show the inverse of the valley-averaged
where Ξ is the effective deformation potential and
eff
relaxationtime as a function of energy for different tem-
the angular part corresponds to the bare spinor overlap
peratures and a carrier density of n = 1012 cm−2 corre-
χ χ of the electronic wave function. In contrast
k+q k
h | i sponding to ε 117 meV and T 57 K for the LA
to the morecomplexangulardependence ofthe coupling F ∼ BG ≈
phonon. Above the BG temperature, the inverse relax-
matrix element predicted by the full microscopic treat-
ation time has the linear energy dependence of Eq. (9)
ment presented here, the angular dependence of the ef-
andaslopeproportionaltothetemperature. Asthetem-
fective coupling matrix element above suppresses only
peratureisdecreasedbelowT ,thefreezingoutofshort
backscattering. In the high-temperature regime where BG
equipartitioning of the acoustic phonons N k T/~ω wavelengthphononsandthesharpeningoftheFermisur-
q B q
∼ face result in limited phase space for phonon scattering
applies, the relaxation time and the resistivity take the
following simples forms20 andanincreasedlifetimeofthecarriersattheFermien-
ergy. In the expressionfor the relaxationtime in Eq.(3)
1 1 Ξ2 k T πΞ2 k T this effect is accounted for by the Fermi and Bose dis-
τ = ~3 4eρffv2Bc2 εk ; ρ= 4e2~effρvB2c2, (9) tribution functions. The limited phase space available
k F F for phonon scattering, manifests itself in the character-
wherethefactorof4inthedenominatorsstemsfromthe istic dip at the Fermi energy that evolves in the inverse
chiral nature of the carriers through the assumed form relaxation time with decreasing temperature.20 For all
of the coupling matrix element in Eq. (8). These ex- temperatures, the linear energy dependence of the high-
pressionsareusedalmostexclusivelytoextractthevalue temperature result in Eq. (9) is recovered in the ε 0
→
of the effective acoustic deformation potential in experi- limit.
mental situations.5,7–9,11 By inspecting the individual contributions, we find
thatthe inverserelaxationtime toalargeextentis dom-
inated by the contribution from the TA phonon both in
IV. RESULTS the high-temperature and the BG regime. The domi-
nation of the TA phonon can be attributed to a num-
In the following, we study the intrinsic acoustic ber of factors. From Eq. (9) it follows directly that the
phonon-limitedmobilityofgrapheneinboththeBGand lower sound velocity of the TA phonon leads to higher
linear resistivity regime using the full coupling matrix scattering rate. Also related to the sound velocity is
elements as given by Eqs. (6) and (7). This allows us the lower BG temperature of TA phonon which allows
5
(dashed lines) reproduces the full calculation extremely
107
11)sV(cid:3)(cid:3) nnnn====1351(cid:4)(cid:4)(cid:4)(cid:4)1111000011112223ccccmmmm(cid:5)(cid:5)(cid:5)(cid:5)2222 wttwiooeitlnahl.mfαrFooo>mrrea1tplhlirneocnatlirhonrueieenaBrcrGedµdern∼µesgi∼itTmie−Tes1.−thhAαietgtmhetme-otmepbmpeilreiptaryetarutsaurhterouedwrseesTpbaee>nthrdaTaevnBnisGcoier-,
2m106 the decrease in the mobility with increasing carrier den-
(c sity stems fromthe linear energydependence ofthe den-
y
t sityofstateswhichprovidesmorephasespaceforphonon
bili scattering for higher Fermi levels. At low temperatures
o
M105 T < TBG where scattering on the full Fermi surface is
frozenout,thelocalvalueofthecouplingmatrixelement
0 50 100 150 200 becomes important and the mobilities for the different
T (K)
carrier densities approach a common value.
The right plot shows the temperature dependence of
5 n=1(cid:8)1012cm(cid:9)2 theexponentαforthesamesetofcarrierdensitieswhich
nn==35(cid:8)(cid:8)11001122ccmm(cid:9)(cid:9)22 tchoerreLsAponpdhotnoonB.GFtreommpetrhaitsurpelsotTBitGis∼m5o7r−e c1l8e0arKthfaotr
ogT4 n=1(cid:8)1013cm(cid:9)2 the departure away from the linear temperature depen-
dl
/ dence happens forT TBG. ForT <TBG, the exponent
og(cid:7) 3 increases monotonica∼lly. Surprisingly, the exponents ob-
dl tainedfromthemobilitycalculatedwiththefullcoupling
(cid:6) 2
matrixelementsdonotsaturateatα=4aspredictedby
the effective couplingmatrix element(dashedlines).20,21
1 Thus, even in the absence of screening, the mobility of
101 102 grapheneshouldtakeonatemperature dependence with
T (K)
α > 4 at sufficiently low temperatures. With carrier
FIG. 3: (Color online) Mobility vs temperature for carrier screeningtakenintoaccount,thisbehaviorisreinforced21
densities1012−1013 cm−2(upperplot). Thelowerplotshows The purely intrinsic mobilities calculated here are sig-
thetemperaturedependenceoftheexponentαinthetemper- nificantlyhigherthanpreviouslyreportedtheoreticalval-
ature dependenceµ∼T−α of themobility. ues.20 This is reflected directly in the extracted defor-
mation potential parameter which is considerably lower
than commonly used values. For a carrier density of
for full backscattering below the BG temperature of the n=1012 cm−2 a room-temperaturemobility inexcessof
LA phonon. Secondly, the coupling matrix element for 105 cm2 V−1 s−1 is here predicted. The associated scat-
the TA phonon allows for backscattering which is sup- teringratewhichisgivenbytherelaxationtimeinEq.(9)
pressed for the LA phonon. In the BG regime, the dom- in the high-temperatureregime, is fromFig. 2 estimated
ination of the TA phonon stems from the suppression of tobe τ−1 1011 s−1 atthe Fermilevelcorrespondingto
∼
the coupling matrix element for the LA phonon in the a mean-free path of λ = vF/τ 1000 nm. Such an ex-
∼
long-wavelength limit (see Fig. 1). The observed domi- tremely large mean-free path may open the opportunity
nance of the TA phonon is in contrast to the often used to study coherent transport in relatively large graphene
assumption that only the LA phonon couples to charge structures. Wenotethatthequasiparticlescatteringrate
carriers in graphene.20 observable in photoemission spectroscopy (ARPES) de-
In order to determine the intrinsic value of the effec- velops a similar dip at the Fermi level for T < TBG re-
tive deformation potential in graphene, we also calcu- sulting in long-lived quasiparticles.
late the relaxation time using coupling matrix element
in Eq. (8). The dashed lines in Fig. 2 show the inverse
relaxation time calculated with an effective deformation V. CONCLUSIONS AND DISCUSSIONS
potentialandsoundvelocityof6.8eVand20.0 103m/s,
×
respectively. It is seen to reproduce the relaxation time In the present study the acoustic electron-phonon in-
basedonthefullmatrixelementsverywellfortheenergy teraction in graphene has been analyzed in detail. The
range shown. While the extracted value for the acoustic exact analytic forms of the coupling matrix elements
deformation potential is much smaller than experimen- in long-wavelength limit were found match the calcu-
tal values,5,7–9,11 it is in better agreement with recently lated first-principles matrix elements almost quantita-
reported ab-initio results yielding 4.5 eV.23 tively even at shorter wavelengths. As previously pre-
Figure 3 summarizes the calculated acoustic phonon- dicted,20,21 the calculated mobilities showed a transition
limited mobility as a function of temperature for car- from a µ T−1 to a µ T−α temperature dependence
rier densities 1012 1013 cm−2. The mobility calculated with α >∼1 below the B∼G temperature. However, con-
−
with the above-mentioned effective coupling parameters trary to earlier studies we found that the full coupling
6
matrix elements cause the temperature dependence of been studied in 2DEGs.42 The relatively large variations
themobilitytoincreasebeyondα=4whichisotherwise in the experimental deformation potentials also indicate
only observed when screening is included.21 that the deformation potential depends on experimental
Byfittingtheresultswithaneffective acousticphonon factors such as e.g. the substrate.
the acoustic deformation potential is found to be Ξ =
eff
6.8 eV. Since this is much lower than the experimen-
tally determined values 18 29 eV, our results suggest
−
that the acoustic phonon-limited transport in substrate-
Acknowledgments
supported graphene is at present not fully understood.
Possibleexplanationsforthelargeexperimentaldeforma-
tion potentials could be (i) substrate-induced modifica- The authors would like to thank T. Markussen and
tionsofthebandstructure40,41thatmodifiesthechirality A.-P.Jauhoforusefulcommentsonthe manuscript. KK
(andtherebytheangulardependenceofthecouplingma- has been partially supported by the Center on Nanos-
trix element) of the electronic states and/or the Fermi tructuring for Efficient Energy Conversion (CNEEC) at
velocity, and (ii) the existence of additional acoustic StanfordUniversity,anEnergyFrontierResearchCenter
phonons not considered in the present work. This could funded by the U.S. Department of Energy, Office of Sci-
beeithertheintrinsicflexuralphononwhichmaybemod- ence,OfficeofBasicEnergySciencesunderAwardNum-
ified when graphene is lying on a substrate or surface- ber de-sc0001060. CAMD is supported by the Lundbeck
acousticphononsonthe substratewhichhavepreviously Foundation.
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