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PHYSI CAL REVI EW LETTERS
PRL 100, 016602 (2008) 11 JANUARY 2008
Giant Intrinsic Carrier Mobilities in Graphene and Its Bilayer
S. V. Morozov,1,2 K. S. Novoselov,1 M. I. Katsnelson,3 F. Schedin,1 D. C. Elias,1 J. A. Jaszczak,4 and A. K. Geim1,*
1
Manchester Centre for Mesoscience and Nanotechnology, University of Manchester, Manchester M13 9PL, United Kingdom
2
Institute for Microelectronics Technology, 142432 Chernogolovka, Russia
3
Institute for Molecules and Materials, University of Nijmegen, 6525 ED Nijmegen, The Netherlands
4
Department of Physics, Michigan Technological University, Houghton, Michigan 49931, USA
(Received 3 November 2007; published 7 January 2008)
We have studied temperature dependences of electron transport in graphene and its bilayer and found
extremely low electron-phonon scattering rates that set the fundamental limit on possible charge carrier
mobilities at room temperature. Our measurements show that mobilities higher than 200 000 cm2=Vs are
achievable, if extrinsic disorder is eliminated. A sharp (thresholdlike) increase in resistivity observed
above 200 K is unexpected but can qualitatively be understood within a model of a rippled graphene
sheet in which scattering occurs on intraripple flexural phonons.
DOI: 10.1103/PhysRevLett.100.016602 PACS numbers: 72.10. d, 72.15.Lh
Graphene exhibits remarkably high electronic quality the values achieved so far. The reported measurements are
such that charge carriers in this one-atom-thick material also important for narrowing dominant scattering mecha-
can travel ballistically over submicron distances [1]. nisms in graphene, which remain hotly debated [3
Electronic quality of materials is usually characterized by 7,11,13].
mobility of their charge carriers, and values of as high The studied devices were prepared from graphene ob-
as 20 000 cm2=Vs were reported for single-layer graphene
tained by micromechanical cleavage of graphite on top of
(SLG) at low temperatures (T) [2 5]. It is also believed
an oxidized Si wafer (usually, 300 nm of SiO2) [14].
that in the existing samples is limited by scattering on
Single- and bilayer crystallites were initially identified by
charged impurities [6] or microscopic ripples [3,7]. Both
their optical contrast [15], verified in some cases by Raman
sources of disorder can in principle be eliminated or re- and atomic-force microscopy [2,14,16] and always cross-
duced significantly. There are, however, intrinsic scatterers
checked by measurements in high magnetic fieldsB, where
such as phonons that cannot be eliminated at roomT and,
SLG and BLG exhibited two distinct types of the quantum
therefore, set a fundamental limit on electronic quality and
Hall effect [2,17]. To improve homogeneity, our standard
possible performance of graphene-based devices. How
Hall bar devices [1 3] were annealed at 200 C in a H2-Ar
high is the intrinsic mobility in for graphene at 300 K?
mixture [18] and, then, inside a measurement cryostat at
This is one of the most important figures of merit for any
400 K in He. To avoid accidental breakdown, gate voltages
electronic material, but it has remained unknown.
Vg were limited to 50 V (n/ Vg with
In this Letter, we show that electron-phonon scatter-
7:2 1010 cm 2=V [1 5]). The measurements discussed
ing in graphene and its bilayer is so weak that, if the
below were carried out by the standard lock-in technique
extrinsic disorder is eliminated, room-T mobilities
and refer to 7 SLG and 5 BLG devices with between
200 000 cm2=Vs are expected over a technologically
3000 and 15 000 cm2=Vs.
relevant range of carrier concentration n. This value ex-
Figure 1 shows a characteristic behavior of Vg in
ceeds in known for any other semiconductor [8]. In
SLG. The device exhibits a sharp peak close to zero Vg
particular, our measurements show that away from the
( 0:2 V), indicating little chemical doping [3].
neutrality point (NP) resistivity of SLG has two compo-
Conductivity 1= is a notably sublinear function of
nents: in addition to the well-documented contribution
Vg in this device. Both linear and sublinear behaviors were
L 1=ne due to long-range disorder [6,7], we have
identified a small but notablen-independent resistivity S reported previously [2 5]. To this end, if we subtract a
constant resistivity S ( 100 in Fig. 1), then L
indicating the presence of short-range scatterers [6,7,9].
1= Vg S becomes perfectly linear over the whole
We have also found that L does not depend onT below
range of positive and negativeVg, except for the immedi-
300 K, whereas S exhibits a sharp rise above 200 K
ate vicinity of NP (< 3 V). This linearization procedure
[10]. The latter contradicts to the existing theories [11] that
was found to work extremely well for all our devices [the
expect a linearTdependence. We attribute this behavior to
only exception was occasional devices with strongly dis-
flexural (out-of-plane) phonons [12] that are excited inside
torted Vg indicating macroscopic inhomogeneity [1]].
ripples. Bilayer graphene (BLG) samples exhibited no
discernible T dependence of away from NP, yielding Furthermore, we digitized a number of curves in recent
even higher in. These findings provide an important literature [4,5] and found the approach equally successful.
benchmark for the research area and indicate that in This shows that resistivity of doped graphene can empiri-
graphene systems can be orders of magnitude higher than cally be described by two contributions: L/1=nand S
0031-9007=08=100(1)=016602(4) 016602-1 © 2008 The American Physical Society
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PHYSI CAL REVI EW LETTERS
PRL 100, 016602 (2008) 11 JANUARY 2008
Á = ÁL+ const Å»# 20K
Á = ÁL+ const Å»# 20K
6
6
Å»# 100K
Å»# 100K
6
6
Å»# 180K
Å»# 180K
Å»# 220K
Å»# 220K
1/ÁL
1/ÁL
1/ÁL Å»# 260K
1/ÁL Å»# 260K
Á
Á
4 4
4 4
1/Á
1/Á
1/Á
1/Á
2
2
2
2
50K
50K
0
0
0
0
-50 -25 0 25 50
-50 -25 0 25 50
-50 -25 0 25 50
-50 -25 0 25 50
Vg (V)
Vg (V)
Vg (V)
Vg (V)
FIG. 2 (color online). Electron transport in graphene below
FIG. 1 (color online). Resistivity (blue curve) and conduc-
300 K can be described by the empirical expression Vg;T
tivity 1= (green curve) of SLG as a function of gate
L Vg S T where S is independent of Vg but varies with
voltage. If we subtract a constant of 100 (used here as a
T. After subtracting S that for this sample changed from
fitting parameter), the remaining part L Vg of resistivity be-
40 at low T to 70 at 260 K, the resulting curves
comes inversely proportional to Vg (red curve). The thin black
L Vg 1= L Vg became indistinguishable (the cluster
line (on top of the red curve for Vg>0) is to emphasize the
marked 1= L consists of 5 such curves). The experiments
linearity (the red curve is equally straight for negativeVg). The
were carried out in a field of 0.5 T to ensure that weak local-
particular device was 1 m wide, andT 50 K was chosen to
ization corrections (rather small [1,22] but still noticeable) do
be high enough to suppress universal conductance fluctuations,
not contribute into the reported T dependences.
still visible on the curves.
Now we turn to BLG. A typical behavior of its conduc-
tivity is shown in Fig. 4. BLG exhibits Vg qualitatively
independent of n due to long- and short-range scatterers,
similar to SLG s: away from NP, L/jVgj yielding a
respectively [3 7,9]. The latter contribution varies from
constant of between 3000 and 8000 cm2=Vs for our
sample to sample and becomes more apparent in high-
devices. This behavior (not reported before) is rather sur-
samples. This observation resolves the controversy about
prising because BLG s spectrum neither is similar to the
the varying (linear vs sublinear) behavior reported in dif-
conical spectrum of SLG [1,17] nor can it be considered
ferent experiments [2 5].
parabolic (the measured cyclotron mass varies strongly
With this procedure in hand, it is now easier to describe
with n [19]). As for T dependence in BLG, its only
T dependence of graphene s conductivity. Figure 2 shows
pronounced feature is a rapid decrease in around NP
that Vg curves become increasingly sublinear with in-
such that the peak value changes by a factor of 3 between
creasing T. However, after linearization, the resulting
liquid helium to room T. The inset of Fig. 3 shows this
curves (with S subtracted) become essentially indistin-
dependence in more detail and compares it with the weak,
guishable away from NP, collapsing onto a single curve
nonmonotonic behavior observed at NP in SLG. The origin
L/jVgjindependently of T (<300 K; at higher T, we
of this pronounced difference between graphene and its
observed clear changes in the shape of L Vg curves,
bilayer lies in their different density of states near NP,
which indicates that the phonon contribution can no longer
which vanishes for SLG but is finite in BLG [1]. For
be described by S independent ofn). The extracted values
SLG, the concentration of thermally excited carriers nT
of S increase with T as shown in Fig. 3 for 4 different
can be estimated as T=@vF 2, whereas in BLG it is
devices. One can see that theirT dependent parts, S
Tm=@2 (vF is the Fermi velocity in SLG and m the
S T S 0 , behave qualitatively similar, despite differ-
effective mass in BLG [17,19]). In the latter case, nT
ent S 0 at liquid-helium T. There is a slow (probably,
1012 cm 2 at roomT, an order of magnitude larger than for
linear) increase in S at lowTbut, above 200 K, it rapidly
SLG. The data in Fig. 3 (inset) are in agreement with this
shoots up (as T5 or quicker). The latter T dependence is
consideration. The weak T dependence at NP is a unique
inconsistent with scattering on acoustic phonons [11]. Note
feature of SLG and can be employed to distinguish SLG
that S does not exceed 50 at 300 K, yielding in from thicker [14,20] crystallites.
between 40 000 and 400 000 cm2=Vs for characteristic
Away from NP, we have never observed any sign of de-
Vg between 5 and 50 V (n between 3 and 30
crease in BLG s conductivity with increasing T. Com-
1011 cm 2). parison of Figs. 2 and 4 clearly illustrates that
016602-2
(1/k
&!
)
(1/k
&!
)
(k
&!
) &
(1/k
&!
)
(k
&!
) &
(1/k
&!
)
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PHYSI CAL REVI EW LETTERS
PRL 100, 016602 (2008) 11 JANUARY 2008
ÁNP (k&!)
ÁNP (k&!)
90
90
6
6
6
6
3
3
4
4
3
3
60
60
2
2
2
2
0
0
0 150 300
0 150 300
T (K)
T (K)
T (K)
T (K)
100 200 300
100 200 300
30
30
1 Å»# 20K
1 Å»# 20K
Å»# 100K
Å»# 100K
Å»# 180K
Å»# 180K
Å»# 260K
Å»# 260K
0
0
0
0
0 100 200 300
0 100 200 300
-50 -25 0 25 50
-50 -25 0 25 50
T (K)
T (K)
Vg (V)
Vg (V)
FIG. 3 (color online). T-dependent part of resistivity for 4 SLG
FIG. 4 (color online). T dependence in bilayer graphene. At
samples (symbols). The accuracy of measuring S was limited
the neutrality point, rapidly increases withTbut, away from it,
by mesoscopic fluctuations at lowTand by gate hysteresis above
no changes are seen. Vg exhibits a small sublinear contribu-
300 K. The hysteresis appeared whenVg was swept by more than
tion that can also be interpreted in terms of a constant S. The
20 V. To find S at higherT, we recorded as a function ofn
inset plots nominal values of found from linear fits of L(Vg)
(found from simultaneous Hall measurements). The solid curve
atVg>20 V away from NP and using S 50 . Here, does
is the best fit by using a combination of T and T5 functions,
not change within 2% and, if anything, shows a slight increase
which serves here as a guide to the eye. One of the samples
at higherT. The measurements were carried out atB 0:5 Tto
(green circles) was made on the top 200 nm of SiO2 covered by
suppress a small contribution of weak localization.
further 100 nm of polymethylmethacrylate (PMMA) and exhib-
ited 7000 cm2=Vs. The same values of for SiO2 and
PMMA substrates probably rule out charged impurities in SiO2
in the limit T qv) and, accordingly, S/T [11],
as the dominant scattering mechanism for graphene. The inset
showsTdependence of maximum resistivity NP (at the neutral- which can explain only our low-Tdata. We also considered
ity point) for SLG and BLG (circles and squares, respectively).
other T-dependent mechanisms such as flexural phonons
Note a decrease in NP with decreasingTbelow 150 K for SLG,
[12,21], electron-electron scattering, and umklapp pro-
which is a generic feature seen in many samples.
cesses, and they cannot explain the experimental behavior.
In the absence of a theory able to describe the rapid
increase in S, we point out that the behavior is consis-
T-dependent scattering in BLG is substantially weaker
tent with scattering on flexural phonons confined within
than in SLG. Further measurements (Fig. 4) have shown
ripples. Ripples are a common feature of cleaved graphene
that phonons do not contribute into BLG s within our
[18,22], suggesting that the atomic sheet is not fully bound
experimental accuracy of <2%. This yields in>
to a substrate (as illustrated in Ref. [23]) and, therefore,
300 000 cm2=Vs and a mean-free path of several microns
may exhibit local out-of-plane vibrations. First, because a
at 300 K.
characteristic size of ripples, d 10 nm, is typically
Let us now try to understand the observedTdependence
smaller than F [18,22], such vibrations induce predomi-
of S (Fig. 3). On one hand, it is partially consistent with
nantly short-range scattering. Second, at low T (qT
scattering by in-plane phonons [11] in the sense that they
2 =d), few flexural modes can be excited inside ripples
lead to resistivity independent ofn. On the other hand, such
but, as T increases and typical wavelengths become
phonons give rise only to S/T, in clear disagreement
shorter, more and more flexural phonons come into play.
with the measurements above 200 K. These are rather
It was suggested [7] that electron scattering in graphene is
general predictions and can be understood as follows.
dominated by static ripples quenched from the flexural-
Because vF=v 103 (v is the speed of sound), the
phonon disorder when graphene was deposited on a sub-
Fermi wavelength F in our experiments exceeds the
strate at roomT(there are also short-range ripples induced
spatial scale associated with thermal phonons, 1=qT, at
by substrate s roughness [18]), which implies that any
T>10 K (qT T=v is the typical wave vector). This
means that the scattering has a short-range character, lead- appreciable number of intraripple phonons start appearing
ing to S independent of n [7,11]. Furthermore, the only around room T. In fact, the observed behavior was
standard momentum and energy conservation considera- predicted by Das Sarma and co-workers who advocating
tions yield that only phonons with wave vectors kF for charged impurities as dominant scatterers in graphene
provide efficient (large-angle) scattering. The number of [6,13] noted that the model of quenched-ripple disorder
such phonons is/T (given by the Boltzmann distribution [7] implied strong temperature dependence (above a cer-
016602-3
3
2
3
2
(10 cm /Vs)
(10 cm /Vs)
S
S
(1/k
&!
)
(1/k
&!
)
"
(
&!
)
"
(
&!
)
week ending
PHYSI CAL REVI EW LETTERS
PRL 100, 016602 (2008) 11 JANUARY 2008
tain quenching temperature of about 100 K) an effect *geim@man.ac.uk
[1] For review, see A. K. Geim and K. S. Novoselov, Nat.
that has not been observed in the experiments [13]. This is
Mater. 6, 183 (2007); A. H. Castro Neto et al.,
exactly the experimental behavior reported here.
arXiv:0709.1163.
To estimate scattering rates 1= for intraripple flexural
[2] K. S. Novoselov et al., Nature (London) 438, 197 (2005);
phonons, one has to take into account two-phonon scatter-
Y. Zhang et al., Nature (London) 438, 201 (2005).
ing processes because out-of-plane deformations modulate
[3] F. Schedin et al., Nat. Mater. 6, 652 (2007).
electron hopping only in the second order [7,12]. For our
[4] Y. W. Tan et al., Phys. Rev. Lett. 99, 246803 (2007).
case ofq kF, we have found [21]
[5] J. H. Chen et al., arXiv:0708.2408.
[6] K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 96,
X
2 !q
t02kFa2 q4
1
e !q 256602 (2006); T. Ando, J. Phys. Soc. Jpn. 75, 074716
8vF q qcM2!2 1 e 2 !q
q
(2006); E. H. Hwang, S. Adam, and S. Das Sarma, Phys.
Rev. Lett. 98, 186806 (2007).
1
; [7] M. I. Katsnelson and A. K. Geim, Phil. Trans. R. Soc. A
e !q 1 2
366, 195 (2008).
[8] T. Durkop et al., Nano Lett. 4, 35 (2004).
where @=T, M is the mass of a carbon atom, a the
[9] E. Fradkin, Phys. Rev. B 33, 3263 (1986); Y. Zheng and
lattice constant, !q/q2 the flexural-phonon frequency,
T. Ando, Phys. Rev. B 65, 245420 (2002).
and t0 the derivative of the nearest-neighbor hopping in-
[10] Weak T dependence in SLG was noted in Refs. [1,2],
tegral with respect to deformation [7]. The integration goes
E. W. Hill et al., IEEE Trans. Magn. 42, 2694 (2006), and
over intraripple phonons that haveqlarger than the cutoff
Y. W. Tan et al., Eur. J. Phys. Special Topics 148, 15
wave vectorqc 2 =dimposed by the quenching. At low
(2007), but neither quantified nor discussed.
T, no flexural vibrations are allowed inside ripples (
[11] T. Stauber, N. M. R. Peres, and F. Guinea, Phys. Rev. B 76,
0), whereas in the high-T limit (qT 2 =d) the above 205423 (2007); F. T. Vasko and V. Ryzhii, Phys. Rev. B 76,
233404 (2007); S. Das Sarma (private communication).
expression allows the estimate @=e2 Td=2 a 2,
[12] E. Mariani and F. von Oppen, arXiv:0707.4350. Small-
yielding 100 to 1000 at 300 K ( 1 eVis the
angle scattering by acoustic phonons (/T4) mentioned in
bending rigidity of graphene [7]). The rapid increase in
the paper is valid in the low-T limit (T<10 K).
S above 200 K can be attributed to transition between
[13] S. Adam, E. H. Hwang, and S. Das Sarma,
the low- and high-Tlimits. The absence of any appreciable
arXiv:0708.0404.
Tdependence in BLG is also consistent with the model, as
[14] K. S. Novoselov et al., Science 306, 666 (2004); Proc.
BLG is more rigid and exhibits weaker rippling [22].
Natl. Acad. Sci. U.S.A. 102, 10451 (2005).
In summary, weakTdependence of electron transport in
[15] P. Blake et al., Appl. Phys. Lett. 91, 063124 (2007).
graphene and its bilayer yields in>200 000 cm2=Vs.
[16] A. C. Ferrari et al., Phys. Rev. Lett. 97, 187401 (2006).
The rapid rise in the smallT-dependent part of in SLG
[17] K. S. Novoselov et al., Nature Phys. 2, 177 (2006);
lends support for the model of quenched-ripple disorder as E. McCann and V. I. Fal ko, Phys. Rev. Lett. 96, 086805
an important scattering mechanism. The model suggests (2006).
[18] M. Ishigami et al., Nano Lett. 7, 1643 (2007).
that the observedTdependence is extrinsic and can proba-
[19] E. V. Castro et al., Phys. Rev. Lett. 99, 216802 (2007).
bly be reduced together with ripples by depositing gra-
[20] S. V. Morozov et al., Phys. Rev. B 72, 201401 (2005).
phene on liquid-nitrogen-cooled substrates. If scattering on
[21] Flexural phonons lead to /T2=n. The expression
in-plane phonons does not increase in a flatter graphene
can be derived by the same technique that was used for
sheet, in could be truly colossal.
two-magnon scattering in half-metallic ferromagnets
We are grateful to M. Dresselhaus who stimulated this
[V. Yu. Irkhin and M. I. Katsnelson, Eur. Phys. J. B 30,
work by repeatedly raising the question about graphene s
481 (2002)]. In Ref. [7], scattering on ripples was consid-
intrinsic mobility. We also thank A. Castro Neto, S. Das
ered by assuming that they were static, resulting from
Sarma, F. Guinea, and V. Falko for useful discussions. This
flexural phonons quenched during graphene s deposition
work was supported by EPSRC (U.K.) and the Royal
on a room-Tsubstrate. A careful quantum analysis leads to
Society. the same result for dynamic ripples (flexural phonons),
if T is much larger than their energy at wave vectors
Note added. After the manuscript was submitted,
kF. The latter is valid for T>1 K. In the opposite
Fratini and Guinea [24] suggested that the observed strong
low-T limit, scattering on flexural phonons was studied
T dependence could alternatively be explained by scatter-
in Ref. [12].
ing on surface phonons in the SiO2 substrate, and this
[22] S. V. Morozov et al., Phys. Rev. Lett. 97, 016801 (2006);
explanation was later used by Chen et al. [25] to analyze
J. C. Meyer et al., Nature (London) 446, 60 (2007);
their experiment. The found agreement between the theory
E. Stolyarova et al., Proc. Natl. Acad. Sci. U.S.A. 104,
and both experiments is striking, but let us note that two of
9209 (2007).
the reported SLG samples were on top of 100 nm of
[23] E. A. Kim and A. H. Castro Neto, arXiv:cond-mat/
PMMA (not SiO2; see Fig. 3), and it would be fortuitous
0702562.
if the materials with so different polarizibility induce the
[24] S. Fratini and F. Guinea, arXiv:0711.1303v1.
same surface phonon scattering. [25] J. H. Chen et al., arXiv:0711.3646.
016602-4
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