Effect of disorder on a graphene p-n junction

0.27 MB·English

by  M. M. Fogler

Effect of disorder on a graphene p-n junction
M. M. Fogler,1,∗ D. S. Novikov,2,3 L. I. Glazman,2,3 and B. I. Shklovskii2
1Department of Physics, University of California San Diego, La Jolla, 9500 Gilman Drive, California 92093
2W. I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455
3Department of Physics, Yale University, New Haven, Connecticut 06511
(Dated: February 2, 2008)
Weproposethetheoryoftransportinagate-tunablegraphenep-njunction,inwhichthegradient
8 of the carrier density is controlled by the gate voltage. Depending on this gradient and on the
0 density of charged impurities, the junction resistance is dominated by either diffusive or ballistic
0 contribution. We find the conditions for observing ballistic transport and show that in existing
2 devices they are satisfied only marginally. We also simulate numerically the trajectories of charge
carriersandillustratechallengesinrealizingmoredelicateballisticeffects,suchasVeselagolensing.
n
a
PACSnumbers: 81.05.Uw,73.63.-b,73.40.Lq
J

8
2 I. INTRODUCTION AND MAIN RESULTS (a) V = V z V =+V
1 2
− b

]
l A. Definition of the model Φ =0
l
a
h d d x
Graphene is a new material whose unique electronic −
s- structure endows it with many unusual properties.1 A (b) n (c) n
e n0 n0
monolayer graphene is a gapless two-dimensional (2D)
m
semiconductor with a massless electron-hole symmet-
.
t ric spectrum near the corners of the Brillouin zone, d d x d d x
a ǫ(k) = ~v k, where v 108cm/s. The concentration − −
m ± | | ≈ n0 n0
of these “Dirac” quasiparticles can be accurately con- − −
- trolled by the electric field effect.2,3 An exciting exper-
d FIG.1: (a)Devicegeometry. Graphene(thinline)liesinthe
n imental development is the ability to apply such fields z = 0 plane. The gates (thick lines) are in the z = b plane.
o locally, by means of submicron gates. Using this tech- (b) Electron density profile for d=0.77b and the symmetric
[c dneiqmuoen,sgtrraaptheden.4e–7p-njunctions(GPNJ)havebeenrecently gate bias V2 =−V1 =V. (c) Samefor d=6.00b.

3 Within idealized treatments that neglect disorder
v and electron interactions, GPNJ were predicted to dis- is the density gradient18 at x=0.
0 play a number of intriguing phenomena. They in-
Ourassumptionsaboutdisorderrequireabriefdiscus-
5 cludeKleintunneling,8–10 Veselagolensing,11microwave-
sion. At present, the nature of disorder in graphene is
1 induced12,13andAndreev14,15reflection,aswellasstrong notcompletely understood.1 Our knowledgeofitderives
2
ballistic magnetoresistance.9,16 Both quantitative and
. mainly from the measurements of the transport mobil-
0 qualitative changes to these phenomena are expected
ityµ. Forasamplewithamacroscopicallyhomogeneous
1
wheninteractionsanddisorderareincludedinthemodel.
carrier concentration n and resistivity ρ, the mobility is
7
For example, long-range Coulomb interactions lead to
0 defined by
nonlinear screening in GPNJ, which can modify its re-
:
v sistance substantially.17 The purpose of this paper is to 1
i µ(n)= . (2)
X investigatehowthejunctionresistanceisaffectedbydis- enρ(n)
order. We show that in existing GPNJ this effect is in- | |
r
a deed strong and suggest what can be done to reduce it. A remarkable fact that holds true for nearly all experi-
WeconsideragenericmodelofanelectrostaticGPNJ, mentsongrapheneisthatµ(n)isobservedtobeapprox-
in which a grounded graphene sheet in the x-y plane imately constant away from the charge-neutrality point
is controlled by two coplanar metallic gates with volt- n=0. Ratherthanenteringadebateonthemicroscopic
ages V1 and V2. The gates are separated by distance b origin of this behavior, we adopt it on phenomenologi-
from graphene and a distance 2d from each other. Un- cal grounds. We can do so because the derivation below
der a symmetric gate bias, V2 = V1 = V (Fig. 1), the applies regardless of the exact microscopic origin of the
−
graphene carrier density n(x) varies linearly in the mid- constant mobility.
dle of the junction (x=0), Itis convenientto define parametern ofdimensionof
i
concentration by
′
n(x) nx, x D max b, d , (1)
≃ | |≪ ≡ { } e
andtends to its limiting values n0 at x D. Here n′ ni = hµ =const, (3)
± | |≫
2
then the resistivity ρ(n) can be written as For small β (high disorder or low density gradient), the
transport is purely diffusive, and the resistance of the
h n
ρ(n)= i , n n . (4) GPNJ is given by
e2 n | |≫ i
| |
h n
Below we will also need the carrier mean free path l, β 1: R 2 i ln β2/3γ , (11)
≪ ≃ e2 n′ W
which is related to the conductivity in a standard way: | |
(cid:0) (cid:1)
e2 whereW isthewidthofthedeviceinthey-directionand
ρ−1 = (2kFl), (5) γ is defined by
h
where k (n) = π n is the Fermi wave vector. Using γ n′ 1/3D 1. (12)
F | | ≡| | ≫
Eq. (4), we find
p The condition γ 1, which is usually satisfied in
k experiment,4–7 ens≫ures that the density n(x) varies
F
l(n)= , n n . (6)
2πn | |≫ i across the GPNJ slowly enough, D = max d, b
i k−1(n ), to justify its evaluation by means of{clas}sic≫al
The inequality n n in Eqs. (4) and (6) is stipulated F 0
| |≫ i electrostatics.17 Equation (11) is written for β2/3γ 1,
by another phenomenological observation: the satura- ≫
i.e., for n n , when the GPNJ is still well-defined de-
tdieonnsiotfieρs.(1n,1)9at a finite value ρmax ∼ h/e2 at low carrier spite rand0o≫m fliuctuations of the electron density n(x,y)
due to disorder.
As we mentioned, our main results can be obtained
In the opposite regime (large β or low disorder) the
without knowing the microscopic origin of Eq. (4) and
GPNJ resistance
ρ . Nevertheless,itisusefultohaveinmindaconcrete
max
model that may clarify the physical meaning of parame- β 1 : R=R +R (13a)
bal dif
ter n . One such actively discussed model assumes that ≫
i
themobilityislimitedbychargedimpuritieslocatedina isthesumoftheballisticandthediffusivecontributions,
close proximity to the graphene sheet.20,21 An impurity
h c
of a unit charge acts as a scatterer with the transport 1
R = , c 1.0; (13b)
cross-section20–23 bal e2 α1/6 n′ 1/3W 1 ≈
| |
h n 4πγ β4/3
Λ=2πc2(α)/kF , (7) Rdif ≃2e2 n′iW ln β4/3 , γ ≫ 4π . (13c)
where c = πα2/2 for α 1 (graphene on large-κ sub- | | (cid:18) (cid:19)
2
strate) and c 0.1 for α≪ 1 (SiO substrate).24 Here Equations (11) and (13c) are valid with logarithmic ac-
2 2
α=e2/κ~v is t∼he dimension≈lessstrengthofCoulombin- curacy26 and match at β 3. The ballistic contribution
∼
teractionsandκistheeffectivedielectricconstant. Ifthe dominates, R Rbal Rdif, provided
≃ ≫
chargedimpuritieshaveanaveragesurfaceconcentration
3/2
Ni, then l=1/(NiΛ). Comparing this with Eq. (6), one 2α1/6 4πγ
indeed arrives at Eq. (3) with β β∗ = ln . (14)
≫ " c1 β∗4/3!#
n =c (α)N . (8)
i 2 i
Realistically, the logarithmically “large” threshold β∗
This argument has a considerable appeal and is sup- here is about 10. In recent experiments5,7 β is of the
ported by recent experiments.25 same order of magnitude. So, they are presumably in
the crossover region R R . To move deeper into
bal dif
∼
the ballistic regime one needs either a larger concentra-
B. Results tion gradient n′ or a higher mobility µ.
| |
The rest of the paper is divided into three sections.
To isolate the transport properties specific to GPNJ In Sec. II we give the analytical derivation of the above
wefollowtheprocedureintroducedbyexperimentalists,5 results. InSec.III weillustratethembynumericalsimu-
andcomputethedifferenceofthe totalresistanceR of lations. Finally, in Sec. IV we discuss their implications
tot
the device in the p-n [Fig. 1(a)] and the n-n states: for ongoing experimental work.
R≡Rtot|V2=−V1=V −Rtot|V2=V1=V . (9)
II. DERIVATION
This allows to largely eliminate the contribution of the
bulk regions x > D. Our results are then as follows.
We find two|qu|alitatively different regimes, depending This section is organized as follows. First, we con-
on magnitude of the dimensionless parameter sider electrostatics of the gate-tunable junction. Next,
westudy separatelythe ballisticandthe diffusivecontri-
′
n
butions to the transport. Finally, we combine them to
β = | | . (10)
n3/2 arrive at total expression for the resistance of a GPNJ.
i
3
β <<1: x densitygradientsharplyincreasesnearthegateedges. In
0 xtun xflc D thoseregions,n(x)isdictatedbythenearestgate(similar
to the case studied in Ref. 17), and one can show that28
x x
flc bal
β >> 1: x dn κ
0 xtun D mxax dx ≃ 27eb2 max{|V1|,|V2|}, d≫b. (21)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
FIG.2: AsketchofthecharacteristiclengthscalesinaGPNJ (cid:12) (cid:12)
for the limiting cases of small and large β. Only the x > 0 B. Ballistic resistance
sideofthejunctionisshown. Thediffusiveregionishatched.
Parameters xtun and xflc are indicated bythedashed linesin The resistanceRbal ofa cleanGPNJ is related9 to the
the regimes β ≪ 1 and β ≫ 1 respectively, as they do not
electric field at the p-n interface. To compute this field
havedirect physical meaning in these cases.
one has to go beyond electrostatics of ideal conductors
and take into account nonlinear screening at the p-n in-
terface. Equation (13b) for R was derived from this
A. Electrostatics bal
analysisin Ref. 17. In the case α 1, the resultfor R
bal
∼
canbequalitativelyunderstoodastheballisticresistance
Electron density in graphene is related to the elec-
of a system with Wk n(x ) transmitting channels:
F tun
trostatic potential Φ(x,z) by the Gauss law, n(x) =
(κ/4πe)∂zΦ(x,+0). To find Φ and n we can treat (cid:0)h 1 (cid:1) h xtun
graphene as an ideal conductor. (For the discussion of Rbal ∼ e2k W ∼ e2 W . (22)
F
this approximation, see Refs. 17 and 27.) The calcula-
tion can be done using the conformal mapping Here the effective “width” of the p-n interface
a+w π x =α−1/6 n′ −1/3 (23)
2w+ln = (x+iz), (15) tun | |
a w b
(cid:18) − (cid:19) is found from the condition that it is of the order of the
which transforms the upper half-plane z > 0 with the quantum uncertainty in the quasiparticle coordinate,
branch cuts along the gates [cf. Fig. 1(a)] to the upper
half-plane of a complex variable w = w(x,z). Here a is xtun ∼kF−1 n(xtun) . (24)
found from
(InRef.17,x wasdenoted(cid:0)byx .)(cid:1)Thequasiparticles
tun TF
a(a+1)+ln(√a+√a+1)=πd/(2b). (16) that manage to get inside the strip x < xtun cross the
p-n boundary without tunneling sup|pr|ession.9
The gpraphene sheet, the left gate, and the right gate Belowweconsidertheresistance(9)ofasymmetrically
are mapped onto the intervals a < w < a, w < a, biased GPNJ, V = V = V. The transport is either
− − 2 − 1
and w > a, respectively, of the real axis. Therefore, the diffusive or ballistic depending on the gradient (10).
sought potential is given by
Φ(x,z)=(1/π)Im[V1ln(a+w) V2ln(a w)]. (17) C. Purely diffusive transport, β≪1
− −
Using these equations and simple algebra, we find
The derivation is based on treating ρ n(x) as the lo-
κ (V +V )a+(V V )w(x) calx-dependent resistivity. This is justified providedthe
2 1 2 1
n(x)= − , (18) (cid:0) (cid:1)
8πeb a(a+1) w2(x) concentration gradient is sufficiently small, such that
−
where w(x) stands for the real quantity w(x,z = 0) de- l(n)∂xn n. (25)
| |≪
finedbyEq.(15). Forthesymmetricgatebiasweobtain
Using Eq. (6), one can easily check that for β 1 the
≪
n(x)= n0(V)w(x) , V = V =V , (19) condition (25) is satisfied at all |x| ≫ xflc where xflc is
a(a+1) w2(x) 2 − 1 defined by
−
′
in which case the density gradient at x=0 is given by xflc =ni/n , (26)
| |
π n (V) κV see also Fig. 2(a). At such distances Eq. (4) is still
′ 0
n = , n (V)= . (20)
2b(1+a)2 0 4πeb valid. On the other hand, in the strip x . xflc, we
have n(x) .n , so that Eq. (4) does not|ap|ply.29 Since
i
| |
Two examples of n(x) computed according to Eqs. (15) thetransportremainsdiffusiveinthestrip x <x (cer-
flc
and (19) are plotted in Fig. 1(b) and (c). In both cases tainly,itcannotbeballisticbecauseofstron|g|disorder30),
′
the linear dependence n nx extends up to x D. we can assume that the corresponding local resistivity is
However,forwidelysepar≃atedgates[Fig.1(c)],|th|e∼local of the order of its bulk value ρ h/e2 at the charge
max
∼
4
neutralitypoint. Thisallowsustoestimatetheresistance Incontrast,withinthe strip x <x the transportis
bal
| |
of this region as ballistic: the local mean-free path l[n(x)] nominally ex-
ceeds x, so that quasiparticles reach the p-n interface
R ρ xflc . (27) largely| w| ithout experiencing impurity scattering. We
flc max
∼ W
now note that the tunneling strip (31) is located deep
inside this ballistic region,
According to our definition (9), the GPNJ resistance is
the difference of the total resistances in the p-n and n-n
4π x
bal
configurations. ItisconvenienttowriteitasR=Rdif(0), xtun ∼ α1/6 β4/3 ≪xbal, (32)
where
∞ see also Fig. 2(b). Therefore, the transmission problem
2 is reduced to the clean case,17 yielding Eq. (13b) for the
(x)= dx˜ ρ(x˜) ρ(x˜) . (28)
Rdif W |V1=−V2 − |V1=+V2 ballistic resistance Rbal. Due to the large logarithmic
Z
x (cid:2) (cid:3) factor in Rdif [Eq. (13c)], the ballistic contribution in
Eq.(13a)starts to dominate the diffusive one only when
Using Eqs. (15), (19), (4), and the expression
β exceeds a logarithmically large threshold β∗, Eq. (14).
n (V)a
0
n(x)= , V =V =V , (29)
a(a+1) w2(x) 1 2
− III. NUMERICAL SIMULATIONS
for the charge profile in the n-n state that follows from
Eq. (18), the integral in Eq. (28) can be transformed to Inthis sectionwe illustrate and supportthe abovean-
alytical results by numerical simulations. In particular,
Rdif(x)≃2eh2 nn′iW ln (a+n10)n′ x , x&xflc. wteeesshoonwlytthhaatttthheectroittaelriorensiβsta≫nceβ∗of[Ethqe. j(u1n4c)]tiognuaRranis-
| | (cid:20) | | (cid:21)
(30) given by the formula derived for a disorder-free GPNJ,
The total resistance is (x )+R , which leads to Eq.(13b). Realizationof other ballistic phenomena may
dif flc flc
R
Eq. (11). Note that the effect of R is only to modify demand cleaner systems, see below.
flc
the numericalfactor in the argumentof the logarithmin To get intuition above the nature of transport at
the final expression. For sufficiently long junctions, this β >β∗ we studied semiclassicaltrajectoriesof the quasi-
logarithm is large, cf. Eq. (11), and so our crude esti- particlesin aGPNJ by numerically solvingthe following
mate of R is quite acceptable. This can be understood relativistic equations of motion
flc
by realizing that in a long junction the resistance of the
r˙ =vp/p , p˙ = Φ(r) . (33)
x < x strip is much smaller than that of the rest of
flc | | ∇| |
| |
the system. In shorter devices, the contribution of this
For illustrative purposes, we adopted the potential
“fluctuating strip” can be significant, and so a more ac-
cFuorraetxeaemvapllue,atoinoenmofayRwdiafnitntoEqp.er(f2o8r)mmthayeibnetengeracetisosnariyn. Φ(r)= sgn(x) πn′ x + Qj . (34)
− | | (r r )2+z2
Eq. (28) numerically using the experimentally measured p Xj − j j
dependence ρ(n) instead of Eq. (4). q
Here the first term models the potential induced by the
gates17 andthesecondtermrepresentsthepotentialcre-
D. Co-existence of ballistic and diffusive transport, ated by impurity charges Qj = 1 with coordinates
β≫1 (rj,zj). This expression assumes α±= e2/κ = ~ = v = 1
and neglects, for simplicity, the screening of these impu-
ritiesbytheelectronsingraphene. Weestimatethatthis
Here the carrier density n(x) varies with x more
entails c 1 in Eq. (8).
rapidly. As a result, the diffusive approximation breaks 2
∼
Other parameters of the simulation were as follows.
down inside the strip x .x whose width is given by
bal
| | The z-coordinates of all the impurities were set to z =
the condition l[n(x )] x , i.e., j
bal bal
∼ 0.01 in some arbitrary length units. The in-plane co-
n′ ordinates of the impurities were chosen randomly in-
xbal ∼ 4|πn|2 . (31) side the square x, y < 100 straddling the p-n in-
i terface. The tot|al|im| p|urity number was 300, so that
sTohethcaatrraietrdxen>sityxatxE=q.x(b4a)l aispsptliilelsh.igTh,hnus(,xbtahle)≫diffnui-, nyi ∼ 83000o/f2F00ig2.=3,01.20607o5f.thInesethiemfipeulrditoiefsvaireews|exe|n.≤T6h0e,
sive contrib|ut|ion tobatlhe resistance is R (x ), |de|n≤sity gradient was set to be n′ = 0.25, which makes
dif dif bal
leading to Eq. (13c). [The extra factor β−≃2Runder the parameter β quite large: β =0.25/n3/2 400.
logarithminEq.(13c)vs. (11)comesfromx β2x . In Fig. 3 we show 51 electron traijec∼tories computed
bal flc
Note, however, that x has no direct physical∼meaning by standard numerical algorithms.31 The energy for all
flc
if β 1.] trajectories was fixed at zero and the starting point was
≫
5
60
Furthermore, scattering of an electron by a Coulomb
impurity typically deflects the electron trajectory by a
40 substantial angle. Therefore, the lensing additionally
requires that a narrow fan of trajectories defined by
Eq. (35) does not undergo impurity scattering. The
20
width of this fan in real space is √x0xtun. Therefore,
∼
the condition on x becomes
0
0
n x √x x .1. (37)
i 0 0 tun
−20
Accordingly, the injection and collection contacts must
be placed no further than the distance
−40
1 4π
x x (38)
−60 lens ∼ n2/3x1/3 ∼ β8/9 bal
i tun
FIG. 3: Semiclassical trajectories of quasiparticles in a dis- from the interface, which may be considerably smaller
ordered GPNJ with β ≫ 1. The trajectories start at the
than x . Indeed, the absence of a discernible Veselago
bal
midpoint of the bottom edge, which belongs to the n-region.
lensinginFig.3isinagreementwithourestimates: since
The p-region (shaded) occupies the upper half of the figure.
x = 60 and x 20 [cf. Eq. (38)], we are not yet in
The open (filled) circles are in-plane Coulomb impurities of 0 lens ∼
the regime x x .
negative (positive) charge. The trajectories that carry cur- 0 ≪ lens
rent across the p-n interface are shown by dark lines on the
n-side and tapering white lines on the p-side. The variable
width of these lines is the local Fermi wavelength 2π~/p(r). IV. DISCUSSION AND CONCLUSIONS
The thin lines are examples of thetrajectories reflected from
theinterface. Inthisfinalsectionofthepaperwediscussgeometrical
requirements imposed by the criterion (14) in actual ex-
periments. Usingarealisticnumberµ 2,500cm2/(V s)
set to x = x = 60, y = 0. The polar angles of the
0 − in Eq. (3), we get n 1.0 1011cm∼−2. Such n c·an
initial velocities formed an equidistant set spanning the i i
∼ ×
be achieved if the transport mobility is limited by, e.g.,
interval ( π/5,π/5).
− charged impurities of concentration N 1012cm−2 [as-
From Eq. (31) we estimate x 350. Therefore, the i
bal ∼
∼ suming c 0.1 in Eq. (8)].
injection point is deep inside the ballistic strip, x 2
0 ∼
≪ We consider first the case of a narrow gap between
x . Simultaneously, x x 2, cf. Eq. (23), so
bal 0 tun
≫ ≈ the gates, b d [Fig. 1(b)], where a 0.4. Taking
that the semiclassical approximation (33) is legitimate.
α 1, b 5≈0nm, and n 2 1012cm≈−2, similar to
As evident from Fig. 3, the average distance between 0
∼ ≈ ∼ ×
those of Ref. 5, for the above n we get β 10. Some
collisions of quasiparticles with impurities exceeds the i
∼
evidencefortheballistictransportwasindeedseenunder
distance x from the injection point to the interface.
0 suchconditions.5 Onthe other hand, observingVeselago
Thus, in agreement with the above estimates, electrons
lensing11seemsratherchallenging: itrequiresplacingthe
can propagate across the interface according to the for-
mulas derived for the disorder-free system.9,17 A closely injection and collection contacts within 10nm from
∼
each other, cf. Eq. (38).
related observation is that for the chosen parameters
Next, in the case of widely separated gates, d = 1µm
there are many points along the interface not “blocked”
(and the same b = 50nm) we get β 0.1 even for a
by the impurities.
veryhighmaximum density n =1013c≈m−2. Inorderto
On the other hand, even for such large β there is no 0
evidence for the recently proposed11 Veselago lensing ef- observe ballistic transport in this device, the suggested
setup should be somewhat modified. For example, using
fect: a self-focusingof holesinto a point(x ,0),a mirror
0
abackgateonecanintroduceauniformoffsetoftheelec-
image of the injection spot ( x ,0). The primary diffi-
0
− tron density n(x), which would shift the location of the
culties with observingthis focusing effect are apparently
p-ninterfaceawayfromthex=0pointandclosertothe
asfollows. First,evenintheabsenceofanydisorder,only
edgeofeitheroneofthegates,asdiscussedinRef.17. In
a small fraction of electrons can penetrate through the
′
this manner the density gradientn at the GPNJ can be
p-n interface: the transverse momenta of such electrons
must satisfy the condition9,17 rampeduptoitsmaximumvalue(21),yieldingβ similar
to that in a narrow-gapdevice.
~
Although we considered a particular junction geome-
p . . (35)
y
| | xtun try, Fig. 1(a), our treatment can be readily extended to
characterize transmission in any GPNJ with a smoothly
Such momenta are much smaller than the typical ones,
varying electron density, γ 1. The basic steps are as
p ~ n′x ~ x0 . (36) follows: (i) findthe carrierd≫ensitygradientn′ atthe p-n
y ∼ 0 ∼ x x interface, (ii) compute β from Eq. (10), (iii) determine,
(cid:18) tun(cid:19)r tun
p
6
′
based on the criterion (14), whether the device is dif- κ. In this case, on the one hand, n is larger for the
fusive or ballistic, and finally (iv) find the diffusive and same gate voltage and, on the other hand, the influence
ballisticcontributionsfromEqs.(11)–(13b). [Formulafor of Coulomb scattering is smaller, c α2 κ−2.
2
∝ ∝
R canbefurtherrefinediftheintegrationinEq.(28)is
dif
done numerically using an accurately measured density
dependence of the bulk resistivity ρ(n).]
To conclude, disorder can strongly inhibit the ballis-
Acknowledgments
tic transport regime in graphene field-effect devices. In
recent experiments4–7 on graphene p-n junctions, this
regime was reached only marginally at best. For bal- This work is supported by the Grants NSF DMR-
listic devices one should aim at larger electron density 0706654, DMR-0749220, and DMR-0754613. We
′
gradients n and higher mobilities to satisfy the condi- are grateful to D. Goldhaber-Gordon, B. Huard,
tion(14). Notethatifthe primarysourceofdisorderare and L. M. Zhang for comments on the manuscript.
′
charged impurities, then the requirement on n becomes M. F. thanks the W. I. Fine TPI for hospitality and
less stringent for substrates of high dielectric constant L. M. Zhang for help with computer simulations.
∗ Electronic address: [email protected] 18 Notethatn′standsforthedensitygradientcomputedclas-
1 A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183 sically. Quantumeffects decrease17 truen′.
(2007); A. H. Castro Neto, F. Guinea, N. M. R. Peres, 19 Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam,
K.S.Novoselov,andA.K.Geim,arXiv:0709.1163 (2007). E. H. Hwang, S. Das Sarma, H. L. Stormer, and P. Kim,
2 K.S.Novoselov, A.K.Geim,S.V.Morozov, D.Jiang, Y. Phys. Rev.Lett. 99, 246803 (2007).
Zhang,S.V.Dubonos,I.V.Grigorieva, andA.A.Firsov, 20 T. Ando, J. Phys. Soc. Jpn. 75, 074716 (2006).
Science 306, 666 (2004). 21 K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 98,
3 K.S.Novoselov,A.K.Geim,S.V.Morozov,D.Jiang, M. 076602 (2007).
I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. 22 E. H. Hwang, S. Adam, and S. Das Sarma, Phys. Rev.
Firsov, Nature(London) 438, 197 (2005). Lett. 98, 186806 (2007).
4 M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, 23 D. S. Novikov, Phys. Rev. B 76, 245435 (2007); Appl.
IEEE Electron DeviceLett. 28, 283 (2007). Phys. Lett.91, 102102 (2007).
5 B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang, 24 TreatingtheimpurityscreeningwithintheRandomPhase
and D. Goldhaber-Gordon, Phys. Rev. Lett. 98, 236803 Approximation(RPA)andcalculatingthescatteringcross-
(2007). sectionperturbativelyyields20,22 c2(0.9)≈0.05.Forα∼1
6 B. O¨zyilmaz, P. Jarillo-Herrero, D. Efetov, D. A. Abanin, this calculation is likely to underestimate c2. Using the
L. S. Levitov, and P. Kim, Phys. Rev. Lett. 99, 166804 RPA at the Dirac point and treating scattering off the
(2007). resultantCoulombpotentialexactlygives23 c2(0.9)≈0.26
7 J. R. Williams, L. DiCarlo, and C. M. Marcus, Science (when averaged over the donors and acceptors). This is
317, 638 (2007). likelytooverestimatetheresistivity.Thus,thetruec2(α∼
8 M.I.Katsnelson,K.S.Novoselov,andA.K.Geim,Nature 1) is probably close to 0.1.
Phys. 2, 620 (2006). 25 J. H. Chen, C. Jang, M. S. Fuhrer, E. D. Williams, and
9 V. V. Cheianov and V. I. Fal’ko, Phys. Rev. B 74, M. Ishigami, arXiv:0708.2408 (2007). See also Ref. 19.
041403(R) (2006). 26 However,wechoosetokeepthenumericalfactor4π inthe
10 J. M. Pereira, Jr., V. Mlinar, F. M. Peeters, and argument of the logarithm in Eq. (13c) because it can be
P. Vasilopoulos, Phys.Rev.B 74, 045424 (2006). significant in practice.
11 V. V. Cheianov, V. I. Fal’ko, and B. L. Altshuler, Science 27 See M. M. Fogler, D. S. Novikov, and B. I. Shklovskii,
315, 1252 (2007). Phys. Rev.B 76, 233402 (2007) and references therein.
12 B.Trauzettel, Ya.M.Blanter, andA.F.Morpurgo,Phys. 28 This follows from Eq. (4) of Zhang and Fogler.17
Rev.B 75, 035305 (2007). 29 The subscript “flc” in xflc stands for “fluctuation” to ex-
13 M.V.FistulandK.B.Efetov,Phys.Rev.Lett.98,256803 press the idea that inside the strip |x| < xflc the local
(2007). electron concentration presumably exhibits spatial fluctu-
14 C. W.J. Beenakker, Phys.Rev.Lett. 97, 067007 (2006). ationsofamplitude∼niduetodisorder,whichoverwhelm
15 A.Ossipov,M.Titov,andC.W.J.Beenakker,Phys.Rev. theaverage linear trend of Eq.(1).
B 75, 241401(R) (2007). 30 Inaddition,weassumethatthetemperatureisnottoolow
16 A. V. Shytov, N. Gu, and L. S. Levitov, arXiv:0708.3081 and neglect any localization effects.
(2007). 31 Details will begiven in L. M. Zhang(in preparation).
17 L. M. Zhang and M. M. Fogler, arXiv:0708.0892 (2007).