Nondiagonal graphene conductivity in the presence of in-plane magnetic fields

Contents lists available atScienceDirect

Physics

Letters

A

www.elsevier.com/locate/pla

Nondiagonal

graphene

conductivity

in

the

presence

of

in-plane

magnetic

fields

R.R. Brandão

a

,

L. Moriconi

b

,

∗

a_{InstitutoNacionaldaPropriedadeIndustrial,RuaSãoBento1,Centro,CEP:20090-010,RiodeJaneiro,RJ,Brazil} b_{InstitutodeFísica,UniversidadeFederaldoRiodeJaneiro,C.P.68528,CEP:21945-970,RiodeJaneiro,RJ,Brazil}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received14January2015

Receivedinrevisedform25March2015 Accepted26March2015

Availableonline30March2015 CommunicatedbyR.Wu

We study the electron/hole transport in puddle-disordered and rough graphene samples which are subject to in-plane magnetic fields. Previous treatments, mostly devoted to regimes where the electron/hole scattering wavelengths are larger than the surface height correlation length, are based onthe useoftransportequations withappropriateformsforthecollisionterm. Wepoint outinthis work, as a counterpoint, that classical Lorentz force effects, which are expected to hold when the Fermi level is far enough away fromthe charge neutralpoint, can be heuristically assessedthrough disordered Boltzmann equations thatcontain magnetic-field dependentmaterialderivatives, and keep thezeromagnetic-fieldstructureofthecollisionterm.Itturnsoutthattheelectricconductivitytensor getsapeculiarnondiagonalcomponent,inducedbythein-planemagneticfieldthatcrossestherough topographyofthegraphenesheet,eveniftheprojectedrandomtransversemagneticfieldvanishesinthe mean.Numericalestimatesofthetransverseconductivitiessuggestthattheyaresuitableofobservation underconditionswhicharewithinthereachofup-to-dateexperimentalmethods.

©2015ElsevierB.V.All rights reserved.

1. Introduction

The high mobility of charge carriers in graphene, related to their pseudo-relativistic Dirac spectrum andsemimetal character, renders it tobe one ofthe mostpromisingmaterials for techno-logical innovation in the field of solid state devices [1]. A flurry ofresearch hasstartedsince thediscovery ofgraphenetenyears ago,markedbyseveralimportantexperimentalandtheoretical ad-vancesalongthe way.Thereis,however,aspreadconsensus that thephysicsofelectrontransportingrapheneisnotyetcompletely understood [2]. The simplest of all approaches, viz., a straight-forward application of linear response theory to the problem of two-dimensionalfreemasslessDiracfermionsisplaguedwith am-biguitiesultimatelyassociatedwiththeevaluationoffermion de-terminants[3], sothat morephysicalingredients arein orderfor proper modeling. A comprehensive study of charge transport in grapheneshouldtake intoaccount,asamatterofprinciple, elec-tron scattering caused by (i) the Coulomb two-body interaction (whichseemstoaffectthefermionspectruminarelevantwayin the vicinity ofthe charge neutralpoint), (ii) the disordered sub-strate doping layer, and (iii) surface roughness. At present, one

*

Correspondingauthor.

E-mailaddress:[email protected](L. Moriconi).

finds in the literature only partial modeling scenarios with vari-abledegreesofsuccess

[4]

.

Asitisusualintransporttheory,thereareessentiallytwomain approachesto thecomputation ofthe conductivitytensor.One is based on the Kubo response formalism [5], while the other re-liesontheanalysisoftransportequations

[2]

.Thelatterapproach, whichistheonetobeadoptedinthiswork,isparticularlysuitable for the investigation of semiclassical regimes. In graphene, these regimes are attained in situations where the Fermi level is far enoughfromthechargeneutralpoint,or,inequivalent words,for larger absolute charge carrier densities, so that the electron/hole wavelengths involved in scattering can be assumed to be much smallerthanthemeanfreepath.

Aninteresting transportproblem,intimatelyrelatedtotheone ofsurface roughnesscharacterization,consistsinthestudyofthe linear response features ofgraphene in the presence of in-plane magneticfields

[6]

.Asthemagneticfieldcrossestherough topog-raphyofthegraphenesample,theelectron/holedynamicscouples tothecomponentofthemagneticfieldthat isnormaltothe sur-face. Thus, in effectiveterms, the electron/hole transport can be modeled as if it would take place in the presence of a random transversemagneticfield,withcorrelationsthatreflecttherandom distribution ofsurface heightsinthe graphenesheet. Ifone then restrictstheanalysistoregimeswherescatteringwavelengthsare takento be largerthan the surface heightcorrelation length, the

http://dx.doi.org/10.1016/j.physleta.2015.03.034 0375-9601/©2015ElsevierB.V.All rights reserved.

physicaleffectsoftherandomtransversemagneticfieldcanallbe encodedinthespecificformofthecollisionoperatorthatis intro-ducedintransportequationsforthechargecarrierdensity.Thisis preciselythepointofviewtakeninRefs.[6–9].

Asacounterpoint,ouraiminthispaperistoexplorethe mag-neticin-plane transport problemin grapheneforthe casewhere the scattering wavelengths are smaller than the mean free path and the surface height correlation length, so that we can work within the framework ofa semiclassical Boltzmannequation ap-proach,in a spiritsimilar to what has beendone inthe context ofelectrongases

[10]

.Accordingly,thereisnoneedtomodifythe structureofthe Boltzmanncollisionterm, oncethe perturbations duetotherandom transversemagneticfield are givenby contri-butions associatedto theclassical Lorentz force,which appearin thelefthandsideoftheBoltzmannequation.Thecentralresultof ouranalysisisthepeculiarnondiagonalstructureofthe conductiv-itytensor,which iscloselyconnectedto thestatisticalproperties ofthegrapheneroughsurface.

Thispaperis organized asfollows.Tostart, aGaussian model of surface roughness is introduced in Section 2, which is then usedtoestablishthestatisticalpropertiesoftheeffectiverandom transversemagnetic field defined on the graphenesheet. In Sec-tion3,wedevelop,followingtheBoltzmannequation approachof Ref.[11],atreatmentofthesemiclassicalelectron/holetransportin graphenesamples disorderedby thepresenceofchargedpuddles

[12] andsubjectto in-plane magnetic fields.We, then, work out numericalestimatesoftheconductivitytensorcomponents,which arenotedto beperfectlywithin thereach ofpresent experimen-talresolution.Finally,inSection4,wesummarizeourfindingsand pointoutdirectionsoffurtherresearch.

2. RandomGaussianmodelofsurfaceroughness

Weareinterestedtostudychargetransportinaroughgraphene sheet which is subject to crossed in-plane electric and mag-neticfields.Surface heightscan berepresented asa realfunction z

=

z

(

r

)

,wherer

= (

x

,

y

)

denotesapointdefinedonthexy plane. Wetake,withoutlossofgenerality,thex directiontobeparallelto theexternal(in-plane)smallelectricfield E thatisusedtoprobe thechargetransportinthesample,thatis,E

≡ (

E

,

0

,

0

)

.For mod-elingpurposes, z

(

r

)

isassumedtobe asmooth Gaussian random field with vanishing expectationvalue,



z

(

r

)

=

0, and two-point correlationfunction



z

(

r

)

z



r



 =



H2 2



1

−

2|r−r|2 L2

+

| r−r|4 L4



,

if

|

r

−

r

| ≤

L

,

0

,

if

|

r

−

r

| >

L

,

(1)

whereH andL parametrizethestandarddeviationandthe corre-lationlengthofheightfluctuationsonthegraphenesheet, respec-tively.Theunitnormalvectoronthegraphenesurface atposition r canbewrittenas

ˆ

n

(

r

)

=

√

1

1

+

a2

₊

_b2

(−

a

,−

b

,

1

)

(−

a

,

−

b

,

1

) ,

(2)

wherea

≡ ∂

xz

(

r

)

,b

≡ ∂

yz

(

r

)

andthe approximation in (2)is re-latedtothesurfacesmoothnessassumption(a

,

b

1).

Consider now, that the in-plane magnetic field B applied on thegraphenesample makesanangle

φ

withtheelectricfield.We have,thus,

B

=

B0

(

cos

(φ),

sin

(φ),

0

) .

(3)

The component of the magnetic field that is parallel to the graphenesurfacehasnotanyroleinthedynamicsofcharge trans-port.Thenormalcomponent,ontheotherhand,

B

(

r

)

≡

B

· ˆ

n

(

r

)

= −

B0



cos

(φ) ∂

xz

+

sin

(φ) ∂

yz



,

(4)

isarandomfieldthatcouplestotheorbitalelectron/holedegrees offreedom.Since z

=

z

(

r

)

isarandomGaussian field,soit isthe transverse magnetic B

=

B

(

r

)

, which vanishes in the mean, i.e.,



B

(

r

)

=

0. We can compute, from Eq.(4), arbitrary N-point ex-pectationvalueslike

GN

(

r1

,

r2

, . . . ,

rN

)

≡ 

B

(

r1

)

B

(

r2

) . . .

B

(

rN

)

 .

(5) ItisclearthatGN

=

0 forN odd.Welist,below,expectationvalues thatareofparticularimportanceinouranalysis:

(

B

(

r

))

2

 =

G2

(

r

,

r

)

=

2H2 L2 B 2 0

,

(6)

(∂

xB

(

r

))

2

 =

lim r→r

∂

x

∂

xG2

(

r

,

r 

₎

=

4H2 L4 B 2 0



sin2

(φ)

+

3 cos2

(φ)



,

(7)





∂

yB

(

r

)



2

 =

lim r→r

∂

y

∂

yG2

(

r

,

r 

₎

=

4H2 L4 B 2 0



cos2

(φ)

+

3 sin2

(φ)



,

(8)

∂

xB

(

r

)∂

yB

(

r

) =

lim r→r

∂

x

∂

yG2

(

r

,

r 

₎

₌

4H2 L4 B 2 0sin

(

2

φ) .

(9)

The effective random magnetic field B

(

r

)

is not statistically isotropic, that is



B

(

r

)

B

(

r

)

is not a function of

|

r

−

r

|

. In fact, wehave,for

|

r

|

<

L,



B

(

0

)

B

(

r

) =

2B 2 0H2 L2



1

−

2 L2



r2

+

2

(

x cos

(φ)

+

y sin

(φ))

2

.

(10) Asderivedinthenextsection,theanisotropiccorrelationfunction

(10) leads, ultimately, to a non-diagonal conductivitytensor that dependsonthegeometricalparameters

φ

,H ,andL.

It isinstructive tobriefly digress on theapparently analogous unusual Hall effect,which ispredicted tooccur, undervery spe-cial circumstances, in the completely different context of topo-logical insulators [13,14], in a model where the magnetic field also vanishes in the mean.As it iswell-known, the existence of Hallresponseisnecessarilyassociatedwithtime-reversal symme-try breaking(in two-dimensional space), whichin the aforemen-tionedmodelisbrokenduetoa specificdistribution ofmagnetic fluxes around lattice links. Inour case, in contrast,the essential point in having non-diagonal conductivities is relatedto a pecu-liar parity-symmetrybreaking mechanism. Actually,one could be puzzledbythefactthatthemagneticfieldvanishesinthemean– asitwouldbeimpliedbyparity-reversalsymmetry.Therefore,the transverse conductivity,whichis oddunderparity-reversal trans-formations,shouldvanishaswell.However,itisimportanttonote that parity-symmetryisbrokeninoursettingonlyatthelevelof second-ordercorrelationfunctions.Infact,parity-symmetry rever-sal canbe implemented inouranalysisvery simplyby means of thereplacement

φ

→ −φ

in

(3)

,modifying,asaconsequence,the expectationvalues

(9)

and

(10)

.

3. Semiclassicaltransportindisorderedgraphene

Intheabsenceofexternalmagneticfields,theusual(one-body) modeling ingredients in the graphene charge transport problem are the electron/hole scattering by substrate impurities and the smoothrandomelectricpotentialassociatedtochargedpuddles.In theBoltzmannequationapproach,oneassumesthatscatteringby impuritiesisencodedintherelaxationtime approximation,while

charged puddles can be modeled as extended subregions of the samplewherethechemicalpotentialisapproximatelyuniform,but randomly fluctuatingfrompuddle to puddle. Inthat way, a phe-nomenologicalrelationbetweentheminimumconductivityvalue, theelectron/hole mobilityparameter(in thesemiclassical region) andthe steepnessoftheconductivityparabolaaround thecharge neutralpointhasbeenpredictedandclearly supported byan ex-tensivecompilationofexperimentaldata

[11]

.

Weextendinthisworkthe Boltzmannequationapproach put forwardin

[11]

tothemoregeneralcontextofgraphenetransport in the presence of in-plane magnetic fields, where,as discussed intheprecedingsection,surfaceroughnessbecomesanadditional sourceofdisorder.Wedeal,morespecifically,withapproximately semiclassicaltransportregimescharacterizedbyameanfreepath



k anda surface correlation height L which are bothlarger than thescatteringwavelength

∼

k−1_{andbothsmallerthanthetypical}

chargedpuddlelinearsizeLp,thatis,

k−1

< 

k

<

Lp

,

k−1

<

L

<

Lp

.

(11) Thepartitionofthesample intoself-correlatedmagneticfield do-mainsandchargedpuddlesisdepictedin

Fig. 1

.

Ofcourse, the Boltzmann equation approach would be firmly grounded if k−1



k. However, as we will see, usual graphene samplesandproperlychosen carrierconcentrationsprovideusat mostwitha“weak”separationofscalesattheborderlineof semi-classicalbehavior.Ourdiscussion,thus,isessentiallyheuristic, hav-ing in mind the present lack of understanding on the crossover region between the semiclassical and the quantum regimes of graphenetransport.

We define, accordingly to the first set of inequalities in (11), a stationary Boltzmann distribution function f±

(

k

,

r

,

ξ )

in each one of the charged puddles, where k and r are, respectively, the wavenumber and position vectors,

ξ

denotes the “puddle-dependent” shift of the chemical potential, andthe positive and negative superscripts refer, respectively, to holes and electrons. Working in the relaxation time approximation, we write, for a givenpuddleofarea A,theBoltzmanntransportequationas



±

e

¯

h



E

+

B

(

r

)

vk

× ˆ

z



· ∇

k

+

vk

· ∇

r

f±

(

k

,

r

, ξ )

= −

1

τ

k

[

f±

(

k

,

r

, ξ )

−

f₀±

(

k

, ξ )

] ,

(12) where f₀±

(

k

, ξ )

≡

4 A−1

(ξ

±

μ

−



k

)

(13)

isthezero-temperatureFermi–Diracdistributionforanidealgasof holesorelectronswithchemicalpotential

ξ

±

μ

,whichalsotakes intoaccountthevalley andthe spindegreesoffreedom (Zeeman splittingeffects are negligiblein ourstudy).The one-particle en-ergy spectrumis givenby



k

=

akα (where

α

=

1 and

α

=

2 are assumed to model monolayer andbilayer graphene systems, re-spectively). The wavenumber dependent particle/hole velocity is, therefore,vk

= ¯

h−1

∇

k



k

=

α

a

¯

h−1kα−2k. Toevaluate a inthe

par-ticular case of monolayer graphene, we recall that the observed FermivelocityisvF

=

a

/

¯

h

106 m

/

s.

StillfollowingRef.[11],therelaxationtimeisdefinedas

τ

k

=

chk

¯

2−α

α

animp

,

(14)

wherenimp is the concentration ofscattering impuritiesand c

1

.

6 is adimensionlessprefactor(theunderlyingmodelistheone ofCoulombian impuritiesspreadoveraSiO2substrate;c isa

func-tion ofWigner–Seitz radius [12]). It isknown that (14) leads, in theabsenceofexternalmagneticfield,toconductivityprofilesthat

Fig. 1. Conceptualscheme ofthegraphene modelforthe analysisofdisordered semiclassicaltransport.Thesampleispartitionedintochargedpuddledomainsi (boundedbysolidlines)andspatiallysmoothsubdomainsi(boundedbydotted lines)oftypicallinearsizesLPandL,respectively.

depend linearlyonthecharge carrierdensityinthesemiclassical regime,ascorroboratedinrealexperiments.

Therandomchemicalpotentialshift

ξ

isassumedtohave zero-mean and standard deviation

δ



0, which introduces an energy

scale into the problem, related to displacements of the energy band acrossthegraphenesampleduetotheexistenceofcharged puddles.IthasbeenputforwardinRef.[11],withreasonable phe-nomenological success, that the probability distribution function (pdf)of

ξ

canbewrittenas

ρ

(ξ )

=

1

δ



0 g

ξ

δ



0



,

(15)

where g

(

·)

isa universal(i.e.,sample independent)pdf withunit variance(aneducatedguessistotakeitasaGaussian).

OnceEq.

(12)

issolved,onemaycompute theelectric conduc-tivitytensorcomponentsas

σ

xx

=

σ

y y

=

e 2

π

2

∂

∂

E

×



d2kd2r

[

f+

(

k

,

r

, ξ )

−

f−

(

k

,

r

, ξ )]ˆ

x

·

vk

E=0

,

σ

xy

= −

σ

yx

=

e 2

π

2

∂

∂

E

×



d2kd2r

[

f−

(

k

,

r

, ξ )

−

f+

(

k

,

r

, ξ )] ˆ

y

·

vk

E=0

,

(16) where the above double-averagenotation stands for the compu-tation ofexpectation values inthe ensemble of external random magnetic fields B

(

r

)

(expression between brackets) and random chemical potential shifts

ξ

(expressionembraced by an overbar). It hasbeentacitly assumed, fromthedefinition

(15)

,that fluctu-ations of the chemical potential (and therefore charged puddles) arenotcorrelatedtocorrugationsofthegraphenesheet.Thisisan interestingissue,stillopentotheoreticalandexperimental discus-sion.Actually,ithasbeensuggestedthatchargedpuddlescouldbe foundasthesoleeffectofgraphenesheetroughness,evenifthey are not coupled to substrates asinsuspended graphene samples

[15]. In this work, we take the much simpler phenomenological view encoded in (15). Otherwise, we should work with conjec-turedjointprobabilitydistributionfunctionsdefinedonthesample space of chemical potential andsurface height fluctuations. As a first approximation,however, itturns out that chemical potential fluctuationsgive subleadingcorrectionstotheconductivitytensor, sointhisfirstapproachtotheproblemwedonothaveworrytoo much withtheprecisemodeling ofthejointrandomfluctuations of

ξ

andz

(

r

)

.

Itispossible,then,toperformtheaverageoverfluctuationsof

ξ

straightforwardly in

(12)

,whichmeans thatwe justreplace,in thatequation, f±

(

k

,

r

,

ξ )

and f₀±

(

k

,

ξ )

by,respectively,

f±

(

k

,

r

)

≡

f±

(

k

,

r

, ξ )

=

∞



−∞ d

ξ

ρ

(ξ )

f±

(

k

,

r

, ξ )

(17) and f₀±

(

k

)

≡

f₀±

(

k

,

r

, ξ )

=

∞



−∞ d

ξ

ρ

(ξ )

f₀±

(

k

, ξ ) .

(18)

TheBoltzmannequationcanbeformallysolvedas

f±

(

k

,

r

)

= [

1

+

D±

(

k

,

r

)]

−1f₀±

(

k

)

=



1

+

∞



n=1

[−

D±

(

k

,

r

)

]

n



f₀±

(

k

) ,

(19) where D±

(

k

,

r

)

= ±

e

τ

k

¯

h



E

+

B

(

r

)

vk

× ˆ

z



· ∇

k

+

τ

kvk

· ∇

r

.

(20)

Relyinguponthescalingformsof

(14)

andoftheenergyspectrum, itisnotdifficulttoshowthat



d2k

[

D±

(

k

,

r

)

]

n



f₀±

(

k

)

=

E n−1 2



p=0 C±

(

n

,

p

)



d2ke

τ

k

¯

hk





k L



H



c



n−1

H



c



p f₀±

(

k

)

+

O

(

E2

) ,

(21)

whereC±

(

n

,

p

)

isdimensionless,



c

= ¯

hk

/

e B0 and



k

= |

vk

|

τ

kare,

respectively,thermscyclotronradiusandthemeanfreepath,both referring to excitations with wavenumber k. In usual graphene samples,we havetypically H

<

0

.

5 nm and L

>

8 nm. A proper choiceoftheFermiwavenumberinthesemiclassical regionleads to



c

>

20 nm,for B0

<

5 T,and



k

<

30 nm in the integrandof

(21).Itturnsoutthat



k L



H



c

<

0

.

6 (22) and H



c

<

2

.

5

×

10−2 (23)

areinfact smallenough tosuggest the(asymptotic)convergence of the perturbative expansion (21), which is then carried up to ordern

=

5, with p

=

0,so that we are ableto find the leading anisotropiccontributionstotheconductivity.Weget

σ

xx

=

A 4

π

∞



0 dk



−

e2ck2

¯

hnimp

+

e4_c5_k4

¯

h3n5_imp



2

[∂

xB

(

r

)

]

2

+ [∂

yB

(

r

)

]

2



+

e4c3k2

¯

h3n3_imp



[

B

(

r

)]

2



_d dk

[

f + 0

(

k

)

−

f0−

(

k

)]

(24) and

σ

xy

=

Ae4c5 4

π

h

¯

3n5_imp



∂

xB

(

r

)∂

yB

(

r

)





∞ 0 dkk4d dk

[

f − 0

(

k

)

−

f0+

(

k

)

] .

(25)

Substituting,now, Eqs. (6)to (9)and

(18)

in

(24)

and

(25)

, it followsthat

σ

xx

=

ce2 2

π

L4_h

_¯

3_n5 imp ∞



−∞ d

ξ

ρ

(ξ )

×



L2n2_imp

μ

+ ξ

a



2/α



L2h

_¯

2n2_imp

−

2B2₀H2c2e2



+

ξ

−

μ

a



2/α



2B2₀H2c2e2



c2

(

cos

(

2

φ)

+

10

)

×

ξ

−

μ

a



2/α

−

L2n2_imp



+

L4h

_¯

2n4_imp



+

μ

+ ξ

a



4/α 2B2₀H2c4e2

(

cos

(

2

φ)

+

10

)



(26) and

σ

xy

=

4B2₀H2e4c5sin

(

2

φ)

π

L4_h

_¯

3_n5 imp ∞



−∞ d

ξ

ρ

(ξ )



ξ

−

μ

a



4/α

+

ξ

+

μ

a



4/α



.

(27)

Eq.

(27)

isodd underparity-reversal transformations,asitshould be.Infact,followingthediscussionofSection2,parity-reversalis implementedin

(27)

bymeansofthesubstitution

φ

→ −φ

,which changes the sign of

σ

xy. The evaluation of (26) and(27) is per-formedwiththehelpof

(15)

andtwousefulrelationstakenfrom Ref.[11],viz., n0

=

1 2

π

δ



0 a



2 α



ξ

α2_g

_{(ξ )}

_d

_ξ



(28) and

μ

2 a2

=

2

π

|

n

−

n0

| ,

(29)

wheren and n0 are thecarrierdensities definedatchemical

po-tential

μ

andat the charge neutral point (

μ

=

0). We note that although

δ



0 itisnotknownapriori,we wouldneedtoknowits

valueonlyineventual contributionsto

(26)

and

(27)

whichareof theorderof

(δ



0/

μ

)

4,and,therefore,areassumedtobenegligible inthesemiclassicalregime.

Foraproperphysicalinterpretationoftheresults

(26)

and

(27)

, we haveto keepin mindthat theconductivitycorrectionsfound hereforthesemiclassicalregimearedominatedbyeffectsrelated totherandomtopographyofthegraphenesheet.Chargedpuddle effects,on theother hand,are morerelevant insituations where thechemicalpotentialiscloseenoughtothecharge-neutralpoint

[11].Thus, if we completely neglect fluctuations ofthe chemical potential, taking the limit

δ



0

→

0 in (27), using (15), it is not

difficulttoshowthatthetransverseconductivity

(27)

isstill non-vanishingandgivenby

σ

xy

=

8B2₀H2e4c5sin

(

2

φ)

π

L4_h

_¯

3_n5 imp



_μ

a



4/α

.

(30)

We report here results for the particular case of monolayer graphene(therearenofurthertechnicaldifficultiesindealingwith bilayer graphene). We plot, in Fig. 2, the corrections



σ

xx and



σ

xy

≡

σ

xy to the longitudinaland transverse conductivities, re-spectively, induced by the in-plane magnetic field, using surface

Fig. 2. Longitudinal(solidlines)andtransverse(dottedlines)conductivityprofiles, fornimp=2×1011_cm−2 _{andcarrierdensityn=5×10}11_cm−2_{,evaluatedwith} inputdataL andH takenfromRefs.[17](A),[18](B),and[8](CandD).

roughnessdatafromthecurrentliterature.Theanglebetweenthe magnetic and the electric field is taken to be

φ

=

π

/

4, in or-der to maximize the transverse conductivity,as givenin (27).In our numerical estimates,we have madeuse of the phenomeno-logical expression n0

nimp/10, which relates the impurity and

carrier concentrations at the charge neutral point [16]. In all of theprofiles,thelongitudinalconductivityintheabsenceofthe in-plane magneticfield is estimatedas

σ

xx

100e2

/

h (observe that graphene conductivities as larger as 300e2

/

h have already been addressedin the literature [19,20]). Also, it islikely that the in-equalities (11) can be experimentally realized, since in our case study (devised for carrier density n

=

5

×

1011_cm−2_{), we have} k−1

5 nm,20 nm

< 

k

<

30 nm,10 nm

<

L

<

30 nm,and, from anumberofscanningtunnelingexperiments,30 nm

<

Lp

<

50 nm

[21–23].

It is clear that the peculiar non-vanishing transverse conduc-tivity

σ

xy predictedandestimatedhere,the central resultof this paper,can be resolved by thepresent conductivitymeasurement techniques,which are able to record fractions of e2

/

h [6–8].We alsostressthattherangeofmagneticfield intensitiesinvestigated in

Fig. 2

isevensmallerthantheonescurrentlyusedingraphene transportexperiments

[6–8]

.

4. Conclusions

Wehavestudied,within thesemiclassical Boltzmannequation approach,chargetransportinusual graphenesampleswhichhave both charged puddle domains (caused by the interaction with a substrate)androughsurface profiles.Ourparticularinterestis re-latedtosituationswhere anin-plane magneticfield isappliedto thesample,insuchawaythatchargetransporttakesplace, effec-tively,underthepresence ofa randomtransverse magneticfield. The key point inour analysisisto consider regimeswhere scat-teringwavelengths aresmallerthanthesurfaceheightcorrelation length, so that all the magnetic field effects can be brought, as Lorentz force contributions, to the left hand side of the

Boltz-mann transport equation. A straightforward perturbative expan-sion, leads,then, to thecorrected conductivitytensor. The some-what surprising result, which to the authors’ knowledge hasnot yet beenexplored inthe grapheneliterature, isthe predictionof a non-vanishing transverse conductivity, without mean external magneticfield.Thisphenomenon,whichisinprinciplewithinthe reach ofcurrent experimental techniques,is likelyto be relevant in studiesof graphenesurface characterization,once the conduc-tivity tensorturns out to dependon combinationsof thesurface statisticalparameters H andL.

It wouldbe interesting, asa topicforfurther research, to ad-dressthepossibleexistenceofsimilarmagnetic-induced anisotro-pic effectsinfullyquantumregimeswherethein-planemagnetic field is assumedto affect only the right hand side of the Boltz-mann equation. Also, one may wonder how the transition from thesemiclassical tothequantumregimeshouldbe modeledwith thehelp oftransportequations,whichis,no doubt,achallenging theoreticalproblem.

Acknowledgements

WearegreatlyindebtedtoCaioLewenkopfforseveral enlight-eningdiscussions andEduardoMarinoforcallingourattentionto Ref. [13]. This work has been partially supported by CNPq and FAPERJ.

References

[1]K.Novoselov,V.Falko,L.Colombo,P.Gellert,M.Schwab,K.Kim,Nature490 (2012)192.

[2]S.DasSarma,S.Adam,E.H.Hwang,E.Rossi,Rev.Mod.Phys.83(2011)407. [3]C.G.Beneventano,P.Giacconi,E.M.Santangelo,R.Soldati,J.Phys.A40(2007)

F435.

[4]E.R.Mucciolo,C.H.Lewenkopf,J.Phys.Condens.Matter22(2010)A263201. [5]S.Ryu,C.Mudry,A.Furusaki,A.W.W.Ludwig,Phys.Rev.B75(2007)205344. [6]M.B.Lundeberg,J.A.Folk,Phys.Rev.Lett.105(2010)146804.

[7]J.Wakabayashi,T.Sano,J.Phys.Conf.Ser.334(2010)012039. [8]K.Genma,M.Katori,arXiv:1211.2046.

[9]R.Burgos,J.Warnes,L.R.F.Lima,C.Lewenkopf,Phys.Rev.B91(2015)115403. [10]P.Hedegaard,A.Smith,Phys.Rev.B51(1995)10869.

[11]L.Moriconi,D.Niemeyer,Phys.Rev.B84(2011)193401.

[12]S.Adam,E.H.Hwang,V.M.Galitski,S.DasSarma,Proc.Natl.Acad.Sci.USA104 (2007)18392.

[13]F.D.M.Haldane,Phys.Rev.Lett.61(1988)2015. [14]A.R.Wright,Sci.Rep.3(2013)2736.

[15]M.Gibertini,A.Tomadin,F.Guinea,M.I.Katsnelson,M.Polini,Phys.Rev.B85 (2012)201405(R).

[16]Y.Zhang,V.W.Brar,C.Girit,A.Zettl,M.F.Crommie,Nat.Phys.5(2009)722. [17]M.Ishigami,J.H.Chen,W.G.Cullen,M.S.Fuhrer,E.D.Williams,NanoLett.7

(2007)1643.

[18]C.H.Lui,L.Liu,K.F.Mak,G.W.Flynn,T.F.Heinz,Nature462(2009)339. [19]Y.-W.Tan,Y.Zhang,K.Bolotin,Y.Zhao,S.Adam,E.H.Hwang,S.DasSarma,H.L.

Stormer,P.Kim,Phys.Rev.Lett.99(2007)246803. [20]R.Anicic,Z.L.Miskovic,Phys.Rev.B88(2013)205412.

[21]J.Martin,N.Akerman,G.Ulbricht,T.Lohmann,J.H.Smet,K.vonKlitzing,A. Ya-coby,Nat.Phys.4(2008)144.

[22]Y.Zhang,V.W.Brar,C.Girit,A.Zettl,M.F.Crommie,Nat.Phys.5(2009)722. [23]A.Deshpande,W.Bao,Z.Zhao,C.N.Lau,B.J.LeRoy,Phys.Rev.B83(2011)