ebook img

Random gauge field effects on the conductivity of graphene sheets with disordered ripples PDF

4.1 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Random gauge field effects on the conductivity of graphene sheets with disordered ripples

Random gauge field effects on the conductivity of graphene sheets with disordered ripples Rhonald Burgos, Jesus Warnes, Leandro R. F. Lima, and Caio Lewenkopf Instituto de F´ısica, Universidade Federal Fluminense, 24210-346 Niter´oi RJ, Brazil (Dated: January 20, 2015) We study the effect of disordered ripples on the conductivity of monolayer graphene flakes. We calculate the relaxation times and the Boltzmann conductivities associated with two mechanisms. First, we study the conductivity correction due to an external in-plane magnetic field B . Due to (cid:107) theirregularlocalcurvaturefoundatgraphenesheetsdepositedoverasubstrate,B canbemapped (cid:107) 5 intoaneffectiverandommagneticfieldperpendiculartothegraphenesurface. Second,westudythe 1 electronmomentumrelaxationduetointrinsicpseudomagneticfieldsoriginatedfromdeformations 0 andstrain. Wefindthatthecompetitionbetweenthesemechanismsgivesrisetoastronganisotropy 2 in the conductivity tensor. This result provides a new strategy to quantitatively infer the strength of pseudo-magnetic fields in rippled graphene flakes. n a PACSnumbers: 72.80.Vp,73.23.-b,72.10.-d J 9 1 I. INTRODUCTION low-energy electronic properties are nicely described by a nearest-neighbor tight-binding model1, the occurrence ] l The electronic transport properties of bulk graphene of ripples change the tight-binding hopping terms1,17,18. l a are quite remarkable1–4. At room temperature graphene For distortions with length scales much larger than the h has mobilities5 as high as µ ≈ 15,000 cm2V−1s−1, that lattice parameter, characteristic to most samples13–16, - s aresignificantlylargerthananyothersemiconductor. At it has been shown that the tight-binding model can e be mapped into an effective Dirac Hamiltonian with a lowtemperature,however,thetypicalmobilitiesincrease m only up to µ ≈ 200,000 cm2V−1s−1, which is a disap- pseudo-magnetic vector potential1,19–21 that depends on t. pointingfigureascomparedwithhighqualityGaAs het- the lattice distortions. Hence, random ripples give rise a to an intrinsic random gauge potential. The experimen- erostructures. These observations triggered an intense m tal evidences of pseudo-magnetic fields are scarce and theoretical and experimental activity to understand the - disorder mechanisms that limit the mobility in graphene indirect, but quite remarkable. Strain fields have been d invoked to associate the local density of states observed (see,forinstance,Refs.1–4forareview). Thetheoretical n in graphene nanobubbles22 to Landau levels with ener- o understanding, acquired from the analysis of the Boltz- c mann equation1,3, numerical simulations2,6,7, and field gies corresponding to very high magnetic fields. To the [ theoretical techniques8 is that extrinsic disorder such as best of our knowledge, no transport experiment has yet ad-atoms absorbed in the graphene surface, substrate observed manifestations of this physical picture. 1 v chargeinhomogeneities,andintrinsicdisordersuchasva- Randommagneticfieldscanalsobeachievedbyrealiz- 9 cancies play a key role. ingthatdisorderedripplesingrapheneandtheroughness 5 Motivated by recent experiments9–12, we study the ef- atsemiconductorinterfaceheterostructuresshareseveral 5 fectofextrinsicandintrinsicrandomgaugepotentialdis- common features. Starting at the late 80’s, a number of 4 order due to strain. We show that, although unlikely to ingenious methods where used to characterize the inter- 0 be dominant in graphene deposited over standard sub- face roughness in the heterostructures23–25. One idea . 1 strates, these kinds of disorder give unique and sizable is particularly suited for graphene studies: By applying 0 contributions to the conductivity. For high-quality sub- a strong magnetic field B(cid:107), aligned to the plane of a 5 strates,thereareevenexperimentalevidences12thatran- heterostructure interface confining the two-dimensional 1 dom strain can be a good candidate for the leading elec- electron gas (2DEG), the (smooth) interface roughness : v tron relaxation mechanism in on-substrate graphene. disorder gives origin to a local random magnetic field Xi Standardelectronictransportexperimentsusesamples perpendicular to the 2DEG surface. The analysis of the where graphene flakes are deposited over an insulating electronic transport properties as a function of the ap- r a substrate. Insuchsetting,ithasbeenexperimentallyes- pliedmagneticfieldgivesquantitativeinformationabout tablished that the graphene surface is characterized by the interface roughness. disordered static ripples13–16. For SiO substrates, the This setting was nicely explored by a recent 2 latter have typical lengths of λ≈5−30 nm and charac- experiment9, that combined information of the average teristic heights of h ≈0.2−0.5 nm. conductivity and its weak localization correction26,27 in rms Latticedeformationsduetorippleschangethedistance graphenetoextractthesamplecharacteristicλandh . rms between the atoms in the graphene sheet. At the quan- This procedure have been also used in the experimental tum level, site lattice displacements change the orbital study of other graphene systems10,11. Theory28 shows bondingbetweenthecorrespondingatoms,modifyingthe that an applied B on a rough surface gives rise to an (cid:107) electronic structure of the material. In graphene, whose effectivedephasing(cid:96) andtothesuppressionoftheweak φ 2 localizationpeak. Inadditiontothisquantumcorrection, where i = 0 stands for scalar disorder (with σ = I ) 0 2 the random magnetic field due to B also contributes to and i = 1,2 for vector potential disorder, while i = 3 (cid:107) theelectronmomentumrelaxation,whichathighdoping represents a mass term. The focus of our study are (in- is accounted for by the Boltzmann theory9. This nice trinsic and extrinsic) disordered gauge fields, associated analysis does not consider the effect of intrinsic pseudo- with V(1) and V(2). In this paper we do not consider magnetic fields due to strain, discussed above. scalar disorder. Our focus is different. We study the combined effect Let us introduce of intrinsic and extrinsic random magnetic fields in the 1(cid:104) (cid:105) Drudeconductivity. WerevisittheanalysisoftheBoltz- (cid:104)k(cid:48)s(cid:48)|V|ks(cid:105)= 1+ss(cid:48)ei(θ−θ(cid:48)) V(0) + (3) 2 k−k(cid:48) mann equation in graphene1,2,4,29 and calculate the con- tributions of random magnetic fields to the Drude con- seiθ(cid:16)V(1) −iV(2) (cid:17)+ s(cid:48)e−iθ(cid:48)(cid:16)V(1) +iV(2) (cid:17), ductivity. Weshowthattheconductivitycorrectionsdue 2 k−k(cid:48) k−k(cid:48) 2 k−k(cid:48) k−k(cid:48) to an applied in-plane magnetic field B on a rippled (cid:107) where the spinor graphene flake depend on the direction of B and are (cid:107) very anisotropic. We find that this result can be recon- 1 (cid:18) 1 (cid:19) |ks(cid:105)= √ eik·r, (4) ciled with the experimental findings of Ref. 9 by theo- 2A seiθ retically treating the effect of strain and B at the same (cid:107) footing. We also show that the combined effect of both is an eigenstate of HK, θ = tan−1(k /k ), s indicates 0 y x sources of random magnetic fields provides a new exper- particle (s=+1) or hole (s=−1) doping, and imental path to quantitatively probe the effects of strain 1 (cid:90) fields in the low-energy electronic dynamics of graphene. V(i) = drei(k−k(cid:48))·rV(i)(r) (5) The paper is structured as follows: In Section II we k−k(cid:48) A presentthedisordermodelweemploytodescriberipples is the momentum representation of V(i). Since we deal and the corresponding effective Hamiltonian. We briefly withelasticprocesses,weassumeintheremainingofthe explain the origin of an effective random field due to an paper that s=s(cid:48). applied in-plane magnetic field B , as well as from the (cid:107) At low temperatures, scalar disorder (short and long intrinsic strain field due to ripples. In Sec. III we cal- ranged) is the main source of momentum relaxation in culate the contributions to the Drude conductivity due graphene systems2,4. In this paper we use a phenomeno- to B and strain using the Boltzmann transport equa- (cid:107) logical transport time τ to account for effects of scalar tion. We show how to account for the disorder potential s disorder in the conductivity. We assume that τ is much anisotropy and discuss its consequences comparing with s shorter than the characteristic transport times due to experiments. Finally, in Sec. IV we present our conclu- random gauge fields. In Sec. IV, where we compare our sions and an outlook. resultstoexperiments,weshowthatτ indeeddominates s theconductivityingraphene,butsometransportproper- tiesareonlyexplainedbyincludingeffectsduetorandom II. MODEL HAMILTONIAN gauge fields. The ripple disorder model employed in this study is In this section we present a model to study the effect defined as follows: We describe the graphene sheet sur- of extrinsic and intrinsic sources of a random magnetic face by z = h(r), where h is the surface displacement fieldsinthedynamicsofelectronsincorrugatedgraphene with respect to the reference plane z = 0 at the posi- monolayer samples. tion r = (x,y). The average of h is set to zero. In Closetothechargeneutralitypoint,theelectronicdis- line with the experiments on graphene deposited over persionrelationofpristinegraphenemonolayersislinear a substrate9,13–16, we further assume that the typical and has two degenerate components, with corresponding heights h are much smaller that the ripple lengths λ. K and K(cid:48) valley indices1. In the presence of a magnetic rms We model the ripple fluctuations in h(r) by the corre- field,theeffectiveelectronicHamiltonianfortheK-valley lation function reads (cid:18)|r−r(cid:48)|(cid:19) HK =vFσ·[p+eA(r)]=H0K +V(r), (1) (cid:104)h(r)h(r(cid:48))(cid:105)=h2rmsF λ , (6) where the vector potential A(r) has been included in HK by minimal coupling. Here v ≈106m/s, σ are the where (cid:104)···(cid:105) denotes an average over disorder. Al- F Pauli matrices acting on the sublattice space, and p is thoughtheorypredictsapowerlawheight-heightcorrela- the electron momentum operator. The Hamiltonian for tionfunctionforfree-standingmembranes30,experiments the K(cid:48) valley has a similar structure1. supportsingle-parametercorrelationsforthe(static)rip- In this description, a generic long-ranged disorder po- ples of graphene deposited over a substrate. For latter tential V(r) is represented in both K and K(cid:48) valleys by convenience, let us define V(r)=(cid:88)σiV(i)(r), (2) h(q)= 1 (cid:90) dreiq·rh(r), (7) A i 3 where A is the sample size. In reciprocal space and is expressed, in a convenient gauge for B = B xˆ, (cid:107) (cid:107) by the vector potential (cid:104)h(q)h(q(cid:48))(cid:105)=h2 F(q)δ , (8) rms q,−q(cid:48) A (r)=0 and A (r)=−B h(r). (13) where F(q) is the Fourier transform of the correlation x y (cid:107) function F(|r−r(cid:48)|). Figure 2(a) illustrates a typical disorder realization We address two mechanisms that generate random of h(r) with fluctuations characterized by the Gaussian magnetic fields. First, we study the case of an external correlation function F(x) = exp(−x2/2λ2). The corre- strongmagneticfieldB appliedparalleltothegraphene (cid:107) spondingmagneticfieldB (r), normaltothegraphene ext sheet. We show that, due to the ripples, B gives rise to (cid:107) sheet, is shown in Fig. 2(b). While h(r) displays an a random effective magnetic field B (r) perpendicular ext isotropic disorder, B (r) is clearly anisotropic. The ext to the graphene surface. Next, we discuss the intrinsic anisotropy direction of B (r) depends on the orienta- ext pseudo-magnetic field B originated by the strain field int tion of B . corresponding to the graphene sheet profile height h(r). (cid:107) Theanisotropyisquantifiedbyinspectingtheautocor- relation function A. Random magnetic field due to ripples and an (cid:28)∂h(r)∂h(r(cid:48))(cid:29) (cid:104)B (r)B (r(cid:48))(cid:105)=B2 , (14) in-plane external B-field ext ext (cid:107) ∂x ∂x(cid:48) Let us first consider the setup of a magnetic field ap- thatcanbeexpressedintermsofF bydirectdifferentia- plied parallel to the sample z = 0, that has been exper- tion. Alternatively, going to reciprocal space, one writes inmoteanttiaolnlyalincvoensvteignaietnedcei,ninawvahraitetfyololofwsysswteemfisx9,2t3h–e25d.irFeocr- (cid:28)∂h(r)∂h(r(cid:48))(cid:29)=h2 (cid:88)q2F(q)e−iq·(r−r(cid:48)) ∂x ∂x(cid:48) rms x tion of B along the x-axis, namely, B =B xˆ. (cid:107) (cid:107) (cid:107) q d2 z =−h2 F(ρ), (15) rmsdx2 B ext with ρ=r−r(cid:48). h(x,y) Let us calculate (cid:104)B (r)B (r(cid:48))(cid:105) for the case where B|| ext ext nˆ h(r) is characterized by a Gaussian correlation function, the same as in Fig. 2(a). The corresponding B (r) au- ext x tocorrelation function reads λ (cid:104)Bext(r)Bext(r(cid:48))(cid:105)=B(cid:107)2hλ2rm2s (cid:20)1−(cid:16)λρ(cid:17)2cos2α(cid:21)e−2ρλ22 (16) FIG. 1. Sketch of h(r) along the x direction. The ripple amplitudes δh are enhanced and made comparable with λ to where α is the angle between B(cid:107) (or the x-axis) and ρ. help the illustration. For convenience, we take B(cid:107) =B(cid:107)xˆ. Figure 3(a) shows the Bext(r) autocorrelation func- tion obtained by averaging over 105 ripple disorder real- As illustrated in Fig. 1, the parallel magnetic field B izations of h(r), as defined by Eq. (6) with a Gaussian (cid:107) has a component perpendicular to the surface z = h(r) correlationfunctionF (formoredetailssee, forinstance, that is given by Ref. 6). As expected, it coincides with Eq. (16) and ex- presses the anisotropy captured by a visual inspection of B (r)=−B ·nˆ(r). (9) ext (cid:107) Fig. 2(b). At the point r = (x ,y ), the surface z = h(r) has a 0 0 0 unit normal vector B. Pseudomagnetic field due to strain  (cid:12) ∂h/∂x (cid:12) 1 (cid:12) nˆ(r0)= (cid:114)  ∂h/∂y (cid:12) . (10) The out of plane deformations of a rippled membrane 1+(cid:0)∂∂hx(cid:1)2+(cid:16)∂∂hy(cid:17)2 −1 (cid:12)(cid:12)r=r0 describedbyh(r)canbeassociatedwiththestraintensor u (r) given by19,31,32 ij We assume that the typical displacement magnitude is characterized by δh. For δh(cid:28)λ, we write 1(cid:18)∂h(cid:19)2 1(cid:18)∂h(cid:19)2 u ≈ , u ≈ , and (cid:12) xx 2 ∂x yy 2 ∂y nˆ(r )≈(∂h/∂x,∂h/∂y,−1)(cid:12) . (11) 0 (cid:12)r=r0 u ≈ 1(cid:18)∂h∂h(cid:19). (17) Hence, the effective local perpendicular magnetic field xy 2 ∂x∂y reads For simplicity we have neglected the effect of in-plane B (r)=−B ·∇h(r), (12) deformations. ext (cid:107) 4 1 0 -0.10 -0.07 -0.05 -0.03 5 -0.01 0.01 0.03 0.06 0.08 l/0 0.10 y 0.12 h (r)/h rm s - 5 (a ) - 1 0 - 1 0 - 5 0 5 1 0 l x / 1 0 -0.09 -0.07 -0.05 -0.03 5 -0.00 0.02 0.04 0.06 0.08 l/0 0.10 y 0.12 -B (r)/B ext // - 5 (b ) - 1 0 - 1 0 - 5 0 5 1 0 l x / 1 0 -3.50 FIG. 3. (color online) Extrinsic and intrinsic magnetic field -2.80 correlations functions: (a) C (r−r(cid:48)) = (cid:104)B (r)B (r(cid:48))(cid:105) -2.10 ext ext ext -1.39 for an external B applied along the x-direction and (b) 5 -00.0.619 Cint(r − r(cid:48)) = (cid:104)B(cid:107)int(r)Bint(r(cid:48))(cid:105) due to strain in units of 0.71 h4 /λ6((cid:126)β/ea)2, both corresponding to a ripple disordered 1.41 rms 2.12 surfaceh(r)characterizedbyaGaussiancorrelationfunction. l/0 2.82 y B (r)/((cid:1)3b.5/2e a ) int - 5 given by (c) (cid:126)βκ A (r)= [u (r)−u (r)], - 1 0 x ea xx yy - 1 0 - 5 0 5 1 0 l (cid:126)βκ x / A (r)=−2 u (r), (19) y ea xy FIG.2. (coloronline)Typicaldisorderrealizationof(a)h(r), where a = 1.42 ˚A is the bond length between nearest characterized by a Gaussian correlation function F, the cor- neighboring carbon atoms, and e is the electron charge. responding (b) B (r) for an external B applied along the ext (cid:107) Hereg ≈4eV,κ≈1/3,andβ =−∂logt/∂loga≈2are xdirection,definedinEq.(12),and(c)B (r)duetolattice int dimensionless material dependent parameters19,31 that deformations, given by Eq. (19). characterizethecouplingbetweentheDiracelectronsand the lattice deformations, where t ≈ 3 eV is the hopping integral between nearest-neighbor π-orbitals. The effect of strain in the low-energy electronic struc- In summary, for any given h(r) one can readily cal- ture of graphene can be accounted for by introduc- culate the pseudo-magnetic field B = ∇×A. Since ing a scalar and a vector gauge potential in the Dirac int equation20,21,30. The scalar term reads Az =0andneitherAxnorAy dependonz,Bint =Bintzˆ. Figure 2(c) shows the B corresponding to the random int V(0)(r)=g[u (r)+u (r)], (18) rippled surface h(r) of Fig. 2(a). Notice that the typical xx yy correlationlengthofB (r)ismuchshorterthanthatof int while, for the K-valley and for an armchair crystallo- h(r). graphic orientation along the x-axis, A = (A ,A ) is Let us calculate (cid:104)B (r)B (r(cid:48))(cid:105) for a random Gaus- x y int int 5 sian correlated h(r), corresponding to Eq. (6) with with F(x) = e−x2/2λ2. To this end, we calculate the Fourier 1(cid:88) transform of the intrinsic pseudo magnetic field, namely u (q)=− q(cid:48)(q −q(cid:48))h(q(cid:48))h(q−q(cid:48)), (21) ij 2 i j j q(cid:48) where i and j label the Cartesian coordinates. We use Eqs. (20) and (21) to write the correlation function of B in momentum space. The evaluation of int (cid:104)B (q)B (−q)(cid:105) amounts to compute the correspond- int int ing (cid:104)u (q)u (−q)(cid:105), that result in four-h correlation ij i(cid:48)j(cid:48) functions. This can be done exactly for Gaussian fluctu- (cid:126)βκ ationsandprovidesagoodqualitativeestimateforother B (q)=i [q u (q)+2q u (q)−q u (q)], (20) cases30. int ea y xx x xy y yy We obtain h4 π (cid:18)(cid:126)βκ(cid:19)2 (cid:34) (cid:35) (cid:104)B (q)B (−q)(cid:105)= rms q2 16+λ4q4sin23θ e−λ2q2/4, (22) int int 32λ2A ea where θ is the angle between q and the x-direction. By Fourier transforming back to coordinate space, we arrive at h4 (cid:18)(cid:126)βκ(cid:19)2(cid:34) ρ2 ρ4 ρ6 (cid:35) C (r−r(cid:48))≡(cid:104)B (r)B (r(cid:48))(cid:105)= rms 8−20 +9 −2 sin23α e−ρ2/λ2, (23) int int int λ6 ea λ2 λ4 λ6 whereαistheanglebetweenρ=r−r(cid:48) andthex-axis.33 tance fluctuations34. In such situations, the Boltzmann The correlation function C (r−r(cid:48)) has 6 symmetry approach is very successful in assessing the conductivity, int axes, reflecting the underlying graphene honeycomb lat- as shown by direct comparison with numerical simula- tice symmetry19. In other words, information about the tionsusinganatomisticbasis6,35. Asoneapproachesthe graphene crystal structure survives disorder averaging. charge neutrality point, and k (cid:96) (cid:46) 1, the semiclassical F Figure3(b)showsC (r−r(cid:48))obtainedfrom105 numer- methodisnolongersuitedandonehastoresorttomore int ical realizations of Gaussian correlated disorder for h(r). sophisticated approaches2,8. Thenumericalsimulationsserveasahelpfultesttocheck We now discuss how to add gauge field disorder in the our analytical results. As in the previous subsection, we Boltzmann approach. For long ranged disorder, some verify an excellent agreement within the statistical pre- authors36–38 arguethatitcanbeadvantageoustoinclude cision. the disorder potential in the classical Liouvillian evolu- tion, that is, to treat V(r) in the left-hand side of the Boltzmann equation. This approach is justified in the III. DRUDE-BOLTZMANN CONDUCTIVITY “classical” regime, where kFλ (cid:29) 1, that is, where the random fields with a characteristic length λ vary slowly in the scale of k−1. In graphene samples with ripple In this Section we use the effective Dirac Hamiltonian F sizes λ of the order of few to ten nanometers13–16, the ofEq.(1)tocalculatethetransporttimeandtheDrude- latter inequality holds for a carrier density |n| (cid:29) 1012 Boltzmann conductivity of graphene monolayers in the cm−2, that is much larger than the doping studied in presence of random gauge fields. most experiments1. Highmobilitygraphenesampleshavetypicalelectronic mean free paths of (cid:96) (cid:38) 50 nm. Recalling1 that the In this study, we calculate transport times for both carrier density is related to the Fermi wave number by short and long ranged disorder by evaluating the corre- k = (cid:112)π|n|, one readily obtains that k (cid:96) (cid:29) 1 al- sponding Boltzmann collision integral (at the right-hand F F ready for a doping where |n| ≈ 1011 cm−2. This indi- side of the equation). As mentioned in the introduction, cates that even for modest carrier densities a semiclas- for graphene on standard substrates (the case of interest sical transport description is justified. For |n| (cid:38) 1011 here), the random gauge field contribution to the con- cm−2 the typical graphene conductivity in good sam- ductivity is not the dominant one. Hence, the electron ples is much larger than e2/h, the order of magnitude meanfreepathduetoripplesislargerthan(cid:96)andthear- of quantum contributions to the electronic transport, guments justifying the semiclassical approximation hold. such as weak localization2,26,27 and universal conduc- TheBoltzmannequationforgrapheneunderauniform 6 electric field E reads1,39 where θ(cid:48) is the angle between k(cid:48) and the x-axis, and −eE· ∂εk,s∂f0 = (cid:88)(g −g )W (24) A (cid:90) ∞ ∂p ∂ε k,s k(cid:48),s(cid:48) k(cid:48),s←k,s W(θ,θ(cid:48))= dk(cid:48)k(cid:48)W k(cid:48),s(cid:48) (2π)2 k(cid:48),s←k,s 0 where εk,s = svF(cid:126)|k|, f0 is the Fermi distribution func- = A|εF| (cid:10)|(cid:104)k(cid:48),s|V|k,s(cid:105)|2(cid:11), (31) tion, gk = fk −f0 is deviation from equilibrium due to 2πvF2(cid:126)3 the electric field, and W is the transition rate k(cid:48),s(cid:48)←k,s from state (k,s) to (k(cid:48),s), that we calculate using Fermi where, due to the zero-temperature limit, k =k(cid:48) =k . F golden rule, namely The matrix element (cid:104)k(cid:48),s|V|k,s(cid:105) depends on q =k− k(cid:48) and ϕ = π/2+(θ+θ(cid:48))/2, the angle between q and Wk(cid:48),s(cid:48)←k,s = 2(cid:126)π (cid:10)|(cid:104)k(cid:48)s(cid:48)|V|ks(cid:105)|2(cid:11)δ(εk,s−εk(cid:48),s(cid:48)), (25) the x-axis. Hence, W(θ,θ(cid:48)) is better cast as W(qζ,ϕ). We use the standard notation ζ = |θ − θ(cid:48)| and q = ζ where V is a generic long-ranged disorder potential 2k sin(ζ/2). F parametrized by Eq. (2). The δ-function reflects the fact By expressing τ and τ in terms of a Fourier series, we are dealing with elastic processes and, hence, s = s(cid:48). x y one transforms Eqs. (29) and (30) into an (infinite) set In our model, the transition rates do not depend on s. of algebraic equations. Using the x-axis symmetry and Accordingly we drop this index whenever its omission that W(θ,θ(cid:48))=W(θ(cid:48),θ)43, one shows that42 does not introduce an ambiguity. The scattering processes we address are anisotropic. ∞ (cid:88) τ (θ)= τ(n)cos[(2n−1)θ] (32) Thecalculationofthetransportpropertiesinthiscaseis x x slightly different40,41 than that of the standard isotropic n=1 case39. In this study, we adapt the nice method devel- (cid:88)∞ τ (θ)= τ(n)sin[(2n−1)θ]. (33) oped by Tokura42 – that is briefly described in what fol- y y lows – to calculate the transport times of massless Dirac n=1 electrons in graphene. By inserting the above relations in Eq. (28), one con- In both situation of interest, the scattering potential cludes that the conductivity tensor is diagonal, with correlation functions have at least one symmetry axis. For convenience, we choose the x-axis along a symmetry e2|ε | e2|ε | σ = F τ(1) and σ = F τ(1), (34) axis and define42 xx (cid:126)2π x yy (cid:126)2π y (cid:18) (cid:19) ∂f g = − 0 ev τ(θ)·E, (26) that supports the interpretation of τ(1) as a transport k ∂ε k k time vector. whereτ(θ)istherelaxationtimevectortobesolved. We The symmetry W(θ,θ(cid:48)) = W(θ(cid:48),θ) implies that recall that θ is the angle between k and the x-axis. Note W(q ,ϕ)=W(q ,ϕ+π)=W(q ,ϕ−π). In turn ζ ζ ζ that τ depends explicitly on θ and implicitly on |k|. The current density (spin and valley degeneracies in- (cid:88)∞ W(q ,ϕ)= W (q )cos(2nϕ), (35) cluded) is given by ζ n ζ n=0 4 (cid:88) j = ev g A k k with an obvious inversion relation. k ByusingtheFourierexpansionsforτ(θ)andW(q ,ϕ), ζ = e2 (cid:90) ∞dkk(cid:18)−∂f0(cid:19)(cid:90) 2πdθv v [τ(θ)·E], (27) Eqs. (29) and (30) can be cast in matrix form42 π2 ∂ε k k 0 k 0 ∞ ∞ (cid:88) (cid:88) from which one obtains the conductivity tensor δ = M− τ(n) and δ = M+ τ(n), (36) l,1 l,n x l,1 l,n y e2|ε |(cid:90) 2π (cid:18)τ (θ)cosθ τ (θ)cosθ (cid:19) n=1 n=1 σ = F dθ x y , (28) (cid:126)2π2 0 τx(θ)sinθ τy(θ)sinθ where the matrix elements of M± are42 where ε is the Fermi energy, measured with respect to the charFge neutrality point energy. For the sake of sim- M± = (−1)l−n(cid:2)(1+δ )J ±J (cid:3), l,n 2 l,n |l−n|,n+l−1 n+l−1,|l−n| plicity, in Eq. (28) we have taken the zero-temperature (37) limit, namely, −∂f /∂ε=δ(ε−ε ). 0 F with Bysubstitutingtheansatz(26)intheBoltzmannequa- tion, Eq. (24), one obtains an integral equation for τ(θ), (cid:90) 2π J = dζW (q )[cos(nζ)−cos(mζ)]. (38) namely n,m n ζ 0 (cid:90) 2π cosθ = dθ(cid:48)[τx(θ)−τx(θ(cid:48))]W(θ,θ(cid:48)) (29) Finally, by inverting M± in Eq. (36), one writes the vec- 0 tor transport time components as (cid:90) 2π sinθ = dθ(cid:48)[τ (θ)−τ (θ(cid:48))]W(θ,θ(cid:48)), (30) y y τ(1) =[(M−)−1] and τ(1) =[(M+)−1] . (39) 0 x 11 y 11 7 Note that for isotropic scattering, all W with n > Inpractice,wecomputeΓ byassumingthatΓ =0and n 3 m 0 are zero and M± is diagonal, with elements K = subsequentiterationof (48). Thechoiceofmdetermines l,l J . Hence, the vector transport time components the precision of the calculation: The larger m, the more 0,2l−1 coincide, τx(1) =τy(1) =τ(1), and read accurate is Γ3. AssumingthatΓ (cid:28)J leadstoaninterestingresult, 3 0,1 1 (cid:90) 2π that is =J = dζW (q )(1−cosζ), (40) τ(1) 0,1 0 0 ζ 1 1 (eB(cid:107))2|εF|(cid:90) 2π = = dζ(1−cosζ)C (q). (49) which is the standard expression for the transport time τy(1) 3τx(1) 8π(cid:126)3 0 h in isotropic systems. In this limit τ(1)/τ(1) =3. In other words, for Γ (cid:28)J y x 3 0,1 the corrections to the conductivity due to B lead to (cid:107) A. Effect of an in-plane magnetic field ∆σyy = 3∆σxx, regardless of the dependence of correla- tion function C (q) on q. h Let us now calculate the effect of an external parallel For Γ3 = 0 and Ch(q) = 2πλ2h2rmse−λ2q2/2, we write magnetic field on the conductivity. From Eq. (13) we τ(1) in closed analytical form, namely x,y write the effective disorder potential for the K-valley as 1 1 (eB )2|ε | = = (cid:107) F (λh )2 Vext(r)=vFeσyAy(r)=−vFeB(cid:107)h(r)σy. (41) τy(1) 3τx(1) 4(cid:126)3 rms Werecallthath(r)variesslowlyinthescaleofthelattice ×e−λ2kF2(cid:2)I0(λ2kF2)−I1(λ2kF2)(cid:3), (50) spacing and, hence, V (r) is long-ranged and does not ext whereI andI aremodifiedBesselfunctionsofthefirst mix valleys. 0 1 (cid:112) kind. We use1 k = π|n| to express the conductivity The momentum relaxation rate W reads F k(cid:48)←k in terms of the charge carrier density n. We conclude 2πe2v (cid:18)θ+θ(cid:48)(cid:19)C (q) thatthecorrectiontotheconductivityduetoanin-plane W =δ(k−k(cid:48)) FB2sin2 h . (42) k(cid:48)←k (cid:126)2 (cid:107) 2 A magnetic field depends quadratically on hrmsB(cid:107) and has a non-trivial dependence on λ2|n|. where In the high doping limit of λ|n|1/2 (cid:29)1, Eq. (50) gives (cid:90) the resistivity contribution of an in-plane magnetic field C (q)= dreiq·r(cid:10)h(0)h(r)(cid:11) (43) h in a rippled graphene sheet 1 h2 is the form factor of the height-height correlation func- ∆ρ =3∆ρ ≈ √ rmsB2|n|−3/2 (51) tion. Here (cid:104)···(cid:105) indicates disorder average. yy xx 2 2(cid:126) λ (cid:107) From Eq. (31) we obtain in agreement with Ref. 9. Figure 4 shows the resistivity ∆ρ versus the car- (eB )2|ε | yy W(q,ϕ)= (cid:107) F C (q)(1+cos2ϕ), (44) rier density n (due to particle hole symmetry, we only 4π(cid:126)3 h show n > 0) for m = 3 and m → ∞. The optimal m that has only 2 non-zero Fourier components, namely, value to obtain convergence depends on λ2n. The inset shows ∆ρ for λ2n values outside the validity range of yy (eB(cid:107))2|εF| the asymptotic expansion. As discussed in the next sec- W (q)= C (q), for n=0,1 (45) n 4π(cid:126)3 h tion, the λ2n range displayed in the inset corresponds to the typical experimental situation. We find that the while W (q)=0 for n≥2. n |n|−3/2 scaling predicted by the asymptotic expansion In this case, the M± matrix is tridiagonal and reads42 (51) is only a rough approximation. (1∓ 12)J0,1 −12J1,2 0 ··· In general, Γ3 is a non-vanishing correction to the M± = −120...J1,2 −J120...J,31,4 −J210...J,51,4 ··..··.··. (46) teτryv(a1e)nr/,sτpxf(oo1rr)tiGstiaamufesusnciaocntmioprniopnopenlnelytsh,oefihgλehnktFcec.oτry(r1e)l/aτtxi(o1n)s(cid:54)=, t3h.eHraotwio- Figure 5 shows τ(1)/τ(1) versus λ2n. It illustrates the y x The inverse transport time components are given by importance of Γ in the calculation of the conductivity 3 corrections. We find that by increasing the carrier con- 1 3 1 1 = J −Γ and = J −Γ , (47) centration the anisotropic conductivity is considerably τx(1) 2 0,1 3 τy(1) 2 0,1 3 favored. ThetransportpropertiespredictedbyEq.(50)areonly where Γ is isotropic and determined by the continued 3 slightly modified for the case of exponential ripple heigh fraction relation correlations, (cid:104)h(0)h(r)(cid:105) = h2 e−r/λ: For λ|n|1/2 (cid:29) 1, rms Γ = (J1,m−1)2 . (48) theresistivitytensorgivenbyEq.(51)ismultiplied44 by m 4(J −Γ ) a prefactor of order of unity times log(λ2|n|). 0,m m+2 8 1.0 0.10 due to strain that has been reported in the literature30 Γ =0 0.8 Γ3∞=0 |n|-3/2 is AnostisnuffiSeccie.nItIfBor, twheecaanlacluylsaitsew(cid:104)eupijr(orp)ousi(cid:48)ej.(cid:48)(r(cid:48))(cid:105) by as- ρ00.01 suming Gaussian correlated ripple height fluctuations, /ρy0 0.6 /∆ρyy |n|-2 Csthra(qig)ht=forw2πaλrd2ha2rlmgseeb−raλ,2qw2/e2.obtAaifnter some lengthy but y 0.00 ρ 0.4 ∆ v2(cid:126)2β2κ2πh4 πλ2n 10 (cid:104)|(cid:104)k(cid:48)s|V |ks(cid:105)|2(cid:105)= F rms (53) 0.2 int 32a2 λ2A (cid:26) (cid:20) (cid:21)(cid:27) 3 × 16+λ4q4cos2 (θ+θ(cid:48)) e−λ2q2/4. 0.0 2 0 1 2 3 4 5 2 πλ n Using Eq. (31) we arrive at (FπIλGh.rm4.sB(cid:107)()c2o/lo2r(cid:126) odnuleinteo)BR(cid:107)esaisstaiviftuync∆tiρoynyofinλ2unn.itIsnsoeft:ρ0Th=e W(q,ϕ)=Wint(cid:20)16+ λ42q4(1+cos6ϕ)(cid:21)e−λ2q2/4, (54) same as in the main plot in log-log scale to illustrate the de- pendence of ∆ρ on |n|. xx where 8.0 β2κ2|ε |h4 W = F rms. (55) Γ =0 int 64a2(cid:126) λ2 7.0 3 Γ =0 (Thenotationisthesameasthatoftheprevioussection.) 6.0 ∞ 1) The only non-zero Fourier components of W(q,ϕ) are ( x τ/ 5.0 1) (cid:18) λ4q4(cid:19) ( y W (q)=W 16+ e−λ2q2/4 and τ 4.0 0 int 2 3.0 W (q)=W λ4q4e−λ2q2/4. (56) 3 int 2 2.0 -5.0 -2.5 0.0 2.5 5.0 In this case, the M± matrix reads42 2 πλ n  J 0 ∓1J −1J ··· aFFcoIcGru.πra5λc.2yn(cfo<orlom5r.0o=,nlt1ihn3ee.)liRmaittiomτy(→1)/∞τx(1)isaastatafiunnedctiwonithoifnπ1λ02−n4. M± =∓−112200...JJ,133,,24 J0,3∓00...12J3,0 J2000...,53,2 J2000...,73,4 ···..···.···, B. Effect of strain fields (57) and the inverse transport time components are given by Let us now consider the effect of the pseudo-magnetic fieldsduetostrain,Bint,intheconductivityofmonolayer 1 =J − (J3,2)2 − (J3,4)2 +··· . (58) graphenesheets. Incontrasttothemechanismdiscussed τ(1) 0,1 4J0,5 4J0,7 above,B (r)issolelydeterminedbyh(r)andthemate- x/y int rialproperties. Hence,itintrinsictoanygraphenesample Sinceτ(1) =τ(1), theconductivitycorrectionsduetothe with disordered ripples. x y strain field are isotropic. This result seems to be at odds We use Eq. (19) to calculate the vector transport time with the fact that the pseudo-magnetic field autocorre- τ fortheintrinsiceffectivevectorpotentialduetostrain. lation function (cid:104)B (r)B (r(cid:48))(cid:105) clearly shows an hexag- For the K-valley V reads int int int onal symmetry, as illustrated by Fig. 3b. As shown by βκ(cid:110) (cid:111) Tokura42, using general arguments, this is a false para- V (r)=(cid:126)v [u (r)−u (r)]σ −2u (r)σ . int F a xx yy x xy y dox: The conductivity tensor becomes anisotropic only (52) forscatteringprocessescharacterizedbyasinglesymme- In contrast with the previous subsection, here it is dif- try axis, like in the B case, analyzed in the previous (cid:107) ficult to make quantitative progress without assuming a subsection. specific form for the ripple height correlation function. Assuming that J vastly dominates the sum in 0,1 The qualitative behavior of the conductivity corrections Eq. (58), we obtain 9 1 (cid:104) (cid:105) τ(1) =Wintπe−λ2kF2/2 (32+8λ2kF2 +16λ4kF4)I0(λ2kF2/2)−(64+24λ2kF2 +16λ4kF4)I1(λ2kF2/2) , (59) x/y whereI andI aremodifiedBesselfunctionsofthefirst temtransportpropertiesaredominatedbyotherscatter- 0 1 kind. ing processes, with a corresponding (isotropic) transport In the limit of λ|n|−1 (cid:29)1, we obtain the correction to time τ . s the resistivity ItiscustomarytouseMatthiessen’srulewhendealing with systems characterized by different competing relax- h 23β2κ2 h4 ∆ρ =∆ρ ≈ rmsλ−3|n|−3/2. (60) ation time mechanisms. In our case, Matthiessen’s rule xx yy e2 32π λ2a2 translates into adding the inverse transport times given by Eqs. (47) and (58), namely The above asymptotic leading order expansion for ∆ρ helps us to develop some insight on the relevant param- eters. However, since the situation of kFλ(cid:29)1 is hardly 1 = 2±1Jext−Γext+Jint−(J3in,2t)2 −(J3in,4t)2 +··· . met in the current experimental situations of interest, it τ(1) 2 0,1 3 0,1 4J0in,5t 4J0in,7t x/y isnecessarytonumericallycalculatetheinversetransport (61) times, as we do in the next subsection. As expected the This naive approach was shown to be inaccurate when straincorrectionstotheconductivitydependonmaterial dealing with anisotropic potentials42. parameters,andareanon-trivialfunctionofλ,h ,and rms Weanalyzethecombinedeffectofintrinsicandextrin- |n|. Since these corrections are small compared to other sic random magnetic fields by considering an effective disorder effects, they are difficult to be noticed in stan- M-matrix given by dard transport experiments. This situation changes if weconsiderthecombinedeffectofintrinsicandextrinsic M± =M± +M±, (62) random magnetic fields, as we discuss in the next. tot ext int where M± and M± are given by Eqs. (46) and (57) ext int respectively. C. Combined effect of extrinsic and intrinsic random magnetic fields The τ components are Weconcludethissectionbyanalyzingthecombinedef- τx(1) =[(Mt−ot)−1]11 and τy(1) =[(Mt+ot)−1]11, (63) fectofbothpreviouslydiscussedsourcesofrandommag- netic field disorder. As before, we assume that the sys- that, for m=5, explicitly read (cid:20) 1 1 1 1 =(1± )Jext+Jint+ − (Jext+Jint)(Jext)2 τ(1) 2 0,1 0,1 (J0e,x3t+J0in,3t∓J3in,0t/2)(J0e,x5t+J0in,5t)−(J1e,x4t)2/4 4 0,5 0,5 1,2 x/y (cid:21) 1 1 1 ∓ JintJextJext− (Jint)2(Jext+Jint∓ Jint) +··· . (64) 4 3,2 1,2 1,4 4 3,2 0,3 0,3 2 3,0 This result is clearly different from Eq. (61), since it M-matrix calculation (indicated as “exact”) shows an mixes intrinsic and extrinsic effects. Let us now dis- overall higher resistivity than that obtained from the cuss the dependence of τ on B ,λ,h , and n. For that Matthiessen rule. It depends linearly on B2 for large (cid:107) rms (cid:107) purpose we numerically invert the matrix M± , at order B and deviates from this dependence only when B tot (cid:107) (cid:107) m ≈ 30−50 to guarantees an accuracy of 10−5 for the becomes small. analyzed parameter range. In contrast, the dependence of ∆ρ with h , λ, and rms The resistivity corrections obtained from the |n| is not trivial. For hrms and λ values taken close to Matthiessen’s rule, Eq. (61) depend quadratically theonesreportedbytopographyexperiments13–16, anu- on B , in line with the experiment9. The full M-matrix merical study using the full M-matrix approach gives (cid:107) analysis does not guarantee this simple dependence. ∆ρyy ∝ λ−α with α ≈ 3···4, ∆ρyy ∝ hβrms with β ≈ 3, In Fig. 6 we plot the resistivity correction ∆ρyy cal- and ∆ρyy ∝ |n|−γ with γ ≈ 2. In summary, ∆ρ is very culated using the full M-matrix and compare it with sensitive on small variations of hrms and λ. the one obtained from the Matthiessen rule, given by In Fig. 7 we compare Eqs. (47), (61), and (64) to gain Eq. (61), for realistic values of h , λ, and n. The full insight on how the strain mechanism affects the ratio rms 10 Ref. 9 reports ∆ρ(70o)/∆ρ(20o) ≈ 0.13−0.26. Using λ 4· 10-4 and hrms values obtained from AFM measurements, we Matthiessen rule obtainτ(1)/τ(1) ≈10forn=1012 cm−2. Thisratioleads y x exact to ∆ρ(70o)/∆ρ(20o) ≈ 0.2 in good agreement with the 3· 10-4 l =30nm, h =0.2nm experiment9. rms 0 r/yy 2· 10-4 Dr 1· 10-4 r/0yy00..4402 l=10nm, hrms=0.4nm IV. CONCLUSIONS AND OUTLOOK Dr 0.38 Inthispaperwestudiedtheeffectofrandommagnetic 0 20 40B260 80 100 0 || fields on the transport properties of a rippled graphene 0 20 40 60 80 100 flake. We used the Boltzmann equation, adapted to the 2 2 B (T ) || case of anisotropic disorder42, to address the case of an external magnetic field applied in-plane, the effect of in- FIG. 6. (color online) Resistivity correction ∆ρyy in units of trinsic strain fields caused by the graphene corrugation, ρ0 =(πλhrmsB(cid:107))2/2(cid:126) as a function of the in-plane magnetic as well as the combination of both. field B for h = 0.2 nm, λ = 30 nm, and n = 1012cm−2. (cid:107) rms We find that an external in-plane magnetic field B Inset: The same as in the main figure for h =0.4 nm and (cid:107) rms gives rise to very anisotropic conductivity corrections. λ=10 nm. By neglecting the effect of strain fields and using a parametrization of the ripple disorder that is consistent τ(1)/τ(1). We find that the strain fields contribute to with experiments, we find conductivity corrections that y x a strong suppression of the anisotropy in the transport scale with B(cid:107)2 and |n|−2, consistent with Ref. 9. In con- timeduetoastrongB(cid:107). However,forrealisticparameter trast, we obtain τy(1)/τx(1) ratios as large as 20···30. values the anisotropy is still very large and of the order In the absence of an external magnetic field, random of τy(1)/τx(1) ≈10 for |n|≈1012 cm−2. gaugefieldsduetorippledisordergiveaisotropiccontri- butiontotheelectronmomentumrelaxationingraphene, 50 inlinewiththeorderofmagnitudeestimatepresentedin m = 3 (no strain) Ref. 30. We find, however, that ripple disorder cannot 40 m = ∞ (no strain) be neglected in the analysis of the conductivity in the Matthiessen (1) x 30 exact panreissoetnrcoepoicfanalatrugreeBof(cid:107).thWeeparolsbolecmon,cMluadteththieasts,ednu’sertuolethies τ / not accurate to address both intrinsic and extrinsic ran- (1) y 20 domfieldsatthesamefooting. Forthatpurposewehave τ to invert the total M-matrix. 10 This approach allows us to successfully describe the corrections to the Drude conductivity reported in the 0-5.0 -2.5 0.0 2.5 5.0 experiment9 using typical λ and hrms parameters taken 12 -2 from the AMF literature. In addition, we also obtain a n (10 cm ) suppression of the resistivity anisotropy (with respect to the case where strain is neglected) that consistent with FIG. 7. (color online) Anisotropy τ(1)/τ(1) as a function y x Ref.9. Webelievethattheanisotropicnatureoftheran- of the carrier concentration n using different approximation dommagneticfielddisorderalsosignificantlychangesthe schemes, for h =0.2 nm, λ=30 nm, and B =8 T. The rms (cid:107) quantum corrections to the conductivity with respect to red and green lines stand for the contribution of B with- (cid:107) the isotropic result28 and deserves further investigation. out accounting for strain fields. The blue line represents the contributionsofbothexternalandstrainfieldsusingthecon- In summary, our results suggest that the investigation ventional Matthiessen’s rule. The black line stands for the of anisotropy corrections to the Drude conductivity can combined effect of intrinsic and extrinsic fields obtained for be a new and insightful path to experimentally quantify the full M-matrix analysis. effects of random pseudo-magnetic fields due to strain. In order to further compare our results with the experiment9, let us introduce the magnetoresistance ∆ρ = E · j/j2, where j = jcosξ, j = jsinξ, and ACKNOWLEDGMENTS x y ξ is the angle between B and j. Using the relation (cid:107) E =ρ J one writes45 WethankEduardoMucciolofornumerousdiscussions. i ij j This work has been supported by the Brazilian funding ∆ρ(ξ)=∆ρ cos2ξ+∆ρ sin2ξ. (65) agencies CAPES, CNPq, and FAPERJ. xx yy

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.