doi: 10.1098/rsta.2010.0242, 5525-5556 368 2010 Phil. Trans. R. Soc. A

Norman J. Morgenstern Horing

Aspects of the theory of graphene

Referencesl.html#ref-list-1http://rsta.royalsocietypublishing.org/content/368/1932/5525.ful This article cites 80 articles, 5 of which can be accessed freeRapid response1932/5525http://rsta.royalsocietypublishing.org/letters/submit/roypta;368/ Respond to this articleSubject collections (41 articles) solid-state physics (62 articles) quantum physics

collectionsArticles on similar topics can be found in the followingEmail alerting service here in the box at the top right-hand corner of the article or click Receive free email alerts when new articles cite this article - sign up http://rsta.royalsocietypublishing.org/subscriptions go to: Phil. Trans. R. Soc. A To subscribe to This journal is 2010 The Royal Society on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from Phil. Trans. R. Soc. A (2010) 368, 55255556doi:10.1098/rsta.2010.0242REVI EWAspects of the theory of grapheneBY NORMAN J. MORGENSTERN HORING*Department of Physics and Engineering Physics, Stevens Instituteof Technology, Hoboken, NJ 07030, USAFollowing a brief review of the device-friendly features of graphene, recent work on itsGreens functions with and without a normal magnetic eld are discussed, for an innitegraphene sheet and also for a quantum dot, with analyses of the Landau-quantized energyspectra of the sheet and dot. The random phase approximation dielectric response ofgraphene is reviewed and discussed in connection with the van der Waals interactionsof a graphene sheet with atoms/molecules and with a second graphene sheet in adouble layer. Energy-loss spectroscopy for a graphene sheet subject to both paralleland perpendicular particle probes of its dynamic, non-local response properties are alsotreated. Furthermore, we discuss recent work on the coupling of a graphene plasmon and asurface plasmon, yielding a collective plasma mode that is linear in wavenumber. Finally,we discuss the unusual aspects of graphene conduction and recent work on diffusive chargetransport in graphene, in both the DC and AC regimes.Keywords: graphene; transport; dielectric properties1. IntroductionOwing to its remarkable properties and its strong potential to provide thematerial basis for a new generation of electronic devices, graphene has been thesubject of massive research throughout the world since the rst experiments in2004 (Das Sarma et al. 2007a; Geim & Novoselov 2007; Castro Neto et al. 2009;Geim 2009). It is one of the most heavily researched materials ever (Barth & Marx2008). Graphene is a single-atom-thick two-dimensional planar layer of carbonatoms in a hexagonal honeycombed array composed of two superposed triangularsub-lattices. The band structure of graphene involves two nodal zero-gap (Dirac)points (K, K

) in the rst Brillouin zone at which the conduction and valencebands touch. In the vicinity of these points, the low-energy electron/hole energydispersion relation is proportional to momentum, rather than its square. This isanalogous to the energy dispersion relation of massless relativistic electrons, sothe electrons/holes of graphene are described as Dirac fermions having no mass.*nhoring@stevens.eduOne contribution of 12 to a Theme Issue Electronic and photonic properties of graphene layersand carbon nanoribbons.This journal is 2010 The Royal Society 5525 on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from 5526 N. J. M. HoringThe exceptional properties of graphene as a device-friendly material includehigh mobility reaching up to 200 000 cm2Vs1, over two orders of magnitudehigher than that of silicon-based materials, over 20 times that of GaAs and overtwice that of InSb. Furthermore, graphene supports a high-electron density, about1013cm2in a single sub-band. It has a long mean-free-path, l 400 nm at roomtemperature, which is promising for ballistic devices based on graphene. Thereis also a quantum Hall effect at room temperature1, and graphene is stable tohigh temperatures of approximately 3000 K. It also has great strength. Finally,graphene has a planar form, suggesting that some variant of the well-developedtop-down complementary metal oxide semiconductor (CMOS) compatible processow may be developed for fabrication of graphene-based devices. This promises asubstantial advantage over carbon nanotubes, which are difcult to integrate intoelectronic devices and are difcult to produce in consistent sizes and electronicproperties.Some graphene-based device fabrication has already been carried out.In particular, a eld-effect transistor was constructed by Walt de Heersgroup Georgia Tech Research News (http://gtresearchnews.gatech.edu/circuitry-based-on-graphite-may-provide-a-foundation-for-devices-that-handle-electrons-as-waves) and it has also been addressed by Lemme (2007). Most recently, XiangfengDuan of the California Nanosystems Institute at UCLA reported the developmentof the fastest transistor to date, which has the potential to operate in theterahertz range (Liao et al. 2010). Furthermore, Schedin et al. (2007) reportedthat graphene-based chemical sensors can detect minute concentrations (1 ppb) ofvarious active gases, even to discern individual events when a molecule attachesto the sensors surface (in this, the high sensitivity is a consequence of the hightwo-dimensional surface/volume ratio, which maximizes the role of adsorbedmolecules as donors/acceptors, coupled with the high conductivity of grapheneand low noise). The advantageous properties of graphene in regard to longspin lifetime, low spin-orbit coupling and high conductivity have facilitated thefabrication of a simple spin valve (Hill et al. 2006) employing it to provide a spintransport medium between ferromagnetic electrodes. Scott Bunch et al. (2007)employed a graphene sheet (in contact with a gold electrode) to construct anelectromechanical resonator actuated electrostatically, with the sheet suspendedover a trench in an SiO2 substrate. The activation mechanisms involve a radiofrequency gate voltage superposed on a DC voltage applied to the graphenesheet; also, optical actuation using a laser focused on the sheet has beenused. Furthermore, a quantum interference device using a ring-shaped graphenestructure was built to manipulate electron wave interference effects. As grapheneresearch is only about 6 years old, this is just the beginning. Other possible usesof graphene are touched upon in recent articles by Geim (2009).This presentation does not purport to be a thorough review of graphenephenomenology or its theory. It addresses selected topics of interest andimportance, providing detailed background of particular instances in the hopethat it may be of some pedagogical value.1In addition to its other exceptional properties, it has just been reported that graphene underappropriate mechanical strain inuences the spectrum of particle states as if they were in a giganticpseudo-magnetic eld of 300 T (Levy et al. 2010).Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from Review. Aspects of the theory of graphene 55272. Graphene Hamiltonian, Greens function for a graphene sheet and quantumdot with and without a magnetic eld(a) Hamiltonian and Greens function in the absence of a magnetic eldThe fundamental low-energy graphene electron/hole dispersion relationproportional to momentum, p =(px, py), which likens graphene carriers tomassless relativistic Dirac fermions, is embodied in the Hamiltonian writtenin pseudo-spin notation ( s =[sx, sy] are Pauli matrices), which distinguishesthe two triangular sub-lattices of the honeycomb lattice on which a graphenequasi-particle can be located,h1=gp s =g_ 0 px sgn(s)ipypx +sgn(s)ipy 0_, (2.1)where the two zero-gap Dirac points correspond tosgn(s) =_ 1 s =K1 s =K

_ (2.2)and g is given in terms of graphene band-structure parameters as g =3 /2 (is the hopping parameter in the tight-binding approximation and is the latticespacing): g plays the role of a constant Fermi velocity independent of density.This Hamiltonian is responsible for features in graphene that are analogous torelativistic phenomena such as Klein tunnelling, zitterbewegung and others.As in the study of massless relativistic neutrino fermions, pseudo-helicity, thecomponent of pseudo-spin in the momentum direction, commutes with h1 andits eigenvectors can be used as a basis in which h1 is diagonal. Introducing thetransformation from pseudo-spin basis to pseudo-helicity basis,U(s)p = 1gp_px sgn(s)ipy px +sgn(s)ipygp gp_, (2.3)h1 can be diagonalized ash1=[U(s)p ]+h1U(s)p =diag[31(p), 32(p)], (2.4)where3m=(1)m+1gp. (2.5)Of course, the propagation of electrons/holes in graphene is described by itsGreens function. As the Hamiltonian in pseudo-spin representation, h1, is a 2 2 matrix and the corresponding Green functions (in the absence, and in thepresence, of a magnetic eld) are also 2 2 matrices. The retarded Green functionmatrix, G0 is dened by the following equation (I is the unit 2 2 pseudo-spinmatrix; no magnetic eld; h 1):_iI vvt h1_ G0(p, t) = I d(t t

). (2.6)Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from 5528 N. J. M. HoringIn positionfrequency representation, this reads as (px X =x x

; pyY =y y

; R=(X, Y); s n =1 for K, K

; gn=g sign(n) =g; T =t t

uunder Fourier transformation)_I u gsx1ivvX gnsy1ivvY_ G0(R, u) = I d(X)d(Y). (2.7)The individual elements satisfyuG011 _g1ivvX gnvvY_G021=d(X)d(Y) (2.8)anduG021=_g1ivvX +gnvvY_G011, (2.9)with similar equations for G022 and G012. The results for the retarded Greenfunction elements in pseudo-spin/momentum representation in the absence ofa magnetic eld areG011(p, u) =G022(p, u) = u(u2g2p2) (2.10)andG012(p, u) =G021(p, u) = g(px ipy)(u2g2p2), (2.11)from which the 2 2 spectral weight matrix, A(p, u), may be obtained usingA(p, u) =2Im[G0(p, u)]. (2.12)(b) Graphene Greens function in a quantizing magnetic eldTo incorporate the magnetic eld, B, (taken normal to the grapheneplane), we make the usual replacement p p eA, where A= 12B r for auniform, constant magnetic eld. The requirement of gauge invariance leads to(Horing 1965)G(r, r

; t, t

) =C(r, r

)G

(r r

; t t

), (2.13)where the factor G

(r r

; t t

) is spatially translationally invariant andgauge invariant, satisfying the equation (Horing & Liu 2009) (R=r r

,T =t t

,h 1)_i vvT gsn_1ivvR e2B R__ G

(R, T) =I d(2)(R)d(T), (2.14)Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from Review. Aspects of the theory of graphene 5529while the factor C(r, r

) embodies all non-spatially translationally invariantstructure and all gauge dependence asC(r, r

) =exp_ ie2hcr B r

f(r) +f(r

)_, (2.15)where f(r) is an arbitrary gauge function.The role of Landau quantization in graphene electron dynamics is embedded inthe solution of equation (2.14) for G

(R, T), which may be written in u-frequencyrepresentation as (dene gn=g sign(n); also h 1 and c 1)[u gsxPXY gnsyPYX]G

(R, u) =I d(X)d(Y), (2.16)where we have denedPXY 1ivvX + eB2 Y and PYX 1ivvY eB2 X. (2.17)The elements of the matrix equation, equation (2.16), yield equations for G

11 andG

21 as (similar results obtained for G

22 and G

12)_u + ggnu(eB)_G

11(X, Y; u) + g2u_ v2vX2 + v2vY2 _eB2_2[X2+Y2]+eBi_X vvY Y vvX__G

11(X, Y; u) =d(X)d(Y) (2.18)anduG

21=[gPXY +ignPYX]G

11. (2.19)Dening the operatorLZ = 1i_X vvY Y vvX_=Lz +Lz , (2.20)where Lz is the angular momentum operator, we note that LZG

(R, T) =0(Horing 1965; Horing & Liu 2009).Therefore, equation (2.18) may be written in the formUG

11(R, U) +_ 12MV2RMU2c8 R2_G

11(R, U) =d(2)(R), (2.21)which is readily recognizable as Greens function equation for an isotropic two-dimensional harmonic oscillator (with the impulsive Dirac d-function driving termhaving its source point at the origin) in positionfrequency representation. Itsretarded solution in positiontime (t) representation may be written as (Horing1965; Horing & Liu 2009) (h+(t) is the Heaviside unit step function)G

11(R; t) =h+(t) MUc4p sin(Uct/2) exp_iMUc[X2+Y2]4 tan(Uct/2)_. (2.22)Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from 5530 N. J. M. HoringConsequently, in frequency (u) representation, we have (note that U is aconvenient parameter here, not the actual frequency, u)G

11(R; U) = MUc4p

0dteiUtsin(Uct/2) exp_iMUc[X2+Y2]4 tan(Uct/2)_, (2.23)with the identications for n =KU=u + ggnueB =u + g2ueB, M= u2g2 and Uc= 2g2ueB. (2.24)Expanding the t-integrand as a generator of Laguerre polynomials, Ln, we obtain(Erdelyi et al. 1953b)G

11(R; u)K = eB2puexp_eB4 [X2+Y2]_

n=0Ln(eB/2[X2+Y2])u22ng2eB . (2.25)Similar treatment for G

22(X, Y; u)K yieldsG

22(R; u)K =G11(R; u)K = eB2puexp_eB4 [X2+Y2]_

n=0Ln(eB/2[X2+Y2])u22ng2eB .(2.26)The energy spectrum of the innite graphene sheet obtained from the frequencypoles of equation (2.26) is given byu

=_2ng2eB, (2.27)as was found earlier by Ando (2005). Moreover, G

21 and G

12 are readily obtainedfrom G

11 and G

22.In the case n =K

, we have identications in equation (2.21) asU=u + ggnueB =u g2ueB, M= u2g2 and Uc= 2g2ueB, (2.28)and, proceeding as above, we obtainG

11(R, u)K =G

22(R, u)K = eB2puexp_eB4 [X2+Y2]_

n=0Ln(eB/2[X2+Y2])u22(n +1)g2eB ,(2.29)which has energy pole positions for the innite graphene sheet atu=_2(n +1)g2eB, (2.30)but the residues representing the relative strengths of the modes differ from thoseobtained above by a unit shift of the index of the Ln(eB/2[X2+Y2]) amplitude.Yet another interesting representation of the Landau-quantized grapheneGreen function can be derived by rewriting equation (2.21) in circular coordinatesPhil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from Review. Aspects of the theory of graphene 5531as (for either K or K

)_ v2vR2 + 1RvvR M2U2cR24 +2MU_G

11(R; U) = Mpd(R)R , (2.31)since there is no angular dependence. For R >0, equation (2.31) has the form ofthe Bessel wave equation (Moon & Spencer 1971)_ v2vR2 + 1RvvR +a2R2+q2 p2R2_Z(R) =0, (2.32)with p =0, a2=M2U2c/4 and q2=2mU. The Bessel wave function solutions(Moon & Spencer 1971) of equation (2.32) for the case at hand, p =0, aredenoted by Z1=J0(a, q, R) having small R behaviour as Z1=1 +0(R2); and thesecond solution is Z2=Z1 ln(R) +0(R2) for small R. Thus, the solution of thehomogeneous equation may be written as a linear combination of Z1 and Z2,G

11(R; U) =AZ1_iMUc2 ,2MU, R_+BZ2_iMUc2 ,2MU, R_, (2.33)subject to the condition at small R 3 0+ arising from the Dirac d(R)-functionof equation (2.31),vv3G

11(3, U)= M2p3or G

11(3, u)= M2pln(3). (2.34)From this, it is clear that the coefcient B in equation (2.33) must be B =M/2.The coefcient A must be chosen to prevent singular behaviour as R .To examine the solutions further for large R, we note that the term UG

11 isnegligible when compared with (MU2cR2/8)G

11 in this limit, and equation (2.31)then becomes _ v2vR2 + 1RvvR M2U2cR24_G

11(R, U) =0. (2.35)Carrying out an inverse Lommel transform (Erdelyi et al. 1953a) onequation (2.35), we obtain a modied Bessel equation of order 0, yielding thelarge-R solution for G

11 asG

11(R, U) AI0_MUc4 R2_+ BK0_MUc4 R2_. (2.36)Here, I0 and K0 are modied Bessel functions of the rst and third kind,respectively, with the latter embodying a typical second solution log-singularityfor nite R but falling off for large R as K0(z) _(p/2z)ez, whereas theformer solution of the rst kind diverges as I0(z) _(p/2z)ez. On the basisof these considerations jointly with the log-requirement of equation (2.35), weconclude thatG

11(R; U) =G

22(R; U) = M2pZ2_iMUc2 ,2MU, R_, (2.37)with Z2 as the second solution of the Bessel wave equation. (G21 and G

12 can bedetermined as indicated above.)Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from 5532 N. J. M. Horing(c) Graphene quantum dot Greens function in a magnetic eldTo represent the presence of a quantum dot on a planar graphene sheet,one may add a potential term of the form U(r) =ad(2)(r) to the Hamiltonianabove. This represents a quantum well in the potential prole at r =0 (witha }0 (p, u) =i_ f0(u)1 +f0(u)_ A(p, u), (3.1)Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from 5534 N. J. M. Horingwhere f0(u) is the FermiDirac distribution function. (In the case of a uniform,constant magnetic eld, this equation applies to G

in place of G.) Withinthe framework of the random-phase approximation (RPA), the polarizability isgiven bya(p, u +i0+) =vc(p)R(p, u), (3.2)where the two-dimensional Coulomb potential isvc(p) = 2pe2p , (3.3)and the RPA density perturbation response function, R =dr/dVeff, is given byR(p, u +i0+) =, (3.4)

>=

0dtei(ui0+)t

d2q(2p)2 Tr[G(q p; t)] (3.5)and

( q; t)G2pF,(3.14)where (k is the static background dielectric constant)pTF= 4e2pF(hkg) (3.15)is the two-dimensional ThomasFermi shielding wavevector for graphene. Theassociated shielded Coulombic impurity potential (in wavenumber representation)vc(p, u=0) = 2pe2p 3(p, u=0) (3.16)has been employed in various transport calculations for graphene.Stauber et al. (2009) have analysed the dynamic polarizability of graphenebeyond the usual Dirac cone low-energy description provided above, deriving anapproximate analytical expression for it. Furthermore, inter-valley plasmons ingraphene have been studied in the tight-binding approximation by Tudorovskiy &Mikhailov (2009), relating them to transitions between the two nearest Diraccones. Moreover, Gangadharaiah et al. (2008) have found a novel plasmon modein graphene based on ladder-type vertex corrections (beyond the RPA), andLiu & Willis (2010) have observed a strongly coupled plasmonphonon mode indispersion measurements in epitaxial graphene. In their analysis of double-layergraphene, Hwang & Das Sarma (2009) have found exotic plasmon modes. Finally,graphene plasmons having a linear dependence on wavenumber are discussedimmediately below (Horing 2009).Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from 5536 N. J. M. Horing4. Linear graphene plasmonsPlasmons in graphene and carbon nanotubes have been carefully examinedexperimentally (Stephan et al. 2002; Taverna et al. 2002, 2008; Eberlein et al.2008; Kramberger et al. 2008; Trevisanutto et al. 2008), and the occurrence ofa plasmon whose frequency is a linear function of q-wavenumber (in contrastto up=lq) has been attributed to local eld effects (Kramberger et al. 2008).Considering other possible sources of linear plasmon dispersion, we found thatsuch a plasmon may arise from the Coulombic interaction between the nativegraphene plasmon (uq) and the surface plasmon of a nearby thick substratehosting a semi-innite plasma.The dynamic, non-local and spatially inhomogeneous screening function,K(r1, t1; r2, t2), which is the spacetime matrix inverse of the direct dielectricfunction 3(r1, t1; r2, t2) of the system (note that r here is three dimensional;r =( r, z) =(x, y, z), with r =(x, y)),

d(3)x

dtK(r1, t1; x, t)3(x, t; r2, t2)=d(3)(r1r2)d(t1t2), (4.1)provides the basic description of longitudinal dielectric response. Its frequencypoles dene the plasmon modes of the system, and the residues describe therelative excitation amplitudes (oscillator strengths) of these modes. Rewritingthis equation in the form of an integral equation, we haveK(r1, t1; r2, t2) +

d(3)x

dtasemi(r1, t1; x, t)K(x, t; r2, t2)=d(3)(r1r2)d(t1t2)

d(3)x

dt a2D(r1, t1; x, t)K(x, t; r2, t2), (4.2)in which we have considered a(r1, t1; r2, t2), the combined polarizability of thesemi-innite and two-dimensional graphene sheet constituents of the system,using an accurate and useful approximation (Horing et al. 2001) in writing thecombined polarizability as the sum of the separate polarizabilities of (i) the semi-innite plasma, asemi(r1, t1; r2, t2), and (ii) the two-dimensional graphene plasma,a2D(r1, t1; r2, t2). Equation (4.2) can also be rewritten in the useful formK(r1, t1; r2, t2) =Ksemi(r1, t1; r2, t2)

d(3)x

dt

d(3)x

dt

Ksemi(r1, t1; x, t)a2D(x, t; x

, t

)K(x

, t

, r2, t2), (4.3)where Ksemi(r1, t1; r2, t2) is the corresponding screening function of the semi-innite medium alone with no graphene sheet.For a semi-innite bulk plasma and a nearby parallel two-dimensional graphenesheet plasma, we Fourier transform in the surface plane of translational invariancePhil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from Review. Aspects of the theory of graphene 5537( r1 r2 q, where r = r, z) and time (t1t2u), obtaining in frequencyrepresentation (suppress q, u),K(z1, z2) =Ksemi(z1, z2)

dz3

dz4Ksemi(z1, z3)a2D(z3, z4)K(z4, z2). (4.4)We have previously shown that the dynamic, non-local and spatiallyinhomogeneous screening function of the semi-innite medium (occupying z =0 ) is given by (Horing et al. 1985) (q(z) =1 for z >0 ; 0 for z 1,p2 4x +3x2p 2x3x21 x3(8 6x2)Re_arctanh_ 11 x2__(1 x2)3/2, for 0 x 1.(8.21)For a typical graphene system on an SiO2 substrate with background dielectricconstant k=2.45 and rs=0.813, the present linear diffusive-transport theory withRPA-screened impurity scattering leads to the minimum conductivity assRPAt=0,min= 4.42e2h , (8.22)at the critical-carrier density valueNRPAe =0.11Ni. (8.23)Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from 5548 N. J. M. Horing14.55.03 2 1 0.42 3 4 5background static dielectric constantmin (e2/h)minNc/NiNc/Ni0.20.10rsFigure 5. Dependencies on rs of the minimum conductivity (in units of e2/h) and the critical-carrierdensity (relative to impurity concentration) for a RPA-screened Coulomb scattering potential.(Reprinted with permission from Liu et al. (2008). Copyright (2008), American Institute ofPhysics). g =1.1 106ms1.Table 1. Critical electron densities and minimum conductivities for the short-range (SR) and RPA-screened Coulomb scattering potentials considered. (Reprinted with permission from Liu et al.(2008). Copyright (2008), American Institute of Physics).Nc/Ni smin/(e2/h)SR p6/32 0.24 26 4.9RPA _F(2rs)G(2rs)/2 2[_F(2rs)/G(2rs)]We again nd almost linear dependence of the conductivity on carrier density, inthis case for Ne >0.11Ni, above the critical density.The minimum conductivity and critical density are presented as functions ofrs for RPA screening in gure 5 (note: NceNc). Furthermore, tables of Nc/Niand smin/(e2/h), exhibited as functions of rs analytically, are presented in table 1for both RPA and SR models of impurity scatterer screening. These results areconsistent with the experimental observations. However, this theory is limited tothe diffusive regime and is not applicable in the limit of very low carrier density.(c) Dynamic conductivityFor the case of an AC electric eld, E=E0eiut, the linearized kinetic equationfor [ r(s)1 ]mm in the diffusive regime may be written as (Liu et al. 2010)iu[ r(s)1 ]mm(u, p) eE0 Vp[ r(s)0 ]mm(p) =[I(1)s ]mm, (8.24)Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from Review. Aspects of the theory of graphene 5549and for [ r(s)1 ]12(off-diagonal), we haveiu[ r(s)1 ]12(u, p) +2igp[ r(s)1 ]12(u, p) =[I(1)s ]12, (8.25)with [ r1]mn(u, p) and [ r0]mm(p), respectively, as the elements of r1(u, p) and r(s)0 (p).The linear electric eld part of the scattering term, I(1)s , is composed of elementsas follows: its diagonal elements, [I(1)s ]mm, are given by[I(1)s ]mm=pNi

k|V(p k)|2d[3(s)m (p) 3(s)m (k)]{[1 +cos(fpfk)]{[ r(s)1 ]mm(u, p) [ r(s)1 ]mm(u, k)}(1)msgn(s) sin(fpfk)Im{[ r(s)1 ]12(u, p) +[ r(s)1 ]12(u, k)}},(8.26)while the off-diagonal element, [I(1)s ]12, takes the form[I(1)s ]12= pNi2

k,m|V(p k)|2d[3(s)m (p) 3(s)m (k)]{(1)m+1isgn(s) sin(fpfk){[ r(s)1 ]mm(u, p) [ r(s)1 ]mm(u, k)}+[1 cos(fpfk)]{[ r(s)1 ]12(u, p) [ r(s)1 ]21(u, k)}}. (8.27)Solution for the diagonal elements of r(s)1 (u, p) can be written as[ r(s)1 ]mm(up) =eE0 v(s)m (p)L(s)m (u, p), (8.28)while the off-diagonal element, [ r(s)1 ]12(u, p), is given by (n is the unit vectornormal to the graphene sheet)[ r(s)1 ]12(u, p) = egp [E0p n]F(s)(u, p), (8.29)as above. However, the functions L(s)m (u, p) and F(s)(u, p) differ from their DCcounterparts and are given in terms of the microscopically determined relaxationtimes t(a,b)m (p) (given above) asRe[F(s)(u, p)] =2t(b)(p)(u +2gp)Im[F(s)(u, p)], (8.30)Im[F(s)(u, p)] =sgn(s)

m_ 12(iut(a)m (p) 1)v{[ r(s)0 ]mm(p)}v3(s)m (p)__2t(b)(p)(u+2gp)2+ 12t(a)m (p)+

m12t(a)m (p)[iut(a)m (p) 1]_1(8.31)Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from 5550 N. J. M. Horing0.1graphene conductivitySR scattering1.0110Re (e2/h)Ne/NiFigure 6. Density dependence of the real part of the dynamic conductivity for various frequenciesof the AC eld in the presence of SR scattering. (Reprinted with permission from Liu et al. (2010).Copyright (2010), IEEE). Ni=1 1016m2; T =0 K. Solid line, 10 THz; dotted line, 7 THz;dashed line, 5 THz; short dashed-dotted line, 3 THz; long dashed-dotted line, 1 THz.andL(s)m (u, p) = 1iut(a)m (p) 1_t(a)m (p)v[[ r(s)0 ]mm(p)]v3(s)m (p)+sgn(s)Im[F(s)(u, p)]_.(8.32)In terms of these functions, the linear AC current of the diffusive regime isgiven byJ(u) = e22 gsg2E0

p,s{[L(s)1 (u, p) +L(s)2 (u, p)] +2sgn(s)Im[F(s)(u, p)]}. (8.33)The conductivity calculated from this result takes account of electronholeinterband coherence, and exhibits a minimum as a function of electron densityfor both SR and long-range RPA-screened electron-impurity scatterings, providedthat the frequency of the AC electric eld is less than a critical value, u0, inthe terahertz regime. For SR scattering, u0=5 THz, while for long-range RPA-screened scattering, u0=3 THz. Again, the existence of such a minimum in theconductivity as a function of electron density is the result of interband coherence(Liu et al. 2010). However, for frequencies greater than the critical value, thedynamic conductivity decreases monotonically with decreasing electron density.These features are exhibited in gures 6 and 7.Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from Review. Aspects of the theory of graphene 55510.1 1.0Re (e2/h)Ne/Ni110graphene conductivityRPA screened scatteringFigure 7. Density dependence of the real part of the dynamic conductivity for various frequenciesof the AC eld in the presence of RPA-screened Coulomb scattering. (Reprinted with permissionfrom Liu et al. (2010). Copyright (2010), IEEE). Ni=1 1016m2; T =0 K. Solid line, 10 THz;dotted line, 7 THz; dashed line, 5 THz; short dashed-dotted line, 3 THz; long dashed-dotted line,1 THz.9. SummaryGraphene holds great promise to become the primary material of a new generationof electronics and sensors. Here, we have touched upon an eclectic selectionof aspects of its theory, including its Greens function with and without amagnetic eld, as well as Greens function for a graphene quantum dot andits Landau-quantized spectrum. Furthermore, its dynamic, non-local dielectricfunction has been discussed in the RPA, along with its application to plasmamodes and some of their interactions. The latter includes discussion of the roleof graphene plasmons in energy-loss probe spectroscopy and vdW interactions(as well as TE and TM electromagnetic modes): moreover, the role of plasmons(and phonons) in graphene self-energy is described. The important issue ofcharge transport in graphene has been addressed for both steady-state DCconduction and dynamic AC conductivity, for both SR and long-range RPAshielded impurity scattering.Having a massless relativistic Dirac-like single-particle Hamiltonian and energyspectrum, graphene is also of great fundamental interest. The introductionof relativistic phenomenology such as Klein tunnelling and zitterbewegungPhil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from 5552 N. J. M. Horingopen the possibility of observing and measuring such relativistic effects underlaboratory conditions that are actually non-relativistic. For example, the niteresidual conductivity of graphene may arise (in part) in connection with therelativistic phenomenon of pair production.It is hardly necessary to point out that there is much, much more literaturepertinent to graphene phenomenology that we have not discussed. Its meteoricrise in the world of science and technology has been documented by Geim &Novoselov (2007) and Barth & Marx (2008). Its extraordinary device-friendlypotential, which we have briey discussed above, is reviewed (along with itsproduction and properties) by Soldano et al. (2010). Of particular importance,the current status of graphene transistors is discussed in detail by Lemme (2010).Finally, another very detailed review article on graphene theory is soon to bepublished by Abergel et al. (2010). The literature on graphene is vast.This article barely scratches the surface of the huge ood of work on thesubject, omitting many very interesting and important studies. However, thisis not intended to be a comprehensive review, and we apologize to all the authorsof such studies that are not cited here.This work was partially supported by DARPA grant no. HR0011-09-1-0008. I am pleased toacknowledge the important contributions of Prof. S. Y. Liu, Shanghai Jiao Tong University andProf. V. Fessatidis, Fordham University in our joint research, as well as valuable conversations andexchange of information with Prof. M. L. Glasser, Clarkson University.ReferencesAbergel, D. S. L., Apalkov, V., Berashevich, J., Ziegler, K. & Chakraborty, T. 2010Properties of graphene: a theoretical prespective. Adv. Phys. 59, 261482. (doi:10.1080/00018732.2010.487978)Adam, A. & Das Sarma, S. 2008 Boltzmann transport and residual conductivity in bilayer graphene.Phys. Rev. B 77, 115436. (doi:10.1103/PhysRevB.77.115436)Ando, T. 2005 Theory of electronic states and transport in carbon nanotubes. J. Phys. Soc. Jpn74, 777817. (doi:10.1143/JPSJ.74.777)Ando, T. 2006 Screening effect and impurity scattering in monolayer graphene. J. Phys. Soc. Jpn75, 074716. (doi:10.1143/JPSJ.75.074716)Barth, A. & Marx, W. 2008 Graphenea rising star in view of scientometrics. (http://arxiv.org/abs/0808.3320).Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. 2009 The electronicproperties of graphene. Rev. Mod. Phys. 81, 109162. (doi:10.1103/RevModPhys.81.109)Chaplik, A. V. 1972 Possible crystallization of charge carriers in low-density inversion layers. Sov.Phys. JETP 35, 395398.Cheianov, V. V., Falko, V. I., Altshuler, B. L. & Aleiner, I. L. 2007 Random resistor networkmodel of minimal conductivity in graphene. Phys. Rev. Lett. 99, 176801. (doi:10.1103/PhysRevLett.99.176801)Cserti, J. 2007 Minimal longitudinal dc conductivity of perfect bilayer graphene. Phys. Rev. B 75,033405. (doi:10.1103/PhysRevB.75.033405)Das Sarma, S., Geim, A. K., Kim, P. & MacDonald, A. K. (eds) 2007a Exploring graphene recentresearch advances. Solid State Commun. 143(12).Das Sarma, S., Hwang, E. H. & Tse, W.-K. 2007b Many-body interaction effects in dopedand undoped graphene: Fermi liquid versus non-Fermi liquid. Phys. Rev. B 75, 121406(R).(doi:10.1103/PhysRevB.75.121406)Das Sarma, S., Adam, S., Hwang, E. H. & Rossi, E. 2010 Electronic transport in two dimensionalgraphene. (http://arxiv.org/abs/1003.4731).Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from Review. Aspects of the theory of graphene 5553Eberlein, T., Bangert, U., Nair, R. R. Jones, R., Gass, A. L., Bleloch, A. L., Novoselov, K. S.,Geim, A. & Briddon, P. R. 2008 Plasmon spectroscopy of free-standing graphene lms. Phys.Rev. B 77, 233406. (doi:10.1103/PhysRevB.77.233406)Erdelyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F. G. 1953a Higher transcendental functions,vol. II, p. 13. Bateman Manuscript Project. New York, NY: McGraw-Hill.Erdelyi, A. et al. 1953b Higher transcendental functions, vol. II, p. 189. Bateman ManuscriptProject. New York, NY: McGraw-Hill.Falkovsky, L. A. & Varlamov, A. A. 2007 Space-time dispersion of graphene conductivity. Eur.Phys. J. B 56, 281284. (doi:10.1140/epjb/e2007-00142-3)Fessatidis, V., Horing, N. J. M. & Sawamura, M. 2010 Graphene energy loss spectroscopy:perpendicular case. Phys. Procedia 3, 12791285. (doi:10.1016/j.phpro.2010.01.176)Gangadharaiah, S., Faird, A. M. & Mischenko, E. G. 2008 Charge response function and a novelplasmon mode in graphene. Phys. Rev. Lett. 100, 166802. (doi:10.1103/PhysRevLett.100.166802)Geim, A. K. 2009 Graphene: status and prospects. Science 324, 15301534. (doi:10.1126/science.115877).Geim, A. K. & Novoselov, K. S. 2007 The rise of graphene. Nat. Mater. 6, 183191. (doi:10.1038/nmat1849)Goerbig, M. O. 2010 Electronic properties of graphene in a strong magnetic eld. (http://arxiv.org/abs/1004.3396).Gorbar, E. V., Gusynin, V. P., Miransky, V. A. & Shovkovy, I. A. 2002 Magnetic eld drivenmetal-insulator phase transition in planar systems. Phys. Rev. B 66, 045108. (doi:10.1103/PhysRevB.66.045108)Gusynin, V. P. & Sharapov, S. G. 2005 Unconventional integer quantum hall effect in graphene.Phys. Rev. Lett. 95, 146801. (doi:10.1103/PhysRevLett.95.146801)Gusynin, V. P. & Sharapov, S. G. 2006 Transport of Dirac quasiparticles in graphene: Hall andoptical conductivities. Phys. Rev. B 73, 245411. (doi:10.1103/PhysRevB.73.245411)Hill, E. W., Geim, A. K., Novoselov, K., Schelin, F. & Bake, P. 2006 Electron transport in graphene.IEEE Trans. Magn. 42, 26942696. (doi:10.1109/TMAG.2006.878852)Horing, N. J. M. 1965 Quantum theory of electron gas plasma oscillations in a magnetic eld. Ann.Phys. (NY) 31, 163. (doi:10.1016/0003-4916(65)90231-9)Horing, N. J. M. 2004 Quantum magnetic eld effects in atom-surface van der Waals interaction.Phys. Rev. A 69, 032901. (doi:10.1103/PhysRevA.69.032901)Horing, N. J. M. 2009 Coupling of graphene and surface plasmons. Phys. Rev. B 80, 193401.(doi:10.1103/PhysRevB.80.193401)Horing, N. J. M. & Chen, L. Y. 2002 Magnetoimage effects in the van der Waals interactionof an atom and a bounded, dynamic, nonlocal plasmalike medium. Phys. Rev. A 66, 042905.(doi:10.1103/PhysRevA.66.042905)Horing, N. J. M. & Fessatidis, V. 2010a Graphene stopping power for a fast charged particle:parallel case. IEEE Sens. J. 10, 674677. (doi:10.1109/JSEN.2009.2039537)Horing, N. J. M. & Fessatidis, V. 2010b Graphene vander Waals interactions with an adatom andwithin a double layer. In Proc. 9th Conf. Quantum Field Theory in External Conditions (ed. K.Milton). Singapore: World Scientic.Horing, N. J. M. & Liu, S. Y. 2009 Greens functions for a graphene sheet and quantum dotin a normal magnetic eld. J. Phys A Math. Theor. 42, 225301. (doi:10.1088/1751-8113/42/22/225301)Horing, N. J. M. & Liu, S. Y. 2010 Energy spectrum and density of states for a graphenequantum dot in a magnetic eld. J. Phys. Condens. Matter 22, 025502. (doi:10.1088/0953-8984/22/2/025502)Horing, N. J. M., Kamen, E. & Cui, H. L. 1985 Inverse dielectric function of a bounded solid-stateplasma. Phys. Rev. B 32, 21842189. (doi:10.1103/PhysRevB.32.2184)Horing, N. J. M., Tso, H. C. & Gumbs, G. 1987 Fast-particle energy loss in the vicinity of atwo-dimensional plasma. Phys. Rev. B 36, 15881594. (doi:10.1103/PhysRevB.36.1588)Horing, N. J. M., Gumbs, G. & Park, T. 2001 Coupling of 2D plasmons to nonlocal bulk plasmons.Physica B 299, 165172. (doi:10.1016/S0921-4526(00)00585-8)Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from 5554 N. J. M. HoringHoring, N. J. M., Fessatidis, V. & Mancini, J. D. In press. Atom/molecule van der Waals interactionwith graphene. In Proc. Nano TR4 Cont., Istanbul, Turkey, 2008.Hwang, E. H. & Das Sarma, S. 2007 Dielectric function, screening, and plasmons in two-dimensionalgraphene. Phys. Rev. B 75, 205418. (doi:10.1103/PhysRevB.75.205418)Hwang, E. H. & Das Sarma, S. 2009 Exotic plasmon modes of double layer graphene.(http://arxiv.org/abs/0906.0771).Isacsson, A., Jonsson, L. M., Kinaret, J. M. & Johnson, M. 2008 Electronic superlattices incorrugated graphene. Phys. Rev. B 77, 035423. (doi:10.1103/PhysRevB.77.035423)Jablan, M., Buljan, H. & Soljacic, M. 2009 Plasmonics in graphene at infra-red frequencies. Phys.Rev. B 80, 245435. (doi:10.1103/PhysRevB.80.245435)Kailasvuori, J. & Lffe, M. C. 2010 Quantum corrections in the Boltzmann conductivity of grapheneand their sensitivity to the choice of formalism. J. Stat. Mech. 2010, P06024. (doi:10.1088/1742-5468/2010/06/P06024)Katsnelson, M. I. 2006 Zitterbewegung, chirality, and minimal conductivity in graphene. Eur. Phys.J. B 51, 157160. (doi:10.1140/epjb/e2006-00203-1)Kramberger, C. et al. 2008 Linear plasmon dispersion in single-wall carbon nanotubes andthe collective excitation spectrum of graphene. Phys. Rev. Lett. 100, 196803. (doi:10.1103/PhysRevLett.100.196803)Lemme, M. C. 2007 Nonvolatile switching in graphene eld-effect devices. IEEE Electron DeviceLett. 28, 282284. (doi:10.1109/LED.2007.891668)Lemme, M. C. 2010 Current status of graphene transistors. Solid State Phenom. 156158, 499509.(doi:10.4028/www.scientic.net/SSP.156-158.499)Levy, N., Burke, S. A., Meaker, K. L., Panlasigui, M., Zettl, A., Guinea, F., Castro Neto, A. H. &Crommie, M. F. 2010 Strain-induced pseudo-magnetic elds greater than 300 Tesla in graphenenanobubbles. Science 329, 544547. (doi:10.1126/science.1191700)Lewkowicz, M. & Rosenstein, B. 2009 Dynamics of particlehole pair creation in graphene. Phys.Rev. Lett. 102, 106802. (doi:10.1103/PhysRevLett.102.106802)Liao, L., et al. 2010 High-speed graphene transistors with a self-aligned nanowire gate. Nature 467,305308. (doi:10.1038/nature09405)Lin, M. F., Huang, C. S. & Chuu, D. S. 1997 Plasmons in graphite and stage-1 graphite intercalationcompounds. Phys. Rev. B 55, 13 96113 971. (doi:10.1103/PhysRevB.55.13961)Liu, Y. & Willis, R. F. 2010 Plasmonphonon strongly-coupled mode in epitaxial graphene. Phys.Rev. B 81, 081406(R). (doi:10.1103/PhysRevB.81.081406)Liu, S. Y., Lei, X. L. & Horing, N. J. M. 2008 Diffusive transport in graphene: the role of interbandcorrelation. J. Appl. Phys. 104, 043705. (doi:10.1063/1.2969787)Liu, S. Y., Horing, N. J. M. & Lei, X. L. 2010 Dynamic conductivity of graphene in a TerahertzAC electric eld. IEEE Sens. J. 10, 681685. (doi:10.1109/JSEN.2009.2039544)Ludwig, A. W. W., Fisher, M. P. A., Shankar, R. & Grinstein, G. 1994 Integer quantum Halltransition: an alternative approach and exact results. Phys. Rev. B 50, 75267552. (doi:10.1103/PhysRevB.50.7526)Martin, J., Akerman, N., Ulbricht, G., Lohmann, T., Smet, J. H., von Klitzing, K. & Yacoby, A.2007 Observation of electronhole puddles in graphene using a scanning single-electrontransistor. Nat. Phys. 4, 144148. (doi:10.1038/nphys781)Matulis, A. & Peeters, F. M. 2008 Quasibound states of quantum dots in single and bilayergraphene. Phys. Rev. B 77, 115423. (doi:10.1103/PhysRevB.77.115423)Mikhailov, S. A. & Ziegler, K. 2007 New electromagnetic mode in graphene. Phys. Rev. Lett. 99,016803. (doi:10.1103/PhysRevLett.99.016803)Mikhailov, S. A. & Ziegler, K. 2008 Nonlinear electromagnetic response of graphene: frequencymultiplication and the self-consistent-eld effects. J. Phys.: Condens. Matter 20, 384204.(doi:10.1088/0953-8984/20/38/384204)Moon, P. & Spencer, E. 1971 Field theory handbook, 2nd edn. Berlin, Germany: Springer.Novoselov, K. S., Geim, A. K., Morozov, S.V., Jiang, D., Zhang, Y., Dubonos, S. V., Grigorieva,I. V. & Firsov, A. A. 2004 Electric eld effect in atomically thin carbon lms. Science 306,666669. (doi:10.1126/science.1102896)Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from Review. Aspects of the theory of graphene 5555Novoselov, K. S., Geim, A. K., Morozov, S. V., Jiang, D., Katsnelson, M. I., Grigorieva, I. V.,Dubonos, S. V. & Firsov, A. A. 2005 Two-dimensional gas of massless Dirac fermions ingraphene. Nature 438, 197200. (doi:10.1038/nature04233)Novoselov, K. S. et al. 2007 Room-temperature quantum Hall effect in graphene. Science 315, 1379.(doi:10.1126/science.1137201)Ostrovsky, P. M., Gornyi, I. V. & Mirlin, A. D. 2006 Electron transport in disordered graphene.Phys. Rev. B 74, 235443. (doi:10.1103/PhysRevB.74.235443)Peres, N. M. R. 2009 The transport properties of graphene. J. Phys.: Condens. Matter 21, 323201.(doi:10.1088/0953-8984/21/32/323201)Peres, N. M. R. 2010 The transport properties of graphene: an introduction.(http://arxiv.org/abs/1007. 2849).Peres, N. M. R., Guinea, F. & Castro Neto, A. H. 2006 Electronic properties of disordered two-dimensional carbon. Phys. Rev. B 73, 125411. (doi:10.1103/PhysRevB.73.125411)Roslyak, O., Iorov, A., Gumbs, G. & Huang, D. 2010 Unimpeded tunneling in graphenenanoribbons. J. Phys.: Condens. Matter 22, 165301. (doi:10.1088/0953-8984/22/16/165301)Roslyak, O., Gumbs, G. & Huang, D. 2010 Tunable band structure effects on ballistic transport ingraphene nanoribbons. Phys. Lett. A 374, 40614064. (doi:10.1016/j.physleta.2010.08.003)Ryu, S., Mudry, C., Furusaki, A. & Ludwig, A. W. W. 2007 Landauer conductance and twistedboundary conditions for Dirac fermions in two space dimensions. Phys. Rev. B 75, 205344.(doi:10.1103/PhysRevB.75.205344)Ryzhii, V. 2006 Terahertz plasma waves in gated graphene heterostructures. Jpn J. Appl. Phys.45, L923. (doi:10.1143/JJAP.45.L923)Ryzhii, V. 2007 Plasma waves in two-dimensional electronhole system in gated grapheneheterostructures. J. Appl. Phys. 101, 024509. (doi:10.1063/1.2426904)Schedin, F., Geim, A. K., Morozov, S. V., Jiang, D., Hill, E. H., Blake, P. & Novoselov, K. S.2007 Detection of individual gas molecules absorbed on graphene. Nat. Mater. 6, 652655.(doi:10.1038/nmat1976)Schmidt, S., Blaschke, D., Rpke, G., Smolyansky, S. A. & Prozorkevich, A. V. 1998 A quantumkinetic equation for particle production in the Schwinger mechanism. Int. J. Mod. Phys. E7, 709722. (doi:10.1142/S0218301398000403)Schwinger, J. 1951 On gauge invariance and vacuum polarization. Phys. Rev. 82, 664679.(doi:10.1103/PhysRev.82.664)Scott Bunch, J., Vanderzande, A. M., Verbridge, S. S., Frank, I. W., Tanenbaum, D. M., Parpia,J. M., Craighead, H. G. & McEuen P. L. 2007 Electromechanical resonators from graphenesheets. Science 315, 490493. (doi:10.1126/science.1136836)Shung, K. W.-K. 1986a Dielectric function and plasmon structure of stage-1 intercalated graphite.Phys. Rev. B 34, 979993. (doi:10.1103/PhysRevB.34.979)Shung, K. W.-K. 1986b Lifetime effects in low-stage intercalated graphite systems. Phys. Rev. B34, 12641272. (doi:10.1103/PhysRevB.34.1264)Soldano, C., Mahmood, A. & Dujardin, E. 2010 Production, properties and potential of graphene.Carbon 48, 21272150. (doi:10.1016/j.carbon.2010.01.058)Stauber, T., Schliemann, J. & Peres, N. M. R. 2009 Dynamical polarizability of graphene beyondthe Dirac cone approximation. Phys. Rev. B 81, 085409. (doi:10.1103/PhysRevB.81.085409)Stephan, O., Taverna, D., Kociak, M., Suegnaga, K., Henrard, L. & Colliex, C. 2002 Dielectricresponse of isolated carbon nanotubes investigated by spatially resolved electron energy-loss spectroscopy: from multiwalled to single-walled nanotubes. Phys. Rev. B 66, 155422.(doi:10.1103/PhysRevB.66.155422)Taverna, D., Kociak, M., Charbios, V., Henrard, L. 2002 Electron energy-loss spectrumof an electron passing near a locally anisotropic nanotube. Phys. Rev. B 66, 235419.(doi:10.1103/PhysRevB.66.235419)Taverna, D. Kociak, M., Stphan, O., Fabre, A., Finot, E., Dcamps, B. & Colliex, C. 2008Probing physical properties of conned uids within individual nanobubbles. Phys. Rev. Lett.100, 035301. (doi:10.1103/PhysRevLett.100.035301)Trevisanutto, P. E., Giorgetti, C., Reining, L., Ladisa, M. & Olevano, V. 2008 Ab Initio GW many-body effects in graphene. Phys. Rev. Lett. 101, 226405. (doi:10.1103/PhysRevLett.101.226405)Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from 5556 N. J. M. HoringTse, W.-K. & Das Sarma, S. 2007 Phonon-induced many-body renormalization of the electronicproperties of graphene. Phys. Rev. Lett. 99, 236802. (doi:10.1103/PhysRevLett.99.236802)Tudorovskiy, T. & Mikhailov, S. A. 2009 Inter-valley plasmons in graphene. (http://arxiv.org/abs/0910.2163)Tworzydlo, J., Trauzettel, B., Titov, M., Rycerz, A. & Beenakker, C. W. J. 2006 Sub-poissonianshot noise in graphene. Phys. Rev. Lett. 96, 246802. (doi:10.1103/PhysRevLett.96.246802)Wnsch, B., Stauber, T., Sols, F. & Guinea, F. 2006 Dynamical polarization of graphene at nitedoping. New J. Phys. 8, 318. (doi:10.1088/1367-2630/8/12/318)Zhang, Y. Tan, Y.-W., Stormer, H. L. & Kim, P. 2005 Experimental observation of the quantumHall effect and Berrys phase in graphene. Nature 438, 201204. (doi:10.1038/nature04235)Zhang, Y., Brar, V. W., Gi rit, C., Zettl, A. & Crommie, M. F. 2009 Origin of spatial chargeinhomogeneity in graphene. Nat. Phys. 5, 722726. (doi:10.1038/nphys1365)Ziegler, K. 1997 Scaling behavior and universality near the quantum Hall transition. Phys. Rev. B55, 10 66110 670. (doi:10.1103/PhysRevB.55.10661)Ziegler, K. 1998 Delocalization of 2D Dirac fermions: the role of a broken supersymmetry. Phys.Rev. Lett. 80, 31133116. (doi:10.1103/PhysRevLett.80.3113)Ziegler, K. 2006 Robust transport properties in graphene. Phys. Rev. Lett. 97, 266802. (doi:10.1103/PhysRevLett.97.266802)Phil. Trans. R. Soc. A (2010) on April 30, 2011 rsta.royalsocietypublishing.org Downloaded from