REVIEWS OF MODERN PHYSICS, VOLUME 83, APRIL–JUNE 2011 Electronic transport in two-dimensional graphene
S. Das Sarma Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
Shaffique Adam Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA and Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-6202, USA
E. H. Hwang Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
Enrico Rossi* Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA (Received 9 March 2010; published 16 May 2011)
A broad review of fundamental electronic properties of two-dimensional graphene with the emphasis on density and temperature-dependent carrier transport in doped or gated graphene structures is provided. A salient feature of this review is a critical comparison between carrier transport in graphene and in two-dimensional semiconductor systems (e.g., heterostructures, quantum wells, inversion layers) so that the unique features of graphene electronic properties arising from its gapless, massless, chiral Dirac spectrum are highlighted. Experiment and theory, as well as quantum and semiclassical transport, are discussed in a synergistic manner in order to provide a unified and comprehensive perspective. Although the emphasis of the review is on those aspects of graphene transport where reasonable consensus exists in the literature, open questions are discussed as well. Various physical mechanisms controlling transport are described in depth including long-range charged impurity scattering, screening, short-range defect scattering, phonon scattering, many-body effects, Klein tunneling, minimum conductivity at the Dirac point, electron- hole puddle formation, p-n junctions, localization, percolation, quantum-classical crossover, midgap states, quantum Hall effects, and other phenomena.
DOI: 10.1103/RevModPhys.83.407 PACS numbers: 72.80.Vp, 81.05.ue, 72.10. d, 73.22.Pr À
CONTENTS 4. Many-body effects in graphene 419 5. Topological insulators 419 I. Introduction 408 F. 2D nature of graphene 419 A. Scope 408 II. Quantum Transport 420 B. Background 408 A. Introduction 420 1. Monolayer graphene 409 B. Ballistic transport 421 2. Bilayer graphene 411 1. Klein tunneling 421 3. 2D Semiconductor structures 412 2. Universal quantum-limited conductivity 422 C. Elementary electronic properties 413 3. Shot noise 423 1. Interaction parameter r 413 s C. Quantum interference effects 423 2. Thomas-Fermi screening wave vector q 414 TF 1. Weak antilocalization 423 3. Plasmons 414 2. Crossover from the symplectic universality class 425 4. Magnetic field effects 415 3. Magnetoresistance and mesoscopic conductance D. Intrinsic and extrinsic graphene 415 fluctuations 426 E. Other topics 417 4. Ultraviolet logarithmic corrections 428 1. Optical conductivity 417 III. Transport at High Carrier Density 428 2. Graphene nanoribbons 417 A. Boltzmann transport theory 428 3. Suspended graphene 418 B. Impurity scattering 430 1. Screening and polarizability 431 2. Conductivity 433 * Present address: Department of Physics, College of William C. Phonon scattering in graphene 439 and Mary, Williamsburg, VA 23187, USA.
0034-6861= 2011=83(2)=407(64) 407 Ó 2011 American Physical Society 408 Das Sarma et al.: Electronic transport in two-dimensional graphene
D. Intrinsic mobility 441 ties, with the emphasis on scattering mechanisms and E. Other scattering mechanisms 442 conceptual issues of fundamental importance. In the context 1. Midgap states 442 of 2D transport, it is conceptually useful to compare and 2. Effect of strain and corrugations 442 contrast graphene with the much older and well established IV. Transport at Low Carrier Density 443 subject of carrier transport in 2D semiconductor structures A. Graphene minimum conductivity problem 443 [e.g., Si inversion layers in metal-oxide-semiconductor-field- 1. Intrinsic conductivity at the Dirac point 443 effect transistors (MOSFETs), 2D GaAs heterostructures, and 2. Localization 444 quantum wells]. Transport in 2D semiconductor systems has 3. Zero-density limit 444 a number of similarities and key dissimilarities with gra- 4. Electron and hole puddles 444 phene. One purpose of this review is to emphasize the key 5. Self-consistent theory 445 conceptual differences between 2D graphene and 2D semi- B. Quantum to classical crossover 445 conductors in order to bring out the new fundamental aspects C. Ground state in the presence of long-range disorder 446 of graphene transport, which make it a truly novel electronic 1. Screening of a single charge impurity 447 material that is qualitatively different from the large class of 2. Density functional theory 447 existing and well established 2D semiconductor materials. 3. Thomas-Fermi-Dirac theory 448 Since graphene is a dynamically (and exponentially) 4. Effect of ripples on carrier density distribution 451 evolving subject, with new important results appearing al- 5. Imaging experiments at the Dirac point 451 most every week, the current review concentrates on only those features of graphene carrier transport where some D. Transport in the presence of electron-hole puddles 452 V. Quantum Hall Effects 455 qualitative understanding, if not a universal consensus, has A. Monolayer graphene 455 been achieved in the community. As such, some active topics, where the subject is in flux, have been left out. Given the 1. Integer quantum Hall effect 455 constraint of the size of this review, depth and comprehension 2. Broken-symmetry states 456 have been emphasized over breadth; given the large graphene 3. The 0 state 458 ¼ literature, no single review can attempt to provide a broad 4. Fractional quantum Hall effect 458 coverage of the subject at this stage. There have already been B. Bilayer graphene 458 several reviews of graphene physics in the recent literature. 1. Integer quantum Hall effect 458 We have made every effort to minimize overlap between our 2. Broken-symmetry states 459 article and these recent reviews. The closest in spirit to our VI. Conclusion and Summary 459 review is the one by Castro Neto et al. (2009) which was written 2.5 years ago (i.e. more than 3000 graphene publica- tions have appeared in the literature since that review was I. INTRODUCTION written). Our review should be considered complimentary to Castro Neto et al. (2009), and we have tried avoiding too A. Scope much repetition of the materials they already covered, con- centrating instead on the new results arising in the literature The experimental discovery of two-dimensional (2D) following the older review. Although some repetition is gated graphene in 2004 by Novoselov et al. (2004) is a necessary in order to make our review self-contained, we seminal event in electronic materials science, ushering in a refer the interested reader to Castro Neto et al. (2009) for tremendous outburst of scientific activity in the study of details on the history of graphene, its band structure consid- electronic properties of graphene, which continued unabated erations, and the early (2005–2007) experimental and theo- up until the end of 2009 (with the appearance of more than retical results. Our material emphasizes the more mature 5000 articles on graphene during the 2005–2009 five-year phase (2007–2009) of 2D graphene physics. period). The subject has now reached a level so vast that no For further background and review of graphene physics single article can cover the whole topic in any reasonable beyond the scope of our review, we mention in addition to the manner, and most general reviews are likely to become Rev. Mod. Phys. article by Castro Neto et al. (2009), the obsolete in a short time due to rapid advances in the graphene accessible reviews by Geim and his collaborators (Geim and literature. The scope of the current review is transport in gated Novoselov, 2007; Geim, 2009), the recent brief review by graphene with the emphasis on fundamental physics and Mucciolo and Lewenkopf (2010), as well as two edited conceptual issues. Device applications and related topics volumes of Solid State Communications (Das Sarma, Geim are not discussed (Avouris et al., 2007), nor are graphene’s et al., 2007; Fal’ko et al., 2009), where the active graphene mechanical properties (Bunch et al., 2007; Lee, Wei et al., researchers have contributed individual perspectives. 2008). The important subject of graphene materials science, which deserves its own separate review, is not discussed at all. B. Background Details of the band structure properties and related phe- nomena are also not covered in any depth, except in the Graphene (or more precisely, monolayer graphene—in this context of understanding transport phenomena. What is cov- review, we refer to monolayer graphene simply as ‘‘gra- ered in reasonable depth is the basic physics of carrier phene’’) is a single 2D sheet of carbon atoms in a honeycomb transport in graphene, critically compared with the corre- lattice. As such, 2D graphene rolled up in the plane is a sponding well-studied 2D semiconductor transport proper- carbon nanotube, and multilayer graphene with weak
Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 409 interlayer tunneling is graphite. Given that graphene is simply approximate analytic formula is obtained for the conduction a single 2D layer of carbon atoms peeled off a graphite (upper, , Ã) band and valence (lower, , ) band: sample, early interest in the theory of graphene band structure þ À 2 2 was all worked out a long time ago. In this review we only 9t0a 3ta 2 E q 3t0 ℏvF q sin 3q q ; consider graphene monolayers (MLG) and bilayers (BLG), Æð Þ Æ j jÀ 4 Æ 8 ð Þj j which are both of great interest. (1.1) with v 3ta=2, arctan 1 q =q , and where t, t are, 1. Monolayer graphene F ¼ q ¼ À ½ x y 0 respectively, the nearest-neighbor (i.e. intersublattice A B) Graphene monolayers have been rightfully described as the and next-nearest-neighbor (i.e. intrasublattice A A orÀB À À ‘‘ultimate flatland’’ (Geim and MacDonald, 2007), i.e., the B) hopping amplitudes, and t 2:5 eV t0 0:1 eV . most perfect 2D electronic material possible in nature, since The almost universally usedð grapheneÞ bandð dispersionÞ at the system is exactly one atomic monolayer thick, and carrier long wavelength puts t0 0, where the band structure for dynamics is necessarily confined in this strict 2D layer. The small q relative to the Dirac¼ point is given by electron hopping in the 2D graphene honeycomb lattice is 2 quite special since there are two equivalent lattice sites [A and E q ℏvFq O q=k : (1.2) Æð Þ¼Æ þ ð Þ B in Fig. 1(a)] which give rise to the ‘‘chirality’’ in the Further details on the band structure of 2D graphene mono- graphene carrier dynamics. layers can be found in the literature (Wallace, 1947; McClure, The honeycomb structure can be thought of as a triangular 1957; Slonczewski and Weiss, 1958; McClure, 1964; Reich lattice with a basis of two atoms per unit cell, with 2D et al., 2002; Castro Neto et al., 2009) and will not be lattice vectors A a=2 3; p3 and B a=2 3; p3 0 ¼ð Þð Þ 0 ¼ð Þð À Þ discussed here. Instead, we provide below a thorough dis- (a 0:142 nm is the carbon-carbon distance). K ffiffiffi ffiffiffi¼ cussion of the implications of Eq. (1.2) for graphene carrier 2= 3a ; 2= 3p3a and K0 2= 3a ; 2= 3p3a transport. Since much of the fundamental interest is in under- ð ð Þ ð ÞÞ ¼ ð ð Þ À ð ÞÞ are the inequivalentffiffiffi corners of the Brillouin zone andffiffiffi are standing graphene transport in the relatively low carrier called Dirac points. These Dirac points are of great impor- density regime, complications arising from the large tance in the electronic transport of graphene, and they play a q K aspects of graphene band structure can be neglected. role similar to the role of À points in direct band-gap semi- ðTheÞ most important aspect of graphene’s energy dispersion conductors such as GaAs. Essentially, all of the physics (and the one attracting the most attention) is its linear energy- discussed in this review is the physics of graphene carriers momentum relationship with the conduction and valence (electrons and/or holes) close to the Dirac points (i.e., within bands intersecting at q 0, with no energy gap. Graphene a 2D wave vector q q 2=a of the Dirac points) just is thus a zero band-gap¼ semiconductor with a linear, rather ¼j j as all the 2D semiconductor physics we discuss will occur than quadratic, long-wavelength energy dispersion for both around the À point. electrons (holes) in the conduction (valence) bands. The The electronic band dispersion of 2D monolayer graphene existence of two Dirac points at K and K0, where the Dirac was calculated by Wallace (1947) and others (McClure, 1957; cones for electrons and holes touch [Fig. 2(b)] each other in Slonczewski and Weiss, 1958) a long time ago, within the momentum space, gives rise to a valley degeneracy gv 2 tight-binding prescription, keeping up to the second-nearest for graphene. The presence of any intervalley scattering¼ neighbor hopping term in the calculation. The following between K and K0 points lifts this valley degeneracy, but
(a) b1 (b) AB ky
δ δ 3 1 K Γ a M 1 δ 2 kx K’ a 2
b2
(c) (d) (e) VSD
S Graphene D
SiO2 Si back−gate
Vbg
FIG. 1 (color online). (a) Graphene honeycomb lattice showing in different colors the two triangular sublattices. Also shown is the graphene Brillouin zone in momentum space. Adapted from Castro Neto et al., 2009. (b) Carbon nanotube as a rolled up graphene layer. Adapted from Lee, Sharma et al., 2008. (c) Lattice structure of graphite, graphene multilayer. Adapted from Castro Neto et al., 2006. (d) Lattice structure of bilayer graphene. 0 and 1 are, respectively, the intralayer and interlayer hopping parameters t, t used in the text. The interlayer hopping ? parameters 3 and 4 are much smaller than 1 t and are normally neglected. Adapted from Mucha-Kruczynski et al., 2010. (e) Typical configuration for gated graphene. ?
Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 410 Das Sarma et al.: Electronic transport in two-dimensional graphene
(a) (b)
(c)E (d)
+σ −σ k
FIG. 2 (color online). (a) Graphene band structure. Adpated from Wilson, 2006. (b) Enlargment of the band structure close to the K and K0 points showing the Dirac cones. Adpated from Wilson, 2006. (c) Model energy dispersion E ℏvF k . (d) Density of states of graphene close to the Dirac point. The inset shows the density of states over the full electron bandwidth.¼ Adaptedj j from Castro Neto et al., 2009.
such effects require the presence of strong lattice scale scat- ℏvFkc and to demand that Ec < 0:4t 1:0 eV , so that one tering. Intervalley scattering seems to be weak and when they can ignore the lattice effects (which leadð to deviationsÞ from can be ignored, the presence of a second valley can be taken pure Dirac-like dispersion). This leads to a cutoff wave vector into account simply via the degenercy factor g 2. given by k 0:25 nm 1. v c À Throughout this introduction, we neglect intervalley scatter-¼ The mapping of graphene electronic structure onto the ing processes. massless Dirac theory is deeper than the linear graphene The graphene carrier dispersion E q ℏvFq explicitly carrier energy dispersion. The existence of two equivalent, Æð Þ¼ depends on the constant vF, sometimes called the graphene but independent, sublattices A and B (corresponding to the (Fermi) velocity. In the literature different symbols (vF, v0, two atoms per unit cell) leads to the existence of a novel =ℏ) are used to denote this velocity. The tight-binding chirality in graphene dynamics where the two linear branches prescription provides a formula for vF in terms of the nearest of graphene energy dispersion (intersecting at Dirac points) become independent of each other, indicating the existence of neighbor hopping t and the lattice constant a2 p3a: ℏvF ¼ ¼ a pseudospin quantum number analogous to electron spin (but 3ta=2. The best estimates of t 2:5 eV and a ffiffiffi 0:14 nm 8 ¼ completely independent of real spin). Thus, graphene carriers give vF 10 cm=s for the empty graphene band, i.e., in the absence of any carriers. The presence of carriers may lead to a have a pseudospin index in addition to the spin and orbital many-body renormalization of the graphene velocity, which index. The existence of the chiral pseudospin quantum num- is, however, small for MLG but could, in principle, be sub- ber is a natural byproduct of the basic lattice structure of stantial for BLG. graphene comprising two independent sublattices. The long- The linear long-wavelength Dirac dispersion, with a Fermi wavelength, low energy effective 2D continuum Schro¨dinger velocity that is roughly 1=300 of the velocity of light, is the equation for spinless graphene carriers near the Dirac point most distinguishing feature of graphene in addition to its therefore becomes strict 2D nature. It is therefore natural to ask about the precise applicability of the linear energy dispersion, since it is ob- iℏvF É r EÉ r ; (1.3) viously a long-wavelength continuum property of graphene À Á r ð Þ¼ ð Þ 1 carriers valid only for q K 0:1 nm À . There are several ways to estimateð theÞ cutoff wave vector where ; is the usual vector of Pauli matrices ¼ð x yÞ (or momentum) kc above which the linear continuum Dirac (in 2D now), and É r is a 2D spinor wave function. dispersion approximation breaks down for graphene. The Equation (1.3) correspondsð Þ to the effective low energy easiest is perhaps to estimate the carrier energy E Dirac Hamiltonian: c ¼
Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 411
FIG. 3 (color online). (a) Energy band of bilayer graphene for V 0. (b) Enlargment of the energy band close to the neutrality point K for different values of V. Adapted from Min et al., 2007. ¼
0 qx iqy by interlayer carbon hopping, it is intermediate between H ℏvF À ℏvF q: (1.4) ¼ qx iqy 0 ¼ Á graphene monolayers and bulk graphite. þ The tight-binding description can be adapted to study the We note that Eq. (1.3) is simply the equation for massless bilayer electronic structure assuming specific stacking of chiral Dirac fermions in 2D (except that the spinor here refers the two layers with respect to each other (which controls to the graphene pseudospin rather than real spin), although the interlayer hopping terms). Considering the so-called A-B it is arrived at starting purely from the tight-binding stacking of the two layers [which is the three-dimensional Schro¨dinger equation for carbon in a honeycomb lattice (3D) graphitic stacking], the low energy, long-wavelength with two atoms per unit cell. This mapping of the low energy, electronic structure of bilayer graphene is described by the long-wavelength electronic structure of graphene onto the following energy dispersion relation (Brandt et al., 1988; massless chiral Dirac equation was discussed by Semenoff Dresselhaus and Dresselhaus, 2002; McCann, 2006; McCann (1984) more than 25 years ago. It is a curious historical fact and Fal’ko, 2006): that although the actual experimental discovery of gated graphene (and the beginning of the frenzy of activities lead- 2 2 2 2 2 2 2 2 2 E q V ℏ vFq t =2 4V ℏ vFq ing to this review) happened only in 2004, some of the key Æð Þ¼½ þ þ ? Æð 2 2 2 2 4 1=2 1=2 theoretical insights go back a long way in time and are as t ℏ vFq t =4 ; (1.5) þ ? þ ? Þ valid today for real graphene as they were for theoretical where t is the effective interlayer hopping energy (and graphene when they were introduced (Wallace, 1947; ? McClure, 1957; Semenoff, 1984; Haldane, 1988; Gonzalez t, vF are the intralayer hopping energy and graphene et al., 1994; Ludwig et al., 1994). Fermi velocity for the monolayer case) (see Fig. 3). We The momentum space pseudospinor eigenfunctions for note that t 0:4 eV Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 412 Das Sarma et al.: Electronic transport in two-dimensional graphene (a) conduction channel E (b) E C ionized donors C Metal E Si C + + E E F F Vgate Oxide E E V V lowest subband E F Al xGa 1 − Asx ionized acceptors GaAs E V FIG. 4 (color online). (a) Diagram showing the bands at the interfaces of a metal-oxide-silicon structure. (b) Band diagram showing the bending of the bands at the interface of the semiconductors and the two-dimensional subband. is m 0:03–0:05 me, which corresponds to a very small the basic electronic structure of 2D semiconductor systems effectiveð mass. Þ which are of relevance in the context of graphene physics, To better understand the quadratic to linear crossover in the without giving much details, which can be found in the effective BLG band dispersion, it is convenient to rewrite the literature (Ando et al., 1982; Bastard, 1991; Davies, 1998). BLG band dispersion (for V 0) in the following hyperbolic There are, broadly speaking, four qualitative differences form: ¼ between 2D graphene and 2D semiconductor systems (see Fig. 4). (We note that there are significant quantitative and E mv2 mv2 1 k=k 2 1=2; (1.7) BLG ¼Ç F Æ F½ þð 0Þ some qualitative differences between different 2D semicon- ductor systems themselves). These differences are suffi- where k t = 2 v is a characteristic wave vector. In this 0 ℏ F ciently important in order to be emphasized right at the form it is¼ easy? toð see thatÞ E k2 k for k 0 for the BLG outset. effective BLG band dispersion! with ðk Þ k (!k ð1kÞ) being 0 0 (i) First, 2D semiconductor systems typically have very the parabolic (linear) band dispersion regimes, k 0 large (> 1 eV) band gaps so that 2D electrons and 2D holes 0:3 nm 1 for m 0:03m . Using the best available estimates À e must be studied using completely different electron-doped or from band structure calculations, we conclude that for carrier hole-doped structures. By contrast, graphene (except biased densities smaller (larger) than 5 1012 cm 2, the BLG sys- À graphene bilayers that have small band gaps) is a gapless tem should have parabolic (linear) dispersion at the Fermi semiconductor with the nature of the carrier system changing level. at the Dirac point from electrons to holes (or vice versa) in a What about chirality for bilayer graphene? Although the single structure. A direct corollary of this gapless (or small bilayer energy dispersion is non-Dirac–like and parabolic, the gap) nature of graphene is of course the ‘‘always metallic’’ system is still chiral due to the A=B sublattice symmetry nature of 2D graphene, where the chemical potential (Fermi giving rise to the conserved pseudospin quantum index. The level) is always in the conduction or the valence band. By detailed chiral 4-component wave function for the bilayer contrast, the 2D semiconductor becomes insulating below a case, including both layer and sublattice degrees of freedom, can be found in the literature (McCann, 2006; McCann and threshold voltage, as the Fermi level enters the band gap. Fal’ko, 2006; Nilsson et al., 2006a, 2006b, 2008). (ii) Graphene systems are chiral, while 2D semiconductors The possible existence of an external bias-induced band are nonchiral. Chirality of graphene leads to some important gap and the parabolic dispersion at long wavelength distin- consequences for transport behavior, as we discuss later in guish bilayer graphene from monolayer graphene, with both this review. (For example, 2kF backscattering is suppressed in possessing chiral carrier dynamics. We note that bilayer MLG at low temperature.) graphene should be considered a single 2D system, quite (iii) Monolayer graphene dispersion is linear, while 2D distinct from ‘‘double-layer’’ graphene (Hwang and Das semiconductors have quadratic energy dispersion. This leads Sarma, 2009a), which is a composite system consisting of to substantial quantitative differences in the transport prop- two parallel single layers of graphene, separated by a distance erties of the two systems. in the z^ direction. The 2D energy dispersion in double-layer (iv) Finally, the carrier confinement in 2D graphene is graphene is massless Dirac-like (as in the monolayer case), ideally two dimensional, since the graphene layer is precisely and the interlayer separation is arbitrary; whereas, bilayer one atomic monolayer thick. For 2D semiconductor struc- graphene has the quadratic band dispersion with a fixed tures, the quantum dynamics is two dimensional by virtue of interlayer separation of 0.3 nm similar to graphite. confinement induced by an external electric field, and as such, 2D semiconductors are quasi-2D systems, and always have an average width or thickness z ( 5 to 50 nm) in the third 3. 2D Semiconductor structures h i direction with z & F, where F is the 2D Fermi wave- Since one goal of this review is to understand graphene length (or equivalentlyh i the carrier de Broglie wavelength). electronic properties in the context of extensively studied (for The condition z <F defines a 2D electron system. more than 40 years) 2D semiconductor systems (e.g., Si The carrierh dispersioni of 2D semiconductors is given by 2 2 inversion layers in MOSFETs, GaAs-AlGaAs heterostruc- E q E0 ℏ q = 2mà , where E0 is the quantum confine- tures, quantum wells, etc.), we summarize in this section mentð Þ¼ energyþ of theð lowestÞ quantum confined 2D state, and Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 413 q q ;q is the 2D wave vector. If more than one quantum ¼ð x yÞ dq 4n 2D level is occupied by carriers (usually called ‘‘subbands’’) n gsgv 2 kF ; (1.9) ¼ q k 2 ! ¼ sffiffiffiffiffiffiffiffiffiffigsgv the system is no longer, strictly speaking, two-dimensional, Zj j F ð Þ and therefore a 2D semiconductor is no longer two- where n is the 2D carrier density in the system. Unless dimensional at high enough carrier density when higher otherwise stated, we will mostly consider electron systems subbands get populated. (or the conduction band side of MLG and BLG). Typical 9 12 2 The effective mass mà is known from band structure experimental values of n 10 to 5 10 cmÀ are achiev- calculations, and within the effective mass approximation able in graphene and Si-MOSFETs;  whereas, in GaAs-based 9 11 2 mà 0:07me (electrons in GaAs), mà 0:19me (electrons 2DEG systems n 10 to 5 10 cmÀ . in Si¼ 100 inversion layers), m 0:38m¼ (holes in GaAs),  à ¼ e and mà 0:92me (electrons in Si 111 inversion layers). In 1. Interaction parameter rs some situations,¼ e.g., Si 111, the 2D effective mass entering the dispersion relation may have anisotropy in the x-y plane The interaction parameter—also known as the Wigner- Seitz radius, the coupling constant, or the effective fine- and a suitably averaged mà pmxmy is usually used. ¼ structure constant—is denoted here by r , which in this The 2D semiconductor wave function is nonchiral, and is s ffiffiffiffiffiffiffiffiffiffiffiffi context is the ratio of the average interelectron Coulomb derived from the effective mass approximation to be interaction energy to the Fermi energy. Noting that the aver- age Coulomb energy is simply V e2= r , where r iq r 1=2 h i¼ h i h i¼ È r;z e Á z ; (1.8) n À is the average interparticle separation in a 2D ð Þ ð Þ systemð Þ with n particles per unit area, and is the background 0 dielectric constant, we obtain rs n for MLG and rs where q and r are the 2D wave vector and position, and z 1=2 ð Þ nÀ for BLG and 2DEG. is the quantum confinement wave function in the z^ direction A note of caution about the nomenclature is in order here, for the lowest subband. The confinement wave function particularly since we have kept the degeneracy factors gsgv in defines the width or thickness of the 2D semiconductor state the definition of the interaction parameter. Putting gsgv 4, with z z2 1=2. The detailed form for z usually ¼ h i¼jh j j ij ð Þ the usual case for MLG, BLG, and Si 100 2DEG, and requires a quantum-mechanical self-consistent local density 2 gsgv 2 for GaAs 2DEG, we get rs e = ℏvF (MLG), approximation calculation using the confinement potential, ¼ 2 2 ¼ ð Þ rs 2me = ℏ pn (BLG and Si 100 2DEG), and rs and we refer the interested reader to the extensive existing me¼2= ℏ2pð n (GaAsÞ 2DEG). The traditional definition of¼ ffiffiffiffiffiffiffi literature for the details on the confined quasi-2D subband the Wigner-Seitzð Þ radius for a metallic Fermi liquid is the ffiffiffiffiffiffiffi structure calculations (Ando et al., 1982; Stern and Das dimensionless ratio of the average interparticle separation to Sarma, 1984; Bastard, 1991; Davies, 1998). 2 2 the effective Bohr radius aB ℏ = me . This gives for the Finally, we note that 2D semiconductors may also in some WS ¼ 2 ð 2 Þ Wigner-Seitz radius rs me = ℏ pn (2DEG and situations carry an additional valley quantum number similar BLG), which differs from¼ the definitionð ofÞ the interaction to graphene. But the valley degeneracy in semiconductor ffiffiffiffiffiffiffi parameter rs by the degeneracy factor gsgv=2. We emphasize structures, e.g., Si-MOSFET 2D electron systems, have noth- that the Wigner-Seitz radius from the above definition is ing whatsoever to do with a pseudospin chiral index. For Si meaningless for MLG, because the low energy linear disper- inversion layers, the valley degeneracy (g 2, 4, and 6, v ¼ sion implies a zero effective mass (or more correctly the respectively, for Si 100, 110, and 111 surfaces) arises from concept of an effective mass for MLG does not apply). For the bulk indirect band structure of Si which has 6 equivalent MLG, therefore, an alternative definition widely used in the ellipsoidal conduction band minima along the 100, 110, and literature defines an effective fine-structure constant ( ) as 111 directions about 85% to the Brillouin zone edge. The the coupling constant e2= ℏv , which differs from the ¼ ð FÞ valley degeneracy in Si MOSFETs, which is invariably definition of r by the factor pg g =2. Putting pg g 2 for s s v s v ¼ slightly lifted ( 0:1 meV), is a well established experimen- MLG gives the interaction parameter r equal to the effective tal fact. ffiffiffiffiffiffiffiffiffiffi s ffiffiffiffiffiffiffiffiffiffi fine-structure constant , just as setting gsgv 2 for GaAs 2DEG gave the interaction parameter equal to¼ the Wigner- C. Elementary electronic properties Seitz radius. Whether the definition of the interaction pa- rameter should or should not contain the degeneracy factor is We describe, summarize, and critically contrast the a matter of taste and has been discussed in the literature in the elementary electronic properties of graphene and 2D context of 2D semiconductor systems (Das Sarma et al., semiconductor-based electron gas systems based on their 2009). long-wavelength effective 2D energy dispersion discussed A truly significant aspect of the monolayer graphene in- in the earlier sections (see Table II). Except where the context teraction parameter, which follows directly from its equiva- is obvious, we abbreviate the following from now on: MLG, lence with the fine-structure constant definition, is that it is a BLG, and semiconductor-based 2D electron gas systems carrier density independent constant, unlike the rs parameter (2DEG). The valley degeneracy factors are typically gv 2 for the 2DEG (or BLG), which increases with decreasing for graphene and Si 100 based 2DEGs, whereas g 1 6¼for carrier density as n 1=2. In particular, the interaction parame- v ¼ ð Þ À 2DEGs in GaAs (Si 111). The spin degeneracy is always ter for MLG is bounded, i.e., 0 rs & 2:2, since 1 , 8 1 gs 2, except at high magnetic fields. The Fermi wave and as discussed earlier, vF 10 cm=s is set by the carbon vector¼ for all 2D systems is given simply by filling up the hopping parameters and lattice spacing. This is in sharp noninteracting momentum eigenstates up to q k : contrast to 2DEG systems where r 13 (for electrons in ¼ F s Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 414 Das Sarma et al.: Electronic transport in two-dimensional graphene 9 2 GaAs with n 10 cmÀ ) and rs 50 (for holes in GaAs We point out two important features of the simple screen- 9 2 with n 2 10 cmÀ ) have been reported (Das Sarma ing considerations described above: (i) In MLG, qs being a et al., 2005 ;ÂHuang et al., 2006; Manfra et al., 2007). constant implies that the screened Coulomb interaction has Monolayer graphene is thus, by comparison, always a exactly the same behavior as the unscreened bare Coulomb fairly weakly interacting system, while bilayer graphene interaction. The bare 2D Coulomb interaction in a back- could become a strongly interacting system at low carrier ground with dielectric constant is given by v q density. We point out, however, that the real low-density 2e2= q and the corresponding long-wavelength screenedð Þ¼ ð Þ 2 regime in graphene (both MLG and BLG) is dominated interaction is given by u q 2e = q qTF . Putting q ð Þ¼ ð þ Þ 1 ¼ entirely by disorder in currently available samples, and there- kF in the above equation, we get u q kF qTF À 10 2 1 1 1 ð Þð þ Þ fore a homogeneous carrier density of n & 10 cmÀ kFÀ 1 qTF=kF À kFÀ for MLG. Thus, in MLG, the func- 9 2 ð þ Þ (10 cmÀ ) is unlikely to be accessible for gated (suspended) tional dependence of the screened Coulomb scattering on the samples in the near future. Using the BLG effective mass carrier density is exactly the same as unscreened Coulomb m 0:03me, we get the interaction parameter for BLG: rs scattering, a most peculiar phenomenon arising from the 68:¼5= pn~ , where n~ n=1010 cm 2. For comparison, the Dirac linear dispersion. (ii) In BLG (but not MLG, see above) ð Þ ¼ À rs parameters for GaAs 2DEG ( 13, mà 0:67me) and Si and in 2DEG, the effective screening becomes stronger as the ffiffiffi ¼ ¼ 1=2 100 on SiO ( 7:7, mà 0:19m , g 2) are r carrier density decreases since q q =k n 0 2 ¼ ¼ e v ¼ s s ¼ TF F À ! 1ð Þ 4=pn~, and rs 13=pn~, respectively. as n 0 . This counterintuitive behavior of 2D screening, For the case when the substrate is SiO , which! is trueð1Þ for BLG systems also, means that in 2D systems ffiffiffi ffiffiffi 2 ¼ð SiO2 þ 1 =2 2:5 for MLG and BLG, we have r 0:8 and r effects of Coulomb scattering on transport properties in- Þ s s 27:4= pn~ , respectively. In vacuum, 1 and rs 2:2 for creases with increasing carrier density, and at very high ð Þ p ¼ density, the system behaves as an unscreened system. This MLG andffiffiffi rs 68:5= n~ for BLG. ð Þ is in sharp contrast to 3D metals where the screening effect ffiffiffi 2. Thomas-Fermi screening wave vector qTF increases monotonically with increasing electron density. Finally, in the context of graphene, it is useful to give a Screening properties of an electron gas depend on the direct comparison between screening in MLG versus screen- density of states D0 at the Fermi level. The simple Thomas- ing in BLG: qBLG=qMLG 16=pn~, showing that as carrier Fermi theory leads to the long-wavelength Thomas-Fermi TF TF density decreases, BLG screening becomes much stronger screening wave vector ffiffiffi than MLG screening. 2e2 q D : (1.10) TF ¼ 0 3. Plasmons The density independence of long-wavelength screening in Plasmons are self-sustaining normal mode oscillations of a BLG and 2DEG is the well-known consequence of the den- carrier system, arising from the long-range nature of the sity of states being a constant (independent of energy); interparticle Coulomb interaction. The plasmon modes are 1=2 defined by the zeros of the corresponding frequency and wave whereas, the property that qTF kF n in MLG is a direct consequence of the MLG density of states being linear vector dependent dynamical dielectric function. The long- in energy. wavelength plasma oscillations are essentially fixed by the A key dimensionless quantity determining the charged particle number (or current) conservation, and can be ob- impurity scattering limited transport in electronic materials tained from elementary considerations. We write down the long-wavelength plasmon dispersion ! : is qs qTF=kF which controls the dimensionless strength of p ¼ 0 quantum screening. From Table I, we have qs n for MLG 2 1=2 1=2 e vFq and qs nÀ for BLG and 2DEG. Using the usual sub- MLG: !p q 0 pngsgv ; (1.12a) stitutions g g 4 2 for Si 100 (GaAs) based 2DEG sys- ð ! Þ¼ ℏ s v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tem, and taking¼ theð Þ standard values of m and for 2ne 1=2 BLG and 2DEG: !p q 0 q : graphene-SiO2, GaAs-AlGaAs, and Si-SiO2 structures, we ð ! Þ¼ m 10 2 get (for n~ n=10 cmÀ ) (1.12b) ¼ A rather intriguing aspect of MLG plasmon dispersion is that MLG: q 3:2; BLG: q 54:8=pn~; (1.11a) s s it is nonclassical [i.e., ℏ appears explicitly in Eq. (1.12), even in the long-wavelength limit]. This explicit quantum nature of n-GaAs: qs 8=pn~;p-GaAs: qs 43=pffiffiffin~: (1.11b) long-wavelength MLG plasmon is a direct manifestation of ffiffiffi ffiffiffi TABLE I. Elementary electronic quantities. Here EF, D E , rs, and qTF represent the Fermi energy, the density of states, the interaction parameter, and theð Thomas-FermiÞ wave vector, respectively. D D E is the density of states at the Fermi energy and q q =k . 0 ¼ ð FÞ s ¼ TF F E D E D D E r q q F ð Þ 0 ¼ ð FÞ s TF s 2 2 2 4n gsgvE pgsgvn e pgsgv p4gsgvne gsgve MLG ℏvF 2 gsgv 2 ℏvF pℏvF ℏvF 2 ℏvF ℏvF 2 ð Þ 2 2 3=2 2 2qℏ ffiffiffiffiffiffiffiffin gsgvm gffiffiffiffiffiffiffiffiffiffisgvm me gffiffiffiffiffiffiffiffisgv gffiffiffiffiffiffiffiffiffiffiffiffiffiffisgvme gsgv me BLG and 2DEG 2 2 2 2 ð 2Þ mgsgv 2ℏ 2ffiffiffiℏ 2ℏ pn ℏ ℏ p4n ffiffiffiffiffi ffiffiffiffiffiffiffi Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 415 its linear Dirac-like energy-momentum dispersion, which has lar to the 2D plane leading to the Landau orbital quantization no classical analogy (Das Sarma and Hwang, 2009). of the system. 4. Magnetic field effects a. Landau level energetics Although magnetic field-induced phenomena in graphene The application of a strong perpendicular external and 2D semiconductors [e.g., quantum Hall (QH) effect and magnetic field (B) leads to a complete quantization of the fractional quantum Hall effect] are briefly covered in Sec. V, orbital carrier dynamics of all 2D systems leading to the we mention at this point a few elementary electronic proper- following quantized energy levels En, the so-called Landau ties in the presence of an external magnetic field perpendicu- levels: MLG: E sgn n v 2eℏB n ; with n 0; 1; 2; ...; (1.13a) n ¼ ð Þ F j j ¼ Æ Æ sgn n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi BLG: E ð Þ 2 n 1 2eBv2 ℏ 4m2v4 2mv2 4 2 2 n 1 2eBv2 ℏ 2mv2 2 2eBv2 ℏ 2 ; n ¼ p ½ð j jþ Þð F Þþ F À ð FÞ þ ð j jþ Þð F Þð FÞ þð F Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi withffiffiffin 0; 1; 2; ...; (1.13b) ¼ Æ Æ eBℏ 2DEG: En n 1=2 ; with n 0; 1; 2; ... (1.13c) ¼ð þ Þ mc ¼ The hallmark of the Dirac nature of graphene is the exis- the interested reader to the recent literature on the subject tence of a true zero-energy [n 0 in Eq. (1.13a)] Landau (Henriksen et al., 2010; Shizuya, 2010. level, which is equally shared by¼ electrons and holes. The experimental verification of this zero-energy Landu level c. Zeeman splitting: in graphene is definitive evidence for the long-wavelength In graphene, the spin splitting can be large since the Lande´ Dirac nature of the system (Miller et al., 2009; Novoselov, g factor in graphene is the same (g 2) as in vacuum. The ¼ Geim et al., 2005; Zhang et al., 2005). Zeeman splitting in an external magnetic field is given by (B is the Bohr magneton) E g B 0:12B T meV, for z ¼ B ¼ ½ b. Cyclotron resonance g 2 (MLG, BLG, Si 2DEG) and Ez 0:03B T meV for¼g 0:44 (GaAs 2DEG). We note¼À that the½ relative External radiation induced transitions between Landau ¼À levels give rise to the cyclotron resonance in a Landau value of Ez=EF is rather small in graphene, Ez=EF 0:01 B T =pn~ 0:01 for B 10 T and n 1012 cm 2. quantized system, which has been extensively studied in 2D ð ½ Þ! ¼ ¼ À semiconductor (Ando et al., 1982) and graphene systems Thus, the spinffiffiffi splitting is only 1% even at high fields. Of (Jiang et al., 2007; Henriksen et al., 2008, 2010). The course, the polarization effect is stronger at low carrier cyclotron resonance frequency in MLG and 2DEG is given by densities, since EF is smaller. D. Intrinsic and extrinsic graphene MLG: ! v p2eℏB pn 1 pn ; (1.14a) c ¼ F ð þ À Þ eB ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi 2DEG: ! : ffiffiffi (1.14b) It is important to distinguish between intrinsic and extrin- c ¼ mc sic graphene because gapless graphene (either MLG or BLG) has a charge neutrality point (CNP), i.e., the Dirac point, For BLG, the cyclotron frequency should smoothly interpo- where its character changes from being electronlike to being late from the formula for MLG for very large n, so that En in holelike. Such a distinction is not meaningful for a 2DEG (or 2 Eq. (1.13) is much larger than 2mvF, to that of the 2DEG for BLG with a large gap) since the intrinsic system is simply an 2 small n so that En 2mvF (where m 0:033 is the ap- undoped system with no carriers (and as such is uninteresting proximate B 0 effective mass of the bilayer parabolic-band from the electronic transport properties perspective). dispersion). Experimental¼ BLG cyclotron resonance studies In monolayer and bilayer graphene, the ability to gate (or (Henriksen et al., 2010) indicate the crossover from the dope) the system by putting carriers into the conduction or quadratic band dispersion (i.e., 2DEG-like) for smaller q to valence band by tuning an external gate voltage enables one the linear band dispersion (i.e., MLG-like) at larger q seems to pass through the CNP where the chemical potential (EF) to happen at lower values of q than that implied by simple resides precisely at the Dirac point. This system, with no free band theory considerations. carriers at T 0, and EF precisely at the Dirac point is called A particularly interesting and important feature of cyclo- intrinsic graphene¼ with a completely filled (empty) valence tron resonance in graphene is that it is affected by electron- (conduction) band. Any infinitesimal doping (or, for that electron interaction effects unlike the usual parabolic 2DEG, matter, any finite temperature) makes the system ‘‘extrinsic’’ where the existence of Kohn’s theorem prevents the long- with electrons (holes) present in the conduction (valence) wavelength cyclotron frequency from being renormalized by band (Mu¨ller et al., 2009). Although the intrinsic system is electron-electron interactions (Kohn, 1961; Ando et al., a set of measure zero (since EF has to be precisely at the 1982). For further discussion of this important topic, we refer Dirac point), the routine experimental ability to tune the Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 416 Das Sarma et al.: Electronic transport in two-dimensional graphene TABLE II. Electronic quantities for monolayer graphene. Note that the graphene Fermi velocity 8 (vF 10 cm=s) and the degeneracy factor g gsgv 4, i.e., the usual spin degeneracy (gs 2) ¼ ¼ ¼ 10 2 ¼ and a valley degeneracy (gv 2), are used in this table. Here n~ n= 10 cmÀ , and B, q, and are measured in T, cm 1, and¼e2=h 38:74 S (or h=e2 25:8¼ k ),ð respectively.Þ À ¼ ¼ Quantity Scale values 5 1 1:77 10 pn~ cmÀ Fermi wave vector (kF)  6 ½ 1 Thomas-Fermi wave vector (qTF) 1:55 10 pn~ffiffiffi= cmÀ  ½ Interaction parameter (rs) 2:19ffiffiffi= 9 1 2 DOS at EF [D0 D EF ] 1:71 10 pn~ meV cm ð Þ Â ½ À À Fermi energy (EF) 11:65pn~ meV ffiffiffi ½ Zeeman splitting (Ez) 0:12BffiffiffimeV Cyclotron frequency (! ) 5:51 1013½pB s 1 c  ½ À Landau level energy (En) sgn l 36:29 B l meV , l 0; 1; 2; ... ð Þ j j ½ ffiffiffiffi ¼ Æ Æ pffiffiffiffiffiffiffiffi 2 Plasma frequency (!p q ) 5:80 10À pn~q= meV ð Þ Â 4 2½ Mobility () 2:42 10 q=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin~ffiffiffi cm =Vs  14 ½ Scattering time ( ) 2:83 10À =pn~ s Level broadening (À) 11:Â63pn~= meV½ ½ ffiffiffi ffiffiffi system from being electronlike to to being holelike by chang- is 300 nm thick, and therefore quantum-capacitance effects ing the external gate voltage, manifestly establishes that one are completely negligible. In this case, a simple capacitance must be going through the intrinsic system at the CNP. If model connects the 2D carrier density (n) with the applied there is an insulating regime in between, as there would be for external gate voltage Vg, n CVg, where C 7:2 10 2  a gapped system, then intrinsic graphene is not being 10 cmÀ =V for graphene on SiO2 with roughly 300 nm accessed. thickness. This approximate value of the constant C seems to Although it is not often emphasized, the achievement of be pretty accurate, and the following scaling should provide n Novoselov et al. (2004) in producing 2D graphene in the for different dielectrics: laboratory is not just fabricating (Novoselov, Jiang et al., 2005) and identifying (Ferrari et al., 2006; Ferrari, 2007) 10 2 t nm n 10 cmÀ 7:2 ½ Vg V ; (1.16) stable monolayers of graphene flakes on substrates, but also ½ ¼  300 3:9 ½ establishing its transport properties by gating the graphene where t is the thickness of the dielectric (i.e., the distance device using an external gate, which allows one to simply from the gate to the graphene layer) and is the dielectric tune an external gate voltage and thereby continuously con- constant of the insulating substrate. trolling the 2D graphene carrier density as well as their nature It is best, therefore, to think of 2D graphene on SiO2 [see (electron or hole). If all that could be done in the laboratory Fig. 1(e)] as a metal-oxide-graphene-field-effect-transistor was to produce 2D graphene flakes, with no hope of doping or similar to the well-known Si-MOSFET structure, with Si gating them with carriers, then the subject of graphene would replaced by graphene where the carriers reside. In fact, this be many orders of magnitude smaller and less interesting. analogy between graphene and Si 100 inversion layer is What led to the exponential growth in graphene literature is operationally quite effective: Both have the degeneracy factor the discovery of gatable and density tunable 2D graphene in gsgv 4 and both typically have SiO2 as the gate oxide 2004. layer.¼ The qualitative and crucial difference is, of course, Taking into account the quantum capacitance in graphene, that graphene carriers are chiral, massless, with linear disper- the doping induced by the external gate voltage Vg is given by sion and with no band gap, so that the gate allows one to go the following relation (Fang et al., 2007; Fernandez-Rossier directly from being n-type to a p-type carrier system through et al., 2007): the charge neutral Dirac point. Thus, a graphene metal-oxide- graphene-field-effect-transistor is not a transistor at all (at CVg CVg least for MLG), since the system never becomes insulating at n nQ 1 1 ; (1.15) ¼ e þ À sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ enQ any gate voltage (Avouris et al., 2007). We will distinguish between extrinsic (i.e., doped) gra- where C is the gate capacitance, e the absolute value of the phene with free carriers and intrinsic (i.e., undoped) graphene 2 2 electron charge, and nQ =2 CℏvF=e . The second with the chemical potential precisely at the Dirac point. All ð Þð Þ term on the right-hand side (r.h.s.) of (1.15) is analogous to experimental systems (since they are always at T Þ 0) are the term due to the so-called quantum capacitance in regular necessarily extrinsic, but intrinsic graphene is of theoretical 2DEG. Note that in graphene, due to the linear dispersion, importance since it is a critical point. In particular, intrinsic contrary to parabolic 2D electron liquids, the quantum ca- graphene is a non-Fermi liquid in the presence of electron- pacitance depends on Vg. For a background dielectric con- electron interactions (Das Sarma, Hwang, and Tse, 2007), stant 4 and gate voltages larger than few millivolts, the while extrinsic graphene is a Fermi liquid. Since the non- second term on the r.h.s. of (1.15) can be neglected for Fermi-liquid fixed point for intrinsic graphene is unstable to thicknesses of the dielectric larger than few angstroms. In the presence of any finite carrier density, the non-Fermi- current experiments on exfoliated graphene on SiO2 the oxide liquid nature of this fixed point is unlikely to have any Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 417 experimental implication. But it is important to keep this non- e2 g g : (1.17b) Fermi-liquid nature of intrinsic graphene in mind when dis- min ¼ s v h cussing graphene’s electronic properties. We also mention (see Sec. IV) that disorder, particularly long-ranged disorder These T 0 results apply to intrinsic graphene, where EF is induced by random charged impurities present in the environ- precisely¼ at the Dirac point. The crossover between these ment, is a relevant strong perturbation affecting the critical two theoretical intrinsic limits remains an open problem Dirac point, since the system breaks up into spatially random (Katsnelson, 2006; Ostrovsky et al., 2006). electron-hole puddles, thus masking its zero-density intrinsic The optical conductivity [Eq. (1.17a)] has been measured nature. experimentally both by infrared spectroscopy (Li et al., 2008) and by measuring the absorption of suspended gra- phene sheets (Nair et al., 2008). In the IR measurements, E. Other topics ! is close to the predicted universal value for a range of ð Þ 1 There are several topics that are of active current research frequencies 4000 Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 418 Das Sarma et al.: Electronic transport in two-dimensional graphene low energy effective theory) gave quantitatively similar re- ribbon widths would then be explained by a dimensional sults for the energy gaps as the tight-binding calculation, crossover as the ribbon width became comparable to the while Son, et al. (2006a) showed that the density functional puddle size. A numerical study including the effect of quan- results could be obtained from the tight-binding model with tum localization and edge disorder was done by Mucciolo some added edge disorder. By considering arbitrary boundary et al. (2009) who found that a few atomic layers of edge conditions, Akhmerov and Beenakker (2008) demonstrated roughness were sufficient to induce transport gaps to appear, that the behavior of the zigzag edge is the most generic for which are approximately inversely proportional to the nano- graphene nanoribbons. These theoretical works gave a simple ribbon width. Two recent and detailed experiments way to understand the gap in graphene nanoribbons. (Gallagher et al., 2010; Han et al., 2010) seem to suggest The first experiments on graphene nanoribbons (Han et al., that a combination of these pictures might be at play (e.g., 2007), however, presented quite unexpected results. As transport through quantum dots that are created by the shown in Fig. 5 the transport gap for narrow ribbons is charged impurity potential), although as of now, a complete much larger than that predicted by theory (with the gap theoretical understanding remains elusive. The phenomenon diverging at widths of 15 nm), while wider ribbons have that the measured transport gap is much smaller than the a much smaller gap than expected. Surprisingly, the gap theoretical band gap seems to be a generic feature in gra- showed no dependence on the orientation (i.e., zigzag or phene, occurring not only in nanoribbons but also in biased armchair direction) as required by the theory. These discrep- bilayer graphene where the gap measured in transport experi- ancies have prompted several studies (Areshkin et al., 2007; ments appears to be substantially smaller than the theoreti- Chen et al., 2007; Sols et al., 2007; Abanin and Levitov, cally calculated, band gap (Oostinga et al., 2008) or even the 2008; Adam, Cho et al., 2008; Basu et al., 2008; Biel, Blase measured optical gap (Mak et al., 2009; Zhang, Tang et al., et al., 2009; Biel, Triozon et al., 2009; Dietl et al., 2009; 2009). Martin and Blanter, 2009; Stampfer et al., 2009; Todd et al., 2009). In particular, Sols et al. (2007) argued that fabrication 3. Suspended graphene of the nanoribbons gave rise to very rough edges breaking the nanoribbon into a series of quantum dots. Coulomb blockade Since the substrate affects both the morphology of gra- of charge transfer between the dots (Ponomarenko et al., phene (Ishigami et al., 2007; Meyer et al., 2007; Stolyarova 2008) explains the larger gaps for smaller ribbon widths. In a et al., 2007) and provides a source of impurities, it became similar spirit, Martin and Blanter (2009) showed that edge clear that one needed to find a way to have electrically disorder qualitatively changed the picture from that of the contacted graphene without the presence of the underlying disorder-free picture presented earlier, giving a localization substrate. The making of ‘‘suspended graphene’’ or length comparable to the sample width. For larger ribbons, ‘‘substrate-free’’ graphene was an important experimental Adam, Cho et al. (2008) argued that charged impurities in the milestone (Bolotin, Sikes, Jiang et al., 2008; Bolotin, vicinity of the graphene would give rise to inhomogeneous Sikes, Hone et al., 2008; Du et al., 2008) where after puddles so that the transport would be governed by percola- exfoliating graphene and making electrical contact, one tion [as shown in Fig. 5, the points are experimental data, and then etches away the substrate underneath the graphene so the solid lines, for both electrons and holes, show fits to that the graphene is suspended over a trench that is approxi- V Vc , where is close to 4=3, the theoretically expected mately 100 nm deep. As a historical note, we mention that ðvalueÀ forÞ percolation in 2D systems]. The large gap for small suspended graphene without electrical contacts was made FIG. 5 (color online). (a) Graphene nanoribbon energy gaps as a function of width. Adapted from Han et al., 2007. Four devices (P1–P4) were orientated parallel to each other with varying width, while two devices (D1–D2) were oriented along different crystallographic directions with uniform width. The dashed line is a fit to a phenomenological model with E A= W W where A and W are fit g ¼ ð À ÃÞ Ã parameters. The inset shows that contrary to predictions, the energy gaps have no dependence on crystallographic direction. The dashed lines are the same fits as in the main panel. (b) Evidence for a percolation metal-insulator transition in graphene nanoribbons. Adapted from Adam, Cho et al., 2008. Main panel shows graphene ribbon conductance as a function of gate voltage. Solid lines are a fit to percolation theory, where electrons and holes have different percolation thresholds (seen as separate critical gate voltages Vc). The inset shows the same data in a linear scale, where even by eye the transition from high-density Boltzmann behavior to the low-density percolation transport is visible. Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 419 earlier by Meyer et al. (2007). Quite surprisingly, the sus- 5. Topological insulators pended samples as prepared did not show much difference There is a deep connection between graphene and topo- from unsuspended graphene, until after current annealing logical insulators (Kane and Mele, 2005a; Sinitsyn et al., (Moser et al., 2007; Barreiro et al., 2009). This suggested 2006). Graphene has a Dirac cone where the ‘‘spin’’ degree of that most of impurities limiting the transport properties of freedom is actually related to the sublattices in real space; graphene were stuck to the graphene sheet and not buried in whereas, it is the real electron spin that provides the Dirac the substrate. After removing these impurities by driving a structure in the topological insulators (Hasan and Kane, large current through the sheet, the suspended graphene 2010) on the surface of BiSb and BiTe (Hsieh et al., 2008; samples showed both ballistic and diffusive carrier transport Chen et al., 2009). Graphene is a weak topological insulator properties. Away from the charge neutrality point, suspended because it has two Dirac cones (by contrast, a strong topo- graphene showed near-ballistic transport over hundreds of logical insulator is characterized by a single Dirac cone on nm, which prompted much theoretical interest (Adam and each surface), but in practice the two cones in graphene are Das Sarma, 2008b; Fogler, Guinea, and Katsnelson, 2008; mostly decoupled and it behaves like two copies of a single Stauber, Peres, and Neto, et al., 2008; Mu¨ller et al., 2009). Dirac cone. Therefore, many of the results presented in this One problem with suspended graphene is that only a small review, although intended for graphene, should also be rele- gate voltage (V 5V) could be applied before the graphene g vant for the single Dirac cone on the surface of a topological buckles due to the electrostatic attraction between the charges insulator. In particular, we expect the interface transport in the gate and on the graphene sheet, and binds to the bottom properties of topological insulators to be similar to the phys- of the trench that was etched out of the substrate. This is in ics described in this review as long as the bulk is a true gapped contrast to graphene on a substrate that can support as insulator. much as Vg 100 V and a corresponding carrier density of 13 2 10 cmÀ . To avoid the warping, it was proposed that one should use a top gate with the opposite polarity, but currently, F. 2D nature of graphene this has yet to be demonstrated experimentally. Despite the As the concluding section of the Introduction, we ask the limited variation in carrier density, suspended graphene has following: what precisely is meant when an electronic system achieved a carrier mobility of more than 200 000 cm2=Vs is categorized as 2D, and how can one ensure that a specific (Bolotin, Sikes, Jiang et al., 2008; Bolotin, Sikes, Hone sample or system is 2D from the perspective of electronic et al., 2008; Du et al., 2008). Recently suspended graphene transport phenomena? bilayers were demonstrated experimentally (Feldman et al., The question is not simply academic, since 2D does not 2009). necessarily mean a thin film (unless the film is literally one atomic monolayer thick as in graphene, and even then, one 4. Many-body effects in graphene must consider the possibility of the electronic wave function The topic of many-body effects in graphene is itself a large extending somewhat into the third direction). Also, the defi- subject, and one that we could not cover in this transport nition of what constitutes a 2D may depend on the physical review. As discussed earlier, for intrinsic graphene the many- properties or phenomena that one is considering. For ex- body ground state is not even a Fermi liquid (Das Sarma, ample, for the purpose of quantum localization phenomena, Hwang, and Tse, 2007), an indication of the strong role the system dimensionality is determined by the width of the played by interaction effects. Experimentally, one can ob- system being smaller than the phase coherence length L (or serve the signature of many-body effects in the compressi- the Thouless length). Since L could be very large at low bility (Martin et al., 2007) and using angle resolved temperature, metal films and wires can, respectively, be photoemission spectroscopy (ARPES) (Bostwick et al., considered 2D and 1D for localization studies at ultralow 2007; Zhou et al., 2007). Away from the Dirac point, where temperature. For our purpose, however, dimensionality is graphene behaves as a normal Fermi liquid, the calculation of defined by the 3D electronic wave function as ‘‘free’’ the electron-electron and electron-phonon contribution to the plane-wave–like (i.e., carrying a conserved 2D wave vector) quasiparticle self-energy was studied by several groups in a 2D plane, while it is a quantized bound state in the third (Barlas et al., 2007, Calandra and Mauri, 2007, E. H. dimension. This ensures that the system is quantum mechani- Hwang et al., 2007a, 2007b; Park et al., 2007, 2009; cally 2D. Polini et al., 2007; Tse and Das Sarma, 2007; Hwang and Considering a thin film of infinite (i.e., very large) dimen- Das Sarma, 2008c; Polini, Asgari et al., 2008; Carbotte sion in the x-y plane and a finite thickness w in the z direction, et al., 2010), and shows reasonable agreement with experi- where w could be the typical confinement width of a potential ments (Bostwick et al., 2007; Brar et al., 2010). For both well creating the film, the system is considered 2D if bilayer graphene (Min, Borghi et al., 2008) and double-layer F 2=kF >w. For graphene, we have F graphene (Min, Bistritzer et al., 2008), an instability towards 350¼=pn~ nm, where n~ n= 1010 cm 2 , and since w ð Þ ¼ ð À Þ an excitonic condensate has been proposed. In general, mono- 0:1 to 0.2ffiffiffi nm (the monolayer atomic thickness), the condition layer graphene is a weakly interacting system since the F w is always satisfied, even for unphysically large 14 2 coupling constant (rs 2) is never large (Muller et al., n 10 cmÀ . 2009). In principle, bilayer graphene could have arbitrarily ¼Conversely, it is essentially impossible to create 2D elec- large coupling at low carrier density where disorder effects tronic systems from thin metal films since the very high are also important. We refer the interested reader to these electron density of metals provides F 0:1 nm, so that works for details on this subject. even for a thickness of w 1 nm (the thinnest metal film Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 420 Das Sarma et al.: Electronic transport in two-dimensional graphene that one can make), F Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 421 noninteracting electrons in graphene. Early studies on the defines the p-n junction (characterized by a length scale ), quantum-mechanical properties of the Dirac Hamiltonian the transmission probability was shown by Cheianov and revealed a peculiar feature—Dirac carriers could not be Fal’ko (2006b) to be T exp k sin2 . This ð Þ¼ ½À ð F Þ Þ confined by electrostatic potentials (Klein, 1929). implies that for both sharp and smooth potential barriers, a An electron facing such a barrier would transmute into a wave packet of Dirac fermions will collimate in a direction hole and propagate through the barrier. In Sec. II.B we study perpendicular to the p-n junction. One can estimate the Klein tunneling of Dirac carriers and discuss how this formal- conductance of a single p-n junction (of width W) to be ism can be used to obtain graphene’s ballistic universal 2 2 minimum conductivity. There is no analog of this type of 4e d kF 1 2e k F Gp-n kFW T W: (2.1) quantum-limited transport regime in two-dimensional semi- ¼ h ð Þ 2 ð Þ! h sffiffiffiffiffiffi conductors. The ‘‘metallic nature’’ of graphene gives rise to Z several interesting and unique properties that we explore in Although the conductance of smooth p-n junctions are this section, including the absence of Anderson localization smaller by a factor of pkF compared to sharp ones, this result suggests that the presence of p-n junctions would make for Dirac electrons and a metal-insulator transition induced ffiffiffiffiffiffiffiffiffi by atomically sharp disorder (such as dislocations). We note a small contribution to the overall resistivity of a graphene that many of the results in this section can be also obtained sample (see also Sec. IV.D), i.e., graphene p-n junctions are using field-theoretic methods (Fradkin, 1986; Ludwig et al., essentially transparent. 1994; Altland, 2006; Ostrovsky et al., 2006; Ryu, Mudry, The experimental realization of p-n junctions came shortly Obuse, and Furusaki, 2007; Fritz et al., 2008; Schuessler after the theoretical predictions (Huard et al., 2007; Lemme ¨ et al., 2009). et al., 2007; Ozyilmaz et al., 2007; Williams et al., 2007). At zero magnetic field, the effect of creating a p-n junction was to modestly change the device resistance. More dramatic was B. Ballistic transport the change at high magnetic field in the quantum Hall regime (see Sec. V). 1. Klein tunneling More detailed calculations of the zero-field conductance of In classical mechanics, a potential barrier, whose height is the p-n junction were performed by taking into account the greater than the energy of a particle, will confine that particle. effect of nonlinear electronic screening. This tends to make In quantum mechanics, the notion of quantum tunneling the p-n junction sharper, and for rs 1, increases the con- 1=6 describes the process whereby the wave function of a non- ductance by a factor rs (Zhang and Fogler, 2008), and relativistic particle can leak out into the classically forbidden thereby further reduces the overall contribution of p-n junc- region. However, the transmission through such a potential tions to the total resistance. The effect of disorder was barrier decreases exponentially with the height and width of examined by Fogler, Novikov et al. (2008) who studied the barrier. For Dirac particles, the transmission probability how the p-n junction resistance changed from its ballistic depends only weakly on the barrier height, approaching unity value in the absence of disorder to the diffusive limit with with increasing barrier height (Katsnelson et al., 2006). One strong disorder. More recently, Rossi, Bardarson, Brouwer, can understand this effect by realizing that the Dirac and Das Sarma (2010) used a microscopic model of charged Hamiltonian allows for both positive energy states (called impurities to calculate the screened disorder potential and electrons) and negative energy states (called holes). While a solved for the conductance of such a disordered n-p-n junc- positive potential barrier is repulsive for electrons, it is tion numerically. attractive for holes (and vice versa). For any potential barrier, The broad oscillations visible in Fig. 6 arise from resonant one needs to match the electron states outside the barrier with tunneling of the few modes with the smallest transverse the hole states inside the barrier. Since the larger the barrier momentum. These results demonstrate that the signatures of is, the greater the mode matching between electron and hole the Klein tunneling are observable for impurity densities as 12 2 states is, the transmission is also greater. For an infinite high as 10 cmÀ and would not be washed away by disorder barrier, the transmission becomes perfect. This is called as long as the impurity limited mean free path is greater than Klein tunneling (Klein, 1929). length of the middle region of the opposite polarity. This By solving the transmission and reflection coefficients for implies that at zero magnetic field, the effects of Klein both the graphene p-n junction (Cheianov and Fal’ko, 2006b; tunneling are best seen with a very narrow top gate. Indeed, Low and Appenzeller, 2009) and the p-n-p junction recent experiments have succeeded in using an ‘‘air bridge’’ (Katsnelson et al., 2006), it was found that for graphene (Gorbachev et al., 2008; Liu et al., 2008) or very narrow top the transmission at an angle normal to the barrier was always gates (Stander et al., 2009; Young and Kim, 2009). The perfect (although there could be some reflection at other observed oscillations in the conductivity about the semiclas- angles). This can be understood in terms of pseudospin sical value are in good agreement with the theory of Rossi, conservation. At normal incidence, the incoming electron Bardarson, Brouwer, and Das Sarma (2010). state and the reflected electron state are of opposite chirality, There is a strong similarity between the physics of phase- resulting in vanishing probability for reflection. coherent ballistic trajectories of electrons and that of light At finite angles of incidence, the transmission depends on waves which is often exploited (Ji et al., 2003; Cheianov, the sharpness of the barrier. In the limit of a perfectly sharp Fal’ko, and Altshuler, 2007; Shytov et al., 2008). In particu- step, the transmission probability is determined only by lar, Liang et al. (2001) demonstrated that one could construct pseudospin conservation and is given by T cos2. a Fabry-Pe´rot resonator of electrons in a carbon nanotube. stepð Þ¼ For a smooth variation in the electrostatic potential that This relies on the interference between electron paths in the Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 422 Das Sarma et al.: Electronic transport in two-dimensional graphene (a) (b) 0.16 ] 2 0.12 R [h/e 0.08 0.04 -3 -2 -1 0 1 2 3 12 -2 ∆ntg [10 cm ] 11 2 FIG. 6 (color online). (a) Disorder averaged resistance as a function of top gate voltage for a fixed back gate density nbg 5 10 cmÀ and several values of the impurity density (from bottom to top n 0, 1, 2.5, 5, 10, and 15 1011 cm 2). Results were obtained¼  using 103 imp ¼  À disorder realizations for a square samples of size W L 160 nm in the presence of a top gate placed in the middle of the sample 10 nm above the graphene layer, 30 nm long, and of width W¼. The¼ charge impurities were assumed at a distance d 1 nm and the uniform dielectric constant was taken equal to 2.5. Adapted from Rossi, Bardarson, Brouwer, and Das Sarma, 2010. (b) Solid¼ line is Eq. (2.4) for armchair boundary conditions showing the aspect ratio dependence of the Dirac point ballistic conductivity (Tworzydło et al., 2006). For W L, the theory approaches the universal value 4e2=h. Circles show experimental data taken from Miao et al. (2007), and squares show the data from Danneau et al. (2008). Inset: Illustration of the configuration used to calculate graphene’s universal minimum conductivity. For Vg > 0, one has a p-p-p junction, while for Vg < 0, one has a p-n-p junction. This illustrates the ballistic universal conductivity that occurs at the transition between the p-p-p and p-n-p junctions when V 0. g ¼ different valleys K and K0. The same physics has been The solution is obtained by finding the transmission prob- observed in ‘‘ballistic’’ graphene, where the device geometry abilities and obtaining the corresponding ballistic conductiv- is constructed such that the source and drain electrodes are ity. This is analogous to the quantum mechanics exercise of closer than the typical electronic mean free path (Miao et al., computing the transmission through a potential barrier, but 2007; Cho and Fuhrer, 2009). now instead for relativistic electrons. Using the noninteract- ing Dirac equation 2. Universal quantum-limited conductivity An important development in the understanding of gra- ℏvF k eV x É r "É r ; (2.2) ½ Á þ ð Þ ð Þ¼ ð Þ phene transport is the use of the formalism of Klein tunneling to address the question of graphene’s minimum conductivity with the boundary conditions corresponding to V x<0 (Katsnelson, 2006; Tworzydło et al., 2006). This of course V x>L V to represent the heavily dopedð leads andÞ¼ considers noninteracting electrons at zero temperature and in ð Þ¼ 1 V x Vg for 0 Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 423 The Landauer conductivity is then given by physical mechanism. Recent numerical studies by San-Jose 2 et al. (2007), Lewenkopf et al. (2008), and Sonin (2008, L gsgve 1 2009) treated the role of disorder to examine the crossover Tn ¼ W h n 0 from the F 1=3 in ballistic graphene to the diffusive X¼ ¼ 2 2 regime (see Sec. IV.B). Within the crossover, or away from 4e 1 L W L 4e : the Dirac point, the Fano factor is no longer universal and h Wcosh2 n 1=2 L=W h ¼ n 0 ! shows disorder dependent deviations. The experimental situ- X¼ ½ ð þ Þ (2.4) ation is less clear. Danneau et al. (2008) measured the Fano factor decrease from 1=3 with increasing carrier density Since at the Dirac point (zero energy) there is no energy scale F to claim agreement with the ballistic theory. While DiCarlo in the problem, the conductivity (if finite) can only depend on et al. (2008) found that was mostly insensitive to carrier the aspect ratio L=W. The remarkable fact is that for W L, F type and density, temperature, aspect ratio, and the presence the sum in Eq. (2.4) converges to a finite and universal of a p-n junction, suggesting diffusive transport in the dirty value—giving for ballistic minimum conductivity min limit. 4e2=h. This result also agrees with that obtained using¼ Since shot noise is, in principle, an independent probe of linear response theory in the limit of vanishing disorder, the nature of the carrier dynamics, it could be used as a suggesting that the quantum-mechanical transport through separate test of the quantum-limited transport regime. evanescent modes between source and drain (or equivalently However, in practice, the coincidence in the numerical value the transport across two p-n junctions with heavily doped of the Fano factor with that of diffusive transport regime leads) is at the heart of the physics behind the universal makes this prospect far more challenging. minimum conductivity in graphene. Miao et al. (2007) and Danneau et al. (2008) probed this ballistic limit experimentally using the two-probe geometry. C. Quantum interference effects Their results, shown in Fig. 6(b), are in good agreement with the theoretical predictions. Although it is not clear what role 1. Weak antilocalization contact resistance (Blanter and Martin, 2007; Giovannetti Over the past 50 years, there has been much progress et al., 2008; Huard et al., 2008; Lee, Balasubramanian towards understanding the physics of Anderson localization et al., 2008; Blake et al., 2009; Cayssol et al., 2009; [for a recent review, see Evers and Mirlin (2008)]. Single Golizadeh-Mojarad and Datta, 2009) played in these two- particle Hamiltonians are classified according to their global probe measurements. symmetry. Since the Dirac Hamiltonian (for a single valley) H ℏv k is invariant under the transformation H ¼ F Á ¼ 3. Shot noise yH Ãy [analogous to spin-rotation symmetry (SRS) in pseudospin space] it is in the AII class (also called the Shot noise is a type of fluctuation in electrical current symplectic Wigner-Dyson class). The more familiar physical caused by the discreteness of charge carriers and from the realization of the symplectic class is the usual disordered randomness in their arrival times at the detector or drain electron gas with strong spin-orbit coupling: electrode. It probes any temporal correlation of the electrons carrying the current, quite distinct from thermal noise (or ℏ2k2 Johnson-Nyquist noise) which probes their fluctuation in H V ; ¼ 2m þ energy. Shot noise is quantified by the dimensionless Fano so (2.6) V V iV k^ 0 k^ ; factor , defined as the ratio between noise power spectrum k; k0 k k0 k k0 F ¼ À À À ð  ÞÁ and the average conductance. Scattering theory gives where and are (real) spin indices, a vector of Pauli (Bu¨ttiker, 1990) matrices. Note that this Hamiltonian is also invariant under Tn 1 Tn SRS, H yH Ãy. We have n ð À Þ ¼ F P : (2.5) ¼ Tn q q0 so so q q0 n V V ð À Þ; V V ð À Þ: (2.7) q q0 q q0 P h i¼ 2 h i¼ 2 so Some well-known limits include F 1 for ‘‘Poisson noise’’ ¼ It was shown by Hikami et al. (1980) that when the classical when Tn 1 (e.g., in a tunnel junction), and F 1=3 for ¼ conductivity is large ( e2=h), the quantum correction to disordered metals (Beenakker and Bu¨ttiker, 1992). For gra- 0 phene at the Dirac point, we can use Eq. (2.3) to get F 1=3 the conductivity is positive for W L (Tworzydło et al., 2006). One should emphasize! e2 that obtaining the same numerical value for the Fano factor ln L=‘ : (2.8) F 1=3 for ballistic quantum transport in graphene as that ¼ h ð Þ ¼ of diffusive transport in disordered metals could be nothing Equivalently, one can define a one-parameter scaling func- more than a coincidence (Dragomirova et al., 2009). tion (Abrahams et al., 1979) Cheianov and Fal’ko (2006b) found that the shot noise of a d ln single p-n junction was F 1 1=2, which is numeri- ; (2.9) ¼ À ð Þ¼d lnL cally quite close to 1=3. pffiffiffiffiffiffiffiffi Since several different mechanisms all give F 1=3, this where for the symplectic class it follows from Eq. (2.8) that makes shot noise a complicated probe of the underlying 1= for large . To have >0 means that the ð Þ¼ ð Þ Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 424 Das Sarma et al.: Electronic transport in two-dimensional graphene conductivity increases as one goes to larger system sizes or one-parameter scaling (i.e., there exists a function) and adds more disorder. This is quite different from the usual case (ii) the function is always positive unlike the spin-orbit of an Anderson transition where a negative function means case. Therefore, Dirac fermions evade Anderson localization that for those same changes, the system becomes more and are always metallic. Similar conclusions were obtained insulating. by Nomura et al. (2007), San-Jose et al. (2007), Titov Since perturbation theory only gives the result for (2007), and Tworzydło et al. (2008). e2=h , the real question becomes what happens to theð The inset of Fig. 7 shows an explicit computation of the functionÞ at small . If the function crosses zero and function comparing Dirac fermions with the spin-orbit model. becomes negative as 0, then the system exhibits the The difference between these two classes of the AII symme- usual Anderson metal-insulator! transition. Numerical studies try class has been attributed to a topological term (i.e., two of the Hamiltonian [Eq. (2.6)] show that for the spin-orbit possible choices for the action of the field theory describing system, the function vanishes at 1:4 and below this these Hamiltonians). Since it allows for only two possibil- à value, the quantum correction to the classical conductivity is ities, it has been called a Z2 topological symmetry (Kane and negative resulting in an insulator at zero temperature. à is an Mele, 2005b; Evers and Mirlin, 2008). The topological term unstable fixed point for the symplectic symmetry class. has no effect at e2=h but is responsible for the differ- As we have seen in Sec. II.B.2, however, graphene has a ences at e2=h and determines the presence or absence of 2 minimum ballistic conductivity min 4e =h and does not a metal-insulator transition. Nomura et al. (2007) presented become insulating in the limit of vanishing¼ disorder. This an illustrative visualization of the differences between Dirac makes graphene different from the spin-orbit Hamiltonian fermions and the spin-orbit symplectic class shown in Fig. 8. discussed above, and the question of what happens with By imposing a twist boundary condition in the wave functions increasing disorder becomes interesting. such that É x 0 exp i É x L and É y 0 Bardarson et al. (2007) studied the Dirac Hamiltonian É y W , oneð ¼ canÞ¼ examine½ the singleð ¼ particleÞ spectrumð ¼ Þ¼ as [Eq. (2.2)] with the addition of a Gaussian correlated disorder a functionð ¼ Þ of the twist angle . For 0 and , the term U r , where phase difference is real and eigenvalues¼ come in Krammer’s¼ ð Þ degenerate pairs. For other values of , this degeneracy is ℏv 2 r r 2 lifted. As seen in the figure, for massless Dirac fermions all U r U r K F exp 0 : (2.10) 0 0 ð Þ2 Àj À2 j energy states are connected by a continuous variation in the h ð Þ ð Þi¼ 2 2 boundary conditions. This precludes creating a localized One should think of K0 as parametrizing the strength of the state, which would require the energy variation with bound- disorder and as its correlation length. If the theory of one- ary condition (also called Thouless energy) be smaller than parameter scaling holds for graphene, then it should be the level spacing. Since this is a topological effect, Nomura possible to rescale the length Là f0 K0 L, where f0 is a et al. (2007) argued that this line of reasoning should be scaling function inversely proportional¼ ð to theÞ effective elec- robust to disorder. tronic mean free path. Their numerical results are shown The situation for the spin-orbit case is very different. The in Fig. 7 and demonstrate that (i) graphene does exhibit same Krammer’s pairs that are degenerate at 0 reconnect at . In this case, there is nothing to prevent¼ localization if the¼ disorder would push the Krammer’s pairs past the mobility edge. Similar considerations regarding the Z2 sym- metry also hold for topological insulators where the metallic surface state should remain robust against localization in the presence of disorder. (a) massless Dirac model (b) random SO model 0.2 0.3 0.1 0 0.25 Energy Energy -0.1 0.2 -0.2 FIG. 7 (color online). Demonstration of one-parameter scaling at the Dirac point. From Bardarson et al., 2007. Main panel shows the -0.3 conductivity as a function of LÃ=, where Là f K0 L is the scaled length and is the correlation length¼ ofð theÞ disorder potential. Note that d ln=d lnL>0 for any disorder strength. ¼ The inset shows explicit comparison of the function for the Dirac FIG. 8 (color online). Picture proposed by Nomura et al. (2007) fermion model and for the symplectic (AII) symmetry class. From to understand the difference in topological structure between the Nomura et al., 2007. massless Dirac model and the random spin-orbit symmetry class. Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 425 2. Crossover from the symplectic universality class obtaining the corrections to the conductivity from the bare Hikami box (see Sec. II.C.3 for a more complete discussion). It is already apparent in the preceding discussion that For the case of no intervalley scattering LR, the each Dirac cone is described by the Dirac Hamiltonian U U resulting Cooperon is ¼ H p. The effective SRS H H is preserved ¼ Á ¼ y à y in each cone, and for most purposes graphene can be viewed 2 LR niu i c k c k 1 as two degenerate copies of the AII symplectic symmetry C Q e ð À Þ ; (2.11) k k ð Þ¼ A v Q 2 class. However, as Suzuura and Ando (2002a) first pointed ð F Þ out, a material defect such as a missing atom would couple i c c with area A LW, Q k k , and e ð k À k Þ 1, the two Dirac cones (and since each cone is located in a ¼ ¼ þ À giving 2e2=2ℏ ln L =‘ . As expected for the different ‘‘valley,’’ this type of interaction is called intervalley LR ¼ð Þ ð Þ scattering). One can appreciate intuitively why such scatter- symplectic class, without intervalley scattering, the quantum ing is expected to be small. The two valleys at points K and correction to the conductivity is positive. With intervalley scattering, U USR calculating the K0 in the Brillouin zone are separated by a large momentum ¼ vector that is inversely proportional to the spacing between same diagrams gives two neighboring carbon atoms. This means that the potential n u2 1 responsible for such intervalley coupling would have to vary SR i i c k c k C Q j j e ð À Þ ; (2.12) appreciably on the scale of 0.12 nm in order to couple the K k k ð Þ¼ A v Q 2 ð F Þ and K0 points. 2 2 We note that while such defects and the corresponding with current j j and SR e = 2 ℏ ln L =‘ . ¼À ¼À ð Þ ð Þ coupling between the valleys are commonly observed in This negative is consistent with the orthogonal symmetry scanning tunneling microscopy (STM) studies on epitaxial class. The explicit microscopic calculation demonstrates the graphene (Rutter et al., 2007), they are virtually absent in all crossover from weak antilocalization to weak localization similar studies in exfoliated graphene (Ishigami et al., 2007; induced by atomically sharp microscopic defects providing Stolyarova et al., 2007; Zhang et al., 2008). In the presence intervalley coupling. of such atomically sharp disorder, Suzuura and Ando (2002a) This crossover was recently observed experimentally proposed a model for the two-valley Hamiltonian that cap- (Tikhonenko et al., 2009). They noted empirically that for tures the effects of intervalley scattering. The particular form their samples the scattering associated with short-range de- of the scattering potential is not important, and in Sec. II.C.3 fects is stronger at high carrier density. In fact, this is what we will discuss a generalized Hamiltonian that includes all one expects from the microscopic theory discussed in Sec. III. nonmagnetic (static) impurities consistent with the honey- Because of the unique screening properties of graphene, long- comb symmetry and is characterized by five independent range scatterers dominate transport at low carrier density parameters (Aleiner and Efetov, 2006; McCann et al., while short-range scatterers dominate at high density. 2006). Here the purpose is simply to emphasize the qualita- Assuming that these short-range defects are also the dominant tive difference between two types of disorder: long-range source of intervalley scattering, one would expect to have (i.e., diagonal) disorder ULR that preserves the effective weak localization at high carrier density (due to the large SRS and a short-range potential USR that breaks this intervalley scattering), and weak antilocalization at low car- symmetry. rier density where transport is dominated by ‘‘atomically We note that with the intervalley term USR, the smooth’’ defects such as charged impurities in the substrate. Hamiltonian belongs to the Wigner-Dyson orthogonal sym- This is precisely what was seen experimentally. Figure 10 metry class, while as discussed in Sec. II.C.1 including only shows a comparison of the magnetoconductance at three the diagonal disorder ULR, one is in the Wigner-Dyson different carrier densities. At the lowest carrier density, the symplectic class. data show the weak antilocalization characteristic of the A peculiar feature of this crossover is that it is governed by symplectic symmetry class, while at high density, one finds the concentration of short-range impurities thus questioning weak localization signaling a crossover to the orthogonal the notion that the universality class is determined only by the universality class. global symmetries of the Hamiltonian and not by microscopic A second crossover away from the symplectic universality details. However, a similar crossover was observed by Miller class was examined by Morpurgo and Guinea (2006). As et al. (2003) where the strength of the spin-orbit interaction discussed, a magnetic field breaks time reversal symmetry was tuned by carrier density, moving from weak localization and destroys the leading quantum corrections to the conduc- tivity B>B 0. This can also be understood as a at low density and a weak spin-orbit interaction, to weak ð ÃÞ¼ antilocalization at high density and a strong spin-orbit crossover from the symplectic (or orthogonal) universality interaction. class to the unitary class. The unitary class is defined by the From symmetry considerations, one should expect that absence of time reversal symmetry and hence vanishing 1 without atomically sharp defects, graphene would exhibit contribution from the Cooperon. weak antilocalization (where > 0) and no Anderson lo- calization (see Sec. II.C.1). However, with intervalley scat- 1The sign of the quantum correction in relation to the global tering, graphene should have weak localization ( < 0) and symmetry of the Hamiltonian can also be obtained from the random be insulating at zero temperature. These conclusions were matrix theory (Beenakker, 1997) =0 1 2= =4, where verified by Suzuura and Ando (2002a) from a microscopic 1, 2, 4 for the orthogonal, unitary, and¼ð symplecticÀ Þ Wigner- Hamiltonian by calculating the Cooperon (see Fig. 9) and Dyson¼ symmetry classes. Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 426 Das Sarma et al.: Electronic transport in two-dimensional graphene 3. Magnetoresistance and mesoscopic conductance fluctuations As discussed, at low energies and in the absence of dis- order, graphene is described by two decoupled Dirac cones located at points K and K0 in the Brillouin zone. Within each cone, one has a pseudospin space corresponding to wave function amplitudes on the A and B sublattice of the honey- comb lattice. The two-valley Hamiltonian is then the outer product of two SU(2) spin spaces KK0 AB. The most generic Hamiltonian in this space of 4 4 Hermitian matri- ces can be parametrized by the generators of the group U(4) (Aleiner and Efetov, 2006; Altland, 2006; McCann et al., FIG. 9. Diagrammatic representation for (a) diffuson, 2006): (b) Cooperon, and dressed Hikami boxes [(c) and (d)]. Adapted from Kharitonov and Efetov, 2008. H ℏvFÆp 14u0 r ÆsÃlusl r ; (2.13) ¼ þ ð Þþs;l x;y;z ð Þ ¼X where Æ Æ ; Æ ; Æ ; ; 1 is ¼ð x y zÞ¼ð z x z y 2 zÞ Similar to short-range impurities inducing a crossover the algebra of the sublattice SU(2) space (recall that the outer from symplectic to orthogonal classes, Morpurgo and product is in the space KK0 AB, and the Æ operator is diagonal in the KK space). Similarly, à ; Guinea (2006) asked if there were other kinds of disorder 0 ¼ð x z y that could act as pseudomagnetic fields and induce a cross- z;z 12 forms the algebra of the valley-spin space (being Þ over to the unitary symmetry class leading to the experimen- diagonal in the AB space). tal signature of a suppression of weak antilocalization. This The Hamiltonian of Eq. (2.13) can be understood in simple was in part motivated by the first experiments on graphene terms. The first term is just two decoupled Dirac cones and is quantum transport showing that the weak localization correc- equivalent to the disorder-free case discussed earlier, but tion was an order of magnitude smaller than expected written here in a slightly modified basis. The second term is (Morozov et al., 2006). They argued that topological lattice identical to ULR and as discussed, it represents any long- defects (Ebbesen and Takada, 1995) (e.g., pentagons and range diagonal disorder. The last term parametrized by the heptagons) and nonplanarity of graphene (commonly referred nine scattering potentials usl r represents all possible types ð Þ to as ‘‘ripples’’) would generate terms in the Hamiltonian that of disorder allowed by the symmetry of the honeycomb looked like a vector potential and correspond to a pseudo- lattice. For example, a vacancy would contribute to all terms magnetic field. (including u0) except uxz and uyz; while bond disorder would In addition, experiments both on suspended graphene contribute to all terms except uzz (Aleiner and Efetov, 2006). (Meyer et al., 2007) and on a substrate showed that graphene The ‘‘diagonal’’ term u r is the dominant scattering 0ð Þ is not perfectly coplanar. It is noteworthy, however, that mechanism for current graphene experiments and originates experiments on a SiO2 substrate showed that these ripples from long-ranged Coulomb impurities, which is discussed in were correlated with the height fluctuations of the substrate more detail in Sec. III. Because of the peculiar screening and varied by less than 1 nm (Ishigami et al., 2007), while properties of graphene, such long-range disorder cannot be graphene on mica was even smoother with variations treated using the Gaussian white noise approximation. To of less than 0.03 nm (Lui et al., 2009). On the other hand, circumvent this problem (for both the long-range u0 and one could deliberately induce lattice defects (Chen, Cullen short-range usl terms), we simply note that for each kind of et al., 2009) or create controlled ripples by straining graphene disorder, there would be a corresponding scattering time before cooling and exploiting graphene’s negative thermal { ; that could, in principle, have very different depen- 0 slg expansion coefficient (Balandin et al., 2008; Bao et al., dence on carrier density. 2009). For the special case of Gaussian white noise, i.e., where 2 1 Just as a real magnetic field, these terms would break the u r u r0 u r r0 , we have ℏ À h slð Þ s0l0 ð Þi¼ sl s;s0 l;l0 ð À Þ sl ¼ time reversal symmetry (TRS) in a single valley (while D E u2 . Moreover, one could assume that after disorder ð FÞ sl preserving the TRS of the combined system). If , averaging, the system is isotropic in the x-y plane. Denoting i then the two valleys are decoupled, these defects would cause x; y , the total scattering time is given by f g? a crossover to the unitary symmetry class, and the resulting 1 1 1 1 1 1 Cooperon (Fig. 9) would vanish. For example, considering À À zzÀ 2 À 2 À 4 À : (2.14) ¼ 0 þ þ z þ z þ the case of lattice defects, the disorder Hamiltonian would be ? ? ?? given by UG 1=4 @ u r @ u r , where These five scattering times could be viewed as independent ¼ð Þ½ x zr½ y xð ÞÀ x yð Þ u r is the lattice strain vector induced by the defect. One microscopic parameters entering the theory (Aleiner and notesð Þ that this term in the Hamiltonian has the form of an Efetov, 2006), or one could further classify scattering times 1 1 1 effective magnetic field B in the K valley and B in the K0 as being either ‘‘intervalley’’ iÀ 4 À 2 zÀ or ‘‘intra- þ À 1 1 1 ¼ ?? þ ? valley (Morpurgo and Guinea, 2006). In the absence of valley’’ À 4 À 2 À . A small contribution from z ¼ z þ zz intervalley coupling, this would suppress weak antilocaliza- trigonal warping (a? distortion to the Dirac cone at the energy tion when the effective magnetic field B is larger than the scale of the inverse lattice spacing) could be modeled by the field B discussed in Sec. II.A. j j perturbative term H Æ Æp Ã Æ Æp Æ , which acts as à w xð Þ z xð Þ x Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 427 an additional source of intravalley scattering (McCann et al., 2006). The transport properties of the Hamiltonian [Eq. (2.13)] are obtained by calculating the two particle propagator. In gen- eral, both the classical contribution (diffusons) and quantum corrections (Cooperons) will be 4 4 matrices defined in terms of the retarded (R) and advanced (A) Green’s functions GR;A as (see also Fig. 9) R A D !; r; r0 G !; r; r0 G ; r0; r ; ð Þ¼h ð þ Þ ð Þi (2.15) C !; r; r GR !; r; r GA ; r; r : ð 0Þ¼h ð þ 0Þ ð 0Þi FIG. 10 (color online). Left panel: Schematic of different magne- toresistance regimes. (i) For strong intervalley scattering (i.e. As discussed in Sec. III.A, the scattering rate is dominated by i and z ), Eq. (2.18) gives weak localization or the diagonal disorder 0. Since both this term and the Dirac part of Eq. (2.13) is invariant under the valley SU(2), B 0 < 0. This is similar to quantum transport in the usual 2DEG.ð ÞÀ (ii)ð ForÞ weak intervalley scattering (i.e., and one can classify the diffussons and Cooperons as ‘‘singlets’’ i z ), one has weak antilocalization, characteristic of the symplectic and ‘‘triplets’’ in the AB-sublattice SU(2) space. Moreover, symmetry class. (iii) For , but , Eq. (2.18) gives a one finds that for both the diffussons and Cooperons, only the i z valley singlets are gapless, and one can completely ignore the regime of suppressed weak localization. Right panel: Experimental realization of these three regimes. Graphene magnetoconductance is valley triplets whose energy gap scales as 1. Considering 0À shown for carrier density (from bottom to top) n 2:2 1010, only the sublattice singlet (j 0) and triplet (j x; y; z), one 12 12 2 ¼  ¼ ¼ 1:1 10 , and 2:3 10 cmÀ , at T 14 K. The lowest carrier finds (McCann et al., 2006; Fal’ko et al., 2007; Kechedzhi density (bottom curve) has a small contribution¼ from short-range et al., 2008; Kharitonov and Efetov, 2008) disorder and shows weak antilocalization [i.e., the zero-field con- 2 ductivity is larger than at finite field. B 0 0 and 1 2 2eA j j ð ¼ Þ¼ þ i! v 0 À D r; r0 r r0 ; B>B , with > 0. B is the phase-breaking field]. In À À 2 F rÀ c þ ð Þ¼ ð À Þ Ã 0 à contrast,ð theÞ¼ highest density data (top curve) has a larger contribu- 2 1 2 2eA j 1 j tion of intervalley scattering and shows weak localization, i.e., i! vF 0 À À C r; r0 À À 2 rþ c þ þ ð Þ < 0. From Tikhonenko et al., 2009. r r ; (2.16) ¼ ð À 0Þ 0 x y 1 1 z 1 with À 0 (singlet), À À iÀ zÀ , and À 2 iÀ (triplet).¼ This equation captures¼ ¼ all theþ differences¼ in the (ii) Weak short-range disorder. For and , i z quantum corrections to the conductivity between graphene all sublattice Cooperons and diffusons remain gapless and usual 2DEGs. at zero magnetic field. One then has > 0 or weak The magnetoresistance and conductance fluctuation prop- antilocalization (symplectic symmetry). This regime erties in graphene follow from this result. The quantum was observed in experiments on epitaxial graphene C C C correction to the conductivity is Nt Ns , where Nt (Wu et al., 2007). The diffuson contribution to the À C is the number of gapless triplet Cooperon modes and Ns is conductance fluctuations is enhanced by a factor of 4 the number of gapless singlet Cooperon modes. In this con- compared with conventional metals. text, gapped modes do not have a divergent quantum correc- (iii) Suppressed localization regime. In the case that there tion and can be neglected. Similarly, the conductance is strong short-range scattering z , but in weak 2 2 x fluctuations are given by G NCD G 2DEG, intervalley scattering i . The Cooperons C and h½ i¼ h½ i y z where NCD counts the total number of gapless Cooperons C will be gapped, but C will remain and cancel the 2 0 and diffusons modes, and G 2DEG is the conductance effect of the singlet C . In this case one would have the h½ i variance for a conventional 2D electron gas. For a suppressed weak localization that was presumably quasi-1D geometry, G 2 1 2e2=h 2 (Lee and seen in the first graphene quantum transport experi- h½ i2DEG ¼ 15 ð Þ Ramakrishnan, 1985). ments (Morozov et al., 2006). We can immediately identify several interesting regimes Although the discussion above captures the main physics, that are shown schematically in Fig. 10: for completeness we reproduce the results of calculating (i) Strong intervalley scattering. Even with strong short- the dressed Hikami boxes in Fig. 9 (Aleiner and Efetov, range disorder (i.e., and ), both the 2006; McCann et al., 2006; Kechedzhi et al., 2008; i z singlet Cooperon C0 and the singlet diffuson D0 remain Kharitonov and Efetov, 2008) and using known results (Lee gapless since À0 0. Contributions from all triplet and Ramakrishnan, 1985). The quantum correction to the Cooperons and difussons¼ vanish. In this situation, the conductance is quantum corrections to the conductivity in graphene are very similar to the regular 2DEG. The Hamiltonian is in the orthogonal symmetry class discussed earlier and 2e2D d2q g Cx Cy Cz C0 ; (2.17) one has weak localization ( < 0). Similarly, for ¼ ℏ 2 2 ð þ þ À Þ conductance fluctuations (typically measured at large Z ð Þ magnetic fields), one would have the same result as the nonrelativistic electron gas. and for the magnetoresistance Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 428 Das Sarma et al.: Electronic transport in two-dimensional graphene e22 B B matic perturbation theory, this is the leading order term in an B 0 F F expansion of ni 0. Typically for other conventional metals ð ÞÀ ð Þ¼À ℏ B À B 2Bi þ and semiconductors,! one makes a better approximation by B 2F ; (2.18) trying to include more diagrams that capture multiple scatter- À B 2B þ z ing off the same impurity. For example, in the self-consistent where B ℏc=4De 1 and F x lnx c 1=2 Born approximation (SCBA) one replaces the bare Green’s ;i;z ¼ð Þ À;i;z ð Þ¼ þ ð þ 1=x , with c the digamma function. The function F x is functions with dressed ones to obtain a self-consistent equa- theÞ same as for 2DEGs (Lee and Ramakrishnan, 1985ð Þ ); tion for the self-energy (Bruus and Flensberg, 2004). however, the presence of three terms in Eq. (2.18) is unique In practice, for graphene, one often finds that attempts to to graphene. The universal conductance fluctuations are go beyond the semiclassical Boltzmann transport theory described in Sec. III.A fare far worse than the simple theory. g g e2 2 3 1 À n2 2 s v 1 1 i x The theoretical underpinnings for the failure of SCBA was G 3 4 4 2 2 h½ i¼ 2ℏ Lx D þ Lx C;D i 0 nx 1 ny 0 pointed out by Aleiner and Efetov (2006), who argued that X X¼ X¼ X¼ the SCBA (a standard technique for weakly disordered n2 2 y À 2 metals and superconductors) is not justified for the Dirac 2 NCD G 2DEG; (2.19) þ Ly ¼ h½ i Hamiltonian. They demonstrated this by calculating terms with only diffusions contributing for B>B B . à to fourth order in perturbation theory, showing that SCBA In this section we assumed Gaussian white noise correla- neglects most terms of equal order. This could have severe tions to calculate the Green’s functions. Since we know that consequences. For example, considering only diagonal dis- this approximation fails for the semiclassical contribution order, the SCBA breaks time reversal symmetry. To further arising from Coulomb disorder, why can we use it success- illustrate their point, Aleiner and Efetov (2006) argued that fully for the quantum transport? It turns out that the quasiu- for the full disorder Hamiltonian [Eq. (2.13)], considering niversal nature of weak localization and conductance three impurity scattering, there are 54 terms to that order, and fluctuations means that the exact nature of the disorder only 6 are captured by the SCBA. potential will not change the result. Several numerical calcu- These terms provide a new divergence in the diagrammatic lations using long-range Coulomb potential have checked this perturbation series, which is distinct from the weak localiza- assumption (Yan and Ting, 2008). Many of the symmetry tion discussed in Sec. II.C.3. Unlike weak localization that for arguments discussed here apply to confined geometries such 2D systems diverges as the size ( ln L=‘ ), this addi- as quantum dots (Wurm et al., 2009). Finally, the diagram- tional divergence occurs at all length scales, ½ and was called matic perturbation theory discussed here applies only away ‘‘ultraviolet logarithmic corrections.’’ The consequences of from the Dirac point. As discussed in Sec. II.C.1, numerically this divergence include the logarithmic renormalization of the calculated weak (anti)localization corrections remain as ex- bare disorder parameters which was studied by Foster and pected even at the Dirac point. However, Rycerz et al. Aleiner (2008) using the renormalization group. For the (2007a) found enhanced conductance fluctuations at the experimentally relevant case of strong diagonal disorder, Dirac point, a possible consequence of being in the ballistic the renormalization does not change the physics. However, to diffusive crossover regime. when all disorder couplings (i.e., intervalley and intravalley) As for the experimental situation, in addition to the obser- are comparable, e.g., relevant for graphene after ion irradia- vation of suppressed localization (Morozov et al., 2006) and tion, the system could flow to various strong coupling fixed antilocalization (Wu et al., 2007), Horsell et al. (2009) made points depending on the symmetry of the disorder potential. a systematic study of several samples fitting the data to In addition to these considerations, interaction effects Eq. (2.18) to extract , i, and z. Their data showed a could also affect quantum transport (e.g., the Altshuler- mixture of localization, antilocalization, and saturation be- Aronov phenomena), particularly in the presence of disorder. havior. An interesting feature is that the intervalley scattering Although such interaction effects are probably relatively 1=2 length Li D i is strongly correlated to the sample small in monolayer graphene, they may not be negligible. width (i.e.,¼ðL W=Þ 2) implying that the edges are the domi- i Interaction effects may certainly be important in determining nant source of intervalley scattering. This feature has been graphene transport properties near the charge neutral Dirac corroborated by Raman studies that show a strong D peak at point (Fritz et al., 2008; Kashuba, 2008; Mu¨ller et al., 2008; the edges, but not in the bulk (Graf et al., 2007; Chen, Cullen Bistritzer and MacDonald, 2009). et al., 2009). The most important finding of Horsell et al. (2009) is that the contribution from short-range scattering ( z) is much larger than one would expect (indeed, compa- III. TRANSPORT AT HIGH CARRIER DENSITY rable to 0). They showed that any predicted microscopic mechanisms such as ripples or trigonal warping that might A. Boltzmann transport theory contribute to z (but not i) were all negligible and could not 1 In this section we review graphene transport for large explain such a large zÀ . It remains an open question why 1 1 carrier densities (n ni, ni is the impurity density), where zÀ iÀ in the experiments. the system is homogeneous. We discuss in detail the microscopic transport properties at high carrier density using 4. Ultraviolet logarithmic corrections the semiclassical Boltzmann transport theory. The semiclassical Boltzmann transport theory treats the It was predicted that the graphene conductivity limited by impurities within the first Born approximation. In a diagram- short-ranged scatterers (i.e., -range disorder) is independent Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 429 of the carrier density, because of the linear-in-energy density phene conductivity by intentionally adding potassium ions to of states (Shon and Ando, 1998). However, the experiments graphene in ultrahigh vacuum, qualitatively observing the (Fig. 11) showed that the conductivity increases linearly in prediction of the transport theory limited by Coulomb disor- the carrier density concentration. To explain this linear-in- der [Fig. 11(c)]. Jang et al. (2008) tuned graphene’s fine- density dependence of experimental conductivity, the long- structure constant by depositing ice on the top of graphene range Coulomb disorder was introduced (Ando, 2006; and observed enhanced mobility, which is predicted in the Cheianov and Fal’ko, 2006a; Nomura and MacDonald, Boltzmann theory with Coulomb disorder [Fig. 11(d) and 2006; Hwang et al., 2007a; Nomura and MacDonald, Chen, Xia, and Tao (2009), Chen, Xia et al. (2009), , and 2007; Trushin and Schliemann, 2008; Katsnelson et al., Kim, Nah et al. (2009)]. The role of remote impurity 2009). The long-range Coulomb disorder also successfully scattering was further confirmed in the observation of drastic explains several recent transport experiments. Tan, Zhang, improvement of mobility by reducing carrier scattering in Bolotin et al. (2007) found the correlation of the sample suspended graphene through current annealing (Bolotin, mobility with the shift of the Dirac point and minimum Sikes, Hone et al., 2008; Du et al., 2008). Recent measure- conductivity plateau width, showing qualitative and semi- ment of the ratio of transport scattering time to the quantum quantitative agreement with the calculations with long-range scattering time by Hong, Zou, and Zhu (2009) also strongly Coulomb disorder [Fig. 11(b)]. Chen, Jang, Adam et al. supports the long-range Coulomb disorder as the main scat- (2008) investigated the effect of Coulomb scatterers on gra- tering mechanism in graphene (Fig. 12). FIG. 11 (color online). (a) The measured conductivity of graphene as a function of gate voltage Vg (or carrier density). The conductivity increases linearly with the density. Adapted from Castro Neto et al., 2009. (b) as a function of Vg for five different samples For clarity, curves are vertically displaced. The inset shows the detailed view of the density-dependent conductivity near the Dirac point for the data in the main panel. Adapted from Tan, Zhang, Bolotin et al., 2007. (c) vs Vg for the pristine sample and three different doping concentrations. Adapted from Chen, Jang, Adam et al., 2008. (d) as a function Vg for pristine graphene (circles) and after deposition of 6 monolayers of ice (triangles). Inset: Optical microscope image of the device. Adapted from Jang et al., 2008. Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 430 Das Sarma et al.: Electronic transport in two-dimensional graphene and the corresponding temperature-dependent resistivity is given by T 1= T . Note that f k ð Þ¼1 ð Þ ð Þ¼ 1 exp k =kBT À , where the finite temperature chemicalf þ ½ð potentialÀ Þ T g is determined self-consistently to conserve the total numberð Þ of electrons. At T 0, f is a ¼ ð Þ step function at the Fermi energy EF T 0 , and we then recover the usual conductivity formula ð ¼ Þ e2v2 F D E E ; (3.3) ¼ 2 ð FÞ ð FÞ where vF is the carrier velocity at the Fermi energy. FIG. 12 (color online). The ratio of transport scattering time ( t) to quantum scattering time ( ) as a function of density for different q B. Impurity scattering samples. Dashed (solid) lines indicate the theoretical calculations (Hwang and Das Sarma, 2008e) with Coulomb disorder (-range The matrix element of the scattering potential is deter- disorder). Adapted from Hong, Zou, and Zhu, 2009. mined by the configuration of the 2D systems and the spatial distribution of the impurities. In general, impurities are lo- The conductivity (or mobility =ne) is calculated cated in the environment of the 2D systems. For simplicity, in the presence of randomly distributed¼ Coulomb impurity we consider the impurities are distributed completely at charges with the electron-impurity interaction being screened random in the plane parallel to the 2D systems located at z d. The location d is a single parameter modeling the by the 2D electron gas in the random phase approximation ¼ (RPA). Even though the screened Coulomb scattering is the impurity configuration. Then the matrix element of the scat- most important scattering mechanism, there are additional tering potential of randomly distributed screened impurity scattering mechanisms (e.g., neutral point defects) unrelated charge centers is given by to the charged impurity scattering. The Boltzmann formalism v q 2 can treat both effects, where zero-range scatterers are treated a 2 i dznið Þ z Vk;k0 z ni ð Þ F q ; (3.4) with an effective point defect density of n . Phonon scattering ð Þjh ð Þij ¼ " q ð Þ d Z ð Þ effects, important at higher temperatures, are treated in the where q k k , , n is the number of impurities next section. We also discuss other scattering mechanisms 0 kk0 i per unit¼ area,j ÀF qj is the form factor associated with the which could contribute to graphene transport. ð Þ We start by assuming the system to be a homogeneous 2D carrier wave function of the 2D system, and vi q 2e2= q e qd is the Fourier transform of the 2D Coulombð Þ¼ carrier system of electrons (or holes) with a carrier density n ð Þ À induced by the external gate voltage V . When the external potential in an effective background lattice dielectric constant g . The form factor F q in Eq. (3.4) comes from the overlap electric field is weak and the displacement of the distribution of the wave function.ð Þ In 2D semiconductor systems it is function from thermal equilibrium is small, we may write related to the quasi-2D nature of systems, i.e., finite width the distribution function to the lowest order in the applied of the 2D systems. The real functional form depends on the electric field (E) f f f , where is the carrier k k k k details of the quantum structures (i.e., heterostructures, energy, f is the equilibrium¼ ð Þþ Fermi distribution function, k square well, etc.). F q becomes unity in the two-dimensional and f ðis proportionalÞ to the field. When the relaxation k limit (i.e., layer).ð However,Þ in graphene the form factor is time approximation is valid, we have f k related to the chirality, not to the quantum structure since =ℏ eE v @f =@ , where v d =dk ¼is k k k k k k graphene is strictly a 2D layer. In Eq. (3.4), " q " q; T is theÀ½ velocityð Þ ofÁ carrier½ ð andÞ is the relaxation¼ time or k the 2D finite temperature static RPA dielectricð Þ (screening)ð Þ the transport scattering time, andð Þ is given by function and is given by 2 1 2 a d k0 2 dznð Þ z V z i 2 k;k0 " q; T 1 v q Å q; T ; (3.5) k ¼ ℏ ð Þ 2 jh ð Þij c ð Þ Xa Z Z ð Þ ð Þ¼ þ ð Þ ð Þ 1 cos ; (3.1) ½ À kk0 ð k À k0 Þ where Å q; T is the irreducible finite temperature polariz- ð Þ where is the scattering angle between the scattering in ability function and vc q is the Coulomb interaction. For kk0 ð Þ a short-ranged disorder, we have and out wave vectors k and k0, nið Þ z is the concentration of the ath kind of impurity, and z representsð Þ the coordinate of normal direction to the 2D plane. In Eq. (3.1) Vk;k z is the a 2 2 0 dznð Þ z Vk;k z ndV F q ; (3.6) matrix element of the scattering potential associatedh ð Þi with i ð Þjh 0 ð Þij ¼ 0 ð Þ Z impurity disorder in the system environment. Within Boltzmann transport theory by averaging over energy, we where nd is the 2D impurity density and V0 is a constant obtain the conductivity short-range (i.e., a function in real space) potential strength. One can also consider the effect on carrier transport by e2 @f dD v2 ; (3.2) scattering from cluster of correlated charged impurities ¼ 2 ð Þ k ð Þ À @ (Katsnelson et al., 2009), as originally done for 2D Z Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 431 semiconductors by Kawamura and Das Sarma (Kawamura where s 1 indicate the conduction ( 1) and valence and Das Sarma, 1996; Das Sarma and Kodiyalam, 1998). ( 1) bands,¼Æ respectively, k k q,þ" sℏv k , À 0 ¼ þ sk ¼ Fj j Without detailed knowledge of the clustering correlations, Fss0 k; k0 1 cos =2, and fsk exp "sk however, this is little more than arbitrary data fitting. 1 1ð withÞ¼ð 1þ=k T. Þ ¼½ f ð À Þgþ À ¼ B Because the screening effect is known to be of vital im- After performing the summation over ss0, it is useful to portance for charged impurities (Ando, 2006; Hwang et al., rewrite the polarizability as the sum of intraband and inter- 2007a), we first provide the static polarizability function. It is band polarizaibility Å q; T Å q; T Å q; T . At ð Þ¼ þð Þþ Àð Þ known that the screening has to be considered to explain the T 0, the intraband (Åþ) and interband (ÅÀ) polarizability density and temperature dependence of the conductivity of 2D becomes¼ (Ando, 2006; Hwang and Das Sarma, 2007) semiconductor systems (Das Sarma and Hwang, 1999, 2005), q and the screening property in graphene exhibits significantly 1 ;q2kF À8kF different behavior (Hwang and Das Sarma, 2007) from that in Å~ q ; (3.8a) þ 8 4k2 ð Þ¼ 1 F q 1 2kF conventional 2D metals. Also, significant temperature depen- >1 1 2 sinÀ ;q>2kF < À2 À q À4kF q dence of the scattering time may arise from the screening q rffiffiffiffiffiffiffiffiffiffiffiffiffiffi Å~ À q :> ; (3.8b) function in Eq. (3.5). Thus, before we discuss the details of ð Þ¼8k conductivity, we first review screening in graphene and in 2D F 2 2 semiconductor systems. where Å~ Æ Å =D , and D gE =2ℏ v is the density ¼ Æ 0 0 F F of states (DOS) at the Fermi level. Intraband Åþ (interband 1. Screening and polarizability ÅÀ) polarizability decreases (increases) linearly as q in- creases, and these two effects exactly cancel out up to q a. Graphene ¼ 2kF, which gives rise to the total static polarizability being The polarizability is given by the bare bubble diagram constant for q<2kF as in the 2DEG (Stern, 1967), i.e., (Ando, 2006; Wunsch et al., 2006; Hwang and Das Sarma, Å q Åþ q ÅÀ q D EF for q 2kF. In Fig. 13 2007) weð showÞ¼ theð calculatedÞþ ð Þ¼ grapheneð Þ static polarizability as a function of wave vector. In the large momentum transfer g fsk fs0k0 Å q; T À Fss0 k; k0 ; (3.7) regime, q>2kF, the static screening increases linearly ð Þ¼ÀA "sk "s k ð Þ kXss0 À 0 0 with q due to the interband transition. In a normal 2D system 1.8 2 (a) 1.0 (c) T/TF =0.0 (e)(a) 1.6 T/TF =1.0 0 F T=0 0 0 0.8 1.4 0.8 1.5 0.6 q,T)/D (q,T)/D 1.2 (q,T)/N T/T =1.0 0.6 0.4 (q,T)/D 0.4 F Π Π ( Π Π 1.0 T=0.1, 0.5, 1, 2T 0.2 1 F 0.2 0.0 0.8 0.0 01234 012345 0 1 2345 q/k q/k F q/k F F 1.5 1.8 (b) (b) (d) 1.0 (f) q=2kF 1.6 F 0 q=2k 0 0 F q=0 1.4 0.8 1.0 q=kF q,T)/D (q,T)/D (q,T)/N (q,T)/D Π Π Π ( 1.2 q=2.0k q=0 Π 0.6 F q=0 1.0 0.5 0.4 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 T/TF T/TF T/TF FIG. 13 (color online). Polarizability Å q; T in units of the density of states at the Fermi level D0. Å q; T of monolayer graphene (a) as a function of wave vector for different temperaturesð Þ and (b) as a function of temperature for different waveð vectors.Þ (c), (d) BLG polarizability. (e), (f) The 2DEG polarizability. Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 432 Das Sarma et al.: Electronic transport in two-dimensional graphene the static polarizability falls off rapidly for q>2k with a (T T ). In Fig. 13 we show the finite temperature polar- F F cusp at q 2kF (Stern, 1967). The linear increase of the izability Å q; T . static polarizability¼ with q gives rise to an enhancement ð Þ of the effective dielectric constant q Ãð ! 1Þ ¼ b. Bilayer graphene 1 gsgvrs=8 in graphene. Note that in a normal 2D systemð þ asÞ q . Thus, the effective interaction in For bilayer graphene, we have the polarizability of à ! !1 Eq. (3.7) with " sk2=2m and F k; k 1 2D graphene decreases at short wavelengths due to interband sk ss0 0 ss cos2 =2 due to the¼ chirality of bilayer graphene.ð Þ¼ð At T þ polarization effects. This large wave vector screening behav- 0 Þ ¼ ior is typical of an insulator. Thus, 2D graphene screening 0, the polarizability of bilayer graphene (Hwang and Das Sarma, 2008d) is given by is a combination of ‘‘metallic’’ screening (due to Åþ) and ‘‘insulating’’ screening (due to ÅÀ), leading overall to rather Å q D0 f q g q q 2kF ; (3.11) strange screening property, all of which can be traced back to ð Þ¼ ½ ð ÞÀ ð Þ ð À Þ where D g g m=2ℏ2 is the BLG density of states at the the zero-gap chiral relativistic nature of graphene. 0 ¼ s v It is interesting to note that the nonanalytic behavior of Fermi level and graphene polarizability at q 2k occurs in the second ¼ F q~2 4 q~ q~2 4 2 2 2 2 derivative, d Å q =dq 1= q 4kF, i.e., the total polar- f q 1 1 2 log À À ; (3.12a) ð Þ / À ð Þ¼ þ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀ q~ þ q~ pq~ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 42 izability, as well as its first derivative,qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi are continuous at q þ À 2k . This leads to an oscillatory decay of the screened¼ 1 1 1 q~p4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi=4 F g q 4 q~4 log ; (3.12b) potential in the real space (Friedel oscillation) which scales þ þ ð Þ¼2 þ À pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 as r cos 2k r =r3 (Cheianov and Fal’ko, 2006a; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ ð F Þ where q~ q=k . Wunsch et al., 2006). This is in contrast to the behavior of ¼ F a 2DEG, where Friedel oscillations scale as r In Fig. 13 the wave vector dependent BLG polarizability is cos 2k r =r2. The polarizability also determinesð Þ the shown. For MLG, intraband and interband effects in polar- ð F Þ izability exactly cancel out up to q 2k , which gives rise to Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction be- ¼ F tween two magnetic impurities as well as the induced spin the total static polarizability being constant for q<2kF. density due to a magnetic impurity, while both quantities are However, for BLG the cancellation of two polarizability proportional to the Fourier transform of Å q . Similar to the functions is not exact because of the enhanced backscattering, ð Þ 3 so the total polarizability increases as q approaches 2kF, screened potential, the induced spin density decreases as rÀ 2 which means screening increases as q increases. Thus BLG, for large distances. Again, this contrasts with the rÀ behav- ior found in a 2DEG. For the particular case of intrinsic in spite of having the same parabolic carrier energy disper- graphene, the Fourier transform of interband polarizability sion of 2DEG systems, does not have a constant Thomas- 3 Fermi screening up to q 2kF (Hwang and Das Sarma, [ÅÀ q ] diverges [even though Å r formally scales as rÀ , ¼ its magnitudeð Þ does not converge],ð whichÞ means that intrinsic 2008d; Borghi et al., 2009), which exists in MLG and graphene is susceptible to ferromagnetic ordering in the 2DEG. In the large momentum transfer regime, q>2kF, presence of magnetic impurities due to the divergent the BLG polarizability approaches a constant value, i.e., Å q N log4, because the interband transition dominates RKKY coupling (Brey et al., 2007). ð Þ! 0 Since the explicit temperature dependence of screening over the intraband contribution in the large wave vector gives rise to significant temperature dependence of the con- limit. For q>2kF, the static polarizability falls off rapidly ( 1=q2) for 2DEG (Stern, 1967) and for MLG it increases ductivity, we consider the properties of the polarizability at finite temperatures. The asymptotic form of polarizability is linearly with q (see Sec. III.B.1). given by The long-wavelength (q 0) Thomas-Fermi screening can be expressed as q !g g me2=ℏ2, which is the TF ¼ s v T q2 T same form as a regular 2D system and independent of Å~ q; T T ln4 F ; (3.9a) F 2 electron concentration. The screening at q 2kF is given ð ÞTF þ 24kF T ¼ by q 2k q p5 log 1 p5 =2 . Screening at T 2 T 2 sð FÞ¼ TF½ À fð þ Þ g Å~ q; T T ð Þ 1 ; (3.9b) q 2kF is about 75% larger than normal 2D Thomas- F ¼ ffiffiffi ffiffiffi ð Þ EF ¼ À 6 TF Fermi (TF) screening, which indicates that in bilayer gra- phene the scattering by the screened Coulomb potential is where T E =k is the Fermi temperature. In addition, the F F B much reduced due to the enhanced screening. finite temperature¼ Thomas-Fermi wave vector in the q 0 A qualitative difference between MLG and BLG polar- long-wavelength limit is given by (Ando, 2006; Hwang! and izability functions is at q 2k . Because of the suppression Das Sarma, 2009b) ¼ F of 2kF backward scattering in MLG, the total polarizability as T well as its first derivative are continuous. In BLG, however, q T T 8 ln 2 r k ; (3.10a) sð FÞ ð Þ s F T large-angle scattering is enhanced due to chirality [i.e., the F overlap factor F in Eq. (3.7)], which gives rise to the 2 T 2 ss0 q T T 4r k 1 : (3.10b) singular behavior of polarizability at q 2kF. Even though sð FÞ s F À 6 T ¼ F the BLG polarizability is continuous at q 2kF, it has a sharp cusp and its derivative is discontinuous¼ at 2k . As q The screening wave vector increases linearly with tempera- F ! 2 2 ture at high temperatures (T T ) but becomes a constant 2kF, dÅ q =dq 1= q 4kF. This behavior is exactly the F ð Þ / À with a small quadratic correction at low temperatures same as that of the regularqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DEG, which also has a cusp at Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 433 q 2kF. The strong cusp in BLG Å q at q 2kF leads to both the long wavelength TF screening and 2kF screening are ¼ ð Þ ¼ 2 2 Friedel oscillations in contrast to the MLG behavior. The same, which is given by qs qTF gsgvme =ℏ . leading oscillation term in the screened potential at large The asymptotic form for¼ the regular¼ 2D polarizability are distances can be calculated as given by 2 ~ TF=T e 4q k sin 2k r Å q 0;T TF 1 eÀ ; (3.17a) r TF F ð F Þ ; (3.13) ð ¼ Þ À ð ÞÀ 2k Cq 2 2k r 2 T q2 T ð F þ TFÞ ð F Þ Å~ q; T T F 1 F : (3.17b) ð FÞ T À 6k2 T where C p5 log 1 p5 =2 , which is similar to the F ¼ À ½ð þ Þ 2DEG except for the additional constant C (C 1 for For q 0, in the T T limit, we get the usual Debye ffiffiffi ffiffiffi ¼ F 2DEG), but different from MLG where Friedel oscillations screening¼ for the regular 2D electron gas system 3 scale as r cos 2kFr =r (Cheianov and Fal’ko, 2006a; ð Þ ð Þ TF Wunsch et al., 2006). qs T TF qTF : (3.18) The enhanced singular behavior of the BLG screening ð Þ T function at q 2kF has other interesting consequences re- A comparison of Eq. (3.18) with Eqs. (3.10) and (3.15) shows lated to Kohn¼ anomaly (Kohn, 1959) and RKKY interaction. that the high-temperature Debye screening behaviors are For intrinsic BLG, the Fourier transform of Å q simply different in all three systems just as the low-temperature becomes a function, which indicates that theð localizedÞ screening behaviors, i.e., the high-temperature screening magnetic moments are not correlated by the long-range wave vector qs in semiconductor 2D systems decreases lin- interaction and there is no net magnetic moment. For extrinsic early with temperature while qs in MLG increases linearly BLG, the oscillatory term in RKKY interaction is restored with temperature and qs in BLG is independent of due to the singularity of polarizability at q 2k , and the temperature. ¼ F oscillating behavior dominates at large kFr. At large dis- In Fig. 13 we show the corresponding parabolic 2D polar- tances 2kFr 1, the dominant oscillating term in Å r is izability normalized by the density of states at Fermi level, 2 ð Þ 2 given by Å r sin 2kFr = kFr . This is the same RKKY D0 gm=ℏ 2. Note that the temperature dependence of 2D interaction asð Þ in/ a regularð Þ 2DEG.ð Þ polarizability¼ at q 2k is much stronger than that of gra- ¼ F In Fig. 13 the wave vector dependent BLG polarizability is phene polarizability. Since in normal 2D systems the 2kF shown for different temperatures. Note that at q 0, scattering event is most important for the electrical resistivity, ¼ Å 0;T NF for all temperatures. For small q, Å q; T the temperature dependence of polarizability at q 2kF ð Þ¼ 4 ð Þ ¼ increases as q . The asymptotic form of polarizability be- completely dominates at low temperatures (T TF). It is comes known that the strong temperature dependence of the polar- izability function at q 2k leads to the anomalously strong 2 ¼ F ~ q TF temperature-dependent resistivity in ordinary 2D systems Å q; T TF 1 2 ; (3.14a) ð Þ þ 6kF T (Stern, 1980; Das Sarma and Hwang, 1999). 1 q4 2 T 2 q4 In the next section the temperature-dependent conductiv- Å~ q; T T 1 : (3.14b) ð FÞ þ 16 k4 þ 16 T k4 ities are provided due to the scattering by screened Coulomb F F F impurities using the temperature-dependent screening prop- More interestingly, the polarizability at q 0 is temperature erties of this section. independent, i.e., the finite temperature Thomas-Fermi¼ wave vector is constant for all temperatures, 2. Conductivity q T q : (3.15) a. Single layer graphene sð Þ¼ TF The eigenstates of single layer graphene are given by the In BLG polarizability at q 0 two temperature effects from plane wave c r 1=pA exp ik r F , where A is the the intraband and the interband¼ transition exactly cancel out, sk sk area of the system,ð Þ¼ðs 1 indicateÞ ð theÁ conductionÞ ( 1) and which gives rise to the total static polarizability at q 0 ¼Æ ffiffiffiffi þ p ik being constant for all temperatures. ¼ valence ( 1) bands, respectively, and Fsyk 1= 2 e ;s with Àtan k =k the polar angle of the¼ð momentumÞð kÞ. k ¼ ð y xÞ ffiffiffi The corresponding energy of graphene for 2D wave vector k c. 2D semiconductor systems is given by sk sℏvF k , and the DOS is given by D The polarizability of ordinary 2D system was first calcu- g = 2ℏ2v2 ,¼ wherej gj g g is the total degeneracyð Þ¼ j j ð FÞ ¼ s v lated by Stern, and all details can be found in the literature (gs 2, gv 2 being the spin and valley degeneracies, (Stern, 1967; Ando et al., 1982). Here we provide the 2D respectively).¼ ¼ The corresponding form factor F q in the polarizability for comparison with graphene. The 2D polar- matrix elements of Eqs. (3.4) and (3.6) arisingð fromÞ the 2 2 izability can be calculated with sk ℏ k =2m and Fss sublattice symmetry (overlap of wave function) (Ando, ¼ 0 ¼ ss0 =2 because of the nonchiral property of the ordinary 2D 2006; Auslender and Katsnelson, 2007) becomes F q systems. Å q at T 0 becomes (Stern, 1967) 1 cos =2, where q k k , . The matrixð Þ¼ ð Þ ¼ 0 kk0 elementð þ ofÞ the scattering¼ potentialj À j of randomly distributed Å q D 1 1 2k =q 2 q 2k ; (3.16) screened impurity charge centers in graphene is given by ð Þ¼ 0½ À Àð F Þ ð À FÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 vi q 1 cos where D0 gsgvm=2ℏ is the 2D density of states at the V 2 ð Þ þ ; (3.19) ¼ sk;sk0 " q 2 Fermi level. Since the polarizability is a constant for q<2kF, jh ij ¼ ð Þ Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 434 Das Sarma et al.: Electronic transport in two-dimensional graphene and the matrix element of the short-ranged disorder is interaction parameter rs 0:8, then the conductivity is given 2 by n 20e =h n=ni (Adam et al., 2007). On the other V 2 V2 1 cos =2; (3.20) ð Þð Þ jh sk;sk0 ij ¼ 0 ð þ Þ hand, the corresponding energy dependent scattering time of short-ranged disorder is where V0 is the strength of the short-ranged disorder potential measured in eV m2. The factor 1 cos in Eq. (3.1) weights À 2 the amount of backward scattering of the electron by the 1 ndV0 EF 2 : (3.23) impurity. In normal parabolic 2D systems (Ando et al., 1982) ¼ ℏ 4 ℏvF the factor 1 cos favors large-angle scattering events. ð Þ À However, in graphene the large-angle scattering is suppressed Thus, the density dependence of scattering time due to the due to the wave function overlap factor 1 cos, which 1=2 þ short-range disorder scattering is given by n nÀ . With arises from the sublattice symmetry peculiar to graphene. Eq. (3.3), we find the conductivity to beð independentÞ/ of The energy dependent scattering time in graphene thus gets density for short-range scattering, i.e., n n0, in contrast weighted by an angular contribution factor of 1 cos to charged impurity scattering which producesð Þ/ a conductivity 1 cos , which suppresses both small-angleð andÀ large-Þ ð þ Þ linear in n. angle scattering contributions in the scattering rate. In Fig. 14(a) the calculated graphene conductivity limited Assuming random distribution of charged centers with by screened charged impurities is shown along with the density n , the scattering time at T 0 is given by i ¼ experimental data (Tan, Zhang, Bolotin et al., 2007; Chen, (Adam et al., 2007; Hwang and Das Sarma, 2008e) Jang, Adam et al., 2008). In order to get quantitative agree- 2 ment with experiment, the screening effect must be included. 1 rs d 2 4 rs g 2rs ; (3.21) The effect of remote scatterers which are located at a distance ¼ 0 2 À drs ½ ð Þ d from the interface is also shown. The main effect of remote impurity scattering is that the conductivity deviates from the where 1 2pn v =pn, and g x 1 x 1 0À ¼ i F ð Þ¼À þ 2 þð À linear behavior with density and increases with both the x2 f x with Þ ð Þ ffiffiffiffi ffiffiffi distance d and n=ni (Hwang et al., 2007a). For very high-mobility samples, a sublinear conductivity, 1 cosh 1 1 for x<1 p1 x2 À x instead of the linear behavior with density, is found in experi- f x À1 1 1 : (3.22) ð Þ¼8 cosÀ for x>1 ments (Tan, Zhang, Bolotin et al., 2007; Chen, Jang, Adam px2 1 x < ffiffiffiffiffiffiffiffiÀ et al., 2008). Such high quality samples presumably have a small charge impurity concentration n , and it is therefore Since r is independent: ffiffiffiffiffiffiffiffi of the carrier density, the scattering i s likely that short-range disorder plays a more dominant role. time is simply given by pn. With Eq. (3.3), we find the Figure 14(b) shows the graphene conductivity calculated density dependence of graphene/ conductivity n n be- ffiffiffi ð Þ/ including both charge impurity and short-range disorder for cause D EF pn. For graphene on SiO2 substrate, the ð Þ/ different values of nd=ni. For small nd=ni, the conductivity is ffiffiffi linear in density, which is seen in most experiments, and for large nd=ni the total conductivity shows the sublinear behav- 120 ior. This high-density flattening of the graphene conductivity (a) 10 (b) 100 is a nonuniversal crossover behavior arising from the com- 2 m /Vs) petition between two kinds of scatterers. In general, this 80 ( µ 1 crossover occurs when two scattering potentials are equiva- 110100 2 60 κ 2 2 e /h) lent, that is, niVi ndV0 . In the inset of Fig. 14(b) the σ ( 40 mobility in the presence of both charged impurities and short-ranged impurities is shown as a function of . As the 20 scattering limited by the short-ranged impurity dominates 0 over that by the long-ranged impurity (e.g., n V2 n V2), 0 1 2345 d 0 i i n/nii the mobility is no longer linearly dependent on the charged impurity and approaches its limiting value FIG. 14 (color online). (a) Calculated graphene conductivity as a 2 function of carrier density (ni is an impurity density) limited by e ℏv 1 ð FÞ : (3.24) Coulomb scattering with experimental data. Solid lines (from ¼ 4ℏ n n V2 bottom to top) show the minimum conductivity of 4e2=h, theory d 0 for d 0 and 0.2 nm. The inset shows the results in a linear scale The limiting mobility depends only on neutral impurity assuming¼ that the impurity shifts by d 0:2 nm for positive voltage concentration n and carrier density, i.e., long-range bias. Adapted from Hwang et al., 2007a¼. (b) Graphene conductivity d calculated using a combination of short- and long-range disorder. In Coulomb scattering is irrelevant in this high-density limit. The temperature-dependent conductivity of graphene aris- the calculation, nd=ni 0, 0.01, and 0.02 (top to bottom) are used. In inset the graphene¼ mobility as a function of dielectric constant ing from screening and the energy averaging defined in () of substrate is shown for different carrier densities n 0:1, 1, Eq. (3.2) is given at low temperatures (T TF) T =0 12 2 ¼ 2 2 2 ð Þ and 5 10 cmÀ (from top to bottom) in the presence of both 1 C1 T=TF , where 0 e vFD EF 0=2 and C1 is a  11 2 À ð Þ ¼ ð Þ long-ranged charged impurity (ni 2 10 cmÀ ) and short- positive constant depending only on rs (Hwang and Das ¼ 10 2 2 ranged neutral impurity (nd 0:4 10 cmÀ ). V0 10 eV nm Sarma, 2009b). The conductivity decreases quadratically is used in the calculation. ¼  ¼ as the temperature increases and shows typical metallic Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 435 Coulomb disorder is given by n n2 in the weak screen- ing limit (q q =2k 1ð)Þ or for the unscreened 0 ¼ TF F Coulomb disorder, and in the strong screening limit (q0 1) n n. In general for screened Coulomb disorder n ð Þn (Das Sarma et al., 2010), where is density dependentð Þ and varies slowly changing from 1 at low density to 2 at high density. Increasing temperature, in general, suppresses screening, leading to a slight enhancement of the exponent . For short-range disorder n n. ð Þ FIG. 15 (color online). Hall mobility as a function of temperature Figure 16 shows the experiment of BLG conductivity. In for different hole densities in (a) monolayer graphene, (b) bilayer Fig. 17(a) the density-dependent conductivities both for graphene, and (c) trilayer graphene. The symbols are the measured screened Coulomb disorder and for short-range disorder are data and the lines are fits. Adapted from Zhu et al., 2009. shown. For screened Coulomb disorder, the conductivity shows superlinear behavior, which indicates that pure Coulomb disorder, which dominates mostly in MLG trans- temperature dependence. On the other hand, at high tempera- port, cannot explain the density-dependent conductivity as 2 tures (T=TF 1), it becomes T =0 C2 T=TF , where seen experimentally (see Fig. 16)(Morozov et al., 2008; ð Þ ð Þ C2 is a positive constant. The temperature-dependent con- Xiao et al., 2010). The density dependence of the conduc- ductivity increases as the temperature increases in the high- tivity with both types of disorder present is approximately temperature regime, which is characteristic of an insulating linear over a wide density range, which indicates that BLG system. s T of graphene due to the short-range disorder carrier transport is controlled by two distinct and independent ð Þ (with scattering strength V0) is given by s T physical scattering mechanisms, i.e., screened Coulomb dis- 2 2 ð Þ¼ s0= 1 eÀ , where s0 e vFD EF s=2 with s order due to random charged impurities in the environment ð þ 2 Þ 2 ¼ ð Þ ¼ nd=4ℏ EFV = ℏvF . In the low-temperature limit the tem- and a short-range disorder. The weaker scattering rate of ð Þ 0 ð Þ perature dependence of conductivity is exponentially sup- screened Coulomb disorder for BLG than for MLG is induced pressed, but the high-temperature limit of the conductivity by the stronger BLG screening than MLG screening, render- approaches =2 as T , i.e., the resistivity at high ing the effect of Coulomb scattering relatively less important s0 !1 temperatures increases up to a factor of 2 compared with in BLG (compared with MLG). the low-temperature limit resistivity. The temperature-dependent conductivity due to screened Recently, the temperature dependence of resistivity of Coulomb disorder (Adam and Stiles, 2010; Das Sarma et al., graphene has been investigated experimentally (Tan, Zhang, 2010; Hwang and Das Sarma, 2010; Lv and Wan, 2010) is Stormer, and Kim, 2007; Bolotin, Sikes, Hone et al., 2008; given by T = 1 C T=T at low temperatures, ð Þ 0 À 0ð FÞ Chen, Jang, Xiao et al., 2008; Zhu et al., 2009). In Fig. 15(a) where C0 4 log2= C 1=q0 with q0 qTF=2kF, and the graphene mobility is shown as a function of temperature. T ¼T=T 2 atð highþ temperatures.Þ When¼ the dimen- ð Þ 1ð FÞ An effective metallic behavior at high density is observed as sionless temperature is very small (T=TF 1), a linear-in-T explained with screened Coulomb impurities. However, it is metallic T dependence arises from the temperature not obvious whether the temperature-dependent correction is quadratic because phonon scattering also gives rise to a temperature dependence (see Sec. III.C). b. Bilayer graphene The eigenstates of bilayer graphene can be written as ikr 2ik c sk e eÀ ;s =p2, and the corresponding energy is given¼ by ð sℏ2kÞ2=2m, where tan 1 k =k and s sk ¼ ffiffiffi k ¼ À ð y xÞ ¼ 1 denote the band index. The corresponding form factor FÆ q of Eqs. (3.4) and (3.6) in the matrix elements arising fromð Þ the sublattice symmetry of bilayer graphene becomes F q 1 cos2 =2, where q k k0 , kk0 . Then theð Þ¼ð matrixþ elementÞ of the scattering¼j À potentialj of randomly distributed screened impurity charge centers in graphene is given by (Koshino and Ando, 2006; Nilsson et al., 2006b; Katsnelson, 2007; Adam and Das Sarma, 2008a; Nilsson et al., 2008) V 2 v q =" q 2 1 cos2 =2: (3.25) jh sk;sk0 ij ¼j ið Þ ð Þj ð þ Þ The matrix element of the short-ranged disorder is given by FIG. 16 (color online). The measured conductivity of bilayer 2 2 Vsk;sk0 V0 1 cos2 =2, and the corresponding en- graphene as a function of gate voltage Vg (or carrier density). jh ij ¼ ð þ Þ 1 ergy dependent scattering time becomes À k The measured conductivity increases linearly with the density. 2 3 ð Þ¼ ndV0 m=ℏ . The density-dependent conductivity for screened Adapted from Morozov et al., 2008. Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 436 Das Sarma et al.: Electronic transport in two-dimensional graphene 150 80 11 -2 (a) based on BLG screening being much stronger than MLG n i =4.5x10 cm 1.2 (b) 2 2 0 n d0V =1.0 (eVA) screening leading to the relative importance of short-range 60 σ / 100 1.0 scattering in BLG. long-range short-range 0.0 0.5 1.0 2 2 40 T/TF (e /h) (e 1.2 (e /h) 0 σ 50 σ c. Semiconductor systems 1.0 20 σ / long range + short-range 0.8 Transport properties of 2D semiconductor-based parabolic 0.0 0.5 1.0 T/T 0 0 F 2D systems (e.g., Si-MOSFETs, GaAs heterostructures, and 01234 012345 quantum wells, SiGe-based 2D structures) have been studied n (1012 cm -2 ) n (10 12 cm -2 ) extensively over the last 40 years (Ando et al., 1982; FIG. 17 (color online). (a) Density dependence of bilayer gra- Abrahams et al., 2001; Kravchenko and Sarachik, 2004). phene conductivity with two scattering sources: screened long-range More recently, 2D transport properties have attracted atten- Coulomb disorder and short-ranged neutral disorder. (b) Density tion because of the experimental observation of an apparent dependence of BLG conductivity for different temperatures: T 0, metallic behavior in the high-mobility low-density electron 50, 100, 150, 200, and 300 K (from bottom to top). Top inset shows¼ inversion layer in Si-MOSFET structures (Kravchenko et al., as a function of T in presence of short-range disorder. Bottom 1994). However, in this review we do not make any attempt at inset shows as a function of T in presence of screened Coulomb reviewing the whole 2D metal-insulator transition (MIT) 11 2 disorder for different densities n 5; 10; 30 10 cmÀ (from literature. Early comprehensive reviews of 2D MIT can be ¼½  bottom to top). Adapted from Das Sarma et al., 2010. found in the literature (Abrahams et al., 2001; Kravchenko and Sarachik, 2004). More recent perspectives can be found dependence of the screened charge impurity scattering, i.e., in Das Sarma and Hwang (2005) and Spivak et al. (2010). the thermal suppression of the 2kF peak associated with Our goal in this review is to provide a direct comparison of backscattering (see Fig. 13). For the short-ranged scattering, the transport properties of 2D semiconductor systems with the temperature dependence only comes from the energy those of MLG and BLG, emphasizing similarities and averaging and the conductivity becomes T 0 1 differences. 1=t ð Þ¼ ð Þ½ þ t ln 1 eÀ , where t T=TF. At low temperatures the It is well known that the long-range charged impurity conductivityð þ Þ is exponentially¼ suppressed, but at high tem- scattering and the short-range surface-roughness scattering peratures it increases linearly. dominate, respectively, in the low and high carrier density Figure 17(b) shows the finite temperature BLG conductiv- regimes of transport in 2D semiconductor systems. In Fig. 18 ity as a function of n. The temperature dependence is very the experimental mobility of Si-MOSFETs is shown as a weak at higher densities as observed in recent experiments function of density. As density increases, the measured mo- (Morozov et al., 2008). At low densities, where T=TF is not bility first increases at low densities, and after reaching the too small, there is a strong insulating-type T dependence maximum mobility it decreases at high densities. This behav- arising from the thermal excitation of carriers (which is ior is typical for all 2D semiconductor systems, even though exponentially suppressed at higher densities) and energy the mobility of GaAs systems decreases very slowly at averaging, as observed experimentally (Morozov et al., high densities. This mobility behavior in density can be 2008). Note that for BLG TF 4:23n~ K, where n~ explained with mainly two scattering mechanisms as shown 10 2 ¼ ¼ n= 10 cmÀ . In the bottom inset the conductivity due to in Fig. 18(b). In the low-temperature region phonons do not screenedð CoulombÞ disorder is shown as a function of tem- play much of a role in resistive scattering. At low carrier perature for different densities. At low temperatures densities long-range Coulomb scattering by unintentional (T=TF 1) the conductivity decreases linearly with tem- random charged impurities invariably present in the environ- perature, but T increases quadratically in high- ment of 2D semiconductor systems dominates the 2D mobil- temperature limit.ð ByÞ contrast, for the short-range disorder ity (Ando et al., 1982). However, at high densities as more always increases with T, as shown in the upper inset of carriers are pushed to the interface the surface-roughness Fig. 17(b). Thus for bilayer graphene, the metallic behavior scattering becomes more significant. Thus transport in 2D due to screening effects is expected at very low temperatures semiconductor systems is limited by the same mechanisms as for low-mobility samples, in which the screened Coulomb in graphene even though at high densities the unknown disorder dominates. In Fig. 15(b) the temperature dependence short-range disorder in graphene is replaced by the surface- of mobility for bilayer graphene is shown. As we expect, the roughness scattering in 2D semiconductor systems. The cru- metallic behavior shows up at very low temperatures cial difference between 2D transport and graphene transport (T<100 K). is the existence of the insulating behavior of 2D semiconduc- We conclude this section by emphasizing the similarity and tor systems at very low densities which arises from the the difference between BLG and MLG transport at high gapped nature of 2D semiconductors. However, the high- densities from the perspective of Boltzmann transport theory density 2D semiconductor transport is not qualitatively differ- considerations. In the MLG the linear density-dependent ent from graphene transport since charged impurity scattering conductivity arises entirely from Coulomb disorder. dominates carrier transport in both cases. However, in the BLG the existence of short-range disorder The experimentally measured conductivity and mobility scattering must be included to explain the linearity because for three different systems as a function of density are shown the Coulomb disorder gives rise to a higher power density in Figs. 18 and 19. At high densities, the conductivity depends dependence in conductivity. The importance of short-range on the density as n with 1 < <2, where n scattering in BLG compared with MLG is understandable depends weakly on the/ density for a given system but variesð Þ Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 437 strongly from one system (e.g. Si-MOSFET) to another (e.g., GaAs). At high densities, before surface-roughness scattering sets in the conductivity is consistent with screened charged impurity scattering for all three systems. As n decreases, n starts decreasing faster with decreasing density and the ex-ð Þ perimental conductivity exponent becomes strongly den- sity dependent with its value increasing substantially, and the conductivity vanishes as the density further decreases. To explain this behavior, a density-inhomogeneity-driven perco- lation transition was proposed (Das Sarma et al., 2005), i.e., the density-dependent conductivity vanishes as n n ð Þ/ð À n p with the exponent p 1:2 being consistent with a pÞ ¼ percolation transition. At the lowest density, linear screening in a homogeneous electron gas fails qualitatively in explain- ing the n behavior; whereas, it gives quantitatively accu- rate resultsð atÞ high densities. As found from direct numerical simulations (Efros, 1988; Nixon and Davies, 1990; Shi and Xie, 2002), homogeneous linear screening of charged impu- rities breaks down at low carrier densities with the 2D system developing strong inhomogeneities leading to a percolation transition at n FIG. 19. (a), (b) Experimentally measured (symbols) and calculated (lines) conductivity of two different n-GaAs samples. The high density conductivity limited by the charged impurities fit well to the experimental data. Adapted from Das Sarma et al., 2005. (c) Mobility of p-GaAs 2D system vs density at fixed temperature T 47 mK. (d) The corresponding conductivity vs density (solid squares) along with the fit generated assuming a percolation transition. The¼ dashed line in (c) indicates the p0:7 behavior. Adapted from Manfra et al., 2007. Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 438 Das Sarma et al.: Electronic transport in two-dimensional graphene 1 2 1 stand the (or ) behavior at high density, we start with the À q0 nÀ , then the conductivity behaves as n 2 / / 1 0 0 ð Þ/ Drude-Boltzmann semiclassical formula, Eq. (3.2), for 2D n . In the weak screening limit À q0 n and n transport limited by screened charged impurity scattering n. These conductivity behaviors are common/ / for 2D systemsð Þ/ (Das Sarma and Hwang, 1999). However, due to the finite with parabolic bands and are qualitatively similar to graphene extent in the z direction of the real 2D semiconductor system, where n behavior is observed. However, due to the the Coulomb potential has a form factor depending on the complicated/ impurity configuration (spatial distribution of details of the 2D structure. For comparison with graphene, we impurity centers) and finite width effects of real 2D semicon- consider the simplest case of 2D limit, i.e., layer. For ductor systems the exponent varies with systems. In gen- layer 2D systems with parabolic band, the scattering times at eral, modulation doped GaAs systems have larger than T 0 for charged impurity centers with impurity density n ¼ i Si-MOSFETs due to the configuration of impurity centers. located at the 2D systems are calculated by An interesting transport property of 2D semiconductor systems is the observation of the extremely strong anomalous 1 1 d 2 2 q f q0 ; (3.26) metallic (i.e., d=dT > 0) temperature dependence of the ¼ À dq ½ 0 ð Þ 0 0 resistivity T in the density range just above a critical 1 2 2 ð Þ where 0À 2ℏ ni=m 2=gsgv q0, q0 qTF=2kF (qTF is carrier density nc where d=dT changes its sign at low a 2D Thomas-Fermi¼ ð waveÞð vector),Þ and¼f x is given in temperatures (see Fig. 20), which is not seen in graphene. Eq. (3.22). Then, the density dependenceð ofÞ conductivity Note that the experimentally measured T of graphene can be expressed as n n with 1 < <2. In the strong shows very weak metallic behavior at highð densityÞ due to screening limit (q ð Þ1/) the scattering time becomes the weak temperature dependence of screening function. It 0 FIG. 20 (color online). (a) Experimental resistivity of Si-MOSFET as a function of temperature at 2D electron densities (from top to bottom) n 1:07; 1:10; 1:13; 1:20; 1:26; 1:32; 1:38; 1:44; 1:50; 1:56; 1:62; and 1:68 1011 cm2. Inset (b) shows for n 1:56; 1:62; and 1:68 ¼½1011 cm2. (c) Theoretically calculated temperature and density-dependent resistivity for sample A for densities¼ ½n 1:26; 1:32; 1:38Â; 1:44; 1:50; 1:56; 1:62; and 1:68 1011 cm2 (from top to bottom). Adapted from Tracy et al., 2009. (e) Experimental ¼½ 9 2  10 10 2 T for n-GaAs (where nc 2:3 10 cmÀ ). The density ranges from 0:16 10 to 1:06 10 cmÀ . Adapted from Lilly et al., 2003. ð Þ ¼    9 9 2 (f) Temperature dependence of the resistivity for p-GaAs systems for densities ranging from 9:0 10 to 2:9 10 cmÀ . Adapted from Manfra et al., 2007.   Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 439 has been suggested (Das Sarma and Hwang, 1999) that the C. Phonon scattering in graphene anomalously strong metallic temperature dependence discov- ered in 2D semiconductor systems arises from the physical In this section we review the phonon scattering limited mechanism of temperature, density, and wave vector carrier transport in graphene. Lattice vibrations are inevitable dependent screening of charged impurity scattering in 2D sources of scattering and can dominate transport near room semiconductor structures, leading to a strongly temperature- temperature. They constitute an intrinsic scattering source, dependent effective quenched disorder controlling T;n at i.e., they limit the mobility at finite temperatures when all low temperatures and densities. Interaction effects alsoð leadÞ extrinsic scattering sources are removed. In general, three to a linear-T conductivity in 2D semiconductors (Zala et al., different types of phonon scattering are considered: intra- 2001). valley acoustic and optical phonon scattering which induce With temperature-dependent screening function " q; T in the electronic transitions within a single valley, and interval- Eq. (3.5), the asymptotic low- (Das Sarma and Hwang,ð 2003Þ ) ley phonon scattering that induces electronic transitions be- and high- (Das Sarma and Hwang, 2004) temperature behav- tween different valleys. iors of 2D conductivity are given by The intravalley acoustic phonon scattering is induced by low energy phonons and is considered an elastic process. The t 1 2D 1 C T=T ; (3.27a) temperature-dependent phonon-limited resistivity (Stauber ð Þ 0 ½ À 1ð FÞ et al., 2007; Hwang and Das Sarma, 2008a) was found to 2D t 1 T=TF 3pq0=4 TF=T ; (3.27b) ð Þ 1 ½ þð Þ be linear (i.e., ph T) for T>TBG, where TBG is the Bloch- qffiffiffiffiffiffiffiffiffiffiffiffi / 2D ffiffiffiffi Gru¨neisen (BG) temperature (Kawamura and Das Sarma, where t T=TF, 0 T 0 , C1 2q0= 1 q0 , and 1992), and T T4 for T Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 440 Das Sarma et al.: Electronic transport in two-dimensional graphene larly since the total phonon contribution itself is much smaller than the total extrinsic contribution). In addition, the experi- mental phonon contribution is obtained assuming Matthiessen’s rule, i.e., , where is the tot ¼ ph þ i tot total resistivity contributed by impurities and defects (i) and phonons (ph), which is not valid at room temperature (Hwang and Das Sarma, 2008a). Thus, two different groups (Chen, Jang, Xiao et al., 2008; Morozov et al., 2008) (Figs. 21 and 22) have obtained totally different behavior of phonon contribution to resistivity. Morozov et al. (2008) found that the temperature dependence is a rather high power (T5) at room temperatures, and the phonon contribution is independent of carrier density. Chen, Jang, Xiao et al. (2008) showed that the extracted phonon contribution is strongly density dependent and is fitted with both linear T from acoustic phonons and Bose-Einstein distribution. Therefore, the phonon contribution, as determined by a simple subtrac- tion, could have large errors due to the dominance of extrinsic scattering. In this section we describe transport only due to the longitudinal acoustic phonons since either the coupling to other graphene lattice modes is too weak or the energy scales of these (optical) phonon modes are far too high for them to FIG. 21 (color online). Temperature-dependent resistivity of gra- provide an effective scattering channel in the temperature phene on SiO2. Resistivity of two graphene samples as a function of range (5 to 500 K) of our interest. Since graphene is a non- temperature for different gate voltages. Dashed lines are fits to the polar material, the most important scattering arises from the linear T dependence with Eq. (3.32). Adapted from Chen, Jang, deformation potential due to quasistatic deformation of the Xiao et al., 2008. lattice. Within the Boltzmann transport theory (Kawamura and Das Sarma, 1990; Kawamura and Das Sarma, 1992), the in the current graphene samples is completely dominated relaxation time due to deformation potential coupled acoustic by extrinsic scattering (impurity scattering described in phonon mode is given by Sec. III.B), even at room temperatures the experimental extraction of the pure phonon contribution to graphene re- 1 1 f "0 1 cos W À ð Þ ; (3.28) sistivity is not unique. In particular, the impurity contribution kk0 kk0 " ¼ k ð À Þ 1 f " to resistivity also has a temperature dependence arising from ð Þ X0 À ð Þ Fermi statistics and screening which, although weak, cannot where kk0 is the scattering angle between k and k0, " be neglected in extracting the phonon contribution (particu- ℏv k , and W is the transition probability from the state¼ Fj j kk0 with momentum k to the state with momentum k0 and is given by 2 W C q 2Á "; " ; (3.29) kk0 ¼ ℏ j ð Þj ð 0Þ Xq where C q is the matrix element for scattering by acoustic phonons,ð andÞ Á "; " is given by ð 0Þ Á "; " N " " ! ð 0Þ¼ q ð À 0 þ qÞ N 1 " " ! ; (3.30) þð q þ Þ ð À 0 À qÞ where ! ℏv q is the acoustic phonon energy with v q ¼ ph ph the phonon velocity and Nq the phonon occupation number N 1= exp ! 1 . The first (second) term is Eq. (3.30) q ¼ ½ ð qÞÀ corresponds to the absorption (emission) of an acoustic pho- non of wave vector q k k0. The matrix element C q is FIG. 22 (color online). Temperature-dependent resistivity for four independent of the phonon¼ À occupation numbers. The matrixð Þ different MLG samples (symbols). The solid curve is the best fit by element C q 2 for the deformation potential is given by using a combination of T and T5 functions. The inset shows T j ð Þj dependence of maximum resistivity at the neutrality point for MLG D2ℏq q 2 and BLG (circles and squares, respectively). Adapted from Morozov C q 2 1 ; (3.31) et al., 2008. j ð Þj ¼ 2Amvph À 2k Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 441 where D is the deformation-potential coupling constant, m is the graphene mass density, and A is the area of the sample. The scattering of electrons by acoustic phonons may be considered quasielastic since ℏ! E , where E is the q F F Fermi energy. There are two transport regimes, which apply to the temperature regimes T TBG and T TBG, depend- ing on whether the phonon system is degenerate (Bloch- Gru¨neisen) or nondegenerate [equipartition (EP)]. The characteristic temperature T is defined as k T BG B BG ¼ 2ℏk v , which is given, in graphene, by T F ph BG ¼ 2vphkF=kB 54pn K with density measured in units of FIG. 23 (color online). (a) Acoustic phonon-limited mobility of 12 2 n-GaAs 2D system as a function of density for two different n 10 cmÀ . Theffiffiffi relaxation time in the EP regime is calculated¼ to be (Stauber et al., 2007; Vasko and Ryzhii, temperatures. (b) Calculated n-GaAs mobility as a function of 2007; Hwang and Das Sarma, 2008a) temperature for different impurity densities. At low temperatures (T<1K) the mobility is completely limited by impurity scattering. 1 1 " D2 Adapted from Hwang and Das Sarma, 2008b. k T: (3.32) " ¼ ℏ3 4v2 v2 B ð Þ F m ph Thus, in the nondegenerate EP regime (ℏ! k T) the the mobility below TBG is completely limited by extrinsic q B scattering rate [1= " ] depends linearly on the temperature. impurity scattering in 2D systems. Above the BG regime (or ð Þ T>4K), the mobility is dominated by phonons. In this limit At low temperatures (TBG T EF=kB) the calculated conductivity is independent of electron density. Therefore, the mobility limited by phonon scattering is much lower than the electronic mobility in graphene is inversely proportional that for charged impurity scattering. Therefore, it will be to the carrier density, i.e., 1=n. The linear temperature impossible to raise 2D mobility (for T>4K) by removing dependence of the scattering/ time has been reported for the extrinsic impurities since acoustic phonon scattering sets nanotubes (Kane et al., 1998) and graphites (Pietronero the intrinsic limit at these higher temperatures (for T> et al., 1980; Woods and Mahan, 2000; Suzuura and Ando, 100 K, optical phonons become dominant) (Pfeiffer et al., 2002b). 1989). In the BG regime the scattering rate is strongly reduced by In Fig. 24, the acoustic phonon-limited graphene mobility 1 the thermal occupation factors because the phonon popula- en À is shown as functions of temperature and car- ð Þ 10 2 2 tion decreases exponentially, and the phonon emission is rier density, which is given by * 10 =D nT~ cm =Vs prohibited by the sharp Fermi distribution. Then, in the where D is measured in eV, the temperature T in K, and n~ 12 2 carrier density measured in units of 10 cmÀ . Thus, the low-temperature limit T TBG the scattering time becomes (Hwang and Das Sarma, 2008a) acoustic phonon scattering limited graphene mobility is in- versely proportion to T and n for T>TBG. Also with the 2 1 1 1 1 D 4! 4 4 generally accepted values in the literature for the graphene ð Þ4 kBT : (3.33) EF kF 2mvph ℏvph ð Þ sound velocity and deformation coupling (Chen, Jang, Xiao h i ð Þ et al., 2008) (i.e., v 2 106 cm=s and deformation po- ph ¼  Thus, the temperature-dependent resistivity in BG regime tential D 19 eV), could reach values as high as becomes T4. Even though the resistivity in the EP regime 5 2 ¼ 12 2 / 10 cm =Vs for lower carrier densities (n & 10 cmÀ ) at is density independent, Eq. (3.33) indicates that the calculated T 300 K (Hwang and Das Sarma, 2008a; Shishir and resistivity in BG regime is inversely proportional to the Ferry,¼ 2009). For larger (smaller) values of D, would be 3=2 1 density, i.e., n since D E . More ex- 2 BG / À /½ ð FÞh iÀ smaller (larger) by a factor of D . It may be important to perimental and theoretical work would be needed for a emphasize here that we know of no other system where the precise quantitative understanding of phonon scattering effect intrinsic room-temperature carrier mobility could reach a on graphene resistivity. value as high as 105 cm2=Vs, which is also consistent with D. Intrinsic mobility Based on the results of previous sections, one can extract the possible (hypothetical) intrinsic mobility of 2D systems when all extrinsic impurities are removed. In Fig. 23 the acoustic phonon-limited mobility is shown for 2D n-GaAs system. For lower temperatures, T increases by a large 7 ð Þ factor ( TÀ for deformation-potential scattering and 5 / / TÀ for piezoelectric scattering) since one is in the Bloch- Gru¨neisen regime where phonon occupancy is suppressed exponentially (Kawamura and Das Sarma, 1990; Kawamura FIG. 24 (color online). Calculated graphene mobility limited by and Das Sarma, 1992). Thus the intrinsic mobility of semi- the acoustic phonon with the deformation-potential coupling con- conductor systems is extremely high at low temperatures stant D 19 eV (a) as a function of temperature and (b) as a ¼ (T Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 442 Das Sarma et al.: Electronic transport in two-dimensional graphene the experimental conclusion by Chen, Jang, Xiao et al. J0 kR0 k 0 1 (2008), Morozov et al. (2008), and Hong et al. (2009). k arctan ð Þ ! ; (3.35) ¼À Y0 kR0 ! À 2 ln kR0 This would, however, require the elimination of all extrinsic ð Þ ð Þ scattering, and first steps in this direction have been taken in where J0 x [Y0 x ] is the zeroth order Bessel function of the fabricating suspended graphene samples (Bolotin, Sikes, first (second)ð Þ kind.ð Þ Expanding for small carrier density, one Jiang et al., 2008; Du et al., 2008). Finally, we point out then finds for the conductivity (Stauber et al., 2007) the crucial difference between graphene and 2D GaAs in 2 phonon-limited mobility. In the 2D GaAS system the acoustic 2e n 2 ln kFR0 ; (3.36) phonon scattering is important below T 100 K and polar ¼ h nd ð Þ optical phonon scattering becomes exponentially¼ more im- portant for T * 100 K; whereas, in graphene a resistivity which other than the logarithmic factor, mimics the behavior linear in T is observed up to very high temperatures of charged impurities, and is linear in carrier density. ( 1000 K) since the relevant optical phonons have very In recent experimental work, Chen, Cullen et al. (2009) high energy ( 2000 K) and are simply irrelevant for carrier irradiated graphene with He and Ne ions to deliberately create transport. large vacancies in the graphene sheet. They further demon- strated that these vacancies induced by ion irradiation gave rise to a strong D peak in the Raman spectra, inferring that the E. Other scattering mechanisms absence of such a D peak in the pristine graphene signalled the lack of such defects (Fig. 25). Moreover, they demon- 1. Midgap states strated that while transport in pristine graphene is dominated The Boltzmann transport theory developed in Sec. III.A is by charged impurities, after ion irradiation the electron scat- considered the limit of weak scattering. One can ask about the tering off these vacancies appears consistent with the theory opposite limit of very strong scattering. The unitarity of the including midgap states [Eq. (3.36)]. In this review, we con- wave functions implies that a potential scatterer can only sider only the case where the disorder changes graphene’s cause a phase shift in the outgoing wave. Standard treatment transport properties without modifying its fundamental of s-wave elastic scattering gives the scattering time chemical structure (Hwang et al., 2007b; Schedin et al., 2007). The subject of transport in graphane (Sofo et al., ℏ 8n d sin2 ; (3.34) 2007; Elias et al., 2009) and other chemical derivatives of ¼ D E ð kÞ graphene is beyond the scope of this work; see, e.g., Robinson k ð kÞ et al.(2008), Bostwick et al. (2009), Cheianov et al. (2009), where the conductivity is then given by the Einstein relation Geim (2009), and Wehling et al. (2009a, 2009b). 2e2=h v k . ¼ð Þ F F kF To model the disorder potential induced by a vacancy, 2. Effect of strain and corrugations Hentschel and Guinea (2007) assumed a circularly symmetric potential with V 0 FIG. 25 (color online). Left panel: Raman spectra (wavelength 633 nm) for (a) pristine graphene and (b) graphene irradiated by 500 eV Neþ ions that are known to cause vacancies in the graphene lattice. Right panel: Increasing the number of vacancies by ion irradiation caused a transition from the pristine graphene (where Coulomb scattering dominates) to the lower curves where scattering from vacancies dominate. Also shown is a fit to Eq. (3.36) from Stauber et al., 2007 that describes scattering off vacancies that have midgap states. From Chen, Cullen et al., 2009. Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 443 2009; Thompson-Flagg et al., 2009). It is nonetheless an IV. TRANSPORT AT LOW CARRIER DENSITY important theoretical question to address the nature of elec- tronic scattering off such ripples. Ripples, by their very A. Graphene minimum conductivity problem nature, are correlated long-range fluctuations across the entire sample (i.e. most experiments measuring ripples calculate a 1. Intrinsic conductivity at the Dirac point height-height correlation function). Yet, for electronic trans- One of the most discussed issues in the context of funda- port, one would like to isolate a ‘‘single ripple’’ and calculate mental graphene physics has been the so-called minimum (or its scattering cross section (assuming that the rest of the minimal) conductivity problem (or puzzle) for intrinsic gra- sample is flat), and then treat the problem of electrons phene. In the end, the graphene minimum conductivity prob- scattering off ripples as that of random uncorrelated impuri- lem turns out to be an ill-posed problem, which can only be ties with the cross section of a single ripple. This was the solved if the real physical system underlying intrinsic (i.e., approach followed by Guinea (2008), Katsnelson and Geim undoped) graphene is taken into account. An acceptable and (2008), and Prada et al. (2010). reasonably quantitatively successful theoretical solution of With this qualitative picture in mind, one could estimate the minimum conductivity problem has only emerged in the the transport time due to ripples as last couple of years, where the theory has to explicitly incorporate carrier transport in the highly inhomogeneous ℏ electron-hole landscape of extrinsic graphene, where density 2D EF VqV q ; (3.37) ð Þh À i fluctuations completely dominate transport properties for actual graphene samples. The graphene minimum conductivity problem is the di- where Vq is the scattering potential caused by the strain fields of a single ripple. chotomy between the theoretical prediction of a universal Introducing a height field h r (that measures displace- Dirac point conductivity D of undoped intrinsic graphene ments normal to the graphene sheet),ð Þ one finds (Katsnelson and the actual experimental sample-dependent nonuniversal and Geim, 2008) minimum of conductivity observed in gated graphene devices at the charge neutrality point with the typical observed mini- mum conductivity being much larger than the universal v 2 ℏ F prediction. VqV q hq q1 hqh q q2 h q2 h À i a q ;q h À À þ À i Unfortunately, is ill-defined, and depending on the X1 2 D q q q q q q ; (3.38) theoretical methods and approximation schemes, many dif-  ½ð À 1ÞÁ 1½ð À 2ÞÁ 2 ferent universal results have been predicted (Fradkin, 1986; Ludwig et al., 1994; Aleiner and Efetov, 2006; Altland, where a is the lattice spacing. Following Ishigami et al. 2006; Peres et al., 2006; Tworzydło et al., 2006; (2007), ripple correlations can be parametrized as h r Bardarson et al., 2007; Fritz et al., 2008; Kashuba, 2008): h 0 2 r2H, where the exponent H provides informationh½ ð ÞÀ ð Þ i¼ about the origin of the ripples. An exponent 2H 1 indicates 4e2 e2 ¼ ; ; 0; ; that height fluctuation domains have short-range correlations, D ¼ h 2h 1 implying that graphene conforms to the morphology of the 1 underlying substrate, while 2H 2 suggests a thermally and other values. The conductivity T;!;F; À; Á;LÀ is excitable membrane only loosely¼ bound by Van der Waals in general a function of many variables:ð temperature (ÞT), forces to the substrate. Ishigami et al. (2007) found experi- frequency (!), Fermi energy or chemical potential (F), mentally that 2H 1:11 0:013, implying that graphene impurity scattering strength or broadening (À), intervalley mostly conforms to the substrate,Æ but with some intrinsic scattering strength (Á), and system size (L). The Dirac point stiffness. Katsnelson and Geim (2008) showed that this has conductivity of clean graphene D 0; 0; 0; 0; 0; 0 is obtained consequences for transport properties, where for 2H 1, in the limit of all the independentð variables beingÞ zero, and 2 2H 1 ¼ 1=ln kFa ; for 2H>1, n n À . For the special the result depends explicitly on how and in which order these case of 2Hð 2Þ (flexural ripples),ð Þ this scattering mimics the limits are taken. For example, ! 0 and T 0 limit is not long-range Coulomb¼ scattering discussed in Sec. III.A. For necessarily interchangeable with the! T 0 and! ! 0 limit. the experimentally relevant case of 2H * 1, electron scatter- In addition, the limit of vanishing impurity! scattering! (À 0) ing off ripples would mimic short-range disorder also dis- and whether À 0 or À Þ 0 may also matter. In the ballistic! cussed in Sec. III.A. Thus, ripple scattering in graphene for limit (À 0),¼ the mesoscopic physics of the system size 2H 1 mimic surface-roughness scattering in Si-MOSFET being finite¼ (1=L Þ 0) or infinite (1=L 0) seems to matter. (Ando et al., 1982). We emphasize that these conclusions are The intervalley scattering being finite¼ (Á Þ 0) or precisely at best qualitative, since the approximation of treating the zero (Á 0) seems to matter a great deal because the scaling ripples as uncorrelated single impurities is quite drastic. A theory of localization predicts radically different results for complete theory for scattering off ripples in graphene is an D, D 0 for Á Þ 0, D for Á 0, in the presence interesting, and at present and open problem. Ripple scatter- of any finite¼ disorder (À Þ 0¼). 1 ¼ ing effects on graphene transport have a formal similarity to A great deal of the early discussion on the graphene the well-studied problem of interface roughness scattering minimum of conductivity problem has been misguided by effects on carrier transport in Si-SiO2 2D electron systems the existing theoretical work which considered the strict (Ando et al., 1982; Adam, Hwang, and Das Sarma, 2008; T 0 limit and then taking the ! 0 limit. Many theories Tracy et al., 2009). claim¼ 4e2= h in this! limit, but the typical D ¼ ð Þ Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 444 Das Sarma et al.: Electronic transport in two-dimensional graphene experimentally measured value is much larger (and sample opening of an intrinsic spectral gap in the graphene band dependent), leading to the so-called problem of the missing structure by using graphene nanoribbons (Han et al., 2007; pi. The limit lim! 0 !; T 0 is, in fact, experimentally Adam, Cho et al., 2008) or biased BLG (Oostinga et al., irrelevant since for! experimentalð ¼ Þ temperatures (even 10 mK), 2008; Zhang, Tang et al., 2009) immediately introduces an k T ℏ!, and thus the appropriate limiting procedure for insulating phase around the charge neutrality point. These B dc conductivity is limT 0 ! 0;T . There is an intuitive two features indicate that the insulating behavior in graphene way of studying this limit! theoretically,ð ¼ Þ which, however, can and 2D semiconductors is connected more with the existence only treat the ballistic (and therefore, the completely unreal- of a spectral gap than with the quantum localization istic disorder-free) limit. We first put ! 0 and assume phenomena. 0, i.e., intrinsic graphene. It is then easy¼ to show that ¼ 2 at T Þ 0, there will be a finite carrier density ne nh T 3. Zero-density limit thermally excited from the graphene valence band¼ to/ the conduction band. The algebraic T2 dependence of thermal It is instructive to think about the intrinsic conductivity as carrier density, rather than the exponentially suppressed ther- the zero-density limit of the extrinsic conductivity for gated mal occupancy in semiconductors, of course follows from the graphene. Starting with the Boltzmann theory high-density nonexistence of a band gap in graphene. Using the Drude result of Sec. III, we see that 2 formula for dc conductivity, we write D ne =m 2 ¼ / 0 Coulomb scattering; T T =m T , where and m are, respectively, the relaxation D n 0 (4.1) ð ! Þ¼ C zero-range scattering; timeð andÞ theð Þ effective mass. Although the graphene effective i mass is zero due to its linear dispersion, an effective definition where the nonuniversal constant Ci is proportional to the of effective mass follows from writing ℏvFk strength of the short-range scattering in the system. We 2 2 ¼ ¼ ℏ kF = 2m , which leads to m pn T (which vanishes note that the vanishing of the Boltzmann conductivity in the asð T Þ ð0) byÞ using k pn. This/ then leads/ to T T . intrinsic zero-density limit for Coulomb scattering is true for ffiffiffi D In the! ballistic limit,/ the only scattering mechanism isð theÞ both unscreened and screened Coulomb impurities. The non- ffiffiffi electron-hole scattering, where the thermally excited elec- vanishing of graphene Boltzmann conductivity for zero-range trons and holes scatter from each other due to mutual function scattering potential in the zero carrier density Coulomb interaction. This inelastic electron-hole scattering intrinsic limit follows directly from the gapless linear disper- rate 1= is given by the imaginary part of the self-energy sion of graphene carriers. We emphasize, however, that D is which, to the leading order, is given by 1= T, leading to nonuniversal for zero-range scattering. D T 1=T const in the ballistic limit. There are loga- For further insight into the zero-density Boltzmann limit ð Þ 1 rithmic subleading terms which indicate that D T 0 T 0 for , consider Eq. (3.3). In general, À E D E grows logarithmically at low temperature in theð ballistic! Þ since¼ the availability of unoccupied states forð scatteringÞ ð Þ limit. The conductivity in this picture, where interaction should be proportional to the density of states. This immedi- effects are crucial, is nonuniversal even in the ballistic limit, ately shows that the intrinsic limit EF n 0 0 is ex- depending logarithmically on temperature and becoming tremely delicate for graphene because Dð E! Þ0! 0, and infinite at T 0. The presence of any finite impurity disorder the product D becomes ill defined at theð Dirac! point.Þ! modifies the¼ whole picture completely. More details along We emphasize in this context, as discussed in Sec. I, that as this idea can be found in the literature (Foster and Aleiner, a function of carrier density (or gate voltage), graphene 2008, 2009; Fritz et al., 2008; Kashuba, 2008; Mu¨ller et al., conductivity (at high carrier density) is qualitatively identical 2008). to that of semiconductor-based 2DEG. This point needs emphasis because it seems not to be appreciated much in 2. Localization the general graphene literature. In particular, n n for both graphene and 2DEG with 1 for grapheneð Þ at inter- A fundamental mystery in graphene transport is the ab- mediate density and 0:3 to 1.5¼ in 2DEG depending on sence of any strong localization-induced insulating phase at the semiconductor system. At a very high density, 0 (or low carrier density around the Dirac point, where kFl 1 even negative) for both graphene and 2DEG. The precise since k 0 at the charge neutrality point and the transport F nature of density dependence (i.e., value of the exponent ) mean free path l is finite (and small). This is a manifest depends strongly on the nature of scattering potential and violation of the Ioffe-Reggel criterion which predicts strong screening, and varies in different materials with graphene localization for kFl & 1. By contrast, 2D semiconductor ( 1) falling somewhere in the middle between Si- systems always go insulating in the low-density regime. It MOSFETs ( 0:3) and modulation doped 2D n-GaAs is conceivable, but does not seem likely, that graphene may ( 1:5). Thus, from the perspective of high-density low- go insulating due to strong localization at lower temperatures. temperature transport properties, graphene is simply a rather Until that happens, the absence of any signature of strong low-mobility (comparable to Si-MOSFET, but much lower localization in graphene is a fundamental mystery deserving mobility than 2D GaAs) 2D semiconductor system. serious experimental attention. Two noteworthy aspects stand out in this context. First, no evidence of strong localization is 4. Electron and hole puddles observed in experiments that deliberately break the A-B sublattice symmetry (Chen, Cullen et al., 2009). Thus, the The low-density physics in both graphene and 2D semi- absence of localization in graphene cannot be attributed to conductors is dominated by strong density inhomogeneity the chiral valley symmetry of the Dirac fermions. Second, the (‘‘puddle’’) arising from the failure of screening. This Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 445 inhomogeneity is mostly due to the random distribution of Jang, Adam et al., 2008). In the next section we describe a unintentional quenched charged impurity centers in the envi- more elaborate density functional theory and an effective ronment. (In graphene, ripples associated with either intrinsic medium approximation to calculate the puddle electronic structural wrinkles or the substrate interface roughness may structure and the resultant transport properties (Rossi and also make contribution to the inhomogeneity.) At low density, Das Sarma, 2008; Rossi et al., 2009). the inhomogeneous puddles control transport phenomena in graphene as well as in 2D semiconductors. Inhomogeneous B. Quantum to classical crossover puddles would also form in doped 3D semiconductors at low carrier densities (Shklovskii and Efros, 1984). The starting point for the quantum transport properties at In Sec. IV.C we discuss the details of electron-hole-puddle the Dirac point discussed in Sec. II.C.1 is the ballistic uni- 2 formation in graphene around the charge neutrality point and versal minimum conductivity min 4e = h for clean describe its implications for graphene transport properties. graphene. The addition of disorder, i.e.,¼ includingð Þ potential Here we emphasize the qualitative difference between gra- fluctuations [given by Eq. (2.10)] that are smooth on the scale phene and 2D semiconductors with respect to the formation of the lattice spacing increases the conductivity through weak of inhomogeneous puddles. In 2D semiconductors, depending antilocalization. This picture is in contrast to the semiclassi- on whether the system is electron doped or hole doped, there cal picture discussed above where the transport properties are are only just electron or just hole puddles. At low density, calculated at high density using the Boltzmann transport n 0, therefore most of the macroscopic sample has little theory and the self-consistent theory is used to handle the finite carrier density except for the puddle regime. From the inhomogeneities of the carrier density around the Dirac point. transport perspective, the system becomes the landscape of This theory predicts that the conductivity decreases with mountains and lakes for a boat negotiating a hilly lake. When increasing disorder strength. Given their vastly different start- percolation becomes impossible, the system becomes an ing points, it is perhaps not surprising that the two approaches insulator. In graphene, however, there is no gap at the Dirac disagree. point, and therefore, the electron (hole) lakes are hole (elec- A direct comparison between the two approaches has not tron) mountains, and one can always have transport even at been possible mainly because the published predictions of the zero carrier density. This picture breaks down when a spectral Boltzmann approach include screening of the Coulomb dis- gap is introduced, and gapped graphene should manifest an order potential; whereas, the fully quantum-mechanical cal- insulating behavior around the charge neutrality point as it culations are for a noninteracting model using Gaussian indeed does experimentally. disorder. Notwithstanding the fact that screening and Coulomb scattering play crucial roles in transport of real 5. Self-consistent theory electrons through real graphene, the important question of the comparison between quantum and Boltzmann theories, The physical puddle picture discussed above enables one even for Gaussian disorder, was addressed only recently by to develop a simple theory for graphene transport at low Adam, Brouwer, and Das Sarma (2009), where they consid- densities using a self-consistent approximation where the ered noninteracting Dirac electrons at zero temperature with graphene puddle density is calculated by considering the potential fluctuations of the form shown in Eq. (2.10). They potential and density fluctuations induced by the charged numerically solved the full quantum problem for a sample of impurities themselves. Such a theory was developed by finite size L [where is the correlation length of the Adam et al. (2007). The basic idea is to realize that at low disorder potential in Eq. (2.10)], for a range of disorder carrier density n < n the self-consistent screening adjust- i strengths parametrized by K .2 ment between thej j impuritiesj j and the carriers could physically 0 Typical results for the quantum transport are shown in lead to an approximate pinning of the carrier density at n Fig. 26. For L & , the transport is ballistic and the conduc- n n . A calculation within the RPA approximation yields¼ à i tivity given by the universal value 4e2= h .For (Adam et al., 2007) min L , one is in the diffusive transport¼ regime.ð ForÞ the n diffusive regime, Adam, Brouwer, Das Sarma (2009) dem- à 2 RPA p 2rs C0 rs;a 4d nà ; (4.2) nimp ¼ ð ¼ Þ onstrated that away from the Dirac point, both the Boltzmann ffiffiffiffiffiffiffiffiffi theory and the full quantum theory agree to leading order with 4E a 2e ar CRPA r ;a 1 1ð Þ À s 0 ð s Þ¼À þ 2 r 2 þ 1 2r ð þ sÞ þ s 2rsa 1 2rsa e E1 2rsa þð þ Þ ð ½ 2Quantum effects are a small correction to the conductivity only if E a 1 2r ; À 1½ ð þ sÞÞ the carrier density n is increased at fixed sample size L. This is the experimentally relevant limit. If the limit L is taken at fixed n, where E z t 1e tdt is the exponential integral func- !1 1ð Þ¼ z1 À À quantum effects dominate [see Eq. (2.8)], where the semiclassical tion. This densityR pinning then leads to an approximately theory does not capture the logarithmic scaling of conductivity with constant minimum graphene conductivity which can be ob- system size. Here, we are not considering the conceptually simple tained from the high-density Boltzmann theory by putting in a question of how quantum transport becomes classical as the phase carrier density of nÃ. This simple intuitive self-consistent coherence length decreases, but the more interesting question of theory is found to be in surprisingly good agreement with how this quantum-Boltzmann crossover depends on the carrier all experimental observations (Adam et al., 2007; Chen, density and disorder strength. Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 446 Das Sarma et al.: Electronic transport in two-dimensional graphene 0.6 5 0.05 parameter range considered, which differs from the ] 2 e /4 Boltzmann theory at small K0. At large K0, the numerical h 4 [ R data follow the trend of the self-consistent theory which 0.4 0 2 1=2 ] 0 5 ] 2 predicts 2e K = h for K 10. This implies that L/ξ h 3 0 0 / e 2 ð Þ /4 e even at the Dirac point, for large enough disorder, the trans- h [ [4 R σ 2 port is semiclassical and described by the self-consistent 0.2 Boltzmann transport theory. 1 For smaller K0, Fig. 27 shows that upon reducing K0 below unity, the conductivity first decreases sharply consistent with 0 0 0 50 100 10 100 a renormalization of the mean free path due to the ultraviolet L/ξ L/ξ logarithmic divergences discussed in Sec. II.C.4. Upon re- ducing K0 further, the Dirac point conductivity saturates at 2 FIG. 26 (color online). Resistance R 1=G (left) and conductiv- the ballistic value min 4e =h (discussed in Sec. II.B.2). ¼ 1 ¼ ity (right) obtained using WdR=dL À as a function of sample In a closely related work, Lewenkopf et al. (2008) nu- ¼½ length L. The three curves shown are for W= 200, K0 2, and merically simulated a tight-binding model to obtain the con- n2 0, 0.25, and 1 [from top to bottom (bottom¼ to top)¼ in left ¼ ductivity and shot noise of graphene at the Dirac point using a (right) panel]. Dashed lines in the right panel show d=d lnL recursive Green’s function method. This method was then 2 ¼ 4e =h. The inset in the left panel shows the crossover to diffusive generalized to calculate the metal-insulator transition in gra- transport (L ). phene nanoribbons where, as discussed in Sec. II C 2, edge disorder can cause the Anderson localization of electrons 2 (Mucciolo et al., 2009). 2pe 2 3=2 2 1=2 n 2n O n : (4.3) The important conclusion of this section is that it provides ð Þ¼ Kffiffiffiffi0h ½ð Þ þ ð Þ the criteria for when one needs a full quantum-mechanical While this agreement is perhaps not surprising, it validates solution and when the semiclassical treatment is sufficient. the assumptions of both theories and demonstrates that they For either sufficiently weak disorder or when the source and are compatible at high carrier density. More interesting are drain electrodes are closer than the scattering mean free path, the results at the Dirac point. Generalizing the self-consistent then the quantum nature of the carriers dominates the trans- Boltzmann theory to the case of a Gaussian correlated dis- port. On the other hand, for sufficiently large disorder, or order potential [Eq. (2.10)], one finds away from the Dirac point, the electronic transport properties 2 1 of graphene are semiclassical and the Boltzmann theory SC 2e K0 K0 À min exp À I1 ; (4.4) correctly captures the most of graphene’s transport properties. ¼ h 2 2 where I is the modified Bessel function. Shown in the left 1 C. Ground state in the presence of long-range disorder panel of Fig. 27 is a comparison of the numerical fully quantum Dirac point conductivity where the weak antilocal- In the presence of long-range disorder that does not mix the ization correction has been subtracted 0 limL L degenerate valleys, the physics of the graphene fermionic 1 ¼ !1½ ð ÞÀ À ln L= with the semiclassical result [Eq. (4.4)]. excitations is described by the following Hamiltonian: Theð rightÞ panel shows the conductivity slightly away from the Dirac point (i.e., at the edge of the minimum conductivity 2 H d rÉy iℏv 1 É plateau). The numerical calculations at the edge of the plateau ¼ r ½À F Árr À r are in good quantitative agreement with the self-consistent Z 2 e 2 2 Boltzmann theory. At the Dirac point, however, the quantum d rd r0Éry Ér V r r0 Éy Ér þ 2 ðj À jÞ r0 0 conductivity K is found to increase with K for the entire 0 0 2 Z ð Þ e 2 d rV r Éy É ; (4.5) 4 þ 2 Dð Þ r r 3 10 Z ] ] h h / 2 / where vF is the bare Fermi velocity, Éry , Ér are the 2 2 e e 1 creation annihilation spinor operators for a fermionic excita- ’ [4 ’ [4 σ σ tion at position r and pseudospin , is the 2D vector 0 1 formed by the 2 2 Pauli matrices and acting in -1  x y pseudospin space, is the chemical potential, 1 is the 110100 1 10 K K 2 2 identity matrix, is the effective static dielectric 0 0 constant equal to the average of the dielectric constants of the materials surrounding the graphene layer, V r r0 FIG. 27 (color online). Semiclassical conductivity 0 ðj À jÞ ¼ 1 ¼ 1= r r0 is the Coulomb interaction, and V r is the bare limL L À ln L= vs disorder strength at the Dirac jj À j Dð Þ !1½ ð ÞÀ ð Þ 2 disorder potential. The Hamiltonian (4.5) is valid as long as point (left) and at carrier density n K0= , corresponding to the edge of the minimum conductivity¼ plateauð Þ of Adam et al. the energy of the fermionic excitations is much lower than the (2007) (right). Data points are from the numerical calculation for graphene bandwidth 3 eV. Using (4.5), if we know V , we D L 50 and the (solid) dashed curves represent the (self- can characterize the ground-state carrier density probability consistent)¼ Boltzmann theory. Adapted from Adam, Brouwer, and close to the Dirac point. In this section we focus on the case Das Sarma, 2009. when VD is a disorder potential whose spatial autocorrelation Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 447 decays algebraically, such as the disorder induced by ripples 2 H d rÉy iℏv 1 É or charge impurities. ¼ r ½À F Árr À r Z 2 d rÉy V n r É : (4.6) 1. Screening of a single charge impurity þ r KS½ ð Þ r Z The problem of screening at the Dirac point of a single The Kohn-Sham potential is given by the sum of the external impurity with charge Ze placed in (or close to) the graphene potential, the Hartree part of the interaction VH and an layer illustrates some of the unique features of the screening exchange-correlation potential Vxc that can only be known properties of massless Dirac fermions. In addition the prob- approximately. In its original form Vxc is calculated within lem provides a condensed matter realization of the QED the local density approximation (LDA) (Kohn and Sham, phenomenon of ‘‘vacuum polarization’’ induced by an exter- 1965), i.e., Vxc is calculated for a uniform liquid of electrons. nal charge (Darwin, 1928; Gordon, 1928; Pomeranchuk and The DFT-LDA approach can be justified and applied to the Smorodinsky, 1945; Case, 1950; Zeldovich and Popov, 1972). study of interacting massless Dirac fermions (Polini, Tomadin In the context of graphene the problem was first studied by et al., 2008). For graphene, the LDA exchange-correlation DiVincenzo and Mele (1984) and recently more in detail by potential within the RPA approximation is given with very several others (Biswas et al., 2007; Fistul and Efetov, 2007; good accuracy by the following expression (Gonza´lez et al., Fogler et al., 2007; Novikov, 2007b; Pereira et al., 2007; 1999; Katsnelson, 2006; Barlas et al., 2007; Hwang et al., Shytov et al., 2007a, 2007b; Terekhov et al., 2008). The 2007a; Mishchenko, 2007; Vafek, 2007; Polini, Tomadin parameter Ze2= ℏv Zr quantifies the strength of ð fÞ¼ s et al., 2008): the coupling between the Coulomb impurity and the massless Dirac fermions in the graphene layer. Neglecting e-e inter- rs 4kc Vxc n n sgn n ln actions for < 1=2 the Coulomb impurity induces a ð Þ¼þ4 j j ð Þ p4n j j pffiffiffiffiffiffiffiffiffiffi screening charge that is localized on length scales of the 2 grs grs ffiffiffiffiffiffiffiffiffi 4kc order of the size of the impurity itself (or its distance d ð Þ n sgn n ln ; (4.7) À 4 j j ð Þ p4n from the graphene layer). Even in the limit < 1=2 the pffiffiffiffiffiffiffiffiffiffi j j inclusion of the e-e interactions induces a long-range tail in where g is the spin and valley degeneracy factorffiffiffiffiffiffiffiffiffi (g 4), and ¼ the screening charge with sign equal to the sign of the charge kc is an ultraviolet wave-vector cutoff, fixed by the range of impurity (Biswas et al., 2007). For > 1=2, the Coulomb energies over which the pure Dirac model is valid. Without j j charge is supercritical, the induced potential is singular loss of generality, we can use kc 1=a, where a is the (Landau and Lifshitz, 1977), and the solution of the problem graphene lattice constant, corresponding¼ to an energy cutoff depends on the regularization of the wave function at the site Ec 3 eV. Equation (4.7) is valid for kF n kc. In of the impurity, r 0. By setting the wave function to be ¼ j j ! Eq. (4.7) is a constant that depends on rs givenpffiffiffiffiffiffiffiffiffiffi by (Polini, zero at r a, the induced electron density in addition to a Tomadin et al., 2008) localized ¼ r term, acquires a long-range tail 1=r2 (with sign opposite½ ð Þ to the sign of the charge impurity) (Novikov, 1 dx gr þ1 : (4.8) 2007a; Pereira et al., 2007; Shytov et al., 2007b) and s 2 2 2 ð Þ¼2 0 1 x p1 x grs=8 marked resonances appear in the spectral density (Fistul Z ð þ Þ ð þ þ Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi and Efetov, 2007; Shytov et al., 2007a) that should also The terms on the r.h.s. of (4.7) are the exchange and corre- induce clear signatures in the transport coefficients. Up until lation potential, respectively. Note that the exchange and now, neither the oscillations in the local density of states nor correlation potentials have opposite signs. In Fig. 28 the the predicted signatures in the conductivity (Shytov et al., exchange and correlation potentials and their sum Vxc are 2007a) have been observed experimentally. It is likely that in plotted as a function of n for rs 0:5. We see that in ¼ the experiments so far the supercritical regime > 1=2 has graphene the correlation potential is smaller than the ex- not been reached because of the low Z of thej barej charge change potential but, contrary to the case of regular impurities and renormalization effects. Fogler et al. (2007) parabolic-band 2DEGs, is not negligible. However, from pointed out, however, that the predicted effects for > 1=2 Eq. (4.7) we have that exchange and correlation scale with are intrinsic to the massless Dirac fermion modelj thatj how- n in the same way. As a consequence in graphene, the ever is inadequate when the small scale cutoff min d; a is correlation potential can effectively be taken into account ½ smaller than arspZ. by simply rescaling the coefficient of the exchange potential. In Fig. 28 the dotted line shows V for a regular parabolic- ffiffiffiffi xc 2. Density functional theory band 2DEG with effective mass 0:067me in a background with 4. The important qualitative difference is that Vxc in Assuming that the ground state does not have long-range graphene¼ has the opposite sign than in regular 2DEGs: due to order (Peres et al., 2005; Dahal et al., 2006; Min, Borghi interlayer processes in graphene the exchange-correlation et al., 2008), a practical and accurate approach to calculate potential penalizes density inhomogeneities contrary to the ground state of many-body problems is the density func- what happens in parabolic-band electron liquids. tional theory (DFT) (Hohenberg and Kohn, 1964; Kohn and Using the DFT-LDA approach, Polini, Tomadin et al. Sham, 1965; Kohn, 1999; Giuliani and Vignale, 2005). In this (2008) calculated the graphene ground-state carrier density approach the interaction term in the Hamiltonian is replaced for single disorder realizations of charge impurities and small by an effective Kohn-Sham (KS) potential VKS that is a samples (up to 10 10 nm). The size of the samples is  functional of the ground-state density n r Éy É : limited by the high computational cost of the approach. For ð Þ¼ r r P Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 448 Das Sarma et al.: Electronic transport in two-dimensional graphene 2p E n ℏv d2rsgn n n 3=2 ½ ¼ F 3 ð Þj j ffiffiffiffi Z rs 2 2 n r n r0 2 d r d r0 ð Þ ð Þ d rVxc n r n r þ 2 r r0 þ ½ ð Þ ð Þ Z Z j À j Z r d2rV r n r d2rn r ; (4.10) þ s Dð Þ ð ÞÀℏv ð Þ Z F Z where the first term is the kinetic energy, the second is the Hartree part of the Coulomb interaction, the third is the term due to exchange and correlation, and the fourth is the term due to disorder. The expression for the exchange-correlation potential is given in Eq. (4.7). The carrier ground-state dis- tribution is then calculated by minimizing E n with respect to n. Using (4.10), the condition E=n 0½requires FIG. 28 (color online). Exchange, solid line, and RPA correlation ¼ potentials, dashed line, as functions of the density n for r 0:5. s rs n r0 The dash-dotted line shows the full exchange-correlation potential¼ sgn n n d2r ð Þ V n r 2 r r xc V . The dotted line is the quantum Monte Carlo exchange- ð Þ j j þ 0 þ ½ ð Þ xc pffiffiffiffiffiffiffiffiffiffi Z j À j correlation potential of a standard parabolic-band 2D electron gas r V r 0: (4.11) þ s Dð ÞÀℏv ¼ (Attaccalite et al., 2002) with effective mass 0:067me placed in F background with dielectric constant 4. Adapted from Polini, Equation (4.11) well exemplifies the nonlinear nature of Tomadin et al., 2008. screening in graphene close to the Dirac point: because in graphene, due to the linear dispersion, the kinetic energy per single disorder realizations, the results of Polini, Tomadin carrier, the first term in (4.11), scales with pn when n 0 et al. (2008) showed that, as predicted (Hwang et al., 2007a), the relation between the density fluctuations n andh i¼ the ffiffiffi at the Dirac point the carrier density breaks up in electron- external disorder potential is not linear even when exchange hole puddles and that the exchange-correlation potential and correlation terms are neglected. suppresses the amplitude of the disorder-induced density We now consider the case when the disorder potential is fluctuations. Given its computational cost, the DFT-LDA due to random Coulomb impurities. In general the charge approach does not allow the calculation of disordered aver- impurities will be a 3D distribution C r , however we can aged quantities. assume to a very good approximation Cð Þr to be effectively ð Þ 2D. The reason is that for normal substrates such as SiO2, the 3. Thomas-Fermi-Dirac theory charge traps migrate to the surface of the oxide; moreover, any additional impurity charge introduced during the gra- An approach similar in spirit to the LDA-DFT is the phene fabrication will be located either on the graphene top Thomas-Fermi theory (Fermi, 1927; Thomas, 1927; Spruch, surface or trapped between the graphene layer and the sub- 1991; Giuliani and Vignale, 2005). Like DFT, the TF theory strate. We then assume C r to be an effective 2D random is a density functional theory: in the Thomas-Fermi theory the distribution located at theð averageÞ distance d from the gra- kinetic term is also approximated via a functional of the local phene layer. An important advantage of this approach is that it density n r . By Thomas-Fermi-Dirac (TFD) theory, we refer ð Þ limits the number of unknown parameters that enter the to a modification of the TF theory in which the kinetic theory to two: charge impurity density n and d. With functional has the form appropriate for Dirac electrons and imp this assumption, we have in which exchange-correlation terms are included via the exchange-correlation potential proper for Dirac electron C r V r dr 0 : (4.12) liquids as described above for the DFT-LDA theory. The D 0 ð2 Þ 2 1=2 ð Þ¼ r r0 d TF theory relies on the fact that if the carrier density varies Z ½j À j þ slowly in space compared to the Fermi wavelength, then the The correlation properties of the distribution C r of the ð Þ kinetic energy of a small volume with density n r is equal, charge impurities are a matter of long-standing debate in with good approximation, to the kinetic energyð ofÞ the same the semiconductor community. Because the impurities are volume of a homogeneous electron liquid with density n charged, one would expect the positions of the impurities to n r . The condition for the validity of the TFD theory is given¼ have some correlation; on the other hand, the impurities are byð theÞ following inequality (Giuliani and Vignale, 2005; Brey quenched (not annealed), they are either imbedded in the and Fertig, 2009b): substrate or between the substrate and the graphene layer or in the graphene itself. This fact makes it very difficult to know rn r the precise correlation of the charge impurity positions, but it jr ð Þj kF r : (4.9) n r ð Þ also ensures that to good approximation the impurity posi- ð Þ Whenever inequality (4.9) is satisfied, the TFD theory is a tions can be assumed to be uncorrelated: computationally efficient alternative to the DFT-LDA ap- C r 0; C r1 C r2 nimp r2 r1 ; (4.13) proach to calculate the ground-state properties of graphene h ð Þi¼ h ð Þ ð Þi¼ ð À Þ in presence of disorder. The energy functional E n in the where the angular brackets denote averaging over disorder TFD theory is given by ½ realizations. A nonzero value of C r can be taken into h ð Þi Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 449 account simply by a shift of the chemical potential . It is exchange-correlation term suppresses the amplitude of the easy to generalize the theory to correlated impurities (e.g., density fluctuations. This fact is clearly visible in Fig. 29(d) impurity clusters) if the correlation function is known. from which we can see that in the presence of exchange The parameters nimp and d that enter the theory are reliably correlation the density distribution is much narrower and fixed by the transport results (see Sec. III.A) at high doping. more peaked around zero. This result, also observed in the Transport results at high density indicate that d is of the DFT-LDA results (Polini, Tomadin et al., 2008), is a con- order of 1 nm; whereas, nimp varies depending on the sample sequence of the the fact that as discussed in Sec. IV.C.2 quality but in general is in the range n 1010–1012 cm 2, the exchange-correlation potential in graphene, contrary to imp ¼ À where the lowest limit applies to suspended graphene. The parabolic-band Fermi liquids, penalizes density distance d is the physical cutoff for the length scale of inhomogeneities. the carrier density inhomogeneities. Therefore, to solve In the presence of disorder, in order to make quantitative Eq. (4.11) numerically, one can use a spatial discretization predictions verifiable experimentally it is necessary to calcu- with unit step of the order of d. For the TFD results presented late disordered averaged quantities. Using TFD, both the below, it was assumed d 1nmand therefore a spatial step disorder average X of a given quantity X and its spatial correlation functionh i Áx Áy 1 nm was used.¼ ¼ ¼ Figure 29 shows the TFD results for the carrier density X2 r X r X X 0 X (4.14) distribution at the Dirac point in the presence of charge ð Þh½ ð ÞÀh i½ ð ÞÀh ii impurity disorder for a single disorder realization with nimp can be efficiently calculated. For conditions typical in experi- 12 2 ¼ 2 10 cmÀ and 2:5 corresponding to graphene on SiO2 ments, 500 disorder realizations are sufficient. From X r , with the top surface¼ in vacuum (or air). It is immediately clear one can extract the following quantities: ð Þ that as predicted (Hwang et al., 2007a), close to the Dirac 2 2 point the disorder induced by the charge impurities breaks up Xrms X 0 ;X FWHM of X r ; the carrier distribution in electron (n>0) and hole (n<0) h½ ð Þ i h½ ð Þ i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (4.15) puddles. The electron-hole puddles are separated by disorder- induced p-n junctions (PNJ). Apart from the PNJ, the carrier respectively the root mean square and the typical spatial density is locally always different from zero even though the correlation of the fluctuations of X. Using the TFD theory, average density n is set equal to zero. For this reason, in the both the spatial correlation function of the screened potential h i presence of disorder it is more correct to refer to the value of Vsc and carrier density are found to decay at long distance as the gate voltage for which n 0 as the charge neutrality 1=r3. This is a consequence of the weak screening properties h i¼ point (CNP) rather than Dirac point: the presence of long- of graphene and was pointed out by Adam et al. (2007) and range disorder prevents the probing of the physical properties Galitski et al. (2007). From the spatial correlation functions, of the Dirac point, i.e., of intrinsic graphene with exactly half n and are extracted. Figures 30(a) and 30(b) show rms n filling, zero density, everywhere. The important qualitative the calculated nrms and at the Dirac point as a function of results that can be observed even for a single disorder by nimp. The disorder averaged results show the effect of the comparing the results of Figs. 29(b) and 29(c) is that the exchange-correlation potential in suppressing the amplitude of the density inhomogeneities nrms, and in slightly increasing −V (eV) 12 −2 (a) D (b) n (10 cm ) (a) 6 No exch. d=1.0 nm (b) No exch. d=1.0 nm With exch. d=1.0 nm 20 With exch. d=1.0 nm No exch. d=0.3 nm No exch. d=0.3 nm 5 With exch. d=0.3 nm With exch. d=0.3 nm 15 4 imp /n 3 [nm] 10 y (nm) (nm) y ξ rms n 2 5 1 0 0 9 10 11 12 9 10 11 12 10 10 10 10 10 10 10 10 x (nm) x (nm) -2 -2 n (10 12 cm− 2 ) n [cm ] (c) (d) nimp (cm ) imp 14 with exch. 2 No exch. (c) (d) No exch. d=1.0 nm 12 Offset With exch. d=1.0 nm ] 0.35 3 No exch. d=0.3 nm 10 1.5 With exch. d=0.3 nm 8 0 6 Q 1 δ A y (nm) 0.3 counts [10 4 0.5 2 No exch. d=1.0 nm With exch. d=1.0 nm 0 No exch. d=0.3 nm With exch. d=0.3 nm 0 -2 -1 0 1 2 0.25 9 10 11 12 9 10 11 12 x (nm) 12 -2 10 10 10 10 10 10 10 10 n (10 cm ) -2 -2 n [cm ] nimp [cm ] imp FIG. 29 (color online). TFD results as a function of position for a single disorder realization with 2:5, d 1 nm, and n FIG. 30 (color online). TFD disorder averaged results for 2:5. imp ¼ 12 2 ¼ ¼ ¼ The solid (dashed) lines show the results obtained including (ne- 10 cmÀ . (a) Bare disorder potential VD. (b) Carrier density obtained neglecting exchange-correlation terms. (c) Carrier density glecting) exchange and correlation terms (a) nrms and (b) as a function of n . (c) Area A over which n r n Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 450 Das Sarma et al.: Electronic transport in two-dimensional graphene their correlation length. The effect of the exchange- with a strong and narrow peak arond zero. The double peak correlation potential increases as nimp decreases. At the structure for finite Vg provides direct evidence for the exist- Dirac point, the quantity can be interpreted as the effective ance of puddles over a finite voltage range. nonlinear screening length. Figure 30 shows that depends n = n decreases with n , a trend that is expected and rms h i h i weakly on nimp. The reason is that only characterizes the that has been observed indirectly in experiments by measur- spatial correlation of the regions in which the density is ing the inhomogenous broadening of the quasiparticle spec- relatively high. If a puddle is defined as a continuous region tral function (Hong et al., 2009). with same sign charges, then at the CNP the puddles have In the limit r 1 it is possible to obtain analytical results s always a size of the same order of the system size. Inside the using the TFD approach (Fogler, 2009). The first step is to puddles there are small areas with high density and size of separate the inhomogeneities of the carrier density and the order of tens of nanometers for typical experimental screened potential in slow, n, Vsc, and fast components, n, conditions, much smaller than the system size L. Vsc: This picture is confirmed in Fig. 30(c) in which the dis- n r n r n r ;Vsc r Vsc r Vsc r ; order averaged area fraction A0 over which n r n < ð Þ¼ ð Þþ ð Þ ð Þ¼ ð Þþ ð Þ j ð ÞÀh ij (4.17) nrms=10 is plotted as a function of nimp. As nimp decreases A0 increases reaching more than 1=3 at the lowest impurity where n and Vsc contain only Fourier harmonics with k<à densities. The fraction of area over which n r n is where 1=à is the spatial scale below which the spatial j ð ÞÀ10 h ij 2 less than 1=5 of nrms surpasses 50% for nimp & 10 cmÀ . variation of n, and Vsc are irrelevant for the physical proper- Figure 30(b) shows the average excess charge Q n 2 ties measured. For imaging experiments, 1=à is the spatial rms at the Dirac point as a function of nimp. Note that as defined resolution of the scanning tip and for transport experiments Q, especially at low n , grossly underestimates the num- 1=à is of the order of the mean free path. Let l imp imp ber of charges both in the electron puddles and in the small 1=2rspnimp and R is the nonlinear screening length. It is regions of size . This is because in the regions of size the assumed that l & 1=à R. With these assumptions and ffiffiffiffiffiffiffiffiffi imp density is much higher than nrms whereas the electron-hole neglecting exchange-correlation terms, from the TFD func- puddles have a typical size much larger than . Using the tional in the limit à kF and small nonlinear screening estimate n r =n 1= for the small regions and the local jr ð Þj ¼ terms compared to the kinetic energy term it follows (Fogler, value of n inside the regions and, for the electron-hole 2009) puddles the estimates V V sgn V V n sc sc sc sc rms n Vsc j 2j ð 2 Þ lnj j ; n nrms; n r ; (4.16) ð Þ’À ℏv À 2‘ ℏvà (4.18) jr ð Þj L ð Þ V ℏvÃ; we find that the inequality (4.9) is satisfied guaranteeing the j scj validity of the TFD theory even at the Dirac point. with Vsc given, in momentum space, by As we move away from the Dirac point more of the area is covered by electron (hole) puddles. However, the density 2rsℏvF Vsc k VD k n k ; (4.19) fluctuations remain large even for relatively large values of ð Þ¼ ð Þþ k ð Þ Vg. This is evident from Fig. 31 where the probability distri- where we have assumed for simplicity d 0. Equation (4.19) bution P n of the density for different values of n is shown can be approximated by the following¼ asymptotic expres- ð Þ h i in Fig. 31(a) and the ratio nrms= n as a function of n is sions: shown in Fig. 31(b). The probabilityh i distribution P n ish non-i Gaussian (Adam et al., 2007; Galitski et al., 2007ð Þ; Adam, VD k ; kR 1; Vsc k ð Þ kR (4.20) Hwang et al., 2009). For density n & n , P n does not ð Þ¼ VD k 1 kR ; kR 1: h i imp ð Þ ð Þ þ exhibit a single peak around n but rather a bimodal structure h i Equations (4.18) and (4.19) define a nonlinear problem that (a) 20 (b) must be solved self-consistently and that in general can only 12 -2 / counts 8 n 1 4 0.5 K ln 1= 4r K : (4.21) 0 ¼ ½ ð s 0Þ 0 0 -4 -2 0 2 4 810 0 12 13 1 6 10 10 K0 is the expansion parameter. To order O K0À , Vsc can be 12 -2 -2 n (10 cm ) Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 452 Das Sarma et al.: Electronic transport in two-dimensional graphene bilayer graphene (Deshpande, Bao, Zhao et al., 2009). In particular, Deshpande, Bao, Miao et al. (2009), starting from the topographic data, calculated the carrier density fluctua- tions due to the local curvature of the graphene layer using Eq. (4.24) and compared them to the fluctuations of the dI=dV map. The comparison shows that there is no corre- spondence between the density fluctuations induced by the curvature and the ones measured directly. This leads to the conclusion that even though the curvature contributes to a variation in the electrochemical potential, it is not the main factor responsible for the features in the dI=dV map. The results for BLG of Fig. 32(b) show that close to the CNP the density inhomogeneities are very strong also in BLG and are in semiquantitative agreement with theoretical pre- dictions based on the TF theory (Das Sarma et al., 2010). Using an SET, Martin et al. (2009) imaged the local density of states also in the quantum Hall regime. The dis- order carrier density landscape has also been indirectly ob- served in imaging experiments of coherent transport FIG. 32 (color online). (a) Carrier density profile map at the CNP (Berezovsky et al., 2010; Berezovsky and Westervelt, 2010). measured with an SET. The contour marks the zero-density contour. Adapted from Martin et al., 2007. (b) Spatial map, on a D. Transport in the presence of electron-hole puddles 80 nm 80 nm region, of the energy shift of the CNP in BLG  from STM dI=dV map. Adapted from Deshpande et al., 2009b. The previous section showed, both theoretically and ex- (c) 60 nm 60 nm constant current STM topography, and perimentally, that close to the Dirac point, in the presence of (d) simultaneous dI=dV map, at the CNP for MLG, V bias long-range disorder, the carrier density landscape breaks up 0:225 V;I 20 pA . Adapted from Zhang, Tang et al.,ð 2009¼. À ¼ Þ in electron-hole puddles. In this situation the transport prob- lem becomes the problem of calculating transport properties of a system with strong density inhomogeneities. The first An indirect confirmation of the existence of electron-hole step is to calculate the conductance of the puddles, Gp, and puddles in exfoliated graphene close to the CNP came from PNJ, GPNJ. the measurement of the magnetic field-dependent longitudi- We have G À, where À is a form factor of order 1 and p ¼ nal and Hall components of the resistivity H and H is the puddle conductivity. Away from the Dirac point (see xxð Þ xyð Þ (Cho and Fuhrer, 2008). Close to the Dirac point, the mea- Sec. III.A) the RPA-Boltzmann transport theory for graphene surements showed that xx H is strongly enhanced and in the presence of random charge impurities is accurate. H is suppressed, indicatingð Þ nearly equal electron and From the RPA-Boltzmann theory, we have xyð Þ ¼ hole contributions to the transport current. In addition, the e n n ;n ;r ; d; T . For the purposes of this section, jh ij ðh i imp s Þ experimental data were found inconsistent with uniformly it is convenient to explicitly write the dependence of on distributed electron and hole concentrations (two-fluid nimp by introducing the function model) but in excellent agreement with the presence of hnimp kF inhomogeneously distributed electron and hole regions of F rs; d; T 2nimp (4.25) equal mobility. ð Þ 2e ¼ vF The first STM experiments on exfoliated graphene were so that we can write performed by Zhang, Brar et al. (2009). The STM experi- ments provided the most direct quantitative characterization 2e2 n jh ij F rs; d; T : (4.26) of the carrier density distribution of exfoliated graphene. ¼ h nimp ð Þ Figure 32(c) shows the topography of a 60 60 nm2 area  Expressions for F r ; d; T at T 0 (or its inverse) were of exfoliated graphene, while Fig. 32(d) shows the dI=dV ð s Þ ¼ map of the same area. The dI=dV value is directly propor- originally given by Adam et al. (2007) [see Eq. (3.21)]. tional to the local density of states. We can see that there is no We can define a local spatially varying puddle conductivity r if r varies on length scales that are larger than the correlation between topography and dI=dV map. This shows ð Þ ð Þ that in current exfoliated graphene samples the rippling of mean free path l, i.e., graphene, either intrinsic or due to the roughness of the r 1 substrate surface, are not the dominant cause of the charge r ð Þ À l: (4.27) r density inhomogeneities. The dI=dV maps clearly reveal the ð Þ presence of high-density regions with characteristic length of By substituting n with n r , we then use Eq. (4.26) to define h i ð Þ 20 nm as predicted by the TFD results. and calculate the local conductivity: Recently, more experiments have been performed to di- 2e2 n r rectly image the electronic structure of both exfoliated single r ð Þ F rs; d; T : (4.28) layer graphene (Deshpande, Bao, Miao et al., 2009) and ð Þ¼ h nimp ð Þ Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 453 2 2 2 Considering that l h= 2e kF and using Eq. (4.28), then e 2e nrms the inequality (4.27¼) takesð the followingÞ form: GPNJ pnrmsp Gp ÀF rs; d; T j j ; ¼ h ¼ ð Þ h nimp ffiffiffiffiffiffiffiffiffiffiffiffi n r 1 F r ; d; T pn (4.32) r ð Þ À ð s Þ : (4.29) n r p nimp ð Þ ffiffiffi i.e., GPNJ Gp. In the limit rs 1 the inequality (4.32) is ffiffiffiffi valid for any value of n (Fogler, 2009). The inequality As shown in the previous sections, at the CNP most of the imp graphene area is occupied by large electron-hole puddles with (4.32) shows that, in exfoliated graphene samples, transport size of the order of the sample size L and density of the order close to the Dirac point is not percolative: the dominant of n n . For graphene on SiO , we have r 0:8 for contribution to the electric resistance is due to scattering rms imp 2 s ¼ which is F 10. Using these facts, we find that the inequality events inside the puddles and not to the resistance of the (4.29) is satisfied¼ when puddle boundaries (Fogler, 2009; Rossi et al., 2009). This conclusion is consistent with the results of Adam, Brouwer, F rs; d; T 1 and Das Sarma (2009) in which the graphene conductivity in L ð Þ ; (4.30) p pnimp the presence of Gaussian disorder obtained using a full quantum-mechanical calculation was found to be in agree- ffiffiffiffi i.e., when the sample isffiffiffiffiffiffiffiffiffi much larger than the typical in-plane ment with the semiclassical Boltzmann theory even at zero distance between charge impurities. Considering that doping provided the disorder is strong enough. Given (i) the in experiments on bulk graphene L>1 m and random position of the electron-hole puddles, (ii) the fact that 10 12 2 nimp 10 –10 cmÀ , we see that the inequality (4.30) is because of the inequality (4.30) the local conductivity is well ½ satisfied. In this discussion we have neglected the presence of defined, and (iii) the fact that G G , the effective PNJ p the small regions of high density and size . For these regions medium theory (EMT) (Bruggeman, 1935; Landauer, 1952; the inequality (4.30) is not satisfied. However, these regions, Hori and Yonezawa, 1975) can be used to calculate the because of their high carrier density, steep carrier density electrical conductivity of graphene. The problem of the mini- gradients at the boundaries, and small size Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 454 Das Sarma et al.: Electronic transport in two-dimensional graphene Figure 33(b) shows the results for as a function of r . 2 r EMT EMT min s d r ð ÞÀ 0 d À P 0; The solid (dashed) line shows the calculated values of r EMT ¼ , EMT ð Þ¼ min Z ð Þþ Z þ including (neglecting) exchange. has a nonmonotonic (4.36) min behavior due to the fact that rs affects both the carrier density where P is the probability for the local value of . Using spatial distribution by controlling the strength of the disorder the relationð Þ between the local value and the local value of potential, screening, and exchange, and the scattering time . the carrier density n, Eqs. (4.28) and (4.36) can be rewritten in The dependence of min on rs has been measured in two the following form: recent experiments (Jang et al., 2008; Ponomarenko et al., 2009). In these experiments the fine-structure constant of n EMT graphene r is modified by placing the graphene on substrates dn ð ÞÀ P n 0; (4.37) s n EMT ½ ¼ with different and/or by using materials with Þ 1 as top Z ð Þþ dielectric layers. Jang et al. (2008) placed graphene on SiO where P n is the density probability distribution that can be 2 ½ and reduced rs from 0.8 (no top dielectric layer) to 0.56 by calculated using the TFD theory, Fig. 31. Using the TFD placing ice in vacuum as a top dielectric layer. The resulting results and Eq. (4.37), the conductivity at the Dirac point and change of is shown in Fig. 33(b) by the two solid its vicinity can be calculated. min squares. As predicted by the theory, when Vxc is included, Figure 33(a) shows Vg as obtained using the TFD ð Þ þ for this range of values of rs, min is unaffected by the EMT theory (Rossi et al., 2009). The theory correctly variation of rs. Overall the results presented by Jang et al. predicts a finite value of very close to the one measured (2008) are consistent with charge impurity being the main experimentally. At high gate voltages, the theory predicts the source of scattering in graphene. Ponomarenko et al. (2009) linear scaling of as a function of Vg. The theory correctly varied rs by placing graphene on substrates with different describes the crossover of from its minimum at V 0 to g ¼ dielectric constants and by using glycerol, ethanol, and water its linear behavior at high gate voltages. Figure 33(a) also as a top dielectric layer. Ponomarenko et al. (2009) found shows the importance of the exchange-correlation term at low very minor differences in the transport properties of graphene gate voltages. The dependence of min on nimp is shown in with different dielectric layers, thus concluding that charge Fig. 33(c). min increases as nimp decreases; the dependence impurities are not the dominant source of scattering. of min on nimp is weaker if exchange-correlation terms are Currently, the reasons for the discrepancy among the results taken into account. Figure 33(d) shows the dependence of of Jang et al. (2008) and Ponomarenko et al. (2009) are not on the inverse mobility 1= n as measured by well understood. The experiments are quite different. It must min / imp Chen, Jang, Adam et al. (2008). In this experiment the be noted that changing the substrate and the top dielectric amount of charge impurities is controlled by potassium layer, in addition to modifying rs, is likely to modify the doping. amount of disorder seen by the carriers in the graphene layer. 35 20 11 -2 (b) (a) nimp=2x10 cm 30 12 -2 nimp=1x10 cm 16 25 12 -2 nimp=2x10 cm /h] 2 12 /h] 20 2 [e [e 15 8 min σ σ 10 4 5 0 0 0 0.4 0.8 1.2 2 0 5 10 15 20 25 1.6 r Vg [V] s (c) (d) 30 /h] 2 20 [e min σ 10 0 9 10 11 12 10 10 10 10 -2 nimp [cm ] FIG. 33 (color online). Solid (dashed) lines show the EMT-TFD results with (without) exchange. (a) as function of Vg for three different values of n . The dots at V 0 are the results presented by Adam et al. (2007) for the same values of n . (b) as a function of r for imp g ¼ imp min s d 1 and n 1011 cm 2. Solid squares show the experimental results of Jang et al. (2008). (c) as a function of n , r 0:8 and ¼ imp ¼ À min imp s ¼ d 1 nm. For comparison the results obtained by Adam et al. (2007) are also shown by the dot-dashed line. (d) min as a function of the inverse¼ mobility as measured by Chen, Jang, Adam et al., 2008. r for d 1 and n 1011 cm 2. (a)–(c) Adapted from Rossi et al., 2009. s ¼ imp ¼ À (d) Adapted from Chen, Jang, Adam et al., 2008. Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 455 By not modifying the substrate and by placing ice in and ultrahigh vacuum, Jang et al. (2008) minimized the change of disorder induced by modifying the top dielectric layer. The approach presented above based on TFD and EMT theories can be used to calculate other transport properties of MLG and BLG close to the Dirac point. Hwang et al. (2009) used the same approach to calculate the thermopower of MLG with results in good agreement with experiments (Checkelsky et al., 2009; Wei et al., 2009; Zuev et al., 2009). Das Sarma et al. (2010) used the TFD EMT approach to calculate the electrical conductivity inþ BLG. Tiwari and Stroud (2009) used the EMT based on a simple FIG. 34. Collapse of the conductivity data obtained for RRNs with various 1=2 i;j 1; i;j p; i;jk;l ikjl; port treatment (Rossi, Bardarson, Fuhrer, and Das Sarma, ¼Æ h i¼ h i¼ 2010). (4.41) and p is proportional to the doping n . For 0, p 0 we V. QUANTUM HALL EFFECTS have percolation. Finite p and areh relevanti perturbations¼ ¼ for the percolation leading to a finite correlation length p; . A. Monolayer graphene On scales much larger than , the RRN is not critical,ð andÞ consists of independent regions of size so that is well 1. Integer quantum Hall effect defined with scaling (Cheianov et al., 2007) p; x ð Þ¼= The unique properties of the quantum Hall effect in gra- a= p; g with p; a À =F p=pà , pà ½and ð 4Þ=3, 1= ðh xÞwhere h ð 7=4 Þand x¼ 0:97 phene are among the most striking consequences of the Dirac are, respectively,¼ ¼ theð fractalþ Þ dimension¼ of the boundaries nature of the massless low energy fermionic excitations in between the electron-hole puddles and the conductance ex- graphene. In the presence of a perpendicular magnetic field B ponent G L a=L xg at the percolation threshold electrons (holes) confined in two dimensions are constrained (Isichenko,h ð 1992Þi¼; Cheianov ð Þ et al., 2007). Figure 34 shows to move in close cyclotron orbits that in quantum mechanics the results obtained solving numerically the RRN defined by are quantized. The quantization of the cyclotron orbits is Eq. (4.40). The numerical results are well fitted using for reflected in the quantization of the energy levels: at finite B F p=p the function F z 1 z2 =2 . Estimating the B 0 dispersion is replaced by a discrete set of energy ð ÃÞ ð Þ¼ð þ Þð Þ ¼ g e2=ℏ ak and g e2=ℏ ak 1=2 (Cheianov and levels, the Landau levels (LL). For any LL, there are N ð Þ F ð Þð FÞ ¼ Fal’ko, 2006b), Cheianov et al. (2007) estimated that BA= 0 degenerate orbital states, where A is the area of the sample and 0 is the magnetic quantum flux. Quantum Hall e2 effects (MacDonald, 1990; Prange and Girvin, 1990; Das a2n 0:41: (4.42) min ℏ ð Þ Sarma and Pinczuk, 1996) appear when N is comparable to the total number of quasiparticles present in the system. In the From the TFD results, one gets n n and so Eq. (4.42) quantum Hall regime the Hall conductivity exhibits well imp xy predicts that min should increase with nimp, a trend that is not developed plateaus as a function of carrier density (or corre- observed in experiments. The reason for the discrepancy is spondingly magnetic field) at which it takes quantized values. Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 456 Das Sarma et al.: Electronic transport in two-dimensional graphene At the same time, for the range of densities for which xy Eq. (5.1) is the hallmark of the chiral nature of the quasipar- is quantized, the longitudinal conductivity xx is zero ticles in graphene. The factor 1=2 in Eq. (5.1) can be under- (Laughlin, 1981; Halperin, 1982). For standard parabolic stood as the term induced by the additional Berry phase that 2DEG (such as the ones created in GaAs and Si quantum the electrons, due to their chiral nature, acquire when com- wells), the LL have energies energy ℏ! n 1=2 , where pleting a close orbit (Mikitik and Sharlai, 1999; Luk’yanchuk cð þ Þ n 0; 1; 2; ...and !c eB=mc, m being the effective mass, and Kopelevich, 2004). Another way to understand its pres- is¼ the cyclotron frequency.¼ Because the low energy fermions ence is by considering the analogy to the relativistic Dirac in graphene are massless, it is immediately obvious that for equation (Geim and MacDonald, 2007; Yang, 2007). From graphene we cannot apply the results valid for standard 2DEG this equation, two main predictions ensue: (i) the electrons (!c would appear to be infinite). In order to find the energy have spin 1=2 and (ii) the magnetic g factor is exactly equal to levels En for the LL the 2D Dirac equation must be solved in 2 for the spin in the nonrelativistic limit. As a consequence the presence of a magnetic field (Jackiw, 1984; Haldane, the Zeeman splitting is exactly equal to the orbital splitting. 1988; Gusynin and Sharapov, 2005; Peres et al., 2006). In graphene the pseudospin plays the role of the spin and The result is given by Eq. (1.13a). Differently from parabolic instead of Zeeman splitting, we have ‘‘pseudospin splitting’’ 2DEG, in graphene we have a LL at zero energy. In addition, but the same holds true: the pseudospin splitting is exactly we have the unconventional Hall quantization rule for xy equal to the orbital splitting. As a consequence the nth LL can (Zheng and Ando, 2002; Gusynin and Sharapov, 2005; Peres be thought as composed of the degenerate pseudospin-up et al., 2006): states of LL n and the pseudospin-down states of LL n 1. For zero mass Dirac fermions, the first LL in the conductionÀ 1 e2 xy g n (5.1) band and the highest LL in the valence band merge contrib- ¼ þ 2 h uting equally to the joint level at E 0, resulting in the half- ¼ compared to the one valid for regular 2DEGs odd-integer quantum Hall effect described by Eq. (5.1). For the E 0 LL, because half of the degenerate states are 2 e already¼ filled by holelike (electronlike) particles, we only xy gn (5.2) ¼ h need 1=2 N electronlike (holelike) particles to fill the level. ð Þ shown in Fig. 35, where g is the spin and valley degeneracy. The quantization rule for xy has been observed experi- Because in graphene the band dispersion has two inequivalent mentally (Novoselov, Geim et al., 2005; Zhang et al., 2005) valleys, g 4 (for GaAs quantum wells we only have as shown in Fig. 35(d). The experimental observation of the spin degeneracy¼ so that g 2). The additional 1=2 in Eq. (5.1) shows clearly the chiral nature of the massless ¼ quasiparticles in graphene. There is another important experimental consequence of (a) (b) the Dirac nature of the fermions in graphene. Because in graphene En scales as pnB [Eq. (1.13a)] rather than linearly as in regular 2DEG [Eq.ffiffiffiffiffiffiffi (1.13c)], at low energies (n) the energy spacing Án En 1 En between LL can be rather þ À large. Because the observation of the quantization of xy relies on the condition Án kBT (T being the temperature), it follows that in graphene the quantization of the LL should be observable at temperatures higher than in regular parabolic 2DEG. This fact has been confirmed by the observation in graphene of the QH effect at room temperature (Novoselov (c) (d) et al., 2007). Graphene is the only known material whose quantum Hall effect has been observed at ambient tempera- ture (albeit at high magnetic fields). By applying a top gate, p-n junctions (PNJ) can be created in graphene. In the presence of strong perpendicular fields graphene PNJ exhibit unusual fractional plateaus for the conductance that have been studied experimentally by O¨ zyilmaz et al. (2007) and Williams et al. (2007) and −4−2 0 2 4 12 −2 theoretically by Abanin and Levitov (2007). Numerical stud- n (10 cm ) ies in the presence of disorder have been performed by Long FIG. 35 (color online). Illustration of the integer QHE found in et al. (2008), Li and Shen (2008), and Low (2009). 2D semiconductor systems, (a), incorporated from MacDonald (1990) and Prange and Girvin (1990). (b) Illustration for SLG. 2. Broken-symmetry states (c) Illustration from BLG. The sequences of Landau levels as a function of carrier concentrations n are shown as dark and light The sequence of plateaus for xy given by Eq. (5.1) peaks for electrons and holes, respectively. Adapted from describes the QH effect due to fully occupied Landau levels Novoselov et al., 2006. (d) xy and xx of MLG as a function of including the spin and valley degeneracy. In graphene for the carrier density measured experimentally at T 4Kand B 14 T. fully occupied LL we have the filling factors gN=N ¼ ¼ ¼ Adapted from Novoselov, Geim et al., 2005. 4 n 1=2 2; 6; 10; .... In this section we study the ð þ Þ¼Æ Æ Æ Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 457 situation in which the spin or valley, or both, degeneracies are which either the real spin or the pseudospin associated with lifted. In this situation QH effects are observable at inter- the valley degree of freedom is polarized. The problem of mediate filling factors 0, 1 for the lowest LL and broken-symmetry states in the QH regime of graphene is 3, 4, 5 for n 1 LL.¼ TheÆ difficulty in observing these¼ analogous to the problem of ‘‘quantum Hall ferromagnetism’’ intermediateÆ Æ Æ QH effects¼Æ is the lower value of the energy gap studied in regular 2DEG in which, however, normally only between successive split LL. If the gap between successive the SU(2) symmetry associated with the spin can be sponta- Landau levels is comparable or smaller than the disorder neously broken [notice however that for silicon quantum strength, the disorder mixes adjacent LL preventing the for- wells the valley degeneracy is also present so that in this mation of well defined QH plateaus for the Hall conductivity. case the Hamiltonian is SU N (N>2) symmetric]. Because For the most part of this section, we neglect the Zeeman in the QH regime the kineticð energyÞ is completely quenched, coupling that turns out to be the lowest energy scale in most the formation of polarized states depends on the relative of the experimentally relevant conditions. strength of interaction and disorder. For graphene, Nomura Koshino and Ando (2007) showed that randomness in the and MacDonald (2006), using the Hartree-Fock approxima- bond couplings and on-site potential can lift the valley degen- tion, derived a ‘‘Stoner criterion’’ for the existence of polar- eracy and cause the appearance of intermediate Landau ized states, i.e., QH ferromagnetism, for a given strength of Levels. Fuchs and Lederer (2007) considered the electron- the disorder. Chakraborty and Pietilainen (2007) numerically phonon coupling as the possible mechanism for the lifting of verified that QH ferromagnetic states with large gaps are the degeneracy. However, in most of the theories the spin and realized in graphene. Sheng et al. (2007), using exact valley degeneracy is lifted due to interaction effects (Alicea diagonalization, studied the interplay of long-range and Fisher, 2006; Goerbig et al., 2006; Gusynin et al., 2006; Coulomb interaction and lattice effects in determining the Nomura and MacDonald, 2006; Yang et al., 2006; Abanin robustness of the 1 and 3 states with respect to et al., 2007b; Ezawa, 2007, 2008; Herbut, 2007), in particular, disorder. Nomura et¼Æ al. (2008) studiedÆ the effect of strong electron-electron interactions. When electron-electron inter- long-range disorder. Wang et al. (2008) performed numerical actions are taken into account, the quasiparticles filling a LL studies that show that various charge density wave phases can can polarize in order to minimize the exchange energy (max- be realized in the partially filled 3 LL. imize it in absolute value). In this case, given the SU(4) Experimentally, the existence of¼Æ broken-symmetry states invariance of the Hamiltonian, the states has been verified by Zhang et al. (2006) [Fig. 36(a)], which showed the existence of the intermediate Landau levels with É0 ck;y 0 ; (5.3) 0, 1 for the n 0 LL and the intermediate level j i¼1 i M k j i ¼ Æ ¼ ¼ Y Y 4 for the n 1 LL that is therefore only partially resolved. where i is the index of the internal states that runs from 1 to GivenÆ that the¼ magnetic field by itself does not lift the valley M 4 n 1=2 4, and 0 is the vacuum, are exact degeneracy, interaction effects are likely the cause for the full eigenstates¼ À ð of theÀ Hamiltonian.Þ Forj i a broad class of repulsive resolution of the n 0 LL. On the other hand, a careful interactions, É is expected to be the exact ground state analysis of the data¼ as a function of the tilting angle of the j 0i (Yang et al., 2006, 2007). The state described by É0 is a magnetic field suggests that the partial resolution of the n 1 ‘‘ferromagnet,’’ sometimes called a QH ferromagnet,j i in LL is due to Zeeman splitting (Zhang et al., 2006). ¼ FIG. 36 (color online). (a) xy as a function of gate voltage at different magnetic fields: 9 T (circle), 25 T (square), 30 T (diamond), 37 T (up triangle), 42 T (down triangle), and 45 T (star). All data sets are taken at T 1:4K, except for the B 9Tcurve, which is taken at T 30 mK. Left upper inset: R and R for the same device measured at B 25¼ T. Right inset: Detailed ¼ data near the Dirac point for ¼ xx xy ¼ xy B 9T(circle), 11.5 T (pentagon), and 17.5 T (hexagon) at T 30 mK. Adapted from Zhang et al., 2006. (b) Longitudinal resistance Rxx as¼ a function of gate voltage V V V at 0.3 K and several¼ values of the magnetic field: 8, 11, and 14 T. The inset shows a graphene g0 ¼ g À 0 crystal with Au leads deposited. The bar indicates 5 m. At V 0, the peak in R grows to 190 k at 14 T. Adapted from Checkelsky g0 ¼ xx et al., 2008. Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 458 Das Sarma et al.: Electronic transport in two-dimensional graphene 3. The 0 state ¼ In the previous section we have seen that in strong mag- netic fields the lowest LL can be completely resolved and the spin and valley degeneracies may be lifted. In particular, an approximate plateau for appear for 0. This state xy ¼ has been experimentally studied by Jiang et al. (2007), Checkelsky et al. (2008, 2009), and Giesbers et al. (2009); see Fig. 36(b). The state is unique in that the plateau of xy corresponds to a maximum of the longitudinal resis- tivity xx in contrast to what happens for Þ 0 where a plateau of xy corresponds to zero longitudinal resistivity. In addition, the 0 edge states are not supposed to carry any FIG. 37 (color online). Graphene fractional quantum Hall data, ¼ charge current, but only spin currents (Abanin et al., 2006; from (a) Du et al., 2009 and (b) Bolotin et al., 2009, observed on Abanin et al., 2007a; Abanin, Novoselov et al., 2007). As two probe suspended graphene samples. pointed out by Das Sarma and Yang (2009), however, the situation is not surprising if we recall the relations between the resistivity tensor and the conductivity tensor: Chakraborty (2006) and Toke et al. (2006). For the ¼ 1=3, the gap has been estimated to be of the order of xx ; xy ; (5.4) xx 2 2 xy 2 2 0:05e2=l , where l ℏc=eB 1=2 is the magnetic length. ¼ xx xy ¼ xx xy B B þ þ Because of the small gapð size, theÞ experimental observation and the fact that the quantization of is associated with the of FQHE requires high quality samples. For graphene, very xy low amount of disorder can be achieved in suspended samples vanishing of xx. This can be seen from Laughlin’s gauge argument (Laughlin, 1981; Halperin, 1982). Using Eq. (5.4), and in these suspended samples two groups (Bolotin et al., a possible resolution of the 0 anomaly is obvious: for any 2009; Du et al., 2009) recently observed signatures of the finite , the vanishing of ¼ corresponds to the vanishing 1=3 fractional quantum Hall state in two-terminal mea- xy xx surements;¼ see Fig. 37. A great deal of work remains to be of ; however, for 0, we have 1= so that xx xy ¼ xx ¼ xx done in graphene FQHE. xx 0 implies xx . This is very similar to the Hall insulator! phase in ordinary!1 2D parabolic-band electron gases. This simple argument shows that the fact that xx seem to B. Bilayer graphene diverge for T 0 for the 0 state is not surprising. However, this argument! may not¼ be enough to explain the 1. Integer quantum Hall effect details of the dependence of 0 on temperature and xxð ¼ Þ In bilayer graphene the low energy fermionic excitations magnetic field. In particular, Checkelsky et al. (2009) are massive, i.e., with good approximation the bands are found evidence for a field-induced transition to a strongly parabolic. This fact would suggest that the bilayer QH effect insulating state at a finite value of B. These observations in graphene might be similar to the one observed in regular suggest that the 0 ground state might differ from the ¼ parabolic 2DEG. There are, however, two important differ- SU(4) eigenstates (5.3) and theoretical calculations proposed ences: the band structure of bilayer graphene is gapless and that it could be a spin-density wave or charge-density wave the fermions in BLG, as in MLG, are also chiral but with a (Herbut, 2007; Jung and MacDonald, 2009). It has also been Berry phase equal to 2 instead of (McCann and Fal’ko, argued that the divergence of xx 0 might be the sig- 2006). As a consequence, as shown in Eq. (1.13c), the energy nature of Kekule instability (Nomurað ¼ Þet al., 2009; Hou, levels have a different sequence from both regular 2DEGs and et al., 2010). MLG. In particular, BLG also has a LL at zero energy, Giesbers et al. (2009) interpreted their experimental data however, because the Berry phase associated with the chiral using a simple model involving the opening of a field- nature of the quasiparticles in BLG is 2, the step between dependent spin gap. Zhang, Camacho et al. (2009) observed the plateaus of xy across the CNP is twice as large as in a cusp in the longitudinal resistance xx for 1=2 and interpreted this as the signature of a transition from a Hall MLG [as shown schematically in Fig. 35(c)]. One way to insulating state for >1=2 to a collective insulator, such as a understand the step across the CNP is to consider that in BLG the n 0 and 1 orbital LL are degenerate. Wigner crystal (Zhang and Joglekar, 2007), for <1=2. No ¼ consensus has been reached so far, and more work is needed The spin and valley degeneracy factor g in BLG is equal to to understand the 0 state in graphene. 4 as in MLG. In BLG the valley degree of freedom can also be ¼ regarded as a layer degree of freedom considering that with- out loss of generality we can use a pseudospin representation 4. Fractional quantum Hall effect in which the K valley states are localized in the top layer and In addition to QH ferromagnetism, the electron-electron the K0 states in the bottom layer. The QH effect has been interaction is responsible for the fractional quantum Hall measured experimentally. Figure 38(a) shows the original effect (FQHE). For the FQHE, the energy gaps are even data obtained by Novoselov et al. (2006). In agreement smaller than for the QH ferromagnetic states. For graphene, with the theory the data show a double size step, compared the FQHE gaps have been calculated by Apalkov and to MLG, for xy across the CNP. Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 Das Sarma et al.: Electronic transport in two-dimensional graphene 459 2. Broken-symmetry states VI. CONCLUSION AND SUMMARY As discussed, the En 0 LL in BLG has an 8-fold degeneracy due to spin degeneracy,¼ valley (layer) degener- In roughly five years, research in graphene physics has acy, and n 0, n 1 orbital LL degeneracy. The En Þ 0 made spectacular advances starting from the fabrication of LL have only¼ a 4-fold¼ degeneracy due to spin and valley gated variable-density 2D graphene monolayers to the obser- degeneracy. As discussed for MLG, it is natural to expect vations of fractional quantum Hall effect and Klein tunneling. that the degeneracy of the full LL will be lifted by external The massless chiral Dirac spectrum leads to novel integer perturbations and/or interactions. Similar considerations to quantum Hall effect in graphene with the existence of a n 0 ¼ the ones made in Sec. V.A.2 for MLG apply here: the quantized Landau level shared equally between electrons and splitting can be due to the Zeeman effect (Giesbers et al., holes. The nonexistence of a gap in the graphene carrier 2009), strain-induced lifting of valley degeneracy (Abanin dispersion leads to a direct transition between electronlike et al., 2007b), or Coulomb interactions. Ezawa (2007) and metallic transport to holelike metallic transport as the gate Barlas et al. (2008) considered the splitting of the En 0 voltage is tuned through the charge neutral Dirac point. By LL in BLG due to electron-electron interactions and calcu-¼ contrast, 2D semiconductors invariably become insulating at lated the corresponding charge gaps and filling sequence. As low enough carrier densities. In MLG nanoribbons and in in MLG, the charge gaps of the splitted LLs will be smaller BLG structures in the presence of an electric field, graphene than the charge gap ℏ!c for the fully occupied LLs and so carrier transport manifests a transport gap because there is an the observation of QH plateaus due to the resolution of the intrinsic spectral gap induced by the confinement and the bias LL requires higher quality samples. This has recently been field, respectively. The precise relationship between the trans- achieved in suspended BLG samples (Feldman et al., 2009; port and the spectral gap is, however, not well understood at Zhao et al., 2010) in which the full resolution of the this stage and is a subject of much current activity. Since eightfold degeneracy of the zero-energy LL has been ob- backscattering processes are suppressed, graphene exhibits served, Fig. 38(c). By analyzing the dependence of the weak antilocalization behavior in contrast to the weak local- maximum resistance at the CNP on B and T, Feldman ization behavior of ordinary 2D systems. The presence of any et al. (2009) concluded that the observed splitting of the short-range scattering, however, introduces intervalley cou- En 0 LL cannot be attributed to the Zeeman effect. pling, which leads to the eventual restoration of weak local- Moreover,¼ the order in magnetic fields in which the ization. Since short-range scattering, arising from lattice broken-symmetry states appear is consistent with the theo- point defects, is weak in graphene, the weak antilocalization retical predictions of Barlas et al. (2008). These facts behavior is expected to cross over to weak localization be- suggest that in BLG the resolution of the octet zero-energy havior only at very low temperatures although a direct ex- LL is due to electron-electron interactions. perimental observation of such a localization crossover is still lacking and may be difficult. The observed sequence of graphene integer quantized Hall conductance follows the expected formula 4e2=h xy ¼ð Þ n 1=2 , indicating the Berry phase contribution and the nð þ0 LandauÞ level shared between electrons and holes. For¼ example, the complete lifting of spin and valley splitting leads to the observation of the following quantized Hall conductance sequence 0; 1; 2; ... with ¼ Æ Æ xy ¼ e2=h; whereas, in the presence of spin and valley degener- acy (i.e., with the factor of 4 in the front) one gets the sequence 2; 6; ... The precise nature of the 0 ¼Æ Æ ¼ IQHE, which seems to manifest a highly resistive (xx ) state in some experiments but not in others, is still an!1 open question as is the issue of the physical mechanism or the quantum phase transition associated with the possible sponta- neous symmetry breaking that leads to the lifting of the degeneracy. We do mention, however, that similar, but not identical, physics arises in the context of ordinary IQHE in 2D semiconductor structures. For example, the 4-fold spin and valley degeneracy, partially lifted by the applied mag- netic field, occurs in 2D Si-(100) based QHE, as already apparent in the original discovery of IQHE by von Klitzing FIG. 38 (color online). (a) Measured Hall conductivity xy in BLG as a function of carrier density for B 12 and B 20 T at et al. (1980). The issue of spin and valley degeneracy lifting T 4K. (b) Measured longitudinal resistivity¼ in BLG at¼T 4K in the QHE phenomena is thus generic to both graphene and and¼B 12 T. The inset shows the calculated BLG bands close¼ to 2D semiconductor systems, although the origin of valley the CNP.¼ Adapted from Novoselov et al., 2006. (c) Two-terminal degeneracy is qualitatively different (Eng et al., 2007; conductance, G, as a function of carrier density at T 100 mK for McFarland et al., 2009) in the two cases. The other similarity different values of the magnetic field in suspended BLG.¼ Adapted between graphene and 2DEG QHE is that both systems tend from Feldman et al., 2009. to manifest strongly insulating phases at very high magnetic Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011 460 Das Sarma et al.: Electronic transport in two-dimensional graphene field when 1. In semiconductor-based high-mobility of Quantum Diffusion in Two Dimensions,’’ Phys. Rev. Lett. 42, 2DEG, typically such a strongly insulating phase occurs 673. (Jiang et al., 1990; Jiang et al., 1991) for <1=5 1=7; Abrahams, E., S. V. Kravchenko, and M. P. Sarachik, 2001, whereas, in graphene the effect manifest near theÀ charge ‘‘Metallic behavior and related phenomena in two dimensions,’’ neutral Dirac point around 0. Whether the same physics Rev. Mod. Phys. 73, 251. controls either insulating phenomena or not is an open Adam, S., P.W. Brouwer, and S. Das Sarma, 2009, ‘‘Crossover from quantum to Boltzmann transport in graphene,’’ Phys. Rev. B 79, question. 201404. Recent experimental observations of 1=3 FQHE in ¼ Adam, S., S. Cho, M. S. Fuhrer, and S. Das Sarma, 2008, ‘‘Density graphene have created a great deal of excitement. These Inhomogeneity Driven Percolation Metal-Insulator Transition and preliminary experiments involve two-probe measurements Dimensional Crossover in Graphene Nanoribbons,’’ Phys. Rev. on suspended graphene samples where no distinction between Lett. 101, 046404. xx and xy can really be made. Further advances in the field Adam, S., and S. Das Sarma, 2008a, ‘‘Boltzmann transport and would necessitate the observation of quantized plateaus in xy residual conductivity in bilayer graphene,’’ Phys. Rev. B 77, with xx 0. Since FQHE involves electron-electron inter- 115436. action effects, with the noninteracting part of the Hamiltonian Adam, S., and S. Das Sarma, 2008b, ‘‘Transport in suspended playing a rather minor role, we should not perhaps expect any graphene,’’ Solid State Commun. 146, 356. dramatic difference between 2DEG and graphene FQHE Adam, S., E. Hwang, and S. Das Sarma, 2008, ‘‘Scattering mecha- since both systems manifest the standard 1=r Coulomb re- nisms and Boltzmann transport in graphene,’’ Physica E (Amsterdam) 40, 1022. pulsion between electrons. Two possible quantitative effects Adam, S., E. H. Hwang, V.M. Galitski, and S. Das Sarma, 2007, ‘‘A distinguishing FQHE in graphene and 2DEG, which should self-consistent theory for graphene transport,’’ Proc. Natl. Acad. be studied theoretically and numerically, are the different Sci. U.S.A. 104, 18 392. Coulomb pseudopotentials and Landau level coupling in the Adam, S., E. H. Hwang, E. Rossi, and S. Das Sarma, 2009, ‘‘Theory two systems. Since the stability of various FQH states de- of charged impurity scattering in two dimensional graphene,’’ pends crucially on the minute details of Coulomb pseudo- Solid State Commun. 149, 1072. pontentials and inter-LL coupling, it is conceivable that Adam, S., and M. D. Stiles, 2010, ‘‘Temperature dependence of the graphene may manifest novel FQHE not feasible in 2D diffusive conductivity of bilayer graphene,’’ Phys. Rev. B 82, semiconductors. 075423. We also note that many properties reviewed here should Akhmerov, A. R., and C. W. J. Beenakker, 2008, ‘‘Boundary con- also apply to topological insulators (Hasan and Kane, 2010), ditions for Dirac fermions on a terminated honeycomb lattice,’’ which have only single Dirac cones on their surfaces. Phys. Rev. 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