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.