The electronic properties of graphene

doi:10.1103/REVMODPHYS.81.109

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The electronic properties of graphene

Journal Article•DOI•

The electronic properties of graphene

A. H. Castro Neto¹, Francisco Guinea², Nuno M. R. Peres³, Kostya S. Novoselov⁴, Andre K. Geim⁴ - Show less +1 more•Institutions (4)

Boston University¹, Spanish National Research Council², University of Minho³, University of Manchester⁴

14 Jan 2009-Reviews of Modern Physics (American Physical Society)-Vol. 81, Iss: 1, pp 109-162

TL;DR: In this paper, the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations, are discussed.

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Abstract: This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.

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Journal Article•DOI•

Nanoscale mapping of electrical resistivity and connectivity in graphene strips and networks.

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Peter N. Nirmalraj¹, Tarek Lutz, Shishir Kumar, Georg S. Duesberg, John J. Boland - Show less +1 more•Institutions (1)

Trinity College, Dublin¹

12 Jan 2011-Nano Letters

TL;DR: It is shown that the transition to bulklike resistivities occurs at seven to eight layers of graphene, and the contact resistance between graphene flakes in a graphene network scales with the flake thickness and the implications for transparent conductor applications are discussed.

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Abstract: In this article we map out the thickness dependence of the resistivity of individual graphene strips, from single layer graphene through to the formation of graphitic structures We report exceptionally low resistivity values for single strips and demonstrate that the resistivity distribution for single strips is anomalously narrow when compared to bi- and trilayer graphene, consistent with the unique electronic properties of single graphene layers In agreement with theoretical predictions, we show that the transition to bulklike resistivities occurs at seven to eight layers of graphene Moreover, we demonstrate that the contact resistance between graphene flakes in a graphene network scales with the flake thickness and the implications for transparent conductor applications are discussed

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168 citations

Journal Article•DOI•

Bloch state tomography using Wilson lines.

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Tracy Li¹, Tracy Li², Lucia Duca², Lucia Duca¹, Martin Reitter¹, Martin Reitter², Fabian Grusdt³, Fabian Grusdt⁴, Fabian Grusdt⁵, Eugene Demler³, Manuel Endres⁶, Manuel Endres³, Monika Schleier-Smith⁷, Immanuel Bloch², Immanuel Bloch¹, Ulrich Schneider¹, Ulrich Schneider⁸, Ulrich Schneider² - Show less +14 more•Institutions (8)

Max Planck Society¹, Ludwig Maximilian University of Munich², Harvard University³, Daimler AG⁴, Kaiserslautern University of Technology⁵, California Institute of Technology⁶, Stanford University⁷, University of Cambridge⁸

27 May 2016-Science

TL;DR: Using an ultracold gas of rubidium atoms loaded in a honeycomb optical lattice, strong-force dynamics in Bloch bands that are described by Wilson lines are realized and an evolution in the band populations that directly reveals the band geometry is observed.

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Abstract: Topology and geometry are essential to our understanding of modern physics, underlying many foundational concepts from high-energy theories, quantum information, and condensed-matter physics. In condensed-matter systems, a wide range of phenomena stem from the geometry of the band eigenstates, which is encoded in the matrix-valued Wilson line for general multiband systems. Using an ultracold gas of rubidium atoms loaded in a honeycomb optical lattice, we realize strong-force dynamics in Bloch bands that are described by Wilson lines and observe an evolution in the band populations that directly reveals the band geometry. Our technique enables a full determination of band eigenstates, Berry curvature, and topological invariants, including single- and multiband Chern and Z_2 numbers.

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168 citations

Cites background from "The electronic properties of graphe..."

...However, in cases when condensed matter properties are determined by multiple bands with degeneracies, such as in topological insulators [4, 5] and graphene [6], they can often not be understood with standard Berry phases....
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Journal Article•DOI•

Interfacial engineering in graphene bandgap

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Xiaozhi Xu¹, Chang Liu¹, Zhanghao Sun¹, Ting Cao², Zhihong Zhang¹, Enge Wang¹, Zhongfan Liu¹, Kaihui Liu¹ - Show less +4 more•Institutions (2)

Peking University¹, University of California, Berkeley²

08 May 2018-Chemical Society Reviews

TL;DR: This review article summarises recent theoretical and experimental advances in interfacial engineering to achieve bandgap opening and indicates that pioneering works have not met all the requirements for electronic applications of graphene at once but hold great promise in this direction and may eventually lead to futureApplications of graphene in semiconductor electronics and beyond.

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Abstract: Graphene exhibits superior mechanical strength, high thermal conductivity, strong light-matter interactions, and, in particular, exceptional electronic properties. These merits make graphene an outstanding material for numerous potential applications. However, a graphene-based high-performance transistor, which is the most appealing application, has not yet been produced, which is mainly due to the absence of an intrinsic electronic bandgap in this material. Therefore, bandgap opening in graphene is urgently needed, and great efforts have been made regarding this topic over the past decade. In this review article, we summarise recent theoretical and experimental advances in interfacial engineering to achieve bandgap opening. These developments are divided into two categories: chemical engineering and physical engineering. Chemical engineering is usually destructive to the pristine graphene lattice via chemical functionalization, the introduction of defects, doping, chemical bonds with substrates, and quantum confinement; the latter largely maintains the atomic structure of graphene intact and includes the application of an external field, interactions with substrates, physical adsorption, strain, electron many-body effects and spin-orbit coupling. Although these pioneering works have not met all the requirements for electronic applications of graphene at once, they hold great promise in this direction and may eventually lead to future applications of graphene in semiconductor electronics and beyond.

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168 citations

Journal Article•DOI•

Ultrafast Dynamics and Nonlinear Optical Responses from sp2- and sp3-Hybridized Domains in Graphene Oxide

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Zhi-Bo Liu¹, Xin Zhao¹, Xiao-Liang Zhang¹, Xiao-Qing Yan¹, Yingpeng Wu¹, Yongsheng Chen¹, Jianguo Tian¹ - Show less +3 more•Institutions (1)

Nankai University¹

21 Jul 2011-Journal of Physical Chemistry Letters

TL;DR: In this paper, the ultrafast relaxation dynamics and nonlinear optical response in single and few-layered graphene oxide (GO) were studied by ultrafast optical differential transmission spectroscopy and Z-scan technique using various pump intensities.

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Abstract: The ultrafast relaxation dynamics and nonlinear optical response in single- and few-layered graphene oxide (GO) were studied by ultrafast optical differential transmission spectroscopy and Z-scan technique using various pump intensities. It was found that charge carriers with subpicosecond-to-picosecond dynamics from sp2-hybridized domains dominate the ultrafast response at low pump intensities, like graphene. Surprisingly, the influence of two-photon absorption from sp3-hybridized domains on the transient absorption signal becomes increasingly strong with pump intensities. On the basis of heterogeneous ultrafast dynamics of GO with saturable absorption in sp2 domains and two-photon absorption in sp3 domains, the nonlinear optical response can be tailored by manipulation of the degree and location of oxidation on GO sheets; this unravels the important role of sp3 domains in graphene optics and will facilitate the potential applications of GO in optoelectronics.

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168 citations

Journal Article•DOI•

Exceptional and Anisotropic Transport Properties of Photocarriers in Black Phosphorus.

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Jiaqi He¹, Jiaqi He², Dawei He², Yongsheng Wang², Qiannan Cui¹, Matthew Z. Bellus¹, Hsin-Ying Chiu¹, Hui Zhao¹ - Show less +4 more•Institutions (2)

University of Kansas¹, Beijing Jiaotong University²

14 May 2015-ACS Nano

TL;DR: It is shown that black phosphorus has room-temperature charge mobilities on the order of 10(4) cm(2) V(-1) s(-1), which are about 1 order of magnitude larger than silicon, and strong anisotropic transport in black phosphorus is demonstrated.

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Abstract: One key challenge in developing postsilicon electronic technology is to find ultrathin channel materials with high charge mobilities and sizable energy band gaps. Graphene can offer extremely high charge mobilities; however, the lack of a band gap presents a significant barrier. Transition metal dichalcogenides possess sizable and thickness-tunable band gaps; however, their charge mobilities are relatively low. Here we show that black phosphorus has room-temperature charge mobilities on the order of 104 cm2 V–1 s–1, which are about 1 order of magnitude larger than silicon. We also demonstrate strong anisotropic transport in black phosphorus, where the mobilities along the armchair direction are about 1 order of magnitude larger than in the zigzag direction. A photocarrier lifetime as long as 100 ps is also determined. These results illustrate that black phosphorus is a promising candidate for future electronic and optoelectronic applications.

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168 citations

Cites background from "The electronic properties of graphe..."

...charge mobilities 2 orders of magnitude higher than silicon.(1,2) However, the lack of a band gap limits its application as a channel material in logic devices....
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Journal Article•DOI•

Electric Field Effect in Atomically Thin Carbon Films

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Kostya S. Novoselov¹, Andre K. Geim¹, Sergey V. Morozov, Da Jiang¹, Y. Zhang¹, S. V. Dubonos, Irina V. Grigorieva¹, A. A. Firsov - Show less +4 more•Institutions (1)

University of Manchester¹

22 Oct 2004-Science

TL;DR: Monocrystalline graphitic films are found to be a two-dimensional semimetal with a tiny overlap between valence and conductance bands and they exhibit a strong ambipolar electric field effect.

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Abstract: We describe monocrystalline graphitic films, which are a few atoms thick but are nonetheless stable under ambient conditions, metallic, and of remarkably high quality. The films are found to be a two-dimensional semimetal with a tiny overlap between valence and conductance bands, and they exhibit a strong ambipolar electric field effect such that electrons and holes in concentrations up to 10 13 per square centimeter and with room-temperature mobilities of ∼10,000 square centimeters per volt-second can be induced by applying gate voltage.

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55,532 citations

"The electronic properties of graphe..." refers background in this paper

...Be ause the DC magnetotransport properties ofgraphene are normally measured with the possibilityof tuning its ele troni density by a gate potential(Novoselov et al., 2004), it is important to ompute the ondu tivity kernel, sin e this has dire t experimentalrelevan e....
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...The same polarizability describes the screening of an external field perpendicular to the layers, like the one induced by a gate in electrically doped systems (Novoselov et al., 2004)....
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...Because the DC magnetotransport properties of graphene are normally measured with the possibility of tuning its electronic density by a gate potential (Novoselov et al., 2004), it is important to compute the conductivity kernel, since this has direct experimental relevance....
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...…studies of graphene sta ks have showed that, within reasing number of layers, the system be omes in reas-ingly metalli ( on entration of harge arriers at zero en-ergy gradually in reases), and there appear several typesof ele tron-and-hole-like arries (Morozov et al., 2005;Novoselov et al., 2004)....
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...The same polarizabilitydes ribes the s reening of an external eld perpendi ularto the layers, like the one indu ed by a gate in ele tri- ally doped systems (Novoselov et al., 2004)....
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Journal Article•DOI•

The rise of graphene

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Andre K. Geim¹, Kostya S. Novoselov¹•Institutions (1)

University of Manchester¹

01 Mar 2007-Nature Materials

TL;DR: Owing to its unusual electronic spectrum, graphene has led to the emergence of a new paradigm of 'relativistic' condensed-matter physics, where quantum relativistic phenomena can now be mimicked and tested in table-top experiments.

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Abstract: Graphene is a rapidly rising star on the horizon of materials science and condensed-matter physics. This strictly two-dimensional material exhibits exceptionally high crystal and electronic quality, and, despite its short history, has already revealed a cornucopia of new physics and potential applications, which are briefly discussed here. Whereas one can be certain of the realness of applications only when commercial products appear, graphene no longer requires any further proof of its importance in terms of fundamental physics. Owing to its unusual electronic spectrum, graphene has led to the emergence of a new paradigm of 'relativistic' condensed-matter physics, where quantum relativistic phenomena, some of which are unobservable in high-energy physics, can now be mimicked and tested in table-top experiments. More generally, graphene represents a conceptually new class of materials that are only one atom thick, and, on this basis, offers new inroads into low-dimensional physics that has never ceased to surprise and continues to provide a fertile ground for applications.

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35,293 citations

"The electronic properties of graphe..." refers background in this paper

...As the current status of the experiment and potential applications have recently been reviewed (Geim and Novoselov, 2007), in this article we mostly concentrate on the theory and more technical aspects of electronic properties of this exciting new material....
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...As the urrent status of the experimentand potential appli ations have re ently been reviewed(Geim and Novoselov, 2007), in this arti le we mostly on entrate on the theory and more te hni al aspe ts ofele troni properties of this ex iting new material....
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...It has also been suggested that Coulomb intera tionsare onsiderably enhan ed in smaller geometries, su has graphene quantum dots (Milton Pereira Junior et al.,2007), leading to unusual Coulomb blo kade e e ts 4(Geim and Novoselov, 2007) and perhaps to magneti phenomena su h as the Kondo e e t....
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...…most versatile systems in ondensedmatter resear h.Besides the unusual basi properties, graphene hasthe potential for a large number of appli ations(Geim and Novoselov, 2007), from hemi al sensors(Chen et al., 2007 ; S hedin et al., 2007) to transistors(Nilsson et al., 2007b; Oostinga et al.,…...
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...Besides the unusual basic properties, graphene has the potential for a large number of applications (Geim and Novoselov, 2007), from chemical sensors (Chen et al....
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Book•

Theory of elasticity

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Stephen P. Timoshenko, James Norman Goodier

01 Jan 1934

TL;DR: The theory of the slipline field is used in this article to solve the problem of stable and non-stressed problems in plane strains in a plane-strain scenario.

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Abstract: Chapter 1: Stresses and Strains Chapter 2: Foundations of Plasticity Chapter 3: Elasto-Plastic Bending and Torsion Chapter 4: Plastic Analysis of Beams and Frames Chapter 5: Further Solutions of Elasto-Plastic Problems Chapter 6: Theory of the Slipline Field Chapter 7: Steady Problems in Plane Strain Chapter 8: Non-Steady Problems in Plane Strain

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20,724 citations

Journal Article•DOI•

Two-dimensional gas of massless Dirac fermions in graphene

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Kostya S. Novoselov¹, A. K. Geim¹, Sergey V. Morozov, Da Jiang¹, Mikhail I. Katsnelson², Irina V. Grigorieva¹, S. V. Dubonos, A. A. Firsov - Show less +4 more•Institutions (2)

University of Manchester¹, Radboud University Nijmegen²

10 Nov 2005-Nature

TL;DR: This study reports an experimental study of a condensed-matter system (graphene, a single atomic layer of carbon) in which electron transport is essentially governed by Dirac's (relativistic) equation and reveals a variety of unusual phenomena that are characteristic of two-dimensional Dirac fermions.

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Abstract: Quantum electrodynamics (resulting from the merger of quantum mechanics and relativity theory) has provided a clear understanding of phenomena ranging from particle physics to cosmology and from astrophysics to quantum chemistry. The ideas underlying quantum electrodynamics also influence the theory of condensed matter, but quantum relativistic effects are usually minute in the known experimental systems that can be described accurately by the non-relativistic Schrodinger equation. Here we report an experimental study of a condensed-matter system (graphene, a single atomic layer of carbon) in which electron transport is essentially governed by Dirac's (relativistic) equation. The charge carriers in graphene mimic relativistic particles with zero rest mass and have an effective 'speed of light' c* approximately 10(6) m s(-1). Our study reveals a variety of unusual phenomena that are characteristic of two-dimensional Dirac fermions. In particular we have observed the following: first, graphene's conductivity never falls below a minimum value corresponding to the quantum unit of conductance, even when concentrations of charge carriers tend to zero; second, the integer quantum Hall effect in graphene is anomalous in that it occurs at half-integer filling factors; and third, the cyclotron mass m(c) of massless carriers in graphene is described by E = m(c)c*2. This two-dimensional system is not only interesting in itself but also allows access to the subtle and rich physics of quantum electrodynamics in a bench-top experiment.

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18,958 citations

"The electronic properties of graphe..." refers background or methods in this paper

...This amazing re-sult has been observed experimentally (Novoselov et al.,2005a; Zhang et al., 2005) as shown in Fig.20....
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...Adapted from(Novoselov et al., 2005a)....
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...Adapted from (Novoselov et al.,2005a).and hen e σxy,inc. = I/VH = ±4Ne2/h, whi h is thenaive expe tation....
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...The period of os illations ∆n = 4B/Φ0,where B is the applied eld and Φ0 is the ux quantum(Novoselov et al., 2005a).or equivalently: (Oσ+ + O†σ−)φ = (2E/ωc)φ , (100)where σ± = σx ± iσy, and we have de ned the dimen-sionless length s ale: ξ = y ℓB − ℓBk , (101)and 1D harmoni os illator operators: O =…...
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...…invery unusual ways when ompared to ordinary ele tronsif subje ted to magneti elds, leading to new physi alphenomena (Gusynin and Sharapov, 2005; Peres et al.,2006 ) su h as the anomalous integer quantum Hall ef-fe t (IQHE) measured experimentally (Novoselov et al.,2005a; Zhang et al., 2005)....
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