Graphene
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225692485
2008-07-14T22:49:22Z
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{{distinguish|Grapheme}}
'''Graphene''' is a one-atom-thick planar sheet of [[sp2 bond|sp<sup>2</sup>-bonded]] [[carbon]] atoms that are densely packed in a honeycomb crystal lattice. It can be viewed as an atomic-scale [[chicken wire]] made of carbon atoms and their bonds. The name comes from <small>[[Graphite|GRAPHITE]]</small> + <small>[[-ene|-ENE]]</small>; [[graphite]] itself consists of many graphene sheets stacked together.
The carbon-carbon bond length in graphene is approximately 1.42 [[Ångström|Å]]. Graphene is the basic structural element of all other graphitic materials including [[graphite]], [[carbon nanotubes]] and [[fullerenes]]. It can also be considered as an infinitely large [[Aromaticity|aromatic]] molecule, the limiting case of the family of flat [[polycyclic aromatic hydrocarbons]] called graphenes.
[[Image:graphene xyz.jpg|right|340px|thumb|Graphene is an [[Chicken wire (chemistry)|atomic-scale chicken wire]] made of carbon atoms]]
==Description==
Perfect graphenes consist exclusively of hexagonal cells; pentagonal and heptagonal cells constitute defects. If an isolated [[pentagon]]al cell is present, then the plane warps into a [[cone (geometry)|cone]] shape; insertion of 12 pentagons would create a [[fullerene]]. Likewise, insertion of an isolated [[heptagon]] causes the sheet to become saddle-shaped. Controlled addition of pentagons and heptagons would allow a wide variety of complex shapes to be made, for instance carbon [[NanoBud]]s. Single-walled [[carbon nanotube]]s may be considered to be graphene cylinders; some have a hemispherical graphene cap (that includes 6 pentagons) at each end.
The [[IUPAC]] compendium of technology states: "previously, descriptions such as [[graphite]] layers, carbon layers, or carbon sheets have been used for the term graphene…it is not correct to use for a single layer a term which includes the term graphite, which would imply a three-dimensional structure. The term graphene should be used only when the reactions, structural relations or other properties of individual layers are discussed". In this regard, graphene has been referred to as an infinite alternant (only six-member carbon ring) [[polycyclic aromatic hydrocarbon]]. The largest molecule of this type consists of 222 atoms and is 10 benzene rings across<ref> C. D. Simpson ''et al''. Synthesis of a Giant 222 Carbon Graphite Sheet. ''Chemistry - A European Journal'', '''6''' 1424 (2002)</ref>. The onset of graphene properties, as compared to those of a [[polycyclic aromatic hydrocarbon]] (PAH) are not known. PAHs of 60, 78, and 120 carbon atoms have UV absorbance spectra that show a discrete PAH electronic structure, but a PAH of 222 carbon atoms has Raman bands similar to those in [[graphite]].
==Occurrence==
It is now presumed that tiny fragments of graphene sheets are produced (along with quantities of other debris) whenever graphite is abraded, such as when drawing a line with a pencil.<ref>"...bits of graphene are undoubtedly present in every pencil mark" —[http://www.sciam.com/article.cfm?id=carbon-wonderland Carbon Wonderland, ''Scientific American'', April 2008 ]</ref> However, it was physicists from the [[University of Manchester]] who first isolated and studied graphene (rather than [[polycyclic aromatic hydrocarbon|PAH]]) in 2004, and defined it in ''[[Science (journal)|Science]]''<ref name ="Nov 04">Novoselov, K.S. ''et al''. Electric Field Effect in Atomically Thin Carbon Films. ''Science'' '''306''', 666 (2004) {{DOI|10.1126/science.1102896}}</ref> as:
<blockquote>Graphene is the name given to a single layer of carbon atoms densely packed into a benzene-ring structure, and is widely used to describe properties of many carbon-based materials, including [[graphite]], large [[fullerenes]], [[nanotubes]], etc. (e.g., carbon nanotubes are usually thought of as graphene sheets rolled up into [[nanometer]]-sized [[cylinder (geometry)|cylinder]]s). Planar graphene itself has been presumed not to exist in the free state, being unstable with respect to the formation of curved structures such as [[soot]], fullerenes, and nanotubes.</blockquote>
The British researchers obtained relatively large graphene sheets (eventually, up to 100 microns in size and visible through a magnifying glass) by mechanical [[exfoliation (material science)|exfoliation]] (repeated peeling) of [[Three-dimensional space|3D]] graphite crystals; their motivation was allegedly to study the electrical properties of thin graphite films and, as purely two-dimensional crystals were unknown before and presumed not to exist, their discovery of individual planes of graphite was presumably accidental. Both theory and experiment previously suggested that perfect [[2D]] structures could not exist in the free state. It is believed that intrinsic microscopic roughening on the scale of 1 nm could be important for the stability of 2D crystals.<ref name ="Meyer07"> Meyer, J. ''et al''. The structure of suspended graphene sheets. ''Nature'' '''446''', 60-63 (1 March 2007){{DOI|10.1038/nature05545}}</ref>
Similar work is ongoing at many universities and the results obtained by the Manchester group in their [[PNAS]] paper "Two-dimensional atomic crystals" have been confirmed by several groups.<ref name ="PNAS05"> Novoselov, K.S. ''et al''. Two-dimensional atomic crystals. ''PNAS'' '''102''', 10451-10453 (January 26, 2005) {{DOI|10.1073/pnas.0502848102}}</ref> For an example of a sample on the order of a monolayer, see figure 1.
Graphene sheets in solid form (e.g. density > 1g/cc) usually show evidence in diffraction for [[graphite]]'s 0.34 nm (002) layering. This is true even of some single-walled carbon nanostructures<ref>D. Kasuya, M. Yudasaka, K. Takahashi, F. Kokai and S. Iijima (2002) ''J. Phys. Chem. B'' '''106''':4947.</ref>. However, unlayered graphene with only (hk0) rings has been found in the core of [[presolar grains|presolar]] graphite onions<ref>T. J. Bernatowicz, R. Cowsik, P. C. Gibbons, K. Lodders, B. Fegley Jr., S. Amari, and R. S. Lewis (1996) [http://www.journals.uchicago.edu/doi/pdf/10.1086/178105 Constraints on stellar grain formation from presolar graphite in the Murchison meteorite], ''Astrophysical Journal'' '''472''':760-782</ref>. [[Transmission electron microscope]] studies show faceting at defects in flat graphene sheets<ref>P. Fraundorf and M. Wackenhut (2002) [http://www.journals.uchicago.edu/doi/abs/10.1086/344633 The core structure of presolar graphite onions], ''Astrophysical Journal Letters'' '''578''':L153-156 ([http://arxiv.org/abs/astro-ph/0110585 astro-ph/0110585],[http://arxiv.org/abs/cond-mat/0606093 cond-mat/0606093])</ref>, and suggest a possible role in this unlayered-graphene for two-dimensional [[dendrite (crystal)|dendritic]] crystallization from a melt<ref>E. S. Mandell (2007) ''Electron beam characterization of carbon nanostructures'', Dissertation in Physics (University of Missouri - Saint Louis and Rolla).</ref>.
Graphene is presently one of the most expensive materials on Earth, with a sample that can be placed at the cross section of a human hair costing more than $1,000 (as of April 2008).<ref name = "SciAm">"Carbon Wonderland", [http://www.sciam.com/article.cfm?id=carbon-wonderland ''Scientific American'', April 2008]</ref> The price may fall dramatically, though, if commercial production methods are developed in the future.
==Properties==
===Atomic Structure===
The atomic structure of isolated, single-layer graphene was studied by [[transmission electron microscope|transmission electron microscopy]] (TEM) on sheets of graphene suspended between bars of a metallic grid.<ref name ="Meyer07"/> Electron diffraction patterns showed the expected hexagonal lattice of graphene. Suspended graphene also showed "rippling" of the flat sheet, with amplitude of about one nanometer. These ripples may be intrinsic to graphene as a result of the instability of two-dimensional crystals<ref name="Carlsson">Carlsson, J. M. Graphene: Buckle or break. ''Nature Materials'', '''6'''(11), 801-802 (2007) </ref><ref name="Fasolino">Fasolino, A., Los, J. H., & Katsnelson, M. I. Intrinsic ripples in graphene. ''Nature Materials'', '''6'''(11), 858-861 (2007)</ref><ref name="RiseGraphene"/>, or may be extrinsic, originating from the ubiquitous dirt seen in all TEM images of graphene. Atomic resolution real-space images of isolated, single-layer graphene on silicon dioxide substrates were obtained<ref name="Ishigami07">{{cite journal
| last = Ishigami
| first = Masa
| coauthors = et al.
| date = [[11 May]] [[2007]]
| volume = 7
| issue = 6
| pages = 1643–1648
| url = http://pubs.acs.org/cgi-bin/article.cgi/nalefd/2007/7/i06/pdf/nl070613a.pdf
| title = Atomic Structure of Graphene on SiO2
| journal = Nano Lett.
| doi = 10.1021/nl070613a
|format=PDF}}</ref><ref name="Stolyarova">{{cite journal
| last = Stolyarova
| first = Elena
| coauthors = et al.
| date = [[21 May]] [[2007]]
| volume = 104
| issue =
| pages = 9209–9212
| url = http://www.pnas.org/cgi/reprint/104/22/9209?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&fulltext=stolyarova&searchid=1&FIRSTINDEX=0&resourcetype=HWCIT
| title = High-resolution scanning tunneling microscopy imaging of mesoscopic graphene sheets on an insulating surface
| journal = Proceedings of the Natioanl Academy of Sciences
| doi = 10.1073/pnas.0703337104
| pmid = 17517635
}}</ref> by [[scanning tunneling microscope|scanning tunneling microscopy]]. Graphene processed using lithographic techniques is covered by [[photoresist]] residue, which must be cleaned to obtain atomic-resolution images <ref name="Ishigami07"/>. Such residue may be the "adsorbates" observed in TEM images, and may explain the rippling of suspended graphene. Rippling of graphene on the silicon dioxide surface was determined by conformation of graphene to the underlying silicon dioxide, and not an intrinsic effect<ref name="Ishigami07"/>.
===Electronic properties===
[[Image:graphene.jpg|right|300px|thumb|Figure 1. This image was captured using a Digital Multimode AFM ([[atomic force microscope]]). Notice the step from the [[Wafer (electronics)|substrate]] at zero height to a graphene flake about 8 [[angstrom]]s high, which is on the order of a [[monolayer]]. <ref>The image source is [http://www.princeton.edu/~pccm/outreach/REU2006/REU2006Presentations/moskowitz.pdf] (slide 17 image b) The height on the image is different. It is equal to 0.549 nm. See also more pictures of graphene [http://www.int.washington.edu/REU/2006/CherianPaper.pdf here].</ref>]]
While electron transport in most condensed matter systems are accurately described by the non-relativistic [[Schrödinger equation]], graphene is quite different from most conventional three-dimensional materials. Intrinsic graphene is a [[semi-metal]] or zero-gap [[semiconductor]]. The E-k relation is linear for low energies near the six corners of the two-dimensional hexagonal [[Brillouin zone]], leading to zero [[effective mass]] for electrons and holes. <ref name="E-Phonon"> J.-C. Charlier, P.C. Eklund, J. Zhu, and A.C. Ferrari, “Electron and Phonon Properties of Graphene: Their Relationship with Carbon Nanotubes,” from Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties and Applications, Ed. By A. Jorio, G. Dresselhaus, and M.S. Dresselhaus, Berlin/Heidelberg: Springer-Verlag, 2008 </ref> Due to this linear “dispersion” relation at low energies, electrons and holes near these six points behave like [[theory of relativity|relativistic]] particles described by the [[Dirac equation]] for spin 1/2 particles. <ref name="Carbon-basedElectronics">Avouris, P., Chen, Z., and Perebeinos, V. Carbon-based electronics. ''Nature Nano''. '''2''' 605-613 (2007)</ref> Hence, the electrons and holes are called Dirac [[fermions]], and the six corners of the Brillouin zone are called the Dirac points.<ref name="E-Phonon"/> The equation describing the E-k relation is <math>E = \hbar v_F\sqrt{k_x^2+k_y^2}</math>; where <math>v_f</math>, the [[Fermi velocity]], is approximately <math>10^6 \mathrm{m}/\mathrm{s}</math>.<ref name="Carbon-basedElectronics"/>
===Optical properties===
Graphene's unique electronic properties produce an unexpectedly high opacity for an atomic monolayer, with a startlingly simple value: it absorbs ''πα'' = 2.3% of white [[light]], where ''α'' is the [[fine-structure constant]].<ref>{{Citation |title=Universal infrared conductance of graphite |first1=A. B. |last1=Kuzmenko |first2=E. |last2=van Heumen |first3=F. |last3=Carbone |first4=D. |last4=van der Marel |journal=Phys. Rev. Lett. | volume=100 |pages=117401(2008)}}</ref><ref>{{Citation |title=Fine Structure Constant Defines Visual Transparency of Graphene |first1=R. R. |last1=Nair |first2=P. |last2=Blake |first3=A. N. |last3=Grigorenko |first4=K. S. |last4=Novoselov |first5=T. J. |last5=Booth |first6=T. |last6=Stauber |first7=N. M. R. |last7=Peres |first8=A. K. |last8=Geim |journal=[[Science (journal)|]] |date=2008-004-03 |doi=10.1126/science.1156965 |volume=320 |pages=1308 |pmid=18388259}}</ref> This has been confirmed experimentally, but the measurement is not precise enough to improve on other techniques for determining the fine-structure constant.<ref>{{Citation |title=Graphene Gazing Gives Glimpse Of Foundations Of Universe |url=http://www.sciencedaily.com/releases/2008/04/080403140918.htm |journal=ScienceDaily |date=2008-04-04 |accessdate=2008-04-06}}</ref>
===Electronic transport===
Experimental results from transport measurements show that graphene has a remarkably high [[electron mobility]] at room temperature, with reported values in excess of <math> 15,000 {cm}^2{V}^{-1}{s}^{-1} </math>. <ref name="RiseGraphene">Geim, A. K. and Novoselov, K. S. The rise of graphene. ''Nature Mater''. '''6''', 183-191 (2007)</ref> Additionally, the symmetry of the experimentally measured conductance indicates that the mobilities for holes and electrons should be nearly the same.<ref name="E-Phonon"/> The mobility is nearly independent of temperature between <math>10</math> and <math>100 K </math> <ref name="2dgasDiracFermions">Novoselov, K. S. ''et al.'' Two-dimensional gas of massless Dirac fermions in graphene. ''Nature'' '''438''', 197-200 (2005)</ref><ref name="GiantMobility">S.V. Morozov ''et al.'' Giant Intrinsic Carrier Mobilities in Graphene and Its Bilayer. ''Phys. Rev. Lett''. '''100''', 016602 (2008)</ref><ref name="GrapheneE-ph">Chen, J. H. ''et al.'' Intrinsic and Extrinsic Performance Limits of Graphene Devices on SiO2. ''Nature Nanotechnology'' '''3''', 206-209 (2008)</ref>, which implies that the dominant scattering mechanism is defect scattering. Scattering by the acoustic [[phonon]]s of graphene limits the room temperature mobility to <math> 200,000 {cm}^2{V}^{-1}{s}^{-1}</math> at a carrier density of <math> 10^{12} cm^{-2}</math><ref name=" GrapheneE-ph"/><ref name="GrapheneMC">Akturk, A. and Goldsman, N. Electron transport and full-band electron-phonon interactions in graphene. ''Journal of Applied Physics'' '''103''', 053702 (2008)</ref>. The corresponding [[resistivity]] of the graphene sheet would be <math> 10^{-6} Ohm-cm</math>, less than the resistivity of [[silver]], the lowest resistivity substance known at room temperature<ref name="UMDnews">Physicists Show Electrons Can Travel More Than 100 Times Faster in Graphene[https://newsdesk.umd.edu/scitech/release.cfm?ArticleID=1621]</ref>. However, for graphene on silicon dioxide substrates, scattering of electrons by optical phonons of the substrate is a larger effect at room temperature than scattering by graphene’s own phonons, and limits the mobility to <math> 40,000 {cm}^2{V}^{-1}{s}^{-1}</math>.<ref name=" GrapheneE-ph"/>
Despite the zero carrier density near the Dirac points, graphene exhibits a minimum [[Electrical conductivity|conductivity]] on the order of <math> {4e^2}/h </math>. The origin of this minimum conductivity is still unclear. However, rippling of the graphene sheet or ionized impurities in the SiO<sub>2</sub> substrate may lead to local puddles of carriers that allow conduction.<ref name="E-Phonon"/> Several theories suggest that the minimum conductivity should be <math>{4e^2}/{h\pi} </math>. However, most measurements are of order <math> {4e^2}/h </math> or greater,<ref name="RiseGraphene"/> and depend on impurity concentration.<ref name="GrapheneK">Chen, J. H. ''et al.'' Charged Impurity Scattering in Graphene. ''Nature Physics'' ''published online April 13, 2008''</ref>
Recent experiments have probed the influence of chemical dopants on the carrier mobility in graphene.<ref name="ChemDoping">Schedin, F. ''et al''. Detection of individual gas molecules adsorbed on graphene. ''Nature Mater''. '''6''', 652-655 (2007)</ref><ref name="GrapheneK"/> Schedin, et al. doped graphene with various gaseous species (some acceptors, some donors), and found the initial undoped state of a graphene structure can be recovered by gently heating the graphene in vacuum. Schedin, et al. reported that even for chemical dopant concentrations in excess of <math>{10}^{12} {cm}^{-2}</math> there is no observable change in the carrier mobility.<ref name="ChemDoping"/>. Chen, et al. doped graphene with [[potassium]] in [[ultra high vacuum]] at low temperature. They found that potassium ions act as expected for charged impurities in graphene<ref name="GrapheneCharge">Adam, S. ''et al''. A self-consistent theory for graphene transport. ''Proc. Nat. Acad. Sci. USA''. '''104''', 18392-18397 (2007)</ref>, and can reduce the mobility 20-fold.<ref name="GrapheneK"/> The mobility reduction is reversible on heating the graphene to remove the potassium.
===Spin Transport===
Graphene is thought to be an ideal material for [[spintronics]] due to small [[spin-orbit interaction]] and near absence of nuclear magnetic moments in carbon. Electrical spin-current injection and detection in graphene was recently demonstrated up to room temperature<ref name="Tombros">
{{cite journal
| last = Tombros
| first = Nikolaos
| coauthors = et al.
| date = [[2 August]] [[2007]]
| volume = 448
| issue =
| pages = 571–575
| url = http://www.nature.com/nature/journal/v448/n7153/pdf/nature06037.pdf
| title = Electronic spin transport and spin precession in single graphene layers at room temperature
| journal = Nature
| doi = 10.1038/nature06037
|format=PDF}}</ref><ref name="ChoSpin">
{{cite journal
| last = Cho
| first = Sungjae
| coauthors = Yung-Fu Chen, and Michael S. Fuhrer
| date = [[19 September]] [[2007]]
| volume = 91
| issue =
| pages = 123105
| url = http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=APPLAB000091000012123105000001&idtype=cvips&prog=normal
| title = Gate-tunable Graphene Spin Valve
| journal = Applied Physics Letters
| doi = 10.1063/1.2784934
}}</ref><ref name="Ohishi">
{{cite journal
| last = Ohishi
| first = Megumi
| coauthors = et al.
| date = [[22 June]] [[2007]]
| volume = 46
| issue =
| pages = L605–L607
| url = http://jjap.ipap.jp/link?JJAP/46/L605/pdf
| title = Spin Injection into a Graphene Thin Film at Room Temperature
| journal = Jpn. J. Appl. Phys.
| doi = 10.1143/JJAP.46.L605
}}</ref>. Spin coherence lengths greater than 1 micron at room temperature were observed<ref name="Tombros"/>, and control of the spin current polarity with an electrical gate was observed at low temperature<ref name="ChoSpin"/>.
===Magnetic effects===
Besides the high mobility and minimum conductivity, graphene shows very interesting behavior in the presence of a magnetic field. Graphene displays an anomalous [[quantum Hall effect]] with the sequence shifted by <math>1/2</math> with respect to the standard sequence. Thus, the Hall conductivity is <math>\sigma_{xy} = \pm {4e^2}/h \left(N + 1/2 \right)</math>, where <math>N</math> is the Landau level index and the double valley and double spin degeneracies give the factor of <math>4 </math>.<ref name="RiseGraphene"/> This remarkable behavior can even be measured at room temperature.<ref name="2dgasDiracFermions"/> Bilayer graphene also shows the quantum Hall effect, but with the standard sequence where <math>\sigma_{xy} = \pm {4Ne^2}/h </math>. Interestingly, the first plateau at <math>N=0</math> is absent, indicating that bilayer graphene stays metallic at the neutrality point.<ref name="RiseGraphene"/>
Unlike normal metals, the longitudinal resistance of graphene shows maxima rather than minima for integral values of the Landau filling factor in measurements of the [[Shubnikov-De Haas effect|Shubnikov-de Haas oscillations]], which show a phase shift of <math>\pi</math>, known as [[Berry’s phase]].<ref name="E-Phonon"/><ref name="2dgasDiracFermions"/> The Berry’s phase arises due to the zero effective carrier mass near the Dirac points.<ref name="Berry'sPhase">Zhang, Y., Tan, Y. W., Stormer, H. L., and Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. ''Nature'' '''438''', 201-204 (2005)</ref> Study of the temperature dependence of the Shubnikov-de Haas oscillations in graphene reveals that the carriers have a non-zero cyclotron mass, despite their zero effective mass from the E-k relation.<ref name="2dgasDiracFermions"/>
===Graphene oxide===
By oxidising and chemically processing graphene, and then floating them in water, the graphene flakes form a single sheet and bond very powerfully. These sheets have a measured [[Tensile Modulus|tensile modulus]] of 32 [[GPa]].<ref>[http://ttp.northwestern.edu/abstracts/viewabs.php?id=316&cat=83 Graphene Oxide Paper, Technology Transfer Program, Northwestern University<!-- Bot generated title -->]</ref>
===Chemical modification===
Soluble fragments of graphene can be prepared in the laboratory<ref>Sandip Niyogi, Elena Bekyarova, Mikhail E. Itkis, Jared L. McWilliams, Mark A. Hamon, and Robert C. Haddon . Solution Properties of Graphite and Graphene. ''[[J. Am. Chem. Soc.]]'' '''128'''(24) pp. 7720–7721 (2006); (Communication) {{DOI|10.1021/ja060680r}}</ref> through chemical modification of graphite. First, microcrystalline graphite is treated with a strongly acidic mixture of [[sulfuric acid]] and [[nitric acid]]. A series of steps involving oxidation and exfoliation result in small graphene plates with [[carboxyl]] groups at their edges. These are converted to [[acid chloride]] groups by treatment with [[thionyl chloride]]; next, they are converted to the corresponding graphene [[amide]] via treatment with octadecylamine. The resulting material (circular graphene layers of 5.3 [[angstrom]] thickness) is soluble in [[tetrahydrofuran]], [[tetrachloromethane]], and [[dichloroethane]].
===Thermal properties===
The near-room temperature [[thermal conductivity]] of graphene was recently measured to be between (4.84±0.44) ×10<sup>3</sup> to (5.30±0.48) ×10<sup>3</sup> Wm<sup>−1</sup>K<sup>−1</sup>. These measurements, made by a non-contact optical technique, are in excess of those measured for carbon nanotubes or diamond. It can be shown by using the [[Wiedemann-Franz law]], that the thermal conduction is phonon-dominated.<ref>Balandi, A.A., Ghosh, S., Bao, W., Calizo, I., Teweldebrahn, D., Miao, F., and Lau, C.N. "[http://pubs3.acs.org/acs/journals/doilookup?in_doi=10.1021/nl0731872 Superior Thermal Conductivity of Single-Layer Graphene]", ''Nano Letters ASAP'' (2008)</ref> However, for a gated graphene strip, an applied gate bias causing a [[Fermi Energy]] shift much larger than k<sub>B</sub>T can cause the electronic contribution to increase and dominate over the [[phonon]] contribution at low temperatures.<ref name="Saito">Saito, K., Nakamura, J., and Natori, A. "Ballistic thermal conductance of a graphene sheet", ''Physical Review B'' '''76''' 115409 (2007)</ref>
Potential for this high conductivity can be seen by considering [[graphite]], a 3D version of graphene that has basal plane thermal conductivity of over a 1000 W/mK (comparable to [[diamond]]). In [[graphite]], the c-axis (out of plane) [[thermal conductivity]] is over a factor of ~100 smaller due to the weak binding forces between basal planes as well as the larger [[lattice spacing]].<ref>"[http://books.google.com/books?id=7p2pgNOWPbEC Graphite and Precursors]", By Pierre Delhaes</ref> In addition, the ballistic [[thermal conductance]] of a graphene is shown to give the lower limit of the ballistic [[thermal conductance]]s, per unit circumference, length of [[carbon nanotubes]].<ref name="mingo">Mingo N., Broido, D.A. "Carbon Nanotube Ballistic Thermal Conductance and Its Limits" ''Physical Review Letters'' '''95''', 096105 (2005)</ref>
Despite its 2-D nature, Graphene has 3 [[acoustic phonon]] modes. The two in-plane modes have a linear [[dispersion relation]], whereas the out of plane mode has a quadratic [[dispersion relation]]. Due to this, the T<sup>2</sup> dependent thermal conductivity contribution of the linear modes is dominated at low temperatures by the T<sup>1.5</sup> contribution of the out of plane mode.<ref name="mingo"/> The ballistic thermal conductance of Graphene is isotropic.<ref name="Saito"/>
===Mechanical properties===
Utilizing an [[atomic force microscope]], research has recently been able to measure the [[spring constant]] of suspended Graphene sheets. Graphene sheets, held together by [[van der Waals forces]], were suspended over [[silicon dioxide]] cavities where an AFM tip was probed to test its mechanical properties. Its spring constant was measured to be on the order of 1-5 N/m and its [[Young's modulus]] was 0.5 TPa, which differs from bulk graphite. These high values make Graphene very strong and rigid. These intrinsic properties could lead to the possibility of utilizing Graphene for [[NEMS]] applications such as pressure sensors, and resonators.<ref>Frank, I. W., Tanenbaum, D. M., Van Der Zande, A.M., and McEuen, P. L. Mechanical properties of suspended graphene sheets. ''J. Vac. Sci. Technol. B'' '''25''', 2558-2561 (2007)</ref>
Graphene is considered to be the first truly 2D crystal. There has been some discrepancy whether this assertion is truly valid or not. While an infinitely-large single layer of graphene would be in direct contradiction to the [[Mermin-Wagner theorem]], a finite-size 2D crystal of graphene could be stable. The [[Mermin-Wagner theorem]] theorem states that a 2D crystal in a 3D environment would not remain ordered over long distances because of long wavelength fluctuations. It is believed that due to this instability, a large 2D structure will fold-up, or crumple to form a more stable 3D structure. Researchers have observed ripples in suspended layers of graphene.<ref name="Meyer07"/> It has been proposed that the ripples are caused by thermal fluctuations in the material. Graphene adjusts to the thermal fluctuations, which could threaten to destroy the structure, by adjusting its bond length to accommodate the fluctuations. Within this framework, it is debatable whether graphene is truly 2D or not, due to its natural tendency to ripple.<ref name="Carlsson"/><ref name="Fasolino"/><ref name="RiseGraphene"/>
==Potential applications==
===Single molecule gas detection===
Graphene makes an excellent sensor due to its 2D structure. The fact that its entire volume is exposed to its surrounding makes it very efficient to detect adsorbed molecules. Molecule detection is indirect: as a gas molecule adsorbs to the surface of graphene, the location of adsorption experiences a local change in [[electrical resistance]]. While this effect occurs in other materials, graphene is superior due to its high electrical conductivity (even when few carriers are present) and low noise which makes this change in resistance detectable.<ref name = "ChemDoping"> </ref>
===GNRs===
[[Graphene nanoribbons]] (GNRs) are essentially single layers of graphene that are cut in a particular pattern to give it certain electrical properties. Depending on how the un-bonded edges are configured, they can either be in a Z (zigzag) or Armchair configuration. Calculations based on tight binding predict that zigzag GNRs are always metallic while armchairs can either can be either metallic or semiconducting, depending on their width. However, recent DFT calculations show that armchair nanoribbons are semiconducting with an energy gap scaling with the inverse of the GNR width. <ref name = "ArmchRibb">Barone, V., Hod, O., and Scuseria, G. E. Electronic Structure and Stability of Semiconducting Graphene Nanoribbons ''Nano Lett.'' '''6''', 2748 (2006)</ref> Indeed, experimental results show that the energy gaps do increase with decreasing GNR width. <ref name = "EgEngGNR">Han., M.Y., Özyilmaz, B., Zhang, Y., and Kim, P. Energy Band-Gap Engineering of Graphene Nanoribbons. ''Phys. Rev. Lett.'' '''98''', 206805 (2007)</ref> However, to date no experimental results have measured the energy gap of a GNR and identified the exact edge structure. <ref>As of Thursday, February 28, 2008</ref> Zigzag nanoribbons are also semiconducting and present spin polarized edges.
Their 2D structure, high electrical and thermal conductivity, and low noise also make GNRs a possible alternative to copper for integrated circuit interconnects. Some research is also being done to create quantum dots by changing the width of GNRs at select points along the ribbon, creating [[quantum confinement]].<ref>Wang, Z. F., Shi, Q. W., Li, Q., Wang, X., Hou, J. G., Zheng, H., et al. Z-shaped graphene nanoribbon quantum dot device. ''Applied Physics Letters'', '''91'''(5), 053109 (2007)</ref>
Due to its high electronic quality, graphene has also attracted the interest of technologists who see them as a way of constructing [[ballistic transistor]]s. Graphene exhibits a pronounced response to perpendicular external electric field allowing one to built FETs ([[field-effect transistor]]s). In their 2004 paper<ref name = "Nov 04"/>, the Manchester group demonstrated FETs with a "rather modest" on-off ratio of ~30 at room temperature. In 2006, [[Georgia Institute of Technology|Georgia Tech]] researchers announced that they had successfully built an all-graphene planar FET with side gates<ref> ''Carbon-Based Electronics: Researchers Develop Foundation for Circuitry and Devices Based on Graphite'' March 14, 2006 gtresearchnews.gatech.edu [http://gtresearchnews.gatech.edu/newsrelease/graphene.htm Link]</ref>. Their devices showed changes of 2% at cryogenic temperatures. The first top-gated FET (on-off ratio of <2) was demonstrated by Aachen researchers in 2007<ref> Lemme, M. C. '' et al'' A graphene field-effect device. '' IEEE Electron Device Lett.'' '''28''', 282 (2007)</ref>. [[Graphene nanoribbons]] may prove generally capable of replacing silicon as a semiconductor in modern technology.<ref name="MIT1">
{{cite journal
| first = Kevin
| last = Bullis
| authorlink = Kevin Bullis
| title = Graphene Transistors
| journal = [[Technology Review]]
| publisher = [[Massachusetts Institute of Technology|MIT]] Technology Review, Inc
| location = Cambridge
| date = [[2008-01-28]]
| url = http://www.technologyreview.com/Nanotech/20119/
| accessdate = 2008-02-18}}</ref>
===Integrated circuits===
Graphene has the ideal properties to be an excellent component of [[integrated circuits]]. Graphene has a high [[carrier mobility]], as well as low noise allowing it to be utilized as the channel in a [[field-effect transistor|FET]]. The issue is that single sheets of graphene are hard to produce, and even harder to make on top of an appropriate substrate. Researchers are looking into methods of transferring single graphene sheets from their source of origin (mechanical exfoliation on SiO<sub>2</sub> / Si or thermal graphitization of a SiC surface) onto a target substrate of interest.<ref>Chen, J., Ishigami, M., Jang, C., Hines, D. R., Fuhrer, M. S., and Williams, E. D. Printed graphene circuits. ''Advanced Materials'', '''19'''(21), 3623-3627 (2007)</ref> In 2008, the smallest transistor so far, one atom thick, 10 atoms wide was made of graphene<ref>[http://news.bbc.co.uk/2/hi/technology/7352464.stm BBC News article of 17 April 2008; Ponomarenko et al, ''Science'', '''320''', 356 (2008)]</ref>.
===Transparent conducting electrodes===
Graphene's high electrical conductivity and high optical transparency make it a candidate for transparent conducting electrodes, required for such applications as [[touchscreen]]s, [[liquid crystal display]]s, [[solar cell|organic photovoltaic cells]], and [[Organic light-emitting diode|OLEDs]]. In particular, graphene's mechanical strength and flexibility are advantageous compared to [[indium tin oxide]], which is brittle, and graphene films may be deposited from solution over large areas<ref name="MPI">
{{cite journal
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| coauthors = et al.
| date = [[11 December]] [[2007]]
| url = http://pubs.acs.org/cgi-bin/abstract.cgi/nalefd/2008/8/i01/abs/nl072838r.html
| title = Transparent, Conductive Graphene Electrodes for Dye-Sensitized Solar Cells
| journal = Nano Letters
| doi = 10.1021/nl072838r
| volume = 8
| pages = 323
}}</ref>
<ref name="Eda">
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| format = {{dead link|date=June 2008}} – <sup>[http://scholar.google.co.uk/scholar?hl=en&lr=&q=author%3AEda+intitle%3ALarge-area+ultrathin+films+of+reduced+graphene+oxide+as+a+transparent+and+flexible+electronic+material&as_publication=Nature+Nanotechnology&as_ylo=&as_yhi=&btnG=Search Scholar search]</sup>
}}</ref>.
==Pseudo-relativistic theory==
The electrical properties of graphene can be described by a conventional [[tight-binding]] model; in this model the energy of the electrons with wavenumber <math>\mathbf{k}</math> is
:<math>E=\pm\sqrt{\gamma_0^2\left(1+4\cos^2{\pi k_ya}+4\cos{\pi k_ya} \cdot \cos{\pi k_x\sqrt{3}a}\right)}</math> <ref>P. R. Wallace, ''The Band Theory of Graphite'', Physical Review '''71''' (1947) 622–634</ref>,
with the nearest-neighbour-hopping energy <math>\gamma_0\approx 2{.}8\ \mathrm{eV}</math> and the [[lattice constant]] <math>a\approx 2{.}46\ \mathrm{\AA}</math>.
[[conduction band|Conduction]]- and [[valence band]], respectively, correspond to the different signs in the above [[dispersion relation]]; they touch each other in six points, the "K-values". However, only two of these six points are independent, whereas the rest is equivalent by symmetry. In the vicinity of the K-points the energy depends ''linearly'' on the wavenumber, similar to a relativistic particle. Since an elementary cell of the lattice has a basis of two atoms, the [[wave function]] even has an effective [[Spinor|2-spinor structure]]. As a consequence, at low energies, the electrons can be described by an equation which is formally equivalent to the [[Dirac equation]]. Moreover, in the present case this pseudo-relativistic description is restricted to the [[Chirality|chiral limit]], i.e., to vanishing rest mass <math>M_0</math>, which leads to interesting additional features:
:<math>v_F\vec\sigma\cdot\vec\nabla \psi(\mathbf{r})\,=\,E\psi(\mathbf{r})</math>
Here <math>v_F\approx 10^6\ \mathrm{m/s}</math> is the [[Fermi distribution|Fermi velocity]] in graphene which replaces the velocity of light in the Dirac theory; <math>\vec{\sigma}</math> is the vector of the [[Pauli matrices]], <math>\psi(\mathbf{r})</math> is the two-component [[wave function]] of the electrons, and <math>E</math> their energy. <ref>A. H. Castro Neto, F. Guinea, N. M. R Peres, K. S. Novoselov, A. K. Geim, ''The electronic properties of graphene'', e-print arXiv.org:0709.1163v2 , (2007), [http://arxiv.org/abs/0709.1163] </ref>
==History and Experimental Discovery==
The term graphene first appeared<ref name=”Mouras87”>Mouras, S. ‘’et al.’’ Synthesis of first stage graphite intercalation compounds with fluorides. ‘’Revue de Chimie Minerale’’ ‘’’24’’’ 572 (1987).</ref> in order to describe single sheets of graphite as one of the constituents of [[graphite intercalation compound]]s (GICs); conceptually a GIC is a crystalline salt of the intercalant and graphene. The term was also used in the earliest descriptions of carbon nanotubes<ref name =”Saito92”>Saito, R. ‘’et al.’’ Electronic structure of graphene tubules based on C60. ‘’Phys. Rev. B’’ ‘’’46’’’ 1804 (1992).</ref>, as well as for epitaxial graphene<ref name=”Forbeaux98”>Forbeaux, I. ‘’et al.’’ Heteroepitaxial graphite on 6H-SiC(0001): Interface formation through conduction-band electronic structure. ‘’Phys. Rev. B’’ ‘’’58’’’ 16396 (1998).</ref>, and polycyclic aromatic hydrocarbons<ref name=”Wang00”>Wang, S. ‘’et al.’’ A new carbonaceous material with large capacity and high efficiency for rechargeable Li-ion batteries. ‘’Journal of the Electrochemical Society’’. ‘’’147’’’, 2498 (2000).</ref>. However, none of these examples constitutes isolated, two-dimensional graphene.
Larger graphene molecules or sheets (so that they can be considered as true isolated 2D crystals) cannot be grown even in principle. An article <ref name ="PhysTod">Geim, A. K. & MacDonald, A. H. Graphene: Exploring carbon flatland. ''Physics Today'', '''60''' 35-41 (Aug 2007)</ref> in [[Physics Today]] reads:
<blockquote> "Fundamental forces place seemingly insurmountable barriers in the way of creating [2D crystals] ... Nascent 2D crystallites try to minimize their surface energy and inevitably morph into one of the rich variety of stable 3D structures that occur in soot.
But there is a way around the problem. Interactions with 3D structures stabilize 2D crystals during growth. So one can make 2D crystals sandwitched between or placed on top of the atomic planes of a bulk crystal. In that respect, graphene already exists within graphite ...
One can then hope to fool Nature and extract single-atom-thick crystallites at a low enough temperature that they remain in the quenched state prescribed by the original higher-temperature 3D growth."</blockquote>
Single layers of graphite were previously (starting from the 1970s) grown epitaxially on top of other materials <ref name="Oshima97">For a review, see Oshima, C. and Nagashima, A. Ultra-thin epitaxial films of graphite and hexagonal boron nitride on solid surfaces. ''J. Phys.: Condens. Matter'' '''9''', 1 (1997)</ref>. This "epitaxial graphene" consists of a single-atom-thick hexagonal lattice of [[sp2 bond|sp<sup>2</sup>-bonded]] carbon atoms, as in free-standing graphene. However, there is significant charge transfer from the substrate to the epitaxial graphene, and, in some cases, hybridization between the d orbitals of the substrate atoms and <math>\pi</math> orbitals of graphene, which significantly alters the electronic structure of the epitaxial graphene.
Single layers of graphite were also observed by [[transmission electron microscopy]] within bulk materials (see section ''Occurrence''), in particular inside soot obtained by chemical exfoliation<ref name="SciAm"/>. There have also been a number of efforts to make very thin films of graphite by mechanical exfoliation (starting from 1990 and continuing until after 2004)<ref name="SciAm"/> but nothing thinner than 50 to 100 layers was produced during these years (read ''discussion'').
The previous efforts did not result in graphene as we know it now, i.e. as "free standing" single-atom-thick crystals of a macroscopic size which are either suspended or interact only weakly with a substrate. It is not important whether graphene is suspended or placed on another (non-binding) substrate. In both cases, it is ''isolated'' and can be studied as such. Within this definition of graphene, it was first isolated by the Manchester group of [[Andre Geim]] who in 2004<ref name="Nov 04"/> finally managed to extract single-atom-thick crystallites from bulk graphite. He provided the first and unexpected proof for the existence of true (free-standing) 2D crystals. Previously, it was assumed that graphene cannot exist in the flat state and should scroll into nanotubes "to decrease the surface energy"<ref> H. Shioyama ''J. Mater. Sci. Lett.'' '''20''', 499 (2001) </ref><ref>L. M. Viculis ''et al. Science''
'''299''', 1361 (2003)</ref> .
This experimental discovery of 2D crystal matter was openly doubted[http://www.esi-topics.com/fmf/2006/november06-AndreGeim.html] until 2005 when in the same issue of ''Nature'' the groups of Andre Geim and Philip Kim of [[Columbia University]] have proved "beyond a reasonable doubt" that the obtained graphitic layers exhibit the electronic properties prescribed by theory. This theory was first developed by Philip R Wallace in 1947 as an approximation trying to understand the electronic properties of more complex, 3 dimensional graphite. He did not use word graphene and referred to "a single hexagonal layer". <ref name=”Wallace47”>Wallace, P. R. The band theory of graphite. ''Physical Review'' '''71''' 622-633 (1947)</ref> Later, graphene crystals obtained by using the Manchester recipe were also made suspended and their thickness proved directly by electron microscopy<ref name ="Meyer07"/>.
==Cultural references==
Graphene was mentioned in comic 1111 of [[Questionable Content]]<ref>{{cite web |author=Jacques, Jeph |year=2008|url=http://questionablecontent.net/view.php?comic=1111|title=Buckeyballs of Flavor(#1111)|work=questionablecontent.net|accessdate=2008-04-01}}</ref>
== References ==
{{reflist|2}}
==See also==
* [[Aromaticity]]
* [[Fullerene]]s
* [[Polycyclic aromatic hydrocarbon]]s
* [[Carbon nanotube]]s
* [[Graphene nanoribbons]]
* [[Graphene Oxide Paper]]
==External links==
* [http://onnes.ph.man.ac.uk/nano/Publications/Naturemat_2007Review.pdf The Rise of Graphene] A review article by A.K. Geim & K.S. Novoselov in Nature Materials 6, 183-191 (2007).
* [http://www.ewels.info/img/science/ Image Gallery - Graphite and Graphene]
* [http://news.bbc.co.uk/2/hi/science/nature/3944651.stm BBC News report]
* [http://news.bbc.co.uk/2/hi/science/nature/7352464.stm BBC News report, latest 18-APR-2008]
* [http://physicsweb.org/articles/news/9/11/6/1 Electrons lose their mass in carbon sheets] Physics Web (09-Nov-2005)
*[http://gtresearchnews.gatech.edu/newsrelease/graphene.htm Potential for graphene computer chips, with explanation of technical issues/problems]
* [http://www.newscientist.com/channel/fundamentals/mg19125591.700 Quantum weirdness on the end of your pencil] Philip Ball, New Scientist Magazine issue 2559, (8 July 2006)
* [http://online.kitp.ucsb.edu/online/graphene_m07/ Talks] at the ''Electronic Properties of Graphene'' conference (January 8-19, 2007)
* [http://physicsworld.com/cws/article/news/31136] Physics World article concerning minimum conductance
* [http://technology.newscientist.com/channel/tech/dn13158-carbon-nanosheets-promise-superfast-chips.html?feedId=online-news_rss20 Carbon nanosheets promise super-fast chips] Graphene has the highest electronic quality among all known materials, ''New Scientist'', 8 January 2008
* [http://onnes.ph.man.ac.uk/nano/Publications.html#Graphene Most of graphene papers published by Andre Geim's group are downloadable here]
* [http://www.abc.net.au/science/articles/2008/01/29/2148939.htm?site=science&topic=latest Researchers leap a nano hurdle - ABC Online (Australia)] Development of a simpler method of graphene production.
* [http://www.nsf.gov/news/news_summ.jsp?cntn_id=111341&org=NSF&from=news Is Graphene the New Silicon?] National Science Foundation, March 27, 2008
* Band structure of graphene [http://www.nanohub.org/resource_files/2005/12/00723/2004.10.20-l21-ece453.pdf].
* [http://heybryan.org/graphene.html General notes on graphene]
[[Category:Aromatic compounds]]
[[Category:Nanomaterials]]
[[Category:Carbon forms]]
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