electronic properties of graphene

Electrons propagating through graphene's honeycomb lattice effectively lose their mass, producing quasi-particles that are described by a 2D analogue of the Dirac equation rather than the Schrödinger equation for spin-{{frac|1|2}} particles.{{cite journal |first1=A Castro |last1=Neto |author-link1=Antonio H. Castro Neto|last2=Peres |first2=N. M. R. |last3=Novoselov |first3=K. S. |author-link3=Konstantin Novoselov|last4=Geim |first4=A. K. |author-link4=Andre Geim|title=The electronic properties of graphene |journal=Rev Mod Phys |volume=81 |issue=1 |year=2009 |pages=109–162 |url=http://onnes.ph.man.ac.uk/nano/Publications/RMP_2009.pdf |archive-url=http://webarchive.nationalarchives.gov.uk/20101115121052/http://onnes.ph.man.ac.uk/nano/Publications/RMP_2009.pdf |url-status=dead |archive-date=2010-11-15 |bibcode=2009RvMP...81..109C |doi=10.1103/RevModPhys.81.109 |arxiv=0709.1163|hdl=10261/18097 |s2cid=5650871 }}{{cite book |url={{google books |plainurl=yes |id=ammoVEI-H2gC}} |last1=Charlier |first1=J.-C. |last2=Eklund |first2=P.C. |last3=Zhu |first3=J. |last4=Ferrari |first4=A.C. |title=Electron and Phonon Properties of Graphene: Their Relationship with Carbon Nanotubes |work=Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties and Applications |editor1-first=A. |editor1-last=Jorio |editor2-first=G. |editor2-last=Dresselhaus and |editor3-first=M.S. |editor3-last=Dresselhaus |editor-link3=Mildred Dresselhaus|location=Berlin/Heidelberg |publisher=Springer-Verlag |year=2008}}

File:Electronic band structure of graphene.svg

When atoms are placed onto the graphene hexagonal lattice, the overlap between the p_z(π) orbitals and the s or the p_x and p_y orbitals is zero by symmetry. The p_z electrons forming the π bands in graphene can be treated independently. Within this π-band approximation, using a conventional tight-binding model, the dispersion relation (restricted to first-nearest-neighbor interactions only) that produces energy of the electrons with wave vector $\backslash mathbf\{k\}=[k\_x,k\_y]$ is{{cite journal |last=Wallace |first=P.R. |author-link=P. R. Wallace|title=The Band Theory of Graphite |doi=10.1103/PhysRev.71.622 |journal=Physical Review |volume=71 |year=1947 |pages=622–634 |bibcode=1947PhRv...71..622W |issue=9}}

: $E(\backslash mathbf\{k\})=\backslash pm\backslash ,\backslash gamma\_0\backslash sqrt\{1+4\backslash cos^2\{\backslash tfrac\{1\}\{2\}a\; k\_x\}+4\backslash cos\{\backslash tfrac\{1\}\{2\}a\; k\_x\}\; \backslash cdot\; \backslash cos\{\backslash tfrac\{\backslash sqrt\{3\}\}\{2\}a\; k\_y\}\}$

with the nearest-neighbor (π orbitals) hopping energy γ₀ ≈ {{val|2.8|u=eV}} and the lattice constant {{nowrap|a ≈ {{val|2.46|u=Å}}}}. The conduction and valence bands, respectively, correspond to the different signs. With one p_z electron per atom in this model the valence band is fully occupied, while the conduction band is vacant. The two bands touch at the zone corners (the K point in the Brillouin zone), where there is a zero density of states but no band gap. The graphene sheet thus displays a semimetallic (or zero-gap semiconductor) character. Two of the six Dirac points are independent, while the rest are equivalent by symmetry. In the vicinity of the K-points the energy depends linearly on the wave vector, similar to a relativistic particle.{{Cite journal |last=Semenoff |first=G. W. |author-link=Gordon Walter Semenoff|title=Condensed-Matter Simulation of a Three-Dimensional Anomaly |doi=10.1103/PhysRevLett.53.2449 |journal=Physical Review Letters |volume=53 |pages=2449–2452 |year=1984 |bibcode=1984PhRvL..53.2449S |issue=26}}{{Cite journal |last1=Avouris |first1=P. |author-link1=Phaedon Avouris|last2=Chen |first2=Z. |last3=Perebeinos |first3=V. |title=Carbon-based electronics |doi=10.1038/nnano.2007.300 |journal=Nature Nanotechnology |volume=2 |year=2007 |pmid=18654384 |issue=10 |bibcode=2007NatNa...2..605A |pages=605–15}} Since an elementary cell of the lattice has a basis of two atoms, the wave function has an effective 2-spinor structure.

As a consequence, at low energies, even neglecting the true spin, the electrons can be described by an equation that is formally equivalent to the massless Dirac equation. Hence, the electrons and holes are called Dirac fermions. This pseudo-relativistic description is restricted to the chiral limit, i.e., to vanishing rest mass M₀, which leads to additional features:{{cite journal |last1=Lamas |first1=C.A. |first2=D.C. |last2=Cabra |first3=N. |last3=Grandi |title=Generalized Pomeranchuk instabilities in graphene |journal=Physical Review B |year=2009 |volume=80 |issue=7 |page=75108 |doi=10.1103/PhysRevB.80.075108 |arxiv=0812.4406 |bibcode=2009PhRvB..80g5108L|s2cid=119213419 }}

: $-\; i\; v\_F\backslash ,\; \backslash vec\; \backslash sigma\; \backslash cdot\; \backslash nabla\; \backslash psi(\backslash mathbf\{r\})\backslash ,=\backslash ,E\backslash psi(\backslash mathbf\{r\}).$

Here v_F ≈ {{val |e=6 |u=m/s}} (0.003 c) is the Fermi velocity in graphene, which replaces the velocity of light in the Dirac theory; $\backslash vec\{\backslash sigma\}$ is the vector of the Pauli matrices; $\backslash psi(\backslash mathbf\{r\})$ is the two-component wave function of the electrons and E is their energy.

The equation describing the electrons' linear dispersion relation is

: $E(k)=\backslash hbar\; v\_\backslash text\{F\}\; k$

where the wavevector $\backslash textstyle\; k=\backslash sqrt\{k\_x^2+k\_y^2\}$ is measured from the Dirac points (the zero of energy is chosen here to coincide with the Dirac points). The equation uses a pseudospin matrix formula that describes two sublattices of the honeycomb lattice.

Graphene's unit cell has two identical carbon atoms and two zero-energy states: one in which the electron resides on atom A, the other in which the electron resides on atom B. However, if the two atoms in the unit cell are not identical, the situation changes. Hunt et al. showed that placing hexagonal boron nitride (h-BN) in contact with graphene can alter the potential felt at atom A versus atom B enough that the electrons develop a mass and accompanying band gap of about {{val|30|ul=meV}}.{{cite journal|last1=Fuhrer|first1=M. S.|year=2013|title=Critical Mass in Graphene|journal=Science|volume=340|issue=6139|pages=1413–1414|bibcode=2013Sci...340.1413F|doi=10.1126/science.1240317|pmid=23788788|s2cid=26403885}}

The mass can be positive or negative. An arrangement that slightly raises the energy of an electron on atom A relative to atom B gives it a positive mass, while an arrangement that raises the energy of atom B produces a negative electron mass. The two versions behave alike and are indistinguishable via optical spectroscopy. An electron traveling from a positive-mass region to a negative-mass region must cross an intermediate region where its mass once again becomes zero. This region is gapless and therefore metallic. Metallic modes bounding semiconducting regions of opposite-sign mass is a hallmark of a topological phase and display much the same physics as topological insulators.

If the mass in graphene can be controlled, electrons can be confined to massless regions by surrounding them with massive regions, allowing the patterning of quantum dots, wires and other mesoscopic structures. It also produces one-dimensional conductors along the boundary. These wires would be protected against backscattering and could carry currents without dissipation.

Electron waves in graphene propagate within a single-atom layer, making them sensitive to the proximity of other materials such as high-κ dielectrics, superconductors and ferromagnetics.

Graphene displays remarkable electron mobility at room temperature, with reported values in excess of {{val|15000 |u=cm²⋅V⁻¹⋅s⁻¹}}.{{sfn|Geim|Novoselov|2007}} Hole and electron mobilities were expected to be nearly identical. The mobility is nearly independent of temperature between {{val|10 |u=K}} and {{val|100 |u=K}},{{cite journal |last1=Novoselov |first1=K. S. |last2=Geim |first2=A. K. |last3=Morozov |first3=S. V. |last4=Jiang |first4=D. |last5=Katsnelson |first5=M. I. |last6=Grigorieva |first6=I. V. |last7=Dubonos |first7=S. V. |last8=Firsov |first8=A. A. |title=Two-dimensional gas of massless Dirac fermions in graphene |doi=10.1038/nature04233 |journal=Nature |volume=438 |pages=197–200 |year=2005 |pmid=16281030 |issue=7065 |arxiv=cond-mat/0509330 |bibcode=2005Natur.438..197N|hdl=2066/33126 |s2cid=3470761 }}{{cite journal |last1=Morozov |first1=S.V. |last2=Novoselov |first2=K. |last3=Katsnelson |first3=M. |last4=Schedin |first4=F. |last5=Elias |first5=D. |last6=Jaszczak |first6=J. |last7=Geim |first7=A. |title=Giant Intrinsic Carrier Mobilities in Graphene and Its Bilayer |doi=10.1103/PhysRevLett.100.016602 |journal=Physical Review Letters |volume=100 |page=016602 |year=2008 |pmid=18232798 |bibcode=2008PhRvL.100a6602M |issue=1 |arxiv=0710.5304|s2cid=3543049 }}{{cite journal |last1=Chen |first1=J. H. |last2=Jang |first2=Chaun |last3=Xiao |first3=Shudong |last4=Ishigami |first4=Masa |last5=Fuhrer |first5=Michael S. |title=Intrinsic and Extrinsic Performance Limits of Graphene Devices on {{chem|SiO|2}} |doi=10.1038/nnano.2008.58 |journal=Nature Nanotechnology |volume=3 |year=2008 |pmid=18654504 |issue=4 |pages=206–9|arxiv=0711.3646 |s2cid=12221376 }} which implies that the dominant scattering mechanism is defect scattering. Scattering by graphene's acoustic phonons intrinsically limits room temperature mobility to {{val|200000 |u=cm²⋅V⁻¹⋅s⁻¹}} at a carrier density of {{val |e=12 |u=cm⁻²}},{{cite journal |last1=Akturk |first1=A. |last2=Goldsman |first2=N. |title=Electron transport and full-band electron–phonon interactions in graphene |doi=10.1063/1.2890147 |journal=Journal of Applied Physics |volume=103 |year=2008 |bibcode=2008JAP...103e3702A |issue=5|pages=053702–053702–8 }} {{val|10 |e=6}} times greater than copper.{{cite arXiv |last1=Kusmartsev |first1=F. V. |last2=Wu |first2=W. M. |last3=Pierpoint |first3=M. P. |last4=Yung |first4=K. C. |title=Application of Graphene within Optoelectronic Devices and Transistors |eprint=1406.0809 |class=cond-mat.mtrl-sci |year=2014}}

The corresponding resistivity of graphene sheets would be {{val |e=-6 |u=Ω⋅cm}}. This is less than the resistivity of silver, the lowest otherwise known at room temperature.[https://newsdesk.umd.edu/scitech/release.cfm?ArticleID=1621 Physicists Show Electrons Can Travel More Than 100 Times Faster in Graphene :: University Communications Newsdesk, University of Maryland] {{webarchive |url=https://web.archive.org/web/20130919083015/https://newsdesk.umd.edu/scitech/release.cfm?ArticleID=1621 |date=19 September 2013 }}. Newsdesk.umd.edu (24 March 2008). Retrieved on 2014-01-12. However, on {{chem|SiO|2}} substrates, scattering of electrons by optical phonons of the substrate is a larger effect than scattering by graphene's own phonons. This limits mobility to {{val|40000 |u=cm²⋅V⁻¹⋅s⁻¹}}.

Charge transport is affected by adsorption of contaminants such as water and oxygen molecules. This leads to non-repetitive and large hysteresis I-V characteristics. Researchers must carry out electrical measurements in vacuum. Graphene surfaces can be protected by a coating with materials such as SiN, PMMA and h-BN. In January 2015, the first stable graphene device operation in air over several weeks was reported, for graphene whose surface was protected by aluminum oxide.{{cite journal |last=Sagade |first=A. A. |title=Highly Air Stable Passivation of Graphene Based Field Effect Devices |doi=10.1039/c4nr07457b |pmid=25631337 |journal=Nanoscale |volume=7 |issue=8 |pages=3558–3564 |year=2015 |display-authors=etal |bibcode=2015Nanos...7.3558S}}{{cite web |url=https://spectrum.ieee.org/graphene-devices-stand-the-test-of-time |title=Graphene Devices Stand the Test of Time|date=2015-01-22}} In 2015 lithium-coated graphene was observed to exhibit superconductivity{{cite web |title=Researchers create superconducting graphene |url=http://www.rdmag.com/news/2015/09/researchers-create-superconducting-graphene |access-date=2015-09-22|date=2015-09-09 }} and in 2017 evidence for unconventional superconductivity was demonstrated in single layer graphene placed on the electron-doped (non-chiral) d-wave superconductor Pr_2−xCe_xCuO₄ (PCCO).{{Cite journal|last1=Di Bernardo|first1=A.|last2=Millo|first2=O.|last3=Barbone|first3=M.|last4=Alpern|first4=H.|last5=Kalcheim|first5=Y.|last6=Sassi|first6=U.|last7=Ott|first7=A. K.|last8=Fazio|first8=D. De|last9=Yoon|first9=D.|date=2017-01-19|title=p-wave triggered superconductivity in single-layer graphene on an electron-doped oxide superconductor|journal=Nature Communications|language=en|volume=8|doi=10.1038/ncomms14024|pmid=28102222|pmc=5253682|issn=2041-1723|page=14024|arxiv=1702.01572|bibcode=2017NatCo...814024D}}

Electrical resistance in 40-nanometer-wide nanoribbons of epitaxial graphene changes in discrete steps. The ribbons' conductance exceeds predictions by a factor of 10. The ribbons can act more like optical waveguides or quantum dots, allowing electrons to flow smoothly along the ribbon edges. In copper, resistance increases in proportion to length as electrons encounter impurities.{{cite web |url=http://www.kurzweilai.net/new-form-of-graphene-allows-electrons-to-behave-like-photons |title=New form of graphene allows electrons to behave like photons |work=kurzweilai.net}}{{cite journal |doi=10.1038/nature12952 |pmid=24499819 |title=Exceptional ballistic transport in epitaxial graphene nanoribbons |journal=Nature |volume=506 |issue=7488 |pages=349–354 |year=2014 |last1=Baringhaus |first1=J. |last2=Ruan |first2=M. |last3=Edler |first3=F. |last4=Tejeda |first4=A. |last5=Sicot |first5=M. |last6=Taleb-Ibrahimi |first6=A. |last7=Li |first7=A. P. |last8=Jiang |first8=Z. |last9=Conrad |first9=E. H. |last10=Berger |first10=C. |last11=Tegenkamp |first11=C. |last12=De Heer |first12=W. A. |arxiv=1301.5354 |bibcode=2014Natur.506..349B|s2cid=4445858 }}

Transport is dominated by two modes. One is ballistic and temperature independent, while the other is thermally activated. Ballistic electrons resemble those in cylindrical carbon nanotubes. At room temperature, resistance increases abruptly at a particular length—the ballistic mode at 16 micrometres and the other at 160 nanometres.

Graphene electrons can cover micrometer distances without scattering, even at room temperature.

Despite zero carrier density near the Dirac points, graphene exhibits a minimum conductivity on the order of $4e^2/h$. The origin of this minimum conductivity is unclear. However, rippling of the graphene sheet or ionized impurities in the {{chem|SiO|2}} substrate may lead to local puddles of carriers that allow conduction. Several theories suggest that the minimum conductivity should be $4e^2/\{(\backslash pi\}h)$; however, most measurements are of order $4e^2/h$ or greater{{sfn|Geim|Novoselov|2007}} and depend on impurity concentration.{{cite journal |last1=Chen |first1=J. H. |last2=Jang |first2=C. |last3=Adam |first3=S. |last4=Fuhrer |first4=M. S. |last5=Williams |first5=E. D. |last6=Ishigami |first6=M. |title=Charged Impurity Scattering in Graphene |doi=10.1038/nphys935 |journal=Nature Physics |volume=4 |pages=377–381 |year=2008 |bibcode=2008NatPh...4..377C |issue=5 |arxiv=0708.2408|s2cid=53419753 }}

Near zero carrier density graphene exhibits positive photoconductivity and negative photoconductivity at high carrier density. This is governed by the interplay between photoinduced changes of both the Drude weight and the carrier scattering rate.[http://www.kurzweilai.net/light-pulses-control-how-graphene-conducts-electricity Light pulses control how graphene conducts electricity]. kurzweilai.net. 4 August 2014

Graphene doped with various gaseous species (both acceptors and donors) can be returned to an undoped state by gentle heating in vacuum.{{cite journal |last1=Schedin |first1=F. |last2=Geim |first2=A. K. |last3=Morozov |first3=S. V. |last4=Hill |first4=E. W. |last5=Blake |first5=P. |last6=Katsnelson |first6=M. I. |last7=Novoselov |first7=K. S. |title=Detection of individual gas molecules adsorbed on graphene |doi=10.1038/nmat1967 |journal=Nature Materials |volume=6 |pages=652–655 |year=2007 |pmid=17660825 |issue=9 |bibcode=2007NatMa...6..652S|arxiv=cond-mat/0610809 |s2cid=3518448 }} Even for dopant concentrations in excess of 10¹² cm⁻² carrier mobility exhibits no observable change. Graphene doped with potassium in ultra-high vacuum at low temperature can reduce mobility 20-fold.{{cite journal |last1=Adam |first1=S. |last2=Hwang |first2=E. H. |last3=Galitski |first3=V. M. |last4=Das Sarma |first4=S. |title=A self-consistent theory for graphene transport |journal=Proc. Natl. Acad. Sci. USA |volume=104 |arxiv=0705.1540 |year=2007 |doi=10.1073/pnas.0704772104 |pmid=18003926 |issue=47 |pmc=2141788 |bibcode=2007PNAS..10418392A |pages=18392–7|doi-access=free }} The mobility reduction is reversible on removing the potassium.

Due to graphene's two dimensions, charge fractionalization (where the apparent charge of individual pseudoparticles in low-dimensional systems is less than a single quantum{{cite journal |first1=Hadar |last1=Steinberg |first2=Gilad |last2=Barak |first3=Amir |last3=Yacoby |title=Charge fractionalization in quantum wires |journal=Nature Physics |volume=4 |issue=2 |year=2008 |pages=116–119 |doi=10.1038/nphys810 |bibcode=2008NatPh...4..116S |arxiv=0803.0744 |s2cid=14581125 |display-authors=etal}}) is thought to occur. It may therefore be a suitable material for constructing quantum computers{{cite journal |arxiv=1003.4590 |title=Dirac four-potential tunings-based quantum transistor utilizing the Lorentz force |first=Agung |last=Trisetyarso |journal=Quantum Information & Computation |url=http://dl.acm.org/citation.cfm?id=2481569.2481576 |volume=12 |year=2012 |page=989 |bibcode=2010arXiv1003.4590T |issue=11–12|doi=10.26421/QIC12.11-12-7 |s2cid=28441144 }} using anyonic circuits.{{cite journal |arxiv=0812.1116 |title=Manifestations of topological effects in graphene |first=Jiannis K. |last=Pachos |journal=Contemporary Physics |doi=10.1080/00107510802650507 |volume=50 |year=2009 |pages=375–389 |bibcode=2009ConPh..50..375P |issue=2|s2cid=8825103 }}
{{cite web |url=http://www.int.washington.edu/talks/WorkShops/int_08_37W/People/Franz_M/Franz.pdf |title=Fractionalization of charge and statistics in graphene and related structures |first=M. |last=Franz |publisher=University of British Columbia |date=5 January 2008 |access-date=6 September 2017 |archive-date=15 November 2010 |archive-url=https://web.archive.org/web/20101115121039/http://www.int.washington.edu/talks/WorkShops/int_08_37W/People/Franz_M/Franz.pdf |url-status=dead }}

In 2018, superconductivity was reported in twisted bilayer graphene.

First-principle calculations with quasiparticle corrections and many-body effects explore the electronic and optical properties of graphene-based materials. The approach is described as three stages.{{cite journal|last1=Onida|first1=Giovanni|last2=Reining|first2=Lucia|last3=Rubio|first3=Angel|author2-link=Lucia Reining|author3-link=Ángel Rubio|year=2002|title=Electronic excitations: Density-functional versus many-body Green's-function approaches|journal=Rev. Mod. Phys.|volume=74|issue=2|pages=601–659|bibcode=2002RvMP...74..601O|doi=10.1103/RevModPhys.74.601|hdl=10261/98472|url=https://digital.csic.es/bitstream/10261/98472/1/Electronic%20excitations.pdf|hdl-access=free}} With GW calculation, the properties of graphene-based materials are accurately investigated, including bulk graphene,{{cite journal|last1=Yang|first1=Li|last2=Deslippe|first2=Jack|last3=Park|first3=Cheol-Hwan|last4=Cohen|first4=Marvin|last5=Louie|first5=Steven|year=2009|title=Excitonic Effects on the Optical Response of Graphene and Bilayer Graphene|journal=Physical Review Letters|volume=103|issue=18|page=186802|arxiv=0906.0969|bibcode=2009PhRvL.103r6802Y|doi=10.1103/PhysRevLett.103.186802|pmid=19905823|s2cid=36067301}} nanoribbons,{{cite journal|last1=Prezzi|first1=Deborah|last2=Varsano|first2=Daniele|last3=Ruini|first3=Alice|last4=Marini|first4=Andrea|last5=Molinari|first5=Elisa|author5-link=Elisa Molinari|year=2008|title=Optical properties of graphene nanoribbons: The role of many-body effects|journal=Physical Review B|volume=77|issue=4|page=041404|arxiv=0706.0916|bibcode=2008PhRvB..77d1404P|doi=10.1103/PhysRevB.77.041404|s2cid=73518107}}{{cite journal|last1=Yang|first1=Li|last2=Cohen|first2=Marvin L.|last3=Louie|first3=Steven G.|year=2007|title=Excitonic Effects in the Optical Spectra of Graphene Nanoribbons|journal=Nano Letters|volume=7|issue=10|pages=3112–5|arxiv=0707.2983|bibcode=2007NanoL...7.3112Y|doi=10.1021/nl0716404|pmid=17824720|s2cid=16943236}}{{cite journal|last1=Yang|first1=Li|last2=Cohen|first2=Marvin L.|last3=Louie|first3=Steven G.|year=2008|title=Magnetic Edge-State Excitons in Zigzag Graphene Nanoribbons|journal=Physical Review Letters|volume=101|issue=18|page=186401|bibcode=2008PhRvL.101r6401Y|doi=10.1103/PhysRevLett.101.186401|pmid=18999843}} edge and surface functionalized armchair oribbons,{{cite journal|last1=Zhu|first1=Xi|last2=Su|first2=Haibin|year=2010|title=Excitons of Edge and Surface Functionalized Graphene Nanoribbons|journal=J. Phys. Chem. C|volume=114|issue=41|pages=17257–17262|doi=10.1021/jp102341b}} hydrogen saturated armchair ribbons,{{cite journal|last1=Wang|first1=Min|last2=Li|first2=Chang Ming|year=2011|title=Excitonic properties of hydrogen saturation-edged armchair graphene nanoribbons|journal=Nanoscale|volume=3|issue=5|pages=2324–8|bibcode=2011Nanos...3.2324W|doi=10.1039/c1nr10095e|pmid=21503364}} Josephson effect in graphene SNS junctions with single localized defect{{cite journal|last1=Bolmatov|first1=Dima|last2=Mou|first2=Chung-Yu|year=2010|title=Josephson effect in graphene SNS junction with a single localized defect|journal=Physica B|volume=405|issue=13|pages=2896–2899|arxiv=1006.1391|bibcode=2010PhyB..405.2896B|doi=10.1016/j.physb.2010.04.015|s2cid=119226501}}{{cite journal|last1=Bolmatov|first1=Dima|last2=Mou|first2=Chung-Yu|year=2010|title=Tunneling conductance of the graphene SNS junction with a single localized defect|journal=Journal of Experimental and Theoretical Physics|volume=110|issue=4|pages=613–617|arxiv=1006.1386|bibcode=2010JETP..110..613B|doi=10.1134/S1063776110040084|s2cid=119254414}} and armchair ribbon scaling properties.{{cite journal|last1=Zhu|first1=Xi|last2=Su|first2=Haibin|year=2011|title=Scaling of Excitons in Graphene Nanoribbons with Armchair Shaped Edges|journal=Journal of Physical Chemistry A|volume=115|issue=43|pages=11998–12003|doi=10.1021/jp202787h|pmid=21939213|bibcode=2011JPCA..11511998Z}}

In 2014 researchers magnetized graphene by placing it on an atomically smooth layer of magnetic yttrium iron garnet. The graphene's electronic properties were unaffected. Prior approaches involved doping.T. Hashimoto, S. Kamikawa, Y. Yagi, J. Haruyama, H. Yang, M. Chshiev, [http://nanojournal.ifmo.ru/en/articles-2/volume5/5-1/paper02/ "Graphene edge spins: spintronics and magnetism in graphene nanomeshes"], February 2014, Volume 5, Issue 1, pp 25 The dopant's presence negatively affected its electronic properties.{{cite news|url=http://www.gizmag.com/magnetized-graphene/35805|title=Scientists give graphene one more quality – magnetism|last=Coxworth|first=Ben|date=27 January 2015|access-date=6 October 2016|publisher=Gizmag}}

In magnetic fields of ~10 tesla, additional plateaus of Hall conductivity at $\backslash sigma\_\{xy\}=\backslash nu\; e^2/h$ with $\backslash nu=0,\backslash pm\; \{1\},\backslash pm\; \{4\}$ are observed.{{cite journal|last1=Zhang|first1=Y.|last2=Jiang|first2=Z.|last3=Small|first3=J. P.|last4=Purewal|first4=M. S.|last5=Tan|first5=Y.-W.|last6=Fazlollahi|first6=M.|last7=Chudow|first7=J. D.|last8=Jaszczak|first8=J. A.|last9=Stormer|first9=H. L.|year=2006|title=Landau-Level Splitting in Graphene in High Magnetic Fields|journal=Physical Review Letters|volume=96|issue=13|page=136806|arxiv=cond-mat/0602649|bibcode=2006PhRvL..96m6806Z|doi=10.1103/PhysRevLett.96.136806|pmid=16712020|last10=Kim|first10=P.|s2cid=16445720}} The observation of a plateau at $\backslash nu=3${{cite journal|last1=Du|first1=X.|last2=Skachko|first2=Ivan|last3=Duerr|first3=Fabian|last4=Luican|first4=Adina|last5=Andrei|first5=Eva Y.|year=2009|title=Fractional quantum Hall effect and insulating phase of Dirac electrons in graphene|journal=Nature|volume=462|issue=7270|pages=192–195|arxiv=0910.2532|bibcode=2009Natur.462..192D|doi=10.1038/nature08522|pmid=19829294|s2cid=2927627}} and the fractional quantum Hall effect at $\backslash nu=1/3$ were reported.{{cite journal|last1=Bolotin|first1=K.|last2=Ghahari|first2=Fereshte|last3=Shulman|first3=Michael D.|last4=Stormer|first4=Horst L.|last5=Kim|first5=Philip|year=2009|title=Observation of the fractional quantum Hall effect in graphene|journal=Nature|volume=462|issue=7270|pages=196–199|arxiv=0910.2763|bibcode=2009Natur.462..196B|doi=10.1038/nature08582|pmid=19881489|s2cid=4392125}}

These observations with $\backslash nu=0,\backslash pm\; 1,\backslash pm\; 3,\; \backslash pm\; 4$ indicate that the four-fold degeneracy (two valley and two spin degrees of freedom) of the Landau energy levels is partially or completely lifted.{{Cite journal |last1=Abergel |first1=D.S.L. |last2=Apalkov |first2=V. |last3=Berashevich |first3=J. |last4=Ziegler |first4=K. |last5=Chakraborty |first5=Tapash |date=July 2010 |title=Properties of graphene: a theoretical perspective |url=https://www.tandfonline.com/doi/full/10.1080/00018732.2010.487978 |journal=Advances in Physics |language=en |volume=59 |issue=4 |pages=261–482 |doi=10.1080/00018732.2010.487978 |arxiv=1003.0391 |bibcode=2010AdPhy..59..261A |s2cid=119181322 |issn=0001-8732}} One hypothesis is that the magnetic catalysis of symmetry breaking is responsible for lifting the degeneracy.{{citation needed|date=December 2013}}

Graphene is claimed to be an ideal material for spintronics due to its small spin–orbit interaction and the near absence of nuclear magnetic moments in carbon (as well as a weak hyperfine interaction). Electrical spin current injection and detection has been demonstrated up to room temperature.{{cite journal |title=Electronic spin transport and spin precession in single graphene layers at room temperature |bibcode=2007Natur.448..571T |last=Tombros |first=Nikolaos |journal=Nature |year=2007 |volume=448 |issue=7153 |pages=571–575 |doi=10.1038/nature06037 |pmid=17632544 |arxiv=0706.1948 |s2cid=4411466 |display-authors=etal}}{{cite journal |first1=Sungjae |last1=Cho |first2=Yung-Fu |last2=Chen |first3=Michael S. |last3=Fuhrer |year=2007 |volume=91 |page=123105 |title=Gate-tunable Graphene Spin Valve |journal=Applied Physics Letters |doi=10.1063/1.2784934 |bibcode=2007ApPhL..91l3105C |issue=12 |arxiv=0706.1597|s2cid=119145153 }}{{cite journal |last=Ohishi |first=Megumi |year=2007 |volume=46 |issue=25 |pages=L605–L607 |title=Spin Injection into a Graphene Thin Film at Room Temperature |journal=Jpn J Appl Phys |doi=10.1143/JJAP.46.L605 |bibcode=2007JaJAP..46L.605O |arxiv=0706.1451 |s2cid=119608880 |display-authors=etal}} Spin coherence length above 1 micrometre at room temperature was observed, and control of the spin current polarity with an electrical gate was observed at low temperature.

Spintronic and magnetic properties can be present in graphene simultaneously.{{cite journal |last1=Hashimoto |first1=T. |last2=Kamikawa |first2=S. |last3=Yagi |first3=Y. |last4=Haruyama |first4=J. |last5=Yang |first5=H. |last6=Chshiev |first6=M. |title=Graphene edge spins: spintronics and magnetism in graphene nanomeshes |journal=Nanosystems: Physics, Chemistry, Mathematics |date=2014 |volume=5 |issue=1 |pages=25–38 |url=http://nanojournal.ifmo.ru/en/wp-content/uploads/2014/02/NPCM51_P25-38.pdf}} Low-defect graphene nanomeshes manufactured using a non-lithographic method exhibit large-amplitude ferromagnetism even at room temperature. Additionally a spin pumping effect is found for fields applied in parallel with the planes of few-layer ferromagnetic nanomeshes, while a magnetoresistance hysteresis loop is observed under perpendicular fields.{{citation needed|date=January 2017}}

Charged particles in high-purity graphene behave as a strongly interacting, quasi-relativistic plasma. The particles move in a fluid-like manner, traveling along a single path and interacting with high frequency. The behavior was observed in a graphene sheet faced on both sides with a h-BN crystal sheet.{{cite web|url=http://newatlas.com/liquid-graphene-dirac-fluid/41801 |first=Dario |last=Borghino|date=February 15, 2016|title=Liquid-like graphene could be the key to understanding black holes|publisher=New Atlas|access-date=February 18, 2017}}

The quantum Hall effect is a quantum mechanical version of the Hall effect, The Hall effect occurs when a magnetic field causes a perpendicular (transverse) current in a material. In the quantum Hall effect, the transverse conductivity $\backslash sigma\_\{xy\}$ is quantized in integer multiples of a basic quantity:

: $e^2/h$

where e is the elementary electric charge and h is the Planck constant. This phenomenon is typically observed in very clean silicon or gallium arsenide solids at temperatures around {{val|3|ul=K}} and high magnetic fields.

Graphenem, a single layer of carbon atoms, exhibits an unusual form of the quantum Hall effect. In graphene, the steps of conductivity quantization are shifted by 1/2 compared to the standard sequence and have an additional factor of 4. This can be expressed as:

: $\backslash sigma\_\{xy\}=\backslash pm\; \{4\backslash cdot\backslash left(N\; +\; 1/2\; \backslash right)e^2\}/h$

where N is the Landau level. The factor of 4 arises due to the double valley and double spin degeneracies of electrons in graphene.{{sfn|Geim|Novoselov|2007}} These anomalies can be observed even at room temperature (about 20 °C or 293 K).

This anomalous behavior is due to graphene's massless Dirac electrons. In a magnetic field, these electrons form a Landau level at the Dirac point with an energy that is precisely zero. This is a result of the Atiyah–Singer index theorem and causes the "+1/2" term in the Hall conductivity for neutral graphene. {{cite journal |last1=Gusynin |first1=V. P. |last2=Sharapov |first2=S. G. |title=Unconventional Integer Quantum Hall Effect in Graphene |doi=10.1103/PhysRevLett.95.146801 |journal=Physical Review Letters |volume=95 |page=146801 |year=2005 |pmid=16241680 |bibcode=2005PhRvL..95n6801G |arxiv=cond-mat/0506575 |issue=14|s2cid=37267733 }}

In bilayer graphene, the quantum Hall effect is also observed but with only one of the two anomalies. The Hall conductivity in bilayer graphene is given by:

: $\backslash sigma\_\{xy\}=\backslash pm\; \{4\backslash cdot\; N\backslash cdot\; e^2\}/h$

In this case, the first plateau at {{nowrap|1=N = 0}} is absent, meaning bilayer graphene remains metallic at the neutrality point.{{sfn|Geim|Novoselov|2007}}

Unlike normal metals, graphene's longitudinal resistance shows maxima, not minima, for integral values of the Landau filling factor in Shubnikov–de Haas oscillations. This is termed the integral quantum Hall effect. These oscillations exhibit a phase shift of π, known as Berry's phase, which is due to the zero effective mass of carriers near the Dirac points.{{cite journal |last1=Zhang |first1=Y. |last2=Tan |first2=Y. W. |last3=Stormer |first3=H. L. |last4=Kim |first4=P. |title=Experimental observation of the quantum Hall effect and Berry's phase in graphene |doi=10.1038/nature04235 |journal=Nature |volume=438 |pages=201–204 |year=2005 |pmid=16281031 |issue=7065 |arxiv=cond-mat/0509355 |bibcode=2005Natur.438..201Z|s2cid=4424714 }} Despite this zero effective mass, the temperature dependence of the oscillations indicates a non-zero cyclotron mass for the carriers.

Graphene samples prepared on nickel films and on both the silicon and carbon faces of silicon carbide show the anomalous quantum Hall effect in electrical measurements.{{cite journal |last1=Kim |first1=Kuen Soo |title=Large-scale pattern growth of graphene films for stretchable transparent electrodes |year=2009 |doi=10.1038/nature07719 |journal=Nature |volume=457 |pmid=19145232 |issue=7230 |bibcode=2009Natur.457..706K |pages=706–10 |last2=Zhao |first2=Yue |last3=Jang |first3=Houk |last4=Lee |first4=Sang Yoon |last5=Kim |first5=Jong Min |last6=Kim |first6=Kwang S. |last7=Ahn |first7=Jong-Hyun |last8=Kim |first8=Philip |last9=Choi |first9=Jae-Young |last10=Hong |first10=Byung Hee|s2cid=4349731 }}{{cite journal |first1=Johannes |last1=Jobst |first2=Daniel |last2=Waldmann |first3=Florian |last3=Speck |first4=Roland |last4=Hirner |first5=Duncan K. |last5=Maude |first6=Thomas |last6=Seyller |first7=Heiko B. |last7=Weber |title=How Graphene-like is Epitaxial Graphene? Quantum Oscillations and Quantum Hall Effect |year=2009 |doi=10.1103/PhysRevB.81.195434 |journal=Physical Review B |volume=81 |issue=19 |page=195434 |arxiv=0908.1900 |bibcode=2010PhRvB..81s5434J|s2cid=118710923 }}{{cite journal |first1=T. |last1=Shen |first2=J.J. |last2=Gu |first3=M |last3=Xu |first4=Y.Q. |last4=Wu |first5=M.L. |last5=Bolen |first6=M.A. |last6=Capano |first7=L.W. |last7=Engel |first8=P.D. |last8=Ye |title=Observation of quantum-Hall effect in gated epitaxial graphene grown on SiC (0001) |doi=10.1063/1.3254329 |journal=Applied Physics Letters |bibcode=2009ApPhL..95q2105S |year=2009 |volume=95 |issue=17 |page=172105 |arxiv=0908.3822|s2cid=9546283 }}{{cite journal |first1=Xiaosong |last1=Wu |first2=Yike |last2=Hu |first3=Ming |last3=Ruan |first4=Nerasoa K |last4=Madiomanana |first5=John |last5=Hankinson |first6=Mike |last6=Sprinkle |first7=Claire |last7=Berger |first8=Walt A. |last8=de Heer |year=2009 |title=Half integer quantum Hall effect in high mobility single layer epitaxial graphene |doi=10.1063/1.3266524 |journal=Applied Physics Letters |volume=95 |issue=22 |page=223108 |arxiv=0909.2903 |bibcode=2009ApPhL..95v3108W|citeseerx=10.1.1.754.9537 |s2cid=118422866 }}{{cite journal |first1=Samuel |last1=Lara-Avila |first2=Alexei |last2=Kalaboukhov |first3=Sara |last3=Paolillo |first4=Mikael |last4=Syväjärvi |first5=Rositza |last5=Yakimova |first6=Vladimir |last6=Fal'ko |first7=Alexander |last7=Tzalenchuk |first8=Sergey |last8=Kubatkin |title=SiC Graphene Suitable For Quantum Hall Resistance Metrology |journal=Science Brevia |date=7 July 2009 |arxiv=0909.1193 |doi= |bibcode=2009arXiv0909.1193L |pmid=}}{{cite journal |first1=J.A. |last1=Alexander-Webber |first2=A.M.R. |last2=Baker |first3=T.J.B.M. |last3=Janssen |first4=A. |last4=Tzalenchuk |first5=S. |last5=Lara-Avila |first6=S. |last6=Kubatkin |first7=R. |last7=Yakimova |first8=B. A. |last8=Piot |first9=D. K. |last9=Maude |first10=R.J. |last10=Nicholas |year=2013 |title=Phase Space for the Breakdown of the Quantum Hall Effect in Epitaxial Graphene |doi=10.1103/PhysRevLett.111.096601 |journal=Physical Review Letters |volume=111 |issue=9 |page=096601 |pmid=24033057 |arxiv=1304.4897 |bibcode=2013PhRvL.111i6601A|s2cid=118388086 }} Graphitic layers on the carbon face of silicon carbide exhibit a clear Dirac spectrum in angle-resolved photoemission experiments. This effect is also observed in cyclotron resonance and tunneling experiments.{{cite journal |first=Michael S. |last=Fuhrer |title=A physicist peels back the layers of excitement about graphene |doi=10.1038/4591037e |journal=Nature |volume=459 |page=1037 |year=2009 |pmid=19553953 |issue=7250 |bibcode=2009Natur.459.1037F|s2cid=203913300 |doi-access=free }}

The Casimir effect is an interaction between disjoint neutral bodies provoked by the fluctuations of the electrodynamical vacuum. Mathematically it can be explained by considering the normal modes of electromagnetic fields, which explicitly depend on the boundary (or matching) conditions on the interacting bodies' surfaces. Since graphene/electromagnetic field interaction is strong for a one-atom-thick material, the Casimir effect is of interest.{{cite journal |last1=Bordag |first1=M. |last2=Fialkovsky |first2=I. V. |last3=Gitman |first3=D. M. |last4=Vassilevich |first4=D. V. |title=Casimir interaction between a perfect conductor and graphene described by the Dirac model |journal=Physical Review B |volume=80 |year=2009 |page=245406 |doi=10.1103/PhysRevB.80.245406 |bibcode=2009PhRvB..80x5406B |issue=24 |arxiv=0907.3242|s2cid=118398377 }}{{cite journal |last1=Fialkovsky |first1=I. V. |last2=Marachevsky |first2=V.N. |last3=Vassilevich |first3=D. V. |title=Finite temperature Casimir effect for graphene |year=2011 |volume=84 |issue=35446 |journal=Physical Review B |arxiv=1102.1757 |bibcode=2011PhRvB..84c5446F |page=35446 |doi=10.1103/PhysRevB.84.035446|s2cid=118473227 }}

The Van der Waals force (or dispersion force) is also unusual, obeying an inverse cubic, asymptotic power law in contrast to the usual inverse quartic.{{cite journal |last1=Dobson |first1=J. F. |last2=White |first2=A. |last3=Rubio |first3=A. |title=Asymptotics of the dispersion interaction: analytic benchmarks for van der Waals energy functionals |journal=Physical Review Letters |volume=96 |year=2006 |page=073201 |doi=10.1103/PhysRevLett.96.073201 |issue=7 |bibcode=2006PhRvL..96g3201D |arxiv=cond-mat/0502422 |pmid=16606085|hdl=10261/97924 |s2cid=31092090 }}

The electronic properties of graphene are significantly influenced by the supporting substrate.{{cite journal|last1=Xu|first1=Yang|last2=He|first2=K. T.|last3=Schmucker|first3=S. W.|last4=Guo|first4=Z.|last5=Koepke|first5=J. C.|last6=Wood|first6=J. D.|last7=Lyding|first7=J. W.|last8=Aluru|first8=N. R.|title=Inducing Electronic Changes in Graphene through Silicon (100) Substrate Modification|journal=Nano Letters|date=2011|doi=10.1021/nl201022t|volume=11|issue=7|pages=2735–2742|pmid=21661740|bibcode=2011NanoL..11.2735X}}

{{cite journal |last1=Pantano |first1=Maria F. |display-authors=et al. |date=July 2019 |title=Investigation of charges-driven interaction between graphene and different SiO2 surfaces |journal=Carbon |volume=148 |pages=336–343 |doi=10.1016/j.carbon.2019.03.071 |url=https://qmro.qmul.ac.uk/xmlui/handle/123456789/57704 |hdl=11572/234972 |s2cid=141310234 |hdl-access=free }}

The Si(100)/H surface does not perturb graphene's electronic properties, whereas the interaction between it and the clean Si(100) surface changes its electronic states significantly. This effect results from the covalent bonding between C and surface Si atoms, modifying the π-orbital network of the graphene layer. The local density of states shows that the bonded C and Si surface states are highly disturbed near the Fermi energy.

{{unreferenced section|date=February 2023}}

If the in-plane direction is confined, in which case it is referred to as a nanoribbon, its electronic structure is different. If it is "zig-zag" (diagram), the bandgap is zero. If it is "armchair" (diagram), the bandgap is non-zero (see figure).

{{Gallery |width=300px |align=center

|file:cnt zz v3.gif|GNR band structure for zig-zag orientation. Tightbinding calculations show that zig-zag orientation is always metallic.

|file:cnt gnrarm v3.gif|GNR band structure for armchair orientation. Tightbinding calculations show that armchair orientation can be semiconducting or metallic depending on width (chirality).

}}

{{reflist|30em}}

• {{cite journal |last1=Geim |first1=A. K. |last2=Novoselov |first2=K. S. |doi=10.1038/nmat1849 |title=The rise of graphene |journal=Nature Materials |volume=6 |issue=3 |pages=183–191 |year=2007 |pmid=17330084|bibcode=2007NatMa...6..183G |arxiv=cond-mat/0702595 |s2cid=14647602 }}

• [https://demonstrations.wolfram.com/GrapheneBrillouinZoneAndElectronicEnergyDispersion/ Wolfram demonstration for graphene BZ and electronic dispersion]

Category:Graphene

Category:Electrical phenomena