allotrope of carbon
Carbon is capable of forming many allotropy, allotropes (structurally different forms of the same element) due to its Valence (chemistry), valency. Well-known forms of carbon include diamond and graphite. In recent decades, many more allotrope ...
consisting of a single layer of atoms arranged in a
hexagonal lattice
The hexagonal lattice or triangular lattice is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° ...
nanostructure
A nanostructure is a structure of intermediate size between microscopic and molecular structures. Nanostructural detail is microstructure at nanoscale.
In describing nanostructures, it is necessary to differentiate between the number of dimens ...
. "Carbon nanostructures for electromagnetic shielding applications",
Mohammed Arif Poothanari, Sabu Thomas, et al., ''Industrial Applications of Nanomaterials'', 2019. "Carbon nanostructures include various low-dimensional allotropes of carbon including carbon black (CB), carbon fiber, carbon nanotubes (CNTs), fullerene, and graphene." The name is derived from "graphite" and the suffix
-ene
The suffix -ene is used in organic chemistry to form names of organic compounds where the -C=C- group has been attributed the highest priority according to the rules of organic nomenclature. Sometimes a number between hyphens is inserted before i ...
, reflecting the fact that the
graphite
Graphite () is a crystalline form of the element carbon. It consists of stacked layers of graphene. Graphite occurs naturally and is the most stable form of carbon under standard conditions. Synthetic and natural graphite are consumed on large ...
allotrope of carbon contains numerous double bonds.
Each atom in a graphene sheet is connected to its three nearest neighbors by a strong
σ-bond
In chemistry, sigma bonds (σ bonds) are the strongest type of covalent chemical bond. They are formed by head-on overlapping between atomic orbitals. Sigma bonding is most simply defined for diatomic molecules using the language and tools of sy ...
, and contributes to a
valence band
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...
one
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
that extends over the whole sheet. This is the same type of bonding seen in
carbon nanotube
A scanning tunneling microscopy image of a single-walled carbon nanotube
Rotating single-walled zigzag carbon nanotube
A carbon nanotube (CNT) is a tube made of carbon with diameters typically measured in nanometers.
''Single-wall carbon na ...
s and
polycyclic aromatic hydrocarbon
A polycyclic aromatic hydrocarbon (PAH) is a class of organic compounds that is composed of multiple aromatic rings. The simplest representative is naphthalene, having two aromatic rings and the three-ring compounds anthracene and phenanthrene. ...
s, and (partially) in
fullerene
A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The molecule may be a hollow sphere, ...
s and
glassy carbon
Glass-like carbon, often called glassy carbon or vitreous carbon, is a non-graphitizing, or nongraphitizable, carbon which combines glassy and ceramic properties with those of graphite. The most important properties are high temperature resis ...
. The valence band is touched by a
conduction band
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...
, making graphene a
semimetal
A semimetal is a material with a very small overlap between the bottom of the conduction band and the top of the valence band.
According to electronic band theory, solids can be classified as insulators, semiconductors, semimetals, or metals. ...
with unusual electronic properties that are best described by theories for massless relativistic particles. Charge carriers in graphene show linear, rather than quadratic, dependence of energy on momentum, and field-effect transistors with graphene can be made that show bipolar conduction. Charge transport is
ballistic
Ballistics may refer to:
Science
* Ballistics, the science that deals with the motion, behavior, and effects of projectiles
** Forensic ballistics, the science of analyzing firearm usage in crimes
** Internal ballistics, the study of the proce ...
over long distances; the material exhibits large
quantum oscillations
In condensed matter physics, quantum oscillations describes a series of related experimental techniques used to map the Fermi surface of a metal in the presence of a strong magnetic field. These techniques are based on the principle of Landau qua ...
and large and nonlinear
diamagnetism
Diamagnetic materials are repelled by a magnetic field; an applied magnetic field creates an induced magnetic field in them in the opposite direction, causing a repulsive force. In contrast, paramagnetic and ferromagnetic materials are attracted ...
. Graphene conducts heat and electricity very efficiently along its plane. The material strongly absorbs light of all visible wavelengths, which accounts for the black color of graphite; yet a single graphene sheet is nearly transparent because of its extreme thinness. The material is about 100 times as strong as would be the strongest steel of the same thickness.
Scientists theorized the potential existence and production of graphene for decades. It has likely been unknowingly produced in small quantities for centuries, through the use of pencils and other similar applications of graphite. It was possibly observed in
electron microscope
An electron microscope is a microscope that uses a beam of accelerated electrons as a source of illumination. As the wavelength of an electron can be up to 100,000 times shorter than that of visible light photons, electron microscopes have a hi ...
s in 1962, but studied only while supported on metal surfaces.
In 2004, the material was rediscovered, isolated and investigated at the
University of Manchester
, mottoeng = Knowledge, Wisdom, Humanity
, established = 2004 – University of Manchester Predecessor institutions: 1956 – UMIST (as university college; university 1994) 1904 – Victoria University of Manchester 1880 – Victoria Univer ...
, by
Andre Geim
, birth_date =
, birth_place = Sochi, Russian SFSR, Soviet Union
, death_date =
, death_place =
, workplaces =
, nationality = Dutch and British
, fields = Condensed matter physics
...
and
Konstantin Novoselov
Sir Konstantin Sergeevich Novoselov ( rus, Константи́н Серге́евич Новосёлов,
p=kənstɐnʲˈtʲin sʲɪrˈɡʲe(j)ɪvʲɪtɕ nəvɐˈsʲɵləf; born 1974) is a Russian-British physicist, and a professor at the ...
. In 2010, Geim and Novoselov were awarded the
Nobel Prize in Physics
)
, image = Nobel Prize.png
, alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
for their "groundbreaking experiments regarding the two-dimensional material graphene". High-quality graphene proved to be surprisingly easy to isolate.
Graphene has become a valuable and useful
nanomaterial
*
Nanomaterials describe, in principle, materials of which a single unit is sized (in at least one dimension) between 1 and 100 nm (the usual definition of nanoscale).
Nanomaterials research takes a materials science-based approach to nan ...
due to its exceptionally high tensile strength, electrical conductivity, transparency, and being the thinnest two-dimensional material in the world. The global market for graphene was $9 million in 2012, with most of the demand from research and development in semiconductor, electronics,
electric batteries
An electric battery is a source of electric power consisting of one or more electrochemical cells with external connections for powering electrical devices.
When a battery is supplying power, its positive terminal is the cathode and its negat ...
IUPAC
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
(International Union for Pure and Applied Chemistry) recommends use of the name "graphite" for the three-dimensional material, and "graphene" only when the reactions, structural relations, or other properties of individual layers are discussed. A narrower definition, of "isolated or free-standing graphene" requires that the layer be sufficiently isolated from its environment, but would include layers suspended or transferred to
silicon dioxide
Silicon dioxide, also known as silica, is an oxide of silicon with the chemical formula , most commonly found in nature as quartz and in various living organisms. In many parts of the world, silica is the major constituent of sand. Silica is one ...
or
silicon carbide
Silicon carbide (SiC), also known as carborundum (), is a hard chemical compound containing silicon and carbon. A semiconductor, it occurs in nature as the extremely rare mineral moissanite, but has been mass-produced as a powder and crystal sin ...
.
History
Structure of graphite and its intercalation compounds
lamellar
A ''lamella'' (plural ''lamellae'') is a small plate or flake, from the Latin, and may also be used to refer to collections of fine sheets of material held adjacent to one another, in a gill-shaped structure, often with fluid in between though s ...
structure of thermally reduced
graphite oxide
Graphite oxide (GO), formerly called graphitic oxide or graphitic acid, is a compound of carbon, oxygen, and hydrogen in variable ratios, obtained by treating graphite with strong oxidizers and acids for resolving of extra metals. The maximally o ...
. In 1916,
Peter Debye
Peter Joseph William Debye (; ; March 24, 1884 – November 2, 1966) was a Dutch-American physicist and physical chemist, and Nobel laureate in Chemistry.
Biography
Early life
Born Petrus Josephus Wilhelmus Debije in Maastricht, Netherlands, D ...
and
Paul Scherrer
Paul Hermann Scherrer (3 February 1890 – 25 September 1969) was a Swiss physicist. Born in St. Gallen, Switzerland, he studied at Göttingen, Germany, before becoming a lecturer there. Later, Scherrer became head of the Department of Physics ...
determined the structure of graphite by
powder X-ray diffraction
Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials. An instrument dedicated to performing such powder measurements is cal ...
. The structure was studied in more detail by V. Kohlschütter and P. Haenni in 1918, who also described the properties of graphite oxide paper. Its structure was determined from
single-crystal
In materials science, a single crystal (or single-crystal solid or monocrystalline solid) is a material in which the crystal lattice of the entire sample is continuous and unbroken to the edges of the sample, with no grain boundaries.RIWD. "Re ...
diffraction in 1924.
The theory of graphene was first explored by
P. R. Wallace
P. R. (Philip Russell; "Phil") Wallace (April 19, 1915 – March 20, 2006) was a Canadian theoretical physicist and long-time professor at McGill University. He was a Fellow of the Royal Society of Canada and Fellow of the National Academy of Scien ...
in 1947 as a starting point for understanding the electronic properties of 3D graphite. The emergent massless
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac part ...
was first pointed out in 1984 separately by Gordon Walter Semenoff, and by David P. DiVincenzo and Eugene J. Mele. Semenoff emphasized the occurrence in a magnetic field of an electronic
Landau level In quantum mechanics, Landau quantization refers to the quantization of the cyclotron orbits of charged particles in a uniform magnetic field. As a result, the charged particles can only occupy orbits with discrete, equidistant energy values, call ...
precisely at the
Dirac point
Dirac cones, named after Paul Dirac, are features that occur in some electronic band structures that describe unusual electron transport properties of materials like graphene and topological insulators.
In these materials, at energies near ...
. This level is responsible for the anomalous integer
quantum Hall effect
The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance exh ...
.
Observations of thin graphite layers and related structures
Transmission electron microscopy
Transmission electron microscopy (TEM) is a microscopy technique in which a beam of electrons is transmitted through a specimen to form an image. The specimen is most often an ultrathin section less than 100 nm thick or a suspension on a g ...
(TEM) images of thin graphite samples consisting of a few graphene layers were published by G. Ruess and F. Vogt in 1948. Eventually, single layers were also observed directly. Single layers of graphite were also observed by
transmission electron microscopy
Transmission electron microscopy (TEM) is a microscopy technique in which a beam of electrons is transmitted through a specimen to form an image. The specimen is most often an ultrathin section less than 100 nm thick or a suspension on a g ...
within bulk materials, in particular inside soot obtained by chemical exfoliation.
In 1961–1962,
Hanns-Peter Boehm
Hanns-Peter Boehm (9 January 1928 – 10 May 2022) was a German chemist and professor emeritus at Ludwig-Maximilians-Universität in Munich, Germany. Boehm is considered a pioneer of graphene research.
Biography
Hanns-Peter Boehm studied chem ...
published a study of extremely thin flakes of graphite, and coined the term "graphene" for the hypothetical single-layer structure. This paper reports graphitic flakes that give an additional contrast equivalent of down to ~0.4 nm or 3 atomic layers of amorphous carbon. This was the best possible resolution for 1960 TEMs. However, neither then nor today is it possible to argue how many layers were in those flakes. Now we know that the TEM contrast of graphene most strongly depends on focusing conditions. For example, it is impossible to distinguish between suspended monolayer and multilayer graphene by their TEM contrasts, and the only known way is to analyze the relative intensities of various diffraction spots. The first reliable TEM observations of monolayers are probably given in refs. 24 and 26 of Geim and Novoselov's 2007 review.
Starting in the 1970s, C. Oshima and others described single layers of carbon atoms that were grown epitaxially on top of other materials. This "epitaxial graphene" consists of a single-atom-thick hexagonal lattice of sp2-bonded carbon atoms, as in free-standing graphene. However, there is significant charge transfer between the two materials, and, in some cases, hybridization between the
d-orbital
In atomic theory and quantum mechanics, an atomic orbital is a Function (mathematics), function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electr ...
s of the substrate atoms and π orbitals of graphene; which significantly alter the electronic structure compared to that of free-standing graphene.
The term "graphene" was used again in 1987 to describe single sheets of graphite as a constituent of
graphite intercalation compound
Graphite intercalation compounds are complex materials having a formula where the ion or is inserted ( intercalated) between the oppositely charged carbon layers. Typically ''m'' is much less than 1. These materials are deeply colored solids t ...
s, which can be seen as crystalline salts of the intercalant and graphene. It was also used in the descriptions of
carbon nanotube
A scanning tunneling microscopy image of a single-walled carbon nanotube
Rotating single-walled zigzag carbon nanotube
A carbon nanotube (CNT) is a tube made of carbon with diameters typically measured in nanometers.
''Single-wall carbon na ...
s by R. Saito and Mildred and Gene Dresselhaus in 1992, and of polycyclic aromatic hydrocarbons in 2000 by S. Wang and others.
Efforts to make thin films of graphite by mechanical exfoliation started in 1990.
Initial attempts employed exfoliation techniques similar to the drawing method. Multilayer samples down to 10 nm in thickness were obtained.
In 2002,
Robert B. Rutherford
Robert B. Rutherford, MD, (1931 – November 22, 2013) was an American vascular surgeon, scientific journal editor, and medical textbook author. He established the definitive textbook in vascular surgery and was a Senior Editor of the '' Journal ...
and Richard L. Dudman filed for a
patent
A patent is a type of intellectual property that gives its owner the legal right to exclude others from making, using, or selling an invention for a limited period of time in exchange for publishing an enabling disclosure of the invention."A p ...
in the US on a method to produce graphene by repeatedly peeling off layers from a graphite flake adhered to a substrate, achieving a graphite thickness of . The key to success was high-throughput visual recognition of graphene on a properly chosen substrate, which provides a small but noticeable optical contrast.
Another U.S. patent was filed in the same year by Bor Z. Jang and Wen C. Huang for a method to produce graphene based on exfoliation followed by attrition.
In 2014, inventor Larry Fullerton patents a process for producing single layer graphene sheets.
Full isolation and characterization
Graphene was properly isolated and characterized in 2004 by
Andre Geim
, birth_date =
, birth_place = Sochi, Russian SFSR, Soviet Union
, death_date =
, death_place =
, workplaces =
, nationality = Dutch and British
, fields = Condensed matter physics
...
and
Konstantin Novoselov
Sir Konstantin Sergeevich Novoselov ( rus, Константи́н Серге́евич Новосёлов,
p=kənstɐnʲˈtʲin sʲɪrˈɡʲe(j)ɪvʲɪtɕ nəvɐˈsʲɵləf; born 1974) is a Russian-British physicist, and a professor at the ...
at the
University of Manchester
, mottoeng = Knowledge, Wisdom, Humanity
, established = 2004 – University of Manchester Predecessor institutions: 1956 – UMIST (as university college; university 1994) 1904 – Victoria University of Manchester 1880 – Victoria Univer ...
. They pulled graphene layers from graphite with a common adhesive tape in a process called either micromechanical cleavage or the Scotch tape technique. The graphene flakes were then transferred onto thin
silicon dioxide
Silicon dioxide, also known as silica, is an oxide of silicon with the chemical formula , most commonly found in nature as quartz and in various living organisms. In many parts of the world, silica is the major constituent of sand. Silica is one ...
(silica) layer on a
silicon
Silicon is a chemical element with the symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a tetravalent metalloid and semiconductor. It is a member of group 14 in the periodic tab ...
plate ("wafer"). The silica electrically isolated the graphene and weakly interacted with it, providing nearly charge-neutral graphene layers. The silicon beneath the could be used as a "back gate" electrode to vary the charge density in the graphene over a wide range.
This work resulted in the two winning the
Nobel Prize in Physics
)
, image = Nobel Prize.png
, alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
in 2010 "for groundbreaking experiments regarding the two-dimensional material graphene." Their publication, and the surprisingly easy preparation method that they described, sparked a "graphene gold rush". Research expanded and split off into many different subfields, exploring different exceptional properties of the material—quantum mechanical, electrical, chemical, mechanical, optical, magnetic, etc.
Exploring commercial applications
Since the early 2000s, a number of companies and research laboratories have been working to develop commercial applications of graphene. In 2014 a
National Graphene Institute
The National Graphene Institute is a research institute and building at the University of Manchester that is focused on the research of graphene. Construction of the building to house the institute started in 2013 and finished in 2015.
Institute ...
was established with that purpose at the University of Manchester, with a 60 million
GBP
Sterling (abbreviation: stg; Other spelling styles, such as STG and Stg, are also seen. ISO code: GBP) is the currency of the United Kingdom and nine of its associated territories. The pound ( sign: £) is the main unit of sterling, and t ...
initial funding. In
North East England
North East England is one of nine official regions of England at the first level of ITL for statistical purposes. The region has three current administrative levels below the region level in the region; combined authority, unitary authorit ...
East Anglia
East Anglia is an area in the East of England, often defined as including the counties of Norfolk, Suffolk and Cambridgeshire. The name derives from the Anglo-Saxon kingdom of the East Angles, a people whose name originated in Anglia, in ...
.
Structure
Graphene is a single layer (monolayer) of carbon atoms, tightly bound in a hexagonal honeycomb lattice. It is an allotrope of carbon in the form of a plane of sp2-bonded atoms with a molecular bond length of 0.142 nanometres.
Bonding
Three of the four outer-
shell
Shell may refer to:
Architecture and design
* Shell (structure), a thin structure
** Concrete shell, a thin shell of concrete, usually with no interior columns or exterior buttresses
** Thin-shell structure
Science Biology
* Seashell, a hard ou ...
electrons of each atom in a graphene sheet occupy three sp2
hybrid orbitals
In chemistry, orbital hybridisation (or hybridization) is the concept of mixing atomic orbitals to form new ''hybrid orbitals'' (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to f ...
– a combination of orbitals s, px and py — that are shared with the three nearest atoms, forming
σ-bond
In chemistry, sigma bonds (σ bonds) are the strongest type of covalent chemical bond. They are formed by head-on overlapping between atomic orbitals. Sigma bonding is most simply defined for diatomic molecules using the language and tools of sy ...
nanometer
330px, Different lengths as in respect to the molecular scale.
The nanometre (international spelling as used by the International Bureau of Weights and Measures; SI symbol: nm) or nanometer (American and British English spelling differences#-re ...
s.
The remaining outer-shell electron occupies a pz orbital that is oriented perpendicularly to the plane. These orbitals hybridize together to form two half-filled bands of free-moving electrons, π and π∗, which are responsible for most of graphene's notable electronic properties. Recent quantitative estimates of aromatic stabilization and limiting size derived from the enthalpies of hydrogenation (ΔHhydro) agree well with the literature reports.
Graphene sheets stack to form graphite with an interplanar spacing of .
Graphene sheets in solid form usually show evidence in diffraction for graphite's (002) layering. This is true of some single-walled nanostructures. However, unlayered graphene with only (hk0) rings has been found in the core of presolar graphite onions. TEM studies show faceting at defects in flat graphene sheets and suggest a role for two-dimensional crystallization from a melt.
Geometry
The hexagonal lattice
structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
of isolated, single-layer graphene can be directly seen with transmission electron microscopy (TEM) of sheets of graphene suspended between bars of a metallic grid Some of these images showed a "rippling" of the flat sheet, with amplitude of about one nanometer. These ripples may be intrinsic to the material as a result of the instability of two-dimensional crystals, or may originate from the ubiquitous dirt seen in all TEM images of graphene.
Photoresist
A photoresist (also known simply as a resist) is a light-sensitive material used in several processes, such as photolithography and photoengraving, to form a patterned coating on a surface. This process is crucial in the electronic industry.
T ...
residue, which must be removed to obtain atomic-resolution images, may be the "
adsorbate
Adsorption is the adhesion of atoms, ions or molecules from a gas, liquid or dissolved solid to a surface. This process creates a film of the ''adsorbate'' on the surface of the ''adsorbent''. This process differs from absorption, in which a fl ...
s" observed in TEM images, and may explain the observed rippling.
The hexagonal structure is also seen in
scanning tunneling microscope
A scanning tunneling microscope (STM) is a type of microscope used for imaging surfaces at the atomic level. Its development in 1981 earned its inventors, Gerd Binnig and Heinrich Rohrer, then at IBM Zürich, the Nobel Prize in Physics in 1986. ...
(STM) images of graphene supported on silicon dioxide substrates The rippling seen in these images is caused by conformation of graphene to the subtrate's lattice, and is not intrinsic.
Stability
Ab initio calculations
''Ab initio'' ( ) is a Latin term meaning "from the beginning" and is derived from the Latin ''ab'' ("from") + ''initio'', ablative singular of ''initium'' ("beginning").
Etymology
Circa 1600, from Latin, literally "from the beginning", from abl ...
show that a graphene sheet is thermodynamically unstable if its size is less than about 20 nm and becomes the most stable
fullerene
A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The molecule may be a hollow sphere, ...
(as within graphite) only for molecules larger than 24,000 atoms.
Properties
Electronic
Graphene is a zero-gap
semiconductor
A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
, because its
conduction
Conductor or conduction may refer to:
Music
* Conductor (music), a person who leads a musical ensemble, such as an orchestra.
* Conductor (album), ''Conductor'' (album), an album by indie rock band The Comas
* Conduction, a type of structured f ...
and
valence bands
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in wh ...
meet at the Dirac points. The Dirac points are six locations in
momentum space
In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension.
Position space (also real space or coordinate space) is the set of all '' position vectors'' r in space, and ...
, on the edge of the
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice i ...
, divided into two non-equivalent sets of three points. The two sets are labeled K and K'. The sets give graphene a valley degeneracy of . By contrast, for traditional semiconductors the primary point of interest is generally Γ, where momentum is zero. Four electronic properties separate it from other
condensed matter
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
systems.
However, if the in-plane direction is no longer infinite, but confined, its electronic structure would change. They are referred to as
graphene nanoribbon
Graphene nanoribbons (GNRs, also called nano-graphene ribbons or nano-graphite ribbons) are strips of graphene with width less than 100 nm. Graphene ribbons were introduced as a theoretical model by Mitsutaka Fujita and coauthors to examin ...
s. If it is "zig-zag", the bandgap would still be zero. If it is "armchair", the bandgap would be non-zero.
Graphene's hexagonal lattice can be regarded as two interleaving triangular lattices. This perspective was successfully used to calculate the band structure for a single graphite layer using a tight-binding approximation.
Electronic spectrum
Electrons propagating through graphene's honeycomb lattice effectively lose their mass, producing
quasi-particle
In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum.
For exa ...
s that are described by a 2D analogue of the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac part ...
rather than the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
for spin- particles.
Dispersion relation
The cleavage technique led directly to the first observation of the anomalous quantum Hall effect in graphene in 2005, by Geim's group and by
Philip Kim
Philip Kim is a South Korean physicist. He is a condensed matter physicist known for study of quantum transport in carbon nanotubes and graphene, including observations of quantum Hall effects in graphene.
Academic career
Kim studied physics a ...
and Yuanbo Zhang. This effect provided direct evidence of graphene's theoretically predicted
Berry's phase In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Ha ...
of massless
Dirac fermion
In physics, a Dirac fermion is a spin-½ particle (a fermion) which is different from its antiparticle. The vast majority of fermions – perhaps all – fall under this category.
Description
In particle physics, all fermions in the standard model ...
s and the first proof of the Dirac fermion nature of electrons. These effects had been observed in bulk graphite by Yakov Kopelevich,
Igor A. Luk'yanchuk
Igor may refer to:
People
* Igor (given name), an East Slavic given name and a list of people with the name
* Mighty Igor (1931–2002), former American professional wrestler
* Igor Volkoff, a professional wrestler from NWA All-Star Wrestling
* ...
, and others, in 2003–2004.
When the 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 therefore be treated independently. Within this π-band approximation, using a conventional
tight-binding
In solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each ...
model, the
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the d ...
(restricted to first-nearest-neighbor interactions only) that produces energy of the electrons with wave vector ''k'' is
:
with the nearest-neighbor (π orbitals) hopping energy ''γ''0 ≈ and the
lattice constant
A lattice constant or lattice parameter is one of the physical dimensions and angles that determine the geometry of the unit cells in a crystal lattice, and is proportional to the distance between atoms in the crystal. A simple cubic crystal has o ...
. The
conduction
Conductor or conduction may refer to:
Music
* Conductor (music), a person who leads a musical ensemble, such as an orchestra.
* Conductor (album), ''Conductor'' (album), an album by indie rock band The Comas
* Conduction, a type of structured f ...
and
valence band
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...
s, 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, although the same cannot be said of a graphene sheet rolled into a
carbon nanotube
A scanning tunneling microscopy image of a single-walled carbon nanotube
Rotating single-walled zigzag carbon nanotube
A carbon nanotube (CNT) is a tube made of carbon with diameters typically measured in nanometers.
''Single-wall carbon na ...
, due to its curvature. 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. Since an elementary cell of the lattice has a basis of two atoms, the
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
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
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac part ...
. Hence, the electrons and holes are called Dirac
fermions
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
. This pseudo-relativistic description is restricted to the chiral limit, i.e., to vanishing rest mass ''M''0, which leads to interesting additional features:
:
Here ''vF'' ~ (.003 c) is the
Fermi velocity
The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi ga ...
in graphene, which replaces the velocity of light in the Dirac theory; is the vector of the
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
, is the two-component wave function of the electrons, and ''E'' is their energy.
The equation describing the electrons' linear dispersion relation is
:
where the
wavevector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
''q'' is measured from the Brillouin zone vertex K, , and the zero of energy is set to coincide with the Dirac point. The equation uses a pseudospin matrix formula that describes two sublattices of the honeycomb lattice.
Single-atom wave propagation
Electron waves in graphene propagate within a single-atom layer, making them sensitive to the proximity of other materials such as
high-κ dielectric The term high-κ dielectric refers to a material with a high dielectric constant (κ, kappa), as compared to silicon dioxide. High-κ dielectrics are used in semiconductor manufacturing processes where they are usually used to replace a silicon dio ...
ferromagnetic
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
s.
Ambipolar electron and hole transport
Graphene displays remarkable
electron mobility
In solid-state physics, the electron mobility characterises how quickly an electron can move through a metal or semiconductor when pulled by an electric field. There is an analogous quantity for holes, called hole mobility. The term carrier mobili ...
at room temperature, with reported values in excess of . Hole and electron mobilities are nearly the same. The mobility is independent of temperature between and , and shows little change even at room temperature (300 K), which implies that the dominant scattering mechanism is
defect scattering Defect types include atom vacancies, adatoms, steps, and kinks that occur most frequently at surfaces due to the finite material size causing crystal discontinuity. What all types of defects have in common, whether surface or bulk defects, is that t ...
. Scattering by graphene's acoustic
phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
s intrinsically limits room temperature mobility in freestanding graphene to at a carrier density of .
The corresponding
resistivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
of graphene sheets would be . This is less than the resistivity of
silver
Silver is a chemical element with the Symbol (chemistry), symbol Ag (from the Latin ', derived from the Proto-Indo-European wikt:Reconstruction:Proto-Indo-European/h₂erǵ-, ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, whi ...
, the lowest otherwise known at room temperature.Physicists Show Electrons Can Travel More Than 100 Times Faster in Graphene :: University Communications Newsdesk, University of Maryland . Newsdesk.umd.edu (24 March 2008). Retrieved on 2014-01-12. However, on 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 .
Charge transport has major concerns due to 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. The protection of graphene surface by a coating with materials such as SiN,
PMMA PMMA may refer to:
* para-Methoxymethamphetamine, a stimulant drug
* Philippine Merchant Marine Academy
The Philippine Merchant Marine Academy ( fil, Akademiya sa Bapor Pangkalakalan ng Pilipinas) also referred to by its acronym PMMA) is a ma ...
, h-BN, etc., have been discussed by researchers. 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. In 2015,
lithium
Lithium (from el, λίθος, lithos, lit=stone) is a chemical element with the symbol Li and atomic number 3. It is a soft, silvery-white alkali metal. Under standard conditions, it is the least dense metal and the least dense solid el ...
-coated graphene exhibited
superconductivity
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
, a first for graphene.
Electrical resistance in 40-nanometer-wide
nanoribbon Nanoribbon may refer to:
* Graphene nanoribbons
* Silicene nanoribbons
* Boron nitride nanoribbons
* Gallium(III) oxide nanoribbons
* titanate nanoribbons - see titanium dioxide
* Phosphorene nanoribbons
{{Short pages monitor
Bor Z. Jang and Wen C. Huang (2002): Nano-scaled graphene plates . US Patent 7071258. Filed on 2002-10-21, granted on 2006-07-04, assigned to Global Graphene Group Inc; to expire on 2024-01-06.Robert B. Rutherford and Richard L. Dudman (2002): Ultra-thin flexible expanded graphite heating element . US Patent 6667100. Filed on 2002-05-13, granted on 2003-12-23, assigned to EGC Operating Co LLC; expired.
The University of Manchester
, mottoeng = Knowledge, Wisdom, Humanity
, established = 2004 – University of Manchester Predecessor institutions: 1956 – UMIST (as university college; university 1994) 1904 – Victoria University of Manchester 1880 – Victoria Univer ...
The Periodic Table of Videos
''Periodic Videos'' (also known as ''The Periodic Table of Videos'') is a video project and YouTube channel on chemistry. It consists of a series of videos about chemical elements and the periodic table, with additional videos on other topics i ...