Graphene Brillouin Zone

Graphene () is an allotrope of carbon consisting of a single layer of atoms arranged in a hexagonal lattice nanostructure.
"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, reflecting the fact that the graphite 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, and contributes to a valence band one electron that extends over the whole sheet. This is the same type of bonding seen in carbon nanotubes and polycyclic aromatic hydrocarbons, and (partially) in fullerenes and glassy carbon. The valence band is touched by a conduction band, making graphene a semimetal 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 over long distances; the material exhibits large quantum oscillations and large and nonlinear diamagnetism. 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 microscopes in 1962, but studied only while supported on metal surfaces.

In 2004, the material was rediscovered, isolated and investigated at the University of Manchester, by Andre Geim and Konstantin Novoselov. In 2010, Geim and Novoselov were awarded the Nobel Prize in Physics 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 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, and composites.

The IUPAC (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 or silicon carbide.

History

Structure of graphite and its intercalation compounds

In 1859, Benjamin Brodie noted the highly lamellar structure of thermally reduced graphite oxide. In 1916, Peter Debye and Paul Scherrer determined the structure of graphite by powder X-ray diffraction. 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 diffraction in 1924.

The theory of graphene was first explored by P. R. Wallace in 1947 as a starting point for understanding the electronic properties of 3D graphite. The emergent massless Dirac equation 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 precisely at the Dirac point. This level is responsible for the anomalous integer quantum Hall effect.

Observations of thin graphite layers and related structures

Transmission electron microscopy (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 within bulk materials, in particular inside soot obtained by chemical exfoliation.

In 1961–1962, Hanns-Peter Boehm 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-orbitals 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 compounds, which can be seen as crystalline salts of the intercalant and graphene. It was also used in the descriptions of carbon nanotubes 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 and Richard L. Dudman filed for a patent 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 and Konstantin Novoselov at the University of Manchester. 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 (silica) layer on a silicon 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 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 was established with that purpose at the University of Manchester, with a 60 million GBP initial funding. In North East England two commercial manufacturers, Applied Graphene Materials and Thomas Swan Limited have begun manufacturing. Cambridge Nanosystems is a large-scale graphene powder production facility in East Anglia.

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 electrons of each atom in a graphene sheet occupy three sp² hybrid orbitals – a combination of orbitals s, p_x and p_y — that are shared with the three nearest atoms, forming σ-bonds. The length of these bonds is about 0.142 nanometers.

The remaining outer-shell electron occupies a p_z 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 (ΔH_hydro) 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 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 residue, which must be removed to obtain atomic-resolution images, may be the " adsorbates" observed in TEM images, and may explain the observed rippling.

The hexagonal structure is also seen in scanning tunneling microscope (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 show that a graphene sheet is thermodynamically unstable if its size is less than about 20 nm and becomes the most stable fullerene (as within graphite) only for molecules larger than 24,000 atoms.

Properties

Electronic

Graphene is a zero-gap semiconductor, because its conduction and valence bands meet at the Dirac points. The Dirac points are six locations in momentum space, on the edge of the Brillouin zone, 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 systems.

However, if the in-plane direction is no longer infinite, but confined, its electronic structure would change. They are referred to as graphene nanoribbons. 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-particles that are described by a 2D analogue of the Dirac equation rather than the Schrödinger equation 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 and Yuanbo Zhang. This effect provided direct evidence of graphene's theoretically predicted Berry's phase of massless Dirac fermions 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, 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 model, the dispersion relation (restricted to first-nearest-neighbor interactions only) that produces energy of the electrons with wave vector ''k'' is

:$E(k_x,k_y)=\pm\,\gamma_0\sqrt$

with the nearest-neighbor (π orbitals) hopping energy ''γ''₀ ≈ and the lattice constant . 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, although the same cannot be said of a graphene sheet rolled into a carbon nanotube, 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 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 ''M''₀, which leads to interesting additional features:

:$v_F\, \vec \sigma \cdot \nabla \psi(\mathbf)\,=\,E\psi(\mathbf).$

Here ''v_F'' ~ (.003 c) is the Fermi velocity in graphene, which replaces the velocity of light in the Dirac theory; $\vec$ is the vector of the Pauli matrices, $\psi(\mathbf)$ is the two-component wave function of the electrons, and ''E'' is their energy.

The equation describing the electrons' linear dispersion relation is

:$E(q)=\hbar v_F q$

where the wavevector ''q'' is measured from the Brillouin zone vertex K, $q=\left, \mathbf-\mathrm\$, 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 dielectrics, superconductors and ferromagnetics.

Ambipolar electron and hole transport

Graphene displays remarkable electron mobility 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. Scattering by graphene's acoustic phonons intrinsically limits room temperature mobility in freestanding graphene to at a carrier density of .

The corresponding resistivity of graphene sheets would be . This is less than the resistivity of silver, 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, 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-coated graphene exhibited superconductivity, a first for graphene.

Electrical resistance in 40-nanometer-wide nanoribbon

HOME


Content is Copyleft
Website design, code, and AI is Copyrighted (c) 2014-2017 by Stephen Payne


Consider donating to Wikimedia



As an Amazon Associate I earn from qualifying purchases