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-1⁄2 particles.[2][3]

Dispersion relation

Electronic band structure of graphene. Valence and conduction bands meet at the six vertices of the hexagonal Brillouin zone and form linearly dispersing Dirac cones.

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 ${\displaystyle \mathbf {k} =[k_{x},k_{y}]}$ is[4][5]

${\displaystyle E(\mathbf {k} )=\pm \,\gamma _{0}{\sqrt {1+4\cos ^{2}{{\tfrac {1}{2}}ak_{x}}+4\cos {{\tfrac {1}{2}}ak_{x}}\cdot \cos {{\tfrac {\sqrt {3}}{2}}ak_{y}}}}}$

with the nearest-neighbor (π orbitals) hopping energy γ₀ ≈ 2.8 eV and the lattice constant a ≈ 2.46 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.[4][6] 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.[4] This pseudo-relativistic description is restricted to the chiral limit, i.e., to vanishing rest mass M₀, which leads to additional features:[4][7]

${\displaystyle -iv_{F}\,{\vec {\sigma }}\cdot \nabla \psi (\mathbf {r} )\,=\,E\psi (\mathbf {r} ).}$

Here v_F ~ 10⁶ m/s (.003 c) is the Fermi velocity in graphene, which replaces the velocity of light in the Dirac theory; ${\displaystyle {\vec {\sigma }}}$ is the vector of the Pauli matrices; ${\displaystyle \psi (\mathbf {r} )}$ is the two-component wave function of the electrons and E is their energy.[2]

The equation describing the electrons' linear dispersion relation is

${\displaystyle E(k)=\hbar v_{F}k}$

where the wavevector ${\displaystyle k={\sqrt {k_{x}^{2}+k_{y}^{2}}}}$ 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.[6]

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 15000 cm²⋅V⁻¹⋅s⁻¹.[9] Hole and electron mobilities were expected to be nearly identical.[3] The mobility is nearly independent of temperature between 10 K and 100 K,[10][11][12] which implies that the dominant scattering mechanism is defect scattering. Scattering by graphene's acoustic phonons intrinsically limits room temperature mobility to 200000 cm²⋅V⁻¹⋅s⁻¹ at a carrier density of 10¹² cm⁻²,[12][13] 10×10⁶ times greater than copper.[14]

The corresponding resistivity of graphene sheets would be 10⁻⁶ Ω⋅cm. This is less than the resistivity of silver, the lowest otherwise known at room temperature.[15] However, on 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 40000 cm²⋅V⁻¹⋅s⁻¹.[12]

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.[16][17] In 2015 lithium-coated graphene was observed to exhibit superconductivity[18] 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).[19]

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.[20][21]

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.[20]

Graphene electrons can cover micrometer distances without scattering, even at room temperature.[2]

Despite zero carrier density near the Dirac points, graphene exhibits a minimum conductivity on the order of ${\displaystyle 4e^{2}/h}$. The origin of this minimum conductivity is unclear. However, rippling of the graphene sheet or ionized impurities in the SiO
₂ substrate may lead to local puddles of carriers that allow conduction.[3] Several theories suggest that the minimum conductivity should be ${\displaystyle 4e^{2}/{(\pi }h)}$; however, most measurements are of order ${\displaystyle 4e^{2}/h}$ or greater[9] and depend on impurity concentration.[22]

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.[23]

Graphene doped with various gaseous species (both acceptors and donors) can be returned to an undoped state by gentle heating in vacuum.[22][24] Even for dopant concentrations in excess of 10¹² cm⁻² carrier mobility exhibits no observable change.[24] Graphene doped with potassium in ultra-high vacuum at low temperature can reduce mobility 20-fold.[22][25] 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[26]) is thought to occur. It may therefore be a suitable material for constructing quantum computers[27] using anyonic circuits.[28]

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.[29] With GW calculation, the properties of graphene-based materials are accurately investigated, including bulk graphene,[30] nanoribbons,[31] edge and surface functionalized armchair oribbons,[32] hydrogen saturated armchair ribbons,[33] Josephson effect in graphene SNS junctions with single localized defect[34] and armchair ribbon scaling properties.[35]

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.[36] The dopant's presence negatively affected its electronic properties.[37]

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.[42][43][44] Spin coherence length above 1 micrometre at room temperature was observed,[42] and control of the spin current polarity with an electrical gate was observed at low temperature.[43]

Spintronic and magnetic properties can be present in graphene simultaneously.[45] 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.

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.[46]

The quantum Hall effect is a quantum mechanical version of the Hall effect, which is the production of transverse (perpendicular to the main current) conductivity in the presence of a magnetic field. The quantization of the Hall effect ${\displaystyle \sigma _{xy}}$ at integer multiples (the "Landau level") of the basic quantity ${\displaystyle e^{2}/h}$ (where e is the elementary electric charge and h is Planck's constant) It can usually be observed only in very clean silicon or gallium arsenide solids at temperatures around 3 K and high magnetic fields.

Graphene shows the quantum Hall effect with respect to conductivity quantization: the effect is anomalous in that the sequence of steps is shifted by 1/2 with respect to the standard sequence and with an additional factor of 4. Graphene's Hall conductivity is ${\displaystyle \sigma _{xy}=\pm {4\cdot \left(N+1/2\right)e^{2}}/h}$, where N is the Landau level and the double valley and double spin degeneracies give the factor of 4.[9] These anomalies are present at room temperature, i.e. at roughly 20 °C (293 K).[10]

This behavior is a direct result of graphene's massless Dirac electrons. In a magnetic field, their spectrum has a Landau level with energy precisely at the Dirac point. This level is a consequence of the Atiyah–Singer index theorem and is half-filled in neutral graphene,[4] leading to the "+1/2" in the Hall conductivity.[47] Bilayer graphene also shows the quantum Hall effect, but with only one of the two anomalies (i.e. ${\displaystyle \sigma _{xy}=\pm {4\cdot N\cdot e^{2}}/h}$). In the second anomaly, the first plateau at N=0 is absent, indicating that bilayer graphene stays metallic at the neutrality point.[9]

Unlike normal metals, graphene's longitudinal resistance shows maxima rather than minima for integral values of the Landau filling factor in measurements of the Shubnikov–de Haas oscillations, whereby the term integral quantum Hall effect. These oscillations show a phase shift of π, known as Berry's phase.[10][3] Berry's phase arises due to the zero effective carrier mass near the Dirac points.[48] The temperature dependence of the oscillations reveals that the carriers have a non-zero cyclotron mass, despite their zero effective mass.[10]

Graphene samples prepared on nickel films, and on both the silicon face and carbon face of silicon carbide, show the anomalous effect directly in electrical measurements.[49][50][51][52][53][54] Graphitic layers on the carbon face of silicon carbide show a clear Dirac spectrum in angle-resolved photoemission experiments. The effect is observed in cyclotron resonance and tunneling experiments.[55]

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.[56][57]

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.[58]

The electronic properties of graphene are significantly influenced by the supporting substrate.[59][60] 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.

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).

• Wolfram demonstration for graphene BZ and electronic dispersion