Graphene: the two-dimensional form of carbon

All carbon atoms of graphene are sp2-hybridized, which means that each carbon atom can form three equivalent σ-bonds to other C atoms. This results in a honeycomb structure also known from the layers of graphite. The carbon-carbon bond lengths are all equal and are 142 pm. The third, non-hybridized 2p orbitals are perpendicular to the graphene plane, as in graphite, and form a delocalized π-bond system.

Consequently, graphene consists of two equivalent sublattices A and B, to which the carbon atoms are assigned. The sublattices are shifted with respect to each other by the bond length The diatomic unit cell is spanned by the lattice vectors and In this case, these point to the neighbors next but one in each case. The length of the vectors and thus the lattice constant can be calculated to

.

Graphene can be understood as a single crystal on the one hand, and as a giant molecule on the other. Similarly, smaller molecules such as benzene, hexabenzocoronene or naphthalene can be seen as hydrogen-substituted graphene fragments.

Graphene has unusual properties that make it interesting for both basic research and applications, especially in physics.

For example, graphene face single crystals are exceptionally stiff and strong within the faces. Its modulus of elasticity of about 1020 GPa is equal to that of normal graphite along the basal planes and almost as large as that of diamond. Its tensile strength of 1.25×1011 Pa is the highest ever determined and about 125 times that of steel. A ribbon of graphene 1 m wide and 3.35 × 10-10 m thick, that is, of one atomic layer, has therefore a tensile strength of 42 N. A ribbon for a space elevator made of graphene with a constant cross-sectional area would only be loaded to 87.3% of its tensile strength at the geostationary orbit altitude of 35,786 km.

Assuming monocrystalline graphite with a density of 2260 kg-m-3 and a layer spacing of 3.35 × 10-10 m, multiplication calculates a mass per unit area for graphene of 7.57 × 10-7 kg-m-2. One square kilometer thus weighs 757 g. "A graphene hammock one square meter in size could support a four-kilogram cat - and itself weigh only as much as a single cat whisker."

Graphene behaves as an electrical conductor due to the lack of a band gap, see the figure below for the energy of electrons in graphene. However, an artificial band gap in graphene can be created by "cutting" a so-called gate into the layer with a maximum width of 10 nm.

Measurements have shown that a single graphene layer attenuates light by about πα ≈ 2.3% (with fine structure constant α), over the entire visible spectrum.

Pseudo-relativistic behaviour

The electrical properties of graphene can be well described by a tight-binding model. Within the framework of this model, the energy of electrons with wavenumber (see wave vector) to

,

With the nearest-neighbor hopping energy γ and the lattice constant . Conduction and valence bands correspond to plus and minus signs, respectively, in the dispersion relation above. They touch each other in graphs exactly in six distinguished points, called K-points, but only two of them are independent (the rest are equivalent to these two by lattice symmetry). In their vicinity, the energy depends linearly on (cf. photon: ). Since the basis is diatomic, the wave function even has a formal spinor structure. This leads to the fact that electrons at low energies can be described by a relation equivalent to the Dirac equation, and additionally in the so-called chiral limit, i.e. for vanishing mass , which gives some special features:

Here denotes the Fermi velocity in graphs, which takes the place of the speed of light; denotes the Pauli matrices, ψ the two-component wave function of electrons, and their energy.

Unusual quantum Hall effect

Because of the peculiarities in dispersion, in this material the staircase structure of the integer quantum Hall plateaus, σ , for all stages exactly "shifted by 1/2", The two-valley structure (formal "pseudospin") and the "real" spin degeneracy together add up to an additional factor of 4. Remarkably, unlike the conventional quantum Hall effect, this can be observed even at room temperature.

Stiffness and temperature dependence

Graphene is extremely stiff in the layer direction because the sp2 bond between neighbouring atoms is comparable in strength to the sp3 bond of diamond. Accordingly, it is generally expected - and this corresponds to the experiment - that the properties of graphene which are of interest for applications are not only valid at absolute temperature zero, i.e. at -273.15 °C, but remain valid at room temperature.

One such property is thermoelectricity: a temperature gradient in graphene planes causes an electric field strength due to a decoupling of the temperature of the electrons from that of the lattice. Electric voltage when graphene is illuminated had been observed before, but the cause was thought to be photovoltaics. The fact that the decoupling can still be observed at room temperature is due to the rigidity of the lattice: the elementary vibrational excitations of the lattice (phonons) are so high-energy that the electrons rarely produce such a phonon.

Elastic behaviour and pseudo magnetic field

In July 2010, a publication in the American scientific journal Science reported extremely strong pseudo-magnetic fields. By elastic deformation, tiny triangular bubbles of 4 nanometers to 10 nanometers in size were created in graphene, in which the electrons moved as if a magnetic field of about 300 teslas was acting on them. It turned out that, unlike the effect of a real magnetic field, the observed effect does not affect the actual spin of the electron, but instead affects the pseudo-spin just mentioned, which is related to the existence of two different equivalent base atoms in the honeycomb structure. This pseudospin has a similar interaction with the pseudomagnetic field as real spins have with real magnetic fields, because these also generate "two-level systems". The experiments around the "pseudo-quantum Hall effect" generated in this way were based on theoretical predictions, which were thus confirmed.

Spin streams

In April 2011, A. Geim and coworkers published an article describing strong spin currents and current-induced magnetism near the Dirac point, i.e., near the meeting point of the conduction band and the valence band. This opens up the prospect of applications in spintronics.

Chemical functionalization and large-scale EU project

In 2013, the EU announced a major new project ("flagship project") on graphene, on which researchers in many Member States will collaborate, especially physicists and chemists. It is possible, for example, to "break" double bonds in graphene - every second bond is such a bond - and replace them with two single bonds, to which various organic molecules can then be attached: this allows the properties of the system to be specifically influenced.

Hydrodynamic behaviour with unusual properties

In 2016, a so-called hydrodynamic ("liquid-like") behaviour of the electrons in graphene was found, but with extraordinary properties: For example. the Wiedemann-Franz law, otherwise valid for metallic systems, that the ratio of thermal and electrical conductivity divided by the Kelvin temperature T is equal to a universal constant in metallic systems, is violated in graphene, in that when the Fermi energy approaches the overlap point, the thermal conductivity becomes much too large (i.e. the specific thermal resistance becomes much too small). Moreover, in this electron fluid (technical term: Dirac fluid) one observes an unusual tendency to form vortex structures.