What is the lattice and band relationship of graphene?

In semiconductors, the energy band structure of an electron determines the range of energies that the electrons allow and is inhibited, and determines the electrical and optical properties of the semiconductor material. The electrons of an isolated atom occupy a certain atomic orbital, forming a series of discrete energy levels.

The lattice structure of graphene is very stable, and the interference of electrons moving in orbit is very small, and has excellent electrical conductivity. This structure leads to the unique electronic band structure of graphene. As shown in Fig. 1(b), the six vertices of the first Brillouin zone are Fermi points (also called Dirac points or K points). The valence band is symmetrical about Dirac, so in pure graphene, electrons and holes have the same properties. That is, near the Dirac point, the energy of the electron is linearly related to the wave vector, E = VFP = VFhk. Among them, VF is the Fermi speed, which is about 1/300 of the speed of light, and k is the wave vector. Therefore, the electrons near the K point are affected by the surrounding symmetric lattice potential field, the effective static mass of the carrier is 0, and the Fermi velocity is close to the speed of light, exhibiting relativistic properties. Therefore, the electronic properties near the K point should be described by the Dirac equation, rather than by the Schrodinger equation. The mobility of graphene carriers exceeds 200,000 cm2*V-1*s-1, and the average free path of electrons in pure graphene is on the order of submicron, which is similar to ballistic transport, which is attractive in manufacturing high-speed devices potential.

Figure 1 (a) Schematic diagram of graphene crystal structure; (b) Three-dimensional band structure of graphene; (c) High symmetry and two-dimensional band structure of graphene in Brillouin zone

(d) Energy band structure near the Dirac point and the movement of the Fermi surface with doping

The carbon outer layer in graphene includes 4 electrons, 3 s electrons (orbital in the plane of graphene), and a p-electron (π electron, orbit perpendicular to the plane of graphene), and the energy band structure of graphene can be tight the bound Hamilton equation approximates the chromatic dispersion relationship of the π band under tight binding conditions:

Where ±1 corresponds to the conduction band and the valence band, kx and ky are the components of the wave vector k, and r0 is the transition energy between the neighboring carbon atoms, usually taking the value of 2.9-3.1 eV, a=sqrt(3)ace, Ace=1.42A is the distance between carbon atoms. Since each carbon atom contributes a π electron, the valence band of graphene just fills up and the conduction band is completely empty. Such a Fermi surface is just at the intersection of the conduction band and the valence band, so that graphene has a singular property different from that of a conventional semiconductor, that is, a material having a zero band gap. Since graphene has a linear dispersion relationship near the intersection point K, the energy and momentum of π electrons are linearly related. The energy eigenvalues of the relativistic particles are obtained according to the Klein-Gauden equation:

Where m0 is the effective mass and the moving speed is constant, very similar to photons. So p electrons are suitable for the relativistic Dirac equation and not for the Schrodinger equation. The p-electron appears as a massless Dirac fermion, and the intersection of the conduction band and the valence band is called the Dirac point. This unique structure causes graphene to exhibit an anomalous semi-integer quantum Hall effect with a Hall point to an odd multiple of quantum conductance and a minimum conductivity of -4e2/h when carriers tend to zero; the speed of electron movement is about 1/300 of the speed of light, which is the highest transmission speed among known materials.

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