The switch effect of conductivity of graphene nanoribbons in an external electric field

The charge carriers in graphene in the low-energy approximation with respect to the Fermi level are described by the Dirac equation, rather than the usual in solid-state physics Schrodinger equation, because of symmetry of the crystal lattice of graphene. Electronic subzones formed by symmetric and antisymmetric combination of wave functions on different sublattices intersect at the edge of the Brillouin zone, which leads to a coneshaped energy spectrum near the “dirac” points.
Influence of tangential fields on the band structure of a two-layer carbon “zig-zag” nanoribbons, calculated from the secular equation. With the increase of tangential electric field changes the transverse electron quasi-momentum, which leads to a shift of the allowed values of the wave vector in the Brillouin zone and their periodic passage through special points $K$ and $K'$. The change of the band structure is reflected on the forbidden band width which varies within a field from $0$ to $1$ eV. The amplitude and period of the change band gap depends on the geometry of the ribbon. Thus, within the approximation of constant relaxation time model in bilayer graphene ribbons a transition “metal-insulator” is observed, that creates “switch effect”: the opening and closing of the band gap by the external transverse electric field.
Calculations have shown that the inclusion of the normal component of the electric field leads to a sharp drop in ribbon conductivity at low temperatures. Increase of the amplitude of the field tangential component leads to a small increase in conductivity over the entire temperature range. However, at low temperatures (below $50$ K), this increase is more pronounced, leading to a local minimum in the temperature dependence. Further increase of the tangential component of the field returns the system to its original state.
By changing the value of the tangential component of the electric field can control the conductive properties of the material that can be used when creating new elements for micro- and nanoelectronics. For example, the effect of switching the electrical conductivity of bilayer graphene at low temperatures under the influence of an external transverse electric field allows to use it as a basis for the creation of the transistor. Identified pattern allows you to create elements of micro- and nanoelectronics with variable electronic characteristics. The predicted effect can be used to create a transistor based on bilayer graphene.