Double injection, resonant-tunneling recombination, and current-voltage characteristics in double-graphene-layer structures

We evaluate the effect of the recombination associated with interlayer transitions in ungated and gated double-graphene-layer (GL) structures on the injection of electrons and holes. Using the proposed model, we derive analytical expressions for the spatial distributions of the electron and hole Fermi energies and the energy gap between the Dirac points in GLs as well as their dependences on the bias and gate voltages. The current-voltage characteristics are calculated as well. The model is based on hydrodynamic equations for the electron and hole transports in GLs under the self-consistent electric field. It is shown that in undoped double-GL structures with weak scattering of electrons and holes on disorder, the Fermi energies and the energy gap are virtually constant across the main portions of GLs, although their values strongly depend on the voltages and recombination parameters. In contrast, the electron and hole scattering on disorder lead to substantial nonuniformities. The resonant inter-GL tunneling enables N-shaped current-voltage characteristics provided that GLs are sufficiently short. The width of the current maxima is much larger than the broadening of the tunneling resonance. In the double-GL structures with relatively long GLs, the N-shaped characteristics transform into the Z-shaped characteristics. The obtained results are in line with the experimental observations [Britnell et al., Nat. Commun. 4, 1794–1799 (2013)] and might be useful for design and optimization of different devices based on double-GL structures, including field-effect transistors and terahertz lasers