The next nearest neighbor effect on the 2D materials properties

In this work, the effect of introducing next nearest neighbor (NNN) hopping to the 2D materials was studied using the graphene 2D honeycomb two sublattice as an example. It is found that introducing NNN to the 2D materials removes the symmetry around the Fermi level and shifts it, at a small value of NNN hopping. This effect increases with increasing NNN hopping. If the NNN hopping becomes competitive with nearest neighbor (NN) hopping, the dispersion relations of the 2D materials changes completely from NN hopping dispersion relations. The results show that the 2D material sensitivity for NNN hopping effect is much larger in the 2D honeycomb lattice than 2D square lattice. This is due to the fact that the number of NNN sites is equal to six, which is the double of NN sites in the 2D honeycomb lattice. Meanwhile, the number of NNN sites is equal to four which is equal to NN sites in 2D square lattice. We therefore conclude that by changing the ratio between NNN and NN sites in the 2D lattice one can tune the sensitivity for NNN hopping effects.