The electronic and optical behaviors of the Ti 2 N graphene-like (GL) materials under the Cr impurity have been investigated based on DFT framework. The band structure and density of state (DOS) diagrams indicated that Ti 2 N:Cr GL has the magnetic metallic behavior about 3.2μ B . The density levels around the Fermi level in the band structure and the continuous DOS of under-Fermi to upper refer to the metallic nature of this compound. Adding Cr impurity has increased the metallic behavior so that the real and imaginary parts of dielectric functions have the red shift and their static values shifted to an infinite amount. Also, other optical parameters such as refraction, extinction and Eloss have been studied from the dielectric functions.Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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In this paper energy bands and Berry curvature of graphene was studied. Desired Hamiltonian regarding the next-nearest neighbors was obtained by tight-binding model. By using the second quantization approach, the transformation matrix is calculated and the Hamiltonian of system is diagonalized. With this Hamiltonian, the band structure and wave function can be calculated. By using calculated wave function the Berry connection and Berry curvature of our system are calculated. Our results are exactly consistent with previous methods and also the Berry curvature throughout the Brillouin zone get zero.