Here we study the electronic properties of a square lattice using Landauer formula and recurssive Green's function. Although a square lattice may not have practical use, the development of the technique for this simple structure is insightful. One can use the Landauer formula and write the system conductance in terms of the scattering matric associated with the sample

Then transmission amplitude can be written in terms of system Green's function via Fisher-Lee relation to write

Quantum Conductance :

"The recursive Green's function method for graphene" Caio H. Lewenkopf, Eduardo R. Mucciolo