Application of the lattice Green’s function for calculating the resistance of an infinite network of resistors

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Abstract
The resistance between two arbitrary grid points of several infinite lattice structures of resistors is calculated by using lattice Green’s functions. The resistance for d dimensional hypercubic, rectangular, triangular, and honeycomb lattices of resistors is discussed in detail. Recurrence formulas for the resistance between arbitrary lattice points of the square lattice are given. For large separation between nodes the asymptotic form of the resistance for a square lattice and the finite limiting value of the resistance for a simple cubic lattice are calculated. The relation between the resistance of the lattice and the van Hove singularity of the tight-binding Hamiltonian is given. The Green’s function method used in this paper can be applied in a straightforward manner to other types of lattice structures and can be useful didactically for introducing many concepts used in condensed matter physics.

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