Equivalence of effective medium and random resistor network models for disorder-induced unsaturating linear magnetoresistance

A linear unsaturating magnetoresistance at high perpendicular magnetic fields, together with a quadratic positive magnetoresistance at low fields, has been seen in many different experimental materials, ranging from silver chalcogenides and thin films of InSb to topological materials like graphene and Dirac semimetals. In the literature, two very different theoretical approaches have been used to explain this classical magnetoresistance as a consequence of sample disorder. The phenomenological random resistor network model constructs a grid of four terminal resistors, each with a varying random resistance. The effective medium theory model imagines a smoothly varying disorder potential that causes a continuous variation of the local conductivity. Here, we demonstrate numerically that both models belong to the same universality class and that a restricted class of the random resistor network is actually equivalent to the effective medium theory. Both models are also in good agreement with experiments on a diverse range of materials. Moreover, we show that in both cases, a single parameter, i.e., the ratio of the fluctuations in the carrier density to the average carrier density, completely determines the magnetoresistance profile.