Theory of negative magnetoresistance in three-dimensional systems

Theory of negative magnetoresistance in two-dimensional systems due to delocalization of electrons by magnetic field by Hikami, Larkin, and Nagoaka is extended to the case of three-dimension. The increase in conductivity by magnetic field is independent of the direction of the current relative to that of the magnetic field, and is proportional to $\text{H}{}^2\text{τ}{}_{\mathrm{?}}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, τ_? being energy relaxation time of electrons, when the magnetic field H is small. If τ_? is large enough at low temperature and H is not too small, it is independent of the parameter characterizing the system, and is of the form $\text{0.918?H mho cm}{}^{\mathrm{?1}}$ (H in kOe).