Low-field negative magnetoresistance in the variable-range-hopping regime in copper indium diselenide

Abstract

Study of the temperature and magnetic-field dependence of the negative magnetoresistance in the Mott variable-range-hopping regime is made on n-type ${\mathrm{CuInSe}}_{2}$. The relative magnetoresistance \ensuremath{\Delta}\ensuremath{\rho}/\ensuremath{\rho}(0) is found to be proportional to ${\mathit{T}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$${\mathit{B}}^{2}$ in the low-field region below about 0.35 T. Below 4 K, where ${\mathit{R}}_{\mathrm{hop}}$/\ensuremath{\xi}\ensuremath{\ge}1.5, it is observed that \ensuremath{\alpha}=3/4. This confirms, to our knowledge for the first time, the theories based on quantum interferences for the negative magnetoresistance in the variable-range-hopping regime. However, \ensuremath{\alpha}=1.22 above 4 K where ${\mathit{R}}_{\mathrm{hop}}$/\ensuremath{\xi}1.5. This is very similar to the values reported for other semiconductors. This leads us to suggest that for the correct application of the theories it is necessary that phase coherence should be maintained over a time scale larger than the hopping time and that the hopping length should be greater than the localization length.