Magnetoconductance due to variable-range hopping in quasi-two-dimensional systems: Application to PrBa 2 Cu 3 O 7  R. B. Thompson and M. Singh Department of Physics and Astronomy, University of Western Ontario, London, Ontario, Canada N6A 3K7 Received 14 May 1997; revised manuscript received 8 August 1997 In this paper, we have developed a theory of magnetoconductance magnetoresistance due to variable-range hopping for quasi-two-dimensional systems. We have included the effect of electric fields on the calculation of the magnetoconductance. The effects of scattering and electron-electron interactions have also been included in our theory. We found analytical expressions for the conductivity for both the scattering and nonscattering cases, and obtained electric- and magnetic-field-dependent power laws in certain approximations. We found that the electric and magnetic-field dependences of the magnetoconductance had different power laws for the scattering and nonscattering cases. We tried to explain the van Ancum et al. magnetoconductance experiments of PrBa 2 Cu 3 O 7- PBCO thin films by using our theory. A good agreement between theory and experiment was found if we included the effect of scattering. In the above PBCO films, it was found that the approximate value of the concentration of localized states lies between 10 11 and 10 12 cm -2 . S0163-18299800602-X I. INTRODUCTION There has been a considerable interest in the study of hopping conduction in low-dimensional systems such as ox- ide superconductors and related materials. 1,2 A material that figures highly in much of this research is PrBa 2 Cu 3 O 7 - PBCO because of its current and potential uses in the tech- nology of high-temperature superconducting junctions. PBCO has been shown to conduct via a variable-range hop- ping VRH mechanism along its CuO 2 planes 1,3 and so quasi-two-dimensional QTD theories are required to ex- plain its properties. Recently, we have developed a theory for variable-range-hopping conductivity in the presence of elec- tric fields for QTD and quasi-one-dimensional systems. 3,4 We have also included the effect of electron-electron inter- actions in our theoretical calculations. We applied our theory to explain the electric-field-dependent conductivity data of Kabasawa et al. 1 for PBCO-based S/N/S junctions and found a good agreement between theory and experiment. Recently van Ancum et al. 5 have measured the magnetic- field-dependent conductivity in PBCO thin films and sug- gested that the magnetoconductance in these films is due to variable-range hopping. They modified three-dimensional expressions of the magnetoconductance 6 for the two- dimensional case and tried to fit their data by using these expressions. They did not include in their expressions how the hopping exponents depended on the material parameters, which allowed them great latitude in fitting the experimental data. The effect of a magnetic field on VRH conduction in three-dimensional systems has been examined using the per- colation method 6 for strong i.e.,  a 0 ) magnetic fields and weak i.e.,  a 0 ) magnetic fields, where  =  / qH , a 0 is the localization length in the absence of a magnetic field, q is the charge of a carrier, and H is the magnetic field. In this paper, we have derived the expressions for the magnetoconductance for QTD systems for weak and strong magnetic fields by using the method developed by us. 3,4 This method allows for the inclusion of all pre-exponential fac- tors, and the inclusion of all relevant material parameters in the exponents. This approach differs significantly from the percolation method in the calculation of mobility and con- ductivity, and is better suited for obtaining analytical results for cases where the electric field is to be included. We have derived formulas for the magnetoconductance both with and without the inclusion of an electric field. In the former case, the magnetic-field dependency has been calculated to a higher order than in any previous work. In the latter case, unified formulas for the conductivity including all tempera- ture, electric field, and magnetic-field dependences have been presented. The effects of scattering and electron- electron interactions have also been included in our theory. The effect of scattering was investigated by Shklovskii in QTD systems by using the percolation method. 6 Our expres- sion of magnetoconductance in the presence of scattering can easily be reduced to that of Shklovskii by making appropri- ate approximations. We found that the logarithm of our expression of the mag- netoconductance for the constant DOS in certain approxima- tions is proportional to H 1/2 for strong magnetic fields with- out scattering. This is in general agreement with the expressions given in Refs. 5 and 7. For electron-electron in- teractions, the result is H 1/3 . For weak magnetic fields with- out scattering we found that the logarithm of the constant density of states DOS magnetoconductivity expression gave an H 2 magnetic-field dependence, consistent with the expression of van Ancum. 5 For electron-electron interactions the result is also an H 2 dependence. We used our theory to explain the PBCO thin-film mag- netoconductance experiments of van Ancum et al. It is found that these experiments cannot be explained by using the ex- pressions of magnetoconductance in the absence of scatter- ing. When the effect of scattering is included in the calcula- tions, a good agreement between theory and experiments is found. One fitting parameter is used to get a good agreement between theory and experiments. From our theoretical calcu- lations, we found that the concentration of localized states PHYSICAL REVIEW B 1 JANUARY 1998-II VOLUME 57, NUMBER 2 57 0163-1829/98/572/12848/$15.00 1284 © 1998 The American Physical Society The author has the right to post and update the article on a free-access e-print server using files prepared and formatted by the author. Any such posting made or updated after acceptance of the article for publication by APS should include a link to the online APS journal article abstract. In all cases, the appropriate bibliographic citation and notice of the APS copyright must be included. DOI: https://doi.org/10.1103/PhysRevB.57.1284