Sign reversal of the Hall resistivity of single-crystalline ${\mathrm{Bi}}_{1.95}{\mathrm{Sr}}_{1.65}{\mathrm{La}}_{0.4}\mathrm{Cu}{\mathrm{O}}_{6+\delta }$ and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$

The Hall effect of single-crystalline ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ and ${\mathrm{Bi}}_{1.95}{\mathrm{Sr}}_{1.65}{\mathrm{La}}_{0.4}\mathrm{Cu}{\mathrm{O}}_{6+\delta }$ has been measured above and below the superconducting transition temperature $T_{c0\mathrm{}}$ by applying magnetic fields $\mathrm{H}\parallel c$ and currents $I\perp c$, where $c$ denotes the direction perpendicular to the Cu-O planes of these materials. For both compounds, the Hall resistivity ${\rho }_{\mathrm{xy}}(T, \mathrm{H})$ reveals similar temperature and field dependences. As $T$ approaches $T_{c0\mathrm{}}$ from higher temperatures, the ${\rho }_{\mathrm{xy}}(T, \mathrm{H})$ versus H curves gradually deviate from linearity and exhibit an unusual sign reversal in the low-field regime. The total Hall resistivity ${\rho }_{\mathrm{xy}}(T, \mathrm{H})$ may be decomposed into three distinctly different terms, i.e., ${\rho }_{\mathrm{xy}}(T, \mathrm{H})={\rho }_{\mathrm{xy}}^0\mathrm{}\left(T\right)+R_H^{\prime }\mathrm{}\left(T\right)\mathrm{}·\mathrm{H}+{\rho }_{\mathrm{xy}}^x(T, \mathrm{H})$. The contribution ${\rho }_{\mathrm{xy}}^x(T, \mathrm{H})$, representing the deviation of ${\rho }_{\mathrm{xy}}(T, \mathrm{H})$ from H linearity, exhibits a universal scaling behavior as a function of $\frac{\mathrm{H}}{{\mathrm{H}}_0\mathrm{}\left(T\right)\mathrm{}}$ for fields between 0 and ${\mathrm{H}}_0\mathrm{}\left(T\right)\mathrm{}$, the field below which ${\rho }_{\mathrm{xy}}(T, \mathrm{H})$ is no longer linear in H.