Sign reversal of the Hall effect in the flux flow region of ${\mathrm{Bi}}_{2+x}{\mathrm{Sr}}_{2-x}{\mathrm{CuO}}_{6+\delta }$

We have measured the longitudinal and transverse resistivity, ${\rho }_{xx}$ and ${\rho }_{xy}$ on the optimally doped ${\mathrm{Bi}}_{2+x}{\mathrm{Sr}}_{2-x}{\mathrm{CuO}}_{6+\delta }$ single crystals with a well-shaped Hall bar structure. It is found that the Hall resistance in the flux flow region exhibits a clear sign-reversal effect. The strongest reversal Hall signal appears in the region of about the half-transition way, either on the temperature dependence of ${\rho }_{xy}$ at a fixed field or the field dependence of ${\rho }_{xy}$ at a fixed temperature. Detailed analysis shows that the theoretically proposed scaling law ${\rho }_{xy}=A{\rho }_{xx}^{\beta }$ with $1.5<\beta <2$ in the small dissipation limit is not obeyed very well. The Hall resistivity is plotted in a $H-T$ phase diagram, and the boundary for the Hall reversal effect to occur follows the theoretical prediction of Feigel'man et al. [JETP Lett. 62, 834 (1995)], which suggests that the anomalous Hall effect may be induced by the properties of a single vortex, rather than by the collective flux motion with pinning and strong thermal fluctuations.