Hall effect in underdoped ${\mathrm{GdBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ thin films: Evidence for a crossover line in the pseudogap regime

We report on measurements of the resistivity and Hall coefficient in underdoped ${\mathrm{GdBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ epitaxial thin films grown by off-axis magnetron sputtering. The films have been lithographically patterned allowing precise measurements of the temperature dependencies of the inverse Hall constant $R_H^{-1}$ and of the Hall angle ${\theta }_H.$ We find that $R_H^{-1}$ is linear in temperature between 300 K and the pseudogap temperature $T^{*},$ whereas $\cot \left({\theta }_H\right)$ displays a perfect $T^2$ temperature dependence typically between 300 and 100 K. We observe for all the samples that the temperature at which the temperature dependence of $\cot \left({\theta }_H\right)$ deviates from the $T^2$ behavior is correlated to the temperature at which $R_H$ displays a peak. This characteristic temperature, found to lie between $T_c$ and $T^{*},$ does not depend markedly on the doping level and defines a crossover line in the temperature versus doping phase diagram. We tentatively relate these findings to recent high-frequency conductivity and Nernst effect experimental results, and we briefly discuss the possible consequences for competing theories for the pseudogap state of the cuprates.