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Crossover in the magnetic response of single-crystalline Ba$_{1-x}$K$_x$Fe$_2$As$_2$ and Lifshitz critical point evidenced by Hall effect measurements

We report the doping evolution of magnetic susceptibility $χ(T)$ and Hall coefficient $R_H$ in high quality Ba$_{1-x}$K$_x$Fe$_2$As$_2$ ($0.13 \leq x \leq 1$) single crystals. It is found that the normal-state magnetic susceptibility of Ba$_{1-x}$K$_x$Fe$_2$As$_2$ compounds undergoes a crossover from linear-$T$ dependence in the undoped and underdoped samples into KFe$_2$As$_2$-type magnetic response in the overdoped samples with increasing K content. Hall coefficient $R_H$ of underdoped sample $x$=0.22 shows a rapid increase above spin-density-wave transition temperature $T_{SDW}$. Below $T_{SDW}$, it increases slowly. $R_H$ of optimally doped and slightly overdoped samples ($0.34 \leq x \leq 0.65$) shows relative weak temperature dependence and saturation tendency below 150 K. However, $R_H$ of K heavily overdoped samples ($0.80 \leq x \leq 1$) increases rapidly below 150 K. Meanwhile, Hall angle $cot θ_H$ displays a concave temperature dependence within the doping range $0.22 \leq x \leq 0.55$, whereas it changes to a convex temperature dependence within the doping range $0.65 \leq x \leq 1$. The dramatic change coincides with Lifshitz transition occurred around the critical doping $x$=0.80, where ARPES measurements had confirmed that electron pocket disappears with excess hole doping in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ system. It is suggested that the characteristic temperature $T$* at around $120 \sim 150$ K observed in susceptibility and Hall coefficient, as well as previously reported resistivity data, may indicate an incoherence-coherence crossover in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ system.