Mixed-state Hall effect and flux pinning in Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ single crystals ($x=0.08$ and $0.10$)

Longitudinal and Hall resistivities, scaling behavior, and magnetizations are examined to study the effect of flux pinning in Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ (BFCA) single crystals with $x$ = 0.08 and 0.01. Larger values of activation energy, critical current density, and pinning force are obtained in BFCA with $x$ = 0.10, indicating relatively strong pinning. The sign reversal of Hall resistivities is clearly observed in BFCA with $x$ = 0.10. The correlation between longitudinal and Hall resistivities shows the scaling behavior of ${\rho }_{\mathit{xy}}\propto ({\rho }_{\mathit{xx}}){}^{\beta }$ with exponents $\beta$ = 3.0–3.4 and 2.0 ± 0.2 for BFCA crystals with $x$ = 0.08 and 0.10, respectively. Furthermore, the normal-state Hall angle is also observed to follow $\cot \hspace{0.5em}{\theta }_H=\Lambda T^2+C$ in BFCA crystals, and is explained by the Anderson theory. The relatively large $c$/Λ value for BFCA with $x$ = 0.10 also implies a larger contribution of impurity scattering due to more Co atoms, which may cause stronger pinning of flux lines. The results are analyzed and coincide with theory, including the pinning-induced backflow effect and plastic flow mechanism in vortex dynamics.

Figure 1

X-ray θ-2θ diffraction spectrum for BFCA with $x$ = 0.10. The inset shows the x-ray θ-2θ diffraction spectra in the region near the (008) peak for BFCA samples with $x$ = 0.08 and 0.10.

Figure 2

Resistivity as a function of temperature with magnetic fields parallel to the crystal $c$ axis for BFCA samples (a) with $x$ = 0.08 and (b) 0.10. Inset of (a): Temperature-dependent resistances normalized to their room-temperature values in zero field. Inset of (b): Field-dependent activation energy for BFCA crystals with $x$ = 0.08 and 0.10. The dashed lines indicate the fitting of $U\propto H^{-\alpha }$.

Figure 3

(a) Magnetization hysteresis loops measured at 0.8$T_c$ for BFCA crystals with $x$ = 0.08 and 0.10 in fields parallel to the $c$ axis. (b) Field dependences of critical current density $J_c$ at 0.8 and 0.9$T_c$ for the corresponding samples. Inset of (a) shows $J_c$ versus the reduced temperature $T/T_c$ in zero field and 1 T for the corresponding samples. Inset of (b) shows the corresponding pinning forces against the magnetic field.

Figure 4

Temperature dependence of ${\rho }_{\mathit{xy}}$ for BFCA samples with (a) $x$ = 0.08 and (b) 0.10. Inset of (a): Hall coefficient versus temperature for BFCA with $x$ = 0.10. (c) Temperature dependence of Hall resistivity ${\rho }_{\mathit{xy}}$ taken on the other two samples as described in the text. Inset of (c): Resistive transition for the corresponding samples.

Figure 5

Magnetic-field dependence of ${\rho }_{\mathit{xy}}$ for BFCA with (a) $x$ = 0.08 and (b) 0.10 at various temperatures. Insets of (a) and (b) demonstrate the plots of $\mathit{log}{}_{10}\left|{\rho }_{\mathit{xy}}\right|$ vs $\mathit{log}{}_{10}\hspace{0.5em}{\rho }_{\mathit{xx}}$ taken respectively at fixed magnetic fields and fixed temperatures for the corresponding samples. The dashed lines indicate the fitting of ${\rho }_{\mathit{xy}}\propto ({\rho }_{\mathit{xx}}){}^{\beta }$.

Figure 6

(a) Temperature dependence of the Hall angle shown as $\cot \hspace{0.5em}{\theta }_H$ vs $T$ $2$ for BFCA with $x$ = 0.08 and 0.10 measured in field of 7 T. The dashed lines indicate the fitting of $\cot \hspace{0.5em}{\theta }_H=\Lambda T^2+C$. (b) ${\rho }_{{\mathit{xy}}_{,max}}$ extracted at different fields and field dependence of normalized pinning force $F_p/F_{p_{,max}}$ for BFCA with $x$ = 0.10 at 0.8$T_c$. Inset:Normalized ${\rho }_{\mathit{xy},max}/(1-t){}^{1.5}$ against the reduced field $h$. Also shown in the inset is a normalized $h(1-h){}^2$ curve for comparison.