Doping evolution of the anisotropic upper critical fields in the iron-based superconductor ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$

In-plane resistivity measurements as a function of temperature and magnetic field up to 35 T with precise orientation within the crystallographic $ac$ plane were used to study the upper critical field $H_{c2}$ of the hole-doped iron-based superconductor ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$. Compositions of the samples studied spanned from under- doped $x=0.17$ $(T_c=12$ K) and $x=0.22$ $(T_c=20$ K), both in the coexistence range of stripe magnetism and superconductivity, through optimal doping $x=0.39$ $(T_c=38.4$ K) and $x=0.47$ $(T_c=37.2$ K), to overdoped $x=0.65$ $(T_c=22$ K) and $x=0.83$ $(T_c=10$ K). We find notable doping asymmetry of the shapes of the anisotropic $H_{c2}\left(T\right)$, suggesting the important role of paramagnetic limiting effects in the $H\parallel a$ configuration in overdoped compositions and multiband effects in underdoped compositions.

Figure 1

Top: Temperature-dependent electrical resistivity of selected representative samples of ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ with $x=0.17$, 0.22, 0.39, 0.47, 0.65, and 0.83. Compositions were chosen to have $T_c$ of about 10 K on the overdoped $\left(x=0.83\right)$ and underdoped $\left(x=0.17\right)$ sides, 20 K $(x=0.65$ and 0.22), and above 35 K $(x=0.39$ and 0.47). Bottom: Doping phase diagram with the position of the samples studied. C2AF corresponds to a range of the stripe antiferromagnet phase, C4AF corresponds to a range of the tetragonal $C_4$ antiferromagnetic phase, C4PM corresponds to the tetragonal paramagnetic state, and SC is the domain of the superconductivity, including ranges of coexistence with C2AF and C4AF phases (SC+AF).

Figure 2

Left: Magnetic-field-dependent resistivity of underdoped sample ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2,x=0.17$, taken in isothermal conditions in magnetic fields oriented along the tetragonal $c$ axis $(H\parallel c$, top panel, temperatures: 16, 14, 12, 10, 8, 7, 6, 5, 4, 3, and 2 K, from left to right) and perpendicular to it $(H\parallel a$, middle panel, temperatures, from left to right: 15, 13, 11, 9, 8, 7, 6, 5, 4, 3, 2, and 1.8 K). The bottom panel shows $H-T$ phase diagrams for two field orientations determined using the midpoint criterion between up- and down-field sweeps. The inset in the top panel shows the sample alignment procedure. Resistivity measurements were taken in field $H$ slightly lower than $H_{c2}^a$, in which sample resistance shows strong angular dependence. The curve was measured in the one-direction motion of the rotator to avoid backlash, with the deep minimum corresponding to $H\parallel ab$. Right: Magnetic-field-dependent resistivity of underdoped sample ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2,x=0.22$, taken in isothermal conditions in magnetic fields oriented along the tetragonal $c$ axis $(H\parallel c$, top panel, temperatures: 16, 14, 12, 10, 8, 7, and 6 K, from left to right) and perpendicular to it $(H\parallel a$, middle panel, temperatures, from left to right: 20, 18, 17, 16, 15, 14, 13, and 11 K). The bottom panel shows $H-T$ phase diagrams for two field orientations determined using the midpoint criterion between up- and down-field sweeps.

Figure 3

Left: Magnetic-field-dependent resistivity of sample ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2,x=0.39$, taken in isothermal conditions in magnetic fields oriented along the tetragonal $c$ axis $(H\parallel c$, top panel, temperatures: 38, 37, 36, 35, 34, 33, 32, 31, 30, and 29 K left to right), and perpendicular to it $(H\parallel a$, middle panel, temperatures, from left to right: 38, 37, 36, 35, and 34 K). The bottom panel shows $H-T$ phase diagrams for two field orientations determined using the midpoint criterion between up- and down-field sweeps (open symbols) and constant resistance criterion (line and solid symbols). Right: Magnetic-field-dependent resistivity of sample ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2,x=0.47$, taken in isothermal conditions in magnetic fields oriented along the tetragonal $c$ axis $(H\parallel c$, top panel, temperatures: 38, 37, 36, 35, 34, 33, 32, 31, 30, and 29 K, from left to right) and perpendicular to it $(H\parallel a$, middle panel, temperatures, from left to right: 38, 37, 36, 35, and 34 K). The bottom panel shows $H-T$ phase diagrams for two field orientations determined using the midpoint criterion between up- and down-field sweeps (open symbols) and constant resistance criterion (lines and solid symbols).

Figure 4

Left: Magnetic-field-dependent resistivity of sample ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2,x=0.65$, taken in isothermal conditions in magnetic fields oriented along the tetragonal $c$ axis $(H\parallel c$, top panel, temperatures: 16, 14, 12, 10, 8, 7, 6, 5, 4, and 2 K, from left to right) and parallel to the plane $(H\parallel a$, middle panel, temperatures, from left to right: 24, 22, 20, 19, 18, 17, 16, 15, 14, 13, and 11 K). The bottom panel shows $H-T$ phase diagrams for two field orientations determined using the midpoint criterion between up- and down-field sweeps. Right: Magnetic-field-dependent resistivity of sample ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ $x=0.83$ taken in isothermal conditions in magnetic fields oriented along the tetragonal $c$ axis $(H\parallel c$, top panel, temperatures: 12, 10, 8, 6, 5, 4, 3, and 2 K, from left to right) and perpendicular to it $(H\parallel a$, middle panel, temperatures, from left to right: 11, 9, 8, 7, 6, 5, 4, 3, and 2 K). The bottom panel shows $H-T$ phase diagrams for two field orientations determined using the midpoint criterion between up- and down-field sweeps.

Figure 5

Left: Comparison of the $H-T$ phase diagrams of a 10 K pair of underdoped $\left(x=0.17\right)$ and overdoped $\left(x=0.83\right)$ samples. Middle: A similar comparison for a 20 K pair of underdoped $\left(x=0.22\right)$ and overdoped $\left(x=0.65\right)$ samples. Right: Data for samples close to optimum doping, $x=0.39$ and $x=0.47$. Note the clear tendency to saturation in both overdoped compositions for the magnetic-field configuration $H\parallel a$.

Figure 6

Comparison of the $H-T$ phase diagrams in parallel magnetic field $H_{c2}^a\left(T\right)$ for samples of ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ for compositions (from left to right) with $x=0.83$ (solid circles), $x=0.17$ (open circles), $x=0.22$ (open squares), $x=0.65$ (solid squares), $x=0.39$ (open triangles), and $x=0.47$ (solid triangles).

Figure 7

Comparison of the $H-T$ phase diagrams of ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ in magnetic field precisely parallel to the plane $H_{c2}^a\left(T\right)$ using normalized temperature, $T/T_{c0}$, and magnetic-field, $H/[T_{c0}d{H}_{c2}\left(T\right)/dT]$, scales. For reference we show expectations for the orbital limiting mechanism in WHH theory [7] (black solid line), the experimentally determined $H_{c2}$ line for the conventional isotropic superconductor NbTi with dominant orbital limiting (dashed brown line), the layered organic superconductor $\kappa$-(BEDT-${\mathrm{TTF})}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]\text{Br}$ $(\kappa$-Br) in magnetic field parallel to the conducting plane in which paramagnetic limiting starts in the close vicinity of $T_c$ [50] (blue line), and paramagnetically limited ${\mathrm{CeCoIn}}_5$ [51] (magenta line). We also plot data for ${\mathrm{KFe}}_2{\mathrm{As}}_2$ as determined from magnetostriction and thermal expansion measurements in $x=1$ by Zocco et al. [15] (green crosses and line) and from resistivity measurements in pulsed field for a sample with $T_c\approx 28$ K corresponding roughly to $x=0.25$ from Yuan et al. (red line with open circles) [5]. Solid symbols represent overdoped compositions with $x=0.83$ (circles), $x=0.65$ (squares), and $x=0.47$ (triangles). Open symbols are used for underdoped compositions with $x=0.17$ (circles) and $x=0.22$ (squares) and for optimally doped $x=0.39$ (triangles).