Competition between superconductivity and magnetic/nematic order as a source of anisotropic superconducting gap in underdoped ${\mathrm{Ba}}_{1-x}$${\mathrm{K}}_x$${\mathrm{Fe}}_2$${\mathrm{As}}_2$

The in-plane London penetration depth $\Delta \lambda \left(T\right)$ was measured using a tunnel diode resonator technique in single crystals of ${\mathrm{Ba}}_{1-x}$${\mathrm{K}}_x$${\mathrm{Fe}}_2$${\mathrm{As}}_2$ with doping levels $x$ ranging from heavily underdoped, $x=0.16$ $(T_c=7\text{K})$, to nearly optimally doped, $x=0.34$ $(T_c=39\text{K})$. Exponential saturation of $\Delta \lambda \left(T\right)$ in the $T\to 0$ limit is found in optimally doped samples, with the superfluid density ${\rho }_s\left(T\right)\equiv {[\lambda \left(0\right)/\lambda \left(T\right)]}^2$ quantitatively described by a self-consistent $\gamma$ model with two nodeless isotropic superconducting gaps. As the doping level is decreased towards the extreme end of the superconducting dome at $x=0.16$, the low-temperature behavior of $\Delta \lambda \left(T\right)$ becomes nonexponential and is best described by the power law $\Delta \lambda \left(T\right)\propto T^2$, characteristic of strongly anisotropic gaps. The change between the two regimes happens within the range of coexisting magnetic/nematic order and superconductivity, $x<0.25$, and is accompanied by a rapid rise in the absolute value of $\Delta \lambda \left(T\right)$ with underdoping. This effect, characteristic of the competition between superconductivity and other ordered states, is very similar to but of significantly smaller magnitude than what is observed in the electron-doped Ba$({\mathrm{Fe}}_{1-x}$${\mathrm{Co}}_x)$${}_2$${\mathrm{As}}_2$ compounds. Our study suggests that the competition between superconductivity and magnetic/nematic order in hole-doped compounds is weaker than in electron-doped compounds, and that the anisotropy of the superconducting state in the underdoped iron pnictides is a consequence of the anisotropic changes in the pairing interaction and in the gap function promoted by both magnetic and nematic long-range orders.

Figure 1

(a) Normalized $\Delta \lambda \left(T\right)$ for samples of ${\mathrm{Ba}}_{1-x}$${\mathrm{K}}_x$${\mathrm{Fe}}_2$${\mathrm{As}}_2$ with $x=0.16$, 0.19, 0.21, 0.3, and 0.34 (left to right). (b) Zoom of the actual $\Delta \lambda \left(T\right)$ plotted vs ${(T/T_c)}^2$ for a range $T/T_c\le 0.3$. The actual data for $x=0.16$ were divided by a factor of 3 $(\times$1/3) and for $x=0.19$ were divided by a factor of 2 $(\times$1/2), respectively, to facilitate the comparison with other compositions; (c) shows the same data over the lowest temperature range $T/T_c\le 0.1$, plotted as a function of ${(T/0.1T_c)}^n$ with $n$ 2.0, 2.3, 2.9, 3.0, and 3.5 $\pm$0.1 for $x$= 0.16, 0.19, 0.21, 0.3, and 0.34, respectively. Note the systematic decrease of $\Delta \lambda \left(T\right)$ on approaching optimal doping.

Figure 2

Superfluid density, ${\rho }_s\left(T\right)\equiv {[\lambda \left(0\right)/\lambda \left(T\right)]}^2$, of the optimally doped sample ${\mathrm{Ba}}_{1-x}$${\mathrm{K}}_x$${\mathrm{Fe}}_2$${\mathrm{As}}_2,x=0.34$, calculated from the data of Fig. 1 using $\lambda \left(0\right)=200$ nm [55] (open symbols). The solid red line on top of the data is the fit using the self-consistent two-gap $\gamma$ model, with the black and green curves in the main panel denoting the partial superfluid densities ${\rho }_1$ (larger gap) and ${\rho }_2$ (smaller gap). The temperature dependence of the corresponding gaps ${\Delta }_1$ and ${\Delta }_2$ is shown in the inset.

Figure 3

(a) Doping phase diagram of ${\mathrm{Ba}}_{1-x}$${\mathrm{K}}_x$${\mathrm{Fe}}_2$${\mathrm{As}}_2$ as determined from resistivity and TDR measurements on single crystals [49] (black circles show $T_{sm}$ and black squares show $T_c)$, matching well the results on polycrystalline samples [57, 58] (green triangles). Open red circles represent the samples used in the present study of the penetration depth. Dashed line shows an extrapolation of the orthorhombic/magnetic transition line, $T_{sm}\left(x\right)$ to $T\to 0$. Panel (b) shows the doping evolution of the exponent of the power-law function $n$ as determined from the data analysis in the temperature range $T/T_c\le 0.1$ (black stars) and $T/T_c\le 0.15$ (red circles). Panel (c) shows the doping evolution of the magnitude of the variation of the London penetration depth at low temperatures, $\Delta \lambda \left(T_{\mathrm{up}}\right)$ with $T_{\mathrm{up}}=0.15T_c$ (red squares) and $T_{\mathrm{up}}=0.3T_c$ (blue circles).

Figure 4

(a) Comparison of the doping phase diagrams of hole-doped ${\mathrm{Ba}}_{1-x}$${\mathrm{K}}_x$${\mathrm{Fe}}_2$${\mathrm{As}}_2$ and of the electron-doped Ba$({\mathrm{Fe}}_{1-x}$${\mathrm{Co}}_x)$${}_2$${\mathrm{As}}_2$. The data are plotted using normalized $x/x_s$ composition scale, where $x_s$ is a doping boundary of $T_s\left(x\right)$ lines with $x_s=0.25$ for BaK122 [57] and $x_s=0.063$ for BaCo122 [36], respectively. (b) Doping evolution of the temperature-dependent part of London penetration depth $\Delta \lambda \left(0.25T_c\right)$ in K-doped (blue circles) and Co-doped (red triangles) samples. Note the three times difference in the magnitude of penetration depth increase in two cases. Inset in (b) shows normalized temperature-dependent resistivity, $R/R\left(300\mathrm{K}\right)$, for underdoped samples with Co-doping $x=0.038,x/x_s=0.6$ and K-doping $x=0.18,x/x_s=0.72$, plotted using normalized temperature scale, $T/T_s$. Note that the resistivity of K-doped samples shows very small change at $T_s$, while that of Co-doped samples increases significantly below $T_s$.