Local characterization of superconductivity in $\mathrm{BaF}{\mathrm{e}}_2{\left(\mathrm{A}{\mathrm{s}}_{1-x}{\mathrm{P}}_x\right)}_2$

We use magnetic force microscopy (MFM) to characterize superconductivity across the superconducting dome in $\mathrm{BaF}{\mathrm{e}}_2{\left(\mathrm{A}{\mathrm{s}}_{1-x}{\mathrm{P}}_x\right)}_2$, a pnictide with a peak in the penetration depth $\left({\lambda }_{ab}\right)$ at optimal doping $\left(x_{\mathrm{opt}}\right)$, as shown in sample-wide measurements. Our local measurements show a peak at $x_{\mathrm{opt}}$ and a $T_C$ vs ${\lambda }_{ab}^{-2}$ dependence similar on both sides of $x_{\mathrm{opt}}$. Near the underdoped edge of the dome ${\lambda }_{ab}$ increases sharply, suggesting that superconductivity competes with another phase. Indeed, MFM vortex imaging shows correlated defects parallel to twin boundaries only in underdoped samples and not for $x\ge x_{\mathrm{opt}}$.

Figure 1

Touchdown curves for the $x=0.29$ sample at one position as a function of temperature. The curves are offset for clarity both horizontally (by $0.1\mu \mathrm{m}$) and vertically (by 0.5 Hz). The left vertical in each curve indicates the surface of the sample (i.e., $z=0\mu \mathrm{m}$ preoffset). Inset: Zoom-in on the $T=28\mathrm{K}$ measurement (no offsets). The absence of a Meissner signal in this touchdown and its presence at 27 K (main panel, red curve) imply that $T_C=27.5\pm 0.5\mathrm{K}$.

Figure 2

(a) Illustration of the model for the tip depicting $\theta$ (the cone half angle), $h$ (the length of the cone truncation, shown out of scale), and $H$ (the effective magnetic height of the tip). Top inset: SEM image of a tip with a $4\mu \mathrm{m}$ scale bar. Bottom inset: An illustration of the tip and the cantilever holding it. The cantilever is $220\mu \mathrm{m}$ long, $30\mu \mathrm{m}$ wide, and approximately $3\mu \mathrm{m}$ thick. (b) The value of ${\lambda }_{ab}$ from fitting as a function of $z_{\mathrm{span}}$ and $H$. The arrow points to the optimal $H=H^{*}=16\mu \mathrm{m}$ for this case. In other cases we get comparable values as well as higher ones, up to many tens of microns [28]. Inset: The raw data and the fit at the optimal $H^{*}=16\mu \mathrm{m}$, which gives ${\lambda }_{ab}=170\pm 5\mathrm{nm}$.

Figure 3

Results of the local measurement of ${\lambda }_{ab}$ and $T_C$ (all lines are guides to the eye). (a) ${\lambda }_{ab}$ (full circles, left axis, light blue line as a guide to the eye) and $T_C$ (triangles, right axis, green line as a guide to the eye) as a function of $x$. The data for $x=0.22$ are shown in red. The error bars are smaller than the symbols. Inset: ${\lambda }_{ab}^2$ as a function of $x$, excluding $x=0.22$, showing the sharp peak at optimal doping. (b) $T_C$ as a function of ${\lambda }_{ab}^{-2}$ for all samples and from all touchdown locations. There are seven blue clusters of data points for $x>0.22$ samples, each marked by its value of $x$ as determined from EDS (cf. table in [28]). In every cluster we show the mean values of $T_C$ and ${\lambda }_{ab}^{-2}$. The error bars take into account both the scatter due to repeat measurements as well as fit errors. The gray line is a guide to the eye. Data for $x=0.22$ is shown in red asterisks with error bars for the value of $T_C$. The errors for ${\lambda }_{ab}^{-2}$ on each asterisk are smaller than the symbols.

Figure 4

Vortex decoration scans. (a) $x=0.33$ sample with $z=600\mathrm{nm}$, $T=4.45\mathrm{K}$, and $B=25\mathrm{G}$. (b) $x=0.26$ sample with $z=650\mathrm{nm}$, $T=4.45\mathrm{K}$, and $B=15\mathrm{G}$. The directions of the crystal $ab$ axes in this sample are shown by black arrows. (c) $x=0.22$ sample with $z=850\mathrm{nm}$, $T=4.5\mathrm{K}$, and $B=35\mathrm{G}$.