Precision global measurements of London penetration depth in FeTe${}_{0.58}$Se${}_{0.42}$

We report tunnel-diode resonator (TDR) measurements of in-plane London penetration depth, $\lambda \left(T\right)$, in optimally-doped single crystals of FeTe${}_{0.58}$Se${}_{0.42}$ with $T_c\sim 14.8$ K. To avoid any size-dependent calibration effect, six samples of different sizes and deliberately introduced surface roughness were measured and compared. The power-law behavior, $\Delta \lambda \left(T\right)=A{T}^n$, was found for all samples with the average exponent $n_{\text{avg}}=2.3\pm 0.1$ and the prefactor $A_{\text{avg}}=1.0\pm 0.2$ nm/K${}^{2.3}$. The average superfluid density is well described by the self-consistent two-gap $\gamma$ model resulting in ${\Delta }_{{}_I}\left(0\right)/k_B{T}_c=1.93$ and ${\Delta }_{{}_{II}}\left(0\right)/k_B{T}_c=0.9$. These results suggest the nodeless two-gap pairing symmetry with strong pair breaking effects. In addition, it is found from comparison among six different samples that, while the exponent $n$ remains virtually unchanged, the prefactor $A$ shows some variation, but stays within a reasonable margin, ruling out some recent suggestions that surface conditions can significantly affect the results. This indicates that the calibration procedure used to obtain $\lambda \left(T\right)$ from the measured TDR frequency shift is robust and that the uncertainty in sample dimensions and the nature of surface roughness play only a minor role.

Figure 1

(a) Sample 2: before roughening and (b) Sample 2-R: after deliberate roughening; see Table .

Figure 2

Temperature-dependent frequency shift, $\Delta f\left(T\right)$, for each sample listed in Table . $T_c$ is consistently determined to be 14.8 K by using onset curves at the phase transition. Inset: $\Delta f\left(T\right)$ normalized based on the value at 16 K. The high-temperature region is zoomed in showing the phase transition.

Figure 3

Temperature-dependent penetration depth, $\Delta \lambda \left(T\right)$, for each sample. (a) High-temperature region and (b) low-temperature region of $\Delta \lambda \left(T\right)$. The arrows indicate the upper limits of fitting. The dashed lines are representative fits for each sample conducted up to $T_c$/3. All the fitting results are summarized in Fig. 4.

Figure 4

Results of the power-law fits to $\Delta \lambda \left(T\right)=A{T}^n$. Four different upper limits, indicated by arrows in Fig. 3b, were used. Panel (a): the exponent $n$, obtained by keeping $A$ and $n$ as free parameters. Panel (b): the prefactor $A$, obtained at a fixed $n_{\text{avg}}=2.3$, which is the average among all six samples shown in panel (a).

Figure 5

Superfluid density ${\rho }_s\left(T\right)={\left[\lambda \left(0\right)/\lambda \left(T\right)\right]}^2$ vs $T/T_c$. $\lambda \left(0\right)=560$ nm obtained from the previous report.[22] (a) ${\rho }_s\left(T\right)$ for all six samples. (b) The average superfluid density for all six samples, ${\rho }_s^{\text{avg}}$ (red circle), is fitted to the two-gap $\gamma$ model.[30] Black solid line is the fitting result with $\rho =\gamma {\rho }_{{}_I}+(1-\gamma ){\rho }_{{}_{II}}$. The inset shows two gaps acquired from the fit, ${\Delta }_{{}_I}$ and ${\Delta }_{{}_{II}}$.