Isotropic single-gap superconductivity of elemental Pb

The unconventional multigap superconductivity in elemental Pb were reported previously by the surface sensitive tunneling experiments as well as predicted by several theory works. To obtain bulk evidence for such multiple gap behavior, the thermodynamic critical-field $B_{\mathrm{c}}$ was measured along three different crystallographic directions ([100], [110], and [111]) in a high-quality Pb single crystal by means of muon spin rotation/relaxation. No difference in temperature evolution of $B_{\mathrm{c}}$ for all three directions was detected. The average reduced gap $\alpha =\mathrm{\Delta }/k_{\mathrm{B}}{T}_{\mathrm{c}}=2.312\left(3\right)$ ($\mathrm{\Delta }$ is the zero-temperature gap value, and $T_{\mathrm{c}}$ is the transition temperature) was further obtained by employing the phenomenological $\alpha$ model. Our results imply that the elemental Pb is an isotropic superconductor with a single energy gap.

Figure 1

Deviation of the thermodynamic critical field $B_{\mathrm{c}}\left(T\right)$ from the parabolic behavior [$D\left(t^2\right)=B_{\mathrm{c}}\left(t^2\right)/B_{\mathrm{c}}\left(0\right)-(1-t^2)$] as expected for the big: ${\alpha }_{\mathrm{big}}={\mathrm{\Delta }}_{\mathrm{big}}/k_{\mathrm{B}}{T}_{\mathrm{c}}=2.21–2.31$ (red segment) and the small: ${\alpha }_{\mathrm{small}}={\mathrm{\Delta }}_{\mathrm{small}}/k_{\mathrm{B}}{T}_{\mathrm{c}}=1.87–2.07$ (violet segment) reduced gaps for elemental Pb reported in Refs. [10, 11, 12] ($t=T/T_{\mathrm{c}}$ is the reduced temperature). Calculations of $D\left(t\right)$ functions are performed by using the $\alpha$ model [23, 24]. Black points are the experimental $D\left(t\right)$ data measured with the external field $B_{\mathrm{ex}}$ applied along three different crystallographic directions: [100], [110], and [111] (see the text for details).

Figure 2

(a) Laue image of a Pb single crystal. The cubic crystal structure of Pb is clearly visible. (b) The schematic of the first set of experiments (expt. No. 1) allowing sample rotation along the 001 crystal axis. In this configuration $B_{\mathrm{ex}}\parallel \left[100\right]$ and $B_{\mathrm{ex}}\parallel \left[110\right]$ sets of measurements were performed. (c) The second set of experiments (expt. No. 2) with the sample rotation along the 011 axis. In these experiments, $B_{\mathrm{ex}}\parallel \left[100\right]$ and $B_{\mathrm{ex}}\parallel \left[111\right]$ configurations were probed.

Figure 3

(a) Temperature dependences of the thermodynamic critical-field $B_{\mathrm{c}}$ of elemental Pb single crystal measured along different crystallographic directions. The solid line is the fit of $B_{\mathrm{c}}\left(T^2\right)$ measured in expt. No. 2 in the $B_{\mathrm{ex}}\parallel \left[100\right]$ configuration by means of Eq. (5). (b) Deviation functions $D\left(T^2\right)$ obtained by subtracting parabolic functions from measured $B_{\mathrm{c}}\left(T^2\right)$ curves, see Eq. (2). The solid line is the same as in panel (a), but after subtracting the parabolic function.