Thermodynamic evidence for a pressure-driven crossover from strong- to weak-coupling superconductivity in Pb

Abstract

The thermodynamic critical field $B_{\rm c}$ provides direct access to the superconducting condensation energy, yet its pressure dependence has been studied much less extensively than that of the transition temperature. Here, muon-spin-rotation/relaxation measurements of the thermodynamic critical field $B_{\rm c}$ of elemental Pb under hydrostatic pressure up to $\simeq2.3$ GPa are reported. From the magnetic-field distribution in the intermediate state, $B_{\rm c}(T)$ is determined and $B_{\rm c}(0)$ is extracted at different pressures. In combination with previously reported high-pressure data for $B_{\rm c}$ and $T_{\rm c}$, it is shown that the pressure dependence of $B_{\rm c}(0)$ follows that of the superconducting gap $Δ(0)$ more closely than that of the transition temperature $T_{\rm c}$. At higher pressures, the logarithmic pressure derivatives of $B_{\rm c}(0)$ and $T_{\rm c}$ are found to converge, indicating that the coupling strengths ratio $α=Δ(0)/k_{\rm B}T_{\rm c}$ becomes nearly pressure independent. This behavior is interpreted as thermodynamic evidence for a pressure-driven crossover from strong- to weak-coupling superconductivity in Pb.

Figures (3)

• Figure 1: (a) Fourier transforms of the muon-time spectra at $p=2.34$ GPa, showing the magnetic-field distribution in the pressure-cell/sample assembly. Three distinct peaks are observed: (i) a peak at $B=0$, corresponding to muons stopping in superconducting domains of the Pb sample (Meissner state); (ii) a peak at the applied field, $B=B_{\rm ap}$, arising from muons stopping in the pressure-cell walls; and (iii) a peak at $B=B_{\rm c}$, corresponding to muons stopping in normal-state domains of the Pb sample. (b) Temperature dependence of the thermodynamic critical field $B_{\rm c}(T)$ measured at pressures $p=0.07$, 1.25, and 2.34 GPa. The solid lines are fits within the $\alpha$ model, while the dashed lines represent parabolic dependences. (c) Temperature dependence of the deviation function $D(T)=B_{\rm c}(T)-B_{\rm c}(0)[1-(T/T_{\rm c})^2]$, highlighting the deviation of $B_{\rm c}(T)$ from parabolic temperature dependence. The solid lines are fits using the $\alpha$ model.
• Figure 2: Pressure dependencies of the superconducting parameters obtained from fits of $B_c(T)$ data using the $\alpha$ model: (a) superconducting transition temperature $T_c(p)$, (b) thermodynamic critical field at zero temperature $B_c(0,p)$, (c) superconducting energy gap $\Delta()0,p)$, and (d) the coupling parameter $\alpha(p)$. Solid lines represent linear fits to the data. The corresponding logarithmic pressure derivatives are indicated.
• Figure 3: Normalized pressure dependences of the superconducting transition temperature $T_c(p)/T_c(p=0)$ and the thermodynamic critical field $B_c(0,p)/B_c(0, p=0)$ for Pb. Closed symbols represent the present $\mu$SR data, while open symbols correspond to literature data from Brandt et al. Ref. Brandt_JETP_1975. The solid lines are empirical fits using a double-exponential form. Inset: Pressure dependences of the logarithmic derivatives $d\;\ln T_c/dp$ and $d\; \ln B_c(0)/dp$ obtained from the fit functions. The slopes approach each other and become nearly equal above $p\sim 8$ GPa, indicating convergence of the pressure responses of $T_c$ and $B_c$.