Superconducting nature of the Bi-II phase of elemental bismuth

The superconductivity in the Bi-II phase of elemental bismuth (transition temperature $T_{\mathrm{c}}\simeq 3.92$ K at pressure $p\simeq 2.80$ GPa) was studied experimentally by means of the muon-spin rotation as well as theoretically by using the Eliashberg theory in combination with density functional theory calculations. Experiments reveal that Bi-II is a type-I superconductor with a zero temperature value of the thermodynamic critical field $B_{\mathrm{c}}\left(0\right)\simeq 31.97$ mT. The Eliashberg theory approach provides a good agreement with the experimental $T_{\mathrm{c}}$ and the temperature evolution of $B_{\mathrm{c}}$. The estimated value for the retardation (coupling) parameter $k_{\mathrm{B}}{T}_{\mathrm{c}}/{\omega }_{\ln }\approx 0.07$ (${\omega }_{\ln }$ is the logarithmically averaged phonon frequency) suggests that Bi-II is an intermediately coupled superconductor.

Figure 1

(a) The distribution of fields in a type-II superconductor in the vortex state (left panel) and the corresponding magnetic field distribution function $P\left(B\right)$ (right panel). (b) The schematic representation of nucleation of a plate-like type-I superconductor in normal state ($B_{\mathrm{N}}=B_{\mathrm{c}}$) and superconducting ($B_{\mathrm{S}}=0$) domains (left panel). An ordered laminar structure is formed with an additional in-plane component $B_i$, after Refs. [23, 24]. The right panel is the $P\left(B\right)$ distribution in type-I superconductor. (c) Fourier transform of TF-$\mu \mathrm{SR}$ time spectra measured at external field $B_{\mathrm{ex}}\simeq 20$ mT reflecting the $P\left(B\right)$ distribution in the Bi-II sample above ($T=2.5$ K) and below ($T=0.5$ and 2.0 K) the superconducting transition temperature [$T_{\mathrm{c}}\left(20\mathrm{mT}\right)\simeq 2.3$ K]. (d) The contour plot of the $P\left(B\right)$ distribution measured at $B_{\mathrm{ex}}\simeq 20$ mT. (e) The temperature dependence of the thermodynamical critical field $B_{\mathrm{c}}$ for the Bi-II sample obtained in $\mu \mathrm{SR}$ experiments with the applied field $B_{\mathrm{ex}}=3$, 10, 20, and 30 mT. The dashed line is the temperature evolution of the “theoretical” $B_{\mathrm{c},\mathrm{T}}$ obtained within the framework of ab initio Eliashberg calculations using density functional theory. The solid line is the same $B_{\mathrm{c},\mathrm{T}}\left(T\right)$ curve with $T_{\mathrm{c}}=3.922$ K and $B_{\mathrm{c}}\left(0\right)=31.97$ mT adjusted from the fit (see text for details).

Figure 2

(a) Calculated phonon density of states. (b) Calculated Eliashberg electron-phonon spectral function (red solid line) and integrated electron-phonon coupling strength ${\lambda }_{\mathrm{el}-\mathrm{ph}}$ (green dashed line). (c) The deviation function $D(T/T_{\mathrm{c}})=B_{\mathrm{c}}\left(T\right)/B_{\mathrm{c}}\left(0\right)-(1-{[T/T_{\mathrm{c}}]}^2)$. The solid blue and dashed lines correspond to the Eliashberg and the BCS approach, respectively.