Critical fields of superconductors with magnetic impurities

The upper critical field $H_{c2}$, the field $H_{c3}$ for nucleation of the surface superconductivity, and the thermodynamic field $H_c$ are evaluated within the weak-coupling theory for the isotropic s-wave case with arbitrary transport, and pair-breaking scattering. We find that, for the standard geometry of a half-space sample in a magnetic field parallel to the surface, the ratio $\mathcal{R}=H_{c3}/H_{c2}$ is within the window $1.55\lesssim \mathcal{R}\lesssim 2.34$, regardless of temperature or the scattering type. While the nonmagnetic impurities tend to flatten the $\mathcal{R}\left(T\right)$ variation, magnetic scattering merely shifts the maximum of $\mathcal{R}\left(T\right)$ to lower temperatures. Surprisingly, while reducing the transition temperature, magnetic scattering has a milder impact on $\mathcal{R}$ than nonmagnetic scattering. The surface superconductivity is quite robust; in fact, the ratio $\mathcal{R}\approx 1.7$ even in the gapless state. We used Eilenberger's energy functional to evaluate the condensation energy $F_c$ and the thermodynamic critical field $H_c$ for any temperature and scattering parameters. By comparing $H_{c2}$ and $H_c$, we find that, unlike transport scattering, the pair-breaking pushes materials toward type-I behavior. We find a peculiar behavior of $F_c$ as a function of the pair-breaking scattering parameter at the low-$T$ transition from gapped to gapless phases, which has recently been associated with the topological transition in the superconducting density of states.

Figure 1

(a) $\mathcal{R}(P,P_m)=H_{c3}/H_{c2}$ vs $T/T_{c0}$ for the scattering parameters indicated. (b) The same plotted vs the actual reduced temperature $T/T_c\left(P_m\right)$.

Figure 2

(a) $\mathcal{R}\left(T\right)=H_{c3}/H_{c2}$ vs $T/T_{c0}$ for strong transport scattering and a set of $P_m$ values for magnetic scattering. (b) $\mathcal{R}$ vs pair-breaking scattering parameter $P_m$ at a low reduced temperature $T/T_c\left(P_m\right)=0.1$ for a set of $P$ values of transport scattering. Note: The maximum $P_m$ possible is 0.1404.

Figure 3

$H_c$ (thin dashed lines) for the indicated values of the Ginzburg-Landau parameter ${\kappa }_0$ and $H_{c2}$ (thick red line) in units $2\pi {\varphi }_0{T}_{c0}^2/{\hslash }^2{v}^2$ vs $T/T_{c0}$. (a) Clean case $P=P_m=0$. (b) Strong magnetic scattering $P_m=0.1$, in the absence of potential scattering $P=0$. When a dashed blue line is above the solid red line, the material is a type-I superconductor.

Figure 4

The summary of our results for $H_{c2}(P,P_m)$ (top left), $H_{c3}(P,P_m)$ (top right), $\mathcal{R}(P,P_m)=H_{c3}(P,P_m)/H_{c2}(P,P_m)$ (lower left), and the ratio $H_{c2}(P,P_m)/H_c(P,P_m)$ (lower right) at the same reduced temperature $T/T_c\left(P_m\right)=0.1$. All fields are in units of $2\pi {\varphi }_0{T}_{c0}^2/{\hslash }^2{v}^2$.

Figure 5

The third derivative of the free energy at a set of low temperatures $t=T/T_{c0}=0.05,0.03,{10}^{-2},{10}^{-3},\mathrm{and}5\times {10}^{-5}$ vs the pair-breaking parameter $\zeta =\hslash /\mathrm{\Delta }{\tau }_m=(2\pi T_{c0}/\mathrm{\Delta })P_m$. In addition, the curve at $T=0$ obtained using the exact Eq. (71) from Maki's review [20]. $\zeta =1$ and $P_m=0.128$ correspond to the transition to the gapless state at $T=0$. In this calculation, the sum over $n$ was extended up to $N_{\text{max}}={10}^5$.