Spectral properties of superconductors with ferromagnetically ordered magnetic impurities

We present a comprehensive theoretical study of thermodynamic properties of superconductors with a dilute concentration of magnetic impurities, with focus on how the properties of the superconducting host change if the magnetic moments of the impurities order ferromagnetically. Scattering off the magnetic impurities leads to the formation of a band of Yu-Shiba-Rusinov states within the superconducting energy gap that drastically influences superconductivity. In the magnetically ordered system, the magnetization displays a sudden drop as a function of the impurity density or magnetic moment amplitude. The drop occurs as the spin-polarized impurity band crosses the Fermi level and is associated with a quantum phase transition first put forward by Sakurai for the single impurity case. Taking into account that the background magnetic field created by the ordered impurity moments enters as a Zeeman shift, we find that the superconducting phase transition changes from second order to first order for high enough impurity concentrations.

Figure 1

We consider a superconducting thin film (S) deposited on a substrate. Magnetic impurities are homogeneously distributed within the sample. Magnetic moments of the impurities are either completely unpolarized (a), or ferromagnetically ordered (b) in the plane of the film.

Figure 2

Example of a transition metal element (Mn) and a rare-earth element (Sm) with their valence electron configurations. When placed in a superconductor, these elements strip the outer $4s^2$ and $6s^2$ shells and behave as localized magnetic moments. The Hund's rule determines the type of local exchange interaction (ferromagnetic or antiferromagnetic), which enters the theory through the sign of the parameter $\alpha$. Itinerant electron is depicted by the dotted red arrow.

Figure 3

Transition temperature as a function of impurity density [see Eq. (11)] for randomly oriented impurity spins. From right to left $u_{\mathrm{S}}=3,5,9$. (Inset) The same data with the horizontal axis rescaled with respect to ${\mathrm{\Gamma }}_c$, critical density at which $T_c=0$ (see text).

Figure 4

Critical temperature as a function of impurity density for ferromagnetically ordered case and $\alpha >0$. From right to left, we have $u_{\mathrm{S}}=3,5,9$. Solid lines are solutions to Eq. (32). Dashed lines correspond to $\delta \mathrm{\Omega }\left(T\right)=0$, see Eq. (36). Dotted lines are found from Eq. (31) requiring ${\mathrm{\Delta }}_0\left(T\right)=0$.

Figure 5

Transition temperature as a function of impurity density for ferromagnetically ordered case with $u_{\mathrm{S}}=3$. The red (right) curve corresponds to $\alpha <0$, while the black (left) one is for $\alpha >0$, see Fig. 4.

Figure 6

Order parameter as a function of impurity density (a) and temperature (b) for randomly oriented impurity spins. (a) From right to left, $u_s=3,5,9$ and $T=0.01T_{c0}$. (b) From top to bottom, $\mathrm{\Gamma }/2\pi k_B{T}_{c0}=0.1,0.2,0.3$ and $u_{\mathrm{S}}=3$.

Figure 7

Order parameter (a) and the free energy difference (b) as a function of impurity density for ferromagnetically aligned impurity spins and $\alpha >0$. From right to left, $u_s=3,5,9$ and $T=0.01T_{c0}$. Two different solutions to Eq. (31) and the corresponding free energy difference are plotted with solid and dotted lines. Thin vertical lines mark the density at which the free energy difference crosses zero.

Figure 8

Order parameter (a)–(c) and Gibbs free energy difference (d)–(f) as a function of temperature for ferromagnetically aligned impurity spins with $u_{\mathrm{S}}=3$ and $\mathrm{\Gamma }/2\pi k_B{T}_{c0}$ equal to $0.04$ for (a) and (d), $0.045$ for (b) and (e), $0.055$ for (c) and (f). The vertical line in (f) depicts the temperature at which the free energy difference crosses zero. The inset in (d) shows a zoom on the dotted line in the main plot. The dotted lines in (b) and (c) depict the second solution to the order parameter equation (31) and the corresponding free energy difference in (e) and (f).

Figure 9

Density of states for randomly oriented impurity spins. (a) Evolution with $u_{\mathrm{S}}$. Black, red, and blue lines correspond to $u_{\mathrm{S}}=2,4,8$ and $\mathrm{\Gamma }/2\pi k_B{T}_{c0}=0.01$. (b) Evolution with $\mathrm{\Gamma }$. Black, red, and blue lines correspond to $\mathrm{\Gamma }/2\pi k_B{T}_{c0}=0.05,0.1,0.2$ and $u_{\mathrm{S}}=5$. For both plots, temperature is $T=0.01T_{c0}$.

Figure 10

Spin-up (full line) and spin-down (dotted line) densities of states for ferromagnetically aligned impurity spins. Black lines correspond to $\alpha >0$ and red ones to $\alpha <0$. (a)–(c) show the effect of increasing $u_{\mathrm{S}}=2,4,8$ with $\mathrm{\Gamma }/2\pi k_B{T}_{c0}=0.01$. (d)–(f) show the effect of increasing $\mathrm{\Gamma }/2\pi k_B{T}_{c0}=0.005,0.01,0.02$ with $u_S=5$. Temperature is $T=0.01T_{c0}$.

Figure 11

(a) Average spin per impurity $\overline{s}$ [see Eq. (37)] as a function of $u_{\mathrm{S}}$ for $\alpha >0$. The impurity density $\mathrm{\Gamma }/2\pi k_B{T}_{c0}$ is equal to $0.001$ (solid line), $0.005$ (dashed line) and $0.01$ (dotted line). (b) The corresponding Gibbs free energy difference.

Figure 12

(a) Difference in the average spin (per impurity) in the superconducting $\overline{s}$ and normal ${\overline{s}}_N$ phases as a function of impurity density for $\alpha >0$. The impurity strength, from right to left, is $u_{\mathrm{S}}=6,7,8,9$. (b) The corresponding Gibbs free energy difference. Thin vertical lines depict the critical density at which it crosses zero.

Figure 13

Diagrammatic representation of Eq. (A2).