Impurity effects on nano-structured dirty superconductors: violation of Anderson’s Theorem

It is shown that in nano-structured s-wave superconductor, transition temperature Tc depends on random impurity potentials. This means violation of the Anderson’s theorem, which states that non-magnetic impurity does not affect the Tc of s-wave superconductor. We determine the impurity effects on Tc for nano-structured superconductor, using the finite element method to solve the Bogoliubov-de Gennes equations under spatially a random impurity potential. We show that some impurity potentials increase Tc but other impurity potentials decrease Tc. We find that the superconductor with localized order parameter shows increased Tc, which is contrary to expectation. Our results show that a dirty superconductor does not always mean weak superconductivity.


Introduction: file preparation and submission
Superconducting transition temperature T c is important for applications of superconductors. Bulk superconductors have their own T c . However, for a nano-structured superconductor, T c depends on the size of the superconductor. Recently, Nishizaki et al found that a bulk nano-structured Nb, which is made by highpressure torsion and contains ultrafine grains [1][2][3][4], shows enhanced T c [5]. However, a bulk nano-structured V shows decrease of T c [6]. They discussed that Oxygen, which is included in V, decreases T c .
There are two mechanisms for size dependences of T c . The first mechanism is due to surface effects. Surface effects come from phonon softening [7,8]. This mechanism can explain enhancement of T c for some nano-size superconductors, such as Al [9,10].
The second mechanism comes from quantum size effects. Anderson showed that superconductivity ceases when the size of superconductor becomes small and the electron discrete energy gap becomes larger than a superconducting energy gap [11]. However, Parmenter found that smaller granular superconductor shows higher T c [12]. Also, investigating parity effects on T c for superconducting particles, von Delft showed that T c is enhanced just before superconductivity ceases with increasing the electron discrete energy gap [13]. Shanenko et al showed that T c ʼs of atomic scale nano-films oscillate with decreasing thickness, and found that the oscillation has a relation with the Fermi level and the Debye energy of phonons, using the Bogoliubov-de Gennes (BdG) equations [14]. Also, Suematsu et al showed that T c ʼs of nano-scaled superconducting square plates become higher than those of bulk superconductors, using the BdG equations [15]. In these two studies [14,15], the spatial dependence of superconducting order parameter is taken into account, and therefore T c is much enhanced than that of previous study [13]. In our previous study [16], we investigated that smaller superconductor shows higher T c , because density of superconducting order parameter increases with decreasing superconductor size.
Effects of impurities on nano-structured superconductors have not been well investigated. For a bulk superconductor, Anderson [11], and Abrikosov and Gor'kov [17] showed that non-magnetic impurities do not affect on s-wave superconductivity, because time-reversal pairing in the s-wave superconductovoty is not affected by non-magnetic impurities. Also, Abrikosov and Gor'kov [18] showed that magnetic impurities decrease T c , because magnetic impurities lift degeneracy of energy levels of electron pairs, which have up and Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. down spins, and break Cooper pairs. Xiang et al [19] showed that a weak impurity potential that is concentrated at the center of a square s-wave superconductor weakly decreases superconducting energy gap, using the BdG equations on a tight binding model. Therefore, how impurities affect T c enhancement in the nano-structured superconductor is still unclear. This is important for searching high T c superconductor in nano-size.
In this paper, we numerically investigate non-magnetic impurity effects on T c for a nano-structured superconductor. In order to take into account spatial variation of the superconducting order parameter, we use the BdG equations, with the finite element method (FEM) [15]. We introduce a random potential as nonmagnetic impurities.

Method
The BdG equations are given as follows, Here u and v are wave functions of particle and hole components of a quasi-particle of the superconductivity, respectively. And Hamiltonian is where μ is the chemical potential and (r V imp ) is a random potential from impurities. The order parameter Δ is determined by following the self-consistent equation, where E c is the cut-off energy of the BCS theory. μ is determined by the electron number conservation law as follows.
. In the FEM, we divide a system into triangular elements, and we define area coordinates z ( ) r i e = ( ) i 1, 2, 3 in e-th element [15]. We expand u, v, V imp , and Δ using the area coordinates.
where i is a node number and e is an element number. Then the BdG equations and the self-consistent equation become  In order to determine T c , we set temperature T tends to T c , and D  0. So we get the equation to determine T c ,  We investigate impurity potentials dependences of T c and mechanism of the impurity depenedences.

Results and discussions
We consider a superconducting rectangular (3.2ξ 0 ×6.4ξ 0 ) plate, where ξ 0 is the coherence length at T=0. Figure 1 shows the FEM model of the superconductor. We set boundary conditions that wave functions, u and v, become zero at edges of the superconductor. Therefore, the order parameter becomes zero at edges of the superconducting plate. We set x = k 3.0   random potential, we change an upper bound of the impurity potential V impMax . Figure 3 shows V impMax dependences of T c for impurity potentials (V1)-(V20). From this figure, we can see that some impurity potentials increase T c , and other impurity potentials decrease T c . To clarify origins of this variation of impurity potential dependence of T c , we investigate ( ) r V imp dependence of eigen-energies E n . Figure 4 shows V impMax dependences of eigen-energies for impurity potentials (a)(V4) and (b)(V5) in figure 3, which increase T c , and (c)(V1) and (d)(V3) in figure 3, which decrease T c . From these figures, we can see that each eigen-energy hardly changes. To magnify these changes of eigen-energies, we consider following energy differences, n n c n c impMax impMax Figure 5 (figure 6) shows ΔE n for each eigen-energy for the four impurity potentials, for which T c increases (descreases) the most, respectively. In order to clarify the effects of ΔE n on T c , we focus on the eigen-energies close to the Fermi energy (−0.2E n /E c 0.2), which have strong effects on T c . From figure 6, we can see that the electron eigen-energies of the superconductor, which shows lower T c , become apart from the Fermi energy.
In the BCS theory, there is a relation between eigen-energies  m = -E k k and T c as follows [20],  where g is an interaction constant, , and μ is a chemical potential. Then, we see that smaller x k has higher contribution to the superconductor. Therefore T c increases, when x | | k becomes smaller. However, in above cases, relation between T c and eigen-energies is opposite.   In order to understand these behaviors, we investigate distributions of the order parameter. Figure 7 shows distributions of the order parameter at V impMax =0.00 (0) and 0.50 (for V1, V3, V4 and V5). From this result, each impurity potential distorts the distribution of the order parameter. In previous study [16], we discussed that the smaller superconductor shows higher T c because of the confinement of the order parameter. So, the distortion of distribution of order parameter may lead to enhancement of T c . In order to evaluate the distortion, we calculate degree of the localization of the order parameter α as follows, where áñ is the average over all nodes in the FEM. Figure 8 shows V impMax dependences of α, for four impurity potentials, which increase T c the best (1) (the worst (2) ). From this result, we can see that the order parameter distribution for the impurity potential that shows higher T c , is more localized. This is because localized order parameter shows larger order parameter density, and leads to higher T c .

Conclusion
In summary, in order to investigate the impurity effects on T c , we have solved the Bogoliubov-de Gennes equations for superconducting rectangular (3.2ξ 0 ×6.4ξ 0 ) plates, with the FEM. We have introduced a random potential as non-magnetic impurities. We have obtained T c for twenty random impurity potentials. We have found that some impurity potentials increase T c , but other impurity potentials decrease T c . To investigate this difference, we have determined eigen-energies and distribution of order parameters. We have found that some of eigen-energies become apart from the Fermi energy for impurity potentials that show higher T c . However, we have found that the distribution of the order parameter is distorted by the impurity potential and the localization of the order parameter leads to higher T c . We may expect impurities or localization of the order parameter lead to weakening of the superconductivity and lowering T c . Our results are contrary to this expectation. Therefore, a dirty superconductor does not always mean weak superconductivity.