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Reentrance and Magnetic-Field-Induced Superconductivity with Kondo Impurities: Bulk and Proximity Systems

Reentrant behavior is known to exist and magnetic-field-induced superconductivity has been predicted in superconductors with Kondo impurities. We present a simple framework for understanding these phenomena and generalize it to explain the long-standing puzzle of paramagnetic reentrance in thick proximity systems as due to small amounts of Kondo impurities.

Introduction. Magnetic impurities act as pair breakers in a superconductor since their interaction with the electrons breaks the time-reversal symmetry of the two members of each Cooper pair. Many works have dealt with the effect of magnetic impurities on superconductivity and the landmark for these is Abrikosov and Grokov's calculation of the depression of the critical temperature of a superconductor in the presence of magnetic impurities [1]. Maki [2] expanded this work to include any pairbreaking mechanism in a universal manner through the pair-breaking energy α. When the pair-breaking energy is of the order of the critical temperature superconductivity will be completely suppressed.
In 1970 Müller-Hartmann and Zittarz [3] expanded Abrikosov and Gorkov's work to include the Kondo effect. They predicted that due to the competition between superconductivity and spin-flip scattering off the Kondo impurities there will be, for certain impurity concentrations, not one but three critical temperatures, that is as the temperature is lowered the superconductor will go into a superconducting state at T C1 , out of it at a lower temperature T C2 and back into it at an even lower temperature T C3 . This happens because the Kondo impurities' spin-flip depairing is maximal around T K , the Kondo temperature, and falls off far from it [4]. This reentrant behavior was confirmed experimentally as far as the existence of T C2 , a reentrance into a normal phase, is concerned [5] but the existence of a third transition and the conditions under which it will be observable are still a matter of debate [6].
In 1989, in an often overlooked set of articles [7], Podmarkov and Sandalov predicted the existence of magnetic-field-induced superconductivity in such systems. They predicted that while for small temperatures and magnetic fields superconductivity will be suppressed by spin-flip scattering off the Kondo impurities the application of magnetic field can polarize the impurities thus reducing the spin-flip scattering and restoring superconductivity.
In this work we shall use a simple interpolation in order to expand Müller-Hartmann and Zittartz's work to include the effect of a finite magnetic field and thus account for Podmarkov and Sandalov's predictions. We shall also see why such behavior should appear in other superconducting systems and since the reentrant phenomenon in thick cylindrical proximity systems, which has intrigued theoreticians since its discovery in 1990 by Visani et al. [8], shows such behavior we will suggest that it arises due to a small concentration of Kondo impurities in the measured samples and estimate the amount of impurities needed to explain it. The Influence of Magnetic Fields on Superconductors with Kondo Impurities. The Abrikosov-Gorkov formula for the depression of the critical temperature due to magnetic impurities is ln with T C0 being the transition temperature without pair breaking, T C the transition temperature, α the pairbreaking energy, and Ψ the digamma function. The critical temperature is completely suppressed when α reaches a critical value α cr ≡ π/2 exp(−γ)T C0 = 0.882 T C0 with γ being the Euler constant. Müller-Hartmann and Zittartz [3], building on the works of Nagaoka [9] and Suhl [10], arrived at an approximate form for the temperature dependence of the pair-breaking energy for small concentrations of Kondo impurities with T K being the Kondo temperature, S being the spin of the magnetic impurities and α max = n s /2πν with n s being the density of the impurities and ν being the density of states at the Fermi level. At low temperatures, T T K , this formula, known as the Nagaoka-Suhl formula, ceases to be valid and more refined methods have to be used (e.g. [4]).
Müller-Hartmann and Zittartz obtained the impurity dependence of the superconductor's critical temperature by plugging α(T C ) from Eq. (2) into Eq. (1) and solving for T C thus resolving self-consistently the interplay between superconductivity and Kondo impurities' pair breaking. The resulting dependence shows a striking reentrant behavior for certain impurity concentrations, see Fig. 1, which was confirmed experimentally [5].  1) and (3) for TK = TC0/100, S = 1/2 and three values of the Zeeman energy gµBH = 0, 5TK and 10TK (in gray, dashed purple and dot-dashed yellow). The reentrant range for which there exists two or three transition temperatures is clearly visible and the magnetic field suppresses this reentrant behavior. The dotted gray line is the value of αmax for which the dependence of the transition temperatures TC1 and TC2 (shown in red and blue measured in units of TC0) on the magnetic field is plotted in Fig. 2 . The third transition temperature, TC3, which might exist at temperatures well below TK is beyond the scope of this work.
As long as the Zeeman interaction is the dominant effect of the magnetic field on the system we can account for the effect of a finite magnetic field by using the interpolation [11] α(T, H) = α max π 2 S(S + 1) with g being the Zeeman g-factor, µ B the Bohr magneton, and H the magnetic field. This equation interpolates between Eq. (2), which is the gµ B H T limit, and the gµ B H T limit where the Zeeman energy, gµ B H, replaces the temperature in that equation [12]. The magnetic field's dominant effect on the system is the Zeeman interaction when it is much smaller than the critical field of the superconductor and other pair-breaking effects [2] are small compared to α max .
There have been several works regarding the crossover between these limits [13] but this simple interpolation, which disregards the pair-breaking energy's dependence on the electrons' energy, allows for an easy analysis of the system's behavior while giving the same qualitative results as other, more sophisticated methods [7] and allowing for easy generalization to other, similar systems (see below).
We can now insert Eq. (3) into Eq. (1), solve for T C and obtain the dependence of the critical temperature on both impurity concentration and magnetic field as seen in Fig. 1 and the phase diagram of our system as can be seen in Fig. 2. For small temperatures and magnetic fields, T, gµ B H T K , the system stops superconducting and goes into a phase dominated by spin-flip scattering off the Kondo impurities. For even smaller temperatures and magnetic fields there might exist another region of superconductivity but since the Nagaoka-Suhl approximation is insufficient in that region we cannot contribute to the debate over its existence. However, we can mention that this extra region of superconductivity's extreme sensitivity to magnetic field might be a contributing factor in its experimental elusiveness.  1) and (3) for TK = TC0/100 and αmax = 1.1TC0. At T = TC1 0.74TC0 the system has a phase transition from a normal phase into a superconducting one which at H HC2 is almost field independent. At a low temperature, TC2 4TK = 0.04TC0, the system undergoes a second phase transition into a non-superconducting phase dominated by spin-flip scattering from the Kondo impurities which may be suppressed by a magnetic field of the order of gµBH = TC2. Thus the application of magnetic field in this phase can induce superconductivity. At even lower temperatures and magnetic fields (T, gµBH TK ) there might exist a transition back into a superconducting state.
In the Kondo phase, application of a magnetic field of the order of T K /µ B will cause the appearance of superconductivity by suppressing the Kondo effect. In this magnetic-field-induced superconductivity the Meissner effect will act to screen the magnetic field from the impurities but since magnetic field is necessary for supercon-ductivity a balance must be obtained and we shall therefore expect to see reduced Meissner screening which will increase at higher fields. There have already been several reports of systems with magnetic-field-induced superconductivity [14] including superconducting systems with magnetic impurities but this type of behavior with both reentrance and magnetic-field-induced superconductivity is yet to have been measured.
In the limit T K < T C T C0 we can obtain approximate analytic expressions for the reentrant temperature and the range of impurity concentrations leading to reentrance which will be valid not only for bulk superconductors but also for other superconducting systems. At and since T K T C0 we can use the temperaturedependent α from Eq. (3) and set α(T C ) = α cr . At zero field this gives us that as long as the maximal spin-flip induced pair breaking exceeds the critical pair breaking energy, that is there exists a transition from a superconducting phase into a non-superconducting phase when the pair breaking exceeds α cr at the temperature When T C2 ∼ T C0 this approximation breaks down but that is exactly the limit where we expect reentrance to cease to exist so we can use the condition T C2 < T C0 with the above expression to obtain an upper bound for the reentrant range α max < α cr π 2 S(S + 1) + ln 2 (T C0 /T K ) π 2 S(S + 1) .
Repeating this procedure at a finite magnetic field we find that and we see that as the field increases T C2 (H) decreases. When T C2 (H) goes below the sample's temperature the system will go out of the phase dominated by spin-flip scattering from the Kondo impurities and become superconducting, see Fig. 2.
The shape of Abrikosov and Gorkov's formula for T C (α), Eq. (1), does not come into play in Eqs. (6), (8) and (9), only the value of α cr does. Since the shape of spin-flip induced Kondo impurity pair breaking is universal [4] then these equations should hold for any system where a critical pair-breaking energy destroys superconductivity and we should expect to see the behavior described above qualitatively in such systems even when Eq. (1) is not valid. Examples of such systems are thin superconducting films, dirty superconductors and proximity induced superconductors (Kondo impurity induced reentrance has already been predicted for thin proximity systems [15]).
Paramagnetic Reentrance in Proximity Systems. We now turn our attention to a system that shows a very similar behavior to the one we have just described, a behavior which has baffled theoreticians for over twenty years. The system in question is the thick proximity cylinders measured by Visani et al. and subsequent experiments [8,[16][17][18] which show both reentrant and magnetic-field-induced superconducting behavior, the two hallmarks of the picture described above.
In 1990, Visani et al. [8] measured the magnetic response of relatively clean thick proximity cylinders of superconducting material (Nb, Ta) coated with normal metal (Ag, Cu). As the samples were cooled the superconductor started to show full diamagnetic screening, that is the Meissner effect, at T T CS ∼ 10K, the critical temperature of the bulk superconductor, as expected. Around T T A ∼ 1K, the Andreev temperature (see definition below), the samples showed, as expected, the proximity effect, that is the normal metal started to show the Meissner effect as well.
In some samples at a lower temperature (∼20 mK) a paramagnetic response came into play that tended to cancel the diamagnetic screening. This response increased gradually as the temperature was decreased until it saturated at T ∼ 1 mK [18]. In some of the samples the normal metal's diamagnetic response was entirely canceled out by the paramagnetic effect and in one sample the paramagnetic effect was reported to cancel out also the superconductor's signal as well. A small magnetic field, of the order of 20 Oe, was shown to be enough to destroy the paramagnetic signal and return the sample to a screening fraction close to that which would have been expected if it wasn't for the reentrant effect. These measurements, which have stimulated a very lively discussion among theoreticians [19][20][21][22][23][24][25], can be simply understood, in light of the theory presented above, as due to the existence of Kondo impurities in the normal metal.
For a thick proximity sample the relevant transition temperature is the Andreev temperature [26], T CS with d being the thickness of the normal metal and v F being the Fermi velocity. It is around this temperature that the normal side of the sample will start showing the Meissner effect and it plays the role of T C0 when applying our results to this system and so α cr ∼ T A . The field in which the supposed Kondo region breaks down is 20 Oe which correlates to a Zeeman energy of ∼ 5 mK, of the same order of the reentrant transition temperature as expected from Eq. (9). We can therefore estimate the Kondo temperature of the impurities in question to be ∼1 mK.
With these parameters we can now use Eq. (8) to assess the range of impurity concentrations needed to explain this phenomenon as 70 ppm < n s < 430 ppm , where we set S = 1/2, T C0 = T A = 500 mK, T K = 1 mK, ν Ag 0.26 atom −1 eV −1 and α cr = T A in order to appreciate the reentrant range. The sample in which the Meissner effect disappeared from the superconducting metal itself must have had a higher amount of impurities, around 2πνT CS ∼ 1000 ppm, such that it managed to affect not only its host metal but the adjacent superconductor as well. These amounts of impurities are very small and might arise from various sources during the samples' preparation and measurement.
Past research into this reentrant phenomenon has led to two main approaches. One approach by Fauchére et al. [20] attributed this phenomenon to repulsive net electron-electron interactions in the normal metal. While attracting much theoretical attention [21][22][23][24] recent experimental results [27,28] indicate that such repulsive interactions do not exist in the relevant metals. Another approach by Bruder and Imry [19] suggested the phenomenon might be due to the contribution of glancing states. Bruder and Imry took the magnetic field to be constant in space in their argument but calculations that treated the magnetic field self-consistently with the Meissner current have shown that the effect of these states is too small to explain this phenomenon [25,29].
The approach presented here explains, at least qualitatively, this phenomenon and can easily be verified experimentally either by characterizing the impurities in the original samples used or by measuring the magnetic response of thick proximity samples with controlled amounts of Kondo impurities. This approach does not rely on geometry and should apply equally to proximity cylinders and slabs.
In summary, we have presented a simple approach for understanding reentrant and magnetic-field-induced superconducting behavior in systems with Kondo impurities and have used it to show that the existence of such impurities can explain the paramagnetic reentrant effect which has been measured in proximty samples.
We thank W. Belzig and C. Bruder for helpful discussions. OEW acknowledges the support of the Albert Einstein Minerva Center for Theoretical Physics, Weizmann Institute of Science. This work was supported by the BMBF within the DIP program, BSF, ISF and its Converging Technologies Program.