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Breakdown by a magnetic field of the superconducting fluctuations in the normal state in Pb

arXiv:cond-mat/0312647v1 [cond-mat.supr-con] 26 Dec 2003

Breakdown by a magnetic ?eld of the superconducting ?uctuations in the normal state in Pb1?xInx alloys

F?lix Soto, Carlos Carballeira, Jes? s Mosqueira, e u Manuel V. Ramallo, Mauricio Ruibal, Jos? A. Veira, F?lix Vidal e e Laboratorio de Baixas Temperaturas e Superconductividade,# Facultade de F? ?sica, Universidade de Santiago de Compostela, Santiago de Compostela, E15782 Spain.
# Unidad Asociada al Instituto de Ciencias de Materiales de Madrid, CSIC, Spain.

Abstract The e?ects induced on the magnetization by coherent ?uctuating Cooper pairs in the normal state have been measured in Pb1?x Inx alloys up to high magnetic ?elds, of amplitudes above HC2 (0), the upper critical ?eld extrapolated to T =0 K. Our results show that in dirty alloys these superconducting ?uctuation e?ects are, in the entire H ? T phase diagram above HC2 (T ), independent of the amount of impurities and that they vanish when H ? 1.1HC2 (0). These striking results are consistent with a phenomenological estimate that takes into account the limits imposed by the uncertainty principle to the shrinkage, when H increases, of the superconducting wave function.


After the pioneering theoretical proposals in the sixties,[1, 2, 3] it was soon realized that in addition to their intrinsic interest the ?uctuating Cooper pairs created in the normal state by the unavoidable thermal agitation provide a useful tool in studying the superconducting transition.[4] Since then, the superconducting ?uctuations (SF) in the normal state have been extensively studied in low- and high-TC superconductors, and at present many of their main aspects are well understood.[4, 5] However, their behaviour under strong magnetic ?elds [of the order of HC2 (0), the upper critical ?eld amplitude extrapolated to T =0 K] is still an open problem. In fact, a central question remains unaddressed, both experimentally and theoretically, until now: Up to what magnetic ?eld amplitudes may the SF in the normal state survive? The interest of this question is enhanced by the fact that it may concern the general behaviour of the Cooper pairs above and below the superconducting transition in the presence of strong “antisymmetric” perturbations (which “act with opposite sign on the two members of a Cooper pair”[6]) and it could then have implications well beyond the SF issue, including the coexistence of magnetism and superconductivity or the interplay between normal-state properties and high temperature superconductivity.[5, 7] In this Letter, we attempt to answer the question stated above by presenting measurements of ?M(T, H), the magnetization induced by SF in the normal state,[8] in Pb1?x Inx alloys with 0≤ x ≤0.45, and up to ?eld amplitudes well above HC2 (0). Our experiments show that the intrinsic SF (not a?ected by dynamic and non-local electrodynamic e?ects) are immune to the presence of non-magnetic impurities and that, independently of the superconductors’ dirtiness, ?M(T, H) vanishes when H becomes close to 1.1HC2 (0). This striking behaviour, not predicted by the existing phenomenological[3, 4, 5, 9, 10, 11, 12] or microscopic[13, 14, 15] approaches for ?M(T, H), is crudely explained here by taking into account the limits imposed by the uncertainty principle to the shrinkage of the superconducting wave function when the magnetic ?eld increases. This will extend to high ?elds our previous proposals for the SF at high reduced temperatures[16], in spite that the magnetic ?eld is an antisymmetric perturbation. Among the available low- and high-TC superconductors, we choose the Pb-In alloys to study the high-?eld behaviour of the SF for four main reasons: i) Its entire H ? T phase diagram is, even for H ? HC2 (0), easily accessible with the existing high resolution, SQUID based, magnetometers. ii) By changing the In concentration, it is possible to cover both type I and type II superconductors and also the range from the clean to the dirty limits (see Table I). This is a crucial advantage because it has allowed us to separate the “universal” magnetic ?eld e?ects on the SF from those associated with the dynamic and non-local electrodynamic e?ects, these last being strongly material-dependent[4, 12, 13, 14, 15, 17, 18]. iii) It is also possible to obtain alloys with high stoichiometric quality. This is another crucial advantage, because it minimizes the spurious magnetization rounding associated with TC inhomogeneities, 2

that otherwise would be entangled with the intrinsic rounding due to the SF. iv) The normal-state magnetic susceptibility of these samples is almost independent of T and H up to, at least, 5TC0 and 5HC2 (0). This allows a very reliable obtainment of the background magnetization around TC (H) by linear extrapolation of the as-measured M(T )H or M(H)T well above TC (H). Moreover, by electrochemically coating with a normal metal (i.e., Au or Cu), it is easy to eliminate in Pb-In alloys the surface superconductivity between HC2 (T ) and HC3 (T ), which otherwise would complicate the analysis above HC2 (T ) for T < TC0 , the zero-?eld critical temperature. The synthesis of this type of alloys and some of their general characteristics (including the values of the critical ?elds) have been already described in various earlier works which address other phenomena in these materials.[19] Other experimental details will be published elsewhere. The presence of ?uctuating Cooper pairs above TC (H) produces a rounding of the as-measured susceptibility versus magnetic ?eld curves, as illustrated in Fig. 1. This example also shows that this rounding is progressively reduced as the applied ?eld increases and it completely vanishes when the reduced ?eld, h ≡ H/HC2 (0), becomes close to 1.1. The ?nite-?eld e?ects may be described quantitatively through the ?M(h)ε curves, as the one presented in Fig. 2, which corresponds to a temperature above TC0 [i.e., ? > 0, where ε ≡ ln(T /TC0 ) is the reduced temperature]. As may be seen in this ?gure, in the low-?eld region ?M(h)ε agrees with the Prange predictions[9] and provides then a direct indication that in this dirty superconductor ?M(h, ε) is not appreciably a?ected by non-local electrodynamic e?ects. The results of Fig. 2 also > > show that when h ? 0.2, ?M(h)ε begins to decrease and for h ? 1.1 the ?uctuationinduced diamagnetism vanishes. The upper horizontal scale in Fig. 2 illustrates that at high √ these ?1/2 ?elds the GL coherence length, which for h ? |ε| behaves as[4] ξ(h)ε ? 2ξ(0)h , becomes of the order of ξ0 , the actual (or Pippard) superconducting coherence length at T =0 K. This scale was obtained by using ξ(0) = 0.74ξ0, which is still a good approximation in the dirty limit.[4] When compared with our previous results at low ?eld amplitudes but at high reduced temperatures[11, 12, 16, 18], this last ?nding already suggests that in spite of the antisymmetric character of the magnetic ?eld the vanishing of ?M(h)ε may also be due to the limits imposed by the uncertainty principle to the shrinkage of the superconducting wave function.[16] When the shrinkage of the superconducting wave function is due to a magnetic ?eld, this leads to: ξ(h)ε
> ?

ξ0 ,


P P where ξ0 for each alloy is related to ξ0 b for pure Pb by[4] ξ0 ? (ξ0 b?)1/2 . The above C inequality directly leads to a critical reduced-?eld, h , given by hC = 2(ξ(0)/ξ0)2 , above which all the SF vanish. By using again ξ(0) = 0.74ξ0, we obtain hC ? 1.1, in excellent agreement with the results of Figs. 1 and 2. As ξ(0)/ξ0 is almost materialindependent,[6, 20] Eq. (1) predicts that the above value of hC will be “universal”. This striking prediction is con?rmed at a quantitative level by the experimental results at TC0 for all the samples studied in this work and summarized in Fig. 3: Independently


of the superconductor dirtiness and also of their type I or type II character, the SF vanish when h ? 1.1. Another central result shown by Fig. 3 is that all the data for the di?erent dirty alloys collapse on the same curve: this provides a direct experimental demonstration that in the absence of non-local electrodynamic e?ects, the SF are not a?ected by impurities. The data summarized in Fig. 3 are also particularly well adapted to probe in the high-?eld region below hC the applicability of the existing theoretical approaches for ?M(h)ε : in the absence of material-dependent e?ects (as dynamic or non-local effects), the Gaussian-Ginzburg-Landau (GGL) approach without any cuto? (as ?rst proposed by Prange[9]) predicts that at TC0 all the data must collapse on the same, ?eld-independent curve (dot-dashed line in Fig. 3). The failure of the Prange approach when h increases, also shown by the results of Fig. 2, was already observed by Gollub and coworkers in their pioneering measurements[4, 17], which did not cover the high-?eld regime (they extended only up to h ? 0.6). Most of these last measurements were done in clean low-TC superconductors and they were entangled with the non-local e?ects that already at low ?elds reduce the ?M(h)TC0 /H 1/2 TC0 amplitude well below the theoretical value. We have also observed this reduction in pure Pb, as illustrated in Fig. 3. The introduction in the GGL Prange approach of di?erent cuto?s, which consider the short-wavelength ?uctuation e?ects at high reduced temperatures and ?elds[10, 11, 12] but do not take into account Eq. (1) (see also below), will lead to a decrease of ?M(h)ε when h increases. However, as illustrated in Fig. 2 for the case of the kinetic-energy or momentum cuto?[11], the resulting ?M(h)ε (dotted line) still > does not account for the experimental decrease with h ? 0.2. The results in Figs. 2 and 3 also show that the existing microscopic approaches for ?M[13, 14, 15], which do not take into account Eq. (1), also fail to account for our experimental results at high ?elds. The dashed lines correspond to the approach proposed by Maki and Takayama[14] and by Klemm, Beasley and Luther[15] (MT-KBL theory) for the dirty limit. This approach generalizes for dirty superconductors the pioneering calculations of Lee and Payne.[13] We have estimated these curves without any adjustable parameter (by using the values of Table I). As may be seen in Figs. 2 and 3, the agreement with the experimental data at low and moderate ?eld amplitudes (up to h ? 0.2) is excellent. However, there is strong disagreement at high ?elds. Could the theoretical approaches summarized above explain the high-?eld behaviour of ?M if Eq. (1) is taken into account? A full answer to this question could be obtained only on the grounds of the microscopic approaches [13, 14, 15, 21]. This task is out of the scope of our present Letter. However, a crude but probably the simplest way to obtain a qualitative answer to this question is to introduce Eq. (1) in the GGL approach, even though far from TC0 or at high ?elds this approach may be not formally applicable.[4, 16] In fact, in this way we may probe if the limitations imposed by the 4

uncertainty principle is the dominant e?ect on the SF at high magnetic ?elds, in spite of the antisymmetric character of the magnetic ?eld. To do that, we ?rst note that in terms of the “total energy” Enkz of the ?uctuating modes of Landau level index n = 0, 1, ... and wave vector parallel to the ?eld kz , this constraint may be written as: where the energies are expressed in units of h2 /2m? ξ 2 (0), and h and m? are, re? ? spectively, the Planck constant and the e?ective mass of the Cooper pairs. To take into account Eq. (2) at high ?elds, we introduce a total-energy-dependent weighting function, W (Enkz ), pondering the contribution of each ?uctuating mode in the free-energy statistical sum. This procedure is similar to the one proposed by Patton, Ambegaokar and Wilkins (PAW).[10] However, PAW’s approach does not take into account the limits imposed by the uncertainty principle to the shrinkage of the superconducting wave function. In fact, PAW’s calculations are equivalent to the choice WPAW (Enkz ) = ln[1 + exp(?Enkz /h0 )]/ ln Enkz , which does not consider the inequality (2). Here the reduced ?eld h0 corresponds to the maximum of the ?M(h)TC0 curve, < < and therefore in our Pb-In alloys it will be 0.2 ? h0 ? 0.25. In our present study, to reproduce the rapid fall-o? of the SF expected when the inequality (2) begins to be violated, we introduce an additional prefactor to the penalization function, using W (Enkz ) = WPAW (Enkz )(1 + exp[(Enkz ? (ξ(0)/ξ0)2 ? δ)/δ])?1 . This additional prefactor has the form of a Fermi-Dirac distribution function, presenting a step-like decay starting at energies ? (ξ(0)/ξ0)2 and with δ as typical half-width. By repeating the standard GGL calculations for ?M(T, H) in isotropic 3D superconductors above the transition[9, 10, 11, 12] but now including the weighting function W (Enkz ), we obtain: ?M = kB T ∞ ∞ ? [W (Enkz )h ln Enkz ] , dkz πφ0 0 n=0 ?h (3)
2 Enkz ≡ ? + (2n + 1)h + ξ 2(0)kz < ?

(ξ(0)/ξ0)2 ,


where kB is the Boltzmann constant and φ0 the ?ux quantum. This formula may be numerically computed thanks to the rapid decay of W (Enkz ) as n or kz increase. In Figs. 2 and 3 we plot the results of that evaluation as a solid line. In making these computations we have used the values of ξ(0) and TC0 given in Table I and (ξ(0)/ξ0)2 = 0.55. We also used h0 = 0.22 and δ = 0.2, which are the values giving a better agreement with our experimental results. As may be seen in Figs. 2 and 3, this agreement is excellent in all the studied h- and ε-range, and it also includes the vanishing of ?M(h)ε at hC ? 1.1. Another well-known procedure to shrink the collective wave function of the ?uctuating Cooper pairs is, as noted before, to increase the temperature well above TC0 in the absence of or under low magnetic ?elds. In fact, we have recently studied in detail the behaviour of the SF in this case in di?erent low-TC and high-TC superconductors through measurements of the paraconductivity or of ?M(ε)h in the low ?eld limit (h/ε ? 1)[12, 16, 18, 22, 23]. When both types of experiments are compared, they further suggest a similar pair breaking mechanism when ε approaches 5

εC ? 0.55 or h approaches hC ? 1.1, despite the magnetic ?eld being an “antisymmetric” perturbation[6]. These results also demonstrate that in the absence of non-local electrodynamic e?ects (which a?ect only the clean and low-κ Pb), the SF in the normal state are una?ected by impurities (as sown in Fig. 3). This immunity of the SF against both impurities and (antisymmetric) magnetic ?eld perturbations, which is also con?rmed by measurements of the paraconductivity at high reduced temperatures in high-temperature cuprate superconductors (HTSC) with di?erent doping levels[16, 23], is to some extent similar to the stability shown by the ?ux quantization below HC2 (T ). Our results suggest then the existence of an unexpected “quantum protectorate”[24] for the coherent ?uctuating Cooper pairs above HC2 (T ), that is only broken by the limits imposed by the uncertainty principle. Its extension in the h ? t phase diagram is represented in Fig. 4, which corresponds to the Pb-45 at. % In alloy. In conclusion, we believe that the measured (h-t) phase diagram shown in Fig. 4 is representative of all type II superconductors una?ected by non-local electrodynamic effects, as the dirty low-TC superconductors. In principle, this (h-t) phase diagram could also apply to HTSC, which are extremely type II superconductors also una?ected by non-local e?ects. However, this last suggestion obviously needs further experimental veri?cation, in particular in view of the recent observation of anomalous thermomagnetic e?ects well above TC (H) in some underdoped cuprates.[25] Other open questions which deserve further studies are the SF in presence of magnetic order (our present results also suggest the robustness of the SF against this antisymmetric perturbation) or the relationships between our crude theoretical analysis and the available microscopic approaches for the SF around TC (H) [13, 14, 15, 21].

We thank Prof. J.B. Goodenough for his careful reading of our manuscript and helpful remarks. We acknowledge the ?nancial support from the ESF “Vortex” Program, the CICYT, Spain, under grants no. MAT2001-3272 and MAT2001-3053, the Xunta de Galicia under grant PGIDT01PXI20609PR, and by Uni?n Fenosa under o grant 220/0085-2002.


[1] D.J. Thouless, Ann. Phys. NY 10, 553 (1960). [2] L.G. Aslamazov, A.I. Larkin, Phys. Lett. 26A, 238 (1968). [3] H. Schmidt, Z. Phys. 216, 336 (1968); A. Schmid, Phys. Rev. 180, 527 (1969). [4] J. Skocpol, M. Tinkham, Repts. Prog. Phys. 38, 1049 (1975); M. Tinkham, Introduction to Superconductivity (McGraw-Hill, New York, 1996), Chaps. 4 and 8. [5] See, e.g., The gap symmetry and ?uctuations in high-Tc superconductors, ed. J. Bok, G. Deutscher, D. Pavuna, and S.A. Wolf (NATO-ASI series, Plenum), 1998. See also, A.A. Varlamov, G. Balestrino, E. Milani, D.V. Livanov, Adv. Phys. 48, 655 (1999). [6] See, e.g., P.G. de Gennes, Superconductivity of Metals and Alloys (W.A. Benjamin, New York) 1966, Chap. 8. [7] See, e.g., J.B. Goodenough, J. Phys.: Cond. Matter 15, R257 (2003); J. Orenstein and A.J. Millis, Science 288, 468 (2000). [8] The ?uctuation-induced magnetization is de?ned as ?M(T, H) ≡ M(T, H) ? MB (T, H), where M(T, H) is the as-measured magnetization and MB (T, H) the so-called background or bare magnetization [which may be approximated by extrapolating the magnetization measured either at T ? TC0 or H ? HC2 (0)]. [9] R.E. Prange, Phys. Rev. B 1, 2349 (1970). [10] B.R. Patton, V. Ambegaokar, J.W. Wilkins, Solid State Comm. 7, 1287 (1969); B.R. Patton and J.W. Wilkins, Phys. Rev. B 6, 4349 (1972). [11] C. Carballeira, J. Mosqueira, A. Revcolevschi, F. Vidal, Phys. Rev. Lett. 84, 3157 (2000); Physica C 384, 185 (2003). [12] J. Mosqueira, C. Carballeira, F. Vidal, Phys. Rev. Lett. 87, 167009 (2001). [13] P.A. Lee, M.G. Payne, Phys. Rev. Lett. 26, 1537 (1971); see also, J. Kurkij¨rvi, a V. Ambegaokar, G. Eilenberger, Phys. Rev. B 5, 868 (1972). [14] K. Maki, H. Takayama, J. Low Temp. Phys. 5, 313 (1971); K. Maki, Phys. Rev. Lett. 30, 648 (1973). [15] R.A. Klemm, M.R. Beasley, A. Luther, Phys. Rev. B 8, 5072 (1973). [16] F. Vidal et al., Europhys. Lett. 59, 754 (2002), and references therein.


[17] J.P. Gollub, M.R. Beasley, R. Callarotti, M. Tinkham, Phys. Rev. B 7, 3034 (1973). [18] J. Mosqueira et al., J. Phys.: Cond. Matter 15, 3283 (2003). [19] See, e.g., F. Vidal, Phys. Rev. B 8, 1982 (1973), and references therein. [20] See, e.g., N.R. Werthamer, in Superconductivity, ed. R.D. Parks, (Marcel Dekker, New York, 1969), chap. 6, p. 338. [21] Z. Te?anovi? et al., Phys. Rev. Lett. 69, 3563 (1992); Z. Te?anovi? and A.V. s c s c Andreev, Phys. Rev. B 49, 4064 (1994). [22] J. Mosqueira et al., Europhys. Lett. 53, 632 (2001); C. Carballeira et al., Phys. Rev. B 63, 144515 (2001). [23] J. Vi? a et al., Phys. Rev. B 65, 212509 (2002); S. R. Curr?s et al., Phys. Rev. n a B 68, 094501 (2003). [24] R. B. Laughlin, D. Pines, Proc. Natl. Acad. Sci. U.S.A. 97, 28 (2000); P. W. Anderson, Science 288, 480 (2000). [25] See, e.g., Z.A. Xu et al., Nature (London) 406, 486 (2000); Y. Wang, Phys. Rev. Lett. 88, 257003 (2002); H. Kontani, Phys. Rev. Lett. 89, 237003 (2002).


Table 1: Main parameters of the Pb-In alloys studied in this work. TC0 was determined from the ?eld-cooled M(T )H curve under a magnetic ?eld of 0.5 mT. HC2 (0) and the Ginzburg-Landau (GL) parameter, κ, were obtained from the reversible M(H)T curves in the mixed state. The GL coherence length amplitude, ξ(0), follows from ξ 2 (0) = φ0 /2π?0HC2 (0). The electronic mean free path, ?, was obtained from meaP surements of the low-temperature residual resistivity. ξ0 b /? is the dirtiness parameP ter, where ξ0 b ? 920 ? is Pippard’s coherence length of pure lead [determined from A Pb Pb ξ0 = 1.35ξ (0)]. In at. % 0 5 8 18 30 45 TC0 ?0 HC2 (0) ξ(0) κ ?) (K) (T) (A 7.16 0.01a 680 0.30a 7.06 0.29 340 1.3 6.99 0.49 260 2.1 6.85 0.85 200 3.4 6.75 1.00 180 4.2 6.43 1.19 170 5.5
P ? ξ0 b/? (?) A ? 50000 ? 0 200 2.7 130 7.1 67 13.7 57 16.1 42 21.9


Extrapolated from the HC2 (0) and κ values of the Pb-In alloys following the Gor’kov theory.[20]


Figure 1: An example, corresponding to the Pb0.55 In0.45 alloy, of the as-measured magnetic susceptibility versus applied magnetic ?eld at constant temperature below TC0 . The solid line is the background susceptibility, obtained by a linear ?t in a region far from the superconducting transition, in this example from 2 Tesla to 3 Tesla [i.e., from ? 5HC2 (T ) to ? 7HC2 (T )].


Figure 2: An example, corresponding to the Pb0.55 In0.45 alloy, of the reduced-magnetic-?eld dependence of the ?uctuation-induced magnetization, at ε = 0.06. > < The upper scale shows that for h ? 0.2, ξ(h)ε ? 2ξ0 and it already suggests that in this high-?eld region the SF behaviour is dominated by the uncertainity principle. The curves correspond to di?erent theoretical approches, as explained in the main text.


Figure 3: ?M(h)/H 1/2 TC0 versus h at TC0 for all the compounds studied here. The upper scale illustrates that for all the compounds the ?uctuation e?ects at TC0 vanish, sharply in this scale, when ξ(h)ε ? ξ0 (which corresponds to h ? 1.1). The data for pure Pb are strongly a?ected by non-local electrodynamic e?ects which, even at low ?elds, decrease the ?M amplitude. The lines correspond to di?erent phenomenological and microscopic approaches, as explained in the main text.


Figure 4: Measured h ? t phase diagram, including the SF above hC2 (t), for the Pb-45 at. % In alloy. The color scale represents the ?uctuation-induced magnetization (scaled by HT ) in units of the Schmid amplitude, AS ≡ π?0 kB ξ(0)/6φ2. 0


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