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Breakdown of the superconducting fluctuations by a magnetic field
ARTICLE IN PRESS
Physica B 378–380 (2006) 393–394 www.elsevier.com/locate/physb
Breakdown of the superconducting fluctuations by a magnetic field F. Soto, C. Carballeira, J. Mosqueira, M.V. Ramallo, M. Ruibal, J.A. Veira, F. Vidal Laboratorio de Baixas Temperaturas e Superconductividade (LBTS), Departamento de Fı´sica da Materia Condensada, Universidade de Santiago de Compostela E15782, Spain
Abstract We have measured the diamagnetism induced in the normal state by the superconducting fluctuations in Pb1x Inx alloys. The experiments were performed with external magnetic fields up to well above H C2 ð0Þ, the upper critical magnetic field extrapolated to T ¼ 0 K. The results show that in dirty alloys the superconducting fluctuation effects are independent of the amount of impurities in all the measured H–T phase diagram above H C2 ðTÞ. Moreover, for all the studied alloys these fluctuation effects vanish when the applied field approaches the corresponding 1:1H C2 ð0Þ. At these high fields the superconducting coherence length becomes of the order of its minimum value, the one at T ¼ 0 K. So, we propose that these striking results could be due to the limits imposed by the uncertainty principle to the shrinkage, when H increases, of the superconducting wave function. r 2006 Elsevier B.V. All rights reserved. PACS: 74.40.+k; 74.25.Ha; 74.70.Ad Keywords: Superconductivity; Magnetic properties; Fluctuations
The normal state of a superconductor may be appreciably affected near the superconducting transition by the presence of Cooper pairs created by the thermal agitation energy [1]. These superconducting fluctuations (SCF) in the normal state were intensively studied in low-T C as well in high-T C superconductors and, nowadays, many of their aspects are already well understood [2]. However, its behavior in presence of a high magnetic field, of the order of the upper critical magnetic field extrapolated to T ¼ 0 K, H C2 ð0Þ, is still an open problem. In this work, we study the SCF at high applied magnetic fields in Pb1x Inx metallic alloys with 0oxo0:45. This family of low-T C superconductors was chosen because their relatively high T C - and low m0 H C2 ð0Þ-values (see below) allowed us to study a great part the superconducting phase diagram with conventional magnetic detection systems (our measurements were performed with an SQUID magnetometer, model MPMS). Moreover, by varying the In concentration it is possible to study the Corresponding author. Tel.: +34 981 563100x14023; fax: +34 981 520 676. E-mail address: [email protected] (J. Mosqueira).
0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.172
SCF behavior for different values of the Ginzburg–Landau (GL) parameter, k, and for different degree of dirtyness (characterized by the ratio x0 =‘, where ‘ is the electronic mean free path and x0 the Cooper pairs size). Details on the samples fabrication and characterization may be found in Ref. [3]. The superconducting parameters related to the SCF analysis are presented in Table 1. The thermal fluctuation effects above T C are studied through the fluctuation-induced magnetic susceptibility which is defined as Dw w wB , where w is the as-measured magnetic susceptibility and wB is the normal-state or background contribution, which is almost T- and H-independent. In Fig. 1(a) we present the reducedmagnetic-field h H=H C2 ð0Þ dependence of Dw for each Pb–In alloy, corresponding to a reduced temperature lnðT=T C Þ ¼ 0:06. To make easier the comparison with the existing theories for Dw (see below) these data are normalized by Txð0Þ, where xð0Þ is the corresponding GL coherence length amplitude (see Table 1). As may be clearly seen, the data for the different alloys collapse into a unique curve, which indicates that the SCF effects on Dw are independent of the amount non-magnetic impurities, and also of the value of k. At very low reduced magnetic
ARTICLE IN PRESS F. Soto et al. / Physica B 378–380 (2006) 393–394
394
Table 1 Superconducting parameters of the Pb–In alloys studied. T C was determined from field-cooled MðTÞH measurements at m0 H ¼ 0:5 mT In (at%)
T C (K)
k
m0 H C2 ð0Þ (T)
xð0Þ (A˚)
x0 =‘
5 8 18 30 45
7.06 6.99 6.85 6.75 6.43
1.32 2.07 3.38 4.17 5.54
0.286 0.487 0.856 0.993 1.19
339 260 196 182 166
2.7 7.1 14 16 22
H C2 ð0Þ and k were obtained from MðHÞT measurements in the mixed state. xð0Þ follows from x2 ð0Þ ¼ f0 =2pm0 H C2 ð0Þ, x0 was obtained from the xð0Þ values through the Gorkov theory, and ‘ from measurements of the residual resistivity just above T C .
The SCF vanishing at high- values was interpreted in terms of the limit imposed by the uncertainty principle to the shrinkage of the superconducting wavefunction, which may be expressed as xðÞ4x0 , where xðÞ 0:74 x0 1=2 is the GL superconducting coherence length. From those expressions, the -value above which the SCF should disappear is C ¼ 0:55. This quantum confinement effect may be formally introduced in the GL theory in the h5 limit through a total-energy cutoff [4], leading to the expression " # C C 1=2 Dw m0 kB atanð Þ1=2 atanð C Þ pffiffi pffiffiffiffiffi ¼ . (1) Txð0Þ 3f20 C
Fig. 1. (a) h- and (b) -dependence of Dw=Txð0Þ for the Pb–In alloys studied. In both figures, the solid line is the result of the GL theory in the limit h5, modified to include the quantum confinement effects at high [Eq. (1)]. In constructing this figure we used the T C , H C2 ð0Þ and xð0Þ values of Table 1.
fields (i.e., for ho), Dw=Txð0Þ is h-independent, and in excellent agreement with the prediction of the GL theory in the Gaussian approximation (see below) [4]. However, for higher h-values the effect of the thermal fluctuations is strongly reduced, to end up by disappearing at h1. This effect is analogous to the SCF vanishing observed at high reduced temperatures in the low-h limit [see Fig. 1(b)], which seems to be present in any kind of superconductor, including the high-T C cuprates [5].
This equation is in excellent agreement with the data of Fig. 1 in the h5 limit. In particular, it reproduces the Dw vanishing at 0:55. This quantum confinement effect may be also invoked to explain the SCF vanishing at high h.pIn ffiffiffi this case the GL coherence length is given by xðhÞ 2 x0 h1=2 (see, e.g., Ref. [2]), and the limit imposed by the uncertainty principle to the shrinkage of the superconducting wavefunction xðhÞox0 implies that the SCF should disappear above hC 1:1, in excellent agreement with the data of Fig. 1(a). In conclusion, the SCF vanishing observed for HH C2 ð0Þ could be a confirmation of the relevance of quantum confinement effects on the SCF when the superconducting wavefunction is shrunk to lengths of the order of the Cooper pairs size. A theory for Dw for these high H values including quantum confinement effects would be desirable to settle these results. This work was supported by the Spanish MEC (MAT2004-04364) and by Unio´n Fenosa (contract 2200085-2002).
References [1] M. Tinkham, Introduction to Superconductivity, MCGraw-Hill, New York, 1996. [2] J. Skocpol, M. Tinkham, Rep. Prog. Phys. 38 (1975) 1049. [3] F. Soto, et al., Phys. Rev. B 70 (2004) 060501. [4] J. Mosqueira, C. Carballeira, F. Vidal, Phys. Rev. Lett. 87 (2001) 167009. [5] F. Vidal, et al., Europhys. Lett. 59 (2002) 754.
Physica B 378–380 (2006) 393–394 www.elsevier.com/locate/physb
Breakdown of the superconducting fluctuations by a magnetic field F. Soto, C. Carballeira, J. Mosqueira, M.V. Ramallo, M. Ruibal, J.A. Veira, F. Vidal Laboratorio de Baixas Temperaturas e Superconductividade (LBTS), Departamento de Fı´sica da Materia Condensada, Universidade de Santiago de Compostela E15782, Spain
Abstract We have measured the diamagnetism induced in the normal state by the superconducting fluctuations in Pb1x Inx alloys. The experiments were performed with external magnetic fields up to well above H C2 ð0Þ, the upper critical magnetic field extrapolated to T ¼ 0 K. The results show that in dirty alloys the superconducting fluctuation effects are independent of the amount of impurities in all the measured H–T phase diagram above H C2 ðTÞ. Moreover, for all the studied alloys these fluctuation effects vanish when the applied field approaches the corresponding 1:1H C2 ð0Þ. At these high fields the superconducting coherence length becomes of the order of its minimum value, the one at T ¼ 0 K. So, we propose that these striking results could be due to the limits imposed by the uncertainty principle to the shrinkage, when H increases, of the superconducting wave function. r 2006 Elsevier B.V. All rights reserved. PACS: 74.40.+k; 74.25.Ha; 74.70.Ad Keywords: Superconductivity; Magnetic properties; Fluctuations
The normal state of a superconductor may be appreciably affected near the superconducting transition by the presence of Cooper pairs created by the thermal agitation energy [1]. These superconducting fluctuations (SCF) in the normal state were intensively studied in low-T C as well in high-T C superconductors and, nowadays, many of their aspects are already well understood [2]. However, its behavior in presence of a high magnetic field, of the order of the upper critical magnetic field extrapolated to T ¼ 0 K, H C2 ð0Þ, is still an open problem. In this work, we study the SCF at high applied magnetic fields in Pb1x Inx metallic alloys with 0oxo0:45. This family of low-T C superconductors was chosen because their relatively high T C - and low m0 H C2 ð0Þ-values (see below) allowed us to study a great part the superconducting phase diagram with conventional magnetic detection systems (our measurements were performed with an SQUID magnetometer, model MPMS). Moreover, by varying the In concentration it is possible to study the Corresponding author. Tel.: +34 981 563100x14023; fax: +34 981 520 676. E-mail address: [email protected] (J. Mosqueira).
0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.172
SCF behavior for different values of the Ginzburg–Landau (GL) parameter, k, and for different degree of dirtyness (characterized by the ratio x0 =‘, where ‘ is the electronic mean free path and x0 the Cooper pairs size). Details on the samples fabrication and characterization may be found in Ref. [3]. The superconducting parameters related to the SCF analysis are presented in Table 1. The thermal fluctuation effects above T C are studied through the fluctuation-induced magnetic susceptibility which is defined as Dw w wB , where w is the as-measured magnetic susceptibility and wB is the normal-state or background contribution, which is almost T- and H-independent. In Fig. 1(a) we present the reducedmagnetic-field h H=H C2 ð0Þ dependence of Dw for each Pb–In alloy, corresponding to a reduced temperature lnðT=T C Þ ¼ 0:06. To make easier the comparison with the existing theories for Dw (see below) these data are normalized by Txð0Þ, where xð0Þ is the corresponding GL coherence length amplitude (see Table 1). As may be clearly seen, the data for the different alloys collapse into a unique curve, which indicates that the SCF effects on Dw are independent of the amount non-magnetic impurities, and also of the value of k. At very low reduced magnetic
ARTICLE IN PRESS F. Soto et al. / Physica B 378–380 (2006) 393–394
394
Table 1 Superconducting parameters of the Pb–In alloys studied. T C was determined from field-cooled MðTÞH measurements at m0 H ¼ 0:5 mT In (at%)
T C (K)
k
m0 H C2 ð0Þ (T)
xð0Þ (A˚)
x0 =‘
5 8 18 30 45
7.06 6.99 6.85 6.75 6.43
1.32 2.07 3.38 4.17 5.54
0.286 0.487 0.856 0.993 1.19
339 260 196 182 166
2.7 7.1 14 16 22
H C2 ð0Þ and k were obtained from MðHÞT measurements in the mixed state. xð0Þ follows from x2 ð0Þ ¼ f0 =2pm0 H C2 ð0Þ, x0 was obtained from the xð0Þ values through the Gorkov theory, and ‘ from measurements of the residual resistivity just above T C .
The SCF vanishing at high- values was interpreted in terms of the limit imposed by the uncertainty principle to the shrinkage of the superconducting wavefunction, which may be expressed as xðÞ4x0 , where xðÞ 0:74 x0 1=2 is the GL superconducting coherence length. From those expressions, the -value above which the SCF should disappear is C ¼ 0:55. This quantum confinement effect may be formally introduced in the GL theory in the h5 limit through a total-energy cutoff [4], leading to the expression " # C C 1=2 Dw m0 kB atanð Þ1=2 atanð C Þ pffiffi pffiffiffiffiffi ¼ . (1) Txð0Þ 3f20 C
Fig. 1. (a) h- and (b) -dependence of Dw=Txð0Þ for the Pb–In alloys studied. In both figures, the solid line is the result of the GL theory in the limit h5, modified to include the quantum confinement effects at high [Eq. (1)]. In constructing this figure we used the T C , H C2 ð0Þ and xð0Þ values of Table 1.
fields (i.e., for ho), Dw=Txð0Þ is h-independent, and in excellent agreement with the prediction of the GL theory in the Gaussian approximation (see below) [4]. However, for higher h-values the effect of the thermal fluctuations is strongly reduced, to end up by disappearing at h1. This effect is analogous to the SCF vanishing observed at high reduced temperatures in the low-h limit [see Fig. 1(b)], which seems to be present in any kind of superconductor, including the high-T C cuprates [5].
This equation is in excellent agreement with the data of Fig. 1 in the h5 limit. In particular, it reproduces the Dw vanishing at 0:55. This quantum confinement effect may be also invoked to explain the SCF vanishing at high h.pIn ffiffiffi this case the GL coherence length is given by xðhÞ 2 x0 h1=2 (see, e.g., Ref. [2]), and the limit imposed by the uncertainty principle to the shrinkage of the superconducting wavefunction xðhÞox0 implies that the SCF should disappear above hC 1:1, in excellent agreement with the data of Fig. 1(a). In conclusion, the SCF vanishing observed for HH C2 ð0Þ could be a confirmation of the relevance of quantum confinement effects on the SCF when the superconducting wavefunction is shrunk to lengths of the order of the Cooper pairs size. A theory for Dw for these high H values including quantum confinement effects would be desirable to settle these results. This work was supported by the Spanish MEC (MAT2004-04364) and by Unio´n Fenosa (contract 2200085-2002).
References [1] M. Tinkham, Introduction to Superconductivity, MCGraw-Hill, New York, 1996. [2] J. Skocpol, M. Tinkham, Rep. Prog. Phys. 38 (1975) 1049. [3] F. Soto, et al., Phys. Rev. B 70 (2004) 060501. [4] J. Mosqueira, C. Carballeira, F. Vidal, Phys. Rev. Lett. 87 (2001) 167009. [5] F. Vidal, et al., Europhys. Lett. 59 (2002) 754.