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Paramagnetic response of NS proximity cylinders
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Physica C 341-348 (2000) 2667-2668
www.elsevier.nl/locate/physc
Paramagnetic Response of NS Proximity Cylinders Kazumi Maki and Stephan Haas
"
Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484 Since the discovery of unusual paramagnetic reentrance behavior at ultra-low temperatures in Nb-Ag, Nb-Au, and Nb-Cu cylinders this phenomenon has remained a puzzle. For the diamagnetic response down to temperatures of the order 15 mK, the standard theory (quasi-classical approximation) for superconductors appears to work very well, assuming that Ag, Au, and Cu remain in the normal state except for the proximity-induced superconductivity. Here we propose that these noble metals may become p-wave superconductors with a transition temperature of order 10 mK. Hence, below T¢ p-wave triplet superconductivity emerges around the periphery of the cylinder. The diamagnetic current flowing in the periphery is therefore compensated by a quantized paramagnetic current in the opposite direction, thus providing a simple explanation for the observed increase in the susceptibility at ultra-low temperatures.
In 1990 Visani et a/.[1] reported an unusual paramagnetic reentrance behavior at ultra-low temperatures. When a Nb cylinder of diameter ,~ 20-100 #m, covered with a thin film of Ag with a thickness of a few #m, is cooled below 10 m K , the expected diamagnetic response decreases instead of increasing as the temperature is lowered. A very similar observation was later reported for an analogous Nb-Cu system[2]. The standard theory for the proximity effect is most conveniently described in terms of the quasiclassical approximation, which is a more refined version of the approach first discussed in Ref. [3]. Within this approach, one can describe the diamagnetic response of an S-N system down to 100 mK perfectly well with only one adjustable parameter, the quasi-particle mean free path in the normal state[4]. The sudden failure of this quasi-classical approach below 100 mK, suggested by the observed reentrance behavior, implies that some crucial and new element is missing from the usual model. Therefore, Bruder and Imry[5] have postulated a new kind of persistent current around the edge of the normal metal, circulating in the direction opposite to the diamagnetic current[5]. Although this current is associated with an extended state in the weak localization theory, it turns out from *We thank A.C. Mota for useful discussions.
a simple estimate that it is of the order of 10 - a smaller that the one required to accurately describe the experiments. More recently, Fauch~re et al.[6] have proposed that the pairing interaction in noble metals, such as Cu, Ag, or Au, is repulsive. This implies that the sign of A(r) changes at the N-S boundary, thus generating an intrinsic ~r-junction at the boundary. As in the model by Bruder and Imry, this r-junction could perhaps generate a current in the direction opposite to the diamagnetic current, resulting in a paramagnetic reentrance effect at ultra-low temperatures. However, a careful inspection of the numbers suggests that the repulsive interaction in the normal state may cause a magnetic transition with T M ,-~ 100 mK. Other sets of experiments[7] on the proximity effect appear to exclude such a large repulsive potential. Furthermore, if the pairing potential is repulsive, we would rather expect p-wave superconductivity in these noble metals if we follow the analysis of Kohn and Luttinger[8]. Let us therefore assume here that p-wave superconductivity is generated in the outer film below a critical temperature of Tc ,-~ 10 - 100 mK. Then, in the presence of the proximity-generated s-wave superconductivity with As(r) penetrating into the outer film, an additional order parameter Ap(r), associated with intrinsic p-wave supercon-
0921-4534/00/$ - see front matter © 2000 Elsevier Science B.E All rights reserved. PII S0921-4534(00)01452-0
K. Maki, S. Haas/Physica C 341-348 (2000) 2667-2668
2668
ductivity in the outer film, has to establish itself below Te. The p-wave superconducting ordering will generate a counter-current, reducing the kinetic energy associated with Ap(r). To make this idea more concrete, let us assume that As(r) is uniform in the outer film, and the phase of A8 (r) is given by ¢(r) = 2eA¢s. Here, A¢ = Br, where B is the external field and r is the radial distance measured from the center of the cylinder, while s is the arc-length measured along the circumference. So in the normal state, ¢(r) = 2eBrs. This ¢(r) gives a contribution to the kinetic energy Ekin =
1 ~2 2 -~(2eBr) ~ , A , ( r ) ,
p
1
2rrn 2 2 2 ¢;Av(r),
(2)
where l is the circumferenceof the thin film, encircling the inner s-wave superconductor. Then the optimal counter-current is obtained by minimizing ( ( 2 e B r - ~.~)2> in the outer film. Since r in the center of the outer film is r0 = l/(2rr), we obtain 2rrn 2>= 2eBl ~ _ 4 e B n + ( ~ _ ~ ) 2 . ( 3 ) This gives n ~_ 2eB(l/(2rr)) 2 = 0.7958B/2, where B and l are expressed in gauss and pm respectively. Hence it is very likely that a spontaneous counter-current with n = 1, 2, 3, ... is generated. This counter-current gives rise to a magnetization with a positive sign (i.e. paramagnetic), and
J M o¢ n[Ap(r)[ 2.
S
N
As(r)
(1)
where ~, is the coherence length for A,(r). A similar expression holds for Ap(r). Now, if Ap(r) generates a counter-current quantized by ~(r) = 2rrns/l, the corresponding kinetic is reduced to Eki . = _ ( 2 e B r - - - - ~ )
temperatures, Tc = 10 - 100 mK. At least, this exciting possibility has not been excluded by experiments so far.
(4)
The paramagnetic current thus jumps discretelyl In order to determine lAp(r)[ ~ itself, the full Ginzburg-Landau equations need to be analyzed. But even from the simple energy considerations presented here, this paramagnetic current is clearly of the same order of magnitude as the diamagnetic current. So perhaps most noble metals become p-wave superconductors at ultra-low
dN Figure 1. Sketch of the radial direction of an NS proximity cylinder.
REFERENCES
1. P. Visani, A.C. Mota, and A. Pollini, Phys. Rev. Lett. 65, 1514 (1990). 2. A.C. Mota, P. Visani, A. Pollini, and K. Aupler, Physica B 197, 95 (1994). 3. Orsay group on Superconductivity in "Quantum Fluids", ed. D. Brewer (North-Holland, Amsterdam, 1966). 4. W. Belzig, C. Bruder, and A. Fauch~re, Phys. Rev. B 58, 14531 (1998); F.B. MiillerAllinger, A.C. Mota, and W. Belzig, Phys. Rev. B 59, 8887 (1999). 5. C. Bruder and Y. Imry, Phys. Rev. Lett. 80, 5782 {1998). 6. A. Fauch~re, W. Belzig, and G. Blatter, Phys. Rev. Lett. 82, 3336 (1999). 7. A.C. Mota, private communications. 8. W. Kohn and J.M. Luttinger, Phys. Rev. Lett. 15,523 (1966).
Physica C 341-348 (2000) 2667-2668
www.elsevier.nl/locate/physc
Paramagnetic Response of NS Proximity Cylinders Kazumi Maki and Stephan Haas
"
Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484 Since the discovery of unusual paramagnetic reentrance behavior at ultra-low temperatures in Nb-Ag, Nb-Au, and Nb-Cu cylinders this phenomenon has remained a puzzle. For the diamagnetic response down to temperatures of the order 15 mK, the standard theory (quasi-classical approximation) for superconductors appears to work very well, assuming that Ag, Au, and Cu remain in the normal state except for the proximity-induced superconductivity. Here we propose that these noble metals may become p-wave superconductors with a transition temperature of order 10 mK. Hence, below T¢ p-wave triplet superconductivity emerges around the periphery of the cylinder. The diamagnetic current flowing in the periphery is therefore compensated by a quantized paramagnetic current in the opposite direction, thus providing a simple explanation for the observed increase in the susceptibility at ultra-low temperatures.
In 1990 Visani et a/.[1] reported an unusual paramagnetic reentrance behavior at ultra-low temperatures. When a Nb cylinder of diameter ,~ 20-100 #m, covered with a thin film of Ag with a thickness of a few #m, is cooled below 10 m K , the expected diamagnetic response decreases instead of increasing as the temperature is lowered. A very similar observation was later reported for an analogous Nb-Cu system[2]. The standard theory for the proximity effect is most conveniently described in terms of the quasiclassical approximation, which is a more refined version of the approach first discussed in Ref. [3]. Within this approach, one can describe the diamagnetic response of an S-N system down to 100 mK perfectly well with only one adjustable parameter, the quasi-particle mean free path in the normal state[4]. The sudden failure of this quasi-classical approach below 100 mK, suggested by the observed reentrance behavior, implies that some crucial and new element is missing from the usual model. Therefore, Bruder and Imry[5] have postulated a new kind of persistent current around the edge of the normal metal, circulating in the direction opposite to the diamagnetic current[5]. Although this current is associated with an extended state in the weak localization theory, it turns out from *We thank A.C. Mota for useful discussions.
a simple estimate that it is of the order of 10 - a smaller that the one required to accurately describe the experiments. More recently, Fauch~re et al.[6] have proposed that the pairing interaction in noble metals, such as Cu, Ag, or Au, is repulsive. This implies that the sign of A(r) changes at the N-S boundary, thus generating an intrinsic ~r-junction at the boundary. As in the model by Bruder and Imry, this r-junction could perhaps generate a current in the direction opposite to the diamagnetic current, resulting in a paramagnetic reentrance effect at ultra-low temperatures. However, a careful inspection of the numbers suggests that the repulsive interaction in the normal state may cause a magnetic transition with T M ,-~ 100 mK. Other sets of experiments[7] on the proximity effect appear to exclude such a large repulsive potential. Furthermore, if the pairing potential is repulsive, we would rather expect p-wave superconductivity in these noble metals if we follow the analysis of Kohn and Luttinger[8]. Let us therefore assume here that p-wave superconductivity is generated in the outer film below a critical temperature of Tc ,-~ 10 - 100 mK. Then, in the presence of the proximity-generated s-wave superconductivity with As(r) penetrating into the outer film, an additional order parameter Ap(r), associated with intrinsic p-wave supercon-
0921-4534/00/$ - see front matter © 2000 Elsevier Science B.E All rights reserved. PII S0921-4534(00)01452-0
K. Maki, S. Haas/Physica C 341-348 (2000) 2667-2668
2668
ductivity in the outer film, has to establish itself below Te. The p-wave superconducting ordering will generate a counter-current, reducing the kinetic energy associated with Ap(r). To make this idea more concrete, let us assume that As(r) is uniform in the outer film, and the phase of A8 (r) is given by ¢(r) = 2eA¢s. Here, A¢ = Br, where B is the external field and r is the radial distance measured from the center of the cylinder, while s is the arc-length measured along the circumference. So in the normal state, ¢(r) = 2eBrs. This ¢(r) gives a contribution to the kinetic energy Ekin =
1 ~2 2 -~(2eBr) ~ , A , ( r ) ,
p
1
2rrn 2 2 2 ¢;Av(r),
(2)
where l is the circumferenceof the thin film, encircling the inner s-wave superconductor. Then the optimal counter-current is obtained by minimizing ( ( 2 e B r - ~.~)2> in the outer film. Since r in the center of the outer film is r0 = l/(2rr), we obtain 2rrn 2>= 2eBl ~ _ 4 e B n + ( ~ _ ~ ) 2 . ( 3 ) This gives n ~_ 2eB(l/(2rr)) 2 = 0.7958B/2, where B and l are expressed in gauss and pm respectively. Hence it is very likely that a spontaneous counter-current with n = 1, 2, 3, ... is generated. This counter-current gives rise to a magnetization with a positive sign (i.e. paramagnetic), and
J M o¢ n[Ap(r)[ 2.
S
N
As(r)
(1)
where ~, is the coherence length for A,(r). A similar expression holds for Ap(r). Now, if Ap(r) generates a counter-current quantized by ~(r) = 2rrns/l, the corresponding kinetic is reduced to Eki . = _ ( 2 e B r - - - - ~ )
temperatures, Tc = 10 - 100 mK. At least, this exciting possibility has not been excluded by experiments so far.
(4)
The paramagnetic current thus jumps discretelyl In order to determine lAp(r)[ ~ itself, the full Ginzburg-Landau equations need to be analyzed. But even from the simple energy considerations presented here, this paramagnetic current is clearly of the same order of magnitude as the diamagnetic current. So perhaps most noble metals become p-wave superconductors at ultra-low
dN Figure 1. Sketch of the radial direction of an NS proximity cylinder.
REFERENCES
1. P. Visani, A.C. Mota, and A. Pollini, Phys. Rev. Lett. 65, 1514 (1990). 2. A.C. Mota, P. Visani, A. Pollini, and K. Aupler, Physica B 197, 95 (1994). 3. Orsay group on Superconductivity in "Quantum Fluids", ed. D. Brewer (North-Holland, Amsterdam, 1966). 4. W. Belzig, C. Bruder, and A. Fauch~re, Phys. Rev. B 58, 14531 (1998); F.B. MiillerAllinger, A.C. Mota, and W. Belzig, Phys. Rev. B 59, 8887 (1999). 5. C. Bruder and Y. Imry, Phys. Rev. Lett. 80, 5782 {1998). 6. A. Fauch~re, W. Belzig, and G. Blatter, Phys. Rev. Lett. 82, 3336 (1999). 7. A.C. Mota, private communications. 8. W. Kohn and J.M. Luttinger, Phys. Rev. Lett. 15,523 (1966).