The Kosztin-Leggett nonlocal effect on the magnetic penetration depth in superconductors with nodes in the gap

Abstract

The effect of nonlocal electrodynamics is considered in finite samples. It is found for such samples that nonlocality modifies not only the in-plane penetration depth lambda||, as predicted by Kosztin and Leggett, but also the out-of-plane penetration depth lambda<FONT FACE=Symbol>^</FONT>. The Sr2RuO4 data are reinterpreted taking into account the contribution from lambda<FONT FACE=Symbol>^</FONT>.

The Kosztin-Leggett nonlocal effect on the magnetic penetration depth in superconductors with nodes in the gap

Ismardo Bonalde

Centro de Física, Instituto Venezolano de Investigaciones Científicas, Apartado 21874, Caracas 1020-A, Venezuela

ABSTRACT

The effect of nonlocal electrodynamics is considered in finite samples. It is found for such samples that nonlocality modifies not only the in-plane penetration depth l_||, as predicted by Kosztin and Leggett, but also the out-of-plane penetration depth l_^. The Sr₂RuO₄ data are reinterpreted taking into account the contribution from l_^.

1 Introduction

Penetration depth measurements have played a very fundamental role in the study of the gap symmetry of all unconventional superconductors. For instance, from magnetic penetration depth measurements it was first suggested the d-wave symmetry for the energy gap in high critical temperature superconductors (HTSC) [1]. It is expected that, for superconducting energy gaps with nodes on the Fermi surface, the deviation of the magnetic penetration depth l(T) from its zero temperature value goes as a power law in temperature, Dl(T) = l(T) - l(0) µ T^a, at temperatures well below the critical temperature T_c. The exponent a depends on gap geometry, dimensionality and the rate at which the gap vanishes near the zeroes. For line nodes in three dimensional systems, like those with a d-wave symmetry, Dl(T) µ T (the same conclusion holds for point nodes in two dimensional systems). In these systems, however, the penetration depth has been found in some cases to cross over from linear to quadratic temperature dependence at a temperature T^* << T_c. Such a crossover has been explained theoretically as due to impurities [2] or nonlocal electrodynamics [3].

The impurity model considers that a dilute concentration of strong scatterers in the unitary limit in unconventional superconductors with line nodes causes a finite residual density of states near the nodes and negligible suppression of T_c. Thus, as T ® 0 a "gapless" temperature behavior [4] Dl(T) µ T² should be expected for such superconductors, and the crossover in Dl(T) is predicted to occur at T^* ~ . Here, G is the scattering rate and D₀ is the energy gap maximum. The nonlocal model supposes that the local condition l(0)/x₀ >> 1, for which the spatial variation of the electromagnetic field over the extent of the Cooper pair is neglected and the pair is taken as a point object, is not satisfied at temperatures well below T_c in most unconventional superconductors. Here x₀º n_F/ pD₀ is the BCS coherence length. The nonlocal model is based on the fact that in superconductors with an anisotropic energy gap with nodes on the Fermi surface the coherence length x₀(k) = n_F/pD(k). Kosztin and Leggett (KL) [3] suggested for the local condition

Thus near the nodes, where D(k) ® 0, the local condition would no longer hold, and the superconductor would become nonlocal. KL predicted for very clean, semi-infinite superconductors with line nodes that nonlocal electrodynamics would affect the magnetic penetration depth only when the exciting field is applied perpendicular to the ab plane; that is to say, only l_||(T) would be perturbed by nonlocality. They proposed that Dl_|| (T) µ T² below a = D₀x₀/l(0) << T_c.

The impurity effect has been confirmed in Zn-doped YBCO [5] and has been used to explain data from organic superconductors [6] and various HTSC [7]. On the other hand, the KL nonlocal effect has been difficult to prove, and only recently some evidence for its existence has been obtained. The importance of testing experimentally the validity of the KL effect is because it would provide a mechanism for the observed crossover from linear to quadratic temperature dependence of the magnetic penetration depth in very clean (impurity effect unapplicable) superconductors with nodes in the energy gap. Here the KL nonlocal effect is discussed in finite superconductors with line nodes. It is found that the out-of-plane component of the penetration depth, l_^, is also affected by nonlocality, and not only the in-plane component as predicted by KL. A discussion is given of some experimental results.

2 The Kosztin-Leggett nonlocal effect

The penetration depth would depend in some cases on the orientation of the sample with respect to the applied magnetic field. If the sample is aligned so that the ab plane is perpendicular to the exciting field, screening currents are generated in the ab plane and one measures the in-plane penetration depth l_||(T). When the field is parallel to the ab plane the measured signal is a combination of l_|| and l_^: . Here l_^, the out-of-plane penetration depth, corresponds to screening currents along the interlayer c direction. The parameters t and w are the thickness and width of the sample, respectively.

To study the effect of nonlocal electrodynamics on the magnetic penetration depth, KL considered a weak, homogeneous magnetic field H applied along the c axis of a semi-infinite HTSC with a plane boundary perpendicular to the b axis (see Fig. 1). For a specular boundary

where K(q,T) is the response function to a weak, transverse vector potential. For a weak coupling, anisotropic superconductor [3]

Here, w_n are the Matsubara frequencies, D_k is the gap function, is the projection of k on the plane boundary, á...ñ is an average over the Fermi surface, and g = (qn_F/2) q·k. The parameter g determines the relevance of nonlocal electrodynamics: if g ¹ 0 (g = 0) nonlocality is (not) important. In the current configuration, screening currents are generated only in the planes, and being g ¹ 0 KL predicted that Dl_||(T) µ Dl_{|| L}(T)T at temperatures T < . Here, Dl_||L is the deviation of the in-plane penetration depth in the local approximation. Thus because Dl_||Lµ T in d-wave superconductors, there should be in such superconductor a crossover from their characteristic linear T behavior at low temperatures to a quadratic T dependence at even lower temperatures. On the other hand, if the exciting field is oriented along the b axis, as shown in Fig. 1, because of the semi-infinite space the only parameter would be again l_||(T). However, in this case g = 0 and the in-plane penetration depth would not be affected by nonlocality. That is, for this orientation Dl_||(T) = Dl_||L(T) µ T.

Consider now what happens in finite samples, which were not evaluated by KL. This is a quite important case, because the experimental samples are in fact small. If H||c, as shown in Fig. 2, the KL prediction for semi-infinite samples in this orientation holds. If H ^ c, as indicated in Fig. 3, the prediction would be different from the one stated by KL for semi-infinite samples. In this orientation, Dl(T) is a combination of l_||(T) and l_^ (T), in the way mentioned above. The deviation of the in-plane penetration depth would follow Dl_||(T) = Dl_||L(T) µ T, because g = 0 for screening currents along the plane. But g ¹ 0 for interlayer currents, so that l_^(T) should be modified by nonlocality. Thus, the statement that the effect of nonlocal electrodynamics is only relevant to l_||(T) is no longer sustained in finite samples. Assuming that for an anisotropic d-wave superconductor (or any superconductor with nodes in the gap) the effect of nonlocal electrodynamics on the temperature dependence of l_^ is similar to that of l_||, one has Dl_^(T) µ Dl_^L(T)T at temperatures T < . It must be pointed out that the for the in- and out-of-plane components should be in general different. This indeed would be the case for anisotropic superconductors.

3 The Sr₂RuO₄ experimental data

Although the KL effect was predicted with HTSC in mind, no studies of it using these superconductors have been reported. This is mainly due to limitations on the experimental resolution for l(T) in HTSC below 1 K ( ~ 2K in YBCO for example).

The Sr₂RuO₄ penetration depth data [8] were the first showing some evidence for the KL nonlocal effect. These data, taken from samples with T_c very close to the impurity-free T_c = 1.5 K, can be reanalyzed by considering the contribution of the out-of-plane penetration depth when H ^ c. It is worth to mention that Sr₂RuO₄ has nodes in the energy gap, although the exact symmetry of the gap is at the present time highly controversial. This material is a marginal local superconductor in the ab plane, which leads to the expected @ 1 K. Measurements of Dl_||(T) down to 30 mK yielded a quadratic T dependence below 0.8 K, as shown in Fig. 4. When the field was applied perpendicular to the c axis, it was found Dl(T) µ T² below 0.8 K (see Fig. 5). Considering that l_^ @ 20l_||, one could suppose that the signal was dominated by the out-of-plane penetration depth. That is, Dl(T) ~ Dl_^(T) µ T². Assuming that the local l_^(T) is linear in T, this result would imply the presence of nonlocal effects. However, using the value l_^(0)/ x_0^ = 1123 = 2.4 mK, which is far below the observed 0.8 K. This suggests that nonlocal effects on the out-of-plane l(T) in Sr₂RuO₄ should not be observed in the temperature range of the experiment (T > 30 mK). Thus, nonlocal effects may be discarded as a possible explanation for the T² dependence of the magnetic penetration depth of Sr₂RuO₄.

4 Conclusions

The effect of nonlocal electrodynamics on the magnetic penetration depth in superconductors with nodes in the gap has been revised for finite samples. It is found that not only the in-plane penetration depth l_||(T) is affected by nonlocal electrodynamics, but also the out-of-plane penetration depth l_^(T). However, the temperature at which the nonlocal effect should be detected in l_^(T) could be much smaller than the one expected for l_||(T).

Received on 1st July, 2003

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Publication Dates

• Publication in this collection
  25 Nov 2005
• Date of issue
  Dec 2003

History

• Received
  01 July 2003