Full text

Turn on search term navigation

ARTICLE

Received 27 Aug 2014 | Accepted 11 Jun 2015 | Published 4 Aug 2015

For any practical superconductor the magnitude of the critical current density, Jc, is crucially important. It sets the upper limit for current in the conductor. Usually Jc falls rapidly with increasing external magnetic eld, but even in zero external eld the current owing in the conductor generates a self-eld that limits Jc. Here we show for thin lms of thickness less than the London penetration depth, l, this limiting Jc adopts a universal value for all superconductorsmetals, oxides, cuprates, pnictides, borocarbides and heavy Fermions.

For type-I superconductors, it is Hc/l where Hc is the thermodynamic critical eld. But surprisingly for type-II superconductors, we nd the self-eld Jc is Hc1/l where Hc1 is the lower critical eld. Jc is thus fundamentally determined and this provides a simple means to extract absolute values of l(T) and, from its temperature dependence, the symmetry and magnitude of the superconducting gap.

DOI: 10.1038/ncomms8820 OPEN

Universal self-eld critical current for thin-lm superconductors

E.F. Talantsev1 & J.L. Tallon1,2

1 Robinson Research Institute, Victoria University of Wellington, PO Box 33436, Lower Hutt 5046, New Zealand. 2 MacDiarmid Institute, Victoria University of Wellington, PO Box 33436, Lower Hutt 5046, New Zealand. Correspondence and requests for materials should be addressed to E.F.T. (email: mailto:[email protected]

Web End [email protected] ) or to J.L.T. (email: mailto:[email protected]

Web End [email protected] ).

NATURE COMMUNICATIONS | 6:7820 | DOI: 10.1038/ncomms8820 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 1

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms8820

Superconductors are characterized by two microscopic length scales: the London penetration depth, l, and the coherence length, x. These strongly inuence both their

fundamental and applied behaviour, especially the critical current density Jcabove which the current becomes dissipative. In practical superconductors, which are all type II, Jc is arguably its most important property and for example, in the high-Tc superconductors, a huge effort has been expended in attempting to maximize Jc as a function of temperature, T, and magnetic eld, H1. Jc falls rapidly with increasing external eld, but even in zero external eld the current owing in the conductor generates a self-eld that itself limits Jc. We refer to this limiting value as Jc(sf).

In a type-II superconductor, Jc is widely thought to be governed by pinning of ux vortices as well as by geometrical factors arising from the detailed pinning microstructure. As a consequence, much of the above-noted effort has been applied to modifying and tuning this microstructure. On the other hand, nearly a century ago Silsbee2 proposed that, for a type-I superconductor, the critical current is that at which the magnetic eld due to the current itself is equal to the critical magnetic eld. In other words, the self-eld Jc is just that which is sufcient to generate a surface eld equal to the critical eld. By this was meant what we now understand to be the thermodynamic critical eld, Hc, given by3

Hc T

f0 2

; 1

where f0 is the ux quantum and m0 is the permeability of free space. Of course for a type-I superconductor, where ux vortices are absent, pinning is irrelevant and Silsbees hypothesis is credible. Jc may depend on geometry but not on microstructure.

But for a type-II superconductor, the general consensus that pinning governs Jc would insist that both geometry and microstructure are key players, and any kind of universal Silsbee criterion is untenable.

Here, by examining a wide range of experimental data, we ask whether this criterion does have any relevance to type-II superconductors. Surprisingly, the answer for conductors of thickness comparable to l is yes, and here the relevant critical eld is the lower critical eld Hc1, given by3

Hc1 T

f0 4pm0l2 T

ln k T

Hl ; 5

where, in the usual notation, B is the magnetic ux density and H the eld intensity, within the conductor surface. We use equation (5) to make an estimate of Jc when bEl. In this case(i) the current penetrates the entire cross-section, so that J is no longer a surface current density but is approximately global across the lm thickness, and (ii) when the x-component of H reaches Hc1, vortices of opposite sign will tend to nucleate at the opposing surfaces and self-annihilate at the centre. They will do so both because of the Lorentz force driving them inwards and because of the attraction of overlapping vortices of opposite sign on opposite faces, which diverges logarithmically when bol. The consequent onset of dissipation denes J Jc where, from equation (5),

JcZHc1/l and equality only applies if this force exceeds surface and bulk pinning forces. In the following, we observe that equality is indeed found for a wide range of superconducting materials and this leads immediately to equation (4).

This very approximate analysis leaves many open questions. For example, for b4l it is usual to discuss Jc in terms of ux entry from the edges. This is discussed later. Our approach is simply to examine the available data from self-eld Jc studies on a wide variety of thin-lm superconductors. If equation (4) does prove to be valid, then we have a simple means to determine absolute values of l (and the superuid density rs l 2) from

measurements of Jc(T). The test of success is how well inferred values of l concur with reported values. Moreover, the magnitude of the superconducting gap may then be determined from tting the low-T behaviour of rs as follows5. For s-wave symmetry:

l T

2 l 0

2 1 2

p pm0x T

l T

0:5

; 2

where k(T) l(T)/x(T) is the GinsburgLandau parameter,

which is effectively constant under the logarithm. With this thickness constraint, we nd for type-I superconductors:

JIc sf

Hcl

pD 0

kBT

s e D 0 =kBT

; 6

while for d-wave symmetry:

l T

2 l 0

2 1

f0k T

2 p pm0l3 T

; 3

; 7

where Dm is the maximum amplitude of the k-dependent d-wave gap, D Dmcos(2y).

London penetration depth. Figure 1 shows normalized plots of reported self-eld Jc(T) values (right-hand scale, arrowed) for a wide range of superconductors including type I, type II, s-wave and d-wave. Also plotted are the inferred values of l(T)

(left-hand scale) calculated from the Jc(T) values by inverting equations (3) or (4). Individual plots are presented and discussed in Supplementary Note 1. Panels (a) and (b) in Fig. 1 are s-wave, while (c) and (d) are d-wave superconductors. For both s- and d-wave cases, the T-dependence of k, calculated from l D

(ref. 6), is weak and, in view of the logarithm and the cube root, we conveniently take k to be constant. The residual effect of a

p kBT

Dm 0

and for type-II superconductors:

JIIc sf

Hc1

f0 4pm0l3 T

ln k 0:5

: 4

As a consequence, Jc(sf) is fundamentally determined just by l and x and independent of both geometry and microstructure. Because of the near constancy of ln(k), for type-II super-conductors Jc(sf) is dependent only on l and this then provides a simple means to extract absolute values of l(T) and, from its temperature dependence, the symmetry and magnitude of the superconducting gap. We present an indicative theoretical justication for this remarkably general and unexpected result, but we recognize that some questions remain to be resolved. We predict the doping and temperature dependence of Jc(sf) for

YBa2Cu3O7d (YBCO) as a test of our hypothesis. Hereafter, we

consider only self-eld Jc values and therefore drop the identier sf, except where we feel it is still needed.

ResultsBasic model. We consider a thin lm of the type-II super-conductor in the form of a long thin tape of rectangular cross-section in the xy plane and of thickness 2b and width 2a, such that booa. Our conductor is of quasi-innite length along the z axis in which a current of magnitude I is owing along its axis. The tape interior is dened by arxr a and bryr b.

According to London, in the Meissner state for small currents the self-eld and transport current penetrate to a depth Bl, and the amplitude of the local surface current density, J, is4

B m0l

2 NATURE COMMUNICATIONS | 6:7820 | DOI: 10.1038/ncomms8820 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms8820 ARTICLE

a b c d

4.0

1.0

3.5

PrOs4Sb12

Jc measured Jc from Hc1

3.0

0.8

2.5

[afii9838] from Hc1

[afii9838](T)/[afii9838](0)

NbIn MgB2

Ba(Fe,Co)2As2 (Ba,K)BiO3

Bi-2212+0.5% Zn Tl2Ba2CaCu2O8

FeTe1/2Se1/2

YBCO(Y,Ca)BCO Jc

J c(T)/J c(0)

0.6

2.0

NbN Amorph-W AlSn YNi2B2C

0.4

1.5

1.0

0.2

0.5 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0

0.0

T /Tc T /Tc T /Tc T /Tc

Figure 1 | Summary of reported Jc and calculated k. The temperature dependence of the normalized penetration depth, l(T)/l(0) (left-hand scale) calculated from the normalized values of self-eld critical current density, Jc(T)/Jc(0) (right-hand scale, as indicated by arrows) for many different type-I and type-II superconductors. Values of l(T) are calculated using equations (3) and (4). (a,b) s-wave superconductors. (c,d) d-wave superconductors, as also seen by their very different low-T behaviour. The dashed red curves are the tted s-wave weak-coupling T-dependence of l(T) and the dashed blue curves are the d-wave counterparts. The dashed black curves are Jc back-calculated from these l(T) curves using equations (3) and (4). The deduced normalization parameters, Tc, Jc0 and l0 l(0) are listed in Supplementary Table 1. (d) A slightly different analysis for PrOs4Sb12. Both Jc and l are

calculated from measurements of Hc1 using equation (4), and the calculated Jc is compared with values measured from remnant magnetization (magenta data points). They are in excellent agreement. The two curves (dashed and solid) are obtained by adding two separate d-wave superuid densities below 0.6 K.

T-dependent k is discussed in Supplementary Note 2. Values of k are sourced from the literature and are listed in Supplementary Table 1.

Our approach is as follows. From Jc(T) data we calculate the l data points as plotted. Using l0 as the only tting parameter, we then t theoretical s-wave (dashed red) or d-wave (dashed blue) curves6. From these curves, we back-calculate Jc(T) to give the dashed black curves, from which Jc0 is found. These deduced values of l0 and Jc0 are listed in Supplementary Table 1. The calculated l(T) data points are then tted at low-T using equations (6) or (7) to determine D0. This is done using the nonlinear curve t routine in the plot package Origin. All data sources, lm thicknesses and results are also summarized in Supplementary Table 1 along with reported values of l0 and D0 for comparison with our inferred values.

In Fig. 1, the radical difference between s-wave and d-wave symmetry at low T is immediately apparent. In the former case Jc(T) exhibits an exponential plateau due to the isotropic gap, while in the latter Jc(T) remains linearly increasing due to the nodal d-wave gap, in either case consistent with equations (6) or(7). Figure 1a shows examples of s-wave superconductors: Nb, In, MgB2, Ba(Fe,Co)2As2 and (Ba,K)BiO3. The t with the weak-coupling s-wave model is excellent though MgB2 and Ba(Fe,-Co)2As2 show small deviations, possibly due to multiple gaps on distinct bands7.

Figure 1b shows the same analysis for ve samples where the t is better using the dirty s-wave model. YNi2B2C will be discussed later. For the type-I elements Al, Sn and In in Fig. 1, we have used equation (3). This is the so-called London depairing current density. We discuss this in relation to the GinzburgLandau depairing current density in Supplementary Note 5. For weak-coupling, s-wave superconductors k1/3 changes by o5% between 0rTrTc so by assuming constancy of k we again infer that l / 1=

p , but here with a different prefactor. The data for Al, Sn and In in Fig. 1a,b strongly support this analysis, and the deduced values of l0 shown in Supplementary Table 1 are in excellent agreement with directly measured values.

Next, Fig. 1c shows ve d-wave examples: FeTe0.5Se0.5, YBa2Cu3O7, 1% Ca-doped YBa2Cu3O7, 0.5% Zn-doped Bi2Sr2-CaCu2O8 and Tl2Ba2CaCu2O8. For each of these samples, the t to the weak-coupling d-wave model is excellent across the entire temperature range. This is surprising because other techniques

such as muon spin relaxation8 suggest that the T-dependence of rs does not always follow the canonical d-wave form.

Finally, for Fig. 1d PrOs4Sb12, we used a different approach. From reported data for Hc1 (ref. 9), we calculated both Jc(T) and l(T) from equation (4) using k 29.7 (refs 10,11). Both

parameters reveal a transition to a second phase below 0.6 K (ref. 9), which results in an additional reduction of l(T). The two distinct curves (dashed and solid) are obtained by adding two separate d-wave superuid densities below 0.6 K. Cichorek et al.9 also determined Jc(T) from remanent magnetization measurements. This is shown by the magenta symbols and, signicantly, Jc(T) shows an enhancement below 0.65 K that almost exactly mirrors our Jc(T) values calculated from Hc1. This represents a quite exacting test of our central thesis.

All our results for calculated absolute values of Jc0, l0 and D0 are plotted versus measured values in Fig. 2. The error bars reect the 2s-uncertainties in measured values of l0 or D0 summarized in Supplementary Table 1. In each case the correlation is excellent and it is this that validates our primary conclusion. Supplementary Note 3 shows how these results validate Silsbees hypothesis. Figure 2a shows that, using equations (3) and (4), Jc scales with l 3 over nearly three orders of magnitude. The only signicant outlier is YNi2B2C (ref. 12), but this is our only example for which b4l. Applying the thickness-correction factor (l/b)tanh(b/l), as introduced below, the calculated Jc0 now falls close to the dashed line as indicated by the curved blue arrow.

Figure 2b shows values of l0 calculated from Jc. As hypothesized, the best Jc values give values of l0 that match the measured values. In other lms, where Jc is low due to impurities, weak links or misalignment, l0 always exceeds the measured values. This is especially notable where a system has been improved over time, as for example, with the six lms shown for MgB2 (magenta data points). Here the inferred values of l descend towards the dashed line, as shown by the magenta arrow, as lms were progressively improved. They do not fall below the line. A similar data progression over time for YBCO is discussed in Supplementary Note 6. Jc(T) is thus fundamentally limited by l and not, for example, by the pinning, as we discuss further below. It is also important to recognize that the close correlation seen in Fig. 2b does not articially arise simply from error reduction by taking the cube root to extract l. This correlation is also seen without the cube root in Fig. 2a over now a much wider range,

NATURE COMMUNICATIONS | 6:7820 | DOI: 10.1038/ncomms8820 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 3

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms8820

a b c

Jc0 Summary

[afii9838]0 Summary

0 Summary

100

600

MgB

2 )

YBa Cu O

500

J c (sf,calc) (MA cm

Tl2212

(Ba,K)BiO

400

Hg2212

Bi2223

Bi2212+Zn

NbN

300

BaFe As

YNi B C

200

Rb C

100

PrOs Sb

FeSe Te

1 10 100

0 0 100 200 300 400 500 600

0.1 0.1 1 10

Jc (sf, meas) (MA cm2)

[afii9838] 0 (calc) (nm)

[afii9838]0 (meas) (nm)

0(calc) (meV)

0 (meas) (meV)

Figure 2 | Summary comparison of calculated and measured values. (a) Comparison of calculated thin-lm self-eld Jc0 values with measured Jc0 values. Jc0 is calculated from reported l values using equations (3) and (4). Red data points are type-I superconductors, black data points are type II. The blue arrow for YNi2B2C shows the effect of the correction factor (l/b)tanh(b/l), discussed in the text, when b4l. Error bars reect the range of reported l0 valuessee Supplementary Table 1. The sample annotation follows the vertical order of the data points. (b) l0 values calculated from reported Jc(T) data and plotted versus independently measured l0 values. The magenta arrow and symbols show the effect of improvement over time in self-eld Jc for six lms of MgB2. The data terminates on the dashed line where Jc is now fundamentally limited by the superuid density. Comparative data for YBCO over time is shown in Supplementary Figure 4. (c) Summary of values of D0 calculated from the low-T behaviour of l(T) using equations (6) and (7), plotted versus measured values of D0.

and also notably, where the relative order of type-I and type-II materials is signicantly altered, but without loss of correlation.

Energy gaps and symmetry. We select two examples to illustrate the calculation of D0. Figure 3a,b shows the contrasting low-T data for l(T) calculated from Jc(T, sf) for (a) YBa2Cu3O7 (this work) representing the d-wave case, and (b) ve lms of NbN (refs 1315), representing the s-wave case. The dashed black curves are the data ts to equations (7) and (6), respectively. For YBa2Cu3O7, the inferred gap value, D0 16.6 meV, compares well

with the tunnelling measurements of Dagan et al.16 (16.7 meV) and gives the ratio 2D0/kBTc 4.26 close to the d-wave weak-

coupling ratio 4.28. It also compares well with the estimate of17.7 meV from the condensation free energy17, but infrared ellipsometry measurements give a higher value of 25 meV (ref. 18). And generally, our calculated values of D0 for the cuprate superconductors tend to be lower than reported values. Partly, this is due to the lack of very low-T data for Jc, but in fact experimental values of gap magnitudes in the cuprates remain contentious. Tunnelling and ARPES data tend to show the presence of the (generally) larger pseudogap, and in our view the most reliable means of distinguishing the two is Raman scattering where B2g symmetry exposes the superconducting gap around the nodes, while B1g exposes the pseudogap around the antinodes19. In the present case, any sample that is optimally doped will have the pseudogap present and this will steepen the slope in l(T), thus reducing the inferred gap magnitude. The red crosses show the low-T penetration depth measurements of Hardy et al.20 and the agreement with ours, determined from Jc, is excellent.

For NbN, we show in Fig. 3b ts to ve data sets1315. Here ol04 is found to be 189 nm with o3% variation. The s-wave ts yield oD04 2.95 meV with o5% variation and we

nd 2D0/kBTc 4.13, somewhat more than the weak-coupling

s-wave limit of 3.53. Independent measurements21 give 2D0/kBTc 4.24, which is rather close to our value.

Figure 2c shows inferred D0 values calculated for all 17 superconductors plotted against measured values as listed in Supplementary Table 1. There is generally good agreement over two orders of magnitude and across all systems. We note that equations (6) and (7) are restricted to the weak-coupling limit and

this does not apply to all the samples investigated. In the case of Bi2Sr2Ca2Cu3O10, the estimated value of D0 is particularly low and this is due to the presence of two-layer intergrowths as is evident in the original paper. This causes l(T) 2 to rise more rapidly below90 K and thus yield a low value for D0. As noted, for the cuprates in general D0 values do tend to be low. This could be an indication of strong coupling but, if the samples are optimally doped then already the competing pseudogap is present22 and this will diminish the inferred D0 values. This can only be claried via the doping dependence of the low-T behaviour of Jc(T) where, in the sufciently overdoped region, the pseudogap is no longer present.

Doping dependence of Jc in YBa2Cu3O7d. We conclude that the self-eld Jc(T) for high-quality, weak-link-free thin lms with brl appears to be a fundamental quantity, governed only by the absolute value of the superuid density. If so we may use the superuid density to predict the evolution of Jc(T, p) with doping, p, for high-Tc cuprates. Figure 4 shows the self-eld Jc calculated

145

210

YBa2Cu3O7

d-wave gap fit [afii9838] from Jc

Hardy et al.

NbN

140

s-wave gap fit [afii9838] from Jc

200

[afii9838] (nm)

135

190

130

125 0 0 2 4 6

10 15 20

180

T (K) T (K)

[afii9838] (nm)

Figure 3 | Low-temperature ts: gap symmetry. (a) The low-T t to l(T) data determined from our measurements of Jc(sf) for YBa2Cu3O7 using equation (7). The t yields l 128.3 nm and D0 16.6 meV. The red

crosses show the low-T penetration depth measurements of Hardy et al.20

for comparison. (b) The low-T t to l(T) for NbN using equation (6) to determine D0. The characteristic at T-dependence of s-wave superconductors at low T is evident. The ts yield l 189 nm and

2.95 meV.

4 NATURE COMMUNICATIONS | 6:7820 | DOI: 10.1038/ncomms8820 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms8820 ARTICLE

100 Y0.8Ca0.2Ba2Cu3Oy

Jc (sf) (MA cm2)

T *

T (K)

0 0.10 0.15 0.20 0.25

Hole concentration, p

Figure 4 | Predicted Jc across the YBCO phase diagram. Map of Jc(sf) across the phase diagram for Y0.8Ca0.2Ba2Cu3Oy calculated from the

superuid density23,24 using equation (4). A sharp peak is centred on the critical doping where the pseudogap T* line falls to zero (solid white curve). The triangles show the low-T data points for T* reported from eld-dependent resistivity29. A second smaller peak is predicted just below PE0.12 where charge ordering has been reported33. The circles are Tc data points.

in this way from the ground-state doping-dependent superuid density, rs(0), reported previously for Y0.8Ca0.2Ba2Cu3O7d (refs 23,24). Using l 0

rs 0

and the theoretical d-wave T-dependence of rs(T)/rs(0) (ref. 6), we generate the full p- and

T-dependence of Jc, which is shown as a false-colour map in the pT plane in Fig. 4.

Predicted values are seen to rise to a sharp maximum of about30 MA cm 2 centred at p 0.19 holes per Cu in the slightly

overdoped region, beyond optimal doping (PE0.16) where Tc reaches its maximum. (Doping with Ca introduces some impurity scattering that lowers the superuid density. As a consequence, the maximum Jc is less than that predicted from the superuid density for Ca-free YBa2Cu3O7: 37 MA cm 2 for nearly fully oxygenated chains, and B42 MA cm 2 for fully ordered chains25.)

It is important to understand the signicance of this skewed behaviour of Jc(p) relative to optimal doping shown in Fig. 4.

High-Tc cuprates are characterized by the opening of a gap in the normal-state excitation spectrum, which is probably associated with reconstruction of the Fermi surface due to short-range magnetic order2628. This phenomenon is associated with the so-called pseudogap that dominates the properties of optimally doped and under-doped cuprates, resulting in weak superconductivity as indicated by a reduction in condensation energy, superuid density and their associated critical elds22.

The link between Jc and the pseudogap is made by plotting in Fig. 4 the previously determined T* line where the pseudogap closes, as determined from eld-dependent resistivity studies on epitaxial thin lms29,30. The T* data extend to much higher temperatures, but some of the lower-T data points are visible in the plot. The key result here is that Jc maximizes just at the point where the pseudogap closes and T*-0. Note, also, how the ridge in Jc(T, p) follows the T* line, inclining towards lower doping at higher T notwithstanding the fact that these two quantities are determined by quite different techniques. Clearly the rapid decline in Jc below T* is due to the opening of the pseudogap and the consequent crossover to weak superconductivity. One is also impressed by the resemblance between this phenomenology and that associated with the presence of a quantum critical point31, where a bubble of high Jc is centred on the point where T*-0.

This sharp peak in Jc(p) at p pcrit is recently conrmed by our

wider group32, but Fig. 4 also predicts a second smaller peak at pE0.12. This second peak is also apparent in the upper critical eld Hc2 of YBa2Cu3O7d (ref. 33), and the search for a second peak in Jc provides a strong test for the present ideas. A similar double peak in rs(p) is found in La2xSrxCuO4 (ref. 34), suggesting that a double peak in Jc(p) will be a common, perhaps universal, cuprate behaviour. Another test is that if Jc(T)

varies as r3=2s, then Jc(0)2/3 should be diminished by impurity scattering in the same canonical way that the superuid density is reduced35. This is distinguished by a much more rapid reduction in rs than in Tc. These ideas can be tested in Zn-substituted

YBa2Cu3O7d and have recently been conrmed by us.

Rened model. We return now to better justify the theoretical basis for our observations. The usual approaches to eld distribution and critical currents are those of Brojeny and Clem36 and Brandt and Indenbom37, where for a very thin lm the y-component of this eld, By(x a), diverges at the lm edges36.

As a consequence, Abrikosov vortices must enter from the edges and there then exists a domain extending in from the edges where the local current density J(x) is constant ( Jc) and in which these

vortices are pinned. At the inner edge of this domain, By falls to zero with innite slope37. The onset of dissipation, dening the overall lm Jc, occurs when this domain extends to the centre of the lm at x 0 and Jc is now the global value not just the local

value.

But, as we have suggested, an alternative approach is to consider the entry of vortices, not from the edges but from the large at surfaces. For a very thin conductor the x-component of this self-eld at the surface, Bx(x, y b), is uniform across the

width and of magnitude m0bJ (ref. 36). Consequently, if the current I is increased until this eld reaches B Bc1, then vortices

will nucleate at the at surface in the form of closed loops around the conductor surface normal to the transport current38. These loops will tend to collapse inwards under the self-imposed Lorentz force (J B) on each vortex. It is only surface and bulk

pinning that will prevent them from migrating to the conductor centre and self-annihilating there. Thus, inverting the above relation

Jc Bc1=m0b Hc1=b: 8 where the inequality arises from the, as yet, indeterminate role of pinning.

Now consider the interesting case when bEl. These vortex loops now experience the additional attractive force of adjacent vortices of opposite sense located just l apart. This force becomes unbounded and thus inevitably overcomes pinning. The vortices mutually annihilate at the centre and the process continues indenitely causing dissipation. This vortex entry from the faces denes a rst critical current density given by the equality sign in equation (8), which is alternative to a second, which is associated with vortex entry from the edges. Which of these has the smaller Jc and therefore is the operative mechanism?

To answer this, let us suppose that J Hc1/b over the full

conductor cross-section and we calculate the degree of ux entry at the edges. Rearranging equation (13) of Brojeny and Clem36 (or indeed equation (8) for a non-vanishingly-thin lm) the y-component of the eld near the edges for a uniform current distribution becomes

By x

Bc1

2p ln

: 9

From this we nd that By falls to the value of Bc1 when x 0.917a, that is, perpendicular vortices can enter at either

edge to only 4.2% of the lm width. Thus the dominant

x a

NATURE COMMUNICATIONS | 6:7820 | DOI: 10.1038/ncomms8820 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 5

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms8820

800

[afii9838]

700

400

YBCO

After irrad

Temperature (K)

Before irrad

600

NbN

[afii9838] L(T) (nm)

J c(MA cm2)

[afii9838] L(T) (nm)

J c(MA cm2)

500

300

[afii9838]

400

[ref. 15] [ref. 13] [ref. 13] [ref. 14] [ref. 14]

300

200

2 6 8

Figure 5 | Thickness and irradiation dependence of Jc. The T-dependence of Jc(sf) and l calculated from equation (4) for thin lms of (a) NbN with different thicknesses, b 4 nm (refs 13,14) and 11 nm (ref. 15) and different bridge widths. (b) YBCO before and after nano-dot irradiation45. The solid red

and blue curves are weak-coupling d-wave ts to the calculated l(T) data, before and after irradiation, respectively. The red and blue dashed curves are the respective back-calculated Jc(T) curves from these d-wave ts. Note in a the small variation in l0 despite the quite large variation in Jc, and l0 is evidently independent of b when bol. Values of D0 2.950.02 meV obtained from tting the low-T behavior of l(T) also reveal little variation. In b, Jc(sf) and l

are independent of irradiation despite the large increase in pinning evidenced by a 60% increase in Jc(T, H) above 1 Tesla.

100 0 4 10 12 14 16

100 0 20 30 40 50 80 90 100

Temperature (K)

mechanism for Jc when bEl is ux entry from the large at surfaces and Jc is given by:

Jc sf Hc1=b: 10

We note that, when b l, equation (10) becomes equivalent to

equation (4), but there is yet one nal ingredient to add. The energy of formation of a vortex/anti-vortex pair on opposite faces of the lm is reduced by an interaction term f20= 2pm0l2

0 2b=l

(Y,Eu) Ba2Cu3O7

J c (sf) (MA cm2 )

J c(sf) (MA cm2 )

(ref. 38). Here K0(x) is the zeroth-

order Bessel function of the second kind, and 2b is the vortex separation. This diverges logarithmically at small b so that Bc1 is reduced as Bc1 B1c1 b=l, where B1c1 is the bulk value at large

b. We adopt a heuristic crossover between the two limits in the form Bc1 B1c1tanhb=l. Combining with equation (10) and

dropping the N sign, we obtain our nal result:

Jc sf

Hc1

l l=b

tanh

b l

: 11

where Hc1 is the bulk value. This concurs with equation (4) but with the additional correction factor (l/b)tanh(b/l). This accounts for all isotropic superconductors in Figs 1 and 2 with bol. In the case of anisotropic superconductors with lxoly (as in the case of high-Tc cuprates), we simply rescale the problem b-b (lx/ly) and Jc in equation (11) becomes:

Jc T; sf

Hc1 T

lab T

0 0 1 2 3 4

: 12

Here we have replaced ly by lc and lx by lab, as is the usual convention for the cuprates, where lc4lab. Equation (12) is the full generalization of equation (4). For bolc, we recover equation (4), while for b4lc we recover the 1/b falloff in Jc. A consequence of this is that if brlc(0), then equation (4) remains applicable to the highest temperatures because lc(T) diverges as T-Tc and the small-b limit is preserved. Let us now compare this functional dependence on b with experimental data.

Figure 5a shows our analysis using equation (4) of ve sets of Jc(T) data for NbN. Four13,14 have b 4 nm and one15 has

b 11 nm, all much less than l 194 nm. Despite these different

thicknesses, all ve data sets yield values of l close to this measured value (see Supplementary Table 1). The implication is that equation (4) is accurate for all brl. In contrast, for the alternative case b4l, Stejic et al.39 report for Nb-Ti lms an inverse thickness dependence of Jc(T), where Jcp1/b. This is precisely what we argue above.

Pursuing this further, Zhou et al.40 report self-eld Jc measurements on many epitaxial thin lms of YBa2Cu3O7d and (Y0.67Eu0.33)Ba2Cu3O7d of varying thickness deposited on

SrTiO3 by pulsed laser deposition. Figure 6a shows the Jc data measured at 75.5 K as a function of lm thickness (data points). Also plotted is equation (12) tted to this datasolid curve. The excellent agreement is misleading. The value of lc 620 nm

Film thickness, 2b (m)

Figure 6 | Thickness dependence of Jc for YBCO. Jc(sf) for epitaxial YBCO lms versus lm thickness. (a) (Y0.67Eu0.33)Ba2Cu3O7d at T 75.5 K, from

Zhou et al.40 The solid curve is equation (12) with lc 620 nm. This value

is too low and reects increasingly non-uniform composition and microstructure across the thickness. (b) YBCO systems with a highly uniform through-thickness microstructure: Feldmann et al.42 (black data points and curve), with lab(77) 3395 nm and lc(77) 1,153238 nm

progressively thinned by ion milling; Zhou et al.43 (blue data points and curve) gives lab(75.6) 2534 nm and lc(75.6) 1,06328 nm;

Feldmann et al.44 (red data points and curve) gives lab(75.6) 2332 nm

and lc(75.6) 1,980790 nm.

lc T =b

tanh

b lc T

6 NATURE COMMUNICATIONS | 6:7820 | DOI: 10.1038/ncomms8820 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms8820 ARTICLE

is too low and reects a non-uniform composition and microstructure, which grows with increasing lm thickness. Fits to other available data including Foltyn et al.1 and Arendt et al.41 give similar lc values, 634 and 654 nm, respectively. Again these are smaller than the expected Z1,100 nm. In contrast,

Fig. 6b shows data for lms with an intentionally highly uniform through-thickness microstructure4244. In the case of Feldmann42, this uniformity was conrmed by progressive ion milling of a single lm. Here the lc values shown in the gure are now indeed realistic.

We conclude that equations (11) and (12) provide a good description of Jc for uniform lms in the general case when bal.

Of course, as b is increased, the alternative Jc mechanism involving ux entry from the edges must eventually become dominant.

Pinning. Our claim is that self-eld Jc is independent of pinning when brl unless, perhaps, very strong pinning is introduced. We illustrate this with one of the few examples available. Lin et al.45 scribed an array of nanocolumns into a YBCO microbridge using a focused electron beam. The array had a lattice constant of 90 nm (corresponding to a matching eld of0.25 Tesla) and diameters about double the coherence length at77 K, while the lm thicknesses were 50 and 90 nm, thus satisfying our condition brl. We show our analysis of the

H 0 self-eld Jc in Fig. 5b. These authors report a 60% increase

in the in-eld Jc(T) above 1 Tesla but, we note, there is essentially no change in Jc(T, sf), indeed a small decrease consistent with a small loss of effective cross-sectional area. There is no change in the inferred l0. Thus the evident increase in bulk pinning and inevitable changes in surface roughness and surface pinning have no apparent effect on Jc(T, sf), consistent with our hypothesis.

Similarly, with 25 MeV 16O ion irradiation of YBCO lms Roas et al.46 report an 80% increase in Jc above 1 Tesla, but a signicant reduction in Jc(sf). On the other hand, a later study by this group47 showed fast neutron irradiation lifted Jc(sf) at 4.2 K from 19.4 to 32 MA cm 2. While this is no more than the Jc(0, sf)

values we quote above for our pristine lms, it suggests that strong pinning could, in the extreme, overtake the Silsbee mechanism we advance here. But here they use an extremely high electric eld criterion of 50 mV cm 1, and the apparent enhancement could simply be caused by a reduction in n-value due to irradiation. Later neutron irradiation studies4850 on epitaxial YBCO lms, reported only a detrimental effect on Jc(sf)

while lifting in-eld performance. On the basis of this, we feel it still remains to be conrmed that pinning centres created by irradiation can improve Jc(sf). Further literature examples supporting the pinning independence of Jc(sf) are given in Supplementary Note 4.

DiscussionWe have shown that for thin lms of thickness bol, the self-eld

Jc is given by Hc/l for type-I superconductors and Hc1/l for type-II superconductors. This provides a simple, direct means to determine the absolute magnitude of the penetration depth and means that, contrary to widespread thinking, Jc(T, sf) is a fundamental property independent of pinning landscape and microstructural architecture. We have thus conrmed Silsbees hypothesis for all the superconductors we have examined. In the case of the cuprates, our prediction of a sharp peak in Jc(p, sf) at the critical doping, where T*-0, is borne out in separate studies32.

To conclude, we suggest the following possible tests of the ideas we have presented. We predict a second peak in Jc(p, sf) near pE0.12, near which charge ordering occurs33, and we suggest

that, for Zn-substituted YBa2Cu3Oy, Jc(sf)2/3 will be suppressed by impurity scattering in the same canonical manner as the superuid density. Along with this the T-dependence of Jc(sf)

should cross over from a linear-in-T behaviour to T2 consistent with the superuid density6. A further key test would be to measure and correlate both superuid density and self-eld Jc in a single lm as a function of progressive disorder via irradiation. Inspection of equation (11) shows that, with increasing lm thickness, Jc(sf) should cross over from a l 3 dependence when bol to a l 2 dependence when b4l. It should be relatively straightforward to test this. The approach reported here should readily translate to superconducting nanowires and could be used to measure the transport mass anisotropy in layered superconductors by comparing Jc(sf) measurements in a-axis-and c-axis-aligned lms. A key challenge will be to treat the crossover from a Silsbee-dominated mechanism in self-eld to a more conventional pinning-dominated mechanism with ux entry from the edges as external eld is increased. The implications for a.c. loss should be explored and, nally, the model we present here lends itself to an error-function onset to resistance (rather than the conventional power law) due to the local distribution of superuid density51.

Methods

Data sourcesself-eld critical current density. Of the vast literature for Jc, surprisingly few data sets are available that meet the collective requirements for the present analysis. These are as follow: we require transport (not magnetization) Jc data; data reported under self-eld conditions; Jc data for weak-link-free thin lms; in which brl; and that extend down to temperatures, Tr0.2Tc. The analyses reported here more or less exhaust the available data.

Data sourcespenetration depth. We have chosen to test equations (3) and (4) using literature data for l0 and not for Hc and Hc1 for which even recent literature shows quite divergent values. An illustrative example reports a breakdown of the Uemura relation between Tc and rs by a factor of eight, based on Hc1 data for

Ba0.6K0.4Fe2As2 (ref. 52). Subsequent measurements of superuid density using muon spin relaxation showed that this system was in fact in full agreement with the Uemura relation5355. Early penetration depth data are variable in quality, and microwave measurements often do not yield absolute values, probing as they do values of Dl l(T)l(0), only. Where possible we have relied on muon spin

relaxation or polarized neutron reectometry for l values, because these directly probe the eld prole and tend to be rather reproducible from one group to another. Where possible we also give multiple sources, and the ranges from these sources are reected in Supplementary Table 1 and the error bars in Fig. 2 and Supplementary Fig. 2.

References

1. Foltyn, S. R. et al. Materials science challenges for high-temperature superconducting wire. Nat. Mater. 6, 631642 (2007).

2. Silsbee, F. B. Note on electrical conduction in metals at low temperatures.J. Franklin Inst. 184, 111 (1917).3. Poole, C. P., Farach, H. A., Creswick, R. J. & Prozorov, R. Superconductivity Chaps 2, 11, 12, 14 (Academic Press, 2007).

4. London, H. Phase-equilibrium of supraconductors in a magnetic eld. Proc. R. Soc. Lond. A 152, 650 (1935).

5. Prozorov, R. & Kogan, V. G. London penetration depth in iron-based superconductors. Rep. Prog. Phys. 74, 124505 (2011).

6. Won, H. & Maki, K. d-wave superconductor as a model of high-Tc superconductors. Phys. Rev. B 49, 13971402 (1994).

7. Kogan, V. G., Martin, C. & Prozorov, R. Superuid density and specic heat within a self-consistent scheme for a two-band superconductor. Phys. Rev. B 80, 014507 (2009).

8. Sonier, J. E. et al. Hole-doping dependence of the magnetic penetration depth and vortex core size in YBa2Cu3Oy: evidence for stripe correlations near 1/8 hole doping. Phys. Rev. B 76, 134518 (2007).

9. Cichorek, T. et al. Pronounced enhancement of the lower critical eld and critical current deep in the superconducting state of PrOs4Sb12. Phys. Rev. Lett.

94, 107002 (2005).10. MacLaughlin, D. E. et al. Muon spin relaxation and isotropic pairing in superconducting PrOs4Sb12. Phys. Rev. Lett. 89, 157001 (2002).

11. Bauer, E. D., Frederick, N. A., Ho, P.-C., Zapf, V. S. & Maple, M. B. Superconductivity and heavy fermion behavior in PrOs4Sb12. Phys. Rev. B 65, 100506(R) (2002).

NATURE COMMUNICATIONS | 6:7820 | DOI: 10.1038/ncomms8820 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 7

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms8820

12. Wimbush, S. C., Schultz, L. & Holzapfel, B. Critical current in YNi2B2C and HoNi2B2C thin lms. Physica C 388-389, 191192 (2003).

13. Bartolf, H., Engel, A., Schilling, A., Ilin, K. & Siegel, M. Fabrication of metallic structures with lateral dimensions less than 15 nm and Jc(T)-measurements in

NbN micro- and nanobridges. Physica C 468, 793796 (2008).14. Engel, A. et al. Temperature- and eld-dependence of critical currents in NbN microbridges. J. Phys. Conf. Ser. 97, 012152 (2008).

15. Clem, J. R., Bumble, B., Raider, S. I., Gallagher, W. J. & Shih, Y. C. Ambegaokar-Baratoff-Ginzburg-Landau crossover effects on the critical current density of granular superconductors. Phys. Rev. B 35, 66376642 (1987).

16. Dagan, Y., Krupke, R. & Deutscher, G. Determination of the superconducting gap in YBa2Cu3O7d by tunneling experiments under magnetic elds. Phys. Rev.

B 62, 146149 (2000).17. Tallon, J. L., Barber, F., Storey, J. G. & Loram, J. W. Coexistence of the superconducting energy gap and pseudogap above and below the transition temperature of cuprate superconductors. Phys. Rev. 87, 140508(R) (2013).

18. Yu, L. et al. Evidence for two separate energy gaps in underdoped high-temperature cuprate superconductors from broadband infrared ellipsometry. Phys. Rev. Lett. 100, 177004 (2008).

19. Sacuto, A. et al. Electronic Raman scattering in copper oxide superconductors: Understanding the phase diagram. C. R. Physique 12, 480501 (2011).

20. Hardy, W. N., Bonn, D. A., Morgan, D. C., Liang, R. & Zhang, K. Precision measurements of the temperature dependence of l in YBa2Cu306.95: strong evidence for nodes in the gap function. Phys. Rev. Lett. 70, 39994002 (1993).

21. Kihlstrom, K. E., Simon, R. W. & Wolf, S. A. Tunneling a2F(o) from sputtered thin-lm NbN. Phys. Rev. B 32, 18431845 (1985).

22. Loram, J. W., Luo, J., Cooper, J. R., Liang, W. Y. & Tallon, J. L. Evidence on the pseudogap and condensate from the electronic specic heat. J. Phys. Chem. Solids 62, 5964 (2001).

23. Tallon, J. L., Loram, J. W. & Cooper, J. R. The superuid density in cuprate high-Tc superconductorsa new paradigm. Phys. Rev. B 68, 180501(R) (2003).

24. Tallon, J. L., Cooper, J. R., Naqib, S. H. & Loram, J. W. Scaling relation for the superuid density in cuprate superconductors. Phys. Rev. B 73, 180504(R) (2006).

25. Tallon, J. L. et al. In-plane anisotropy of the penetration depth due to superconductivity on the CuO chains in YBa2Cu3O7d, Y2Ba4Cu7O15d and

YBa2Cu4O8. Phys. Rev. Lett. 74, 10081011 (1995).26. Storey, J. G. & Tallon, J. L. Two-component electron uid in underdoped high-Tc cuprate superconductors. Europhys. Lett. 98, 17011 (2012).

27. Yang, K.-Y., Rice, T. M. & Zhang, F.-C. Phenomenological theory of the pseudogap state. Phys. Rev. B 73, 174501 (2006).

28. Fujita, K. et al. Simultaneous transitions in cuprate momentum-space topology and electronic symmetry breaking. Science 344, 612616 (2014).

29. Naqib, S. H., Cooper, J. R., Tallon, J. L., Islam, R. S. & Chakalov, R. A. Doping phase diagram of Y1xCaxBa2(Cu1yZny)3O7d from transport measurements: tracking the pseudogap below Tc. Phys. Rev. B 71, 054502 (2005).

30. Tallon, J. L., Barber, F., Storey, J. G. & Loram, J. W. Coexistence of the superconducting gap and pseudogap above and below the transition temperature in cuprate superconductors. Phys. Rev. B. 87, 140508(R) (2013).

31. Tallon, J. L. et al. Critical doping in overdoped high-Tc superconductors: a quantum critical point? Phys. Stat. Sol. (b) 215, 531540 (1999).

32. Talantsev, E. F. et al. Hole doping dependence of critical currents in YBCO conductors. Appl. Phys. Lett. 104, 242601 (2014).

33. Grissonnanche, G. et al. Direct measurement of the upper critical eld in a cuprate superconductor. Nat. Commun. 5, 3280 (2014).

34. Panagopoulos, C. et al. Evidence for a generic quantum glass transition in high-Tc cuprates. Phys. Rev. B 66, 064501 (2002).

35. Bernhard, C. et al. Suppression of the superconducting condensate in the high-Tc cuprates by Zn substitution and overdoping: evidence for an unconventional pairing state. Phys. Rev. Lett. 77, 23042307 (1996).

36. Brojeny, A. B. & Clem, J. R. Self-eld effects upon critical current density of at superconducting strips. Supercond. Sci. Technol. 18, 888895 (2005).

37. Brandt, E. H. & Indenbom, M. Type-II-superconductor strip with current in a perpendicular magnetic eld. Phys. Rev. B 48, 1289312906 (1993).

38. Tinkham, M. Introduction to Superconductivity Ch. 5 (McGraw-Hill, 1996).39. Stejic, G. et al. Effect of geometry on the critical currents of thin lms. Phys. Rev. B 49, 12741288 (1994).

40. Zhou, H. et al. Improved microstructure and enhanced low-eld Jc in

(Y0.67Eu0.33)Ba2Cu3O7d lms. Supercond. Sci. Technol. 21, 025001 (2008).

41. Arendt, P. N. et al. High critical current YBCO coated conductors based on IBAD MgO. Physica C 412-414, 795800 (2004).

42. Feldmann, D. M. et al. Through-thickness superconducting and normal-state transport properties revealed by thinning of thick lm ex situ YBa2Cu3O7x coated conductors. Appl. Phys. Lett. 83, 39513953 (2003).

43. Zhou, H. et al. Thickness dependence of critical current density in YBa2Cu3O7d lms with BaZrO3 and Y2O3 addition. Supercond. Sci. Technol. 22, 85013 (2009).

44. Feldmann, D. M. et al. 1000 A cm 1 in a 2mm thick YBa2Cu3O7xlm with BaZrO3 and Y2O3 additions. Supercond. Sci. Technol. 23, 115016 (2010).

45. Lin, J.-Y. et al. Flux pinning in YBa2Cu3O7d thin lms with ordered arrays of columnar defects. Phys. Rev. B 54, R12717R12720 (1996).

46. Roas, B., Hensel, B., Saemann-Ischenko, G. & Schultz, L. Irradiation-induced enhancement of the critical current density of epitaxial YBa2Cu3O7x thin lms.

Appl. Phys. Lett. 54, 10511053 (1989).47. Schindler, W., Roas, B., Saemann-Ischenko, G., Schultz, L. & Gerstenberg, H. Anisotropic enhancement of the critical current density of epitaxial YBa2Cu3O7x lms by fast neutron irradiation. Physica C 169, 117122 (1990).

48. Vostner, A. et al. Neutron irradiation studies on Y-123 thick lms deposited by liquid phase epitaxy on single crystalline substrates. Physica C 399, 120128 (2003).

49. Withnell, T. D., Schppl, K. R., Durrell, J. H. & Weber, H. W. Effects of irradiation on vicinal YBCO thin lms. IEEE Trans. Appl. Supercon. 19, 29252928 (2009).

50. Eisterer, M., Fuger, R., Chudy, M., Hengstberger, F. & Weber, H. W. Neutron irradiation of coated conductors. Supercond. Sci. Technol. 23, 014009 (2010).

51. Luan, L. et al. Local measurement of the superuid density in the pnictide superconductor Ba(Fe1xCox)2As2 across the superconducting dome. Phys. Rev.

Lett. 106, 067001 (2011).52. Ren, C. et al. Evidence for two energy gaps in superconducting Ba0.6K0.4Fe2As2

single crystals and the breakdown of the Uemura plot. Phys. Rev. Lett. 101, 257006 (2008).53. Khasanov, R. et al. Two-gap superconductivity in Ba1xKxFe2As2: a

complementary study of the magnetic penetration depth by muon-spin rotation and angle-resolved photoemission. Phys. Rev. Lett. 102, 187005 (2009).54. Goko, T. et al. Superconducting state coexisting with a phase-separated static magnetic order in (Ba,K)Fe2As2, (Sr,Na)Fe2As2, and CaFe2As2. Phys. Rev. B 80, 024508 (2009).

55. Guguchia, Z. et al. Muon-spin rotation measurements of the magnetic penetration depth in the iron-based superconductor Ba1xRbxFe2As2. Phys. Rev.

B 84, 094513 (2011).

Acknowledgements

J.L.T. thanks the Marsden Fund of New Zealand and the MacDiarmid Institute for Advanced Materials and Nanotechnology for nancial support. We also thank N.M. Strickland, S.C. Wimbush and J.G. Storey for developing the HTS-based Jc measuring rig, and Professor A.M. Campbell, S.C. Wimbush, A. Malozemoff and the referees for helpful comments on this manuscript.

Author contributions

Both authors contributed equally to all aspects of this work.

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/naturecommunications

Web End =http://www.nature.com/ http://www.nature.com/naturecommunications

Web End =naturecommunications

Competing nancial interests: The authors declare no competing nancial interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/

Web End =http://npg.nature.com/ http://npg.nature.com/reprintsandpermissions/

Web End =reprintsandpermissions/

How to cite this article: Talantsev, E. F. & Tallon, J. L. Universal self-eld critical current for thin-lm superconductors. Nat. Commun. 6:7820 doi: 10.1038/ncomms8820 (2015).

This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the articles Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Web End =http://creativecommons.org/licenses/by/4.0/

8 NATURE COMMUNICATIONS | 6:7820 | DOI: 10.1038/ncomms8820 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

Word count: 8266

Show less

Abstract

Translate

For any practical superconductor the magnitude of the critical current density, J_c , is crucially important. It sets the upper limit for current in the conductor. Usually J_c falls rapidly with increasing external magnetic field, but even in zero external field the current flowing in the conductor generates a self-field that limits J_c . Here we show for thin films of thickness less than the London penetration depth, λ, this limiting J_c adopts a universal value for all superconductors--metals, oxides, cuprates, pnictides, borocarbides and heavy Fermions. For type-I superconductors, it is H_c /λ where H_c is the thermodynamic critical field. But surprisingly for type-II superconductors, we find the self-field J_c is H_c1 /λ where H_c1 is the lower critical field. J_c is thus fundamentally determined and this provides a simple means to extract absolute values of λ(T) and, from its temperature dependence, the symmetry and magnitude of the superconducting gap.

Details

Title

Universal self-field critical current for thin-film superconductors

Author

Talantsev, E F; Tallon, J L

Pages

7820

Publication year

2015

Publication date

Aug 2015

Publisher

Nature Publishing Group

e-ISSN

20411723

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1038/ncomms8820

ProQuest document ID

1700960643

Universal self-field critical current for thin-film superconductors

Jump to:

Full text

Abstract

Details

Suggested sources