Anomalous critical fields in quantum critical superconductors

Abstract

Fluctuations around an antiferromagnetic quantum critical point (QCP) are believed to lead to unconventional superconductivity and in some cases to high-temperature superconductivity. However, the exact mechanism by which this occurs remains poorly understood. The iron-pnictide superconductor BaFe₂ (As_1-x P_x )₂ is perhaps the clearest example to date of a high-temperature quantum critical superconductor, and so it is a particularly suitable system to study how the quantum critical fluctuations affect the superconducting state. Here we show that the proximity of the QCP yields unexpected anomalies in the superconducting critical fields. We find that both the lower and upper critical fields do not follow the behaviour, predicted by conventional theory, resulting from the observed mass enhancement near the QCP. Our results imply that the energy of superconducting vortices is enhanced, possibly due to a microscopic mixing of antiferromagnetism and superconductivity, suggesting that a highly unusual vortex state is realized in quantum critical superconductors.

ARTICLE

Received 20 Feb 2014 | Accepted 28 Oct 2014 | Published 5 Dec 2014

C. Putzke1, P. Walmsley1, J.D. Fletcher2, L. Malone1, D. Vignolles3, C. Proust3, S. Badoux3, P. See2, H.E. Beere4,

D.A. Ritchie4, S. Kasahara5, Y. Mizukami5,6, T. Shibauchi5,6, Y. Matsuda5 & A. Carrington1

Fluctuations around an antiferromagnetic quantum critical point (QCP) are believed to lead to unconventional superconductivity and in some cases to high-temperature superconductivity. However, the exact mechanism by which this occurs remains poorly understood. The iron-pnictide superconductor BaFe2(As1 xPx)2 is perhaps the clearest example to date of a high-temperature quantum critical superconductor, and so it is a particularly suitable system to study how the quantum critical uctuations affect the superconducting state. Here we show that the proximity of the QCP yields unexpected anomalies in the superconducting critical elds. We nd that both the lower and upper critical elds do not follow the behaviour, predicted by conventional theory, resulting from the observed mass enhancement near the QCP. Our results imply that the energy of superconducting vortices is enhanced, possibly due to a microscopic mixing of antiferromagnetism and superconductivity, suggesting that a highly unusual vortex state is realized in quantum critical superconductors.

DOI: 10.1038/ncomms6679 OPEN

Anomalous critical elds in quantum critical superconductors

1 H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK. 2 National Physical Laboratory, Hampton Road, Teddington TW11 0LW, UK. 3 Laboratoire National des Champs Magntiques Intenses (CNRS-INSA-UJF-UPS), 31400 Toulouse, France. 4 Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, UK. 5 Department of Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan. 6 Department of Advanced Materials Science, University of Tokyo, Kashiwa 277-8561, Japan. Correspondence and requests for materials should be addressed to A.C. (email: mailto:[email protected]

Web End [email protected] ).

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Quantum critical points (QCPs) can be associated with a variety of different orderdisorder phenomena, however, so far superconductivity has only been found close to

magnetic order. Superconductivity in heavy fermions, iron pnictides and organic salts is found in close proximity to antiferromagnetic order1,2, whereas in the cuprates the nature of the order (known as the pseudogap phase) is less clear3. The normal state of these materials has been widely studied and close to their QCPs non-Fermi liquid behaviour of transport and thermodynamic properties are often found, however, comparatively little is known about how the quantum critical uctuations affect the superconducting state4. This is important as it is the difference in energy between the normal and superconducting state that ultimately determines the critical temperature Tc.

Among the various iron-pnictide superconductors,

BaFe2(As1 xPx)2 has proved to be the most suitable family for

studying the inuence of quantum criticality on the super-conducting state. This is because the substitution of As by P introduces minimal disorder as it tunes the material across the phase diagram from a spin-density wave antiferromagnetic metal, through the superconducting phase to a paramagnetic metal5. The main effect is a compression of the c axis arising from the smaller size of the P ion compared with As, which mimics the effect of external pressure6. Normal state properties such as the temperature dependence of the resistivity7 and spin-lattice relaxation rate8 clearly point to a QCP at x 0.30. Measurements

of superconducting state properties that show signatures of quantum critical effects include the magnetic penetration depth l and the heat capacity jump at Tc, DC9,10. Both of these quantities show a strong increase as x tends to 0.30, and it is shown that this could be explained by an underlying approximately sixfold increase in the quasiparticle effective mass m* at the QCP10.

In the standard single-band GinzburgLandau theory, the upper critical eld is given by

Hc2

f0 2pm0x2GL

T (K)

0H (T)

10 15 20 25

30 10 20 40

30 50 60

0

60

0.38 0.35 0.30

0.43

5 x = 0.49

15

00

50

40

; 1

where f0 is the ux quantum and xGL is the GinzburgLandau coherence length. In the clean limit at low temperature, xGL is

usually well approximated by the BCS coherence length, which results in Hc2p(m*D)2, where m* is the mass of the quasiparticles and D is the superconducting gap. This simplied analysis is borne out by the full strong coupling BCS theory11. Hence, a strong peak in m* at the QCP should result in a corresponding increase in Hc2 as well as the slope of Hc2 at

Tc h0 dHc2=dT

Tc

. This latter quantity is often more easily

accessible experimentally because of the very high Hc2 values in compounds such as iron pnictides for TooTc and also because the values of Hc2 close to Tc are not reduced by the effect of the magnetic eld on the electron spin (Pauli limiting effects).

For the lower critical eld Hc1, standard GinzburgLandau theory predicts that

Hc1

f0 4pm0l2 ln k

0H c2(T)

10

30

20

10

(a.u.)

0

10

Crystal

Centre

Edge

0.3

20

0.2

|B r0.5| (mT0.5)

15

B(mT)

Hp

Hirr

0.1

10

0.0

5

5

10

15

20

0 5 10 15

0.1

0H (mT)

0H (mT)

0:5

; 2

where k l/xGL, and so the observed large peak in l at the

QCP9 should result in a strong suppression of Hc1. Here we show that the exact opposite, a peak in Hc1 at the QCP, occurs

in BaFe2(As1 xPx)2, and in addition the expected sharp increase

in Hc2 is not observed. This suggests that the critical elds of quantum critical superconductors strongly violate the standard theory.

ResultsUpper critical eld Hc2. We measured Hc2 parallel to the c axis, in a series of high-quality single-crystal samples of BaFe2(As1 xPx)2 spanning the superconducting part of the phase

diagram using two different techniques. Close to Tc(H 0), we

measured the heat capacity of the sample using a micro-calorimeter in elds up to 14 T (see Fig. 1a). This gives an unambiguous measurement of Hc2(T) and the slope h0, which unlike transport measurements is not complicated by contributions from vortex motion12. At a lower temperature, we used micro-cantilever torque measurements in pulsed magnetic elds up to 60 T. Here an estimate of Hc2 was made by observing the eld where hysteresis in the torque magnetization loop closes (see Fig. 1b). Although, strictly speaking, this marks the irreversibility line Hirr, this is a lower limit for Hc2(0) and in superconductors with negligible thermal uctuations and low anisotropy such as BaFe2(As1 xPx)2 Hirr should coincide approximately with Hc2.

Indeed, in Fig. 2 we show that the extrapolation of the high-temperature-specic heat results, using the HelfandWerthamer (HW) formula13, to zero temperature are in good agreement with the irreversibility eld measurements showing both are good estimates of Hc2(0).

In the clean limit we would expect (Hc2(0))1/2/Tc to be proportional to the renormalized effective mass m*. Surprisingly, we show in Fig. 2 that this quantity increases by just B20% from x 0.47 to x 0.30, whereas m* increases by B400% for the

same range of x.

Lower critical eld Hc1. We measured Hc1 in our BaFe2 (As1 xPx)2 samples using a micro-Hall probe array. Here the

0

Figure 1 | Determination of critical elds. (a) Hc2(T) data close to

Tc(H 0) from heat capacity measurements for different samples of

BaFe2(As1 xPx)2. (b) Magnetic torque versus rising and falling eld for a

sample with x 0.40 at T 1.5 K. The irreversibility eld Hirr is marked. (c) Magnetic ux density B versus applied eld H as measured by the micro-Hall sensors, for x 0.35 and T 18 K at two different sensor

positions: one at the edge of the sample and the other close to the centre (schematic inset). (d) Remnant eld Br after subtraction of the linear term due to ux leakage around the sample. |Br|0.5 versus m0H is plotted as this best linearizes Br(H)14. Note that the changes in linearity of B(H) evident in d are not visible by eye in c.

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60

40

20

10

5

0

0.5 K1 )

0H c2(0) (T)

m* / m b

0.5 /T c (T

0 0.3 0.5 0.5

0.3

0.2

0.1

0.0

( 0H c2 (0))

0.4 0.3

0.4

X

X

Figure 2 | Upper critical eld as a function of concentration x. (a) Hc2(0) in BaFe2(As1 xPx)2 estimated from the slope of Hc2(T) close to Tc using

Hc2 0 0:73Tc dHc2=dT

j

Tc (squares) 13, and also estimates of Hc2(0) from the irreversibility eld at low temperature (T 1.5 K) measured by torque

magnetometry (circles). Error bars on Hc2 (circles) represent the uncertainties in locating Hirr and (squares) in extrapolating the values close to Tc to T 0.

Error bars on x represent s.d. (b) The same data plotted as (Hc2(0))0.5/Tc, which, in conventional theory, are proportional to the mass enhancement m*. The mass renormalization m*/mb derived from specic heat measurements is shown for comparison (triangles) 10. The dashed line is a guide to the eye and solid lines in both parts are linear ts to the data.

60 x = 0.29 x = 0.30 x = 0.31 x = 0.36 x = 0.39 x = 0.47

magnetic ux density B is measured at several discrete points a few microns from the surface of the sample. Below Hc1, B increases linearly with the applied eld H due to incomplete shielding of the sensor by the sample. Then, as the applied eld passes a certain eld Hp, B increases more rapidly with H indicating that vortices have entered the sample (see Fig. 1c,d). Care must be taken in identifying Hp with Hc1 because, in some cases, surface pinning and geometrical barriers can push Hp well above Hc1. However, in our measurements, several different checks, such as the equality of Hp for increasing and decreasing eld14, and the independence of

Hp on the sensor position15, rule this out (see Methods).

The temperature dependence of Hc1 is found to be linear in T at low temperature for all x (Fig. 3), which again is indicative of a lack of surface barriers that tend to become stronger at low temperature causing an upturn in Hc1(T)16. Extrapolating this linear behaviour to zero temperature gives us Hc1(0), which is plotted versus x in Fig. 4a. Surprisingly, instead of a dip in Hc1(0)

at the QCP predicted by equation (2) in conjunction with the observed behaviour of l(x)9, there is instead a strong peak. To resolve this discrepancy we consider again the arguments leading to equation (2).

In general Hc1 is determined from the vortex line energy Eline, which is composed of two parts17,

Hc1 Eem Ecore=f0: 3 The rst, Eem is the electromagnetic energy associated with the magnetic eld and the screening currents, which in the high k approximation is given by

Eem

f20

0H c1 (mT)

50

40

30

20

10

0 0 4 8 12

T (K)

16

20 24 28

Figure 3 | Temperature dependence of Hc1 in samples of BaFe2(As1 xPx)2. The lines show the linear extrapolation used to

determine the value at T 0. Error bars represent the uncertainty in

locating Hc1 from the raw B(H) data.

4pm0l2 ln k: 4 The second contribution arises from the energy associated with creating the normal vortex core Ecore. In high k superconductors,

Ecore is usually almost negligible and is accounted for by the additional constant 0.5 in equation (2). However, in superconductors close to a QCP we argue this may not be the case.

In Fig. 4b,c we use equations (3) and (4) to determine Eem and

Ecore. Away from the QCP, Ecore is approximately zero

and so the standard theory accounts for Hc1(0) well. However, as the QCP is approached there is a substantial increase in Ecore as determined from the corresponding increase in Hc1.

We can check this interpretation by making an independent estimate of the core energy from the condensation energy Econd, which we estimate from the experimentally measured specic heat (see Methods). The core energy is then Econdpx2e, where xe is the effective core radius that may be estimated from the coherence length xGL derived from Hc2 measurements using equation (1). In Fig. 4, we see that Econdpx2e has a similar dependence on x as Ecore and is in approximate

quantitative agreement if xeC4.0xGL for all x. Hence, this suggests that the observed anomalous increase in Hc1 could be caused by the high energy needed to create a vortex core close to the QCP.

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0H c1(0)(mT)

Energy(eV/m)

Energy(eV/m)

60

50

40

30

20

10

30

20

10

T c(K)

were u1,2 are the in-

plane Fermi velocities on the two sheets. So if the velocity was strongly renormalized on one sheet only (u1-0) then Hc2 would be determined mostly by u2 on the second sheet and hence would not increase with m* in accordance with our results. However, in this case the magnetic penetration depth l, which will also be dominated by the Fermi surface sheet with the largest u, would not show a peak at the QCP in disagreement with experiment9. In fact, the numerical agreement between the increase in m* with x as determined by l or specic heat, which in contrast to l is dominated by the low Fermi velocity sections, rather suggests that the renormalization is mostly uniform on all sheets10. In the opposite limit, appropriate to the prototypic multiband superconductor MgB2, where intraband pairing dominates over interband, Hc2 will be determined by the band with the lowest u (ref. 18) and again an increase in m* should be reected in Hc2. So

these multiband effects cannot easily explain our results.

Another effect of multiband superconductivity is that it can modify the temperature dependence of Hc2 such that it departs from the HW model. For example, in some iron-based super-conductors a linear dependence of Hc2(T) was found over a wide temperature range20. For BaFe2(As1 xPx)2, however, the

coincidence between the HW extrapolation of the Hc2 data

close to Tc and the pulsed eld measurement of Hirr for T5Tc for all x, would appear to rule out any signicant underestimation of

Hc2(0). In Supplementary Fig. 3 we show that Hirr for a sample with x 0.51 ts the HW theory for Hc2(T) over the full

temperature range. There is no reason why Hirr would

underestimate Hc2(0) by the same factor as the HW extrapolation. Even in cuprate superconductors where, unlike here, there is evidence for strong thermal uctuation effects, Hirr

has been shown to agree closely with Hc2 in the low-temperature limit21. The magnitude of the discrepancy between the behaviour of Hc2(0) and m* discussed above (see Fig. 2) also makes an explanation based on an experimental underestimate of Hc2(0)

implausible.

Another possibility is that in heavy fermion superconductors the mass enhancement is often reduced considerably at high elds and therefore m* could be reduced at elds comparable to Hc2. In

BaFe2(As1 xPx)2, however, a signicantly enhanced mass in

elds greater than Hc2 can be inferred from the dHvA measurements10 and low temperature, high eld, resistivity22. Although very close to the QCP the mass inferred from these measurements is slightly reduced from the values inferred from the zero eld specic heat measurements10 this cannot account for the lack of enhancement of Hc2 shown in Fig. 2.

Our results are similar to the behaviour observed in another quantum critical superconductor, CeRhIn5. Here the pressure tuned QCP manifests a large increase in the effective mass as measured by the dHvA effect and the low-temperature resistivity. Tc is maximal at the QCP but Hc2 displays only a broad peak, inconsistent with the mass enhancement shown by the other probes23. We should note that in this system Hc2 at low temperatures is Pauli limited. However, close to Tc, Hc2 is always

orbitally limited and as neither h0 or Hc2(0) are enhanced in

0 QCP

0

600

400

200

0

600

Eline

Eem

Ecore

400

200

0 0.3 0.4 0.5

X

Econd 2e

Figure 4 | Concentration x dependence of lower critical eld and associated energies for BaFe2(As1 xPx)2. (a) Lower critical eld Hc1

extrapolated to T 0 and Tc. The location of the QCP is indicated. Error

bars on Hc1 represent the combination of uncertainties in extrapolating Hc1(T) to T 0 and in the demagnetizing factor. Error bars on x are s.d.

(b) Vortex line energy Eline Eem Ecore at T 0 from the Hc1(0) data and equations (4) and (3) shown as squares. The electromagnetic energy calculated using equation (4) and different estimates of l are also shown. The triangles are direct measurements from ref. 9, and the circles are estimates derived by scaling the band-structure value of l by the effective mass enhancement from specic heat 10. Error bars on Eem (circles) are calculated from the uncertainty in jump size in heat capacity at Tc. (c) Vortex core energy Ecore Eline Eem along with an alternative

estimate derived from the specic heat condensation energy (Econd) and the

effective vortex area (pxe2). The uncertainties are calculated from a combination of those in the other panels. The dashed lines in all panels are guides to the eye.

DiscussionIn principle, the relative lack of enhancement in Hc2 close to the

QCP could be caused by impurity or multiband effects, although we argue that neither are likely explanations. Impurities decrease xGL and in the extreme dirty limit Hc2pm*Tc/c, where c is the electron mean-free-path11. Hence, even in this limit we would

expect Hc2 to increase with m* although not as strongly as in the clean case. Impurities increase Hc2 and as the residual resistance increases close to x 0.3 (ref. 7) we would actually expect a larger

increase in Hc2 than expected from clean-limit behaviour. dHvA measurements show that c44xGL at least for the electron bands and for x40.38, which suggest that, in fact, our samples are closer to the clean limit.

To discuss the effect of multiple Fermi surface sheets on Hc2,

we consider the results of Gurevich18 for two ellipsoidal Fermi surface sheets with strong interband pairing. This limit is probably the one most appropriate for BaFe2(As1 xPx)2

(ref. 19). In this case for H||c, h0c= u21 u22

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BaFe2(As1 xPx)2 or CeRhIn5 (ref. 23), Pauli limiting can be ruled

out as the explanation.

A comparison with the behaviour observed in cuprates is also interesting. Here two peaks in Hc2(0) as a function of doping p in

YBa2Cu3O7 d have been reported21, which approximately

coincide with critical points where other evidence suggests that the Fermi surface reconstructs. Quantum oscillation measurements indicate that m* increases close to these points24, suggesting a direct link between Hc2(0) and m* in the cuprates in contrast to our nding here for BaFe2(As1 xPx)2. However, by

analysing the data in the same way as we have done here, it can be seen25 that Hc2(0)0.5/Tc for YBa2Cu3O7 d is independent of p

above pC0.18 and falls for p below this value, reaching a minimum at pC1/8. This suggests that at least the peak at higher p is driven by the increasing gap value rather than a peak in m*, in agreement with our results here, and that the minimum in Hc2(0)0.5/Tc coincides with the doping where charge order is strongest at pC1/8 (ref. 26).

The lack of enhancement of Hc2(0) in all these systems suggests a fundamental failure of the theory. One possibility is that this may be driven by microscopic mixing of super-conductivity and antiferromagnetism close to the QCP. In the vicinity of the QCP, antiferromagnetic order is expected to emerge near the vortex core region where the superconducting order parameter is suppressed27,28. Such a eld-induced anti-ferromagnetic order has been observed experimentally in cuprates29,30. When the QCP lies beneath the superconducting dome, as in the case of BaFe2(As1 xPx)2 (refs 4,9), anti-

ferromagnetism and superconductivity can coexist on a microscopic level. In such a situation, as pointed out in ref. 28, the eld-induced antiferromagnetism can extend outside the effective vortex core region where the superconducting order parameter is nite. Such an extended magnetic order is expected to lead to further suppression of the superconducting order parameter around vortices. This effect will enlarge the vortex core size, which in turn will suppress the upper critical eld in agreement with our results. We would expect this effect to be a general feature of superconductivity close to an antiferromagnetic QCP, but perhaps not relevant to the behaviour close to p 0.18 in the cuprates.

To explain the Hc1 results we postulate that the vortex core size is around four times larger than the estimates from Hc2. This is in fact expected in cases of multiband superconductivity or super-conductors with strong gap anisotropy. In MgB2 (refs 31,32) and also in the anisotropic gap superconductor 2H-NbSe2 (ref. 33) the effective core size has been found to be around three times xGL,

similar to that needed to explain the behaviour here. BaFe2(As1 xPx)2 is known to have a nodal gap structure34,

which remains relatively constant across the superconducting dome9 and so we should expect the core size to be uniformly enhanced for all x. The peak in Hc1(x) at the QCP is then,

primarily caused by the uctuation-driven enhancement in the normal-state energy, but the effect is magnied by the nodal gap structure of BaFe2(As1 xPx)2.

We expect the observed anomalous increase in Hc1 to be a

general feature of quantum critical superconductors as these materials often have nodal or strongly anisotropic superconducting gap structures and the increase in normal state energy is a general property close to a QCP. The relative lack of enhancement in Hc2 also seems to be a general feature, which may be linked to a microscopic mixing of antiferromagnetism and superconductivity.

Methods

Sample growth and characterization. BaFe2(As1 xPx)2 samples were grown

using a self-ux technique as described in ref. 7. Samples for this study were screened using specic heat and only samples with superconducting transition

width o1 K were measured (see Supplementary Fig. 1). To determine the phosphorous concentration in the samples we carried out energy-dispersive X-ray analysis on several randomly chosen spots on each crystal (Hc1 samples) or measured the c axis lattice parameter using X-ray diffraction (Hc2 samples), which scales linearly with x. For some of the Hc2 samples measured using high-eld torque magnetometry the measured de Haasvan Alphen frequency was also used to determine x as described in ref. 10.

Measurements of Hc2. Close to Tc the upper critical eld was determined using heat capacity. For this a thin lm microcalorimeter was used10. We measured the superconducting transition at constant magnetic eld up to 14 T (see Supplementary Fig. 2). The midpoint of the increase in C at the transition denes Tc(H). At low temperatures (TooTc) we used piezo-resistive microcantilevers to measure the magnetic torque in pulsed magnetic eld and hence determine the irreversibility eld Hirr. The crystals used in the pulsed eld study were the same as those used in ref. 10 for the de Haasvan Alphen effect (except samples for xC0.3). By taking the difference between the torque in increasing and decreasing eld we determined the point at which the superconducting hysteresis closes as Hirr (see

Fig. 1b). For some compositions we measured Hirr in d.c. eld over the full

temperature range and found it to agree well with the HW model and also the low-temperature measurements in pulsed eld on the same sample (Supplementary Fig. 3). Our heat capacity measurements of Hc2 close to Tc(H 0) are in good

agreement with those of ref. 35.

Measurements of Hc1. The measurements of the eld of rst ux penetration Hp have been carried out using micro-Hall arrays. The Hall probes were made with either GaAs/AlGaAs heterostructures (carrier density ns 3.5 1011cm 2) or

GaAs with a 1 mm thick silicon doped layer (concentration ns 1 1016cm 3).

The latter had slightly lower sensitivity but proved more reliable at temperatures below 4 K. The measurements were carried out using a resistive magnet so that the remanent eld during zero eld cooling was as low as possible. The samples were warmed above Tc after each eld sweep and then cooled at a constant rate to the desired temperature.

When strong surface pinning is present Hp may be pushed up signicantly beyond Hc1. In this case there will also be a signicant difference between the critical eld Hp measured at the edge and the centre of the sample (for example see ref. 15) and also a difference between the eld where ux starts to enter the sample and the eld at which it leaves. Some of our samples, also showing signs of inhomogeneity, such as wide superconducting transitions, showed this behaviour. An example is shown in Supplementary Fig. 4. In this sample the sensor at the edge shows rst ux penetration at HpE5 mT, whereas the value is B3 times higher at the centre. For decreasing elds, the centre sensor shows a similar value to the edge sensor. All the samples reported in this paper showed insignicant difference between Hp at the centre and the edge and also for increasing and decreasing elds.

Hence, we conclude that Hc1 in our samples is not signicantly increased by pinning.

As our samples are typically thin platelets, demagnetization effects need to be taken into account for measurement of Hc1. Although an exact solution to the demagnetization problem is only possible for ellipsoids and innite slabs, a good approximation for thin slabs has been obtained by Brandt36. Here Hc1 is related to the measured Hp, determined from H using

Hc1

Hptanh

0:36lc=la p

5

where lc is the sample dimension along the eld and la perpendicular to the eld.

All samples in this study had lcoola. To ensure that the determination of the effective eld is independent of the specic dimension we have carried out multiple measurements on a single sample cleaved to give multiple ratios of lc/la. The results

of this study (Supplementary Fig. 5) show that Hc1 determined by this method are independent of the aspect ratio of the sample. Furthermore, the samples used all had similar lc/la ratios (see Supplementary Table 1), and so any correction would not give any systematic errors as a function of x.

Calculation of condensation energy. The condensation energy can be calculated from the specic heat using the relation

Econd Z

1 CsT CnT

dT: 6

To calculate this, we rst measured a sample of BaFe2(As1 xPx)2 with x 0.47,

using a relaxation technique in zero eld and m0H 14 T, which is sufcient at this

doping to completely suppress superconductivity and thus reach the normal state. We used this 14 T data to determine the phonon heat capacity and we then subtract this from the zero eld data to give the electron specic heat of the sample. We then tted this data to a phenomenological nodal gap, alpha model (with variable zero temperature gap) similar to that described in ref. 37 (see Supplementary Fig. 6). We then integrated this t function using equation (6) to give Econd for this

value of x. For lower values of x (higher Tc) the available elds were insufcient to suppress superconductivity over the full range of temperature, so we assumed that

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the shape of the heat capacity curve does not change appreciably with x but rather just scales with Tc and the jump height at Tc. This is implicitly assuming that the superconducting gap structure does not change appreciably with x, which is supported by magnetic penetration depth l measurements which show that normalized temperature dependence l(T)/l(0) is relatively independent of x9. With this assumption we can then calculate

Econdx

Econd xref TcxDCx

Tc xref DC xref

;

where xref 0.47.

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Acknowledgements

We thank Igor Mazin and Georg Knebel for useful discussions and A.M. Adamska for experimental help. This work was supported by the Engineering and Physical Sciences Research Council (Grant No. EP/H025855/1), National Physical Laboratory Strategic Research Programme, EuroMagNET II under the EU Contract No. 228043 and KAKENHI from JSPS.

Author contributions

A.C. and C. Putzke. conceived the experiment. C. Putzke performed the high-eld torque measurements (with D.V., C. Proust and S.B.) and the Hall probe measurements. P.W. and L.M. performed heat capacity measurements. The Hall probe arrays were fabricated by J.D.F., P.S., H.E.B and D.A.R. Samples were grown and characterized by S.K., Y. Mizukami, T.S. and Y.Matsuda. The manuscript was written by A.C. with input fromC. Putzke, C. Proust, P.W., L.M., J.D.F, T.S and Y. Matsuda.

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How to cite this article: Putzke, C. et al. Anomalous critical elds in quantum critical superconductors. Nat. Commun. 5:5679 doi: 10.1038/ncomms6679 (2014).

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