Introduction

Quantum critical points (QCPs) can be associated with a variety of different order–disorder 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 fluctuations 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 influence of quantum criticality on the superconducting 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 λ and the heat capacity jump at Tc, ΔC9,10. 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 Ginzburg–Landau theory, the upper critical field is given by

where φ0 is the flux quantum and ξGL is the Ginzburg–Landau coherence length. In the clean limit at low temperature, ξGL is usually well approximated by the BCS coherence length, which results in Hc2∝(m*Δ)2, where m* is the mass of the quasiparticles and Δ is the superconducting gap. This simplified 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 . This latter quantity is often more easily accessible experimentally because of the very high Hc2 values in compounds such as iron pnictides for T≪Tc and also because the values of Hc2 close to Tc are not reduced by the effect of the magnetic field on the electron spin (Pauli limiting effects).

For the lower critical field Hc1, standard Ginzburg–Landau theory predicts that

where κ=λ/ξGL, and so the observed large peak in λ 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 fields of quantum critical superconductors strongly violate the standard theory.

Results

Upper critical field 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 microcalorimeter in fields up to 14 T (see Fig. 1a). This gives an unambiguous measurement of Hc2(T) and the slope h′, 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 fields up to 60 T. Here an estimate of Hc2 was made by observing the field 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 fluctuations 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-specific heat results, using the Helfand–Werthamer (HW) formula13, to zero temperature are in good agreement with the irreversibility field measurements showing both are good estimates of Hc2(0).

Figure 1: Determination of critical fields.
figure 1

(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 field for a sample with x=0.40 at T=1.5 K. The irreversibility field Hirr is marked. (c) Magnetic flux density B versus applied field 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 field Br after subtraction of the linear term due to flux leakage around the sample. |Br|0.5 versus μ0H 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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Figure 2: Upper critical field as a function of concentration x.
figure 2

(a) Hc2(0) in BaFe2(As1−xPx)2 estimated from the slope of Hc2(T) close to Tc using (squares) 13, and also estimates of Hc2(0) from the irreversibility field 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 specific 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 fits to the data.

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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 ~20% from x=0.47 to x=0.30, whereas m* increases by ~400% for the same range of x.

Lower critical field Hc1

We measured Hc1 in our BaFe2(As1−xPx)2 samples using a micro-Hall probe array. Here the magnetic flux 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 field H due to incomplete shielding of the sensor by the sample. Then, as the applied field passes a certain field 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 field14, 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 λ(x)9, there is instead a strong peak. To resolve this discrepancy we consider again the arguments leading to equation (2).

Figure 3: Temperature dependence of Hc1 in samples of BaFe2(As1−xPx)2.
figure 3

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.

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Figure 4: Concentration x dependence of lower critical field and associated energies for BaFe2(As1−xPx)2.
figure 4

(a) Lower critical field 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 λ are also shown. The triangles are direct measurements from ref. 9, and the circles are estimates derived by scaling the band-structure value of λ by the effective mass enhancement from specific 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 specific heat condensation energy (Econd) and the effective vortex area (πξe2). The uncertainties are calculated from a combination of those in the other panels. The dashed lines in all panels are guides to the eye.

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

The first, Eem is the electromagnetic energy associated with the magnetic field and the screening currents, which in the high κ approximation is given by

The second contribution arises from the energy associated with creating the normal vortex core Ecore. In high κ 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 specific heat (see Methods). The core energy is then , where ξe is the effective core radius that may be estimated from the coherence length ξGL derived from Hc2 measurements using equation (1). In Fig. 4, we see that has a similar dependence on x as Ecore and is in approximate quantitative agreement if ξe≅4.0ξGL 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.

Discussion

In 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 ξGL and in the extreme dirty limit Hc2∝m*Tc/ℓ, where ℓ 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 ℓ>>ξGL at least for the electron bands and for x>0.38, 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, were υ1,2 are the in-plane Fermi velocities on the two sheets. So if the velocity was strongly renormalized on one sheet only (υ1→0) then Hc2 would be determined mostly by υ2 on the second sheet and hence would not increase with m* in accordance with our results. However, in this case the magnetic penetration depth λ, which will also be dominated by the Fermi surface sheet with the largest υ, 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 λ or specific heat, which in contrast to λ 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 υ (ref. 18) and again an increase in m* should be reflected 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 superconductors 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 field measurement of Hirr for T≪Tc for all x, would appear to rule out any significant underestimation of Hc2(0). In Supplementary Fig. 3 we show that Hirr for a sample with x=0.51 fits 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 fluctuation 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 fields and therefore m* could be reduced at fields comparable to Hc2. In BaFe2(As1−xPx)2, however, a significantly enhanced mass in fields greater than Hc2 can be inferred from the dHvA measurements10 and low temperature, high field, resistivity22. Although very close to the QCP the mass inferred from these measurements is slightly reduced from the values inferred from the zero field specific 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 h′ or Hc2(0) are enhanced in 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−δ 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 finding 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−δ is independent of p above p≅0.18 and falls for p below this value, reaching a minimum at p≅1/8. 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 p≅1/8 (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 superconductivity 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 field-induced antiferromagnetic 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), antiferromagnetism and superconductivity can coexist on a microscopic level. In such a situation, as pointed out in ref. 28, the field-induced antiferromagnetism can extend outside the effective vortex core region where the superconducting order parameter is finite. 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 field 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 superconductors 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 ξGL, 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 fluctuation-driven enhancement in the normal-state energy, but the effect is magnified 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-flux technique as described in ref. 7. Samples for this study were screened using specific heat and only samples with superconducting transition width <1 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-field torque magnetometry the measured de Haas–van Alphen frequency was also used to determine x as described in ref. 10.

Measurements of Hc2

Close to Tc the upper critical field was determined using heat capacity. For this a thin film microcalorimeter was used10. We measured the superconducting transition at constant magnetic field up to 14 T (see Supplementary Fig. 2). The midpoint of the increase in C at the transition defines Tc(H). At low temperatures (T≪Tc) we used piezo-resistive microcantilevers to measure the magnetic torque in pulsed magnetic field and hence determine the irreversibility field Hirr. The crystals used in the pulsed field study were the same as those used in ref. 10 for the de Haas–van Alphen effect (except samples for x≅0.3). By taking the difference between the torque in increasing and decreasing field we determined the point at which the superconducting hysteresis closes as Hirr (see Fig. 1b). For some compositions we measured Hirr in d.c. field over the full temperature range and found it to agree well with the HW model and also the low-temperature measurements in pulsed field 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 field of first flux 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 μm 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 field during zero field cooling was as low as possible. The samples were warmed above Tc after each field sweep and then cooled at a constant rate to the desired temperature.

When strong surface pinning is present Hp may be pushed up significantly beyond Hc1. In this case there will also be a significant difference between the critical field Hp measured at the edge and the centre of the sample (for example see ref. 15) and also a difference between the field where flux starts to enter the sample and the field 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 first flux penetration at Hp≈5 mT, whereas the value is ~3 times higher at the centre. For decreasing fields, the centre sensor shows a similar value to the edge sensor. All the samples reported in this paper showed insignificant difference between Hp at the centre and the edge and also for increasing and decreasing fields. Hence, we conclude that Hc1 in our samples is not significantly 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 infinite slabs, a good approximation for thin slabs has been obtained by Brandt36. Here Hc1 is related to the measured Hp, determined from H using

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

All samples in this study had lc≪la. To ensure that the determination of the effective field is independent of the specific 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 specific heat using the relation

To calculate this, we first measured a sample of BaFe2(As1−xPx)2 with x=0.47, using a relaxation technique in zero field and μ0H=14 T, which is sufficient 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 field data to give the electron specific heat of the sample. We then fitted 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 fit function using equation (6) to give Econd for this value of x. For lower values of x (higher Tc) the available fields were insufficient to suppress superconductivity over the full range of temperature, so we assumed that 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 λ measurements which show that normalized temperature dependence λ(T)/λ(0) is relatively independent of x9. With this assumption we can then calculate

where xref=0.47.

Additional information

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