Abstract
Fluctuations around an antiferromagnetic quantum critical point (QCP) are believed to lead to unconventional superconductivity and in some cases to hightemperature superconductivity. However, the exact mechanism by which this occurs remains poorly understood. The ironpnictide superconductor BaFe_{2}(As_{1−x}P_{x})_{2} is perhaps the clearest example to date of a hightemperature 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.
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 order^{1,2}, whereas in the cuprates the nature of the order (known as the pseudogap phase) is less clear^{3}. The normal state of these materials has been widely studied and close to their QCPs nonFermi liquid behaviour of transport and thermodynamic properties are often found, however, comparatively little is known about how the quantum critical fluctuations affect the superconducting state^{4}. This is important as it is the difference in energy between the normal and superconducting state that ultimately determines the critical temperature T_{c}.
Among the various ironpnictide superconductors, BaFe_{2}(As_{1−x}P_{x})_{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 spindensity wave antiferromagnetic metal, through the superconducting phase to a paramagnetic metal^{5}. 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 pressure^{6}. Normal state properties such as the temperature dependence of the resistivity^{7} and spinlattice relaxation rate^{8} 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 T_{c}, ΔC^{9,10}. 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 QCP^{10}.
In the standard singleband Ginzburg–Landau theory, the upper critical field is given by
where φ_{0} is the flux quantum and ξ_{GL} is the Ginzburg–Landau coherence length. In the clean limit at low temperature, ξ_{GL} is usually well approximated by the BCS coherence length, which results in H_{c2}∝(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 theory^{11}. Hence, a strong peak in m* at the QCP should result in a corresponding increase in H_{c2} as well as the slope of H_{c2} at . This latter quantity is often more easily accessible experimentally because of the very high H_{c2} values in compounds such as iron pnictides for T≪T_{c} and also because the values of H_{c2} close to T_{c} are not reduced by the effect of the magnetic field on the electron spin (Pauli limiting effects).
For the lower critical field H_{c1}, standard Ginzburg–Landau theory predicts that
where κ=λ/ξ_{GL}, and so the observed large peak in λ at the QCP^{9} should result in a strong suppression of H_{c1}. Here we show that the exact opposite, a peak in H_{c1} at the QCP, occurs in BaFe_{2}(As_{1−x}P_{x})_{2}, and in addition the expected sharp increase in H_{c2} is not observed. This suggests that the critical fields of quantum critical superconductors strongly violate the standard theory.
Results
Upper critical field H_{c2}
We measured H_{c2} parallel to the c axis, in a series of highquality singlecrystal samples of BaFe_{2}(As_{1−x}P_{x})_{2} spanning the superconducting part of the phase diagram using two different techniques. Close to T_{c}(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 H_{c2}(T) and the slope h′, which unlike transport measurements is not complicated by contributions from vortex motion^{12}. At a lower temperature, we used microcantilever torque measurements in pulsed magnetic fields up to 60 T. Here an estimate of H_{c2} 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 H_{irr}, this is a lower limit for H_{c2}(0) and in superconductors with negligible thermal fluctuations and low anisotropy such as BaFe_{2}(As_{1−x}P_{x})_{2} H_{irr} should coincide approximately with H_{c2}. Indeed, in Fig. 2 we show that the extrapolation of the hightemperaturespecific heat results, using the Helfand–Werthamer (HW) formula^{13}, to zero temperature are in good agreement with the irreversibility field measurements showing both are good estimates of H_{c2}(0).
In the clean limit we would expect (H_{c2}(0))^{1/2}/T_{c} 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 H_{c1}
We measured H_{c1} in our BaFe_{2}(As_{1−x}P_{x})_{2} samples using a microHall probe array. Here the magnetic flux density B is measured at several discrete points a few microns from the surface of the sample. Below H_{c1}, 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 H_{p}, B increases more rapidly with H indicating that vortices have entered the sample (see Fig. 1c,d). Care must be taken in identifying H_{p} with H_{c1} because, in some cases, surface pinning and geometrical barriers can push H_{p} well above H_{c1}. However, in our measurements, several different checks, such as the equality of H_{p} for increasing and decreasing field^{14}, and the independence of H_{p} on the sensor position^{15}, rule this out (see Methods).
The temperature dependence of H_{c1} 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 H_{c1}(T)^{16}. Extrapolating this linear behaviour to zero temperature gives us H_{c1}(0), which is plotted versus x in Fig. 4a. Surprisingly, instead of a dip in H_{c1}(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).
In general H_{c1} is determined from the vortex line energy E_{line}, which is composed of two parts^{17},
The first, E_{em} 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 E_{core}. In high κ superconductors, E_{core} 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 E_{em} and E_{core}. Away from the QCP, E_{core} is approximately zero and so the standard theory accounts for H_{c1}(0) well. However, as the QCP is approached there is a substantial increase in E_{core} as determined from the corresponding increase in H_{c1}. We can check this interpretation by making an independent estimate of the core energy from the condensation energy E_{cond}, which we estimate from the experimentally measured specific heat (see Methods). The core energy is then , where ξ_{e} is the effective core radius that may be estimated from the coherence length ξ_{GL} derived from H_{c2} measurements using equation (1). In Fig. 4, we see that has a similar dependence on x as E_{core} and is in approximate quantitative agreement if ξ_{e}≅4.0ξ_{GL} for all x. Hence, this suggests that the observed anomalous increase in H_{c1} 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 H_{c2} close to the QCP could be caused by impurity or multiband effects, although we argue that neither are likely explanations. Impurities decrease ξ_{GL} and in the extreme dirty limit H_{c2}∝m*T_{c}/ℓ, where ℓ is the electron meanfreepath^{11}. Hence, even in this limit we would expect H_{c2} to increase with m* although not as strongly as in the clean case. Impurities increase H_{c2} and as the residual resistance increases close to x=0.3 (ref. 7) we would actually expect a larger increase in H_{c2} than expected from cleanlimit behaviour. dHvA measurements show that ℓ>>ξ_{GL} 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 H_{c2}, we consider the results of Gurevich^{18} for two ellipsoidal Fermi surface sheets with strong interband pairing. This limit is probably the one most appropriate for BaFe_{2}(As_{1−x}P_{x})_{2}(ref. 19). In this case for Hc, were υ_{1,2} are the inplane Fermi velocities on the two sheets. So if the velocity was strongly renormalized on one sheet only (υ_{1}→0) then H_{c2} would be determined mostly by υ_{2} 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 experiment^{9}. 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 sheets^{10}. In the opposite limit, appropriate to the prototypic multiband superconductor MgB_{2}, where intraband pairing dominates over interband, H_{c2} will be determined by the band with the lowest υ (ref. 18) and again an increase in m* should be reflected in H_{c2}. So these multiband effects cannot easily explain our results.
Another effect of multiband superconductivity is that it can modify the temperature dependence of H_{c2} such that it departs from the HW model. For example, in some ironbased superconductors a linear dependence of H_{c2}(T) was found over a wide temperature range^{20}. For BaFe_{2}(As_{1−x}P_{x})_{2}, however, the coincidence between the HW extrapolation of the H_{c2} data close to T_{c} and the pulsed field measurement of H_{irr} for T≪T_{c} for all x, would appear to rule out any significant underestimation of H_{c2}(0). In Supplementary Fig. 3 we show that H_{irr} for a sample with x=0.51 fits the HW theory for H_{c2}(T) over the full temperature range. There is no reason why H_{irr} would underestimate H_{c2}(0) by the same factor as the HW extrapolation. Even in cuprate superconductors where, unlike here, there is evidence for strong thermal fluctuation effects, H_{irr} has been shown to agree closely with H_{c2} in the lowtemperature limit^{21}. The magnitude of the discrepancy between the behaviour of H_{c2}(0) and m* discussed above (see Fig. 2) also makes an explanation based on an experimental underestimate of H_{c2}(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 H_{c2}. In BaFe_{2}(As_{1−x}P_{x})_{2}, however, a significantly enhanced mass in fields greater than H_{c2} can be inferred from the dHvA measurements^{10} and low temperature, high field, resistivity^{22}. 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 measurements^{10} this cannot account for the lack of enhancement of H_{c2} shown in Fig. 2.
Our results are similar to the behaviour observed in another quantum critical superconductor, CeRhIn_{5}. Here the pressure tuned QCP manifests a large increase in the effective mass as measured by the dHvA effect and the lowtemperature resistivity. T_{c} is maximal at the QCP but H_{c2} displays only a broad peak, inconsistent with the mass enhancement shown by the other probes^{23}. We should note that in this system H_{c2} at low temperatures is Pauli limited. However, close to T_{c}, H_{c2} is always orbitally limited and as neither h′ or H_{c2}(0) are enhanced in BaFe_{2}(As_{1−x}P_{x})_{2} or CeRhIn_{5} (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 H_{c2}(0) as a function of doping p in YBa_{2}Cu_{3}O_{7−δ} have been reported^{21}, 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 points^{24}, suggesting a direct link between H_{c2}(0) and m* in the cuprates in contrast to our finding here for BaFe_{2}(As_{1−x}P_{x})_{2}. However, by analysing the data in the same way as we have done here, it can be seen^{25} that H_{c2}(0)^{0.5}/T_{c} for YBa_{2}Cu_{3}O_{7−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 H_{c2}(0)^{0.5}/T_{c} coincides with the doping where charge order is strongest at p≅1/8 (ref. 26).
The lack of enhancement of H_{c2}(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 suppressed^{27,28}. Such a fieldinduced antiferromagnetic order has been observed experimentally in cuprates^{29,30}. When the QCP lies beneath the superconducting dome, as in the case of BaFe_{2}(As_{1−x}P_{x})_{2} (refs 4, 9), antiferromagnetism and superconductivity can coexist on a microscopic level. In such a situation, as pointed out in ref. 28, the fieldinduced 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 H_{c1} results we postulate that the vortex core size is around four times larger than the estimates from H_{c2}. This is in fact expected in cases of multiband superconductivity or superconductors with strong gap anisotropy. In MgB_{2} (refs 31, 32) and also in the anisotropic gap superconductor 2HNbSe_{2} (ref. 33) the effective core size has been found to be around three times ξ_{GL}, similar to that needed to explain the behaviour here. BaFe_{2}(As_{1−x}P_{x})_{2} is known to have a nodal gap structure^{34}, which remains relatively constant across the superconducting dome^{9} and so we should expect the core size to be uniformly enhanced for all x. The peak in H_{c1}(x) at the QCP is then, primarily caused by the fluctuationdriven enhancement in the normalstate energy, but the effect is magnified by the nodal gap structure of BaFe_{2}(As_{1−x}P_{x})_{2}.
We expect the observed anomalous increase in H_{c1} 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 H_{c2} also seems to be a general feature, which may be linked to a microscopic mixing of antiferromagnetism and superconductivity.
Methods
Sample growth and characterization
BaFe_{2}(As_{1−x}P_{x})_{2} samples were grown using a selfflux 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 energydispersive Xray analysis on several randomly chosen spots on each crystal (H_{c1} samples) or measured the c axis lattice parameter using Xray diffraction (H_{c2} samples), which scales linearly with x. For some of the H_{c2} samples measured using highfield torque magnetometry the measured de Haas–van Alphen frequency was also used to determine x as described in ref. 10.
Measurements of H_{c2}
Close to T_{c} the upper critical field was determined using heat capacity. For this a thin film microcalorimeter was used^{10}. 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 T_{c}(H). At low temperatures (T≪T_{c}) we used piezoresistive microcantilevers to measure the magnetic torque in pulsed magnetic field and hence determine the irreversibility field H_{irr}. 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 H_{irr} (see Fig. 1b). For some compositions we measured H_{irr} in d.c. field over the full temperature range and found it to agree well with the HW model and also the lowtemperature measurements in pulsed field on the same sample (Supplementary Fig. 3). Our heat capacity measurements of H_{c2} close to T_{c}(H=0) are in good agreement with those of ref. 35.
Measurements of H_{c1}
The measurements of the field of first flux penetration H_{p} have been carried out using microHall arrays. The Hall probes were made with either GaAs/AlGaAs heterostructures (carrier density n_{s}=3.5 × 10^{11}cm^{−2}) or GaAs with a 1 μm thick silicon doped layer (concentration n_{s}=1 × 10^{16}cm^{−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 T_{c} after each field sweep and then cooled at a constant rate to the desired temperature.
When strong surface pinning is present H_{p} may be pushed up significantly beyond H_{c1}. In this case there will also be a significant difference between the critical field H_{p} 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 H_{p}≈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 H_{p} at the centre and the edge and also for increasing and decreasing fields. Hence, we conclude that H_{c1} 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 H_{c1}. 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 Brandt^{36}. Here H_{c1} is related to the measured H_{p}, determined from H using
where l_{c} is the sample dimension along the field and l_{a} perpendicular to the field.
All samples in this study had l_{c}≪l_{a}. 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 l_{c}/l_{a}. The results of this study (Supplementary Fig. 5) show that H_{c1} determined by this method are independent of the aspect ratio of the sample. Furthermore, the samples used all had similar l_{c}/l_{a} 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 BaFe_{2}(As_{1−x}P_{x})_{2} with x=0.47, using a relaxation technique in zero field and μ_{0}H=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 E_{cond} for this value of x. For lower values of x (higher T_{c}) 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 T_{c} and the jump height at T_{c}. 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 x^{9}. With this assumption we can then calculate
where x_{ref}=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).
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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.
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A.C. and C. Putzke. conceived the experiment. C. Putzke performed the highfield 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 from C. Putzke, C. Proust, P.W., L.M., J.D.F, T.S and Y. Matsuda.
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Correspondence to A. Carrington.
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Putzke, C., Walmsley, P., Fletcher, J. et al. Anomalous critical fields in quantum critical superconductors. Nat Commun 5, 5679 (2014). https://doi.org/10.1038/ncomms6679
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Physical Review Applied (2019)

Ultrahigh critical current densities, the vortex phase diagram, and the effect of granularity of the stoichiometric high Tc superconductor CaKFe4As4
Physical Review Materials (2018)
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