Introduction

Unconventional superconductivity often arises around the point where some kind of magnetic order (either antiferromagnetic or ferromagnetic) is suppressed by a tuning parameter, such as pressure or chemical substitution1,2,3. This led to the idea that superconducting pairing in these materials may result from the low-energy magnetic fluctuations surrounding the quantum critical point where magnetism is suppressed to zero temperature4. The magnetic ground state of typical iron-based superconducting parent compounds is a stripe-type antiferromagnetic spin-density wave (SDW) phase, which breaks the C4 symmetry of the high-temperature tetragonal paramagnetic phase and is closely related to a tetragonal-to-orthorhombic structural distortion of the lattice5. An important debate, triggered by the observation that this orthorhombic structural distortion sometimes occurs slightly above the magnetic transition6, concerns the respective roles of spin and orbital degrees of freedom7,8. In the orbital picture, the structural transition is due to orbital ordering and can trigger a, secondary, magnetic transition8. In the so-called spin-nematic scenario, on the other hand, magnetism is essential and the structural transition is really just the first step of the magnetic transition, induced by an Ising-nematic degree of freedom associated with the emerging stripe-type magnetic order9. Importantly, both orbital and spin fluctuations are candidates for the superconducting pairing7,8.

A surprising recent result is the observation of a C4-symmetric tetragonal magnetically ordered phase in Ba1−xNaxFe2As2 using neutron scattering10, which was taken as evidence for the spin scenario because the existence of such a phase can hardly be reconciled with orbital order being a prerequisite for magnetism10. Up to now, a similar C4-magnetic phase has not been observed in the closely related Ba1−xKxFe2As2 system, which was the first iron-based superconductor of 122-type stoichiometry to be discovered11 and has the highest Tc=38 K within this family. However, an unidentified phase transition in resistivity measurements under pressure12 hints that an additional instability is close by. Earlier theoretical work indicates the close proximity in energy between C4- and C2-symmetric magnetic structures13.

Here we re-examine the phase diagram of Ba1−xKxFe2As2 in unprecedented detail using thermal-expansion and specific-heat measurements of very high-quality single crystals. Within a narrow composition range, we observe a C4-symmetric magnetic phase, which apparently has been missed in previous investigations12,14,15,16. In strong contrast to Na-doped BaFe2As2, the C4-phase is, however, not stable to zero temperature and reverts back to the C2 phase in the vicinity of the onset of superconductivity. This preference of superconductivity for the stripe-type C2- over the C4-magnetic state in this system may provide important clues about the superconducting pairing mechanism17.

Results

Orthorhombic distortion

Figure 1 presents our first main result, the orthorhombic distortion δ=(a−b)/(a+b) (a and b are the in-plane lattice constants) versus temperature of underdoped Ba1−xKxFe2As2 obtained using our high-resolution capacitance dilatometer18. Even though dilatometry is a macroscopic probe, the thermal expansion of the individual a and b axes, as well as the distortion δ, can be derived using the difference between ‘twinned’ and ‘detwinned’ data sets, as shown previously for FeSe (ref. 19) and detailed in the supplementary material (Supplementary Figs 1 and 2). The reliability of this method, which has higher resolution by several orders of magnitude than either neutron or X-ray diffraction experiments, is demonstrated by the close match to results of neutron powder diffraction14 (see Fig. 1e). In Ba1−xKxFe2As2, the structural and magnetic transitions are coincident and first-order over the entire phase diagram14. Our curves in Fig. 1d, indeed, exhibit the well-known increase of δ at Ts,N and the subsequent suppression of δ below Tc (ref. 20) for 22% K content. In the small concentration range ∼24.7%–27.6% K content we find a surprising new result, namely, a sudden reduction of the orthorhombic distortion at T1<Ts,N, followed by a sudden increase at T2 to a value slightly below its maximum value. This sudden reduction of δ strongly suggests that these Ba1−xKxFe2As2 samples undergo a similar transition to a C4-magnetic phase as found in Ba1−xNaxFe2As2 (refs 10, 21). The increase of δ at T2, on the other hand, is an indication for re-entrance of the C2-SDW phase. To prove that the intermediate phase is truly tetragonal, we have corrected the data of the 24.7% K crystal for the finite stress applied during the measurement, which induces non-zero distortion δ even in C4-symmetric phases (Fig. 1f, see Supplementary Figs 1 and 2). The corrected value, δ0, indeed vanishes completely (to within the extremely small value ∼0.01 × 10−3, related to our error bar) in the intermediate phase, as expected for a truly tetragonal C4 phase.

Figure 1: Orthorhombic distortion of Ba1−xKxFe2As2 as measured using a capacitance dilatometer.
figure 1

(a) Schematic representation of the tetragonal C4 high-temperature phase. (b) Representation of the structural domains formed in the orthorhombic C2 phase (‘twins’, indicated by different colours). The domain structure is unaffected by the force from the spring-loaded dilatometer if it is applied along the tetragonal [100] in-plane direction (purple arrows). (c) Representation of the mostly ‘detwinned’ state, achieved by applying the dilatometer force along [110], which selects the domains with their (shorter) orthorhombic b axis along the direction of the applied force. (d) Temperature dependence of the orthorhombic distortion δ=(a−b)/(a+b) of underdoped Ba1−xKxFe2As2 obtained using difference of ‘twinned’ and ‘detwinned’ data from our high-resolution capacitance dilatometer. Abrupt changes of δ mark phase transitions, examples of which are indicated by vertical arrows. (e) Good agreement between our results (continuous lines), and results from neutron powder diffraction14 (symbols) for the thermal expansion of the a and b axis demonstrates the reliability of our technique. (f) Orthorhombic distortion corrected for the effect of the applied force, δ0, for the sample with 24.7% K content, distinguishing tetragonal and orthorhombic phases (see Supplementary Figs 1 and 2 for details on the measurement of the orthorhombic distortion).

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Similar to inverse melting22, the counterintuitive transition at T1 into a state of seemingly higher symmetry on lowering of the temperature does not violate the laws of thermodynamics. Rather, it convincingly demonstrates that the structural distortion is not the primary order parameter and that other degrees of freedom must be involved. The absolute measure for disorder is the entropy of the isolated system, which never increases with decreasing temperature. In the following we investigate the novel re-entrant transitions at T1 and T2 and their relation to superconductivity by examining the electronic entropy/heat capacity of the system. Moreover, the fact that δ below T2 is still reduced from the value expected by extrapolation from the C2-SDW phase (dashed black line for the 24.7% sample, red curve in Fig. 1d) suggests that the superconducting transition lies somewhere within the intermediate C4 phase. Since no clear signature of superconductivity can be observed in the δ (T) curves, heat-capacity data are also crucial to nail down the location of the bulk superconducting transition.

Specific heat and electronic entropy

Figure 2 presents our electronic heat capacity Ce/T and electronic entropy divided by temperature , complemented by δ, and the temperature-dependent in-plane length change ΔLab/Lab for ‘twinned’ samples (c axis data and thermal-expansion coefficients are given in the Supplementary Fig. 3) for various compositions. In a Fermi liquid, Se/T is expected to be constant, and how the entropy is reduced on cooling through the transitions provides important information about the strength of the competing orders. For the 23.2% sample, the usual first-order tetragonal-to-orthorhombic transition occurs at Ts,N=81 K and results in a sizable reduction of Se/T. The remaining Se/T is lost through superconductivity with an onset at Tc=24 K (see Fig. 2a,f,k,p).

Figure 2: Phase transitions in thermal expansion and electronic specific heat for 5 samples of different K content.
figure 2

(a–e) Orthorhombic distortion δ, (f–j) average in-plane length change ΔLab/Lab for ‘twinned’ samples (k–o) electronic specific heat Ce/T, (p–t) electronic entropy divided by temperature Se/T. Horizontal lines indicate the value of Se/T of the high-temperature paramagnetic phase which is lost on cooling through the various transitions. (l–n) also show data measured in a large magnetic field and the insets show a magnification of the low-temperature region. Vertical lines mark the transitions temperatures labelled in panels (a–e).

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The lowest K concentration for which the C4-magnetic phase is observed is 24.7% (Fig. 2b,g,l,q). Surprisingly, we do not observe a clear signature of a superconducting transition neither in Ce/T nor in Lab for this crystal, while strong first-order peaks indicate Ts,N, T1 and T2 unambiguously. Nevertheless, the heat-capacity data show that the Fermi-surface is completely gapped at low temperature, implying a superconducting ground state. Tc is located by applying a large magnetic field. Notably, in a magnetic field of 12 T, the sharp peak at T2 disappears and is replaced by a broadened step-like anomaly, which is slightly shifted downward in temperature from the peak at T2, while the transitions at T1 and Ts,N are hardly affected. The step-like anomaly, the shift and the broadening in field are expected for a superconducting transition, and we therefore identify this transition with Tc. This result implies that the strong specific-heat peak in zero-field at T2 is a novel combined structural, magnetic and superconducting first-order transition, in which a re-entrance of the C2-SDW phase occurs. How exactly a magnetic field tunes the system from a first- to second-order transition at Tc is a direction for future studies.

Increasing the K-content by just over 1% (to 26.2% K) drastically changes the behaviour. Superconductivity at Tc=26 K can now easily be identified by the clear second-order specific-heat anomaly and a small kink in Lab (Fig. 2h,m). The specific-heat anomaly is again broadened and shifted to lower temperatures by a high magnetic field. Superconductivity now coexists with the intermediate C4 phase and reentrance of the C2 phase occurs at T2<Tc, as evidenced by the increase of δ and Lab (Fig. 2c,h). Surprisingly, there is only a very small anomaly in the heat capacity at T2 (Fig. 2m). At still a slightly higher K content (27.6% K, Fig. 2d,i,n,s), no more anomaly that could be associated with T2 is observed in either the heat capacity or in Lab. The weak re-emergence of the orthorhombic distortion at low temperature is likely induced by the stress applied for detwinning. For this sample, the reduction of Se/T is mainly due to superconductivity, and the magnetic and structural phase transitions at Ts,N and T1 play only a very minor role. Finally, Fig. 2e,j,o,t show the results for a sample with 30% K content, which undergoes only a superconducting transition. Note the larger specific-heat anomaly, which implies a considerably larger superconducting condensation energy than for the other samples.

Our heat-capacity data also show a striking low-temperature contribution to the electronic specific heat, or equivalently to the entropy, in the superconducting state for samples with 26.2 and 27.6% K content (see Fig. 2m,n,r,s). This feature, which is very reminiscent of the very small superconducting gaps found in KFe2As2 (ref. 23), is a sign of excited quasiparticles far below Tc and seems to occur only when the structural-magnetic transitions are weak, that is, induce only a small entropy change. From the position of the maximum in Ce/T, we estimate for the size of the smallest superconducting gap ΔSC∼0.07kBTc in the multigap system, which is even smaller than the ‘lilliputian’ gaps in KFe2As2 (ref. 23). The occurrence of such an extremely small gap may be related to peculiar features of the Fermi surface resulting from a reconstruction at Ts,N and T1 (see below). When this reconstruction becomes weaker on K doping, some parts of the reconstructed Fermi surface may move to within the superconducting gap ΔSC of the Fermi level and can contribute to the superconducting condensate, as recently argued by Koshelev et al.24. This would explain why this low-temperature feature suddenly disappears once Ts,N and T1 are suppressed by doping (see Fig. 2o,t).

Phase diagram

The transition temperatures from Fig. 2 are summarized in the phase diagram of Fig. 3a together with additional thermodynamic data covering the whole phase diagram25, (F. Hardy et al., manuscript in preparation). We find five distinct thermodynamic ordered phases, which all compete for the electronic entropy provided by the high-temperature C4-paramagnetic phase: C2 SDW, C4 magnetic, C2 SDW coexisting with superconductivity, C4 magnetic coexisting with superconductivity and C4-superconducting. Strikingly, Tc drops by about 7 K on going from the C2- to the C4-magnetic phase and, similarly, Tc increases by about 6 K at the boundary from the C4-magnetic to the C4-paramagnetic state. This points to a much stronger competition between superconductivity and the C4-magnetic phase than between superconductivity and the C2-magnetic phase, which is probably due to the additional pronounced suppression of entropy at T1 (see Fig. 2q). In particular, it would be clearly thermodynamically advantageous for superconductivity if the system would revert back to the C2-magnetic phase with the higher electronic entropy available for superconducting pairing. The system apparently does just this via a peculiar first-order magnetic, structural and superconducting transition at T2 for the crystal with 24.7% K content. On further K doping, T1 increases slightly, even though the entropy reduction at T1 becomes significantly weaker. Apparently, this renders the coexistence of superconductivity and the C4-magnetic phase possible, diminishing the driving force of the re-entrant transition at T2.

Figure 3: Detailed phase diagram of Ba1−xKxFe2As2.
figure 3

(a) Phase diagram of Ba1−xKxFe2As2 spanning the whole substition range. Ts,N, T1, T2 and Tc (determined from the data shown in Fig. 2 and additional thermodynamic data) are shown by symbols. Lines and coloured areas are a guide to the eye indicating the five distinct thermodynamically ordered phases. A narrow region of a C4-symmetric (tetragonal) magnetic phase, occurring within the usual C2-symmetric (orthorhombic) SDW phase, is observed. It coexists with superconductivity with a reduced Tc. Superconductivity-induced reentrance of the C2-SDW phase occurs at T2. The inset shows an enlarged view of the region containing the C4-magnetic phase. Vertical error bars indicate the uncertainty in the transition temperatures and horizontal error bars indicate the statistical error in the sample composition. (b) Predicted effect of uniaxial in-plane pressure pab on the phase diagram, from thermodynamic relations using our thermal-expansion and heat-capacity data. The direction of the arrows indicates the sign of the pressure derivative of the transition temperature and the length is proportional to its absolute value (see scale). Lines are reproduced from panel (a).

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The interplay between different thermodynamic phases is also reflected in the effect of pressure on the phase diagram. We infer the effect of uniaxial, in-plane average, pressure pab on the phase diagram from our thermal-expansion and specific-heat data using thermodynamic relations (see Methods) and the results are shown in Fig. 3b. For example, when superconductivity coexists with the C2-SDW phase the uniaxial-pressure dependences dTs,N/dpab<0 and dTc/dpab>0 are of opposite sign and rather large showing the well-known competition of these two phases. Interestingly, a similar competition between the two magnetic phases is implied by the derivatives dTs,N/dpab<0 and dT1/dpab>0. However, dTc/dpab≈−1 K GPa−1 is relatively small within the C4 magnetic phase, even though T1 and T2 are very sensitive to pab. Note that the strong competition between superconductivity and the C2-SDW phase manifests itself by a strong coupling of both order parameters to the orthorhombic distortion, and clearly such a coupling is not possible in a tetragonal state, which might explain this small value. Finally, it is interesting that, within the whole range from 28 to 100% K content, the Tc line is very smooth and dTc/dpab≈−(2−3) K GPa−1 changes only weakly with doping. This suggests that the superconducting state does not change over the whole phase diagram and, in particular, does not undergo a symmetry change26.

Discussion

The C4 state which we observe in K-doped BaFe2As2 is very reminiscent of the magnetic C4 phase observed in Na-doped BaFe2As2 (ref. 10), and is also probably closely related to the pressure-induced phase found in Ba1−xKxFe2As2 (ref. 12) (see Supplementary Fig. 4). This suggests that the occurrence of the C4 phase is a universal property of the hole-doped 122-materials, which are expected to be 'cleaner' than the electron-doped ones. The most striking difference to Ba1−xNaxFe2As2 is the sudden re-emergence of δ, that is, the re-entrance of the C2-SDW phase, below T2≤Tc in Ba1−xKxFe2As2. One explanation for this difference may be the lower stability of the C4 phase, indicated by considerably lower values of T1. It is possibly related to a chemical pressure effect, since our results show that the magnetic C4 phase in Ba1−xKxFe2As2 is extremely sensitive to (uniaxial) pressure (Fig. 3b). In particular, in-plane pressure is expected to strongly increase the extent of the C4-magnetic phase, similar to hydrostatic pressure12.

Several authors10,17,27,28,29,30 have suggested that the detailed nature of the additional magnetic phase can hold important clues regarding the spin-orbital debate. As pointed out in ref. 10, the existence of a C4-symmetric magnetic phase shows that the orthorhombic distortion is not necessary for magnetism which strongly supports the spin-nematic scenario. Concerning the magnetic structure, such a phase is possible when there are two antiferromagnetic propagation vectors and the iron magnetic moments are either non-collinear or non-uniform13,27,31. On the other hand, magnetic neutron scattering results on Ba1−xNaxFe2As2 single crystals point to a re-orientation of the magnetic moments from in-plane to c axis orientated as the main element of the transition at T1 and call for further study on whether the magnetic structure really has C4 symmetry. Incidentally, the observed spin-re-orientation demonstrates that spin-orbit coupling cannot be neglected, and more theoretical work explicitly including this coupling is needed. Using the specific heat data, we can address the nature of these phase transitions from a thermodynamic point of view. In particular, the large entropy jump at the T1 transition is not expected for a simple spin reorientation. Rather, our data imply that a significant change of the band structure occurs, which may be more in line with the emergence of a second SDW order parameter in the itinerant spin-nematic viewpoint12,17. Moreover, our high-resolution data demonstrate that the orthorhombic distortion vanishes in the second magnetic phase with an error bar of less than 1% of its maximum value, which strongly suggests that the phase is truly C4-symmetric and that spin re-orientation is not the only element of the transition. Our data also show that superconductivity has a significant impact and can tip the balance between the two magnetic states in favour of the C2-symmetric one. Very recent theoretical works based on three-band itinerant fermionic or five-band tight-binding approaches in fact find surprisingly good qualitative agreement with our experimental phase diagram, suggesting that the physics of the hole-doped materials can be understood fairly well within these frameworks17,30. In particular, the suppression of superconductivity within the C4 phase and the re-entrance back into the C2 phase is captured in the work of ref. 30.

In summary, a new and very detailed phase diagram of Ba1−xKxFe2As2, revealing intriguing new interactions between superconductivity and magnetism, has been derived from thermodynamic measurements. In particular, we find evidence for a narrow region of a C4-symmetric magnetic phase which competes more strongly with superconductivity than the C2-symmetric SDW phase. This competition between the two magnetic phases and superconductivity for the electronic entropy of the system results in a novel re-entrance of the C2-symmetric phase either as a magneto-structural transition within the superconducting phase or as a peculiar first-order concomitant superconducting, structural and magnetic transition, both of which are exceptional for strongly correlated systems. The strong reduction of entropy associated with the second magnetic phase suggests it to be a C4-type SDW in an itinerant scenario, rather than a simple spin re-orientation transition, in good agreement with recent theoretical work17,30. There are, however, still some unresolved issues, including the exact magnetic structure and the role of orbitals in this C4 SDW phase. The existence of the C4 phase at ambient pressure in high-quality single crystals opens up the possibility for studies using a large variety of more microscopic probes, such as magnetic neutron scattering, nuclear magnetic resonance, and angle-resolved photoemission studies, which will hopefully unravel some of these mysteries. Finally, we note that our preliminary thermodynamic studies on the Ba1−xNaxFe2As2 system show that this system is even more complicated than thought at present10, and a detailed comparison between the K- and Na-doped systems will be very useful (L. Wang et al., manuscript in preparation).

Methods

Sample preparation and characterization

High-quality Ba1−xKxFe2As2 samples were grown by a self-flux technique in alumina crucibles sealed in iron cylinders using very slow cooling rates of 0.2–0.4 °C per h. They were in situ annealed by further slow cooling to room temperature. Single crystals with a mass of 1–3 mg were chosen for measurement and the composition of the five main samples of this article was determined by refinement of four-circle single-crystal X-ray diffraction patterns of a small piece of each crystal to be x=0.232(3), 0.247(2), 0.262(3), 0.276(2), 0.302(35). Good homogeneity is attested by the sharp thermodynamic transitions. The phase diagram in Fig. 3 is compiled using thermodynamic measurements of samples whose K content was determined by either single-crystal X-ray diffraction refinement or energy dispersive X-ray spectroscopy.

Thermal-expansion and specific-heat measurements

Uniaxial thermal expansion was measured in a home-made capacitance dilatometer18 and specific heat in a Physical Property Measurement System from Quantum Design. The electronic specific heat was obtained by subtracting (F. Hardy et al., manuscript in preparation), from the raw data, a concentration-weighted sum of the lattice heat capacity of KFe2As2 and Ba(Fe0.85Co0.15)2As2 derived from refs 23 and 32.

Determination of uniaxial pressure derivatives

Uniaxial-pressure derivatives were obtained from the Clausius–Clapeyron relation dT/dpab=Vm(ΔLab/Lab)/ΔS for first-order phase transitions (Ts,N, T1 and T2) and from the Ehrenfest relation, dT/dpab=VmΔαab/Δ(C/T) for second-order phase transitions (Tc). Here, pab is the average in-plane pressure, αab=1/LabdLab/dT the thermal expansion coefficient and ΔLab, ΔS and Δαab are the discontinuities at the phase transition and Vm=61.4 cm3 mol−1 is the molar volume, which hardly changes on K substitution.

Additional information

How to cite this article: Böhmer, A.E. et al. Superconductivity-induced reentrance of the orthorhombic distortion in Ba1−xKxFe2As2. Nat. Commun. 6:7911 doi: 10.1038/ncomms8911 (2015).