Sudden reversal in the pressure dependence of T_c in the iron-based superconductor KFe₂As₂

Abstract

Proximity to an antiferromagnetic phase suggests that pairing in iron-based superconductors is mediated by spin fluctuations^1,2,3,4, but orbital fluctuations have also been invoked⁵. The former typically favour a pairing state of extended s-wave symmetry with a gap that changes sign between electron and hole Fermi surfaces^6,7,8,9 (s_±), whereas the latter yield a standard s-wave state without sign change⁵ (s₊₊). Here we show that applying pressure to KFe₂As₂ induces a sudden change in the critical temperature T_c, from an initial decrease with pressure to an increase above a critical pressure P_c. The smooth evolution of the resistivity and Hall coefficient through P_c rules out a change in the Fermi surface. We infer that there must be a change of pairing symmetry at P_c. Below P_c, there is compelling evidence for a d-wave state^{10,11,12,13,14}. Above P_c, the high sensitivity to disorder rules out an s₊₊state. Given the near degeneracy of d-wave and s_± states found theoretically^{15,16,17,18,19}, we propose an s_± state above P_c. A change from d-wave to s-wave would probably proceed through an intermediate s+i d state that breaks time-reversal symmetry^20,21,22.

Main

KFe₂As₂ is a stoichiometric iron arsenide with a superconducting critical temperature T_c = 4 K. It is a member of the extensively studied 122 family of iron-based superconductors²³. Single crystals can be grown with very high purity, making it by far the cleanest of the iron-based superconductors. Its high hole concentration is such that its Fermi surface does not contain the usual electron pocket at the X point (of the unfolded Brillouin zone); it consists mainly of three hole-like cylinders: two located at the zone centre (Γ) and one at the corner (M; Fig. 1a). There is no antiferromagnetic order, but there are antiferromagnetic spin fluctuations, detected by inelastic neutron scattering²⁴. In iron-based superconductors, spin fluctuations generally favour the s_± pairing state in which the gap changes sign between hole and electron pockets^1,2,3,4 (Fig. 1b). In the absence of the electron pocket at X, this mechanism becomes much less effective, and functional-renormalization-group calculations find that a d-wave state (Fig. 1c) is the most stable state in KFe₂As₂ (ref. 15). Other theoretical methods find that s_± and d-wave states are very close in energy^17,18. Experimentally, thermal conductivity studies in KFe₂As₂ make a compelling case for d-wave symmetry^10,11,12,13: line nodes are found to be vertical and present on all Fermi surfaces, and the thermal conductivity is independent of impurity scattering, as expected of symmetry-imposed line nodes²⁵. A d-wave state is also consistent with penetration depth data¹⁴. However, in a recent angle-resolved photoemission spectroscopy (ARPES) study of KFe₂As₂, vertical line nodes in the gap were observed on only one of the three Fermi surfaces²⁶. To explain this, a particular kind of s_± state was proposed²⁷ where the sign change is between the two Γ-centred hole pockets (Fig. 1d).

Figure 1: Fermi surface of KFe₂As₂ and possible superconducting states.

a, Schematic of the main Fermi surface sheets of KFe₂As₂, in the k_z = 0 plane, shown in the unfolded Brillouin zone (with one Fe per unit cell)³. It consists of two zone-centred hole pockets (h₁ and h₂) and one hole pocket at M (h₃). b, Sketch of the main Fermi surface sheets of K_1−xBa_xFe₂As₂, at x = 0.4, with an electron pocket (e) at X (and no hole pocket at M)³. The standard s_± pairing state (type I) involves full gaps on each pocket, with a sign change from + on the hole pockets (h₁, h₂) to − on the electron pocket (e). c, d-wave pairing state¹⁶, where the gap changes sign as the azimuthal angle crosses the zone diagonals (dashed lines). This symmetry forces the gap to have nodes (zeros) on all Fermi surface sheets that cut those diagonals (small open circles). d, Illustration of an s_± pairing state (type II) where the gap changes sign from + on the inner Γ-centred hole pocket (h₁) to − on the outer Γ-centred hole pocket (h₂) (see ref. 27).

To help clarify the situation, we have studied the effect of hydrostatic pressure on KFe₂As₂, by measuring the resistivity and Hall effect in two single crystals, labelled sample A and sample B (Supplementary Information). As seen in Fig. 2a, we find that T_c decreases initially with pressure, as found elsewhere²⁸, but once the pressure is increased above a critical value P_c = 17.5 kbar, it suddenly starts to rise. T_c varies linearly on either side of P_c, producing a V-shaped dependence of T_c on P. This sharp inversion in the effect of pressure on T_c is our central finding, reproduced in both samples (Supplementary Fig. S1). There are two possible mechanisms: a Lifshitz transition, whereby the Fermi surface undergoes a sudden change; or a phase transition with broken symmetry. In Fig. 3a, we see that the Hall coefficient R_H in the T = 0 limit remains completely unchanged by pressure, right through P_c (Fig. 3b). A Lifshitz transition, such as the appearance of an electron pocket, would produce a sudden change in R_H(0). It can therefore be excluded as a possible cause for the rise of T_c beyond P_c.

Figure 2: Pressure dependence of T_c in KFe₂As₂.

a, Pressure dependence in KFe₂As₂, with filled circles representing sample A (from colour-coded resistivity data in b,c) and open circles sample B. The critical pressure P_c = 17.5 kbar marks the transition from a decreasing to an increasing T_c. The dashed lines are linear fits to the data within 10 kbar on either side of P_c. b, Isobars of ρ(T) in sample A, normalized to unity at T = 4 K, for three pressures below P_c, as indicated. c, The same as in b, for pressures above P_c, with ρ normalized at T = 2.5 K. T_c is the temperature below which ρ(T) = 0. The relative accuracy on T_c is better than ±2% (Supplementary Fig. S2); the relative uncertainty on the value of pressure is ±0.2 kbar. The grey arrows show how T_c moves with pressure.

Figure 3: Hall coefficient and resistivity of KFe₂As₂ under pressure.

a, Hall coefficient R_H of KFe₂As₂ as a function of temperature, plotted as R_H versus T², measured in sample A for a magnetic field of 13 T along the c axis of the tetragonal lattice, at different values of the applied pressure P, as indicated (see also Supplementary Fig. S4). The lines are linear fits to the data, extrapolated to T = 0 to yield the residual values, R_H(0). b, Pressure dependence of R_H(0), seen to remain completely unchanged throughout the range investigated. c, Pressure dependence of ρ measured in sample A at T = 20 K, seen to decrease monotonically through P_c. The solid line is a smooth fit through all data points.

We deduce that a phase transition occurs at P_c. Any density-wave or structural transition that breaks translational symmetry would reconstruct the Fermi surface, and cause associated anomalies in the transport properties. Such transitions are therefore ruled out by the absence of any anomaly in R_H (Fig. 3b) and in the electrical resistivity ρ (Fig. 3c) at P_c. Note that density-wave phases such as the antiferromagnetic phase in Ba_1−xK_xFe₂As₂ with x<0.4 generally compete with superconductivity and so produce a dome-shaped curve of T_c versus P or x (refs 1, 2, 12), not a V-shaped curve as seen here (Fig. 2a). We conclude that what occurs at P_c is not a transition in the normal-state electronic properties, but a transition to a superconducting phase of a different symmetry.

For the phase above P_c, the effect of impurity scattering on T_c rules out the standard s₊₊ state. At ambient pressure, 4% Co impurities in KFe₂As₂ suppress T_c to zero¹³—the critical value of the residual resistivity being ρ₀^crit = 4.5 μΩ cm (refs 11, 13). This is consistent with a d-wave state, whose T_c is expected to vanish when the scattering rate is of the order of T_c (ref. 11). We measured the resistivity of a sample of KFe₂As₂ with 3.4% Co impurities, in which ρ₀ = 3.8 μΩ cm and T_c = 1.7 K at ambient pressure (Supplementary Fig. S3). Under pressure, the T_c of this Co-doped sample is suppressed to zero and does not re-emerge above P_c (Fig. 4). This shows that the superconducting state above P_c cannot be s₊₊, a state that is insensitive to non-magnetic impurities. Neither could this high-pressure state be the same s_± state as in the usual BaFe₂As₂-based superconductors (with an electron pocket at X), because T_c in these materials is very robust against impurity scattering²⁹.

Figure 4: Effect of impurity scattering on T_c in KFe₂As₂.

Pressure dependence of T_c in KFe₂As₂ for a nominally pure sample (open circles, sample B) and a sample with 3.4% Co impurities (filled squares, K(Fe_1−xCo_x)₂As₂ with x = 0.034, from resistivity data in Supplementary Fig. S3). The dotted lines are linear fits to the pure data within 10 kbar on either side of P_c. The solid green line is a linear fit to the first two data points of the impure sample. Note how T_c is suppressed by the addition of impurities both below and above P_c, showing that the pairing states on both sides of the transition are highly sensitive to impurity scattering.

A plausible candidate for the phase above P_c is the s_± state proposed in ref. 27 (for ambient pressure), with a sign change between the two hole pockets at Γ (Fig. 1d). Given the similarity of these two pockets, inter-band scattering is likely to be significant and T_c is therefore expected to be rather sensitive to disorder, as observed. The fact that, above P_c, T_c rises even though ρ continues to decrease (Fig. 3c) is consistent with calculations for this type of s_± state, which require that the interaction between fermions be largest at small momentum transfer²⁷. As mentioned above, this type of s_± state is consistent with the ARPES study that finds a large, angle-dependent gap on the two Γ-centred hole pockets and a small nodeless gap on the M-centred hole pocket²⁶. We suggest that the state measured by ARPES at ambient pressure is this s_± state, stabilized at the cleaved (polar) surface of KFe₂As₂ even though the bulk is in a d-wave state.

In iron-based superconductors with a Fermi surface that contains hole and electron pockets (Fig. 1b), the natural proximity of s_± and d-wave states was nicely revealed by calculations¹⁹ where the strength of the (π, π) spin fluctuations (connecting two nearby electron pockets) was gradually increased for a fixed strength of (π, 0) spin fluctuations (connecting hole and electron pockets). A V-shaped variation of T_c is obtained, as the superconducting phase goes from s_± to d-wave. Impurity scattering suppresses T_c on both sides of the transition and opens up an intermediate region without superconductivity¹⁹. This is the phenomenology we observe in KFe₂As₂, where an analogous mechanism of competing interactions may be at play even if the Fermi surface does not contain an electron pocket.

Our proposal is that the V-shaped dependence of T_c on pressure in KFe₂As₂ reflects a change of pairing symmetry, most likely from a d-wave state below P_c to an s_± state above P_c. To confirm this interpretation, measurements that probe the superconducting state below T_cwill be needed, especially above P_c. Note that when d-wave and s-wave phases come together, an intermediate s+i d phase that breaks time-reversal symmetry is likely to intervene^{19,20,21,22,30}. One signature of such a phase is a spontaneous internal magnetic field that appears below T_c, which could in principle be detected with muons, in clean samples of KFe₂As₂ at pressures near P_c.