Nematic superconducting state in iron pnictide superconductors

Nematic order often breaks the tetragonal symmetry of iron-based superconductors. It arises from regular structural transition or electronic instability in the normal phase. Here, we report the observation of a nematic superconducting state, by measuring the angular dependence of the in-plane and out-of-plane magnetoresistivity of Ba0.5K0.5Fe2As2 single crystals. We find large twofold oscillations in the vicinity of the superconducting transition, when the direction of applied magnetic field is rotated within the basal plane. To avoid the influences from sample geometry or current flow direction, the sample was designed as Corbino-shape for in-plane and mesa-shape for out-of-plane measurements. Theoretical analysis shows that the nematic superconductivity arises from the weak mixture of the quasi-degenerate s-wave and d-wave components of the superconducting condensate, most probably induced by a weak anisotropy of stresses inherent to single crystals.


Supplementary Note 1. Simulation of and magnetoresistivity using solutions of Eilenberger equations
We used Eilenberger equations in the clean limit approximation to calculate all necessary properties of the superconducting condensate. The Eilenberger equations [3] and gap equation are We note that near c2 the induced currents are very small and can be neglected, thus, the magnetic field in the superconductor is equal to the applied magnetic field ( ), i.e. × = .

Supplementary Note 2. Parameterization of the pairing potential
The parameterization of the pairing potential was constructed using a procedure similar to the ones found in Ref. [4] and Ref. [5]. We decomposed the pairing potential, where is the number of components of the superconducting condensate. We note these functions are orthogonal and normalized, using the norm Near we can approximate the pairing potential, ( , ′), by the products of the form ̅ ̅ ( ) ̅ ( ′) where ̅ correspond to critical temperatures, c , in the neighborhood of the material's critical temperature c . Taking into account experimental evidences found in literature, we assumed a pairing potential with two dominant components, s±-wave and 2 − 2 -wave, for which we have assigned the functions, ̅ 1 ( ) and ̅ 2 ( ), respectively. In this case, the pairing potential can be approximated by ( , ′) = ∑ 2 =1 ̅ ̅ ( ) ̅ ( ′), (11) and the expansion of the superconducting condensate function by where Δ ̅ 1 ( ) and Δ ̅ 2 ( ) are the spatial components of the gap function associated with the symmetries s±-and 2 − 2 -wave, respectively. Additionally, we inserted couplings between the different components that might arise from small anisotropic distribution of the doping atoms or small strains in the sample. In this case the pairing potential transforms into and the superconducting condensate function maintain the same form in respect to the new variables, i.e Δ( , ) = ∑ 2 =1 ( )Δ ( ) , where the Δ 1 ( ) and Δ 2 ( ) are linear combinations of Δ ̅ 1 ( ) and Δ ̅ 2 ( ) and 1 and 2 are constants associated with new 1 and 2 functions, respectively. Inserting these approximations into the linearized Eilenberger equations, we obtain To obtain ̅ ( ) for Ba1-xKxFe2As2, we expanded it in harmonic functions centered in each pocket of the Fermi surface, in a similar procedure to the one described in Ref. [5]. The selected harmonic functions are compliant with the symmetry of pocket's geometry and of the corresponding ̅ ( ), i.e.
We assume that ̅ ( ) is evenly distributed over the Fermi surfaces of the central holes and electron pockets, which is a good approximation since the interband scattering is stronger than the intra-band scattering between the central hole pockets and the electron pockets. However, we made a broader assumption, which was to consider that ̅ ( ) is evenly distributed over all pockets of the Fermi surface.
Usual BCS equation (which can be obtained from Supplementary Eq. (3)) for a homogeneous single component superconductor (this means that Δ = . and By considering that → c then Δ → 0 and as usual we obtain Using it, the following sum can separated into This sum can also be approximated by and thus from which we get Supplementary Eq. (16), can be approximated to and combining it with Supplementary Eq. (29) we obtain For simplicity we took the same cutoff energy, Ω BCS , for all components.

Supplementary Note 3. Expansion into Landau levels and of two-component superconductors
We made a perturbation expansion of the first equation with respect to the operator ⋅ . The -th order of perturbation of the anomalous Green function is, (1) ( , , ).
Inserting this expansion from the zero-th to the fourth order into the gap equation Afterwards, we expressed the operator ⋅ in terms of ladder operators associated with the Landau levels of Δ 1 and Δ 2 , i.e.
The values of variables 1,1 , 2,1 , 1,2 and 2,2 are arbitrary and we have chosen these such that 2,2 ( 1 2 ) = 0 and 2,2 ( 2 2 ) = 0. Projecting the first and second equation into the lowest Landau levels Δ 1 0 and Δ 2 0 , respectively, we simplified the previous equations to where |0〉 1 and |0〉 2 are the Dirac ket's directly corresponding to the Landau levels of Δ 1 0 and Δ 2 0 , respectively, and was replaced by c2 since, by definition, it is the second critical field corresponding to a given .

Supplementary Note 4. of three-component superconductors
The experimental curves display other features rather that the strong anisotropy close to Fe-Fe bond direction that cannot be explained within the two-component model constructed previously. To refined our model we add a third component ̅ 3 ( ) with a symmetry different from the one on ̅ 1 ( ) and ̅ 2 ( ) into the pairing potential which becomes and can be transformed into the form and, in the case of -wave symmetry, ̅ 3 ( ) is given by where |0〉  Tinkham's models, it is required to calculate the pinning barrier energy that is proportional to the free energy density of the system. Moreover, the free energy in singleband superconductors can be related to the second critical magnetic field using the Abrikosov expression [6]. In Anderson-Kim's model, the expression for resistivity is given by where 0 is the characteristic frequency of the flux-line vibration (from 10 5 to 10 11 s -1 ), and 0 is its activation energy (or barrier's height), is the Boltzmann constant, and is the temperature. In Tinkham's model, the resistivity is given by where is the resistivity of the normal state, 0 is the modified Bessel function, and 0 is the activation energy (or barrier's height). Both these models were applied by us to capture the phenomenological features of the experimental magnetoresistivity curves.
However except in this subsection, we only considered Tinkham's model within the rest of the publication (including all presented plots of magnetoresistivity) since it gives a better fitting to experimental results.
In both models, 0 can be expressed in terms of the free energy and characteristic lengths, i.e., 0 ∝ Δ Φ 0 ( ) where Δ is the Gibbs free energy, is the coherence length of the condensate, Φ 0 is the magnetic flux quantum, and is the magnitude of the induced magnetic field. We note that where c2 is the magnetic induction associated with the second critical magnetic field c2 (note that we are using the CGS unit system where c2 = c2 if we neglect the magnetic field developed by the induced currents), is the magnitude of the magnetic field, is a characteristic parameter of the material, and A is the Abrikosov parameter that for a triangular lattice is A = 1.16. This expression is valid near c2 , and it is obtained by considering first perturbation in the wave function and in the magnetic field to the solution of linearized Ginzburg-Landau (GL) equation in which the magnetic field is equal to the applied magnetic field as-described in the sections above. We note that we do not observe in our compound a shift in onset value of the transition (upper part of the resistivity curves in the normal-superconducting state transition) with changes in the magnetic field magnitude, as it is expected in conventional superconductors. The same behavior is observed in other high c superconductors like copper-based superconductors.
To take this aspect into account, we made one last approximation in Abrikosov expression, presented in Supplementary Eq. (59), as it is done in Ref. [7]. when considering three components, we still cannot obtain the correct absolute value of c2 , probably due to renormalization of the effective mass of electrons or the effect of more bands. To account for this discrepancy, we introduced a renormalization factor into the Fermi velocities consistent with a renormalization of the effective mass of ≈ 2.0. We note that a small measuring error in the assessment of the superconducting critical temperature Tc will have a strong impact on estimated the renormalization of the effective mass. For example if Tc shifts to a higher temperature just by 0.1 K, the renormalization value will be strongly reduced. Furthermore, the broad normal-superconductor transition displayed in the magnetoresistivity curves does not allow a very accurate determination of its onset value.

Supplementary
We have to distinguish two distinct sets of simulations: the first for c2 displayed in  However, according to theoretical study in Ref. [9] the value of the strongest coupling constant for a similar doping levels is ≈ 2, consequently we took = 2.
The set of parameters, ̅ 1 , ̅ 2 , ̅ 3 , ̅ 12 , ̅ 23 and ̅ 13 is not unique since, the values of the coupling constants, ̅ 12 , ̅ 23 and ̅ 13 , between different components can be changed arbitrarily such that simulation results of c2 and IMR remain almost unchanged, if we also change the critical temperatures or the parameters ̅ associated with ̅ , where ∈ {1,2,3}. Additionally, the highest , designated by ̅ is defined such that c ′ = c . and ,2 can be found by the normalization of ̅ . We varied this ratio value and found that it only influences very weakly the c2 and magnetoresistivity results.
In the calculation of magnetoresistivity 0 = AΔ proportionality factor that was extracted by fitting the simulated to the experimental curves, A = 0.01. The value of ( ) = 0 (1 − / c ) was set to 0 = 3.53 nm according to Ref. [2].

Supplementary Note 7. Estimation of the impact of misalignment in
We can assess the impact of misalignment in c2 by calculating the relative deviation of the magnetic critical field, i.e.
( ) = c2 ( ,0)− c2 (0,0) For this calculation we will consider only a single -wave component since magnetic field cannot the mix the -wave and the -wave components. We will present here mathematical expressions that are shown, below, in theoretical description. We can obtain c2 ( , ) using Supplementary Eq. (44).
If we only take into account the 1 component, it becomes: In the case of a -wave component 1 = 1.

=0
(2 ) 2 ℏ 2 The relative deviation of c2 becomes: Temperature dependence of magnetoresistivity for the optimal-doped single crystal Ba0.5K0.5Fe2As2 under magnetic fields from 0 to 9 T. Here the magnetic fields were applied within the ab-plane with angle θ = 135º. A superconducting transition shows an onset temperature of 37.6 K and a zero-R temperature of 37.3 K without a magnetic field.
This sharp transition suggests that the quality of the single crystal is high, although the transition became wider with increasing magnetic field. The angle θ is defined as that between the magnetic field and the a(b)-axis of the lattice, as indicated in Fig. 1a. The dependence of ρab on θ was measured by rotating the ab-plane around the c-axis in a fixed magnetic field parallel to the ab-plane and at a fixed temperature. left to right). The thickness of the base crystal was larger than 20 μm, and the in-plane geometry was considerably larger as few hundred micrometers. Therefore, the resistance of the base crystal was extremely lower than that of the mesa.

Supplementary
Supplementary Figure 6. Out-of-plane magnetoresistivity under in-plane fields.
Angular dependence of out-of-plane magnetoresistivity for Ba0.5K0.5Fe2As2 in an in-plane magnetic field of 5 T for (a) and the corresponding polar plots for (b). The c measurements were conducted at initial angles (ϕ) of 0 and 45° between the field and a(b)-axis. As expected, the c-T curve is slightly different from the ab-T curve. The anisotropy parameter is 2.3 at 300 K, suggesting a quite weak electronic dimensionality of two dimensions. The angle-dependent out-of-plane magnetoresistivity shows an anomaly similar to that observed in the IMR measurements. Supplementary Figure 6 shows the out-of-plane magnetoresistivity at 37.6 and 37.8 K under a magnetic field of 5 T. The c-θ curves are observed to undergo sinusoidal oscillation with the maximum at the angle where H is parallel to the ГM direction, comparable to the IMR results. In addition, we tested the dependence of θ on the initial angle. Again, we confirmed that the angular dependence of the magnetoresistivity within the ab-plane truly reflects the inplane nature.