A Analysis of \lambda^{-2}_{ab}(T)

# Strong Competition of Superconducting and Magnetic Order Parameters in Ba1-xNaxFe2As2

## Abstract

We study the interplay of magnetic and superconducting order in single crystalline hole doped BaNaFeAs using muon spin relaxation. We find microscopic coexistence of magnetic order and superconductivity. In a strongly underdoped specimen the two forms of order coexist without any measurable reduction of the ordered magnetic moment by superconductivity, while in a nearly optimally doped sample the ordered magnetic moment is strongly suppressed below the superconducting transition temperature. This coupling can be well described within the framework of an effective two-band model incorporating inter- and intra-band interactions. In optimally doped BaNaFeAs we observe no traces of static or dynamic magnetism and the temperature dependence of the superfluid density is consistent with two s-wave gaps without nodes.

###### pacs:
74.70.Xa, 61.05.C-, 74.62.Dh, 76.75.+i

Strongly correlated electron systems like the cuprate high-, organic, and heavy-fermion superconductors exhibit a delicate interplay of superconductivity and antiferromagnetic order. Heavy-fermion superconductivity often takes place in the vicinity of a magnetic quantum critical point Mathur et al. (1998). Similarly, magnetic order lies nearby and competes with superconductivity in the phase diagrams of the cuprate high- superconductorsSachdev (2000); Orenstein and Millis (2000); Lee et al. (2006). In the ferropnictide superconductors the interplay of magnetism and superconductivity leads to a new phase that is both homogeneously superconducting and magnetically ordered Parker et al. (2009); Fernandes and Schmalian (2010); Vorontsov et al. (2010); Fernandes and Schmalian (2010); Fernandes et al. (2010a)—i.e., magnetic order and superconductivity coexist microscopically Pratt et al. (2009); Fernandes and Schmalian (2010); Mazin (2010). Still, both instabilities compete for the same electron states around the Fermi surface, which causes a reduction of the ordered magnetic moment upon cooling below . In some materials electronic phase separation appears instead of microscopic coexistence. Experiments with electron doped BaFeCoAs suggest microscopic coexistence Pratt et al. (2009); Julien et al. (2009); Nandi et al. (2010), nevertheless a recent report by Bernhard et al. (2012) indicates electronic inhomogeneity are important near optimal doping. Electronic phase separation, on the other hand, commonly occurs in hole doped BaKFeAs Park et al. (2009); Aczel et al. (2008); Goko et al. (2009); Julien et al. (2009). Only recently Wiesenmayer et al. (2011) showed a reduction of the ordered moment by superconductivity in polycrystalline underdoped BaKFeAs—an unambiguous proof of microscopic coexistence—consistent with other reports Rotter et al. (2009); Urbano et al. (2010); Avci et al. (2011).

In this Letter we report a strong interplay of the order parameters of superconductivity and magnetic order in BaNaFeAs. In a nearly optimally doped compound, the ordered magnetic moment of the Fe spin density wave order, that appears below  K, is reduced by % upon cooling below  K.

We examined platelike BaNaFeAs single crystals (growth conditions and characterization are described in Ref. Aswartham et al. (2012)) with the -axis oriented along the muon beam. The muon-spin relaxation was measured in zero (ZF) and transverse (TF) magnetic fields of up to 0.64 T using the DOLLY and GPS instruments at the Paul Scherrer Institute equipped with He flow-cryostats. To analyze the SR data, we used the musrfit framework Suter and Wojek (2012). The stoichiometry of each crystal was determined with electron-dispersive x-ray spectroscopy (EDX). Magnetization, electrical resistivity, angle-resolved photoemission spectroscopy (ARPES), Hall-effect, and specific-heat measurements of single crystals from the same batches are published in Refs. Pramanik et al. (2011); Aswartham et al. (2012); Pramanik et al. (2012).

Fig. a shows the muon spin polarization in ZF perpendicular to the beam. For , the relaxation of is temperature independent and characteristic for a paramagnetic material (down to the lowest measured temperature of 2 K). For , and 0.30 we find two precession signals with frequencies that indicate two magnetically nonequivalent muon sites, and prove long range magnetic order with , and 40.0(5) K, respectively. We analyze with two damped-cosine oscillations, and a nonrelaxing signal due to paramagnetic and non-oscillating signals.

The muon-spin precession frequencies (which are proportional to the magnetic order parameter) of , and 0.30 are shown in Figs. b, and c. For we find no anomaly at  K on cooling ( from Ref. Aswartham et al. (2012)). A fit with a general order parameter temperature dependence , shown in Fig. b, to the data above reproduces the data also below . By setting we estimate , consistent with the 2D-Ising universality class (), and similar to results for BaFeAs Wilson et al. (2010).

We conducted SR “pinning experiments” to verify bulk superconductivity: We cooled the sample with to  K in a transverse magnetic field  mT, recorded a time histogram (Fourier transform shown in Fig. a as blue line), then isothermally reduced the field to  mT and recorded another histogram (red line). The magnetic order caused four precession signals in the external magnetic field: , and , as indicated by the Fourier transform in Fig. a, in addition to the background signal due to muons stopped by the cryostat walls or the detector. The signal fraction that followed the change of the external magnetic field was % of the total signal, which is consistent with a background signal. The remainder of the signal maintained its average internal field after we reduced the field, which indicates strong flux pinning due to superconductivity in the majority of the sample. Instead, caused by enhanced vortex lattice disorder, only the damping of the spin precession increased (spectral lines broadened). This evidence for bulk superconductivity taken together with the bulk magnetic order supports microscopic coexistence of magnetic order and superconductivity, albeit with a coupling of the order parameters that is too small to be detectable Fernandes et al. (2010b).

The pinning experiment for , shown in Fig. b, also indicates bulk superconductivity. In contrast to , the magnetic order parameter for (see Fig. c) was suppressed by % on cooling below  K, which proves microscopic coexistence of magnetic order and superconductivity and strong coupling of both order parameters.

Theoretical studies show that microscopic coexistence as well as phase separation arise in itinerant multi-band models as a function of Fermi surface topology, band dispersion, and symmetry of the superconducting order parameter Parker et al. (2009); Fernandes and Schmalian (2010); Vorontsov et al. (2010); Fernandes and Schmalian (2010); Fernandes et al. (2010a). Within these models, Cooper-pairing, in which the gap changes sign between the hole and electron pockets, has the largest tendency towards microscopic coexistence. In order to show that such an approach can account for our measurements, we calculate the temperature dependence of the ordered magnetic moment within a simplified two-band model incorporating inter- and intra-band interactions that drive the spin density wave order (SDW) as well as superconductivity (SC) Brydon and Timm (2009); Knolle et al. (2011): , with the dispersion of the electron () and of the hole () band. Decoupling the interactions in mean-field we diagonalize the Hamiltonian by sequential Bogolyubov transformations that lead to the eigenenergies where and . We use the following parameters appropriate for a two-band model with hole doping: . Here is the antiferromagnetic (SDW) order parameter and are the two SC gaps (see Ref. Knolle et al. (2011) for details). We solve the self-consistency equations in order to obtain the temperature dependence of the magnetic order parameter , and the gaps , of superconductivity.

The results for and the superconducting gaps are shown as a solid/dashed lines in Fig. c. The dip in coincides with the onset of SC at . In particular, the relative reduction of the magnetic order parameter below is mostly determined by the magnitude of the superconducting order parameter that competes for the same Fermi surface points. More elliptical electron pockets reduce the magnitude of in comparison to the superconducting order parameter, thus increasing the relative suppression. The agreement between the calculated temperature dependence and the experimental data is very good for  K. At lower temperatures the experimental data drop somewhat faster than the theoretical temperature dependence. The discrepancy may be connected to a change of magnetic fluctuations that change the magnetic moment at the SC due to gapping of particle-hole excitations.

For , the temperature independent total oscillation amplitude and TF-SR show that 100% of the sample is magnetically ordered (not shown). For , the remaining precession amplitude in 5 mT TF SR indicates a paramagnetic phase with a volume fraction of % below , shown in Fig. a. The diamagnetic shift of the average internal field of the paramagnetic phase already for  K indicates that this phase becomes superconducting below  K. The precession amplitude in zero field (see Fig. b) is temperature independent, which indicates that the magnetic volume fraction reaches % and remains temperature independent below  K. The sharp increase of the magnetic volume fraction in Fig. b shows that, even though the transition temperature is reduced to  K, the transition remains sharp—much sharper than in other ferropnictides (measured from onset to saturation of the magnetic volume fraction it is not broader than 5(1) K) Wiesenmayer et al. (2011); Bernhard et al. (2012); Shiroka et al. (2011); Sanna et al. (2009). The small damping rate of the muon spin precession also indicates well defined magnetic order: it did not exceed for in the well-ordered magnetic state but also increased with doping, i.e., for at  K (visible in Fig. a). The increase of the damping rate with doping indicates that Na doping indirectly causes disorder in the FeAs layers, albeit much weaker than Co doping that causes overdamped muon spin precession close to optimal doping Bernhard et al. (2012). Because Na is introduced between the FeAs layers, whereas Co resides directly in the magnetic FeAs layer, such behavior be expected. We summarize the results of TF- and ZF-SR for in a phase diagram shown in Fig. b.

The temperature dependence of the precession frequencies , shown in Figs. b, LABEL:sub@img.35-freq are identical and the temperature dependencies of the precession amplitudes, shown in Fig. b for (scaled to the magnetic volume fraction), are constant below for both magnetic compounds. This observation indicates a magnetic structure that remains unaffected by superconductivity. In particular, it indicates that the reduction of the precession frequencies for below is solely due to the reduction of the ordered moment and not due to a reduction of the magnetic volume. With a local probe like SR this distinction is possible, whereas scattering techniques are only sensitive to the product of both quantities.

The magnetic order for , and 0.30 prohibits the study of the penetration depth by SR. For we measure the probability distribution of local magnetic fields in the vortex state by TF-SR with  mT parallel to the crystallographic -axis. Fig. 4 shows , which is proportional to the standard deviation of , and the average local magnetic field Brandt (1988). We find that is best described by two independent s-wave gaps without nodes , 3.0(4) meV,  K, the weight of the large gap is , and for the small gap and  nm (see supplemental material for details). Vortex lattice disorder would artificially reduce (broaden ), therefore  nm is a lower limit for the in-plane penetration depth. These values are in good agreement with , 3 meV determined ARPES on a single crystal from the same batch with the same composition Aswartham et al. (2012). Specific heat measurements Pramanik et al. (2011) on a crystal with 35% Na doping and  K revealed , 2.68 meV, which indicates that the small gap increases linearly with ( for both compositions), whereas the large gap grows non-linear with ( for , 0.40, respectively).

In summary, we have conducted muon-spin relaxation experiments on BaNaFeAs, with , 0.30, and 0.40. The Na doping reduces the magnetic ordering temperature to , and 40.0(5) K for , and 0.30, respectively, but the magnetic transitions remains sharp and the magnetic order well defined. For , magnetic order and slow fluctuations were absent down to  K, as opposed to optimally doped BaFeCoAs Bernhard et al. (2012). All three specimens are bulk superconductors with critical temperatures , 34.5(13), and 33.5(2) K for , 0.30, and 0.40, respectively ( for from Ref. Aswartham et al. (2012)). For , and 0.30 we presented unambiguous evidence for microscopic coexistence of the two phases. The reduction of the magnetic order parameter by superconductivity was strongly dependent on doping: it was too weak to detect for , but amounted to % for . For underdoped BaKFeAs, on the other hand, Wiesenmayer et al. (2011) reported a much smaller reduction of the ordered magnetic moment below that did not increase significantly with doping. Within our itinerant model this can be accounted for by a much weaker superconducting order parameter relative to the magnetic one, i.e., better nesting of the electron and hole bands. On the other hand, the non-monotonic doping dependence points to disorder effects which could suppress magnetism and superconductivity differently Fernandes et al. (2012). The presence of a second, purely superconducting minority phase for suggests chemical inhomogeneity—nevertheless, microscopic coexistence occurred in the majority of the sample volume (77(2)%), suggesting that this phenomenon is more robust against disorder and doping in this system than in BaKFeAs. The symmetry and size of the superconducting gaps, as well as for is similar to those of BaKFeAs Khasanov et al. (2009). Considering the evidence for microscopic coexistence presented by Wiesenmayer et al. (2011) in underdoped BaKFeAs polycrystals and in this Letter for under- and nearly optimally doped BaNaFeAs, we conclude that this phenomenon is intrinsic to hole doped BaFeAs and can be described by an itinerant two-band model as for BaFeCoAs Fernandes et al. (2010b).

###### Acknowledgements.
HM thanks R. Fernandes for helpful discussions. This work was funded by the German Research Foundation (DFG) within the priority program SPP1458, and the graduate school GRK1621. JK acknowledges support from the German National Merit Foundation and the IMPRS Dynamical Processes in Atoms, Molecules and Solids. SW acknowledges support by the DFG under the Emmy-Noether program (Grant No. WU595/3-1). Part of this work has been performed at the Swiss Muon Source at the Paul Scherrer Institute, Switzerland.

## Appendix A Analysis of λ−2ab(T)

In analogy to Ref. Khasanov et al. (2009) we analyzed assuming two independent contributions to the superfluid density, arising from two -wave gaps without nodes with different gap sizes and but identical . We let the gap size independent of , which is the so called -model Padamsee et al. (1973). We fit the following expression to the data Carrington and Manzano (2003); Tinkham (2004)

 λab(T)−2=λab(0)−2[ω(1−D(Δ1(T),T))+(1−ω)(1−D(Δ2(T),T))],D(Δ(T),T)=2∫∞Δ−∂f(E)∂EE√E2−Δ(T)2dE, (1)

where is the Fermi-Dirac distribution, the weighting factor of the gaps, and the temperature dependence given by Carrington and Manzano (2003).

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