Interplay between superconductivity and itinerant magnetism in underdoped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ ($x=$ 0.2) probed by the response to controlled point-like disorder

The response of superconductors to controlled introduction of point-like disorder is an important \emph{phase sensitive} tool to probe the electronic and magnetic structures. In the case of iron-based superconductors (IBS), magnetic fluctuations presumably play an important role in inducing high-temperature superconductivity. Therefore, studying compositions where superconductivity coexists with the long-range magnetic order is of particular importance. In one of the most studied pnictide family, hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ (BaK122), this coexistence occurs over a wide range of doping levels, 0.16~$\lesssim x \lesssim $~0.25 . We used relativistic 2.5 MeV electrons to induce vacancy-interstitial (Frenkel) pairs that act as efficient point-like scattering centers. Upon increasing dose of irradiation, the superconducting transition temperature $T_c$ decreases dramatically. In the absence of nodes in the order parameter this provides a strong support for a sign-changing $s_{\pm}$ pairing. Simultaneously, in the normal state, there is a strong violation of the Matthiessen's rule and a decrease (surprisingly, at the same rate as $T_c$) of the magnetic transition temperature $T_{sm}$, which indicates the itinerant nature of the long-range magnetic order. Comparison of the hole-doped BaK122 with electron-doped Ba(Fe$_x$Co$_{1-x}$)$_2$As$_2$ (FeCo122) with similar $T_{sm}\sim$110~K, $x=$0.02, reveals significant differences in the normal states, with no apparent Matthiessen's rule violation above $T_{sm}$ on the electron-doped side. We interpret these results in terms of the distinct impact of impurity scattering on the competing itinerant antiferromagnetic and $s_{\pm}$ superconducting orders.


I. INTRODUCTION
The use of controlled disorder is a powerful phasesensitive way to study the nature of the superconducting state without affecting the chemical composition [1][2][3][4][5][6][7][8][9][10].According to Anderson's theorem [1], conventional isotropic s−wave superconductors are not affected by the scalar potential (i.e.non spin-flip) scattering, but are sensitive to spin-flip scattering due to magnetic impurities (for recent theoretical results on the impact of impurities on T c , see for example Refs.[11][12][13]).In singleband high superconducting transition temperature (high-T c ) cuprates, both magnetic and non-magnetic impurities cause a rapid suppression of T c , consistent with the nodal d−wave pairing [14].In multi-band iron-based superconductors (IBS), a sign-changing order parameter between the electron-like and hole-like Fermi sheets, s ± , is the most plausible pairing state [15][16][17][18][19][20].Although its response to non-magnetic scattering depends sensitively on the multi-band structure of the pairing interaction, on the chemical potential, and on the gap anisotropy, it is generally expected that interband scattering is much less efficient in causing pair-breaking than intraband scattering [7,18,[21][22][23][24][25][26][27][28].Additionally, the orbital content of the bands can also affect the suppression of T c [23,24,[29][30][31][32]. We note that the multi-band character of the superconducting state alone is not sufficient to have T c suppression [33].For instance, in the known two-gap s ++ superconductor, MgB 2 , where the gap does not change sign, electron irradiation resulted only in a small change due to gap magnitude difference between two bands [34].
While the effect of scattering induced by various means from chemical substitution to irradiation with various particles on T c has been studied in many IBS, there is limited experimental information on the effects of point-like disorder simultaneously on superconducting and magnetic transitions in the regime where superconductivity and antiferromagnetism coexist.The expected physics, however, is very intriguing.Assuming an itinerant nature for long-range magnetic order (LRMO), it has been shown that T c may actually increase upon the introduction of disorder due to the stronger effect on magnetism quantified via the suppression of the magnetic transition temperature, T sm .(Here "sm" is used to indicate simultaneous structural and magnetic transitions in underdoped BaK122) [26].However, this is not a universal trend, as it depends on the relative ratio of the magnetic and superconducting state energies and on the relative strength of the intraband and interband scattering rates.
Irradiation of relatively thin crystals (∼20 µm in our case) with 2.5 MeV relativistic electrons is known to produce vacancy -interstitial Frenkel pairs, which act as efficient point-like scattering centers [35,36].In the arXiv:1808.09532v1[cond-mat.supr-con]28 Aug 2018 high-T c cuprates these defects are known to be strong unitary scatterers causing significant suppression of T c [37].There is a growing number of studies of the effects of electron irradiation not only on T c (see [8] and references therein), but on other properties, such as vortex pinning and creep [38] and London penetration depth [9,[39][40][41][42].In a previous study of electron irradiated Ba 1−x K x Fe 2 As 2 we focused on the evolution of the superconducting gap structure with the potassium concentration and found noticeable changes in the behavior, such as increasing gap anisotropy [40,41,43].
In this paper, we focus on the effects of electron irradiation simultaneously on T c , T sm and normal state resistivity in an underdoped composition of (Ba 1−x K x )Fe 2 As 2 , x = 0.2 [44], in which LRMO coexists with superconductivity.In the normal state, we find strong violation of the Matthiessen's rule below T sm , which is expected in the magnetically ordered state and above T sm in the whole studied temperature range, which is unexpected at least in the simple picture.Moreover, this behavior is in a stark contrast with the electrondoped Ba(Fe 1−x Co x ) 2 As 2 (x = 0.02) with similar T sm in which the Matthiessen's rule is expectedly violated below T sm but obeyed above T sm .At a first sight this could be understood that in this case additional disorder is not so effective, because despite notably lower substitution level, x, required to suppress magnetism, doping into Fe-As planes introduces much higher scattering ratse as evidenced by notably higher residual resistivity values.This argument, however, does not hold, because (1) the magnetic transition temperature, T sm , changes by a similar abount as in BaK122 and (2) the same compliance with the Matthiessen's rule above T sm is observed in isovalently substituted Ba(Fe 1−x Ru x ) 2 As 2 [45] and very clean BaFe 2 (As 1−x P x ) 2 [10].Therefore, the difference is likely in the electronic structure of BaK122 and specifics of its inter-and intra-band interactions and scattering channels [26].

II. EXPERIMENTAL
Single crystals of (Ba 1−x K x )Fe 2 As 2 were synthesized using high temperature FeAs flux method [46].Electrical resistivity was measured on ∼ 1×1 mm 2 sample with four contacts soldered with Sn [47] at the corners in van der Pauw configuration [48].The resistivity of the sample at room temperature, ρ(300K), before irradiation was set to 300 µΩcm [49], the value determined from measurements on big arrays of crystals, with actually measured value being within 10 % uncertainty of geometric factor determination. Single crystals of Ba(Fe 1−x Co x ) 2 As 2 were grown from FeAs/CoAs flux from a starting load of metallic Ba, FeAs, and CoAs, as described in detail elsewhere [50].The composition of the sample was determined using wavelength dispersive spectroscopy (WDS) version of electron probe microanalysis as x=0.02±0.002.Electrical resistivity of the sample was measured in four-probe configuration.Similar to hole-doped sample, resistivity of the sample before irradiation was set as 300 µΩcm [52].The samples were mounted on a thin mica plate in a a hollow Kyocera chip, so that they could be moved between irradiation chamber and resistivity and Hall effect setups in a different 4 He cryostat without disturbing the contacts.
The low-temperature 2.5 MeV electron irradiation was performed at the SIRIUS Pelletron linear accelerator operated by the Laboratoire des Solides Irradiés (LSI) at the Ecole Polytechnique in Palaiseau, France [51].The Kyocera chip was mounted inside the irradiation chamber and was cooled by a flow of liquid hydrogen to T ≈ 22 K to remove excess heat produced by relativistic electrons upon collision with the ions.The flux of electrons amounted to about 2.7 µA of electric current through a 5 mm diameter diaphragm.This current was measured with the Faraday cage placed behind the hole in the sample stage, so that only transmitted electrons were counted.The irradiation rate was about 5 × 10 −6 C/(cm 2 •s) and large doses were accumulated during several irradiation runs.Sample resistance at 22 K was monitored in-situ during irradiation, revealing linear increase with irradiation dose [8], one segment of the broken line in Fig. 1a.Periodically samples were extracted from irradiation chamber and the effect of irradiation was characterized by ex-situ measurements of electrical resistivity as function of temperature, Fig. 1(b), and of the Hall effect (see Fig. 2 below).Warming the sample to room temperature leads to partial defect annealing, as can be seen as the down-steps in the dose dependence of resistivity at 22 K at the start of the next irradiation run.This annealing is incomplete, as evidenced by gradual increase of resistivity for subsequent runs.The resistivity of the sample at room temperature remained stable for a period of at least several months, unless the sample was further warmed above room temperature.

III. RESULTS AND DISCUSSION
The panel (b) in Fig. 1 shows the evolution of the temperature dependent resistivity of under-doped Ba 1−x K x Fe 2 As 2 , x=0.20, with increase of irradiation dose/disorder.
We zoom on the low-temperature range revealing features in ρ(T ) curves at the structural/magnetic, T sm , and superconducting, T c , transitions.The resistivity of the samples at temperatures just above T c follows well a ρ(0) + AT 2 dependence, similar to previous reports [53], (lines in panel (b)), allowing easy extrapolation of ρ(0) and tracking its evolution with disorder.The residual resistivity of the pristine samples was about 40 µΩcm, and residual resistivity ratio ρ(300K)/ρ(0) >7.Residual resistivity increased up to approximately 120 µΩcm at the highest dose of 6 C/cm 2 .On structural/magnetic ordering, resistivity of the sample shows small down-turn on cooling, due to a loss of spin-disorder scattering.The structural transition tem- perature was determined using temperature dependent resistivity derivative and peak position as a criterion, as shown in Fig. 1 (c).The T sm is monotonically suppressed with increase of sample residual resistivity as shown in Fig. 1 panel (d), right scale.The superconducting transition temperature was determined using zero resistivity criterion, it shows monotonic decease with irradiation from above 16 to 9 K, Fig. 1 (d).Interestingly, the decrease of both temperatures in absolute numbers is almost the same and the two are linearly proportional to each other, see Fig. 1 panel (e).
In the top panel (a) of Fig. 2 we show resistivity data for the sample of Ba 1−x K x Fe 2 As 2 , x=0.20, in pristine state and after high dose irradiation (3.08 C/cm 2 ).Nonparallel shift of the curves provides clear evidence for Matthiessen's rule violation.This violation is natural for temperatures below T sm , where due to formation of the band-folding gaps the carrier density changes.Close to room temperatures the Matthiessen's rule is valid, with the violation being closely linked with a crossover feature in the temperature-dependent resistivity at around 200 K.For comparison in inset in panel (a), Fig. 2, we show temperature-dependent resistivity of slightly elec-tron doped Ba(Fe 1−x Co x ) 2 As 2 , x =0.02, in pristine state and after 3.4 C/cm 2 electron irradiation.Irradiation leads to a comparable increase of the room temperature resistivity, ρ(300K), in both electron and hole-doped compositions, 3 µΩcm per C/cm 2 .The increase remains roughly constant above T sm in electron-doped composition, similar to the behavior of hole-doped composition near room temperature, above the crossover feature and to P-substituted samples [10].The step-like increase of resistivity on cooling though T sm is in contrast with slight downturn in the hole-doped composition.In both electron and hole-doped compositions resistivity increase with irradiation proceeds much faster below T sm , consistent with loss of the carrier density.
For this comparison it is important to note that orbitals of iron and arsenic in FeAs layer are contributing the most to the density of states at the Fermi level in BaFe 2 As 2 based materials.Therefore disorder introduced by random positions of substitutional Co atoms in the FeAs layer, affects electron scattering significantly stronger than substitutional disorder of K on Ba cite.This can be directly seen in notably lower residual resistivity in Ba 1−x K x Fe 2 As 2 , x=0.20, ρ(0) ∼40 µΩcm than in Ba(Fe 1−x Co x ) 2 As 2 , x =0.02, ρ(0) ∼170 µΩcmBa, despite five times smaller level of substitution in the latter case.Because of this high level of substitutional disorder in Co-doped material, additional disorder introduced by electron irradiation plays relatively smaller role than in K-doped compound.This different level of background disorder leads to different resistivity behavior on passing T sm in pristine samples.Loss of the carrier density below T sm due to partial gap opening in conditions when carrier mean free path is controlled by disorder and is essentially temperature-independent, gives resistivity increase in disordered Co-doped material.Same loss is compensated by notable increase of mean free path due to the loss of spin-disorder scattering in hole-doped compositions.These considerations were directly illustrated recently in irradiation study on BaFe 2 As 2 with P substitution [10].Note, however, that these considerations do not explain violation of Matthiessen rule above T sm in hole-doped as opposed to its validity in electron-doped compositions.The difference is not related to the level of substitutional disorder in two cases, since both absolute increase of resistivity above T sm and suppression rate of T sm with disorder are very similar on both sides.
In the bottom panel of Fig. 2 we plot temperature dependent Hall coefficient, R H , in the sample of Ba 1−x K x Fe 2 As 2 , x=0.20 before and after irradiation.For reference we show data in other hole-doped samples, x=0.3 and x=0.4, in all cases normalizing data at 20 K, the lowest temperature of our Hall effect measurements.Irradiation does not change either magnitude or temperature dependence of the Hall effect in sample with x=0.20, despite three-fold variation of sample resistivity.On the other hand doping clearly changes magnitude and temperature dependence of the Hall effect.These observations clearly show that defects introduced by irradiation  are not doping the system.Of interest, independence of Hall coefficient on residual resistivity is possible only if all types of carriers have the same mobility.Fig. 3 summarizes our observations.Left panel shows temperature dependent difference in resistivities of Kand Co-doped samples, revealing contrasting behavior with respect to Matthiessen's rule violation and with respect to the rate of T sm suppression on electron and hole doped sides of the doping phase diagram.Both features, which are the main results of our paper, provide inter- esting insights into the nature of magnetism and its interplay with superconductivity.
If magnetism was due to localized spins, one would expect that disorder, as introduced in our experiment, would affect T sm primarily via the effect of random dilution.If magnetism however arises from a Fermi surface instability, the change in the lifetime of the electronic states will affect T sm .Indeed, Ref. [26], studying a simplified two-band model for the interplay between superconductivity and magnetism, showed that both intraband and interband impurity scattering suppress T sm .This is to be contrasted with the case of s ± superconductivity, in which T c is only affected by interband impurity scattering.Because long-range magnetic order competes with superconductivity, depending on how strong this competition is, it is possible that the net effect of disorder is to increase T c in the coexistence region.This seems to be the case in P-doped Ba122 [10], but not in BaK122, were we find T c to also be suppressed.One possible explanation for this difference would be that the competition between superconductivity and magnetism is not as strong in K-doped systems as in P-doped systems, or that the intraband scattering is dominant over the interband scattering in the system studied here.
As for Matthiessen's rule, a known scenario in which it is violated is when impurities are added in a system whose main scattering mechanism is strongly anisotropic in momentum space [54].In BaK122, a natural candidate is the scattering by spin fluctuations, which in this system are strongly peaked at the finite wave-vectors (π, 0) and (0, π).In this case, the violation of Matthiessen's rule would imply that the resistivity of the normal state is dominated by magnetic fluctuations.Such a preponderance of magnetic fluctuations could in principle favor a higher T c state, if indeed pairing is mediated by spin fluctuations.
It should be noted, however, that the position of hot spots as well as the strength of inelastic scattering are very similar on electron and hole doped sides, while the effect of disorder (as seen in direct comparison Fig. 3) is dramatically different.Alternative explanation for strong Matthiessen rule violation was suggested in a recent study of the effect of natural growth disorder on properties of BaFe 2 As 2 with Ru substitution [45].Here it was assigned to predominant suppression of high mobility carriers with disorder.

IV. CONCLUSIONS
In conclusion, irradiation with relativistic 2.5 MeV electrons leads to rapid suppression of both superconducting T c and the temperature of concomitant orthorhombic/antiferromagnetic transition T sm .In the absence of nodes in the superconducting order parameter, observation of rapid suppression of T c provides a strong support for a sign-changing s ± pairing.Rapid suppression of T sm , surprisingly at the same rate as T c , indicates the itinerant nature of the long-range magnetic order.Comparison of the hole-doped BaK122 with electron-doped Ba(Fe x Co 1−x ) 2 As 2 (FeCo122) with similar T sm ∼110 K, x =0.02, reveals significant differences in the normal states, with no Mathiessen rule violation above T sm on the electron-doped side and strong violation on the hole-doped side.Our results provide strong evidence of the itinerant nature of the AFM phase and non-trivial influence of disorder on competing superconductivity and magnetism in iron based superconductors.

FIG. 1 .
FIG. 1. (Color online) (a) Dose dependence of electrical resistivity at 22 K measured in-situ during electron irradiation.The resistivity increases linearly during single irradiation run, steps in the curve result from partial defect annealing on warming sample to room temperature between runs for characterization.(b) Temperature-dependent resistivity, ρ(T ), measured after subsequent irradiation runs and room-temperature annealing.(c) Temperature-dependent resistivity derivative, dρ/dT , revealing a sharp peak at Tsm .Peak position was used as a criterion for Tsm determination.(d) Variation of the superconducting transition temperature, Tc (left axis) and magnetic/structural transition, Tsm (right axis) offset vertically to match each other (see text).(e) Variation of Tc plotted as a function of the variation of Tsm.

3 FIG. 2 .
FIG. 2. (Color online) (a) Comparison of the effect of irradiation on the normal-state resistivity of hole (BaK122, main panel) and electron (BaCo122, inset) -doped Ba122.The character of the magnetic transition indicates much more background scattering in the electron-doped compound.(b) Normalized Hall coefficient as function of temperature before (open circles) and after (filled circles) electron irradiation for x =0.2 showing no change.This is compared to the data for x =0.3 (triangles) and 0.4 (squares) showing clear doping dependence.
FIG. 3. (Color online) Left panel -substantial temperature dependence of the resistivity difference before and after irradiation signifying gross violation of the Matthiessen's rule in BaK122 at all temperatures, including temperatures much greater than Tsmcompared to practically no violation in BaCo122 in this regime.Right panel -summary phase diagram of electron and hole-doped BaK122 and positions of corresponding concentrations shown in the left panel.