Stoner factors of doped 122 Fe-based superconductors: First principles results

TL;DR

This study uses first principles calculations to analyze how doping affects the Stoner factor and magnetic fluctuations in 122 Fe-based superconductors, revealing correlations with structural parameters and Fermi surface nesting.

Contribution

It provides a comprehensive first-principles analysis of doping effects on the Stoner factor in 122 Fe-based superconductors, linking electronic structure, doping type, and magnetic properties.

Findings

Co/Ru and P doping reduce the Stoner factor with increased doping.

Na/K doping enhances the Stoner factor at higher doping levels.

The Stoner factor correlates with pnictide height and Fermi surface nesting.

Abstract

A comprehensive study on the evolution of Stoner factor with doping concentration for various doped 122 systems (like BaFe$_2$As$_2$, SrFe$_2$As$_2$) of Fe-based superconductors is presented. Our first principles electronic structure calculations reveal that for Co/Ru (electron or iso-electronic) doping at Fe sites or P doping at As sites result in a reduction of Stoner factor with increasing doping concentration. On the contrary, in case of Na/K (hole) doping at the Ba sites, Stoner factor is enhanced for higher doping concentrations. This may be considered as an indicator of elevation of "magnetic fluctuation" in these systems. We find that the Stoner factor uniquely follows the variation of the pnictide height z$_{As}$/Fe-As bond length with various kinds of doping. Our calculated Fermi surfaces explicate the diversities in the behaviour of Stoner factors for various doped 122 systems ; larger degree of Fermi surface nesting, larger the value of Stoner factor and vice versa.

Full text

I Introduction

Invention of superconductivity in Fe-based compounds not only introduced a drastic change in the belief that Fe is inimical to superconductivity due to the presence of strong local magnetic moment associated with Fe atom but also recreate a glimpse of hope for developing a complete generalized theory of high temperature superconductivity. One of the most interesting aspects of these Fe-based superconductors (SCs) from a fundamental point of view, is that superconductivity may arise from magnetic fluctuation or orbital fluctuation 1; 2; 3; 4; 5. However, the glue to the electron-electron attraction or more precisely pairing mechanism in these Fe-based SCs is far from being settled 5; 6; 7; 8. Phase diagrams of Fe-based SCs establish the manifestation of various exotic phases like spin density wave (SDW), orbital order, nematic order, structural transition etc. 9; 10; 11; 12; 13; 14; 15; 16. Various physical properties including superconductivity of these Fe-based SCs are very much sensitive to temperature, pressure as well as doping concentration as evident from phase diagrams 17; 18; 19; 20. Influence of structural parameters on the superconducting as well as on other exotic phases is well established through extensive theoretical and experimental investigations 21; 22; 20. Magnetic (antiferromagnetic spin) fluctuation in high Tc cuprates provide pairing in d${}_{x^{2}-y^{2}}$ channel whereas in Fe-based superconductors it is believed to provide s*±* pairing symmetry 6. The quantity that may be used as an indicator of magnetic fluctuation in metallic systems, is Stoner factor. Although Stoner theory is formulated for ferromagnetic instability, it can also be applied for anti-ferromagnetic systems 23.

Broadly, there exists about six invented families of Fe-based SCs 1. Among all these families 122 family is the most studied one both theoretically and experimentally largely because of availability of high quality single crystals in this family. The generalized chemical formula of 122 family is MFe2X2, where M$=$Ba, K, Ca, Sr and X$=$As, P. The phase diagrams of 122 family exhibit a large number of diversities in the physical properties 1. For example, superconductivity emerges in the electron doped systems (e.g., replacing Co by Fe) with a very small doping concentration (\sim$$2%) 13; 14; 15 whereas the same arises in hole doped systems with higher doping concentration (\sim$$25%) 9. On the other hand, for the case of iso-electronic Ru doping at the Fe site, superconductivity appears at about 50% Ru doping concentration 24; 25 which is quite high compared to other doped 122 systems. Another qualitatively distinct diversity in the phase diagram of 122 system is observed in case of Na doped Ba122 system 26. Avci et al., displayed the appearance of an additional C4 (tetragonal) phase well inside the orthorhombic phase in Ba1-xNaxFe2As2 system at the boundary between superconductivity and SDW (stripe antiferromagnetism) which is different from the previous observations of re-entrant tetragonal phase in electron-doped compounds 15 and has many important consequences. The role of various structural parameters specially zAs (fractional z co-ordinate of As atom) or anion height (height of the As atom from the Fe plane) in superconductivity, magnetism and structural transition is indispensable 27; 28; 29; 30; 31; 22; 20. However the inter-relationship among superconductivity, magnetic order, nematic order, orbital order and structural transition is under thorough investigations both theoretically as well as experimentally 32; 3; 5; 33; 34. The presence of magnetic fluctuation in these Fe-based systems is probed by magnetic susceptibility ($\chi$). A large number of theoretical and experimental works are available in the literature addressing the role of magnetic fluctuation in superconductivity and other observed physical properties of Fe-based SCs 35; 32. An important finding of the present work is that the Stoner factor of various 122 material (that may model magnetic fluctuation) varies with doping in the same fashion as that of the pnictide height or zAs.

Density functional theory (DFT), in its most simple implementation [the local-(spin)-density-approximation (L(S)DA)] is in principle an exact theory to estimate the ground state properties of any system. It is now well established that LSDA and generalized gradient approximation (GGA) approach, which does not take into account the effect of near-critical fluctuations on the long-range magnetism, fails to predict accurate electronic structures for itinerant magnetic system near quantum critical point. A significant amount of effort has been made to introduce these fluctuations within first principles DFT method 36; 37. Stoner factor is a parameter which mimic the presence of magnetic instability in a system and can be calculated within LSDA approach as shown in ref 23. However no thorough comprehensive investigations, either theoretically or experimentally exist as to how Stoner factor is being modified due to various kinds of dopings and temperature. In this paper we present a comprehensive picture of Stoner factor for various 122 Fe-based systems as a function of doping concentration from first principles simulations.

We calculate Stoner factor from first principles study of density of states (DOS) for these 122 compounds as a function of doping concentrations. We have found that Stoner factors get modified qualitatively differently for different kinds of doping. For example, in case of electron doping (Co doping at the Fe site) and iso-valent doping (Ru doping at Fe site as well as P doping at As site), Stoner factor gradually decreases with increasing doping concentration. On the other hand, exactly opposite trend in the behaviour of Stoner factor is observed in case of hole doped systems (K/Na doping at the Ba site). This diversity in the behaviour of Stoner factor is then analysed in the light of calculated nature of the FSs of these systems for various doped cases. The method of calculations, including the evaluation of Stoner factor for various doped systems require magnitude of DOS at the Fermi level which are obtained through detailed first principles simulation and are discussed in the forthcoming theoretical method section. An important speciality of our simulation results is that we use exact experimentally determined structural parameters of various doped systems as input which are being kept intact. As a consequence of that, presented results are very much realistic. In the succeeding section we present the variation of Stoner factor with doping concentration for many doped materials of 122 family of Fe-based SCs. Modifications of the FSs are also depicted to elucidate the observed anomalies in the behaviour of Stoner factor with doping. In the last section, we finally conclude and summarize our results.

II Theoretical Methods

Our first principles electronic structure calculations are performed by implementing ultrasoft pseudopotential with plane wave basis set based on density functional theory 38. Electronic exchange correlation is treated under the generalised gradient approximation (GGA) using Perdew-Burke-Ernzerhof (PBE) functional 39. It should be mentioned here that density funtional theory within local density approximation (LDA) as well as generalized gradient approximation (GGA) was unable to optimize the experimental value of $z_{As}$ (fractional co-ordinate of As) with desired accuracy 22; 20; 40; 41; 42. In fact, the optimized value of $z_{As}$ is about 0.1 to 0.15 Å smaller than that of the experimental one. The origin of this discrepancy is the presence of strong magnetic fluctuation, associated with Fe atoms in these materials 43. Electronic structures (band structure, Fermi surface etc.), calculated using optimized lattice parameters ($a$, $b$, $c$ and $z_{As}$) do not resemble with that of the experimentally measured one 53; 21. This insist us to employ experimental lattice parameters i.e., $a$, $b$, $c$ and $z_{As}$ 9; 44; 22; 19; 13; 45; 46 instead of geometry optimized (total energy minimization) lattice parameters as inputs of our first principles electronic structure calculations. We use experimental orthorhombic (low temperature) as well as tetragonal (high temperature) lattice parameters $a$, $b$, $c$ and $z_{As}$ as input of our ab-initio electronic structure calculations 9; 44; 22; 19; 13; 45; 46. Various modern X-ray diffraction techniques e.g., using Synchrotrons radiation sources etc. that determines crystallographic information at different external perturbations (like temperature, chemical doping etc.) are essentially result of diffraction from various atomic charge densities (Bragg’s diffraction). Considering experimentally determined structural parameters at different temperatures, doping as input thus in turn provides temperature/doping dependent correct densities to our first principles calculation. These input structural parameters are kept fixed through out the calculation for a fixed doping. This is how we use the DFT formalism to bring out realistic doping dependent observable with the help of experimental input (energy being functional of electron density E $\equiv E[\rho(r,T,x)]\equiv E[\rho(a(x,T),b(x,T),c(x,T))$]. The main effect on the electronic structure from finite temperature/doping is the underlying crystal structure, and the average crystal structure can usually be reliably determined from the diffraction experiment at a given doping.

In order to dope the system, we use two theoretical methods: (1) Virtual crystal approximation (VCA) 47 (2) Super-cell. These two methods are used to prepare various doped Ba122 and Sr122 systems as per requirement. For example, we use super-cell method for calculating the DOS as well as the FSs of Ru doped 122 systems because the VCA method is unable to produce accurate electronic structures in the higher doping regime for these systems. On the contrary, we employed VCA method to dope K, Na and P in Ba122 system. The detailed discussion about the application of VCA and super-cell methods for various doped 122 systems can be found in ref 48. In this work, we use the VCA method, developed by Bellaiche and Vanderbilt 47 based on weighted averaging of pseudopotentials. It should be noted that although DOS of Co doped 122 systems are calculated using super-cell method, FSs are calculated within VCA method to handle the small percentage of doping in the primitive unit cell. Moreover, size of the super-cells are different for different doping cases. For example, in Ru doped Ba122 and Sr122 systems, we use $\sqrt{2}\times\sqrt{2}\times 1$ super-cells, whereas $2\times 2\times 1$ super-cell is used for Co doped 122 materials. Non-spin-polarized and spin polarised single point energy calculations were carried out for tetragonal phase with space group symmetry I4/mmm (No. 139) and orthorhombic phase with space group symmetry Fmmm (No. 69) respectively using ultrasoft pseudo-potentials and plane-wave basis set with energy cut off 500 eV and higher as well as self-consistent field (SCF) tolerance as $10^{-6}$ eV/atom. Brillouin zone is sampled in the k space within Monkhorst–Pack scheme and grid size for SCF calculation is chosen as per requirement of the calculation for various systems and approaches. For simulating Fermi surfaces, grid size of SCF calculation is chosen as $26\times 26\times 31$. LDA+U calculations have also been performed for orthorhombic Na and P doped Ba122 systems with SDW spin configuration 49, where doping has been implemented within super-cell method. Stoner factor of a compound like doped 122, can be defined as $I^{Fe}\times[N^{Fe}(E_{F})]^{2}+I^{Ru/Co}\times[N^{Ru/Co}(E_{F})]^{2}$, where $N^{Fe}(E_{F})$ and $N^{Ru/Co}(E_{F})$ are the density of states at the Fermi level from Fe and Ru/Co atoms respectively 50; 23. The values of Stoner parameters $I^{Fe}$ and $I^{Ru/Co}$ are taken from ref 50; 51; 52.

III Results and discussions

We have studied many doped samples (hole doping, electron doping and iso-valent doping) of 122 Fe-based SCs to study the evolution of Stoner factor with doping concentration. As a representative of hole doped and electron doped systems, we consider Na/K doping (at the Ba site) and Co doping (at the Fe site) of BaFe2As2 (Ba122) systems respectively. Isovalent or iso-electronic doping at Fe site (Ru doping) for BaFe2As2 and SrFe2As2 (Sr122) systems as well as in As site (P doping) for BaFe2As2 system are also considered in our investigation. In the theoretical method section we have already described the method of calculation of Stoner factor of a compound system within density functional theory. Virtual crystal approximation (VCA) and super-cell methods are used to mimic various doped 122 Fe-based superconducting systems. We employed experimental lattice parameters $a$, $b$, $c$ and $z_{As}$ in both the orthorhombic and tetragonal phases for our first principles electronic structure (DOS and FSs) calculations. It is well known that in 122 Fe-based SCs, DOS at the Fermi level is dominated by Fe-d orbital and to some extent by As-p orbitals. It is for this reason, the particular case of doping at the Fe site is treated within super-cell method in order to calculate the atom projected DOS for Fe and Co/Ru atoms separately with desirable precision. It should be mentioned here that within VCA method, the atom projected DOS of Fe and Ru/Co atoms can not be estimated separately. And for other doping cases (K/Na doping at the Ba site and P doping at As site) we employ VCA for implementation of doping (in which Fe atom projected DOS can be obtained). We have also employed super-cell method for implementing K/Na/P doping in 122 system, but the results of which are not presented here, are very similar to that of the VCA results.

First, we calculate the atom projected DOS for Co doped Ba122 systems (electron doping) for several Co doping concentrations in the tetragonal phase. In FIG.1, DOS contribution of Fe and Co atoms are separately presented for various doping concentrations, indicated in the figures. As discussed earlier, for electron doped system (like Co doping at Fe site) superconducting as well as other exotic phases appear in the system within 10% to 15% doping concentration which is smaller compared to that for other doped 122 systems (for example, superconductivity arises in Ru doped Ba122 system at a very high doping concentration). That is why we restrict our calculation in the low doping regime in the case of Co doped Ba122 system. We depict atom projected DOS in the tetragonal phase for Ru doped Sr122 and Ba122 systems in FIG.2 and FIG.3 respectively. In those figures, DOS contribution of Ru and Fe atoms are presented separately as a function of Ru doping concentration. In this case our study is no longer restricted to low doping regime but extended our investigation for whole regime of doping concentrations (low to high). On the other hand, we exhibit atom projected DOS in the orthorhombic phase for Fe atom in case of K, Na and P doped Ba122 systems in FIG.4, FIG.5 and FIG.6 respectively. Moreover, atom projected DOS in the tetragonal phase for K, Na and P doped Ba122 systems are also calculated but not presented here. In these cases we employ VCA method for handling doped systems as Fe site is not directly affected by K and Na doping at the Ba site (hole doping) and P doping at the As site (iso-electronic doping). However, we also calculate Fe atom projected DOS for these systems using super-cell approach. We find DOS calculated within super-cell method is qualitatively similar to that using VCA method.

Using this atom projected DOS, we calculate Stoner factor using the method, described in the theoretical method section for various doped 122 systems. In FIG.7a, we display the Stoner factor as well as atom projected DOS of Fe and Co atoms at Fermi level as a function of Co doping concentration in the tetragonal phase. In FIG.7b we provide estimated zAs (also use as an input of our first principles calculation) as a function of Co doping concentration. We observe that with increasing Co doping concentration, both the Stoner factor as well as zAs decreases. We have also calculated the FSs of Co doped Ba122 system within VCA method. Our calculated FSs also provide a clear indication of more 3D like FSs with higher Co doping concentration. This 3D like FSs work against nesting and results in suppression of magnetic order (SDW). Since Stoner factor is intimately related to magnetic fluctuation or magnetic instability, reduction of Stoner factor with increasing Co doping concentration gives a clear indication of depletion of possible “magnetic fluctuation” in the system with the introduction of Co. This is consistent with the evolution of calculated FSs as a function of doping. Now we move to the case of iso-electronic doping at Fe site (Ru doping at Fe site of Ba122 and Sr122 systems). For both the systems with increasing doping concentration zAs decreases as shown in FIG.8b and FIG.8d. In FIG.8a and FIG.8c atom projected DOS at the Fermi level for Ru and Fe atoms as well as Stoner factor are shown as a function of Ru doping concentration for tetragonal Sr122 and Ba122 systems respectively. For both the cases Stoner factor decreases with increasing Ru doping concentration in a very similar manner. We have also performed the FS calculation for Ru doped Sr122 system within supercell method for 50% Ru doping case. We find that for 122 system 50% Ru doping modifies the FSs from 2D like to more 3D like, which is consistent with previous theoretical and experimental investigations 53; 54. This in-turn degrades the nesting of FSs and triggers superconductivity in the systems by means of suppression of SDW order or magnetic order. This again creates an impression of curtailment of magnetic order with increasing Ru doping concentration, which is also consistent with the calculated variation of Stoner factor with Ru doping. As far as our FS calculation is concerned, upto 50% Ru doping in Sr122 system, no significant moderation is observed in the FS topology in contrast to Ba122 system. However SrRu2As2 has completely 3D like FSs just like BaRu2As2 system, presented in FIG.13c. In FIG.13d we depict the calculated FSs of 50% Ru doped Sr122 system within VCA method. The topology of FSs calculated within VCA method, certainly do not match with the experimental one as reported in literature 53.

Next we calculate the evolution of Stoner factor for K, Na and P doped Ba122 system with increasing doping concentration. In FIG.9a, c the variation of Stoner factor as a function of doping concentration has been shown for K and P doped tetragonal Ba122 systems respectively. In FIG.9b, 9f the variation of zAs with doping concentration is also presented for the above mentioned systems respectively. Calculated evaluation of Stoner factors in the orthorhombic Ba122 systems as a function of K, Na and P doping concentrations also show similar trends as that of the tetragonal K, Na and P doped Ba122 systems, shown in FIG.10. In FIG.10b and 10f, the variation of zAs with doping concentration is also presented for the same systems respectively. In case of K and Na doped systems (hole doped) Stoner factor increases with the increase of doping concentration which also corroborates well with the nature of the calculated FSs, shown in FIG.14 and 15. Although exactly opposite trends in the behaviour of zAs with doping concentration (experimental) is observed in case of K and Na doping (see FIG10b and d) which makes the Na doped system different from other hole doped 122 systems. For P doped system both zAs and Stoner factor decreases with increasing doping concentration, which indicate reduction of magnetic instability with P doping. So in general, the Stoner factors of various 122 systems follow the variation of zAs with doping except in the case of K doped Ba122 system where the variation of zAs with doping is not well defined. Experimental data of zAs for K doped Ba122 is not as consistent (probably accurate) compared to that for the other doped 122 family of compounds (see FIG. 10b, d). Since our theoretical calculation is based on the experimental crystallographic data as input, the calculated electronic structure differs from general observation (actually the experimental inconsistency in zAs with doping is reflected in the calculated results). In case of Ba1-xKxFe2As2, thus, comparison of Stoner factor with zAs would not be a correct quantity to look into but the Fe-As bond length as a function of doping concentration. Actually, Stoner factor universally follows the variation of Fe-As bond length for various types of doping. In case of hole doping, Stoner factor inversely follows Fe-As bond length with the variation of doping concentration. On the other hand, in the case of electron doping, Stoner factor follows the doping dependent behaviour of Fe-As bond length. In order to understand universality in the behaviour of Stoner factor and zAs/Fe-As bond length of hole doped Ba122, we shed more light into the other structural parameters. To elucidate the inconsistency in the behaviour of Stoner factor and zAs in K doped Ba122, we introduce in plane and out of plane As-As distances which also play very important and crucial role. Size of the Ba atom (2.22Å) [atomic radius in metallic bonding] is larger than Na atom (1.86Å) but smaller than K atom (2.27Å). Because of this reason, with the substitution of K atom in place of Ba atom, out of plane As-As distance increases and exactly opposite behaviour in the out of plane As-As distance is observed in the case of Na substitution at Ba site. It should also be noted that the c-axis increases with the increasing doping concentration for both Na and K doping (in case of Na doped Ba122 materials c-axis increases with doping up to certain doping concentration and after that it decreases). This qualitatively explains the observed behaviour of zAs with doping for K and Na doped Ba122 systems 55. But in both the cases Fe-As bond length decreases with increasing doping concentration. Calculated Stoner factors for K/Na/P doped 122 systems using super-cell approach also follow the same trends as that of the calculated one within VCA method. Calculated FSs also become more and more 3D like with increasing P doping concentration, which is consistent with the suppression of magnetic order as observed in experiments (see FIG.16). On the contrary to all other cases of doping, hole doping makes the FSs more 2D like which actually helps having larger degree of nesting of FSs. This increasing trend of Stoner factor with doping, may be related to the more 2D like FSs of the hole doped 122 systems that enhances nesting segments. In all these 122 systems, superconductivity emerges into the system with certain percentage of doping concentration (small or large). On the contrary to other 122 Fe-based SCs, Stoner factor of hole doped 122 systems (K/Na doped) display distinctly different behaviour. In these cases only, the Stoner factor increases with increasing doping concentration which indicate that, these systems become more and more unstable against magnetic instability. So in these hole doped 122 systems, superconductivity arises at the same moment when magnetic instability also dominates. We have also studied the behaviour of Stoner factor in presence of electron correlation which can be introduced through LDA+U calculation within DFT. Since Fe based SCs specially 122 systems are weakly correlated, we restrict our LDA+U calculation for a small value of U, which defines the strength of correlation (U=0.5 and U=1). In FIG.11 the variation of Stoner factor with doping concentration for various U values (U=0.5 and U=1) are depicted for BaFe2(As1-xPx)2 and Ba1-xNaxFe2As2 systems in the orthorhombic phase. It is to be noted here that with the introduction of U, Stoner factor decreases for both the cases but shows very similar trends with doping concentration as that in the case of U=0. This study (of LDA+U) has also been extended to all other materials presented in this work. The reduction of Stoner factor with correlation is consistent with the current literature 23. It should also be noted here that K-doped Ba122 systems have the highest superconducting transition temperature (Tc) among all other Fe-based SCs in ambient condition. This may give some impression that superconductivity in these systems is intimately related to magnetic order and magnetic fluctuations. On the whole, our study provides a clear view about the variation of Stoner factor as a function of doping for all variety of doped 122 Fe-based SCs. We also explain the observed diversities in the behaviour of various doped 122 systems by presenting calculated FSs.

The phase diagram of the iron pnictides reveals that the superconducting transition temperature Tc usually shows a dome-like relation with doping. Should the Stoner factor vary with doping in the fashion as Tc in phase diagrams ? One should be aware of the fact that both temperature and doping plays a very important role in any phase diagram. In fact in phase diagram, both temperature and doping dependencies of various phases like spin density wave, superconducting phase etc., are known for various Fe based superconductors. However, in our first principles electronic structure calculations, we use experimentally measured doping dependent lattice parameters ($a$, $b$, $c$, zAs) at a fixed temperature as input. That is why our calculated Stoner factor does not contain any temperature dependencies (only doping dependency). In general, zAs (as well as other structural parameters) has a very sensitive temperature dependencies. For example, in our earlier work we have shown in Ru doped Ba122 system, structural parameters including zAs are highly sensitive to temperature 22. When such a temperature dependency was included in the electronic structure calculation by us in the past, the temperature dependency of Stoner factor did reproduce the experimental magnetic fluctuation behaviour 32; 56. In the current perspective where we discuss detailed behaviour of Stoner factor over a wide doping ranges for a number of doped-122 family, clean crystallographic data, particularly zAs as a function of both, doping and temperature is not available. As a result such a calculation of Stoner factor as a function doping as well as temperature is only a future possibility. That is why the Stoner factor does not show dome like behaviour.

IV Conclusion

It is well documented that the magnetic fluctuation plays a significant role in superconductivity of Fe-based SCs. Stoner factor is the precursor of magnetic fluctuation or magnetic instability in a system. We have presented in detail the modification of Stoner factor with doping concentration for various doped 122 Fe-based SCs. VCA as well as super-cell methods are employed to dope these materials theoretically to calculate DOS and FSs of doped 122 compounds. Diversities in the behaviour of Stoner factor with varying doping concentrations for various nature of doped 122 systems are discussed. In case of Ru (iso-electronic)/Co (electron) doped systems, Stoner factor decreases with increasing Ru/Co concentration which is compatible with the calculated FSs. Same trends in the behaviour of Stoner factor and FSs are also observed in the case of other iso-electronic doping, P doping at the As site. On the other hand, in case of hole doped systems (Na/K doping at the Ba site) Stoner factor increases with increasing doping concentration which is also consistent with the corresponding modifications in the calculated FSs for various doping concentrations. Remarkably, we find that the Stoner factor follows the variation as that of the zAs with doping. Larger the pnictide height larger is the Stoner factor as well as larger degree of the Fermi surface nesting, larger the value of Stoner factor and vice versa. Sensitive dependence of superconducting Tc with anion height was established earlier 27 but our present work clearly establishes the same for Stoner factor or magnetic fluctuation. As a whole this work provide a comprehensive study of Stoner factor (magnetic instability) for various doped 122 Fe-based SCs.

Acknowledgements

We thank Dr. A. Bharathi and Dr. A. K. Sinha for discussion on experimental aspects. We thank Dr. P. A. Naik and Dr. P. D. Gupta for their encouragement in this work. One of us (SS) acknowledges the HBNI, RRCAT for financial support and encouragements.

Additional information

Competing financial interests: The authors declare no competing financial interests.