Decoding 122-Type Iron-Based Superconductors: A Comprehensive Simulation of Phase Diagrams and Transition Temperatures

Iron-based superconductors, a cornerstone of low-temperature physics, have been the subject of numerous theoretical models aimed at deciphering their complex behavior. In this study, we present a comprehensive approach that amalgamates several existing models and incorporates experimental data to simulate the superconducting phase diagrams of the principal 122-type iron-based compounds. Our model considers a multitude of factors including the momentum dependence of the superconducting gap, spin-orbital coupling, antiferromagnetism, spin density wave, induced XY potential on the tetrahedral structure, and electron-phonon coupling. We have refined the electron-phonon scattering matrix using experimental angle-resolved photoemission spectroscopy (ARPES) data, ensuring that all electrons pertinent to iron-based superconductivity are accounted for. This innovative approach allows us to calculate theoretical critical temperature Tc values for Ba1-xKxFe2As2, CaFe2As2 and SrFe2As2 as functions of pressure. These calculated values exhibit remarkable agreement with experimental findings. Furthermore, our model predicts that MgFe2As2 remains non-superconducting irrespective of the applied pressure. Given that 122-type superconductivity at low pressure or low doping concentration has been experimentally validated, our combined model serves as a powerful predictive tool for generating superconducting phase diagrams at high pressure. This study underscores that the high transition temperatures and the precise doping and pressure dependence of iron-based superconductors are intrinsically linked to an intertwined mechanism involving a strong interplay between structural, magnetic and electronic degrees of freedom.


Introduction
The '122' family of iron-based superconductors, represented by AFe2As2 (where A is Ba, Sr, or Ca), has been the subject of extensive study.BaFe2As2 (Ba122), under ambient pressure, exhibits a stripe-type antiferromagnetic spin density wave (SDW) order without superconductivity.However, the introduction of external hydrostatic pressure, internal chemical pressure (e.g., isovalent doping 1,2 ), or ionic substitution in systems such as Ba1-xKxFe2As2, Ba1-xNaxFe2As2, or Ba(Fe1-xCox)2As2 can induce superconductivity.Substitutions such as replacing Ba 2+ with K + or Na + introduce holes 3 , while substituting Fe 2+ with Co 3+ introduces electrons 4 .The application of pressure or doping gradually suppresses the SDW transition and gives rise to a superconducting phase.The SDW transition is accompanied by a structural change from a tetragonal hightemperature to an orthorhombic low-temperature structure 5 , known as a nematic electronic transition.
Hole-doped Ba122 presents a rich phase diagram, including a re-entrant tetragonal C4 phase region that restores the fourfold symmetry in the basal plane.This phase is characterized by electron spin rearrangement and absence of electronic nematic ordering 6,7 and somewhat suppresses the superconducting transition 7,8 .SrFe2As2 and CaFe2As2 also exhibit iron-based superconductivity under pressure 9,10 .Interestingly, MgFe2As2 does not exhibit superconductivity despite magnesium's position in the 2 nd column of the periodic table 5 .This anomaly underscores the complex interplay between structural, magnetic and electronic degrees of freedom in these materials.The coexistence of superconductivity with a momentum-dependent superconducting gap with other electronically ordered phases such as antiferromagnetic SDW and nematicity 11 , and the presence of strong spin-orbital coupling further highlights the complex unconventional nature of these materials.
While it is widely accepted that antiferromagnetism enhances electron-phonon coupling on the Fermi surface in unconventional superconductors, recent studies suggest that the significance of electron-phonon coupling in unconventional superconductivity may have been underestimated.Li et al. 12 demonstrated that phonon softening in AFeAs compounds amplifies electron-phonon coupling by a factor of approximately 1.6.Deng et al. 13 interpreted out-of-plane lattice vibrations as a phonon softening phenomenon, which they incorporated into their calculations to enhance electron-phonon scattering.Coh et al. further refined these models and proposed that the electronphonon scattering matrix in iron-based superconductors was underestimated by a factor of approximately 4 10,11 .They attributed this underestimation to the presence of nearest-neighbor interactions under an antiferromagnetic SDW, which resulted in a first amplification factor of 2, and the vertical displacement of lattice Fe caused charge transfer to induce xy potential in tetrahedral regions under an antiferromagnetic SDW situation, leading to another factor of approximately 2 increase in the electron-phonon scattering matrix 14 .S.-F.Wu et al. 15 also observed a significant increase in intensity for the emergent As phonon mode in the XY plane geometry.Moreover, a discernible shift of the spectral weight between the normal and the superconducting state is evident in the photoemission spectra below the superconducting energy gap of various ironbased compounds in an energy range of approximately 30-60meV below the Fermi energy [16][17][18][19] .This shift, observed in the ARPES range, suggests that the involvement of superconducting electrons in iron-based superconductivity may have been underestimated.This underestimation could potentially account for the discrepancy between theoretical and experimental Tc values based on the electron-phonon coupling method.The ARPES data [16][17][18][19] can be utilized to revise the electron-phonon scattering matrix.Given the high transition temperatures of Fe-based superconductors, it is crucial to consider the full electronic density of states (DOS) in a range of to F E and not only the Fermi level value, where Debye E represents the upper limit of the phonon energies that can be transferred to electrons.In contrast to classical low-Tc superconductors, at the high transition temperatures of Fe-based superconductors, contributions of high-energy phonons in the electron-phonon scattering mechanism become significant.This approach, which is a direct consequence of energy conservation, is corroborated by ARPES experiments [16][17][18][19] where the energy range of the spectral weight shift is approximately in the order of the Debye energy.
In our quest to decode the intricate electronic phenomena in 122-type iron-based superconductors, we initially turn our attention to Ba1-xKxFe2As2.This compound, boasting the highest Tc among the 122-type superconductors, presents a particularly rich phase diagram [20][21][22][23] and becomes superconducting above 0.8GPa at x = 0 18 .Our investigation will delve into and compare the influence of both first-order and higher-order antiferromagnetic fluctuations on the Tc calculations.This comparative analysis will encompass the compounds Ba1-xKxFe2As2, CaFe2As2, and SrFe2As2.Furthermore, we will explore whether the application of pressure could potentially induce superconductivity in MgFe2As2.This comprehensive approach aims to shed light on the complex interplay of factors governing superconductivity in these materials.

Computational algorithm
Our preliminary Tc calculations include at least 6 components.
(1) Exchange factor: The pressure dependence of the antiferromagnetic interaction can be used to monitor the variation of the exchange interaction.We define (2) Coh factor: It is well accepted that the antiferromagnetic (AFM) state usually increases the electron-phonon scattering matrix by a ratio of AF R when the ab-initio calculations change from spin-restricted to spin-unrestricted mode.To encounter the SDW, Coh et al 14  .Using the Coh factor brings the simulation results more in line with the experimental observations 14 .However, the appearance of the induced xy potential requires calibrating the GGA+A functional, which is a time-consuming experimental effort and a computationally expensive mission 14  , is used to mimic the effect of an anisotropic momentum space when there is 4-fold symmetry (as illustrated by the two overlapping red ellipses in Figure 1).The major is the area of an isotropic s-wave momentum space.
angular f equals to 1 if it is an isotropic s-wave superconductor.
(5) Spin-orbital coupling SOC factor: The spin-orbital coupling of iron-based superconductors 26 is typically ~10meV, where the SOC energy can be comparable to the ARPES energy range of spectral weight shift [16][17][18][19] .The effect of SOC should be included in the calculation of the pairingstrength.
(6) Electron-phonon factor: The electron-phonon coupling is F  is the phonon density of states as a function of frequency  .Taking into account the above factors, the 2 () PS F  becomes 27 where ( , ) is the velocity in the ARPES range of spectral weight shift and () In the case of a strong coupling, the pairing strength and the Coulomb pseudopotential are renormalized to * PS  and *  , respectively 27 .
The electronic and dielectric properties of the samples are computed by CASTEP at the GGA-PBE level [28][29] .The maximum SCF cycle is 100 with the tolerance of 2 × 10 −6 eV/atom.The reciprocal k-space interval is 0.025(1/Å).The norm-conserving potential is used for SOC calculations instead of the ultrasoft pseudopotential.The finite displacement method is used to calculate the phonon data at the LDA level, where the supercell defined by cutoff radius is 0.5nm and the interval of the dispersion is 0.04(1/Å).The exchange factor based on the mean field approach is calculated by the spin-unrestricted GGA-PW91 functional and the 1 st and 2 nd AFM fluctuations are investigated separately.To avoid that all simulation parameters are dependent on our computation, we use the lattice parameters and Debye temperature from the literature if available.Otherwise, the Debye temperature is taken from the CASTEP platform.While the strongly correlated electron-electron interaction is not easily calculated from the electronic DOS, Debye temperature and Fermi energy 32 , the consideration of pseudopotentials in the range of 0.1 to 0.2 should make sense.All Coulomb pseudopotentials  are set to 0.15 for fair comparison.The anisotropic factor is adjusted to demonstrate how it affects the theoretical Tc.The increase in electron-phonon coupling resulting from the exchange enhancement can be represented as a separable variable 30 .The separable variable representing the increase in electron-phonon coupling due to exchange enhancement, can be obtained by multiplying the electron-phonon coupling by the exchange enhancement factor 30 .
It should be noted that this is not restricted to the first-order exchange interaction 30 .The parameters influencing AMF under pressure can be described by , as we have manually removed one of the tetrahedral planes to mimic the out-of-plane phonon that appears at the spinunrestricted GGA+A level 14 .The pairing strength is substituted into the McMillian Tc formula 27 .Only Fe and As atoms are imported in the ab-initio calculation to showcase the bare pairing strength.

Results
Fig. 1a displays our theoretical Tc of BaFe2As2 under external pressure compared to experimental literature data 20 .The combined model based on the 2 nd AFM fluctuation shows a reasonable accuracy in the superconducting phase diagram simulation.The enhanced electron-phonon coupling, and the exchange factor are optimized at 1.3GPa, but they are drastically reduced at higher pressures, as shown in Fig. 1b.The Debye temperatures of uncompressed BaFe2As2 at the low and high temperature limits are 379K and 470K, respectively 31 .Our computed BaFe2As2 at 0.8GPa only increases from 0.33 to 0.37 after activation of the spin-orbital coupling.However, the Coh factor and ARPES factors are the main ingredients to increase the pairing strength to ~0.9, allowing the theoretical Tc to occur above 30K.In contrast, the combined model makes use of 1st AFM fluctuation results in a significant discrepancy between the calculated and experimental Tc at high pressures. is 0.88 and the literature value is 0.9 33 .When the doping concentration is increased from 0.4 to 0.6, the * PS  decreases slightly.The pressure effect on the dielectric constant of BaFe2As2 and Ba1-xKxFe2As2 under spin-unrestricted conditions is small.
By performing calculations, we estimate the Tc of SrFe2As2 under compression.The theoretical Tc values of SrFe2As2 at low pressures are in good agreement with the experimental data 9 presented in Table 1, regardless of whether the first or second AFM fluctuation is used.However, significant discrepancies between the theoretical and experimental Tc values are observed for SrFe2As2 under high pressure conditions 9 .On the other hand, the theoretical Tc value of CaFe2As2 as a function of pressure 10 does not vary significantly when either the first or second-order AFM fluctuation method is used.Furthermore, our spin-unrestricted calculation reveals that the magnetic moment of MgFe2As2 remains at zero when the pressure exceeds 3GPa, as indicated in Table 2.In addition to these results, we also investigated the relationship between the momentum space and the superconducting gap, as depicted in Figure 3.It is worth noting that the calculated values of Tc exhibit minimal error (δTc ~ 2-4K), regardless of the presence of gap anisotropy.The theoretical Tc towards an isotropic superconducting gap is slightly higher.

Discussion
While the here presented combined model has demonstrated efficacy in bridging the gap between theoretical and experimental Tc, a comprehensive theory of IBSC remains elusive.This paper does not aim to reevaluate the six models outlined in the methodology section, as their scientific validity has been established in peer-reviewed literature.We do not seek to validate the theory of ironbased superconductivity in this work.Instead, our objective is to amalgamate Tc calculations from these validated studies and systematically integrate each sub-model.This approach allows us to examine its potential for predicting the theoretical superconducting phase diagram.Our focus in this section is primarily on data analysis, rather than conjecture about the triggers of iron-based superconductivity.If a proposed model of iron-based superconductors is deemed incorrect, a universal theory of iron-based superconductors would need to be already in place.However, at present, no universally accepted or fully established unified theory for iron-based superconductors exists.
Our theoretical analysis indicates that solely considering the first-order AFM fluctuation may not suffice to simulate the complete superconducting phase diagram of BaFe2As2.Antiferromagnetic fluctuations typically diminish under pressure.The pressure-dependent theoretical Tc (secondorder AFM) in BaFe2As2 is significantly enhanced, as the first-order AFM fluctuation does not decrease as rapidly as the second-order AFM fluctuation at high pressures.This aligns with the results of Coh et al 14 when a higher-order AFM fluctuation is employed.Both first and secondorder AFM fluctuations can yield accurate theoretical Tc values at low pressures, as it is feasible to fit a dependent variable (Y) linearly within the high-order term when the independent variable (X) is near zero or small.A similar scenario is observed in SrFe2As2, where only the second-order AFM fluctuations can accurately calculate Tc at high pressure (the exchange factor drops by ~25% rapidly).However, for CaFe2As2, the first-order AFM fluctuation maintains precision in calculating Tc, as its exchange factor only decreases by less than 4% from 0.1GPa to 1.2GPa.
Elevating the Debye frequency permits higher energy phonons to interact with a greater number of electrons within the ARPES range of spectral weight shift [16][17][18][19] .This interaction results in an augmentation of the RARPES value, as the effective electronic DOS increases.However, the () ex fE effects are typically counterbalanced by high pressure, which mitigates the impact of RAF and Rph.
The implementation of the second-order AFM fluctuation is crucial to decrease the Tc of BaFe2As2 above ~2GPa.The RARPES factor accounts for electron energies situated down to roughly 30meV (equivalent to ~350K) below the Fermi level.This method continues to adhere to the hyperbolic tangent shape of the Fermi-Dirac statistics across the Fermi level at a finite temperature 13  The reported experimental Tc of Ba0.8K0.2Fe2As2 by Rotter et al. 32 and Böhmer et al. 8 are 26K and 20K, respectively.The slight discrepancy in the experimental Tc values may be attributed to the experimental methodologies employed or the uncertainty in the tetrahedral Fe-As-Fe angle.
Utilizing the tetrahedral angle of BaFe2As2 measured by R. Mittal et al. 34  The factors contributing to electron participation within the ARPES range of spectral weight shift in IBSC remain ambiguous, potentially involving a myriad of complex elements such as spinorbital coupling, nematicity, antiferromagnetism, the competition between s-wave and d-wave pairing, and electron-phonon interaction [36][37][38][39][40][41] .In certain iron-based superconductors, the Fermi surface displays nematic order 41 , which influences the superconducting order parameter.The precise effect of nematic order on the pairing symmetry on the Fermi surface is still an open question.However, the intricate interactions between electrons on the Fermi surface can be simplified by considering all high-energy electrons within the ARPES range of spectral weight shift.Here, high-energy electrons are not anticipated to involve either nematicity or 4-fold symmetry in the superconducting gap.The electron-phonon coupling on the Fermi surface typically weakens by ~20-30% when transitioning to 4-fold or d-wave like symmetry 42 , which validates our calculated value of ~0.8 angular f .In our Tc calculation, we considered the 4-fold pairing symmetry across the entire ARPES range of spectral weight shift below the Fermi level.However, the change in Tc is only a few Kelvin, as depicted in Figure 3.We have not examined the case of 0.5 angular f  because reducing the pairing strength by gap anisotropy necessitates the formation of a p-wave order parameter 43 , and none of our studied compounds are expected to be p-wave superconductors.
Another potential source of error in Tc could be the approximation of the Debye energy in the ARPES factor.A trend can be observed between the ARPES range and the Debye energy when comparing the ARPES data of different materials.An examination of the ARPES data of LiFeAs, FeSe, and FeSe/SrTiO3 provides insights into this observation.In the case of FeSe/SrTiO3, the Tc is approximately 100K 16 , and the interfacial phonon energy is around 1200K 44 .Concurrently, the ARPES range in this context is measured to be between 0.1-0.3eV 16.Conversely, for LiFeAs or FeSe, the Tc is lower 12 , around 10K, and the phonon energy is approximately 300K.Correspondingly, the ARPES ranges for LiFeAs and FeSe are much narrower, around 0.03-0.06eV.
These data suggest a trend where an increase in Debye energy corresponds to an increase in the ARPES range, indicating a broader energy range for electronic excitations.Therefore, we select the Debye energy as an approximated energy range in the ARPES factor, but it may not precisely correspond to the actual energy range below the Fermi level.This approximation can lead to either an overestimation or underestimation of Tc.To rectify this issue, one could determine the exact ARPES range for each IBSC and use that value instead of an approximation.This approach would yield a more accurate estimation of the ARPES factor.However, scanning all ARPES data for every discovered IBSC would indeed require significant experimental effort and pose practical challenges and limitations.
Upon initial observation, the anisotropy factor associated with nematicity appears to exert minimal influence on the calculated Tc values in Figure 3.However, this does not suggest that they are inconsequential to the mechanism of IBSC.In fact, they may play a pivotal role in initiating IBSC 45 .
Once iron-based superconductivity is triggered, the impact of the anisotropy factor diminishes due to the renormalization of the strong pairing strength in compressed IBSC.Therefore, it would be unjust to assess the significance of the factor or nematicity based solely on Figure 3.
Conversely, the combined model may not be intended for discovering new IBSC compounds.Rather, it may be suitable for predicting Tc at higher pressures or heavy doping levels if superconductivity has already been confirmed at ambient pressure, low pressure, or low doping concentration.Despite bridging the gap between theoretical and experimental Tc values, the combined model does not explicitly provide a definitive statement about the pairing mechanism of iron-based superconductors.The unified theory of iron-based superconductors remains an open question necessitating further investigation and research.The possibility of other models capable of producing accurate theoretical Tc of iron-based superconductors via entirely different mechanisms is not excluded.

Conclusions
We have successfully established a framework for simulating the superconducting phase diagrams of major 122-type intercalated IBSC with commendable accuracy.Our observations indicate that certain IBSC require the consideration of higher-order antiferromagnetic fluctuations, particularly under high-pressure conditions.This is a pivotal step towards integrating various effects into the pairing mechanism of IBSC, including superconducting gap anisotropy, spin-orbital coupling, high-energy electrons, abnormal phonon behavior, screening effects, spin density wave, and antiferromagnetism.Our findings pave the way for a deeper understanding of the complex interplay of factors governing superconductivity in these materials.

FeM
and co E as the magnetic moment of the Fe atom and the exchange-correlation energy, respectively.The exchange factor becomes or chemical pressure P25 .

 3 ).
. To solve this problem, we define within the ARPES range, where the shift range of spectral weight occurs.XY upper DOS represents the average electronic density of states for the structure containing only the upper tetrahedral plane, while XY lower DOS indicates the average electronic DOS for the structure that only contains lower tetrahedral planes.XY both DOS is the average electronic DOS representing the original structure coexisting upper and lower tetrahedral regions.The ionic interaction XY ion V on the XY plane in the presence of charge fluctuations in the tetrahedral regions is XY APRES factor: To include all relevant electrons in iron-based superconductivity in the calculations, the average electron-phonon scattering matrix ' The dielectric constant '  controls the screening effect 25 when all the relevant electrons interact with XY ion V .Including the ARPES factor increases the electron-phonon scattering matrix by and minor b axes control the anisotropy of the gap.The energy change due to the formation of an anisotropic momentum space is

Fv
is the Fermi velocity.The velocity Debye v can be interpreted from the Debye energy.A is a material constant.is the Planck constant over 2π and ' p  is the wave function of the electrons.The2 st AFM fluctuation.It is not necessary to switch on the spin-unrestricted mode to calculate

Fig 1 ..
Fig 1.The Tc distribution and the corresponding interaction terms of BaFe2As2 as a function of hydrostatic pressure.(a) Theoretical Tc of BaFe2As2 under compression (red and blue squares) together with experimental data (open circles) 20 .The 4-fold symmetry is outlined by two overlapped red ellipses in which blue lines are used to split it into 8 regions in equal partition, where the area of each region is

Fig 2 .
Fig 2. (a) The doping dependence of the theoretical and experimental 8,32 Tc of Ba1-xKxFe2As2.The theoretical Tc based on the 1 st and 2 nd AFM fluctuations are marked by squares and triangles, respectively.(b) The individual interaction terms of Ba1-xKxFe2As2 as a function of K content.The renormalized * PS  is calculated under 2 nd AFM fluctuations.

Figure 3 :
Figure 3: Effect of the momentum dependence of the superconducting gap on the theoretical Tc in the studied compounds under the 2 nd AFM fluctuation.
. The correlation between Tc and AFM fluctuations, as depicted in Fig 1b and Fig 2b, underscores the significance of AFM fluctuations in the superconducting pairing process.

Table 2 :
Magnetic analysis of MgFe2As2 under pressure.
P(GPa) Fe moment (μB) P(GPa) 32 low temperatures as a reference, the theoretical Tc of Ba0.8K0.2Fe2As2reverts to ~20K.However, we employ data from Rotter et al., as it offers a more systematic Tc calculation, where all lattice parameters, bond lengths, and bond angles are derived from a single experimental setup32.While our theoretical Tc values of Ba1-xKxFe2As2 are not devoid of errors, the Tc profiles depicted in Fig 2a align more closely with the experimental data.The dopant-induced pressure in Ba1-xKxFe2As2 is relatively weak, hence the combined model based on first-order AFM fluctuation remains accurate.The same pairing strength formulae are applied for BaFe2As2, Ba1-xKxFe2As2, CaFe2As2, SrFe2As2, MgFe2As2 with reasonably well-calculated Tc, suggesting that the pairing mechanism of these four undoped ironbased superconductors may share similar components from a statistical perspective.No Tc has been reported for MgFe2As2; this may be due to the fact that as the () ex fE drops to zero above 3GPa, the pairing strength becomes zero and we propose that pressure is not an effective method to induce iron-based superconductivity in MgFe2As2.