Preliminary Tc Calculations for Iron-Based Superconductivity in NaFeAs, LiFeAs, FeSe and Nanostructured FeSe/SrTiO3 Superconductors

Many theoretical models of iron-based superconductors (IBSC) have been proposed, but the superconducting transition temperature (Tc) calculations based on these models are usually missing. We have chosen two models of iron-based superconductors from the literature and computed the Tc values accordingly; recently two models have been announced which suggest that the superconducting electron concentration involved in the pairing mechanism of iron-based superconductors may have been underestimated and that the antiferromagnetism and the induced xy potential may even have a dramatic amplification effect on electron–phonon coupling. We use bulk FeSe, LiFeAs and NaFeAs data to calculate the Tc based on these models and test if the combined model can predict the superconducting transition temperature (Tc) of the nanostructured FeSe monolayer well. To substantiate the recently announced xy potential in the literature, we create a two-channel model to separately superimpose the dynamics of the electron in the upper and lower tetrahedral plane. The results of our two-channel model support the literature data. While scientists are still searching for a universal DFT functional that can describe the pairing mechanism of all iron-based superconductors, we base our model on the ARPES data to propose an empirical combination of a DFT functional for revising the electron–phonon scattering matrix in the superconducting state, which ensures that all electrons involved in iron-based superconductivity are included in the computation. Our computational model takes into account this amplifying effect of antiferromagnetism and the correction of the electron–phonon scattering matrix, together with the abnormal soft out-of-plane lattice vibration of the layered structure. This allows us to calculate theoretical Tc values of LiFeAs, NaFeAs and FeSe as a function of pressure that correspond reasonably well to the experimental values. More importantly, by taking into account the interfacial effect between an FeSe monolayer and its SrTiO3 substrate as an additional gain factor, our calculated Tc value is up to 91 K and provides evidence that the strong Tc enhancement recently observed in such monolayers with Tc reaching 100 K may be contributed from the electrons within the ARPES range.


Introduction
The pairing mechanism of unconventional high-temperature superconductors (HTSCs) remains one of the greatest unsolved mysteries of physics. All unconventional superconductors, including cuprates [1,2] and iron-based HTSCs [3,4] but also heavy fermions [5] and organic superconductors [6], have in common that the superconducting phase occurs near a magnetic phase. Furthermore, their phase diagrams typically show at least one other form of electronic order, e.g., charge or orbital order [7,8], a pseudogap phase [2], stripe order [2] or nematic order [9]. The proximity of the magnetic phases naturally suggests the involvement of magnetism [10]. In most theoretical approaches, spin fluctuations play a leading high T c [31]. According to the experiment by S. Zhang et al. [31], the F-K phonons of the FeSe/SrTiO 3 surface show new energy loss modes, and the line width is widened compared with bare SrTiO 3 .
In this article, we revise the superconducting electron concentration and use an ab initio approach to examine if the T c values of LiFeAs, NaFeAs and FeSe as a function of pressure can be calculated reasonably by taking into account the R ph and R SDW factors, etc. If successful, we use this model to test whether such an approach can be applied to thẽ 100 K superconductivity in the nanostructured FeSe/SrTiO 3 . Not all mechanisms of ironbased superconductivity have been encountered in this work, because the unified theory of iron-based superconductors remains an open question. We only apply mathematical techniques to convert the two models from the literature into T c values, which may be important to find out the possible mechanism of iron-based superconductors.

Computational Methods
As a starting point, the electronic properties of all compounds investigated in this article are computed by the spin-unrestricted Generalized Gradient Approximation of the Perdew-Burke and Ernzerhof (GGA-PBE) functional (unless otherwise specified) [31][32][33][34][35] in Wien2K. The SCF tolerance is 1 × 10 −5 eV, and the interval of the k-space is 0.025(1/Å). The maximum SCF cycle is 1000. The magnetism and phonon data are calculated by CASTEP. Finite displacement mode is chosen where the supercell defined by cutoff radius is 5 Å and the interval of the dispersion is 0.04(1/Å). Ultrasoft pseudopotential is assigned, and density mixing is chosen to be the electronic minimizer [31][32][33][34][35]. The experimental lattice parameters are used [36,37]. In this article, only Fe and As atoms are imported for the 111-type compounds.
Instead of calibrating "A" in the GGA+A functional, which entails an enormous computational cost and time-consuming experimental effort [21,38,39], we propose a twochannel model to more easily model the induced xy potential, where the upper tetrahedral plane is called channel 1 and the lower tetrahedral plane is called channel 2, respectively. We apply the superposition principle to separately calculate the induced xy potentials induced by channels 1 and 2. Our two-channel model has fulfilled an assumption that the probability of finding an Fe atom moving in the +z and −z directions is equal, but their vibrational amplitudes never cancel each other out. This assumption is justified by Coh et al., whose explicit calculation confirms that the iron-based system consists of an out-of-phase vertical displacement of iron atoms, with the first adjacent iron atoms moving in opposite directions [21]. We define R ph = 0.5(DOS XY 1 + DOS XY 2 ) DOS XY 12 . In the ARPES range, DOS XY 1 represents the average electronic density of states for the structure that exclusively contains upper tetrahedral planes. Similarly, DOS XY 2 indicates the average electronic density of states within the ARPES range for the structure that only contains lower tetrahedral planes. Meanwhile, DOS XY 12 corresponds to the average electronic density of states within the ARPES range for the original structure that has coexisting upper and lower tetrahedral planes.
F(ω) is the phonon density of states as a function of frequency ω, and the integral d 2 p F is taken over by the Fermi surface with the Fermi velocity v F . The Eliashberg function is written as [40] The electron-phonon matrix elements are given by g pp v = Cω p−p v g v (p, p ), where ψ * p u i · ∇V XY ψ p dr is abbreviated as g v (p, p ), ψ p is the wavefunction of electron, is the Planck constant divided by 2π and C is the material constant related to lattice [40]. u i and V XY represent the displacement of the ion relative to its equilibrium position and the ionic potential. ψ * p ψ p is the electronic probability density in the nonmagnetic state.
The resultant ionic interaction V XY ion on the XY plane, due to the abnormal phonon, is calculated by multiplying the ionic potential by R ph , i.e., V XY ion = V XY · R ph . Moreover, the antiferromagnetic interaction along the XY plane modifies the electronic wavefunction φ p , and the probability density fulfills φ * p φ p ∼ ψ * p R SDW ψ p . The spin density wave factor R 2 SDW can be considered as the amplification factor for electron-phonon scattering under an antiferromagnetic SDW state, relative to a nonmagnetic state [21]. Rearranging the mathematical terms yields the electron-phonon matrix element as To derive a superconducting transition temperature from the simulation parameters, we use the McMillan T c formula [40]. Due to the high transition temperatures, the electron-phonon scattering matrix takes into account the full electronic DOS in a range from E F − E Debye to E F and not only the value at Fermi level (i.e., increasing the effective electronic DOS). Here, we consider that E Debye represents the upper limit of the phonon energies that can be transferred to electrons, and at the high transition temperatures of Fe-based superconductors, contributions from high-energy phonons become important in the electron-phonon scattering mechanism, as opposed to classical low-T c superconductors. Although this approach is a simple consequence of the conservation of energy, it is supported by experiments: a shift of the spectral weight between the normal and the superconducting state is clearly visible in the photoemission spectra below the superconducting energy gap of various iron-based compounds in an energy range of~30-60 meV below the Fermi energy [18][19][20]. This energy range is approximately on the order of Debye energy.
In Bardeen-Cooper-Schrieffer (BCS) superconductors, the electrons on the Fermi surface condense into the Bose-Einstein superconducting state, where the total number of electrons on the Fermi surface equals the total number of electrons on the superconducting state. Hence, the theoretical T c of BCS superconductors remains the same if we substitute either the electronic DOS on the Fermi level or the electronic DOS of the condensed Bose-Einstein state. However, the situation is different in iron-based superconductivity, where the electrons located between E F − E Debye and E F transfer energy to the electrons in the Bose-Einstein superconducting states. When this happens, we have to revise the resultant electron-phonon scattering matrix in the condensed Bose-Einstein state. The Bose-Einstein statistic favors more electrons occupying the superconducting state. The electrons within the ARPES range increases the effective electronic DOS in the condensed Bose-Einstein state indirectly. The electrons within the ARPES range cannot be excited to the Fermi surface due to electrostatic repulsion. However, these electrons have another route to follow the Bose-Einstein distribution, which can be argued as a reason why these electrons disappear below the Fermi level.
The computation of band structure produces discrete (E,k) points, where E and k are the energy and the wavevector of the electron, respectively. The ratio of the electron- , which is abbreviated as the gives the total number of (E, k) points in the range is the percentage of electrons contributed to the R g term. To make a fair comparison, the intervals of K space in the numerator and denominator of R g are essentially the same. The R g term controls the proportion of electrons scattered below the Fermi level.
Due to the fact that the superconducting transition temperatures are low, we calculate the mean occupation number f (E) in the Fermi-Dirac statistic at low tempera-tures (T < 100 K), where f (E) and f (E F − E Debye ) are 0.5 and~0.5005, respectively. If the tiny offset in the mean occupation number may allow the Eliashberg function to approximately obey the following form.
, and the velocity v Debye is converted from the Debye energy.
the ARPES range. The form of the antiferromagnetically amplified electron-phonon coupling is expressed as λ Coh The α E F is the average square of the electron-phonon scattering matrix on the Fermi surface [40]. In the case of strong coupling, the renormalized electron-phonon coupling is expressed as When all the terms in the pairing strength at any pressure are entirely calculated by the spin-unrestricted GGA-PBE functional [33], this approach is defined as a "traditional combination of DFT functional", in which R 2 SDW may be neglected, as the effect of SDW should be included in α 2 E F , R 2 Ph , R 2 g and F(ω) automatically in the spin-unrestricted mode. On the other hand, we propose an "empirical combination of DFT functional" which imposes the antiferromagnetic effect on the pairing strength separately. In this case, the antiferromagnetically amplified pairing strength is separately calculated by multiplying the nonmagnetic pairing strength with the antiferromagnetic factors. α 2 E F and F(ω) are computed by spin-restricted mode, but R 2 SDW always needs an operation of the spinunrestricted mode in order to add the effect of SDW. As the two-channel model has already mimicked the contributions of the abnormal phonon under antiferromagnetism manually, it is recommended to apply the spin-restricted mode to calculate R 2 Ph . Otherwise, the effect of antiferromagnetism on the abnormal phonon may be overestimated.
For the "empirical combination of DFT functional", the pairing strength is further corrected by the AF Ising Hamiltonian in the presence of pressure. To include the magnetic effect, this AF Ising Hamiltonian is acquired by the spin-unrestricted GGA-PW91 functional. The pairing strength formulas of LiFeAs (111-type), NaFeAs (111-type) and FeSe (11-type) under pressure are given as λ 111 . The ratio f 111 11 (E ex ) monitors the pressure dependence of the AF energy at each external pressure P, and E co is the exchange-correlation energy. We use f 111 11 (E ex ) to correct the antiferromagnetism under pressure instead of recalculating the R 2 SDW . The Debye temperature of the FeSe/SrTiO 3 is replaced by the vibrational energy of the F-K phonon across the interface [31]. The pairing strength is substituted into the McMillian T c formula [27], which includes the en- .

Results
The atomic spring constants between the FeFe bond k FeFe and FeSe bond k FeSe in the iron-based superconductors are compared. Our DFT calculation shows that k FeSe /k FeFe is 0.25, while the k FeAs is almost 2 times stronger than k FeSe . As the atomic spring constants of the tetrahedral bonds are comparable to the FeFe bond, the orthogonal phonon appearing is feasible. Our two-channel model demonstrates that the induced xy potential is good enough to be emerged at the "GGA-PBE" level. We calculated that the electron-phonon scattering matrix of FeSe under the induced xy potential amplified by R ph = 2.8. While the accuracy of our two-channel model is comparable to the R ph = 2.2 obtained from the calibrated GGA + A functional [21], we determine the R ph of NaFeAs and LiFeAs to be 1.97 and 1.8, respectively. The pressure dependence on R ph is less than~5% due to c >> a.
A critical parameter in any ab initio approach is the value of the renormalized Coulomb pseudopotential. Figure 1 estimates the error of the theoretical T c by tuning µ*. Despite that the calculation of µ* as a function of Debye temperature and Fermi level [41] may not be very accurate in such a strongly correlated electron system [42], it has been argued that for the most Fe-based superconductors, µ* should be 0.15-0.2 [12]. In this paper, we choose the value (µ* = 0.15) of the Coulomb pseudopotential to calculate the T c of LiFeAs, NaFeAs and FeSe to make a fair comparison. Our calculated µ* value of the uncompressed NaFeAs is 0.13. The error of our T c calculation due to the uncertainty of µ* between µ* = 0.15 and µ* = 0.13 is within~15%.  Figure 2a shows that our approach can generate the theoretical Tc values in an appropriate range. The ARPES data confirm that LiFeAs and FeSe require the use of the Rg term, while the NaFeAs does not [18,20,43]. The theoretical Tc of NaFeAs at 0 GPa and 2 GPa are 11 K and 12.5 K, respectively [44]. The antiferromagnetically enhanced electronphonon interaction on the Fermi surface and the AF exchange Hamiltonian compete in the compressed NaFeAs, as illustrated in Figure 2b. We observe that the antiferromagnetism is slightly weaker at finite pressure, but the antiferromagnetically assisted electron-phonon coupling on the Fermi layer is increased almost linearly at low pressure. We show the steps to estimate the Tc of NaFeAs at 0 GPa as an example. After activating the spin-unrestricted mode, the The theoretical T c of NaFeAs varies slightly with the Coulomb pseudopotential. Our calculated µ*-value of the uncompressed NaFeAs is 0.13. Figure 2a shows that our approach can generate the theoretical T c values in an appropriate range. The ARPES data confirm that LiFeAs and FeSe require the use of the R g term, while the NaFeAs does not [18,20,43]. The theoretical T c of NaFeAs at 0 GPa and 2 GPa are 11 K and 12.5 K, respectively [44]. The antiferromagnetically enhanced electron-phonon interaction on the Fermi surface and the AF exchange Hamiltonian compete in the compressed NaFeAs, as illustrated in Figure 2b. We observe that the antiferromagnetism is slightly weaker at finite pressure, but the antiferromagnetically assisted electron-phonon coupling on the Fermi layer is increased almost linearly at low pressure. We show the steps to estimate the T c of NaFeAs at 0 GPa as an example. After activating the spin-unrestricted mode, the R 2 SDW is 1.625. The antiferromagnetically assisted electron-phonon coupling on the Fermi surface is  Figure 2a shows that our approach can generate the theoretical Tc values in appropriate range. The ARPES data confirm that LiFeAs and FeSe require the use o Rg term, while the NaFeAs does not [18,20,43]. The theoretical Tc of NaFeAs at 0 GPa 2 GPa are 11 K and 12.5 K, respectively [44]. The antiferromagnetically enhanced elect phonon interaction on the Fermi surface and the AF exchange Hamiltonian compe the compressed NaFeAs, as illustrated in Figure 2b. We observe that the antiferrom netism is slightly weaker at finite pressure, but the antiferromagnetically assisted tron-phonon coupling on the Fermi layer is increased almost linearly at low pressure show the steps to estimate the Tc of NaFeAs at 0 GPa as an example. After activating spin-unrestricted mode, the  The DFT parameter can be found in Table 1.
According to the McMillian T c Formula, the T c becomes We compare our theoretical T c by substituting the raw data of other groups [15,21]; their calculated λ AF E F is 0.39 [15], and the induced xy potential by the out-of-plane phonon reinforces the electron-phonon coupling matrix by 2.2 [21].
After renormalization, these two couplings are softened to λ 111 11 = * λ Coh PS = 1.88/ (1.88 + 1) = 0.652, and the renormalized Coulomb pseudopotential µ * Based on the data in other groups [15,21], the theoretical T c becomes Our calculated value of the electron-phonon coupling on the Fermi surface of the uncompressed LiFeAs is~0.1 [45], but the magnetic amplification factors increase the pairing strength to 0.82, remarkably. The Debye temperature T Debye of LiFeAs remains at 385 K below 8 GPa [46], as shown in Table 2. A reduction in the theoretical T c is also observed in the compressed LiFeAs, and the weakening effect of * λ Coh PS and f 111 11 (E ex ) under pressure is identified, as shown in Figure 3b. In compressed FeSe [24], however, a gain in f 111 11 (E ex ) is observed that triggers the increase in T c under pressure (Figure 4). It should be noted that our approach is a mean field approach, and we treat the spin fluctuations as being proportional to the mean field Hamiltonian. The vanishing of the macroscopic AF order observed in real samples is due to the strong fluctuation effects in these layered compounds. The magnetism considered here in the nonmagnetic regimes of the phase diagrams is of a fluctuating microscopic nature. The optimized pairing strength of LiFeAs and FeSe is achieved at a pressure of 0 GPa and 0.7 GPa, respectively. The differences between DOS(E F -E Debye ) and DOS(E F ) in LiFeAs and FeSe are less than 4%. The R g term in LiFeAs is reduced with pressure, but the R g term of FeSe is optimized at medium pressure (see Tables 2 and 3).
Based on the data in other groups [15,21]  It should be noted that our approach is a mean field approach, and we treat the spin tuations as being proportional to the mean field Hamiltonian. The vanishing of the m roscopic AF order observed in real samples is due to the strong fluctuation effects in t layered compounds. The magnetism considered here in the nonmagnetic regimes o phase diagrams is of a fluctuating microscopic nature. The optimized pairing streng LiFeAs and FeSe is achieved at a pressure of 0 GPa and 0.7 GPa, respectively. The di ences between DOS(EF-EDebye) and DOS(EF) in LiFeAs and FeSe are less than 4%. Th term in LiFeAs is reduced with pressure, but the Rg term of FeSe is optimized at med pressure (see Tables 2 and 3).    Based on the successful Tc calculation of the bulk FeSe, LiFeAs and NaFeAs, we start our journey to acquire the theoretical Tc of the FeSe monolayer on a SrTiO3 substrate step by step using the model of an antiferromagnetically enhanced electron-phonon coupling. The flowchart is shown in Figure 5. After the geometric relaxation of FeSe/SrTiO3, the angles of the unit cell are 89.81°, 90.88° and 89.05°, with a tiny internal shear force being captured. The relaxed tetrahedral angle of Fe-Se-Fe is 108 degrees. The antiferromagnetic energy of FeSe can be amplified by low dimensionality when it is deposited in the form of a monolayer on SrTiO3 [26]. Compared with a FeSe monolayer without substrate, the FeSe film on SrTiO3 shows an increased exchange correlation energy of ~16% on FeSe. Apart from this, the local Fe moment in the isolated FeSe monolayer is only ~0.5 µB. However, contact with SrTiO3 amplifies the local Fe moment of the FeSe film up to ~1.3 µB. Our calculated electron-phonon coupling on the Fermi surface without any amplification  Based on the successful T c calculation of the bulk FeSe, LiFeAs and NaFeAs, we start our journey to acquire the theoretical T c of the FeSe monolayer on a SrTiO 3 substrate step by step using the model of an antiferromagnetically enhanced electron-phonon coupling. The flowchart is shown in Figure 5. After the geometric relaxation of FeSe/SrTiO 3 , the angles of the unit cell are 89.81 • , 90.88 • and 89.05 • , with a tiny internal shear force being captured. The relaxed tetrahedral angle of Fe-Se-Fe is 108 degrees. The antiferromagnetic energy of FeSe can be amplified by low dimensionality when it is deposited in the form of a monolayer on SrTiO 3 [26]. Compared with a FeSe monolayer without substrate, the FeSe film on SrTiO 3 shows an increased exchange correlation energy of~16% on FeSe. Apart from this, the local Fe moment in the isolated FeSe monolayer is only~0.5 µ B . However, contact with SrTiO 3 amplifies the local Fe moment of the FeSe film up to~1.3 µ B . Our calculated electron-phonon coupling on the Fermi surface without any amplification factor is λ Fermi = 0.12. Based on our simulation, the antiferromagnetism of FeSe/SrTiO 3 is still as strong as of the FeSe monolayer without substrate. Hence, the simultaneous occurrence of antiferromagnetism and tetrahedral atoms makes the Coh factor unavoidable. The analytical result of C AF = 2 is used [21], and our calculated C Ph in FeSe/SrTiO 3 is 2.9. After amplification of the Coh factor, the theoretical T c is only 14 K. However, a massive enhancement of the pairing strength can be observed when the interfacial F-K phonon is involved [31]. The F-K phonon actuated via the interface contributes the vibrational energy of~100 meV (~1159 K) [31]. With this enormous Debye temperature, the theoretical T c is increased to 69 K, although the electron-phonon interaction is limited to the Fermi energy. In the ARPES data, it is evident that a shift in spectral weight occurs in the superconducting state 0.1~0.3 eV below the Fermi level [19], which means that electrons in this energy range are affected by electron-phonon scattering as a result of the high phonon frequencies. This means that electrons in this energy range contribute to superconductivity, since the high phonon frequencies can scatter them up to the Fermi energy and need to be considered in the McMillan formula, and not only those at the Fermi energy, as in the usual approximation applied to classical low-T c superconductors. The superconducting electron concentration is thus corrected, and the average electron-phonon scattering matrix in these multienergy layers is 1.96 times higher than the matrix considering only the Fermi level. This is the last factor with which our theoretical T c can reach 91 K, which corresponds quite well to the experimental T c of 100 K. . Based on our simulation, the antiferromagnetism of FeSe/SrTiO3 is still as strong as of the FeSe monolayer without substrate. Hence, the simultaneous occurrence of antiferromagnetism and tetrahedral atoms makes the Coh factor unavoidable. The analytical result of CAF = 2 is used [21], and our calculated CPh in FeSe/SrTiO3 is 2.9. After amplification of the Coh factor, the theoretical Tc is only 14 K. However, a massive enhancement of the pairing strength can be observed when the interfacial F-K phonon is involved [31]. The F-K phonon actuated via the interface contributes the vibrational energy of ~100 meV (~1159 K) [31]. With this enormous Debye temperature, the theoretical Tc is increased to 69 K, although the electron-phonon interaction is limited to the Fermi energy. In the ARPES data, it is evident that a shift in spectral weight occurs in the superconducting state 0.1~0.3 eV below the Fermi level [19], which means that electrons in this energy range are affected by electron-phonon scattering as a result of the high phonon frequencies. This means that electrons in this energy range contribute to superconductivity, since the high phonon frequencies can scatter them up to the Fermi energy and need to be considered in the McMillan formula, and not only those at the Fermi energy, as in the usual approximation applied to classical low-Tc superconductors. The superconducting electron concentration is thus corrected, and the average electron-phonon scattering matrix in these multienergy layers is 1.96 times higher than the matrix considering only the Fermi level. This is the last factor with which our theoretical Tc can reach 91 K, which corresponds quite well to the experimental Tc of 100 K. The pairing strength is renormalized as  The pairing strength is renormalized as The pseudopotential is diluted as We substitute all parameters into the McMillian T c formula: The pure FeAs layer in the 111-type, 1111-type and 122-type Fe-based superconductors is believed to trigger superconductivity [50,51]. The investigation of the pure FeAs layer without the Li and Na atoms in the simulation can show the bare pairing strength. The T c vs. pressure of the NaFeAs is not as sensitive as for the other materials. The reason for this is that the increase in * λ Coh PS and the decrease in f 111 11 (E ex ) almost cancel out the variation in the pairing strength. The unusually high T c in the LiFeAs and FeSe at 0 GPa is mainly due to the R ph , R SDW and R g terms (Coh factor: R ph and R SDW ; ARPES factor: R g ). Our approach confirms that the reduction in T c in compressed LiFeAs is mainly due to the decreases in * λ Coh PS and AF energy as a function of pressure. Conversely, the magnetic moment of Fe in FeSe increases under compression, resulting in an increase in AF energy under pressure. As a result, the increase in T c in compressed FeSe is observed. The R g term is minimized at high pressure, since the kinematics of electrons below the Fermi level is more restricted under pressure. Our simulation shows that the variation in the induced xy potential is less than~3% for the electrons at~100 meV below the Femi level, and therefore, the use of the APRES factor (or R g ) in LiFeAs and FeSe is justified.
We correct the pairing strength at high pressures with the help of the AF Ising Hamitonian. In the following, we compare the T c when R g and R ph are calculated by the spin-unrestricted GGA-PBE functional at high pressures, simply called the "traditional combination of DFT functional". Despite the "traditional combination of DFT functional" providing an accurate theoretical T c at ambient pressure, the error of T c is significant at high pressures. We demonstrate this for the case of FeSe in Table 4. In this approach, we do not use the AF Ising Hamiltonian at finite pressure because magnetism is already considered. Since 2008, the ARPES factor (R g ) has been missing in the calculation of the electron-phonon coupling constant. However, Table 5 confirms that the consideration of the electron-phonon coupling on the Fermi surface is not sufficient to argue whether iron-based superconductivity is mediated by phonons. If the ARPES factor (R g ) really participates in iron-based superconductivity, the abnormal distribution of electrons below the Fermi level should be given a larger range when the T c of the iron-based superconductor is higher. This argument is supported by the ARPES data of the 100 K 2D FeSe/SrTiO 3 [19] with the parameters shown in Table 6. For these~10-30 K iron-based superconductors, the electrons located at 0.03-0.06 eV below the Fermi level are affected by superconductivity [22,29]. However, the electrons in the 100 K 2D FeSe/SrTiO 3 , which are located in a much wider range of 0.1-0.3 eV below the Fermi level, participate in superconductivity [19]. The theoretical T c of the 2D FeSe/SrTiO 3 reaches 91 K only if the ARPES factor (R g ) is considered.   An empirical rule is that the T c of the iron-based superconductor is optimized when the tetrahedral angle is close to 109.5 degree [52]. When the FeSe monolayer is attached to the SrTiO 3 , the tetrahedral angle is changed from 103 degrees to 108 degrees, and T c benefits. However, all these antiferromagnetic and tetrahedral effects cannot explain the high T c near 100 K until the interface properties are considered [31]. Despite the Debye temperature of the FeSe phonons (~250 K) showing no significant size effect, an energetic F-K phonon carrying an energy of 100 meV (~1159 K) was observed at the interface between the FeSe film and SrTiO 3 [31]. Since the 3D and 2D FeSe phonon are almost identical [31], the out-of-plane phonon from the tetrahedral sites should amplify the electron-phonon coupling of FeSe/SrTiO 3 by the same R Ph factor = 2. Assuming that the F-K phonon and FeSe phonon interact with electrons simultaneously, two Debye energies, i.e., from the FeSe phonons and the F-K phonons, may influence the Cooper pairs. The two-fluid model, however, ensures that the onset T c is always related to the mechanism that gives the strongest pairing strength, and therefore, choosing 1159 K as the Debye temperature is justified.
The ARPES data of FeSe/SrTiO 3 show that the electrons in a wide range below the Fermi level (∆E~0.1-0.3 eV) participate in superconductivity [19]. A question may be asked: Which energy source causes this shift of spectral weight? The F-K phonon may be one of the options since the E Debye is~0.1 eV [31]. Would it be exchange coupling? The exchange-correlation energy E co of FeSe/SrTiO 3 is also~0.1-0.2 eV. However, we believe that the F-K phonon is the energy source to generate this shift of spectral weight in FeSe/SrTiO 3 . To support our argument, we revisit the ARPES results [18,20], where the bulk iron-based superconductors carrying E co~0 .1 eV displayed a shift in spectral weight at ∆E~30-60 meV below the Fermi level. If the shift is caused by the exchange-correlation energy, ∆E and E co should be comparable in the bulk iron-based superconductors, but this is not the case. If the exchange correlation energy is not the correct answer, we reinvestigate the magnitude of E Debye . Interestingly, the narrower range ∆E~30-60 meV is comparable to the Debye temperature [53,54] of bulk iron-based superconductors. With this, we believe that ∆E~E Debye is unlikely to be a coincidence. The shift of spectral weight in ARPES in iron-based superconductors is thus likely triggered by phonon-mediated processes. After revising the electron concentration in the superconducting state, our calculated T c is further increased to 91 K. We verified that the Coh factor is only reduced by~3% at E F -100 meV.

Would the Errors in T c Be Rescued by Nematicity and Spin-Orbital Coupling?
On the Fermi surface, a nematic order may be observed in various iron-based superconductors [52,55], and the electron-electron interaction should be influenced accordingly. Although our approach does not consider the nematic order, our approach averages the electron-phonon coupling between E F -E Debye and E F , which pales the contribution from the nematic order on the Fermi surface. The numerator of R g contains the average electronphonon scattering matrix in multienergy layers, where the Fermi energy is only one of them. Under these circumstances, the error of α PS from neglecting the nematic effect may be relatively small (the variation of T c enhanced by nematic phase in S-doped FeSe is just a few kelvins! If the nematic phase is encountered in our approach, this may help increase the calculated T c to 100 K; however, the T c calculation based on the concept of the nematic phase is still an open question), and our T c calculation should remain accurate. The spin-orbital coupling SO may be a reason for triggering the unusually high T c in FeSe/SrTiO 3 due to the heavy elements in SrTiO 3 [56,57]. If the effect of SO is taken into account, the calculated T c may move even closer to the experimental value. Additionally, another source of error in the T c of FeSe/SrTiO 3 may be caused by the thickness of SrTiO 3 used in the simulation. The theoretical T c of FeSe may increase as the thickness of SrTiO 3 is increased in the simulation in the future.

The Universal Theory of IBSC Remains an Open Question
The T c acquired by the "traditional combination of DFT functional" fails at high pressures, mainly because R g is excessively suppressed. To monitor electron-phonon coupling under pressure, the use of the "empirical combination of DFT functional" is a better choice. Although the accuracy of the GGA-PBE/PW91 functional may not be perfect, we empirically correct the numerical output value λ 111 11 directly via the AF Ising Hamiltonian and the two-channel model. On one hand, the two-channel model corrects the effect of the out-of-plane phonon at a low computational cost. On the other hand, the introduction of the induced xy potential in the electron-phonon calculation indirectly corrects the effect of the band diagram. The λ 111 11 is controlled by the band diagram, which contains the information about the effective mass. The numerator and denominator in R g are obtained from the same band diagram, so that the error due to the effective mass in these three nonheavy fermion superconductors can almost be cancelled.
It is still an open question which DFT functional is the best for iron-based superconductors. From an empirical point of view, the one-body Green's function and the dynamically screened Coulomb interaction (GW), or screened hybrid functional, are likely suitable for unconventional bismuthate and transition metal superconductors [58]. The modeling of the Hubbard potential in the GGA+U approach provides good agreement with the experimental results of BaFe 2 As 2 and LaFeAsO [38]. Since the electron-electron interaction in the iron-based superconductors is complicated, the use of the highly correlated DFT functional should be reasonable. However, the T c calculated with the screened hydrid functional HSE06 convinces us to use a different approach. We calculate the T c of these three materials by the HSE06 functional, which is a class of approximations to the exchange-correlation energy functional in density functional theory, which includes a part of the exact exchange item from the Hartree-Fock theory with the rest of the exchange-correlation energy from other sources [38]. However, the exchange-correlation energy considered by the screened hydrid functional HSE06 does not suit the NaFeAs, LiFeAs and FeSe materials, whose calculated T c values become less than 0.1 K. The more advanced approaches, such as GW or dynamical mean-field theory (DMFT), can simulate most of the electronic properties of bulk FeSe closer to the experimental values, but the major drawback is that the calculation of the electron-phonon coupling with these methods is based on a simplified deformation potential approximation, since electron-phonon coupling matrix elements are difficult to obtain [39].
The induced xy potential was rarely reported at the GGA level. If the channels where the out-of-plane phonon cannot be hidden are considered separately, the GGA functional is already good enough to generate the induced xy potential. If the lattice Fe moves orthogonally away from the xy plane in the iron-based superconductors, the electric charges in the xy plane are disturbed. Since the electronegativity of the tetrahedral atom (Se or As) is stronger, the electron will populate the FeSe or FeAs bonds more [21]. For example, when the Fe moves along the +z axis, the local electron density in the xy plane changes. The induced charges have two possible paths, i.e., the electrons are shifted either above or below the xy plane to the FeSe (or FeAs) bond [21]. However, the upward displacement of the Fe atom, which emits the electric field, confines the electrons more covalently in the upper tetrahedral region. The more covalently bonded FeSe (or FeAs) interaction allows electrons to move out of the FeSe or (FeAs) bond below the plane [21]. A charge fluctuation is created and generates the induced xy potential. Since the out-of-plane phonon is simulated by the two-channel model, the occurrence of the induced xy potential at the GGA level means that the two-channel model has already taken the AF into account.
The McMillian formula takes into account the distribution of electrons in the form of a hyperbolic tangent (tanh) function across the Fermi level [40]. At finite temperature, the Fermi-Dirac statistics fit the shape of the hyperbolic tangent function with the mean occu-pation number f (E F ) = 0.5. For example, elemental aluminum holds the superconducting transition temperature at 1.
In addition, the offset f (E F − E Debye ) − f (E F + E Debye ) of elemental tin is 0.0028 at~3 K. The McMillian formula provides the theoretical T c of aluminum and tin correctly with the tiny offsets of 0.0056 and 0.0028, respectively. The relevant electrons in the studied superconductors may be located in the energy range between E F − E Debye and E F + E Debye , but their offsets f (E F − E Debye ) − f (E F + E Debye ) at low temperatures are as small as~0.005. If f (E F − E Debye ) − f (E F + E Debye ) in the iron-based superconductors are comparable to BCS superconductors, the numerical error due to the fitting of the relevant electrons indicated by the energy range we extracted from ARPES data as input in the McMillian formula and the Eliashberg function may not be obvious. If the APRES factor (R g ) is introduced in a narrow energy range below the Fermi level, it fits even better with the tanh function. Furthermore, the AF Ising model shows that the energy of the spin fluctuations is smaller than the Debye energy, and hence, the maximum integral in the McMillian derivation [40] cannot exceed the Debye temperature. Finally, none of the amplified electron-phonon couplings exceed the limit of the straight-line fit for determining the empirical parameters [40]. Therefore, the McMillian formula becomes applicable in these three iron-based superconductors.
After we consider all electrons taking part in iron-based superconductivity between E F and E F -E D , the calculated T c of the above samples are much closer to experimental values. We thus suggest that given the relatively high transition temperatures of Fe-based superconductors at which a considerable amount of high energy phonons are excited, it is absolutely required to consider the entire energy range of electrons that can scatter up to the Fermi energy through these phonons, in contrast to the traditional low-T c approaches, where the electronic density of the states at the Fermi level can be used as an approximation. For a proposed theory of iron-based superconductors to be deemed incorrect, an unified theory of iron-based superconductors would need to have already existed. However, what is the unified theory of iron-based superconductor? It is still an open question. Despite our algorithm producing reasonable theoretical T c for these four samples, this article only combines several proposed mechanisms of IBSCs instead of presenting a comprehensive theory. However, our research provides optimism for scientists that accurate T c calculations in iron-based superconductors may be possible. It is crucial to conduct further theoretical work to develop a unified theory of iron-based superconductors that can accurately predict the theoretical T c of all iron-based superconductors.

Conclusions
After revising the superconducting electron concentration in the McMillan T c formula, we could show that when the conduction electrons interact with local Fe moments in Fe-based superconductors, the coexistence of superconductivity with local fluctuating antiferromagnetism together with the abnormal lattice vibration, which can lead to an enormous increase in the electron-phonon coupling, is sufficient to predict the high T c values. Our ab initio approach can generate theoretical T c values of NaFeAs, LiFeAs and FeSe close to the experimental values. When the model is applied to monolayered FeSe on a SrTiO 3 substrate, we find that the interfacial phonons are of major importance to explain the high-temperature superconductivity.