**2. Calculation methods**

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

mathematical rigor is required for implementation.

ment to approximations of the Eliashberg model [19, 20].

computational resolution [26].

CASTEP and ADF consistently report a value for EF of ~8.4 eV [9–11]. Calculated variations of this magnitude have garnered limited attention [8] for SCs due to an underlying assumption that EF/kθ > > 1 (where k is Boltzman's constant and θ is the Debye temperature). Due to this assumption and the deceptive influence of average values for phonon frequencies, a value for EF is not considered in the simplified McMillan version of the Eliashberg model for superconductivity [12]. However, as noted by Malik [8], equations that explicitly include EF and/or critical current (jo) values may provide clues on how to increase or modify Tc. Malik suggests that regardless of physical attributes, SCs may be distinguished by their values of EF [8]. Approximations to the Eliashberg model that minimize computational cost require estimates of the electron–phonon interaction, λ, and the Coulomb strength, μ\* [13, 14]. For many conventional SCs, these parameters are limited to a narrow range of values and provide reasonable estimates for superconducting properties of known materials [15–18]. More recently, Sanna *et al*. [19] show that a fully *ab initio* Eliashberg approach provides good estimates of superconducting properties including Tc for a range of compounds without invoking estimates of free parameters such as μ\*. This work by Sanna *et al*. [19], as well as development and use of the Superconducting Density Functional Theory (SCDFT) [20, 21], are elegant computational approaches to the Eliashberg model that have successfully predicted superconducting properties of new materials such as H3S [19, 22]. Nevertheless, these codes are not universally available to materials researchers particularly if deep

In our search for new SCs, we have evaluated the computational resolution and electron–phonon detail possible with DFT codes readily available in well-known software packages such as CASTEP or Quantum Espresso, to name a couple of examples. Using MgB2 and similar Bardeen-Cooper-Schrieffer (BCS) compounds, we have systematically explored the sensitivity and use of PD and EBS constructs to calculate key superconducting properties without recourse to free parameter estimates or modification of functionals. We initially explored use of a phonon anomaly to estimate Tc [23, 24]; an approach that appears effective for strong phonon mediated superconductivity including for metal substituted MgB2 [25]. In other work, we extended this systematic approach to evaluate PDs and EBSs for a wide range of metal diboride compounds (*e.g.* ScB2, YB2, TiB2) using DFT at appropriate

More recently, we have examined the link between PDs and EBSs and, in particular, the topology of the Fermi Surface (FS) with pressure [27] and the change in electron density distributions as MgB2 transitions to the superconducting state [28]. In both approaches, we are able to confirm experimentally determined superconducting properties for a range of conventional (BCS) compounds and then, to predict Tc for new metal substituted analogues of MgB2 [23, 25]. These approaches are, in essence, empirical methods, which systematically identify regular dispersion patterns in calculated PDs and EBSs, based entirely on accepted codification of the DFT [9, 10] and, equally, a clear understanding of input parameter limitations that determine computational resolution. Thus, this use of DFT software, underpinned by elegant formalism and constructs by Kohn and colleagues [29, 30], is a comple-

In this work, we show why computational resolution for DFT models of EBSs and PDs for conventional SCs is critical. In addition, we delineate a third approach to estimate the superconducting gap using parabolic, or higher order quartic, approximations to key bands in the EBS. This approach requires examination of an extended Brillouin zone (BZ) schema and demands lower computational cost compared to equivalent PD calculations for similar outcome(s). When applied to MgB2, at sufficiently fine k-grid and reciprocal space cut-off, this approach directly

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EBS calculations are undertaken using DFT as implemented in the Cambridge Serial Total Energy Package (CASTEP) of Materials Studio (MS) 2017 and 2018 [9, 10]. All structures are optimized for geometry, including cell parameters, starting with crystal information files (.cif) available in standard databases. In general, the local density approximation (LDA) and generalized gradient approximation (GGA), with norm-conserving pseudopotentials, are used in DFT calculations. The typical setup for calculations uses a k-grid ranging from 0.06 Å−1 to 0.02 Å−1 or smaller, with a plane wave basis set cut-off of 990 eV, ultra-fine (or better) customized setup to ensure total energy convergence of less than 5 × 10−6 eV/atom, a maximum force of less than 0.01 eV/Å, a maximum stress of less than 0.02 GPa and maximum displacements of less than 5 × 10−4 Å.

All outputs meet convergence criteria at the same fine tolerance level for geometry optimization. The effects of input parameters to DFT calculations described in this work are not related to differences in calculation convergence, but critically, are due to the discreteness, or the finite number of the reciprocal space points, used to select plane waves as basis functions. To illustrate particular points, we also vary specific parameters such as basis set cut-off values, Δk values or software versions as identified in the text.

We also perform numerical interpolations of EBSs to validate higher order trends and to delineate fine structure in computed outcomes. To obtain parabolic, or higher order polynomial, approximations to the electronic bands, the DFT calculated data from MS is exported in csv/excel format. For MgB2, sections of particular bands in the energy range − 14 to 4 eV along the Γ-M (and Γ-K) directions are selected and mirrored across the vertical axis at Γ. Individual parabolic, or higher order quartic, trendline fittings are obtained and used to overlay for comparison with the (periodically repeated) extended BZ scheme of the DFT calculated EBS. Effective masses are also calculated and evaluated for parabolic approximations of different branches of the EBS.
