**Abstract**

We present three systematic approaches to use of Density Functional Theory (DFT) for interpretation and prediction of superconductivity in new or existing materials. These approaches do not require estimates of free parameters but utilize standard input values that significantly influence computational resolution of reciprocal space Fermi surfaces and that reduce the meV-scale energy variability of calculated values. Systematic calculations on conventional superconductors show that to attain a level of resolution comparable to the energy gap, two key parameters, Δk and the cut-off energy, must be optimized for a specific compound. The optimal level of resolution is achieved with k-grids smaller than the minimum reciprocal space separation between key parallel Fermi surfaces. These approaches enable estimates of superconducting properties including the transition temperature (Tc) *via* (i) measurement of the equivalent thermal energy of a phonon anomaly (if present), (ii) the distribution of electrons and effect on Fermi energy (EF) when subjected to a deformation potential and (iii) use of parabolic, or higher order quartic, approximations for key electronic bands implicated in electron–phonon interactions. We demonstrate these approaches for the conventional superconductors MgB2, metal substituted MgB2 and boron-doped diamond.

**Keywords:** Fermi energy, Fermi level, Fermi surface, reciprocal space, density functional theory, parabolic equations, phonon dispersions, transition temperature, magnesium diboride

## **1. Introduction**

Design and synthesis of new materials requires translation of reciprocal space detail in electronic band structures (EBSs) and phonon dispersions (PDs) to equivalent real space representations [1–4]. We have shown that for conventional superconductors (SCs), the format and depth of modes in PDs associated with a Kohn anomaly are strongly influenced by the computational resolution of DFT models [5]. Our view is that EBS calculations, performed with appropriate resolution, may also provide critical information on the superconducting gap, which for many SC materials, is in the meV range.

The Fermi level, Fermi energy (EF), and the density of nearly free-electron carriers calculated by DFT are key values that, to date, have been variously reported with differences of many hundreds of meV for the same compound. For example, the value of EF for a well studied compound such as MgB2 has been variously reported as "several eV" [6], 0.55 eV or 0.122 eV [7], and more recently, as 0.428 eV [8]. In comparison, many publications on MgB2 and software packages such as

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 mathematical rigor is required for implementation.

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 computational resolution [26].

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

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

**115**

*Insights from Systematic DFT Calculations on Superconductors*

maximum displacements of less than 5 × 10−4 Å.

estimates the superconducting gap and assists identification of valence bands and the origin for EF. In combination, these three approaches provide reliable property predictions for unknown, or theoretical, structures for materials researchers.

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

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

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

We provide outputs from a series of *ab initio* DFT calculations on two SC compounds with substantially different experimentally determined Tc values (*i.e.* MgB2 Tc ~ 39.5 K; B-doped diamond Tc ~ 4.0–7.5 K depending on level of doping) [31–33]. For both compounds, when the value of k-grid is varied in the examples below, all other parameters are maintained the same for all calculations. A range of k-grid values are exemplified in order to highlight differences in sensitivity of EBS and PD outputs for SC compounds. These examples highlight the key role computational

We have been intrigued by the potential to directly determine the superconducting gap energy for a BCS SC using an appropriate resolution EBS. In this regard, MgB2 offers good opportunity to evaluate this potential due to well defined

*DOI: http://dx.doi.org/10.5772/intechopen.96960*

**2. Calculation methods**

as identified in the text.

different branches of the EBS.

**3. Computational resolution**

resolution can play with interpretation of SC properties.

**3.1 Band structure – variation with k-grid**

estimates the superconducting gap and assists identification of valence bands and the origin for EF. In combination, these three approaches provide reliable property predictions for unknown, or theoretical, structures for materials researchers.
