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

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

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

crystallography and the key role ascribed to σ bands and superconductivity [12, 34]. DFT calculations for a compositional suite, or structure type, produce EBSs that show, in general, similar formats even when two different functional approximations are used [26]. A typical outcome for MgB2 using the LDA approximation in the CASTEP module of Materials Studio using a k-grid value of 0.008 Å−1 is shown in **Figure 1**. These band structures convey useful information for elucidation of potential for superconductivity. For example, the σ bands appear as approximate inverted parabolas (in red and blue lines; green dotted box) near the Γ center point and cross the Fermi level on either side of Γ (**Figure 1**). These bands display a strong electron–phonon coupling to the E2g phonon modes and are implicated in superconductivity for MgB2 *via* both theoretical and experimental analyses [35–38].

**Figure 1.**

*Electronic band structure for MgB2 calculated with the LDA approximation for k = 0.008 Å−1 using the CASTEP module of materials studio [9]. The green boxes enclose sections of* σ *bands.*

### **Figure 2.**

*Schematic showing the direct relationship between* σ *bands crossing the Fermi level around* Γ *and the topology of the FS.*

**117**

for MgB2.

**Figure 3.**

*Insights from Systematic DFT Calculations on Superconductors*

For MgB2, the relationship of these σ bands near the Fermi level on either side of Γ is shown in the schematic in **Figure 2**. The reciprocal space projection of degenerate σ bands at the Fermi level correspond to the three-dimensional reciprocal space representations of two FSs at Γ parallel with the *k*z direction. That is, except for the direction along *k*z, the band is a 2D projection at *k*z = 0 of the 3D Fermi surface representation. The region between these two FSs is sensitive to the reciprocal space projection(s) at the Fermi level and, as noted in previous work, is key to the super-

*Electronic band structure for MgB2 calculated with the LDA approximation for k-grid values 0.02 Å−1 (red), 0.04 Å−1 (blue) and 0.06 Å−1 (black) using the CASTEP module of materials studio. Notice the substantial* 

*differences in band energies, particularly in regions associated with the* σ *bands.*

Variations in the calculated energies of specific bands, for example in MgB2 where electron–phonon coupling is predominantly linked to the σ bands [17, 39, 40], are strongly influenced by the k-grid value used in DFT computations. **Figure 3** demonstrates the effect of k-grid value, or the sensitivity of DFT calculations, on the EBS for MgB2 using the LDA functional for a series of Δk values 0.02 Å−1, 0.04 Å−1 and 0.06 Å−1. The k-grid value affects calculated energies for bands near the Fermi level particularly those σ bands associated with superconductivity highlighted in **Figure 1** for MgB2. The differences in energy at Γ or A between calculations range from tens of meV to hundreds of meV for three different k-grid values as shown in **Figure 3**. In **Table 1**, we show substantial meV shifts in enthalpy and EF for MgB2 calculated at different k-grid values using the same functional and the same ultra-fine tolerance for geometry optimization convergence. For MgB2, **Table 1** shows the difference in energy, ΔEv (in eV), between the Fermi level and the vertex of the parabola at Γ for different values of Δk. Differences in lattice parameters (*i.e.* ~0.01 Å), enthalpy values (*i.e.* ~10–20 meV) and EF values (*i.e.* ~200 meV) are evident for geometry optimized calculations with different k-grids. **Table 1** also shows that as the Δk value is reduced, values for enthalpy achieve a consistent value

We also show calculated values for B-doped diamond using different k-grid values in **Table 1**. In this case, k-grid intervals are smaller (0.005 Å−1 < Δk < 0.020 Å−1) than those used for MgB2 with corresponding smaller shifts in enthalpy and EF. This lower magnitude impact of the k-grid is in part due to fewer degrees of freedom (*e.g.* cubic symmetry compared to hexagonal) and a significantly lower value for Tc [32, 33], with corresponding FSs in closer reciprocal space proximity. Nevertheless, the EF value

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

conducting mechanism in MgB2 [27, 28].

*Insights from Systematic DFT Calculations on Superconductors DOI: http://dx.doi.org/10.5772/intechopen.96960*

### **Figure 3.**

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

ductivity for MgB2 *via* both theoretical and experimental analyses [35–38].

*Electronic band structure for MgB2 calculated with the LDA approximation for k = 0.008 Å−1 using the* 

*Schematic showing the direct relationship between* σ *bands crossing the Fermi level around* Γ *and the topology* 

*CASTEP module of materials studio [9]. The green boxes enclose sections of* σ *bands.*

crystallography and the key role ascribed to σ bands and superconductivity [12, 34]. DFT calculations for a compositional suite, or structure type, produce EBSs that show, in general, similar formats even when two different functional approximations are used [26]. A typical outcome for MgB2 using the LDA approximation in the CASTEP module of Materials Studio using a k-grid value of 0.008 Å−1 is shown in **Figure 1**. These band structures convey useful information for elucidation of potential for superconductivity. For example, the σ bands appear as approximate inverted parabolas (in red and blue lines; green dotted box) near the Γ center point and cross the Fermi level on either side of Γ (**Figure 1**). These bands display a strong electron–phonon coupling to the E2g phonon modes and are implicated in supercon-

**116**

**Figure 2.**

*of the FS.*

**Figure 1.**

*Electronic band structure for MgB2 calculated with the LDA approximation for k-grid values 0.02 Å−1 (red), 0.04 Å−1 (blue) and 0.06 Å−1 (black) using the CASTEP module of materials studio. Notice the substantial differences in band energies, particularly in regions associated with the* σ *bands.*

For MgB2, the relationship of these σ bands near the Fermi level on either side of Γ is shown in the schematic in **Figure 2**. The reciprocal space projection of degenerate σ bands at the Fermi level correspond to the three-dimensional reciprocal space representations of two FSs at Γ parallel with the *k*z direction. That is, except for the direction along *k*z, the band is a 2D projection at *k*z = 0 of the 3D Fermi surface representation. The region between these two FSs is sensitive to the reciprocal space projection(s) at the Fermi level and, as noted in previous work, is key to the superconducting mechanism in MgB2 [27, 28].

Variations in the calculated energies of specific bands, for example in MgB2 where electron–phonon coupling is predominantly linked to the σ bands [17, 39, 40], are strongly influenced by the k-grid value used in DFT computations. **Figure 3** demonstrates the effect of k-grid value, or the sensitivity of DFT calculations, on the EBS for MgB2 using the LDA functional for a series of Δk values 0.02 Å−1, 0.04 Å−1 and 0.06 Å−1. The k-grid value affects calculated energies for bands near the Fermi level particularly those σ bands associated with superconductivity highlighted in **Figure 1** for MgB2. The differences in energy at Γ or A between calculations range from tens of meV to hundreds of meV for three different k-grid values as shown in **Figure 3**.

In **Table 1**, we show substantial meV shifts in enthalpy and EF for MgB2 calculated at different k-grid values using the same functional and the same ultra-fine tolerance for geometry optimization convergence. For MgB2, **Table 1** shows the difference in energy, ΔEv (in eV), between the Fermi level and the vertex of the parabola at Γ for different values of Δk. Differences in lattice parameters (*i.e.* ~0.01 Å), enthalpy values (*i.e.* ~10–20 meV) and EF values (*i.e.* ~200 meV) are evident for geometry optimized calculations with different k-grids. **Table 1** also shows that as the Δk value is reduced, values for enthalpy achieve a consistent value for MgB2.

We also show calculated values for B-doped diamond using different k-grid values in **Table 1**. In this case, k-grid intervals are smaller (0.005 Å−1 < Δk < 0.020 Å−1) than those used for MgB2 with corresponding smaller shifts in enthalpy and EF. This lower magnitude impact of the k-grid is in part due to fewer degrees of freedom (*e.g.* cubic symmetry compared to hexagonal) and a significantly lower value for Tc [32, 33], with corresponding FSs in closer reciprocal space proximity. Nevertheless, the EF value


**Table 1.**

*Parameters calculated for MgB2 and for B-doped diamond.*

differs by ~35 meV and the difference in energy to the vertex of the band at Γ (*i.e.* ΔEv) differs by up to ~35 meV depending on the Δk value.

We show a more detailed systematic comparison of calculated enthalpies as a function of the k-grid value for MgB2 in **Figure 4**. As noted above, all calculations are converged to the same ultra-fine criteria, or tolerance, where self-consistency is achieved. The variability of results shown in **Figure 4** is due to the discreteness of functions and values used to derive the full solution of the Schroedinger equation. The variability is not due to lack of convergence which in all cases is defined in Section 2 above.

**Figure 4a** shows that the value of enthalpy for MgB2 oscillates around a consistent minimal value of −1,748.502 eV as the k-grid value is decreased to <0.015 Å−1. The context for this variation in enthalpy is shown in **Figure 4b** where the k-grid value is extended to 0.2 Å−1– a value that has been used in some DFT calculations as criterion for machine learning algorithms [41]. At these higher values for Δk, enthalpy calculations do not provide useful information on subtle structural

### **Figure 4.**

*Systematic comparison of calculated enthalpies for MgB2 (a) for fine values of k-grid (i.e. < 0.03 Å−1) and (b) for coarser grid values including those utilized for machine learning searches (arrowed) of materials databases [41]. Enthalpies shown in Figure 4(a) are reproduced in (b) for reference. The lightly shaded region in Figure 2(b) delineates the k-grid values used for EBS calculation in Figure 3.*

**119**

**Figure 5.**

*superconductivity [26].*

*Insights from Systematic DFT Calculations on Superconductors*

variations due to superlattices or of order/disorder. For example, we have shown using DFT calculations with appropriate k-grid values for (Mg1-xAlx)B2 that ordered motifs with adjacent Al-layers are thermodynamically favored by ~0.15 eV over more complex, disordered configuration(s) [42]. A discrepancy of ~0.2 eV due to incorrect choice of k-grid value (**Figure 4**) does not enable such distinction to be

In a dynamic system, other factors may also influence the position of key electronic bands with respect to the Fermi level. For example, atoms in all solids at temperatures above absolute zero vibrate [43] and in some cases, the resulting phonons may align with specific crystallographic real space features such as inter-atom bonds. This circumstance occurs for MgB2 in which one of the dominant E2g phonon modes – shown to be intimately involved in electron–phonon coupling at the onset of superconductivity [17, 34] – aligns with B–B bonds in the *ab* plane [36]. Using DFT, we can model the effect of bond deformation along specific planar orientations by displacing atoms from their structural equilibrium positions consistent with the direction of the E2g phonons [26, 28]. Under different extents of displacement, electron density distributions along the B–B bond and the corresponding

**Figure 5** shows the effect on the EBS for MgB2 of atom displacement along the

B–B bond by ~0.6% (*i.e.* a shift of ~0.063 Å) from equilibrium [28]. **Figure 5** shows that the E2g phonon, which is degenerate at Γ with a peak parabola at 398 meV, splits into two separate non-degenerate bands above and below the equilibrium condition. The upper σ band - which we attribute to the heavy effective mass - has a calculated energy 813 meV above the Fermi level. Thus, parallel or nearly parallel FSs attributable to the superconducting condition [44], no longer exist with a 0.6% shift in atom position(s) [28]. A shift of atom position(s) is also reflected in the form and energy of key phonon modes in the corresponding PD for MgB2 [26]. An atom displacement of 0.6% along B–B for MgB2 is not unrea-

*Enlarged view of EBS around* Γ *for MgB2 using LDA functional and* Δ*k = 0.018 Å−1 showing (a) degenerate*  σ *bands at equilibrium (blue lines) and (b) after atom displacement Dx = 0.063 Å (red lines) along the E2g mode direction; note the split of* σ *bands causing loss of degeneracy which coincides with loss of* 

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

**3.2 Influence of atom displacements**

EBS, can be determined [28, 36].

sonable at temperatures >40 K [28].

made with confidence.

variations due to superlattices or of order/disorder. For example, we have shown using DFT calculations with appropriate k-grid values for (Mg1-xAlx)B2 that ordered motifs with adjacent Al-layers are thermodynamically favored by ~0.15 eV over more complex, disordered configuration(s) [42]. A discrepancy of ~0.2 eV due to incorrect choice of k-grid value (**Figure 4**) does not enable such distinction to be made with confidence.
