Electronic Structure analysis of Si using DFT (Quantum ESPRESSO, PBE)! This project aims at proving the claimed underestimation of PBE exchange correlation functional in calculating the observable 'band-gap'.
Perform a complete Density Functional Theory (DFT) workflow for bulk silicon to:
- obtain the electronic ground state
- optimize the crystal structure
- compute band structure and density of states (DOS)
- extract and interpret the band gap
- analyze orbital contributions via PDOS
- investigate bonding using charge density
Solved the Kohn–Sham equations:
Ĥ ψ_i = ε_i ψ_i
Result:
- Converged electron density ρ(r)
- Total energy: −93.45138 Ry
Minimized total energy with respect to atomic positions and cell:
F_i = −∂E/∂R_i
σ = ∂E/∂cell
Result:
- Lattice parameter ≈ 5.47 Å
- Negligible forces and stress
Computed E(k) along high-symmetry path:
L → Γ → X → W → K
Concepts:
- Bloch theorem
- Brillouin zone sampling
Eg = E_CBM − E_VBM
Results:
- VBM ≈ 0 eV (Γ)
- CBM ≈ 0.609 eV (Γ–X)
- Eg ≈ 0.61 eV (indirect)
Computed using NSCF + dos.x.
Observations:
- Valence band: populated states below 0 eV
- Band gap: D(E) ≈ 0 region
- Conduction band: states above ~0.6 eV
Result:
- DOS confirms Eg ≈ 0.6 eV
Computed using projwfc.x.
Key findings:
- Lower valence region → stronger s-character
- Upper valence + conduction → dominant p-character
- Significant orbital mixing across energy range
Löwdin charges per Si atom:
- total ≈ 3.96
- s ≈ 1.17
- p ≈ 2.79
Spilling parameter:
- **0.009
Interpretation:
- Electronic states are hybridized
- Consistent with sp³ bonding
- Charge density forms continuous, non-spherical regions
- Density extends between neighboring atoms
- Plane passes through Si–Si bond
- Shows continuous charge density between atoms
Interpretation:
- electrons are shared between atoms
- bonding is directional and covalent
Figure: Electronic band structure of silicon along the high-symmetry path L → Γ → X → W → K. The valence band maximum occurs at Γ, while the conduction band minimum lies along Γ–X, confirming an indirect band gap of ~0.61 eV.
Figure: Total density of states (DOS) of silicon. A clear energy region with zero states is observed between the valence and conduction bands, confirming a band gap of ~0.6 eV consistent with the band structure.
Figure: Projected density of states showing s and p orbital contributions. The lower valence region contains stronger s character, while the upper valence and conduction regions are dominated by p states, indicating sp³ hybridization.
Figure: Isosurface of the total charge density of silicon. The continuous electron density between neighboring atoms indicates shared electrons and directional covalent bonding in the diamond cubic structure.
Figure: Two-dimensional slice of the charge density through a Si–Si bond. The continuous electron density between atoms forms a charge bridge, providing direct real-space evidence of covalent bonding and sp³ hybridization.
- Silicon is an indirect band gap semiconductor
- PBE underestimates the experimental band gap (~1.1 eV)
- PDOS reveals s–p hybridization
- Charge density confirms directional covalent bonding
Combined analysis:
Energy-space (PDOS) + real-space (charge density) → consistent sp³ hybridization picture
- inputs/ → QE input files
- outputs/ → raw outputs
- pseudo/ → pseudopotentials
- results/ → plots and figures, detailed results
- scripts/ → plotting scripts
- Quantum ESPRESSO (pw.x, bands.x, dos.x, projwfc.x, pp.x)
- Python (NumPy, Matplotlib)
- XCrySDen (visualization)
- Bash utilities
- SCF convergence and numerical stability
- Structural optimization using BFGS
- k-point sampling in reciprocal space
- Interpretation of band structure and DOS
- Orbital-resolved analysis via PDOS
- Real-space bonding analysis via charge density
- PBE underestimates band gap
- PDOS depends on projection basis
- Charge density is total (not difference density)
For step-by-step analysis and interpretation:
- Band Structure
- Density of States (DOS)
- Projected Density of States (PDOS)
- Charge Density & Bonding Analysis
Md. Saidul Islam