Density functional theory has become an indispensable computational tool for investigating the electronic, mechanical, and structural properties of graphene and other two-dimensional nanomaterials. By solving the Kohn-Sham equations, DFT provides insights into quantum mechanical phenomena at the atomic scale, enabling accurate predictions of material behavior without empirical fitting. The method's ability to handle periodic boundary conditions makes it particularly suitable for studying crystalline 2D systems where long-range order dominates material properties.

The electronic band structure of graphene reveals its unique Dirac cone formation at the K-point in the Brillouin zone. DFT calculations demonstrate how the linear dispersion relation around this point leads to massless Dirac fermion behavior, with Fermi velocities approaching 10^6 m/s. These simulations accurately reproduce the touching of valence and conduction bands at the Dirac point, showing zero bandgap characteristics. For bilayer graphene, DFT predicts the opening of a tunable bandgap under perpendicular electric fields due to broken inversion symmetry, with calculated gaps reaching 250 meV for field strengths around 1 V/nm. The method captures the transition from parabolic to linear dispersion as layer stacking changes from AA to AB configurations.

Defect engineering studies using DFT reveal how atomic-scale modifications alter graphene's properties. Single vacancies introduce mid-gap states 0.7 eV below the Dirac point, while Stone-Wales defects modify the local density of states without creating gap states. DFT simulations of nitrogen doping show that pyridinic configurations shift the Fermi level 0.5 eV below the Dirac point, whereas graphitic doping creates n-type behavior with the Fermi level moving 0.3 eV above. Sulfur-doped graphene models indicate strong spin polarization around defect sites, with magnetic moments reaching 0.6 μB per sulfur atom. These calculations guide the design of doped graphene systems for specific electronic applications.

Van der Waals interactions in layered materials present a challenge for conventional DFT functionals. The development of non-local correlation functionals such as vdW-DF and DFT-D methods has enabled accurate modeling of interlayer bonding. For bilayer graphene, these approaches reproduce the experimental interlayer spacing of 3.35 Å with binding energies of 20-25 meV/atom. In transition metal dichalcogenide heterostructures like MoS2/WS2, DFT-D3 calculations predict interlayer distances within 0.1 Å of experimental values and correctly identify stacking-dependent band alignment. The method captures the layer-dependent bandgap evolution in MoS2, showing direct-to-indirect transitions from monolayer (1.8 eV) to bulk (1.2 eV).

Stress-strain response analysis through DFT reveals the mechanical limits of 2D materials. Graphene exhibits linear elasticity up to 15% strain, with DFT-calculated Young's modulus of 1.0 TPa and intrinsic strength of 130 GPa. The method predicts anisotropic mechanical behavior in phosphorene, with armchair direction stiffness (166 GPa) exceeding zigzag direction values (44 GPa). Nonlinear effects become significant beyond 5% strain, where DFT shows bond angle distortion dominating over bond stretching in energy storage. Fracture patterns in defective graphene sheets demonstrate stress concentration factors of 2-3 around vacancy clusters, reducing failure strains by 40% compared to pristine sheets.

Heterostructure interface studies employ DFT to probe interlayer charge transfer and band alignment. In graphene/hexagonal boron nitride systems, calculations reveal a 0.3 eV electron transfer from graphene to hBN, creating a small interfacial dipole. Moiré superlattices in twisted bilayer graphene show DFT-predicted flat bands at magic angles (1.1°), with bandwidths narrowing below 10 meV. For vertical heterostructures like WSe2/MoS2, DFT identifies type-II band alignment with 0.4 eV conduction band offset, enabling charge separation. The method quantifies interlayer coupling strengths, showing 50 meV hybridization energies in graphene/MoS2 systems.

Doping effects in 2D materials beyond graphene are extensively studied with DFT. Boron nitride monolayers with carbon impurities exhibit deep acceptor levels 1.5 eV above the valence band maximum. Transition metal doping in MoS2 introduces mid-gap states that enhance catalytic activity, with Co-doped systems showing hydrogen adsorption free energies within 0.1 eV of ideal catalysts. DFT predicts that phosphorus substitution in ReS2 creates p-type conductivity with hole effective masses of 0.8 m0. For silicene, alkali metal adsorption induces semiconductor-to-metal transitions through charge transfer doping, shifting the Dirac cone 1.2 eV below the Fermi level.

Phase transitions in 2D materials are investigated through DFT-based nudged elastic band calculations. The energy barrier for the 2H to 1T' transition in MoTe2 is calculated as 0.7 eV per formula unit, with the intermediate state stabilized by tensile strain. In black phosphorus, DFT predicts a reversible orthorhombic-to-berlinite transition under 8 GPa pressure, accompanied by a semiconductor-to-metal transition. The method identifies critical strains for phase transformations in ReS2, showing layer-dependent transition pressures decreasing from 15 GPa (monolayer) to 8 GPa (bulk).

Thermal properties of 2D nanomaterials are accessible through DFT-based lattice dynamics. Phonon dispersion calculations for graphene reproduce the out-of-plane acoustic (ZA) mode's quadratic dispersion, crucial for understanding its high thermal conductivity (3000-5000 W/mK). The method identifies anomalous thermal expansion in graphene, with negative coefficients below 700 K due to ZA mode dominance. For hBN, DFT predicts anisotropic thermal conductivity with armchair direction values (600 W/mK) exceeding zigzag directions (400 W/mK). Temperature-dependent studies employ density functional perturbation theory to track phonon softening effects that reduce thermal conductivity by 30% between 100-500 K.

DFT investigations of edge states in nanoribbons reveal width-dependent electronic properties. Armchair graphene nanoribbons show bandgap oscillations with period 3, where DFT-calculated gaps follow 1/W scaling (W=width). Zigzag ribbons exhibit spin-polarized edge states with exchange splittings of 0.2-0.3 eV, stabilized by Hubbard U corrections. Transition metal dichalcogenide nanoribbons display edge reconstruction effects, with Mo-terminated edges in MoS2 nanoribbons showing metallic character regardless of width. The method quantifies edge formation energies, showing sulfur-rich edges are energetically favorable under most conditions.

Advanced DFT approaches address electronic correlation effects in 2D materials. GW corrections to bandgaps improve agreement with experiment, increasing MoS2 monolayer gaps from 1.8 eV (DFT) to 2.5 eV (GW). For magnetic systems like CrI3, DFT+U calculations reproduce the experimentally observed 45 meV bandgap and predict layer-dependent magnetic ordering. Time-dependent DFT studies of excitonic effects in WS2 show binding energies exceeding 0.7 eV due to reduced dielectric screening in 2D. These methods extend DFT's predictive power to strongly correlated phenomena in reduced dimensions.

The continuous development of exchange-correlation functionals and computational algorithms ensures DFT remains at the forefront of 2D materials research. Recent progress in machine-learning accelerated DFT and high-throughput screening approaches enables rapid exploration of novel 2D systems while maintaining quantum mechanical accuracy. These computational tools provide essential guidance for understanding and designing next-generation nanomaterials with tailored electronic, optical, and mechanical properties.