Theoretical modeling of hexagonal boron nitride (hBN) plays a crucial role in understanding its fundamental properties and guiding experimental investigations. As a layered material with a structure analogous to graphene, hBN exhibits unique electronic, thermal, and mechanical characteristics that make it valuable for various applications. Computational approaches, including density functional theory (DFT), molecular dynamics (MD), and continuum models, have been extensively employed to predict and analyze these properties. This article explores these theoretical frameworks and their predictions, comparing them with experimental observations where applicable.

Density functional theory (DFT) is one of the most widely used methods for studying the electronic structure of hBN. DFT simulations provide insights into the bandgap, density of states, and charge distribution in hBN. The material is known to exhibit a wide bandgap, typically calculated to be around 5.9 eV to 6.0 eV using the generalized gradient approximation (GGA) or hybrid functionals. This large bandgap makes hBN an excellent insulator, which aligns well with experimental measurements using techniques such as ultraviolet photoelectron spectroscopy (UPS) and optical absorption spectroscopy. DFT also reveals that hBN has a non-magnetic ground state, with boron and nitrogen atoms contributing differently to the valence and conduction bands due to their electronegativity differences. Additionally, DFT-based studies have examined the effects of defects, such as vacancies and substitutions, on electronic properties. For instance, boron vacancies introduce acceptor states near the valence band edge, while nitrogen vacancies create donor states near the conduction band edge, as confirmed by scanning tunneling microscopy (STM) studies.

Molecular dynamics (MD) simulations complement DFT by providing dynamic and temperature-dependent insights into hBN's thermal and mechanical behavior. Classical MD, often employing Tersoff or reactive force fields, has been used to study thermal conductivity, phonon transport, and mechanical deformation. hBN exhibits high in-plane thermal conductivity, with simulations predicting values ranging from 250 to 400 W/mK at room temperature, depending on sample quality and isotopic purity. These predictions are consistent with experimental measurements using time-domain thermoreflectance (TDTR) and Raman thermometry. MD simulations also reveal the anisotropic nature of thermal transport in hBN, with out-of-plane conductivity being significantly lower due to weak van der Waals interactions between layers. Mechanical properties, such as Young's modulus and tensile strength, have also been investigated using MD. The in-plane Young's modulus is estimated to be approximately 800 GPa, while the out-of-plane modulus is much lower, around 30 GPa. These values agree well with nanoindentation and atomic force microscopy (AFM) experiments.

Continuum models provide a macroscopic perspective on hBN's behavior, particularly in multilayer systems or large-scale structures. Elastic continuum theory, for example, has been applied to study bending rigidity, interlayer shear, and wrinkling phenomena in hBN sheets. The bending rigidity of monolayer hBN is estimated to be about 1.2 eV, which is slightly higher than that of graphene due to its ionic character. Continuum approaches also model thermal expansion, predicting a negative thermal expansion coefficient in the in-plane direction at low temperatures, transitioning to positive values as temperature increases. Experimental measurements using X-ray diffraction (XRD) confirm this behavior, with the in-plane coefficient ranging from -2.9 × 10^-6 K^-1 at 100 K to +1.5 × 10^-6 K^-1 at 800 K. For multilayer hBN, continuum models describe interlayer interactions using van der Waals forces, explaining the material's lubricating properties and layer-dependent thermal conductivity reduction.

Theoretical models have also been employed to investigate defect dynamics and their impact on hBN's properties. For example, DFT and MD simulations show that grain boundaries and dislocations can significantly alter mechanical strength and thermal transport. Grain boundaries in hBN reduce thermal conductivity by introducing phonon scattering centers, with simulations predicting a 30-50% decrease compared to pristine samples. Experimentally, this is observed in polycrystalline hBN films, where thermal conductivity measurements show a strong dependence on grain size. Similarly, mechanical strength is affected by defect density, with MD simulations indicating a 20-40% reduction in tensile strength for highly defective samples, consistent with AFM-based mechanical testing.

Phonon dispersion and vibrational properties of hBN have been extensively studied using DFT-based lattice dynamics calculations. These simulations predict the existence of optical and acoustic phonon modes, including the high-frequency in-plane optical (IPO) mode at approximately 1370 cm^-1 and the out-of-plane optical (OPO) mode near 780 cm^-1. Raman spectroscopy experiments confirm these predictions, with observed peaks at 1366 cm^-1 (IPO) and 770 cm^-1 (OPO) for bulk hBN. The strong ionic character of hBN leads to large LO-TO splitting in the optical phonon branches, which is well-captured by DFT simulations using density functional perturbation theory (DFPT). These phonon properties are critical for understanding thermal transport, as phonon-phonon scattering rates calculated using perturbation theory reveal the dominance of three-phonon processes in hBN's thermal conductivity.

Theoretical models have also explored the effects of strain and external perturbations on hBN's properties. DFT calculations predict that uniaxial tensile strain can reduce the bandgap linearly, with a rate of approximately 0.1 eV per 1% strain. This trend has been corroborated by photoluminescence (PL) studies on strained hBN membranes. Similarly, electric field effects have been investigated, showing that perpendicular electric fields can induce a Stark effect in few-layer hBN, slightly modifying the bandgap and excitonic transitions. These predictions align with experimental observations in field-effect devices incorporating hBN as a dielectric or insulating layer.

In summary, theoretical modeling approaches, including DFT, MD, and continuum models, provide a comprehensive understanding of hBN's electronic, thermal, and mechanical properties. These simulations not only predict material behavior with reasonable accuracy but also guide experimental efforts by identifying key parameters and phenomena. The consistency between theoretical predictions and experimental data underscores the reliability of these computational tools in advancing the study of hBN and related layered materials. Future refinements in modeling techniques, such as incorporating higher-level electron correlations or more accurate interatomic potentials, will further enhance the predictive power of these approaches.