The optimized lattice structure of pristine crystalline silicon obtained within the PBE [8] functional is found to be in good agreement with reported theoretical and experimental data [5][9]. Upon substitutional doping, both boron and phosphorus atoms induce only localized distortions of the silicon lattice, while the global diamond structure is preserved. Boron doping leads to a slight contraction of the nearest-neighbor Si-B bonds due to the smaller atomic radius of boron compared to silicon. In contrast, phosphorus substitution results in a minor expansion of the local Si-P bonds, consistent with the larger atomic radius of phosphorus. These structural relaxations remain confined to the immediate vicinity of the dopant atoms, confirming the dilute nature of the doping regime.

Ab initio molecular dynamics simulations performed at 300 K further demonstrate the thermal stability of both doped systems. The total energy and temperature fluctuations remain stable throughout the simulation time, and no atomic diffusion or structural degradation is observed. This indicates that substitutional boron and phosphorus doping does not compromise the structural integrity of crystalline silicon under ambient conditions.

Although defect formation energies were not explicitly calculated in this work, previous first-principles studies have shown that substitutional boron and phosphorus dopants in crystalline silicon exhibit low formation energies under typical growth conditions, confirming their thermodynamic stability [6][9].

The electronic band structure of pristine silicon calculated within the PBE [8] approximation exhibits an indirect band gap of approximately 0.62 eV, with the valence band maximum located at the Γ point and the conduction band minimum near the X point of the Brillouin zone. This value is significantly lower than the experimental band gap of 1.12 eV, reflecting the well-known limitation of semilocal exchange-correlation functionals [7][8]. The densities of states reveal that the valence band is dominated by Si 3p states, while the conduction band has mixed s and p character.

Hybrid HSE06 [2][3] calculations substantially improve the description of the electronic structure by increasing the band gap to approximately 1.12 eV, in close agreement with experimental measurements [5][9]. Importantly, the overall dispersion and topology of the band structure remain unchanged, indicating that the PBE [8] functional provides a reliable qualitative description of the electronic states.

Figure 1 compares the electronic band structure of pristine silicon calculated

Figure 1. Electronic band structure of pristine crystalline silicon calculated along the high-symmetry path Γ-X-M-Γ-R-X/M-R using Density Functional Theory. (a) PBE exchange-correlation functional and (b) HSE06 hybrid functional. The PBE functional reproduces correctly the overall band dispersion but significantly underestimates the band gap. In contrast, the HSE06 functional yields an improved band gap value (≈ 1.1 eV), in close agreement with experimental data, while preserving the indirect nature of the band gap.

using the PBE and HSE06 functionals in order to assess the accuracy of the electronic gap description.

For the boron-doped system, the calculated density of states reveals the presence of a shallow acceptor level located approximately 0.08 eV above the valence band maximum. This behavior indicates that boron acts as an efficient dopant introducing mobile hole carriers without significantly altering the electronic framework of the host lattice. When boron substitutes a silicon atom, the resulting electron deficiency leads to an incomplete bonding configuration and the formation of a hole-like state. This state is primarily associated with B 2p orbitals hybridized with neighboring Si 3p states. The shallow energetic position of the acceptor level indicates weak localization and strong coupling to the valence band, which explains the efficient activation of holes at room temperature.

As a consequence, the Fermi level shifts toward lower energies, confirming the p-type character of boron-doped silicon.

The global band structure of the boron-doped system remains very similar to that of pristine silicon, with only minor perturbations near the valence band edge.

In the phosphorus-doped system, an additional donor state appears approximately 0.05 eV below the conduction band minimum. This donor behavior can be physically understood by considering the presence of an extra valence electron introduced by the phosphorus atom compared to silicon. When phosphorus substitutes a silicon atom, this additional electron occupies an energy level slightly below the conduction band minimum.

The weak binding of this donor state reflects its delocalized nature and strong coupling with the conduction band states of silicon. The shallow nature of the donor level facilitates thermal ionization at room temperature, making phosphorus an effective n-type dopant toward the conduction band, consistent with n-type conductivity.

Similar to the boron-doped case, the overall electronic band structure of phosphorus-doped silicon closely resembles that of pristine silicon.

The calculated ionization energies of 0.08 eV for boron and 0.05 eV for phosphorus are in good agreement with experimental values, typically reported around 0.045 - 0.065 eV for boron and ~0.045 eV for phosphorus in crystalline silicon [4][5].

From a broader physical perspective, the shallow nature of both acceptor and donor states reflects the fact that substitutional boron and phosphorus act as nearly ideal dopants in crystalline silicon. Their impurity states remain strongly coupled to the band edges rather than forming deep, highly localized defect levels within the band gap. Consequently, the global electronic structure of silicon is preserved while the carrier concentration and Fermi level position are effectively tuned, which underpins the technological relevance of doped silicon in microelectronic applications.

Hybrid HSE06 [2][3] calculations performed for the doped systems confirm that the inclusion of exact exchange primarily affects the band gap magnitude while preserving the relative position of dopant-induced states. The acceptor and donor levels introduced by boron and phosphorus remain shallow with respect to the corrected band edges. This result supports the reliability of combining PBE-based [8] trends with HSE06 [2][3] quantitative corrections.

Overall, the results demonstrate that substitutional boron and phosphorus doping effectively tunes the electronic properties of silicon by introducing shallow impurity states without significantly perturbing the host band structure. The consistency between structural, electronic, and thermal analyses highlights the robustness of the present first-principles approach.

Figure 2 illustrates the total density of states of pristine, boron-doped, and phosphorus-doped silicon calculated using the HSE06 hybrid functional, highlighting the effect of substitutional doping on the electronic states near the Fermi level.

In the dilute doping limit considered here, the shift of the Fermi level relative to the band edges is subtle due to the low dopant concentration and large supercell size. Consequently, the p-type and n-type character is inferred from the appearance of shallow acceptor and donor states near the valence and conduction band edges, rather than from a pronounced macroscopic Fermi level displacement.