An accurate DFT-1/2 approach for shallow defect states: Efficient calculation of donor binding energies in silicon

TL;DR

Shallow donor binding energies in silicon are difficult to predict accurately with standard DFT due to band-gap underestimation and delocalization, and beyond-DFT methods are computationally expensive for large systems. The authors develop a unified DFT-1/2 correction workflow that simultaneously fixes the host band structure and the donor self-interaction within a single calculation, using $E_b = \varepsilon^{CBM}_\Gamma - \varepsilon^{donor}_\Gamma + e \Delta V$ and extrapolating to the dilute limit, enabling large embedded supercells. Applied to P, As, Sb, and Bi donors (with SOC for Bi), the method achieves near-experimental binding energies: As ≈ $54.04$ meV (exp $53.77$ meV), Bi with SOC ≈ $66.25$ meV (exp $70.88$ meV), Sb ≈ $37.5$ meV, and P ≈ $37.4$ meV (exp ≈ $45.58$ meV); As matches tandem-HSE closely, while Bi requires SOC to reach accuracy. An embedding-based strategy allows large-scale calculations up to 4096 atoms, and the approach is transferable to other shallow impurities, offering a cost-effective alternative to hybrid functionals for defect energetics in semiconductors.

Abstract

Accurate prediction of shallow-donor electron binding energies is critical for device modeling, dopant activation, and donor-based quantum technologies. Traditional beyond-DFT approaches (e.g., hybrid functionals, GW) are prohibitively expensive for the large supercells needed to capture the extended, hydrogenic wavefunctions, while semi-local DFT underestimates band gaps and suffers from delocalization errors. We present a simple, practical protocol for shallow donors based on the DFT-1/2 approximate quasiparticle correction that maintains the computational cost of standard DFT and enables supercells up to thousands of atoms. This approach provides a straightforward and reproducible workflow that delivers reliable donor binding energies with minimal computational overhead. Applied to group-V donors in Si, Si:X (X= P, As, Sb, Bi), the method yields binding energies in close agreement with experiment. We found that, for Si:Bi, it is essential to include spin-orbit coupling to achieve near-experimental values with a difference of only $\sim$ 4 meV. For arsenic, the method yields excellent agreement with experiment, with a difference of only ~0.3 meV. For antimony, the results match experiment to within ~5 meV, and for phosphorus, the deviation is within ~8 meV. Beyond its high accuracy, DFT-1/2 offers a significant practical advantage, providing a straightforward, reproducible, and transferable workflow that is less demanding than hybrid functional approaches while remaining fully generalizable to other shallow impurities in semiconductors.

Figures (12)

• Figure 1: Eigenvalues of a $\Gamma$-point DFT calculation for a shallow Sb donor in Si using high smearing (0.05 eV) for the $4\times 4 \times 4$ supercell. Blue and red bands represent spin-up and spin-down states, respectively. Numbers indicate fractional occupations, while arrows mark fully occupied bands. The figure shows the $A_1$, $T_2$, and $E$ shallow levels along with the VBM and CBM.
• Figure 2: A projection of the total force on every atom in the $xy$-plane for the $5\times 5 \times 5$ fully relaxed supercell and the larger supercells, $6\times 6 \times 6$ and $7\times 7 \times 7$, with embedded $5\times 5 \times 5$ . The forces on each atom are represented by both the color and circle size, with larger circles indicating larger forces.
• Figure 3: The PDOS for all the group V donors in silicon for a $4\times 4 \times 4$ supercell. For the creation of the PDOS the smearing was increased to 0.05 eV and a DFT-$\frac{1}{2}$ correction was applied to the silicon atoms. Due to the shallow donor the fermi level is close to the CBM.
• Figure 4: The DFT-$\frac{1}{2}$ cutoff optimization for As and Bi donor in silicon in both case half an electron is removed from the $s$-orbital. The optimized energy separation $\varepsilon_C-\varepsilon_D$, represents the energy difference between the first empty conduction band and the occupied shallow level of the same spin. Optimization using a pseudopotential with $d$-valence orbitals is indicated by the labels As_d and Bi_d.
• Figure 5: The binding energy for group V donors in silicon at various supercell sizes across different levels of the DFT-$\frac{1}{2}$ correction along with fit lines is shown. The fits in these figures are made with all binding energies of supercell sizes of $5\times 5 \times 5$ and larger. Binding energies calculated with only the bulk DFT-$\frac{1}{2}$ correction are denoted by Si$-\frac{1}{4}$. The plots for arsenic and bismuth incorporate the HSE data and fit from Swift2020. The experimental results are obtained from Tab. 1 in Ref. Saraiva_2015.