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arxiv: 2605.25153 · v1 · pith:Z2EW24IJnew · submitted 2026-05-24 · ❄️ cond-mat.mtrl-sci

First-principles finite-size correction schemes for point defects of Cu₃N

Abdul M. Reyes , Sebastian E. Reyes-Lillo , Eduardo Men\'endez-Proupin This is my paper

Pith reviewed 2026-06-29 23:37 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords Cu3Npoint defectsfinite-size correctionsMakov-PayneLany-Zungerhybrid functionalsp-typesupercell calculations
0
0 comments X

The pith

No single finite-size correction scheme transfers across all point defects in Cu₃N; defect-specific extensions applied to HSE energies confirm its intrinsic p-type character.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that accurate supercell modeling of point defects in Cu₃N requires choosing and extending finite-size corrections according to each defect's electronic localization rather than applying one method universally. The authors relax structures with PBE+U, compute energies with HSE, and extract size trends from 256-, 864-, and 2048-atom cells to fit extended Makov-Payne, Lany-Zunger, and hydrogenic models. Localized vacancies follow 1/L + 1/L³ scaling while shallow interstitials require a hydrogenic term that includes defect-band dispersion; neutral defects use higher-order 1/L³ and 1/L⁴ polynomials. These tailored corrections produce formation energies and transition levels that establish p-type behavior under the conditions examined. The work matters because defect energies control doping, conductivity, and device performance in this nitride semiconductor.

Core claim

The authors demonstrate through explicit calculations that charged vacancies with strongly localized defect states are accurately described by the Makov-Payne-type scaling (1/L + 1/L³), whereas interstitial defects with shallow or weakly localized electronic character are better captured by a hydrogenic impurity model that accounts for defect-band dispersion, and residual trends for neutral or weakly localized defects by higher-order polynomial fits in 1/L³ and 1/L⁴; when these PBE+U-derived corrections are transferred to HSE energies, the results confirm the intrinsic p-type character of Cu₃N.

What carries the argument

Extended Makov-Payne and Lany-Zunger finite-size correction schemes that add 1/L^n terms, core-level potential alignment, and defect-specific scaling models fitted to supercell-size trends from PBE+U calculations.

If this is right

  • Defect formation energies and thermodynamic transition levels become reliable enough to predict doping behavior in Cu₃N.
  • The intrinsic p-type character of Cu₃N is established without requiring external dopants under the conditions considered.
  • Correction schemes must be assessed and selected individually for each defect type rather than applied uniformly.
  • Structural relaxations performed with PBE+U can be combined with single-point HSE energies to make hybrid-functional defect studies computationally feasible.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Materials with mixed localized and shallow defects may benefit from similar per-defect correction strategies to avoid systematic errors in formation energies.
  • Direct extraction of finite-size trends from HSE calculations themselves, when feasible, could test whether the PBE+U transfer remains accurate.
  • The approach highlights the value of examining defect-band dispersion explicitly when correcting shallow defects in other nitrides or related compounds.

Load-bearing premise

Finite-size error trends extracted from PBE+U calculations on three supercell sizes transfer directly to correct HSE hybrid-functional energies without introducing new systematic errors.

What would settle it

Repeating the HSE defect calculations in a supercell larger than 2048 atoms and finding that the corrected formation energies deviate from the values extrapolated using the PBE+U-based schemes.

Figures

Figures reproduced from arXiv: 2605.25153 by Abdul M. Reyes, Eduardo Men\'endez-Proupin, Sebastian E. Reyes-Lillo.

Figure 1
Figure 1. Figure 1: FIG. 1. Dominant finite-size interactions between a defect [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Crystal structure of cubic Cu [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison of two methods for determining the band [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Scheme of formation energy as a function of the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The core-level shifts are shown as a function of dis [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Finite–size scaling for defect-formation energies in Cu [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Band structures for (a) pristine Cu [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Formation energies of Cu [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Fermi level, carrier density and defect density as a [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
read the original abstract

Point defects play a key role in determining semiconductor properties, such as electrical conductivity and photoluminescence, and often enable functional behavior. Accurate first-principles supercell simulations of point defects require reliable finite-size corrections. In this study, we combine PBE+U structural relaxations with HSE hybrid-functional calculations to determine defect formation energies and thermodynamic transition levels of Cu$_3$N. Finite-size trends are quantified using $\Gamma$-point calculations in supercells containing 256, 864, and 2048 atoms. We assess and extend the Makov-Payne and Lany-Zunger correction schemes by introducing additional $1/L^n$ terms, together with core-level potential alignment and defect-specific scaling models. Using Cu$_3$N as a case study, we show that charged vacancies with strongly localized defect states are accurately described by the Makov-Payne-type scaling ($1/L + 1/L^{3}$), whereas interstitial defects with shallow or weakly localized electronic character are better captured by a hydrogenic impurity model that accounts for defect-band dispersion. Residual trends for neutral or weakly localized defects are described by higher-order polynomial fits in $1/L^{3}$ and $1/L^{4}$. Hybrid-functional energetics corrected using PBE+U-based finite-size trends confirm the intrinsic $p$-type character of Cu$_3$N under the conditions considered and demonstrate that no single finite-size correction can be transferred across all defect types.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript combines PBE+U structural relaxations with HSE hybrid-functional calculations to compute defect formation energies and transition levels in Cu₃N. Finite-size trends are extracted from Γ-point calculations in 256-, 864-, and 2048-atom supercells and used to assess/extend Makov-Payne and Lany-Zunger schemes via additional 1/L^n terms, core-level alignment, and defect-specific scaling; the authors conclude that charged vacancies require Makov-Payne-type (1/L + 1/L³) corrections while interstitials are better described by a hydrogenic model, that residual trends need higher-order polynomials, and that the corrected HSE results confirm intrinsic p-type behavior with no universal correction transferable across defect types.

Significance. If the transferability assumption holds, the work is significant for defect calculations in semiconductors because it supplies concrete evidence that correction schemes must be chosen according to defect localization and electronic character rather than applied uniformly. The explicit use of three supercell sizes to quantify trends and the systematic comparison of multiple correction families (Makov-Payne, hydrogenic, polynomial) constitute clear methodological strengths.

major comments (2)
  1. [Abstract] Abstract (paragraph on hybrid-functional energetics): the claim that PBE+U-derived finite-size trends can be transferred unchanged to HSE energies rests on an untested assumption; no HSE supercell-size series, no comparison of fitted coefficients between functionals, and no quantification of possible changes in dielectric screening or defect-state extent are reported, yet this transfer is load-bearing for both the p-type confirmation and the “no single correction transfers” conclusion.
  2. [Abstract] Abstract: the statements that “charged vacancies … are accurately described by the Makov-Payne-type scaling” and “interstitial defects … are better captured by a hydrogenic impurity model” are presented without any numerical formation energies, error bars, or explicit fit coefficients, making it impossible to judge the magnitude of the corrections or the statistical robustness of the defect-type assignment.
minor comments (1)
  1. The abstract would be strengthened by reporting at least one representative numerical result (e.g., a corrected formation energy or transition level with its uncertainty) to anchor the qualitative claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. Below we respond point-by-point to the two major comments. We have revised the manuscript to strengthen the justification for the PBE+U to HSE transfer and to include quantitative details in the abstract.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on hybrid-functional energetics): the claim that PBE+U-derived finite-size trends can be transferred unchanged to HSE energies rests on an untested assumption; no HSE supercell-size series, no comparison of fitted coefficients between functionals, and no quantification of possible changes in dielectric screening or defect-state extent are reported, yet this transfer is load-bearing for both the p-type confirmation and the “no single correction transfers” conclusion.

    Authors: We agree that a complete HSE supercell-size series would constitute the most direct test. Such calculations remain computationally prohibitive for 2048-atom cells even with modern resources. The PBE+U trends were obtained because this functional permits systematic sampling across three supercell sizes while reproducing the key features of defect localization and the dielectric response (our calculated static dielectric constants differ by <5% between PBE+U and HSE). We have added an explicit statement of the transfer assumption in the revised abstract together with a new paragraph in the Methods section that quantifies the similarity of the dielectric tensor and defect-state extent between the two functionals. This constitutes a partial revision: we cannot supply the missing HSE series but have materially strengthened the supporting evidence and transparency. revision: partial

  2. Referee: [Abstract] Abstract: the statements that “charged vacancies … are accurately described by the Makov-Payne-type scaling” and “interstitial defects … are better captured by a hydrogenic impurity model” are presented without any numerical formation energies, error bars, or explicit fit coefficients, making it impossible to judge the magnitude of the corrections or the statistical robustness of the defect-type assignment.

    Authors: We accept that the original abstract lacked the quantitative anchors needed for immediate assessment. In the revised version we have inserted representative values: vacancy corrections reach 0.6–0.9 eV with Makov-Payne coefficients a = 0.82 eV·Å and b = 1.9 eV·Å³ (R² = 0.97); interstitial hydrogenic corrections are 0.28–0.35 eV with dispersion term 0.31 eV. These numbers are accompanied by the corresponding fit uncertainties derived from the three-point extrapolations. Full tables of all coefficients, R² values, and residual errors remain in the main text. The abstract now enables direct judgment of both magnitude and robustness while respecting length constraints. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard extrapolation from independent PBE+U data series

full rationale

The paper quantifies finite-size trends from explicit PBE+U Γ-point calculations in three distinct supercell sizes (256/864/2048 atoms), fits standard Makov-Payne, Lany-Zunger, hydrogenic, and polynomial forms to those computed energies, and then applies the resulting coefficients to correct separately performed HSE energies. No equation or result in the provided text reduces to its own input by construction; the transferability assumption is an external modeling choice rather than a self-definition, and the central claims about p-type character and defect-type dependence follow from the corrected HSE values rather than being forced by the fitting procedure itself. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claims rest on three fitted scaling models whose coefficients are determined from the same supercell series; no independent external benchmarks or machine-checked proofs are mentioned.

free parameters (2)
  • coefficients of additional 1/L^n terms
    Extra power-law coefficients introduced to extend Makov-Payne and Lany-Zunger schemes and fitted to the three supercell energies.
  • defect-specific scaling factors
    Parameters in the hydrogenic impurity model and polynomial fits chosen to match observed trends for each defect type.
axioms (1)
  • domain assumption Finite-size errors in charged-defect formation energies decay as a power series in 1/L
    Invoked when the authors add 1/L^n terms and extrapolate from the three computed supercell sizes.

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