Substrate effects on charged defects in two-dimensional materials

Dan Wang; Ravishankar Sundararaman

REVIEW 3 major objections 5 minor 47 references

Substrate effects on charged defects in two-dimensional materials

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A calibrated continuum dielectric profile reproduces explicit-substrate charged-defect energies in 2D materials to within 0.16 eV.

desk verdict A pragmatic continuum-solvation extension that passes its own benchmarks, but the single-Gaussian calibration makes the blanket generality claim thinner than the abstract suggests. read the letter →

arxiv 1908.05208 v1 pith:J4VORRRL submitted 2019-08-14 cond-mat.mtrl-sci physics.comp-ph

Dan Wang , Ravishankar Sundararaman This is my paper · ORCID

classification cond-mat.mtrl-sciphysics.comp-ph PACS 71.55.-i71.15.Mb

keywords chargeddefectstwo-dimensionalmaterialscontinuumdielectricmodelsubstrateeffectsdensity-functionaltheorychargetransitionlevelshexagonalboronnitridemolybdenumdisulfide

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Charged defects in two-dimensional materials are routinely calculated as free-standing slabs, yet real devices place the monolayer on a substrate whose dielectric screening shifts defect levels by hundreds of meV. This paper argues that the substrate can be removed from the DFT supercell and replaced by a smooth continuum dielectric profile, calibrated with a single DFT calculation of the bare substrate responding to one Gaussian test charge. With that calibration, solving a modified Poisson equation inside the DFT self-consistency loop reproduces explicit-substrate ionization energies to within 0.05 eV for MoS2/SiO2 and 0.10–0.16 eV for hBN on SiO2 and diamond, at 40–200x lower cost. If the method holds, high-throughput screening of quantum defects with realistic substrate environments becomes practical, and a further simplification (Eq. 7) shifts all free-standing defect levels by a single defect-independent number.

What carries the argument

The load-bearing object is the dielectric shape function $s(z) = \frac{1}{2}\mathrm{erfc}\left(\frac{z_0 - z}{\sigma \sqrt{2}}\right)$ and the associated modified Poisson equation solved in cylindrical coordinates within the DFT self-consistency loop; it converts the substrate's many atoms into a smooth dielectric slab whose bound surface charge screens the defect. Its boundary position $z_0$ is calibrated from one Gaussian test charge, and $\sigma$ is fixed at 0.2 Å with results insensitive below 0.3 Å. A secondary mechanism, Eq. 7, uses the isolated self-energy of the same Gaussian model charge in the 2D-only versus 2D-plus-substrate dielectric environments, plus band-edge offsets, to give a defect-independent estimate of the ionization-energy shift.

What would settle it

Compute, with explicit substrate DFT, a charged defect whose charge density is strongly delocalized or shaped very differently from a Gaussian (for example, a shallow donor with a diffuse orbital, or a defect whose wavefunction hybridizes with the substrate), and compare its ionization energy to the continuum-model prediction. A disagreement larger than the 0.05–0.16 eV band found for the localized defects studied here would show that a single Gaussian test charge does not calibrate the substrate for all defects.

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Extended reading notes

Core claim

The central claim is that a continuum model of the substrate, given by a dielectric profile $\epsilon(z) = 1 + (\epsilon_b - 1)s(z)$ with $s(z) = \frac{1}{2}\mathrm{erfc}\left(\frac{z_0 - z}{\sigma \sqrt{2}}\right)$, contains the decisive electrostatic physics of charged defects in 2D materials on substrates. The boundary position $z_0$ is not guessed; it is fixed by requiring the interaction energy of a Gaussian test charge at the monolayer center with the continuum substrate to equal the corresponding DFT interaction energy of the explicit substrate. Once calibrated, the modified Poisson equation replaces the Hartree term in the DFT calculation, yielding charged-defect total energies whose ionization energies agree with explicit substrate calculations. For band edges, an additional explicit calculation of the perfect 2D material on the substrate supplies the non-electrostatic VBM/CBM offsets, which are small relative to the roughly 0.94–1.00 eV rigid electrostatic shift. The paper also shows the ionization-energy reduction is nearly identical across defects, so a single Gaussian self-energy shift (Eq. 7) converts all free-standing defect levels to substrate values within about 0.02 eV.

Load-bearing premise

The load-bearing premise is that one dielectric profile, fitted to the response of the bare substrate to a single Gaussian test charge at the monolayer center, stays accurate for every defect's charge distribution, position, and orbital character, and also for the band-edge shifts.

Editorial extensions

If this is right

Charged-defect transition levels in 2D materials on realistic substrates can be computed from free-standing DFT plus a calibrated dielectric environment, at 40–200x lower cost than explicit substrate calculations.
Substrate screening lowers donor and acceptor ionization energies: by 0.10–0.15 eV for ReMo and NbMo in MoS2/SiO2, by 0.27–0.33 eV for hBN defects on SiO2, and by 0.47–0.64 eV on diamond.
All ten hBN defects studied remain deep on SiO2 and diamond even after substrate screening, so they are expected to retain the long excited-state lifetimes needed for quantum applications.
Because the ionization-energy reduction is nearly independent of the defect, a single Gaussian-charge shift computed once per material/substrate pair can update every free-standing defect level with about 0.02 eV accuracy.
The continuum construction applies to arbitrary combinations of defects, 2D materials, and substrates, enabling screening of both defect candidates and material–substrate pairs for targeted properties.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One testable extension: because the shift is nearly defect-independent for localized defects, the method's range could be mapped by testing delocalized or substrate-hybridized charge states, where the single-Gaussian calibration is likely to be least accurate.
The paper leaves implicit that the same dielectric-profile construction applies to encapsulated multilayers and heterostructures by superposing profiles for each interface; the calibration cost would be paid once per material pair, not per defect.
A direct prediction the benchmarks do not test: replacing the optical dielectric constant with the low-frequency value should reproduce explicit calculations in which substrate atoms relax, a comparison that would validate the model's treatment of substrate degrees of freedom.
The explicit band-edge offset calculation remains a one-time DFT run per (2D material, substrate) pair, so the 40–200x speedup applies to the per-defect screening loop rather than the whole pipeline; the economics favor many defects sharing one substrate.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things up front. First, this paper does something useful: it adapts continuum solvation to charged defects in 2D materials on substrates, and benchmarks it against explicit DFT for MoS2/SiO2 and hBN on SiO2/diamond. The agreement is honestly good—0.05 eV for the MoS2 cases, 0.10–0.16 eV for hBN—for a 40–200x cost cut. Second, the weakest link is the calibration: they fit the substrate cavity boundary z0 to the response of a single Gaussian test charge at the monolayer center, then use that same z0 for every defect and for band-edge shifts. That transferability is asserted more than demonstrated.

What's genuinely new: a calibrated dielectric shape function for substrates, the explicit band-edge shift procedure (electrostatic shift from continuum plus core-level-aligned DOS corrections), and the Eq. 7 one-number estimator for ionization-energy reduction. Table I shows that estimator works to ~0.02–0.04 eV for the tested defects, which is a nice practical tool. The paper is also honest about its limits—semi-local DFT, optical dielectric constant, no Pauli repulsion in the continuum—and the calibration target is an auxiliary DFT quantity, not the defect energies themselves, so the circularity burden is low.

Now the soft spots. The single-Gaussian fit is the load-bearing assumption. E_int^DFT in Eq. 6 includes not just the image-charge response but also the test charge's interaction with the bare substrate potential. A dielectric-only continuum can't separate those, so z0 can absorb a substrate-potential term. The band-edge shifts are handled separately, but the cancellation between defect and band-edge states is not automatic, especially for acceptors versus donors or defects with out-of-plane charge centroids. The benchmarks don't exercise that: all tested defects are roughly in-plane and localized, so they probe the same monopole response used to fit. This doesn't invalidate the method for the tested cases—it clearly works there—but it does undercut the abstract's "arbitrary combinations" claim. The supplemental data (input files, Table S1–S5) is referenced but not included in the arXiv version, so the reproducibility claim can't be checked from the manuscript alone.

The citation pattern is appropriate; it builds on the authors' prior work without hiding that. This is a solid methods paper for defect engineering and quantum-defect screening in 2D materials. I'd send it to a serious referee. My recommendation: accept conditionally, asking for either more diverse validation (different substrate strongly varying potential, defect with out-of-plane charge) or a careful statement limiting the regime of validity to localized, in-plane defects.

Referee Report

3 major / 5 minor

Summary. The paper presents a continuum dielectric model for including substrate effects in DFT calculations of charged defects in 2D materials. The substrate is represented by a smooth dielectric profile with parameters sigma and z0; z0 is calibrated against the DFT interaction energy of a Gaussian test charge with the explicit substrate. Defect ionization energies are then computed in the free-standing 2D material with the continuum substrate, plus band-edge shifts from an explicit perfect-layer/substrate calculation. The method is benchmarked against explicit substrate DFT for ReMo/NbMo in MoS2/SiO2 and for selected hBN defects on SiO2 and diamond, with reported agreement of 0.05 eV and 0.10-0.16 eV, respectively. The paper also introduces Eq. 7, a defect-independent estimate of the ionization-energy shift, and applies the method to predict substrate effects on ten hBN defects.

Significance. The work addresses a real bottleneck: charged-defect supercells with explicit substrates are costly. If the method holds, it enables high-throughput screening of defects in 2D materials on substrates at 40-200x reduction in defect-supercell cost. Strengths: benchmarks against explicit DFT for two material/substrate combinations; the sigma independence is demonstrated; Eq. 7 is a simple, falsifiable prediction; calculations use open-source software. Main limitation: the calibration and defect-independence assumptions are tested only on in-plane, fairly localized defects, leaving transferability to general defects unproven.

major comments (3)

[II.B, Eq. 6] The calibration condition E_int^s(z0) = E_int^DFT fits z0 to the total DFT interaction energy of one Gaussian test charge. That energy includes the electrostatic interaction of the test charge with the bare substrate's potential, not just the dielectric image-charge response. A dielectric-only continuum model cannot distinguish these contributions, so z0 can absorb a substrate-potential term that may not transfer to defects with different charge centroids or orbital character. The current benchmarks are all in-plane localized defects, so they mainly probe the same monopole-like response used for calibration. To support the general claim, the authors should test z0 transferability, for example by calibrating with test charges at different positions or with a dipole/quadrupole charge distribution, and by benchmarking one defect with significant out-of-plane charge density.
[III.C, Eq. 7 and Table I] The claim that Eq. 7 estimates ΔIE with ~0.02 eV accuracy is supported only by comparison with the continuum model's own self-consistent calculations. Both columns in Table I derive from the same Gaussian self-energy in the same dielectric profile, so agreement is partially by construction. The authors should compare Eq. 7 directly with the explicit DFT ΔIE values (e.g., for ReMo/NbMo in MoS2/SiO2 and the explicit hBN cases) or clearly state that Eq. 7 is a fast approximation within the continuum model whose accuracy relative to explicit DFT has not been separately benchmarked.
[Abstract and Section III.B] The claim of "computational expediency of calculating defects in free-standing 2D materials" should be qualified, because the method still requires one explicit DFT calculation of the perfect 2D material on the substrate to determine ΔεVBM and ΔεCBM (Fig. 1(d), Section III.B). The 40-200x speedup applies only to the defect-containing supercells; the required substrate-plus-perfect-layer calculation remains an explicit, nontrivial cost. This should be stated clearly so readers can assess the true computational savings.

minor comments (5)

[II.B] The notation for the continuum interaction energy is inconsistent; "Es(z)int" should be defined (e.g., E_int^s(z)) and distinguished from E_int^DFT.
[Table I] Headers "Self-consistent" and "Continuum Model" in the two sub-tables are inconsistent; use the same labels for the same quantity.
[III.A] The comparison for MoS2/SiO2 is against previous explicit calculations (Ref. 40), not new explicit calculations; this should be stated explicitly in the text.
[III.B] The sentence "The defects are all deep in hBN with ionization energies in the range of 2.14-4.01 eV" appears to refer to both donors and acceptors; specify which panel and whether this range includes both.
[II.B] The text says "we select the cavity parameters such that E_int^s(z)=E_int^DFT" and then mentions two parameters but fixes sigma; please clarify that only z0 is fitted.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the continuum substrate model is calibrated to a DFT test-charge response and its predictions are benchmarked against independent explicit-substrate DFT calculations.

full rationale

The paper's central parameter z0 is fit to E_DFT_int, the interaction of a Gaussian test charge with the explicit DFT substrate (Eq. 6), which is not a defect ionization energy. The claimed predictions (ionization-energy reductions) are then compared with explicit substrate DFT: previously published MoS2/SiO2 calculations (Ref. 40) and the authors' own explicit DFT for selected hBN defects. Thus the headline agreement is not forced by the fitting target. The continuum correction scheme is taken from prior published work (Refs. 17, 18, including the second author), but this is supporting methodology, not the novel substrate model, and it is not the source of the benchmark agreement. Eq. 7 reuses the Gaussian self-energy to estimate ionization-energy shifts and agrees with the self-consistent continuum model, but that agreement is a model-internal consistency check, while the self-consistent model itself is validated externally. No equation reduces to the calibration input by construction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central method rests on treating the substrate as a continuum dielectric and calibrating its boundary from a test-charge calculation, plus rigid band-edge shifts from one explicit combined DFT run. These are modeling assumptions, not new physical entities; they are disclosed and partly validated by benchmarks.

free parameters (2)

z0 (substrate dielectric transition center) = Not reported in main text; listed in Table S1
Chosen so the continuum interaction energy of a Gaussian test charge matches an explicit DFT calculation of the substrate; controls how close the dielectric response approaches the 2D material.
sigma (shape function width) = 0.2 Å recommended
Fixed by hand at 0.2 Å after checking predictions vary below 0.01 eV for sigma less than 0.3 Å; not fitted to defect data.

assumptions (5)

domain assumption Substrate electrostatic response can be represented by a local dielectric function varying only along z, with an erfc shape function s(z).
Introduced in Eq. 5 and Section II.B; this is the core continuum approximation and ignores nonlocal and chemical effects, acknowledged as a source of 0.03 to 0.16 eV errors.
domain assumption A dielectric profile calibrated against one Gaussian test charge at the monolayer center transfers to all defect charge distributions.
Section II.B calibrates z0 so E_int^substrate matches a single test charge; all defect ionization energies and Eq. 7 rely on this transferability.
domain assumption Substrate-induced band edge shifts are rigid shifts determined by core-level alignment, with no other electronic structure changes.
Section II.B and Fig. 1(d): VBM/CBM shifts are split into an electrostatic rigid shift plus offsets from DOS core-level alignment; this ignores band hybridization effects beyond shifts.
domain assumption Using the optical dielectric constant of the substrate is adequate because substrate atoms are held fixed.
Section II.B states epsilon_infinity is used when substrate atoms are not relaxed; full relaxation would require the low-frequency constant.
domain assumption Semi-local DFT predicts correct trends in defect transition levels.
Stated in Computational details; absolute ionization energies may change with hybrid functionals or many-body methods.

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Cite this review

Pith. "Pith review of Substrate effects on charged defects in two-dimensional materials." pith.science (2026). https://pith.science/paper/J4VORRRL

@misc{pith2026190805208,
  author       = {Pith},
  title        = {Pith review of: Substrate effects on charged defects in two-dimensional materials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J4VORRRL}},
  note         = {Machine review of arXiv:1908.05208}
}

abstract

Two-dimensional (2D) materials are strongly affected by the dielectric environment including substrates, making it an important factor in designing materials for quantum and electronic technologies. Yet, first-principles evaluation of charged defect energetics in 2D materials typically do not include substrates due to the high computational cost. We present a general continuum model approach to incorporate substrate effects directly in density-functional theory calculations of charged defects in the 2D material alone. We show that this technique accurately predicts charge defect energies compared to much more expensive explicit substrate calculations, but with the computational expediency of calculating defects in free-standing 2D materials. Using this technique, we rapidly predict the substantial modification of charge transition levels of two defects in MoS$_2$ and ten defects promising for quantum technologies in hBN, due to SiO$_2$ and diamond substrates. This establishes a foundation for high-throughput computational screening of new quantum defects in 2D materials that critically accounts for substrate effects.

Figures

Figures reproduced from arXiv: 1908.05208 by the authors.

**Figure 2.** (b) shows the variation of the results with the one free parameter σ that sets the smoothness of the transition from vacuum to substrate dielectric constant. (As discussed above, z0, which sets the center of the transition of s(z), is constrained using the response of a DFT substrate to a Gaussian test charge, for a given σ.) The results are insensitive to σ as long as it is small enough, with variations in the pr… view at source ↗

**Figure 3.** FIG. 3. Optimized atomic configurations of promising hBN [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5. (a) Atomic configuration of hBN/diamond in top and [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 7.** Figure 7: FIG. 7. The stable charge state of all hBN defects shown in [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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