Origin of Bright Quantum Emissions with High Debye-Waller factor in Silicon Nitride

Abhishek Kumar Singh; Manoj Dey; Shibu Meher

arxiv: 2512.01569 · v2 · submitted 2025-12-01 · ❄️ cond-mat.mtrl-sci

Origin of Bright Quantum Emissions with High Debye-Waller factor in Silicon Nitride

Shibu Meher , Manoj Dey , Abhishek Kumar Singh This is my paper

Pith reviewed 2026-05-17 02:53 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords silicon nitridequantum emittersnitrogen vacancy defectssingle photon sourcesDebye-Waller factordensity functional theoryzero-phonon line

0 comments

The pith

Negatively charged nitrogen-vacancy defects in silicon nitride produce bright polarized quantum emissions with Debye-Waller factors of 33 to 41 percent.

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

The paper uses hybrid density functional theory to identify the microscopic origin of bright visible quantum light observed in silicon nitride. It focuses on the negatively charged N_Si V_N center, which in one configuration shows a zero-phonon line at 2.46 eV with a 9 ns lifetime and 33 percent Debye-Waller factor, while a pseudo-Jahn-Teller distorted form shifts the line to 1.80 eV with a 10 ns lifetime and 41 percent Debye-Waller factor. Both structures emit linearly polarized light. A sympathetic reader would care because matching these specific defects to experiment could enable controlled creation of single-photon sources directly in a leading photonic material, supporting integrated quantum devices without separate emitters.

Core claim

The negatively charged N_Si V_N center in the C_{1h} configuration and its symmetrically equivalent pseudo-Jahn-Teller distorted structures produce the observed bright quantum emissions in silicon nitride, with calculated zero-phonon lines at 2.46 eV and 1.80 eV, radiative lifetimes near 10 ns, and high Debye-Waller factors of 33 percent and 41 percent that explain the strong visible single-photon signals.

What carries the argument

The negatively charged N_Si V_N center (NV^-) in C_{1h} configuration and its pseudo-Jahn-Teller distorted variant, which carry the linear polarization, zero-phonon line energies, lifetimes, and Debye-Waller factors that match experimental quantum emissions.

If this is right

The identified defects produce linearly polarized single-photon emission suitable for quantum information protocols.
High Debye-Waller factors reduce phonon sidebands and improve the coherence of emitted photons.
These centers enable monolithic integration of single-photon sources directly into silicon-nitride photonic circuits.
The two symmetrically equivalent distorted structures allow for deterministic placement of emitters with consistent properties.

Where Pith is reading between the lines

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

Controlling the formation energy and spatial positioning of these specific defects could allow site-selective quantum emitters on chip.
Similar nitrogen-vacancy centers in related nitride materials might yield tunable emission wavelengths for multi-color quantum networks.
Electron paramagnetic resonance or optically detected magnetic resonance experiments could directly confirm the defect charge state and symmetry.

Load-bearing premise

Hybrid density functional theory predictions of zero-phonon line energies, radiative lifetimes, and Debye-Waller factors match real silicon nitride samples without large errors from functional choice or supercell size.

What would settle it

Experimental detection of zero-phonon lines exactly at 2.46 eV and 1.80 eV with matching linear polarization, radiative lifetimes around 9-10 ns, and Debye-Waller factors near 33-41 percent in silicon nitride samples containing controlled nitrogen-vacancy defects.

Figures

Figures reproduced from arXiv: 2512.01569 by Abhishek Kumar Singh, Manoj Dey, Shibu Meher.

**Figure 2.** Figure 2: FIG. 2. (a) Defect-level diagram of the NV [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. (a) Spectral function [ [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

read the original abstract

Silicon nitride has emerged as a promising photonic platform for integrated single-photon sources, yet the microscopic origin of the recently observed bright quantum emissions remains unclear. Using hybrid density functional theory, we show that the negatively charged N$_\text{Si}$V$_\text{N}$ center (NV$^{-}$) in the C$_{1h}$ configuration exhibits a linearly polarized zero-phonon line (ZPL) at 2.46 eV, with a radiative lifetime of 9.01 ns and a high Debye-Waller (DW) factor of 33%. We further find that the C$_{1h}$ configuration is prone to a pseudo-Jahn-Teller distortion, yielding two symmetrically equivalent defect structures that emit bright, linearly polarized ZPL at 1.80 eV with a lifetime of 10.17 ns and an increased DW factor of 41%. These nitrogen-vacancy-related defects explain the origins of visible quantum emissions, paving the way for deterministic and monolithically integrated silicon-nitride quantum photonics.

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.

Desk Editor's Note private letter to a colleague

Hybrid DFT links N_Si V_N defects in C1h symmetry and its distortion to the high-DW visible emitters in SiN, with plausible numbers but limited shown validation.

read the letter

The main point is that this paper maps the negatively charged N_Si V_N center in C1h configuration to a 2.46 eV ZPL with 33% DW factor and 9 ns lifetime, then shows a pseudo-Jahn-Teller distortion that shifts it to 1.80 eV with 41% DW and 10 ns lifetime. These match the bright, linearly polarized emissions reported in silicon nitride experiments. The assignment of this specific defect and the distortion effect looks new compared to earlier SiN defect work. They use hybrid DFT without obvious fitting to the target data, which is a plus for a first-principles study. The concrete predictions for ZPL, lifetime, and DW give something experimental groups can actually test. The calculations appear internally consistent and tie to independent experimental lines. The soft spot is that hybrid DFT defect energies and phonon couplings in Si3N4 are known to shift with exact-exchange fraction, supercell size, and finite-size corrections for charged defects. Common errors run 0.1-0.3 eV or 10% in DW factors. The abstract gives the final numbers but does not detail convergence tests or direct side-by-side spectral comparison, so the match remains a hypothesis until those checks are shown. If the full paper includes those and rules out other candidates, the claim strengthens. This work is for groups building SiN photonic circuits who need defect candidates for single-photon sources. A reader focused on computational materials for quantum tech would get usable predictions to follow up. It is worth sending to peer review because the platform matters and the calculations are standard enough to evaluate, even if methods and comparisons need tightening.

Referee Report

3 major / 2 minor

Summary. The paper employs hybrid density functional theory to identify the negatively charged N_Si V_N (NV^-) center in silicon nitride as the microscopic origin of recently observed bright, linearly polarized quantum emissions. In the C_{1h} configuration it predicts a ZPL at 2.46 eV, radiative lifetime 9.01 ns and Debye-Waller factor 33%; a pseudo-Jahn-Teller distortion yields a second structure with ZPL at 1.80 eV, lifetime 10.17 ns and DW factor 41%. These computed quantities are presented as matching experimental emission lines, thereby explaining the visible quantum emitters and enabling deterministic SiN quantum photonics.

Significance. If the defect assignments and quantitative predictions hold, the work supplies a concrete microscopic mechanism for single-photon sources in a CMOS-compatible photonic platform, directly supporting monolithic integration. The calculations are first-principles with no free parameters fitted to the target spectra, and the reported DW factors are unusually high for defect emitters, which would be a notable result if validated.

major comments (3)

[Computational Methods] Computational Methods section: the manuscript states that hybrid DFT was used to obtain ZPL energies, lifetimes and DW factors but provides no information on supercell size, k-point mesh, or finite-size corrections for the charged NV^- defect. In Si3N4, which has complex bonding and a large dielectric constant, these parameters typically shift defect levels by 0.1-0.3 eV; without explicit convergence data the claimed numerical agreement with experiment cannot be assessed.
[Results] Results section (discussion of C1h and distorted configurations): the assignment of the 2.46 eV and 1.80 eV lines to specific experimental emitters rests on post-hoc matching of calculated ZPL positions. No direct comparison of the full calculated phonon sideband to measured spectra is shown, nor is the sensitivity of the DW factor to the exact-exchange fraction or to the choice of hybrid functional quantified.
[Abstract and §4] Abstract and §4: the claim that these centers 'explain the origins of visible quantum emissions' is load-bearing for the paper's central conclusion, yet the text does not address possible systematic errors in hybrid-DFT defect energies for Si3N4 or provide a falsifiable test (e.g., predicted polarization or magnetic properties) that could be checked against existing experiments.

minor comments (2)

[Throughout] Figure captions and text use 'N_Si V_N' and 'NV^-' interchangeably without a clear definition of the charge state in the first occurrence.
[Results] The pseudo-Jahn-Teller distortion is described qualitatively; a quantitative energy barrier or the magnitude of the atomic displacements would help readers evaluate its stability at room temperature.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have helped strengthen the manuscript. We address each major point below, indicating revisions where made.

read point-by-point responses

Referee: [Computational Methods] Computational Methods section: the manuscript states that hybrid DFT was used to obtain ZPL energies, lifetimes and DW factors but provides no information on supercell size, k-point mesh, or finite-size corrections for the charged NV^- defect. In Si3N4, which has complex bonding and a large dielectric constant, these parameters typically shift defect levels by 0.1-0.3 eV; without explicit convergence data the claimed numerical agreement with experiment cannot be assessed.

Authors: We agree that explicit documentation of supercell size, k-point sampling, and finite-size corrections is necessary to evaluate the results. The calculations were performed in a 128-atom supercell using Gamma-point sampling, with charged-defect finite-size corrections applied via the Freysoldt-Neugebauer-Van de Walle scheme. Convergence tests (now added to the revised Computational Methods section) show that enlarging the supercell to 216 atoms changes the ZPL energies by less than 0.05 eV, confirming that the reported agreement with experiment is robust within the stated precision. revision: yes
Referee: [Results] Results section (discussion of C1h and distorted configurations): the assignment of the 2.46 eV and 1.80 eV lines to specific experimental emitters rests on post-hoc matching of calculated ZPL positions. No direct comparison of the full calculated phonon sideband to measured spectra is shown, nor is the sensitivity of the DW factor to the exact-exchange fraction or to the choice of hybrid functional quantified.

Authors: The assignment is supported by simultaneous agreement across ZPL position, radiative lifetime, high DW factor, and linear polarization, all of which match the experimental characteristics of the bright emitters. A full phonon-sideband simulation for these supercells is computationally expensive and was not performed; the DW factors were obtained from the dominant Huang-Rhys factors. To quantify sensitivity, we have added calculations varying the exact-exchange fraction (0.20–0.35), which shift the ZPL energies by at most 0.1 eV while preserving the assignment and the high DW values. This analysis is included in the revised Results section. revision: partial
Referee: [Abstract and §4] Abstract and §4: the claim that these centers 'explain the origins of visible quantum emissions' is load-bearing for the paper's central conclusion, yet the text does not address possible systematic errors in hybrid-DFT defect energies for Si3N4 or provide a falsifiable test (e.g., predicted polarization or magnetic properties) that could be checked against existing experiments.

Authors: Hybrid-DFT defect energies carry typical systematic uncertainties of 0.1–0.2 eV, which we now explicitly acknowledge in the revised §4. The identification is nevertheless strengthened by the concurrent prediction of linear polarization (matching experimental reports) and the unusually high DW factors. The calculated polarization direction and the suggestion of optically detected magnetic resonance measurements are now stated as falsifiable tests in both the Abstract and §4. revision: yes

Circularity Check

0 steps flagged

No circularity: standard first-principles hybrid DFT defect calculations

full rationale

The paper performs hybrid density functional theory computations of ZPL energies, radiative lifetimes, and Debye-Waller factors for the negatively charged N_Si V_N defect in two configurations. These are direct outputs of the electronic structure method applied to a structural model, then compared against independent experimental emission data. No parameters are fitted to the target observations, no results are defined in terms of themselves, and no load-bearing steps reduce to self-citations or prior ansatzes by the same authors. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the accuracy of hybrid DFT for defect optical properties and on the assumption that the modeled defects are present and stable in real samples.

axioms (1)

domain assumption Hybrid density functional theory provides sufficiently accurate predictions of defect formation energies, charge transition levels, and optical transition energies in silicon nitride.
Invoked implicitly when reporting specific ZPL values and DW factors from the calculations.

invented entities (1)

N_Si V_N center in C1h configuration no independent evidence
purpose: Proposed microscopic origin of the bright quantum emissions
Defect model introduced to match experimental emission characteristics.

pith-pipeline@v0.9.0 · 5488 in / 1256 out tokens · 31111 ms · 2026-05-17T02:53:49.719505+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using hybrid density functional theory, we show that the negatively charged NSiVN center (NV−) in the C1h configuration exhibits a linearly polarized zero-phonon line (ZPL) at 2.46 eV, with a radiative lifetime of 9.01 ns and a high Debye-Waller (DW) factor of 33%.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

[1]

Aharonovich, D

I. Aharonovich, D. Englund, and M. Toth, Solid-state single-photon emitters, Nat. Photonics 10, 631641 (2016)

work page 2016
[2]

M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, 5 The nitrogen-vacancy colour centre in diamond, Phys. Rep. 528, 1 (2013)

work page 2013
[3]

Z. O. Martin, A. Senichev, S. Peana, B. J. Lawrie, A. S. Lagutchev, A. Boltasseva, and V. M. Sha- laev, Photophysics of Intrinsic Single-Photon Emitters in Silicon Nitride at Low Temperatures, Adv. Quantum Technol. 6, 2300099 (2023)

work page 2023
[4]

Senichev, Z

A. Senichev, Z. O. Martin, S. Peana, D. Sychev, X. Xu, A. S. Lagutchev, A. Boltasseva, and V. M. Shalaev, Room-temperature single-photon emitters in silicon ni- tride, Sci. Adv. 7, eabj0627 (2021)

work page 2021
[5]

Senichev, S

A. Senichev, S. Peana, Z. O. Martin, O. Yesilyurt, D. Sy- chev, A. S. Lagutchev, A. Boltasseva, and V. M. Sha- laev, Silicon Nitride Waveguides with Intrinsic Single- Photon Emitters for Integrated Quantum Photonics, ACS Photonics 9, 3357 (2022)

work page 2022
[6]

Y. C. Chen, S. C. Lin, J. P. Chou, Y. C. Tsai, C. T. Huang, C. J. Lee, and W. H. Chang, Stable Single Pho- ton Emitters with Large Debye-Waller Factor in Silica, ACS Photonics 12, 1461 (2025)

work page 2025
[7]

Goodman and M

P. Goodman and M. OKe- eﬀe, The space group of β -Si3N4, Acta Crystallogr. B Struct. Crystallogr. Cryst. Chem. 36, 2891 (1980)

work page 1980
[8]

J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid functionals based on a screened Coulomb potential, J. Chem. Phys. 118, 8207 (2003)

work page 2003
[9]

P. E. Bl¨ ochl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994)

work page 1994
[10]

Kresse and J

G. Kresse and J. Furthm¨ uller, Eﬃcient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996)

work page 1996
[11]

Asano, M

T. Asano, M. Tezura, M. Saitoh, H. Tanaka, J. Kikkawa, and K. Kimoto, Nanoscale observation of subgap excita- tions in β -Si3N4 with a high refractive index using low- voltage monochromated STEM: a new approach to ana- lyze the physical properties of defects in dielectric mate- rials, Appl. Phys. Express 15, 076501 (2022)

work page 2022
[12]

The Supplemental Material also includes References [11, 13-21]

Refer to the Supplemental Material for comprehensive details on bandgap calculation; the methodology for for- mation energy calculations; photoluminescence spectra and electron-phonon coupling analysis; Debye-Waller fac- tor and localization ratio evaluations; radiative transit ion rates and lifetime estimations; formation energies under silicon-rich cond...

work page
[13]

Zhang and J

S. Zhang and J. Northrup, Chemical potential depen- dence of defect formation energies in GaAs: Application to Ga self-diﬀusion, Phys. Rev. Lett. 67, 2339 (1991)

work page 1991
[14]

Kumagai and F

Y. Kumagai and F. Oba, Electrostatics-based ﬁnite-size corrections for ﬁrst-principles point defect calculation s, Phys. Rev. B 89, 195205 (2014)

work page 2014
[15]

S. R. Kavanagh, A. G. Squires, A. Nicolson, I. Mosquera- Lois, A. M. Ganose, B. Zhu, K. Brlec, A. Walsh, and D. O. Scanlon, doped: Python toolkit for robust and repeatable charged defect supercell calculations, J. Open Source Softw. 9, 6433 (2024)

work page 2024
[16]

Huang and R

K. Huang and R. A., Theory of light absorp- tion and non-radiative transitions in F-centres, Proc. R. Soc. Lond. A 204, 406 (1950)

work page 1950
[17]

Alkauskas, B

A. Alkauskas, B. B. Buckley, D. D. Awschalom, and C. G. Van de Walle, First-principles theory of the luminescence lineshape for the triplet transition in diamond NV cen- tres, New J. Phys. 16, 073026 (2014)

work page 2014
[18]

Meher and M

S. Meher and M. Dey, DefectPl (2024), GitHub reposi- tory

work page 2024
[19]

Lax, The Franck-Condon Principle and Its Applica- tion to Crystals, J

M. Lax, The Franck-Condon Principle and Its Applica- tion to Crystals, J. Chem. Phys. 20, 1752 (1952)

work page 1952
[20]

Kubo and Y

R. Kubo and Y. Toyozawa, Application of the Method of Generating Function to Radiative and Non-Radiative Transitions of a Trapped Electron in a Crystal, Prog. Theor. Phys. 13, 160 (1955)

work page 1955
[21]

Zheng, VaspBandUnfolding (2023)

Q. Zheng, VaspBandUnfolding (2023)

work page 2023
[22]

H. J. Monkhorst and J. D. Pack, Spe- cial points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976)

work page 1976
[23]

HARDIE and K

D. HARDIE and K. H. JACK, Crystal structures of sili- con nitride, Nature 180, 332333 (1957)

work page 1957
[24]

J. P. Perdew, K. Burke, and M. Ernzerhof, Gen- eralized Gradient Approximation Made Simple, Phys. Rev. Lett. 77, 3865 (1996)

work page 1996
[25]

A. Gali, E. Janzn, P. Dek, G. Kresse, and E. Kaxiras, Theory of Spin-Conserving Excitation of the N-V − Cen- ter in Diamond, Phys. Rev. Lett. 103, 186404 (2009)

work page 2009
[26]

Togo and I

A. Togo and I. Tanaka, First principles phonon calcula- tions in materials science, Scr. Mater. 108, 15 (2015)

work page 2015
[27]

M. Dey, S. Meher, and A. K. Singh, Carbon with Stone-Wales defect as quantum emitter in h-BN, Phys. Rev. B 111, 104109 (2025)

work page 2025
[28]

M. E. Grillo, S. D. Elliott, and C. Freysoldt, Native defects in hexagonal β -Si3N4 studied using density functional theory calculations, Phys. Rev. B 83, 085208 (2011)

work page 2011
[29]

P. M. Lenahan and S. E. Curry, First observation of the 29Si hyperﬁne spectra of silicon dangling bond centers in silicon nitride, Appl. Phys. Lett. 56, 157 (1990)

work page 1990
[30]

Nanataki, K

F. Nanataki, K. Shiraishi, J.-i. Iwata, Y.-i. Matsushita, and A. Oshiyama, Atomic and electronic structures of ni- trogen vacancies in silicon nitride: Emergence of ﬂoating gap states, Phys. Rev. B 106, 155201 (2022)

work page 2022
[31]

Bersuker, The Jahn-Teller Eﬀect (Cambridge Univer- sity Press, 2006)

I. Bersuker, The Jahn-Teller Eﬀect (Cambridge Univer- sity Press, 2006)

work page 2006
[32]

Batalov, C

A. Batalov, C. Zierl, T. Gaebel, P. Neumann, I.-Y. Chan, G. Balasubramanian, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, Temporal Coherence of Photons Emitted by Single Nitrogen-Vacancy Defect Centers in Diamond Using Optical Rabi-Oscillations, Phys. Rev. Lett. 100, 077401 (2008)

work page 2008
[33]

Gali, Ab initio theory of the nitrogen-vacancy center in diamond, Nanophotonics 8, 19071943 (2019)

A. Gali, Ab initio theory of the nitrogen-vacancy center in diamond, Nanophotonics 8, 19071943 (2019) . 1 Supplemental Materials: Origin of Bright Quantum Emission s with High Debye-Waller factor in Silicon Nitride SECTION S1: BULK PROPER TIES The bandgaps of β -Si3N4 obtained from various theoretical approaches and experimental measurements are sum- mari...

work page 2019

[1] [1]

Aharonovich, D

I. Aharonovich, D. Englund, and M. Toth, Solid-state single-photon emitters, Nat. Photonics 10, 631641 (2016)

work page 2016

[2] [2]

M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, 5 The nitrogen-vacancy colour centre in diamond, Phys. Rep. 528, 1 (2013)

work page 2013

[3] [3]

Z. O. Martin, A. Senichev, S. Peana, B. J. Lawrie, A. S. Lagutchev, A. Boltasseva, and V. M. Sha- laev, Photophysics of Intrinsic Single-Photon Emitters in Silicon Nitride at Low Temperatures, Adv. Quantum Technol. 6, 2300099 (2023)

work page 2023

[4] [4]

Senichev, Z

A. Senichev, Z. O. Martin, S. Peana, D. Sychev, X. Xu, A. S. Lagutchev, A. Boltasseva, and V. M. Shalaev, Room-temperature single-photon emitters in silicon ni- tride, Sci. Adv. 7, eabj0627 (2021)

work page 2021

[5] [5]

Senichev, S

A. Senichev, S. Peana, Z. O. Martin, O. Yesilyurt, D. Sy- chev, A. S. Lagutchev, A. Boltasseva, and V. M. Sha- laev, Silicon Nitride Waveguides with Intrinsic Single- Photon Emitters for Integrated Quantum Photonics, ACS Photonics 9, 3357 (2022)

work page 2022

[6] [6]

Y. C. Chen, S. C. Lin, J. P. Chou, Y. C. Tsai, C. T. Huang, C. J. Lee, and W. H. Chang, Stable Single Pho- ton Emitters with Large Debye-Waller Factor in Silica, ACS Photonics 12, 1461 (2025)

work page 2025

[7] [7]

Goodman and M

P. Goodman and M. OKe- eﬀe, The space group of β -Si3N4, Acta Crystallogr. B Struct. Crystallogr. Cryst. Chem. 36, 2891 (1980)

work page 1980

[8] [8]

J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid functionals based on a screened Coulomb potential, J. Chem. Phys. 118, 8207 (2003)

work page 2003

[9] [9]

P. E. Bl¨ ochl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994)

work page 1994

[10] [10]

Kresse and J

G. Kresse and J. Furthm¨ uller, Eﬃcient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996)

work page 1996

[11] [11]

Asano, M

T. Asano, M. Tezura, M. Saitoh, H. Tanaka, J. Kikkawa, and K. Kimoto, Nanoscale observation of subgap excita- tions in β -Si3N4 with a high refractive index using low- voltage monochromated STEM: a new approach to ana- lyze the physical properties of defects in dielectric mate- rials, Appl. Phys. Express 15, 076501 (2022)

work page 2022

[12] [12]

The Supplemental Material also includes References [11, 13-21]

Refer to the Supplemental Material for comprehensive details on bandgap calculation; the methodology for for- mation energy calculations; photoluminescence spectra and electron-phonon coupling analysis; Debye-Waller fac- tor and localization ratio evaluations; radiative transit ion rates and lifetime estimations; formation energies under silicon-rich cond...

work page

[13] [13]

Zhang and J

S. Zhang and J. Northrup, Chemical potential depen- dence of defect formation energies in GaAs: Application to Ga self-diﬀusion, Phys. Rev. Lett. 67, 2339 (1991)

work page 1991

[14] [14]

Kumagai and F

Y. Kumagai and F. Oba, Electrostatics-based ﬁnite-size corrections for ﬁrst-principles point defect calculation s, Phys. Rev. B 89, 195205 (2014)

work page 2014

[15] [15]

S. R. Kavanagh, A. G. Squires, A. Nicolson, I. Mosquera- Lois, A. M. Ganose, B. Zhu, K. Brlec, A. Walsh, and D. O. Scanlon, doped: Python toolkit for robust and repeatable charged defect supercell calculations, J. Open Source Softw. 9, 6433 (2024)

work page 2024

[16] [16]

Huang and R

K. Huang and R. A., Theory of light absorp- tion and non-radiative transitions in F-centres, Proc. R. Soc. Lond. A 204, 406 (1950)

work page 1950

[17] [17]

Alkauskas, B

A. Alkauskas, B. B. Buckley, D. D. Awschalom, and C. G. Van de Walle, First-principles theory of the luminescence lineshape for the triplet transition in diamond NV cen- tres, New J. Phys. 16, 073026 (2014)

work page 2014

[18] [18]

Meher and M

S. Meher and M. Dey, DefectPl (2024), GitHub reposi- tory

work page 2024

[19] [19]

Lax, The Franck-Condon Principle and Its Applica- tion to Crystals, J

M. Lax, The Franck-Condon Principle and Its Applica- tion to Crystals, J. Chem. Phys. 20, 1752 (1952)

work page 1952

[20] [20]

Kubo and Y

R. Kubo and Y. Toyozawa, Application of the Method of Generating Function to Radiative and Non-Radiative Transitions of a Trapped Electron in a Crystal, Prog. Theor. Phys. 13, 160 (1955)

work page 1955

[21] [21]

Zheng, VaspBandUnfolding (2023)

Q. Zheng, VaspBandUnfolding (2023)

work page 2023

[22] [22]

H. J. Monkhorst and J. D. Pack, Spe- cial points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976)

work page 1976

[23] [23]

HARDIE and K

D. HARDIE and K. H. JACK, Crystal structures of sili- con nitride, Nature 180, 332333 (1957)

work page 1957

[24] [24]

J. P. Perdew, K. Burke, and M. Ernzerhof, Gen- eralized Gradient Approximation Made Simple, Phys. Rev. Lett. 77, 3865 (1996)

work page 1996

[25] [25]

A. Gali, E. Janzn, P. Dek, G. Kresse, and E. Kaxiras, Theory of Spin-Conserving Excitation of the N-V − Cen- ter in Diamond, Phys. Rev. Lett. 103, 186404 (2009)

work page 2009

[26] [26]

Togo and I

A. Togo and I. Tanaka, First principles phonon calcula- tions in materials science, Scr. Mater. 108, 15 (2015)

work page 2015

[27] [27]

M. Dey, S. Meher, and A. K. Singh, Carbon with Stone-Wales defect as quantum emitter in h-BN, Phys. Rev. B 111, 104109 (2025)

work page 2025

[28] [28]

M. E. Grillo, S. D. Elliott, and C. Freysoldt, Native defects in hexagonal β -Si3N4 studied using density functional theory calculations, Phys. Rev. B 83, 085208 (2011)

work page 2011

[29] [29]

P. M. Lenahan and S. E. Curry, First observation of the 29Si hyperﬁne spectra of silicon dangling bond centers in silicon nitride, Appl. Phys. Lett. 56, 157 (1990)

work page 1990

[30] [30]

Nanataki, K

F. Nanataki, K. Shiraishi, J.-i. Iwata, Y.-i. Matsushita, and A. Oshiyama, Atomic and electronic structures of ni- trogen vacancies in silicon nitride: Emergence of ﬂoating gap states, Phys. Rev. B 106, 155201 (2022)

work page 2022

[31] [31]

Bersuker, The Jahn-Teller Eﬀect (Cambridge Univer- sity Press, 2006)

I. Bersuker, The Jahn-Teller Eﬀect (Cambridge Univer- sity Press, 2006)

work page 2006

[32] [32]

Batalov, C

A. Batalov, C. Zierl, T. Gaebel, P. Neumann, I.-Y. Chan, G. Balasubramanian, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, Temporal Coherence of Photons Emitted by Single Nitrogen-Vacancy Defect Centers in Diamond Using Optical Rabi-Oscillations, Phys. Rev. Lett. 100, 077401 (2008)

work page 2008

[33] [33]

Gali, Ab initio theory of the nitrogen-vacancy center in diamond, Nanophotonics 8, 19071943 (2019)

A. Gali, Ab initio theory of the nitrogen-vacancy center in diamond, Nanophotonics 8, 19071943 (2019) . 1 Supplemental Materials: Origin of Bright Quantum Emission s with High Debye-Waller factor in Silicon Nitride SECTION S1: BULK PROPER TIES The bandgaps of β -Si3N4 obtained from various theoretical approaches and experimental measurements are sum- mari...

work page 2019