Defect-Controlled Multiferroicity via Stacking Control in Nonmagnetic van der Waals Bilayers

Andrew M. Rappe; Bumseop Kim; Sayed Ali Akbar Ghorashi

REVIEW 4 major objections 5 minor 45 references

Defect-Controlled Multiferroicity via Stacking Control in Nonmagnetic van der Waals Bilayers

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A vacancy pair turns nonmagnetic bilayer hBN into an electrically switchable antiferromagnet.

desk verdict A clean symmetry story and internally consistent DFT data that deserve referee time, but the 10.9% magnetic contrast needs a spin-density partition check before I'd bet on it. read the letter →

arxiv 2608.07666 v1 pith:G7S6OSXB submitted 2026-08-07 cond-mat.mtrl-sci

Bumseop Kim , Sayed Ali Akbar Ghorashi , Andrew M. Rappe This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords hexagonalboronnitrideslidingferroelectricityvacancydefectslocalizedmagnetismNéelorderelectric-fieldcontrolvanderWaalsbilayersstackingregistry

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

The paper aims to establish that the sliding registry of a nonmagnetic van der Waals bilayer can serve as a local control parameter for magnetism once vacancy defects are introduced. In bilayer hexagonal boron nitride, a nitrogen vacancy carries a spin-1/2 moment and a boron vacancy a spin-1 moment; a single vacancy lifts the degeneracy between the two polar sliding states (AB and BA), and in interlayer pairs the hosting sublattice dictates whether the coupling is ferrimagnetic or antiferromagnetic. For the antiferromagnetic nitrogen-vacancy pair, an out-of-plane electric field selects the preferred registry and changes the magnitude of the layer-resolved Néel order by about 10.9% at $|E_z| = 0.10\ \mathrm{V/\AA}$. The reason to care is that this is a purely electrostatic route to magnetoelectric control in a material with no magnetic atoms, avoiding the usual scarcity of single-phase two-dimensional multiferroics.

What carries the argument

The load-bearing object is a symmetry-adapted free energy for the bilayer built from the sliding coordinate $Q_s$ (BA = +1, AB = -1), the defect-layer variable $\eta_{\mathrm{def}}$, the defect-species variable $\tau_{\mathrm{def}}$ ($V'_B$, pristine, $V_N^\times$), and the layer-resolved Néel order $L_z=(M_{\mathrm{top},z}-M_{\mathrm{bottom},z})/2$. Symmetry of the AA reference structure admits $Q_s E_z L_z^2$ as the lowest-order field-linear invariant that distinguishes AB from BA while leaving the two time-reversed Néel states degenerate; this term is what converts an out-of-plane electric field into a change in $|L_z|$ rather than a spin reorientation. The single-defect couplings $Q_s\eta_{\mathrm{def}}\tau_{\mathrm{def}}$ and $Q_s\tau_{\mathrm{def}}^2$ encode the vacancy-selected registry preference and the registry-even polarization offset, and all coefficients are fitted to first-principles total energies, Berry-phase polarizations, and field-dependent layer-resolved moments.

What would settle it

Prepare bilayer hBN with one nitrogen vacancy per layer, apply out-of-plane electric fields of $+0.10$ and $-0.10\ \mathrm{V/\AA}$, and measure the layer-resolved magnetization: if the magnitude of the Néel order does not change by roughly 10.9% between the stacking registries selected by the two field directions, the central claim of electric-field amplitude control fails. A separate check is whether the Fermi level can actually be pinned in the $3.48$–$4.59\ \mathrm{eV}$ window in which both required vacancy charge states are stable.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that atomic vacancies in bilayer hBN are registry sensors, not passive impurities: a single neutral nitrogen vacancy $V_N^\times$ or negatively charged boron vacancy $V'_B$ breaks the AB–BA equivalence of the sliding-ferroelectric bilayer and produces a registry-even polarization offset that reverses when the vacancy moves to the opposite layer. In interlayer pairs, the hosting sublattice fully dictates the exchange: $V'_B$–$V'_B$ moments remain nearly decoupled, $V'_B$–$V_N^\times$ couples antiparallel into a ferrimagnet, and $V_N^\times$–$V_N^\times$ forms a compensated antiferromagnet stabilized by kinetic exchange. For the $V_N^\times$–$V_N^\times$ pair, a finite out-of-plane electric field lifts the AB–BA degeneracy approximately linearly, and the field-favored registry exhibits the larger defect-localized moment, with a registry-dependent contrast of about 10.9% in the layer-resolved Néel order $L_z=(M_{\mathrm{top},z}-M_{\mathrm{bottom},z})/2$ at $|E_z|=0.10\ \mathrm{V/\AA}$. The field modulates the amplitude of the Néel vector, not its orientation, and the time-reversed states remain degenerate, so the result is amplitude control of a compensated antiferromagnetic order parameter without an external magnetic field or spin-orbit coupling.

Load-bearing premise

The scenario depends on the Fermi level in real hBN being pinned near $3.6\ \mathrm{eV}$ above the valence-band maximum so that neutral nitrogen vacancies and negatively charged boron vacancies are stable in the same sample, and on the layers sliding rigidly without relaxing around the vacancies.

Editorial extensions

If this is right

A single vacancy can act as a local, nondestructive readout of the stacking registry, since its formation energy and polarization offset distinguish AB from BA.
An out-of-plane electric field gives nonvolatile, purely electrostatic control over a compensated antiferromagnetic order parameter in a nonmagnetic parent material.
The hosting sublattice of the vacancy, not just the presence of a vacancy, determines the magnetic ground state of an interlayer pair, providing a selection rule for ferrimagnetic versus antiferromagnetic coupling.
The same registry–defect coupling should occur in other polar sliding bilayers, moiré domains, and transition-metal dichalcogenide heterostructures that support localized moments.

Reading between the lines

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

If the mechanism generalizes as claimed, a moiré superlattice formed by twisting two defective hBN layers should show a spatial map of registry-dependent Néel amplitude, making the stacking texture electrically readable at the nanoscale.
Because the $V_N^\times$–$V_N^\times$ exchange is kinetic-exchange-like, interlayer pressure or twist angle should tune both the antiferromagnetic stabilization energy and the field response; this is a testable prediction beyond the paper's explicit calculations.
The invariant $Q_s E_z L_z^2$ is fixed by symmetry and does not depend on the specific defect chemistry, so the same amplitude modulation should appear in any polar sliding bilayer with localized vacancy moments, provided the two layers carry identical defects.

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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

Short version: this is a serious computational prediction with a clean symmetry story, and I think it deserves referee time. But the headline 10.9% effect is softer than the abstract suggests: it comes from a half-cell spin-density partition, and the total-energy fit only constrains a combination of direct-polar and magnetic coefficients. The authors know this and say so, which helps.

What's new: a single vacancy breaks the AB-BA degeneracy in bilayer hBN, and a pair of nitrogen vacancies gives a compensated Néel order whose amplitude can be modulated by an out-of-plane field through the registry. The sublattice-selective exchange (BB weak, BN ferrimagnetic, NN antiferromagnetic) is a concrete pattern that could be tested in other vdW bilayers. The symmetry analysis rooted in the AA reference structure is the paper's own, and the Landau expansion is used consistently with it.

What's good: raw DFT numbers are in tables, the charged-supercell treatment is at least careful, and the authors explicitly acknowledge that p_eff cannot be separated into polar and magnetic shares from energy data, and that the AFM state is a broken-symmetry solution. That disclosure makes me trust the numbers.

Soft spots: first, the 10.9% registry contrast is defined via L_z obtained from a planar cut at the interlayer midpoint. That is a legitimate definition of a layer moment, but it is not a defect-localized moment, and no check is given for the cut position or an alternate projection (Bader, atomic spheres, Wannier). The spin density is likely well localized on the three boron neighbors, so I suspect the effect is real, but it is a missing sanity check. Second, no convergence tests (k-points, cutoff, supercell size) are shown for meV-scale splittings, and the sliding paths are rigid with fixed intralayer coordinates. Local relaxation around a vacancy could alter the registry coupling. Third, the simultaneous stability of V_N and V'_B rests on E_F - E_VBM = 3.6 eV from the literature; that is a physically motivated assumption but still an assumption.

Net: this is a solid, internally consistent prediction, and the Q_s E_z L_z^2 symmetry argument is correct. The main risk is that the quantitative 10.9% effect is partition-dependent. I would send it to a referee, with the expectation that they ask for a spin-density partition check and convergence tests before publication.

Referee Report

4 major / 5 minor

Summary. The paper proposes a general mechanism that couples vacancy-localized magnetic moments to sliding ferroelectricity in nonmagnetic van der Waals bilayers, using bilayer hBN as a prototype. The authors perform first-principles DFT calculations for single nitrogen and boron vacancies and for interlayer vacancy pairs. They show that a single vacancy lifts the AB/BA registry degeneracy and creates a registry-even polarization offset, and that interlayer vacancy pairs exhibit sublattice-dependent exchange: V'_B-V'_B is nearly magnetically decoupled, V'_B-V_N forms a ferrimagnetic state, and V_N-V_N forms a compensated antiferromagnetic state. The central result is that for the V_N-V_N pair, an out-of-plane electric field selects the polar registry and changes the layer-resolved Néel order magnitude by about 10.9% at |E_z| = 0.10 V/Å. This behavior is rationalized with a Landau free-energy expansion in which Q_s E_z L_z^2 is the lowest-order invariant coupling the sliding coordinate, the field, and the squared Néel amplitude. The paper also reports Berry-phase polarizations and fully relativistic noncollinear calculations to support the symmetry arguments.

Significance. If the reported amplitude modulation is robust, this work provides a conceptually new route to magnetoelectric functionality in nominally nonmagnetic 2D systems, with potential for spintronic readout of sliding ferroelectricity. The symmetry-based Landau framework is elegant and well matched to the DFT results, and the authors are unusually transparent about the limitations of their model: they explicitly disclose the underdetermination of p_eff, the Fermi-level dependence of the vacancy charge states, and the rigid-sliding approximation. The paper also reports internal consistency checks, including time-reversal degeneracy of the Néel states and polarization invariance under L_z reversal. However, the quantitative centerpiece of the work, the 10.9% contrast in |L_z|, depends on a particular half-cell partition of the spin density and on calculations for which no convergence tests are shown. The significance is therefore conditional on additional validation of this order parameter and on the numerical precision of the small energy scales involved.

major comments (4)

[SM, Eqs. (S10)-(S12); main text Fig. 3(e)] The central quantitative result of a ~10.9% registry-dependent contrast in |L_z| is obtained by partitioning the out-of-plane spin density at the interlayer midpoint (SM Eqs. S10-S12). This half-cell integral captures all magnetization in each half of the cell, including delocalized tails and field-induced charge redistribution, rather than isolating the vacancy-localized moment on the three B atoms neighboring each V_N. The main text refers to this quantity as the 'defect-localized moment' and interprets the field-induced change as amplitude modulation of the Néel order. Because the Q_s E_z L_z^2 symmetry invariant is compatible with any continuous registry-dependent spin-density redistribution, the physical interpretation is only as strong as the partition. Please provide a partition-independence test (e.g., varying the dividing plane, atomic-basin/Bader projection, or Wannier-based local moments) or revise the language to describe the change in the layer-projected magnetization density rather than the defect-localized moment.
[SM Eq. (S19) and Eq. (S15); main text Eq. (3)] The total-energy fit determines only the combined coefficient p_eff = A P_pair + gamma L_0^2, not the magnetic coefficient gamma separately. Thus the registry preference under an electric field can be explained entirely by the direct polar term -A P_pair Q_s E_z, with no magnetic contribution. The claim that the electric field modulates the amplitude of the Néel order relies on the L_z values from the half-cell partition (Table S4) and on the fit of Eq. (S22). I ask the authors to either compute the pair polarization P_pair independently (e.g., from Berry-phase polarization of the pair system as a function of E_z) or perform constrained-L_z calculations that separate the two contributions. The manuscript already discloses this degeneracy, but the magnetoelectric attribution is a central claim and should be supported by data that do not presuppose the partition.
[SM, Computational Methods; Tables S2 and S4] No convergence tests with respect to plane-wave cutoff, k-point grid, or supercell size are reported. The decisive quantities are small: the AB-BA energy splittings for single vacancies are about 1-4 meV, the pair magnetic splittings range down to 0.07 meV, and the magnetic anisotropy energies in Table S2 are 0.1-4.9 µeV, close to total-energy noise. The 10.9% contrast and the linear slopes in Eq. (S22) are quantitative claims that require at least a subset of convergence checks (e.g., a 7x7x1 grid or an 8x8 supercell for the V_N-V_N pair at one or two fields). Without such checks, the numerical precision of the central result is not established.
[SM, 'Selection of Vacancy Charge States', Table S1] The adopted Fermi-level window E_F - E_VBM ≈ 3.6 eV is motivated by N-rich, impurity-containing hBN. In clean or differently doped samples, the stable charge states in Table S1 change, so the neutral V_N species and the negatively charged V'_B species are not simultaneously present, and the specific vacancy pairs discussed here would not form. The authors disclose this assumption but the abstract and conclusion state the mechanism as 'robust' and 'general'. Please state more explicitly in the main text that the demonstrated AFM V_N-V_N pair requires this specific Fermi-level window, and discuss how the proposed mechanism would be modified if adjacent charge states become stable under gating.

minor comments (5)

[Main text, End-matter Fig. 4] The figure labeled 'Fig. 4' appears after the references and is described only in an unnumbered paragraph; it should be placed in the main text or cited with a proper figure environment.
[Main text after Eq. (2)] The notation for τ_def is introduced only implicitly; please state explicitly that τ_def = +1, 0, -1 for V'_B, pristine, and V_N^×, respectively, in the same sentence as the definition of τ_def.
[SM, Eq. (S14)] There is a formatting error: '2X i=1' should read the summation over i = 1 to 2; as printed it is not readable.
[Abstract and Introduction] The phrase 'nonmagnetic van der Waals bilayers' is potentially confusing because the bilayers are nonmagnetic only in their pristine form; consider using 'nonmagnetic host' or 'nominally nonmagnetic'.
[SM, Table S1] The superscript symbols for the vacancy charge states (e.g., V_N^•, V_N^', V'_B) are not defined in text; a one-sentence explanation of the charge-state notation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the DFT data are the evidence, and the Landau model is a transparent post hoc fit whose bundled coefficient is explicitly not decomposed.

full rationale

The paper's central claims rest on raw first-principles quantities: total energies, Berry-phase polarizations, and layer-resolved magnetization densities. The Landau free energy is introduced after those quantities are computed and is fitted to them, so it is not used to derive a prediction independent of its own inputs. The authors state this explicitly in the SM: "The total-energy data determine only the combined coefficient p_eff. Separating AP_pair from γL_0^2 would require an independent determination of the pair polarization contribution or calculations at constrained L_z" and "The present data determine the ratios γ/b and ζ/b, but not the individual Landau coefficients." The 10.9% registry contrast is computed directly from the DFT values in Table S4 via Eq. (S24), not obtained from a Landau fit. The Q_s E_z L_z^2 invariant is justified by the symmetry classification of the AA reference structure, which is independent of the fitted coefficients. No load-bearing self-citation appears: the only author-cited item is the dataset [38], and the charge-state selection is supported by external thermodynamic studies [19,32,33]. Concerns about the half-cell partition used for L_z or about the assumed charge-state stability are validity or robustness issues, not circularity, because no step equates a claimed result to a definition or to a fitted parameter renamed as a prediction.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claims depend on a set of fitted free-energy coefficients and on assumptions about DFT accuracy, rigid-layer sliding, charge-state stability, and the truncation of the Landau expansion. The phenomenological model does not add independent predictive power beyond the DFT data it is fit to.

free parameters (7)

c20, c40, c11 (pristine sliding free-energy coefficients) = -88.63 meV/f.u., 55.43 meV/f.u., 1.143 pC/m
Fitted to the DFT relative-energy and Berry-phase polarization profiles of pristine bilayer hBN along AA-AB-AA-BA-AA.
c1110, c1120, c1011, c1021, c0111, c0121 (single-defect coupling coefficients) = -0.70, -1.30 meV/f.u.; -0.061, -0.214, 0.079, 0.319 pC/m
Simultaneous fit to DFT total-energy and polarization for single-vacancy AB/BA splittings; determine registry-defect coupling terms in Eq. (2).
c121(d) Lz^2 (magnetoelectric coefficient times squared Neel order) = 0.0196 e Å
Extracted from the 3.92 meV AB-BA splitting at E_z = 0.10 V/Å for the V_N-V_N pair; sets the Q_s E_z L_z^2 coupling strength.
L0 = 0.4495 μB
Zero-field magnitude of the layer-resolved Neel order from fit of |L_z|^2(E_z) in SM Eq. (S22).
gamma/(2b) = 0.20056 μB^2 Å/V
Fit coefficient of linear E_z dependence of |L_z|^2; drives the registry-dependent amplitude modulation.
zeta/(2b) = 0.51977 μB^2 Å^2/V^2
Fit coefficient of quadratic E_z dependence of |L_z|^2 in SM Eq. (S22).
p_eff = 0.01965 e Å
Slope of E_BA - E_AB versus E_z; combines direct polar contribution A P_pair and magnetic renormalization gamma L0^2; the two cannot be separated from total-energy data.

assumptions (6)

domain assumption PBE-GGA exchange-correlation with DFT-D3 dispersion and norm-conserving pseudopotentials accurately describes vacancy magnetism and interlayer energetics in hBN.
All energies and polarizations are computed with this DFT setup (SM Computational Methods); no benchmark against higher-level theory is provided.
domain assumption Rigid-layer sliding keeps intralayer coordinates and interlayer separation fixed.
SM: 'interlayer sliding paths were constructed by rigidly translating one hBN layer relative to the other'; defect-induced local relaxation is not included.
domain assumption E_F - E_VBM = 3.6 eV is a physically appropriate Fermi level for stabilizing V'_B and V_N.
SM Selection of Vacancy Charge States, based on N-rich impurity-containing hBN from refs [32,33]; if the environment differs, these charge states may not coexist.
ad hoc to paper Truncation of the free-energy expansion at lowest-order allowed invariants such as Q_s E_z L_z^2 is sufficient.
Eq. (3) and SM Eq. (S15) retain only leading terms; higher-order couplings such as L_z^4 E_z^2 are neglected without quantitative justification.
standard math Time-reversal symmetry forbids linear-in-L_z terms.
Used to restrict the free energy to even powers of L_z and to argue P_z is invariant under Neel reversal.
domain assumption DFT broken-symmetry solutions represent the AFM singlet state.
SM acknowledges the AP solution is a broken-symmetry representation, not an exact many-body singlet.

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Cite this review

Pith. "Pith review of Defect-Controlled Multiferroicity via Stacking Control in Nonmagnetic van der Waals Bilayers." pith.science (2026). https://pith.science/paper/G7S6OSXB

@misc{pith2026260807666,
  author       = {Pith},
  title        = {Pith review of: Defect-Controlled Multiferroicity via Stacking Control in Nonmagnetic van der Waals Bilayers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G7S6OSXB}},
  note         = {Machine review of arXiv:2608.07666}
}

read the original abstract

We present a general paradigm that directly couples vacancy-localized magnetism to interfacial sliding ferroelectricity in nonmagnetic van der Waals (vdWs) bilayers. Utilizing bilayer hexagonal boron nitride (hBN) as a prototypical vdWs host system, we use first-principles calculations to show that a single vacancy acts as a local registry sensor, lifting the degeneracy between polar sliding partners via a defect-centered polarization offset. For interlayer vacancy pairs, we discover a defect selectivity where the hosting sublattice fully dictates the interlayer exchange, stabilizing either ferrimagnetic or antiferromagnetic configurations. Applying an out-of-plane electric field selects the polar registry and drives an amplitude modulation of the compensated N\'eel order parameter. These findings establish a robust, sublattice-dependent engineering of multiferroic functionality via stacking control across a wide class of nonmagnetic 2D heterostructures.

Figures

Figures reproduced from arXiv: 2608.07666 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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**Figure 4.** Figure 4: FIG. 4. Out-of-plane Berry-phase polarization [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

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Works this paper leans on

45 extracted references · 28 canonical work pages

[1]

N. A. Spaldin and M. Fiebig, The renaissance of magne- toelectric multiferroics, Science309, 391 (2005)

2005
[2]

Eerenstein, N

W. Eerenstein, N. D. Mathur, and J. F. Scott, Multi- ferroic and magnetoelectric materials, Nature442, 759 (2006)

2006
[3]

Cheong and M

S.-W. Cheong and M. Mostovoy, Multiferroics: A mag- netic twist for ferroelectricity, Nature Materials6, 13 (2007)

work page 2007
[4]

Fiebig, T

M. Fiebig, T. Lottermoser, D. Meier, and M. Trassin, The evolution of multiferroics, Nature Reviews Materials 1, 16046 (2016)

2016
[5]

A. K. Geim and I. V. Grigorieva, Van der waals het- erostructures, Nature499, 419 (2013)

2013
[6]

K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. Castro Neto, 2d materials and van der waals heterostruc- tures, Science353, aac9439 (2016)

2016
[7]

Huang, G

B. Huang, G. Clark, D. R. Klein, D. MacNeill, E. Navarro-Moratalla, K. L. Seyler, N. Wilson, M. A. McGuire, D. H. Cobden, D. Xiao, W. Yao, P. Jarillo- Herrero, and X. Xu, Electrical control of 2d magnetism in bilayer CrI3, Nature Nanotechnology13, 544 (2018)

2018
[8]

Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun, Y. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen, and Y. Zhang, Gate-tunable room-temperature ferro- magnetism in two-dimensional Fe 3GeTe2, Nature563, 94 (2018)

work page 2018

Show all 45 references

[9]

Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song, J. Sanchez, W. Yao, D. Xiao, X. Zhu, A. F. May, W. Wu, D. H. Cobden, J.-H. Chu, and X. Xu, Two-dimensional itinerant ferromagnetism in atomically thin Fe 3GeTe2, Nature Materials17, 778 (2018)

2018
[10]

C. Gong, E. M. Kim, Y. Wang, G. Lee, and X. Zhang, Multiferroicity in atomic van der waals heterostructures, Nature Communications10, 2657 (2019)

2019
[11]

Y. Gao, M. Gao, and Y. Lu, Two-dimensional multifer- roics, Nanoscale13, 19324 (2021)

2021
[12]

Y. Wu, Z. Zeng, H. Lu, X. Han, C. Yang, N. Liu, X. Zhao, L. Qiao, W. Ji, R. Che, L. Deng, P. Yan, and B. Peng, Coexistence of ferroelectricity and antiferroelectricity in 2d van der waals multiferroic, Nature Communications 15, 8616 (2024)

2024
[13]

Z. Tang, Y. Hu, L. Rog´ ee, D. Liu, and S. P. Lau, To- wards multiferroicity in two-dimensional van der waals materials: Challenges and opportunities, Chemistry of Materials37, 4925 (2025)

2025
[14]

M. S. Si and D. S. Xue, Magnetic properties of vacancies in a graphitic boron nitride sheet by first-principles pseu- dopotential calculations, Physical Review B75, 193409 (2007)

2007
[15]

Yasuda, X

K. Yasuda, X. Wang, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Stacking-engineered ferroelectricity in bilayer boron nitride, Science372, 1458 (2021)

2021
[16]

Huang and H

B. Huang and H. Lee, Defect and impurity properties of hexagonal boron nitride: A first-principles calculation, Physical Review B86, 245406 (2012)

2012
[17]

J. Yang, D. Kim, J. Hong, and X. Qian, Magnetism in boron nitride monolayer: Adatom and vacancy defect, Surface Science604, 1603 (2010)

2010
[18]

Ouyang and J

B. Ouyang and J. Song, Strain engineering of magnetic states of vacancy-decorated hexagonal boron nitride, Ap- plied Physics Letters103, 102401 (2013)

2013
[19]

Weston, D

L. Weston, D. Wickramaratne, M. Mackoit, A. Alka- uskas, and C. G. Van de Walle, Native point defects and impurities in hexagonal boron nitride, Physical Review B97, 214104 (2018)

2018
[20]

Sajid and K

A. Sajid and K. S. Thygesen,V NCB defect as source of single-photon emission from hexagonal boron nitride, 2D Materials7, 031007 (2020)

2020
[21]

Gottscholl, M

A. Gottscholl, M. Kianinia, V. Soltamov, S. Orlin- skii, G. Mamin, C. Bradac, C. Kasper, K. Krambrock, A. Sperlich, M. Toth, I. Aharonovich, and V. Dyakonov, Initialization and read-out of intrinsic spin defects in a van der waals crystal at room temperature, Nature Ma- terial...

2020
[22]

H. L. Stern, Q. Gu, J. Jarman, S. Eizagirre Barker, N. Mendelson, D. Chugh, S. Schott, H. H. Tan, H. Sirringhaus, I. Aharonovich, and M. Atat¨ ure, Room- temperature optically detected magnetic resonance of single defects in hexagonal boron nitride, Nature Com- munications13, ...

2022
[23]

Mathur, A

N. Mathur, A. Mukherjee, X. Gao, J. Luo, B. A. McCul- lian, T. Li, A. N. Vamivakas, and G. D. Fuchs, Excited- state spin-resonance spectroscopy ofV − B defect centers in hexagonal boron nitride, Nature Communications13, 3233 (2022)

2022
[24]

N.-J. Guo, S. Li, W. Liu, Y.-Z. Yang, X.-D. Zeng, S. Yu, Y. Meng, Z.-P. Li, Z.-A. Wang, L.-K. Xie, R.-C. Ge, J.-F. Wang, Q. Li, J.-S. Xu, Y.-T. Wang, J.-S. Tang, A. Gali, C.-F. Li, and G.-C. Guo, Coherent control of an ultra- bright single spin in hexagonal boron nitride at ro...

2023
[25]

Udvarhelyi, T

P. Udvarhelyi, T. Clua-Provost, A. Durand, J. Li, J. H. Edgar, B. Gil, G. Cassabois, V. Jacques, and A. Gali, A planar defect spin sensor in a two-dimensional material susceptible to strain and electric fields, npj Computa- tional Materials9, 150 (2023)

2023
[26]

Durand, T

A. Durand, T. Clua-Provost, F. Fabre, P. Kumar, J. Li, J. H. Edgar, P. Udvarhelyi, A. Gali, X. Marie, C. Robert, J.-M. G´ erard, B. Gil, G. Cassabois, and V. Jacques, Optically active spin defects in few-layer thick hexago- nal boron nitride, Physical Review Letters131, 116902 (2023)

2023
[27]

Sortino, A

L. Sortino, A. Gale, L. K¨ uhner, C. Li, J. Biechteler, F. J. Wendisch, M. Kianinia, H. Ren, M. Toth, S. A. Maier, I. Aharonovich, and A. Tittl, Optically addressable spin defects coupled to bound states in the continuum meta- surfaces, Nature Communications15, 2008 (2024)

2024
[28]

Vizner Stern, Y

M. Vizner Stern, Y. Waschitz, W. Cao, I. Nevo, K. Watanabe, T. Taniguchi, E. Sela, M. Urbakh, O. Hod, and M. Ben Shalom, Interfacial ferroelectricity by van der waals sliding, Science372, 1462 (2021)

2021
[29]

Wu and J

M. Wu and J. Li, Sliding ferroelectricity in 2d van der waals materials: Related physics and future opportuni- ties, Proceedings of the National Academy of Sciences 118, e2115703118 (2021)

2021
[30]

C. R. Woods, P. Ares, H. Nevison-Andrews, M. J. Holwill, R. Fabregas, F. Guinea, A. K. Geim, K. S. Novoselov, N. R. Walet, and L. Fumagalli, Charge- polarized interfacial superlattices in marginally twisted 7 hexagonal boron nitride, Nature Communications12, 347 (2021)

2021
[31]

See Supplemental Material for computational details, fully relativistic magnetic-anisotropy calculations, and the derivation of the two-site Hubbard model
[32]

Maciaszek, L

M. Maciaszek, L. Razinkovas, and A. Alkauskas, Ther- modynamics of carbon point defects in hexagonal boron nitride, Phys. Rev. Mater.6, 014005 (2022)

2022
[33]

Maciaszek and B

M. Maciaszek and B. Baur, Cbvb-n h complexes as preva- lent defects in metal-organic vapor-phase epitaxy-grown hexagonal boron nitride, npj 2D Materials and Applica- tions10, 39 (2026)

2026
[34]

Iv´ ady, G

V. Iv´ ady, G. Barcza, G. Thiering, S. Li, H. Hamdi, J.- P. Chou, ¨O. Legeza, and A. Gali, Ab initio theory of the negatively charged boron vacancy qubit in hexagonal boron nitride, npj Computational Materials6, 41 (2020)

2020
[35]

Li and M

L. Li and M. Wu, Binary compound bilayer and multi- layer with vertical polarizations: Two-dimensional ferro- electrics, multiferroics, and nanogenerators, ACS Nano 11, 6382 (2017)

2017
[36]

A. Gale, D. Scognamiglio, I. Zhigulin, B. Whitefield, M. Kianinia, I. Aharonovich, and M. Toth, Manipulat- ing the charge state of spin defects in hexagonal boron nitride, Nano Letters23, 6141 (2023)

2023
[37]

X. Wang, K. Yasuda, Y. Zhang, S. Liu, K. Watanabe, T. Taniguchi, J. Hone, L. Fu, and P. Jarillo-Herrero, In- terfacial ferroelectricity in rhombohedral-stacked bilayer transition metal dichalcogenides, Nature Nanotechnology 17, 367 (2022)

2022
[38]

B. Kim, S. A. A. Ghorashi, and A. M. Rappe, Defect- controlled multiferroicity via stacking control in nonmag- netic van der waals bilayers (2026), dataset

2026
[39]

Giannozzi, S

P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ- cioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. ...

2009
[40]

Giannozzi, O

P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavaz- zoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carn- imeo, A. Dal Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio Jr., A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R...

2017
[41]

J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Physical Review Letters77, 3865 (1996)

1996
[42]

Grimme, J

S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, A con- sistent and accurate ab initio parametrization of den- sity functional dispersion correction (DFT-D) for the 94 elements h–pu, The Journal of Chemical Physics132, 154104 (2010)

2010
[43]

H. J. Monkhorst and J. D. Pack, Special points for brillouin-zone integrations, Physical Review B13, 5188 (1976)

1976
[44]

R. D. King-Smith and D. Vanderbilt, Theory of polar- ization of crystalline solids, Physical Review B47, 1651 (1993). END MA TTER FIG. 4. Out-of-plane Berry-phase polarizationP z for the AFM (VN-VN) bilayer hBN depending on E-field To verify the self-consistency of this framew...

1993
[45]

The corresponding thermodynamically stable vacancy states are summarized in Table S1

= 4.90 eV forV B, andε(+1/0) = 3.48 eV andε(0/−1) = 4.59 eV forV N, where the Fermi level is referenced to the valence-band maximum and the calculated band gap is 5.94 eV [19]. The corresponding thermodynamically stable vacancy states are summarized in Table S1. In a real hBN ...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

N. A. Spaldin and M. Fiebig, The renaissance of magne- toelectric multiferroics, Science309, 391 (2005)

2005

[2] [2]

Eerenstein, N

W. Eerenstein, N. D. Mathur, and J. F. Scott, Multi- ferroic and magnetoelectric materials, Nature442, 759 (2006)

2006

[3] [3]

Cheong and M

S.-W. Cheong and M. Mostovoy, Multiferroics: A mag- netic twist for ferroelectricity, Nature Materials6, 13 (2007)

work page 2007

[4] [4]

Fiebig, T

M. Fiebig, T. Lottermoser, D. Meier, and M. Trassin, The evolution of multiferroics, Nature Reviews Materials 1, 16046 (2016)

2016

[5] [5]

A. K. Geim and I. V. Grigorieva, Van der waals het- erostructures, Nature499, 419 (2013)

2013

[6] [6]

K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. Castro Neto, 2d materials and van der waals heterostruc- tures, Science353, aac9439 (2016)

2016

[7] [7]

Huang, G

B. Huang, G. Clark, D. R. Klein, D. MacNeill, E. Navarro-Moratalla, K. L. Seyler, N. Wilson, M. A. McGuire, D. H. Cobden, D. Xiao, W. Yao, P. Jarillo- Herrero, and X. Xu, Electrical control of 2d magnetism in bilayer CrI3, Nature Nanotechnology13, 544 (2018)

2018

[8] [8]

Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun, Y. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen, and Y. Zhang, Gate-tunable room-temperature ferro- magnetism in two-dimensional Fe 3GeTe2, Nature563, 94 (2018)

work page 2018