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Danis I. Badrtdinov, Carlos Rodriguez‐Fernandez, Magdalena Grzeszczyk, Zhizhan Qiu, Kristina Vaklinova, Pengru Huang, Alexander Hampel, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Lu Jiong, Marek Potemski, Cyrus E. Dreyer, Maciej Koperski, Malte Rösner

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[Dielectric Environment Sensitivity of Carbon Centers in Hexagonal Boron Nitride](https://mdr.nims.go.jp/datasets/eb5fb797-38b0-422d-8033-c85047bba757)

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Dielectric Environment Sensitivity of Carbon Centers in Hexagonal Boron NitrideRESEARCH ARTICLEwww.small-journal.comDielectric Environment Sensitivity of Carbon Centers inHexagonal Boron NitrideDanis I. Badrtdinov, Carlos Rodriguez-Fernandez, Magdalena Grzeszczyk, Zhizhan Qiu,Kristina Vaklinova, Pengru Huang, Alexander Hampel, Kenji Watanabe, Takashi Taniguchi,Lu Jiong, Marek Potemski, Cyrus E. Dreyer, Maciej Koperski,* and Malte Rösner*A key advantage of utilizing van-der-Waals (vdW) materials as defect-hostingplatforms for quantum applications is the controllable proximity of the defectto the surface or the substrate allowing for improved light extraction,enhanced coupling with photonic elements, or more sensitive metrology.However, this aspect results in a significant challenge for defect identificationand characterization, as the defect’s properties depend on the the atomicenvironment. This study explores how the environment can influence theproperties of carbon impurity centers in hexagonal boron nitride (hBN). Itcompares the optical and electronic properties of such defects betweenbulk-like and few-layer films, showing alteration of the zero-phonon lineenergies and their phonon sidebands, and enhancements of inhomogeneousbroadenings. To disentangle the mechanisms responsible for these changes,including the atomic structure, electronic wavefunctions, and dielectricscreening, it combines ab initio calculations with a quantum-embeddingapproach. By studying various carbon-based defects embedded in monolayerand bulk hBN, it demonstrates that the dominant effect of the change in theenvironment is the screening of density–density Coulomb interactionsbetween the defect orbitals. The comparative analysis of experimental andtheoretical findings paves the way for improved identification of defects inlow-dimensional materials and the development of atomic scale sensors fordielectric environments.D. I. Badrtdinov, M. RösnerInstitute for Molecules and MaterialsRadboud UniversityHeijendaalseweg 135, 6525 AJ Nijmegen, NetherlandsE-mail: m.roesner@science.ru.nlC. Rodriguez-FernandezFaculty of Engineering and Natural SciencesTampere UniversityTampere 33720, FinlandM. Grzeszczyk, K. Vaklinova, P. Huang, M. KoperskiInstitute for Functional Intelligent MaterialsNational University of SingaporeSingapore 117544E-mail: msemaci@nus.edu.sgThe ORCID identification number(s) for the author(s) of this articlecan be found under https://doi.org/10.1002/smll.202300144© 2023 The Authors. Small published by Wiley-VCH GmbH. This is anopen access article under the terms of the Creative Commons AttributionLicense, which permits use, distribution and reproduction in anymedium, provided the original work is properly cited.DOI: 10.1002/smll.2023001441. IntroductionPoint defects in semiconductors and insu-lators have emerged as robust and manip-ulatable quantum systems for applicationssuch as qubits for quantum computers,[1–4]single-photon emitters (SPEs) for quan-tum communication,[5,6] and nanoprobesfor quantum metrology.[7] In this context,2D vdW bonded compounds have been pro-posed as promising host materials. One keybenefit arises from the potentially atom-ically perfect surfaces of these materials.Quantum defects may reside in close prox-imity to the surface without suffering in-stabilities in their optoelectronic and/or co-herence properties due to surface danglingbonds, adsorbates, or local charge varia-tions. This has several advantages for creat-ing sensors of local fields at the nanoscale,better extraction efficiency for defect SPEs,and direct imaging or manipulating de-fects by scanning probe techniques.[8,9] Themost developed of such host material ishexagonal boron nitride (hBN).[10,11] It iswidely available[12] and its large bandgap isZ. Qiu, L. JiongDepartment of ChemistryNational University of Singapore117543, SingaporeP. Huang, M. KoperskiDepartment of Materials Science and EngineeringNational University of SingaporeSingapore 117575, SingaporeP. HuangGuangxi Key Laboratory of Information MaterialsGuilin University of Electronic TechnologyGuilin 541004, ChinaA. Hampel, C. E. DreyerCenter for Computational Quantum PhysicsFlatiron Institute162 5 th Avenue, New York NY 10010, USASmall 2023, 19, 2300144 © 2023 The Authors. Small published by Wiley-VCH GmbH2300144 (1 of 13)http://crossmark.crossref.org/dialog/?doi=10.1002%2Fsmll.202300144&domain=pdf&date_stamp=2023-06-17www.advancedsciencenews.com www.small-journal.comconducive to the formation of deep quantum levels. There havebeen many reports indicating hBN hosts SPEs[13–16] with someallowing for spin manipulations.[17–19] Specifically, it has beenshown experimentally[20–25] and theoretically[26–37] that carbonimpurities in hBN can give rise to a plethora of defect centerswith a wide variety of properties attractive for quantum applica-tions.The existence of defects in proximity to surfaces and inter-faces also presents a significant challenge. Quantum technolo-gies often require arrays of defects with identical properties andthough the surfaces and interfaces of the substrate can be atom-ically clean, proximity to them may change the properties of thedefect, including emission energies and linewidths.[38] For ex-ample, hBN films are usually multilayer and range from tens ofnanometers to a few atomic layers. Even a defect with the samechemical composition may exhibit different experimental signa-tures depending on its position with respect to the interface, sur-face, or edge of the film. A quantitative understanding of theseeffects is necessary for defect identification and quantum appli-cations.In order to understand the influence of surfaces and interfaceson the properties of defects in hBN and other 2DvdW materials,several effects must be disentangled. First, the different distancesto surrounding atoms or vacant spaces may change the structureof the defect, especially if it involves displacements of atoms outof the 2D plane. Such structural modifications will also impactthe electron-lattice coupling responsible for the phonon broad-ening of optical transitions. The electronic wave functions asso-ciated with the defect states may also evolve according to the vari-ations in symmetry and environment. Additionally, the alterationof the dielectric screening environment experienced by the defectwill modify the Coulomb interactions between electrons occupy-K. WatanabeResearch Center for Functional MaterialsNational Institute for Materials ScienceTsukuba 305-0044, JapanT. TaniguchiInternational Center for Materials NanoarchitectonicsNational Institute for Materials ScienceTsukuba 305-0044, JapanL. JiongCentre for Advanced 2D MaterialsNational University of SingaporeSingapore 117546, SingaporeM. PotemskiLaboratoire National des Champs Magnétiques IntensesCNRS-UGA-UPS-INSA-EMFL25 Av. des Martyrs, 38042 Grenoble, FranceM. PotemskiCENTERA LabsInstitute of High Pressure PhysicsPAS PL-01-142, Warsaw PolandM. PotemskiFaculty of PhysicsUniversity of Warsawul. Pasteura 5, 02-093 Warszawa, PolandC. E. DreyerDepartment of Physics and AstronomyStony Brook UniversityStony Brook, New York 11794-3800, USAing the defect levels. The latter has been recognized before as animportant contribution to correlation effects in layered materi-als, such as for exciton formation[39–41] and magnetism.[42] Therelative importance of all of these factors is expected to differ be-tween defects with different atomic and electronic structures ofthe ground and excited states.In this study, we use a combination of experimental charac-terization (optical and scanning probe spectroscopy) and first-principle theoretical analysis to elucidate how the surroundingsof defects in hBN influence their properties. Experimentally, weinspect this via studying the sensitivity of optical defect proper-ties to the thickness of the hBN samples. This is enabled by in-tentional carbon doping, which allows us to create reproducibledefect centrers with stable photoluminescence (PL) resonances.Thus, we can control the defect density and the proximity ofthe defects to the surface by isolating hBN films of varied thick-nesses. We demonstrate the evolution of the optical spectra fromensembles of specific defect types in ≈50 nm thick hBN films toindividual defects in ≈10 atomic layers of hBN. We interpret themodification of the PL spectra based on a combination of density-functional theory (DFT) calculations and a recently-developedquantum embedding approach.[33] Using the experimental char-acterization as a guide to the types of defects to analyze, we con-sider theoretically the two extremes in defect surroundings: de-fects in a free-standing monolayer of hBN, and defects in infi-nite bulk hBN. We calculate the changes to the atomic and elec-tronic structure of the defects relevant to, e.g., the optical mea-surements.We find that the dominant effect of the environment is thechange in dielectric screening, which affects the intra- and in-terorbital density–density Coulomb interaction between elec-trons in defect states. The enhanced screening in the bulk formof the crystal significantly reduces the energy of intradefect opti-cal transitions if they depend on such interactions. Based on this,we propose that this sensitivity to the dielectric environment canbe utilized as a sensor for local dielectric constants when broughtinto contact with surfaces, adsorbates, or liquids. We find that theoptical transitions within the substitutional carbon dimer havethe potential for such applications.The paper is organized as follows. In Section 2 we sum-marize our optical and scanning-probe measurements, whichserves as the experimental motivation for the systematic the-oretical study of environmental effects on defects given inSection 3. In Section 4 we discuss the main results andthe implications for defect identification/characterization, aswell as the use of defects to detect changes in a dielectricenvironment.2. Experimental Results: Characterizationof Carbon-Enriched hBNWe characterize ultra-pure hBN crystals grown via high-pressure-temperature gradient methods. Carbon doping is achieved post-growth by annealing the crystals in a graphite furnace at 2000°C for one hour, which gives rise to multiple optically ac-tive defect centres[24] in few-layer and bulk hBN. Details onthe growth and sample fabrication are given in the Methodssection.Small 2023, 19, 2300144 © 2023 The Authors. Small published by Wiley-VCH GmbH2300144 (2 of 13) 16136829, 2023, 41, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202300144 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.small-journal.comFigure 1. STS spectra for defects located in three-layer thick carbon-dopedhBN film, which were initially identified through STM. The STM imagesare presented next to the corresponding STS curves. All experiments wereperformed at the temperature T = 77 K, except for the defect #4, whichwas inspected at T = 4.7 K. STM images of defects #1−#3 were measuredunder the conditions Vs = 5∼V, It = 30∼pA. For defect #4, STM image wasobtained under the conditions Vs = 4.5∼V, It = 100∼pA. The STS #0 wasmeasured on a pristine hBN area free of defects. The inset constitutes themagnified image of the STS curve of defect #4, which demonstrates thefine structure of the vibrionic response arising due to the electron-phononcoupling. The scale bar in the STM images corresponds to 2 nm.With these experimental characterizations we aim to a) deter-mine the general properties of carbon defects in hBN hosts, b)motivate the choices of defects for our theoretical study, and c) in-spect how the local impurity environment controlled by the hostthickness changes the defect properties.In Figure 1 we depict representative scanning-tunneling mi-croscopy/spectroscopy (STM/STS) measurements on a three-layer-thick hBN sample exemplifying the huge variety of carbon-related defects that may occur in hBN.[27,35–37] The STM data (pan-els on the right) demonstrates well-isolated defect centers that ap-pear as symmetric or asymmetric features in the color maps. Inthe dI/dV STS spectra we see that several of these defects (labelled#1 to #3), involve resonances near the hBN conduction or valencebands, consistent with simple defects, such as carbon substitu-tions for boron or nitrogen vacancies.[23,26,32] Accordingly, we startwith simple defects in our theoretical modelling discussed inSection 3.In Figure 2 we present PL and second-order photon corre-lation data for carbon enriched bulk-like (ca. 50 nm) and thin-film (about 10 layers) hBN hosts. In Figure 2a, we show thePL spectrum together with a comparative analysis of the fourmost pronounced defect centers given in Figure 2b–e in the bulk-like hBN host. Defect A1 displays emission in form of a broadasymmetric band indicative of strong electron-phonon coupling.The emission spectra of defects A2 and A4 are dominated bynarrow resonances characteristic of centres exhibiting atomicTable 1. Experimentally measured Debye-Waller and Huang-Rhys factorsof defects A1-A4 responsible for the luminescence peaks in Figure 2a.Defect ZPL Debye-Waller factor Huang-Rhys factorA1 1.38 eV ≤0.032 ± 0.006 ≥3.4 ± 0.2A2 1.54 eV 0.29 ± 0.08 1.2 ± 0.3A3 2.00 eV 0.12 ± 0.03 2.1 ± 0.2A4 2.31 eV 1.0 ≈0.0 (atomic)character.[24] Defect A3 displays a spectrum typically observedfor more complex defect structures such as nitrogen-vacancycenters in diamond, where the well-pronounced dominant zero-phonon line (ZPL) is accompanied by lower-energy phononsidebands.[24]We can characterize the luminescence signatures of the bulkhBN defect centers in the framework of a simple 1D Franck-Condon model.[43] In that model, the lineshape is governed bythe coupling of the electronic transition to a single effective vi-brational mode and is parameterized by the Huang-Rhys factorSHR, which gives the average number of phonons emitted in anoptical transition. SHR depends on the difference in atomic struc-ture between the electronic states involved in the transition, andthe curvature of the potential energy surface for each state withrespect to displacements of the effective vibrational mode. Exper-imentally, SHR can be determined from the ratio of the spectrallyintegrated intensity of the ZPL and the intensity of the total defectemission (i.e., the Debeye-Waller factor w = IZPL/Itotal) using thephenomenological relation SHR = −ln (w). In Table 1 we presentw and SHR for the defects in bulk-like hBN. We see that theyare characterized by small to medium strength of the electron-phonon coupling. For A1, the value of SHR should be taken asa lower bound since the ZPL is not discernible. Similarly, thephonon sideband for A4 is not distinctive, and thus we reportSHR near zero.Since these defect centers yield reproducible and well-recognizable spectral characteristics, we can directly compare de-fects in the bulk-like and few-layer samples. In Figure 2f,g weshow the corresponding PL spectra for energies around A3 andA4 in bulk and ten-layer thick hBN:C films (see Supplemen-tal Information for further examples). Importantly, we find forboth hBN thicknesses PL resonances at similar but not identicalemission energies. In the A3 energy range, we see that the ZPLblueshifts by about 50 meV from the bulk to the ten-layer sam-ple, which is accompanied by a reduction of the ZPL linewidth(reduction of IZPL) indicative of the removal of inhomogeneousbroadening and by an increasing Huang-Rhys factor SHR from2.1 to 3.2. As the phonon sideband is not drastically affected,we attribute the enhanced SHR to the reduced inhomogeneousZPL broadening.The two-peak structure of A4 is visible for both hBN thick-nesses and shows a linewidth reduction, without clear signs ofa blueshift and the same value of SHR ≃ 0. For both defect types,in the few-layer sample, we confirm that we are probing individ-ual defects[15,44,45] by measuring the antibunching in the secondorder correlation function g(2)(𝜏) [see Figure 2h,i].These optical measurements explicitly reveal the significantchanges in defect properties, but the particulars vary from de-fect to defect. This observation provides the motivation for ourSmall 2023, 19, 2300144 © 2023 The Authors. Small published by Wiley-VCH GmbH2300144 (3 of 13) 16136829, 2023, 41, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202300144 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.small-journal.comFigure 2. The PL spectra of bulk (about 50 nm thick) hBN:C film measured at low temperature (1.6 K) in microscopic backscattering geometry (a).Four distinct spectral features can be identified, which we attribute to specific defect centres active in near-infrared (A1 and A2) and visible (A3 and A4)spectral regions. The spectra of the specific defects (b–e) are normalized toward uniform maximum intensity for ease of comparison. A comparativerepresentation of PL spectra of hBN:C in the bulk limit (≈50 nm thick) and a few layers’ limit (about ten layers) is presented for defects A3 (f) and A4(g). The second-order photon correlations g(2)(𝜏) measurements done for the emission resonances in ten-layer thick films demonstrate an antibunchingindicative of single photon emission for defects A3 (h) and A4 (i).theoretical investigation of environmental effects on different de-fects in the next section. Since the attribution of such optical sig-nals with specific defects is still controversial, we will discuss gen-eral defects that are likely to exist in hBN:C samples,[27,35–37] andprovide further commentary on how our results could be used toaid defect identification in Section 4.3. Theoretical Modelling ResultsWe now analyze theoretically how changes to the structural andscreening environment caused by different hBN host thicknessesaffect intradefect excitations and their associated Huang-Rhysfactors. We consider a variety of charge-neutral native and C-containing defects in three different situations: bulk (P63/mmc),“constrained” monolayer, and free-standing monolayer hBN, asdepicted in Figure 3. The bulk and monolayer structures are fullyrelaxed, while the constrained monolayer structure is taken fromthe bulk structure without any further lattice relaxations. Withthese three structures, we are able to theoretically disentangle theeffects of screening and atomic relaxations going from bulk tomonolayer hBN hosts. We note that well-controlled impurities infree-standing monolayer hBN are difficult to realize in realisticsamples; nevertheless, the monolayer hosts serve as a theoreti-cal limit with the strongest environmental impact to the defectcenters. We combine first-principles DFT calculations with thequantum embedding scheme described in Ref. [33], which allowsus to systematically study how changes in the environment andmodifications in the impurity structure in the ground and excitedstates affect the impurity properties.3.1. Computational ApproachFor each of the cases depicted in Figure 3, the atomic relaxationsaround the defect is treated at the DFT level, and constrained DFTcalculations are used to relax the defect structure in excited elec-tronic states. We furthermore estimate how the electron-phononcoupling associated with defect transitions is affected in each caseFigure 3. Schematic of theoretical situations used to isolate different environmental effects to impurity complexes (vacancies and C substitutions, thelatter in red) embedded in 2DvdW hBN hosts (black). a) Bulk: defect in the infinite bulk environment. b) Constrained monolayer: defect in a monolayerconstrained to the monolayer structure as in the bulk case. c) Monolayer: defect with fully relaxed structure in monolayer. In (a) and (b) we sketchintradefect electric field lines to highlight the difference in the screening environment. Note in (c) that the impurity positions are different.Small 2023, 19, 2300144 © 2023 The Authors. Small published by Wiley-VCH GmbH2300144 (4 of 13) 16136829, 2023, 41, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202300144 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.small-journal.comby evaluating the Huang-Rhys factor SHR, introduced in the pre-vious section. A full picture of the vibrational coupling requiresdetermining SHR for all relevant modes in the system, whichis a challenging task for transitions with moderate SHR ≈ 3 asthe defects observed in PL;[46,47] however, often a qualitative pic-ture of the strength of the electron-phonon coupling can be ob-tained by considering a single effective mode corresponding tothe structural difference between the ground and excited elec-tronic states.[48,49] Under this assumption, and presuming the vi-brational coupling is equal in the excited and ground state, SHR =EFC/ℏ𝜔, where EFC = (Eabs − Eem)/2 is the Frank-Condon re-laxation energy (i.e., half of the difference between vertical ab-sorption and emission energies), and 𝜔 is the frequency of themode.[43,48,50] In addition, if we assume that the mode is har-monic, we can write SHR = ΔQℏ√EFC2, where ΔQ is the absolutechange of the nuclear coordinate between the ground and exciteddefect state indicative of the renormalization of the inter-atomicbond strength.The impact of changing the defect environment from mono-layer to bulk hBN to the electronic properties of the de-fects will be elucidated using an embedding approach withinwhich we map the complicated many-electron problem ofthe defect in hBN to an effective minimal Hamiltonian withonly a small number of defect-related states i, j, k, l of theform:H = −∑ij,𝜎(tijc†i𝜎cj𝜎 + H.c.) + 12∑ijkl,𝜎𝜎′Uijklc†i𝜎c†j𝜎′cl𝜎′ck𝜎 (1)−HDC − 𝜇∑i,𝜎c†i𝜎ci𝜎where 𝜇 is the chemical potential to control the occupation ofthe defect states, 𝜎 is the spin, and c†i and ci are correspondingelectronic creation and annihilation operators. HDC denotes theso-called double counting correction term.[33] Diagonalizing thisHamiltonian results in many-body energies and wavefunctionsof intradefect excitations.The “active space” of defect-related states is isolated from thehost hBN electronic structure via the construction of Wannierfunctions, which allows us to determine the hopping matrixelements tij. Changes in the hopping matrix elements reflectchanges in the properties of the defect states on the DFT level.The screened Coulomb matrix elements Uijkl are computed inthe static constrained random-phase approximation (cRPA),[51]which takes the environmental screening into account. Compar-ing the unscreened (bare) Coulomb matrix elements vijkl (whichare only sensitive to changes in defect Wannier functions) withUijkl gives us insight into how the dielectric environment changesfrom bulk to monolayer hBN hosts and how this affects the in-tradefect transition energies. For computational details, see theMethods section.We can also obtain EFC from the intradefect transitions calcu-lated from the many-body energies of Equation (1). To this end,we calculate the transition energy in both the ground state ge-ometry and the excited state geometry. Under the assumptionsdiscussed above, half of their difference gives EFC.Table 2. Bare (v = viiii) and screened (U = Uiiii) Coulomb matrix elements(in eV) of single C impurities. 𝜖 = v/U is the effective dielectric constant.monolayer bulkv U 𝜖 v U 𝜖CB 6.27 2.65 2.4 6.54 1.70 3.9CN 7.25 2.85 2.5 7.04 1.87 3.83.2. Single Carbon ImpuritiesWe start our theoretical discussion with single carbon impuritieseither replacing a boron (CB) or a nitrogen (CN) site, as depictedin Figure 4. Both yield single occupied in-gap states, close to thehBN conduction band (CB) or valence bands (CN), in good quali-tative agreement with the STS/STM data shown in Figure 1 andas previously reported.[22,23]With only one electron in a single state, there are no intra-impurity excitations possible.[20] Nevertheless, these most simpledefects already allow us to study the impact of the environmenton the impurity orbitals. For both of these cases, the change inatomic structure between bulk and monolayer is negligible, withC-N (C-B) bond lengths changing by only 0.003 Å (0.004 Å) ingood agreement with previous data on the difference betweenmono- and bilayer hosts.[31]We construct localized Wannier orbitals for both C impurities,as depicted in Figure 4b. These pz-like Wannier functions allowus to calculate the on-site energies tii, bare (v = viiii) and cRPAscreened (U = Uiiii) Coulomb matrix elements, which we list inTable 2. We see that the difference in the bare Coulomb matrixelements between monolayer and bulk hBN hosts is quite small(≈ 4%), indicating that the single-particle electronic structure[i.e., the shape of the pz-like Wannier functions, see Figure 4b]does not change significantly between monolayer and bulk. How-ever, U changes significantly, increasing by more than 50% in themonolayer case compared to bulk. We can cast this into a changein the effective dielectric constants 𝜖 = v/U felt by the electrons inthe defect state; for bulk and monolayer hBN we find 𝜖 ≈ 3.8 and≈2.4, respectively. Note that this trend follows the out-of-planedielectric constant decrease from bulk to monolayer hBN.[52]3.3. Boron and Nitrogen VacanciesBoron and nitrogen vacancies, VB and VN, have been discussed inthe the literature.[53–55] On the DFT level they yield more compleximpurity states, as illustrated in Figure 4. The VN defect gives riseto a half-filled level below the conduction band accompanied bytwo additional nearly degenerate empty defect levels within theedge of the conduction band, which is also in qualitative agree-ment with the STS/STM data of impurity #1 in Figure 1a andRef. [13]. The VB defect creates three levels originating from thedangling bonds due to the missing B atom. Two of these statesare in close vicinity to the valence band edge, nearly degenerate,and host together one electron, while the third level is deeplyburied within the valence band.[18] In these cases, the hoppingmatrix tij has a 3 × 3 form, while the screened Coulomb interac-tion Uijkl is a rank-4 tensor with three elements per dimension.To compare the Coulomb interactions, we define the averageSmall 2023, 19, 2300144 © 2023 The Authors. Small published by Wiley-VCH GmbH2300144 (5 of 13) 16136829, 2023, 41, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202300144 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.small-journal.comFigure 4. a) DFT band structure of hBN bulk in the presence of carbon substitution centers and vacancies. Defect states are highlighted with red colorsand are parametrized using Wannier functions (b).intra-orbital density-density interaction U0 = N−1orb∑i Uiiii,inter-orbital density–density interaction U1 = [Norb(Norb −1)]−1 ∑i≠j Uijij, and exchange interaction J = [Norb(Norb −1)]−1 ∑i≠j Uijji, where Norb is the number of Wannier orbitals. Fur-thermore, we define the effective dielectric constant 𝜀 = v(d)l ∕U(d)lusing the leading eigenvalues of the bare and cRPA screeneddensity–density Coulomb matrices.[56]In Table 3 we compare all these parameters together withthe single-particle energy separation ΔE = t00 − t11 betweenKohn-Sham eigenvalues for monolayer and bulk hBN hosts. Wefind that single-particle energies associated with these defects[and thus the hoppings in Equation (1)] change by 2% for VBand 4% for VN. In contrast, the screened density–density in-teractions decrease significantly from monolayer to bulk hostsby more than 50%. Similar to the screening effects for the Csubstitutional defects, this is driven by the effective dielectricconstants decrease from ≈3 in the bulk to ≈2 in the mono-Table 3. Coulomb matrix elements (in eV) of single B/N vacancies. SeeSection 3.3 for the definition of variables.monolayer bulkU0 U1 𝜖 J ΔE U0 U1 𝜖 J ΔEVB 5.04 2.17 2.4 0.03 1.86 3.45 1.31 3.6 0.03 1.82VN 3.50 3.00 2.2 0.23 1.60 2.47 2.00 3.2 0.23 1.54layer. Notably, the exchange elements J are nearly unaffectedby the changes in the environment as a result of their dipolarcharacter. Indeed, macroscopic dielectric environmental screen-ing mostly affects density–density Coulomb interactions in lay-ered materials and does not significantly affect non-density-density elements.[42,56] Thus, many-body properties which areSmall 2023, 19, 2300144 © 2023 The Authors. Small published by Wiley-VCH GmbH2300144 (6 of 13) 16136829, 2023, 41, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202300144 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.small-journal.comTable 4. Many-body states of single B/N vacancies.Impurity spin En monolayer (in eV) En bulk (in eV)VB Q0 0 0Q1 0.51 0.63V Q2 2.82 2.31D1 3.18 2.61D2 3.63 3.00Q3 4.63 4.04VN D0 0 0Q1 1.48 1.37strongly affected by density–density Coulomb interactions, suchas charge excitations, will be in general most prone to changesin the dielectric environment, e.g., resulting from the tran-sition from bulk to monolayer hBN, while exchange interac-tion driven many-body states, such as spin excitations, will beless affected.This behavior is visible in the many-body energies of VB,as summarized in Table 4. For both monolayer and bulk hBNhosts, the VB ground state corresponds to four-times degener-ate quadruplet state Q0, which is approximately given by threeelectrons fully occupying the lowest and partially occupying thedegenerate single-particle states. From the corresponding excita-tion energies, we find that nearly all excited states decrease inenergy by about 500 to 600 meV (with Q1 as the sole exception)in bulk. This demonstrates that these transitions are decisivelyaffected by the density–density Coulomb interactions.For VN we find a similar trend with doublets as the ground stateand the following quadruplet excited state. However, all thesestates are occupied by a single electron, which eliminates anyCoulomb contribution. The differences we see in the excited en-ergies stem from the slightly altered single-particle properties asindicated in Table 3.3.4. Impurity ComplexesThe picture that emerges from the simple defects dis-cussed in the previous two sub-sections outlines the keyrole of differential dielectric screening, which influencesdensity–density Coulomb interactions within and betweendefect orbitals, inducing a change in the properties uponvarying the defect environment from bulk to monolayerhBN.We now consider more complex defect/impurity structures,which have in-gap intradefect transitions with possible techno-logical relevance, and that could be detected by optical means, asexemplified in the previous section. Specifically, we consider acarbon dimer (CBCN)[29,31,33,34] replacing nearest-neighbor B andN atoms with carbon, a carbon-vacancy complex (CBVN),[28,30,57]and a combination of a dimer with a neighboring vacancy(CBCNVN).[37] Especially CBCN has recently attracted significantattention, as it was proposed as the origin of the 4.1 eV ZPLsingle-photon emitter[20,22,58] observed in hBN based on the en-ergetics of emission[29] and calculations of photoluminescencelineshapes.[59,60] All these impurity complexes have modest for-mation energies[36,37] and are thus likely to occur in both, mono-layer and bulk hBN upon C exposure. In Figure 5 we summarizeall model details.From the DFT band structure calculations presented inFigure 5a–c we see that all three impurity complexes form in-gap defect states with varying occupations (two electrons in CBCNand CBVN and three electrons in CBCNVN). We use these statesas the active space for our minimal modelling (see Section S3,Supporting Information for details). The resulting lowest excitedmany-body levels are shown in Figure 5d–f with indications ofthe ground and excited state types, singlets (S), doublets (D),triplets (T), and quadruplets (Q). We show these energies cal-culated for the three cases schematically depicted in Figure 3.Comparing the “constrained monolayer” case to the fully re-laxed bulk and monolayer cases allows us to isolate the effectsof the dielectric environment from changes in structure fromthe atomic environment. We thus first comment on the compar-ison between bulk and “constrained monolayer” hBN hosts, andthen on the effect of relaxations between bulk and monolayerhosts.Throughout all impurity complexes, we see a consistent trendof decreasing optical transition energies when changing from theconstrained monolayer to bulk hBN. Upon allowing for furtherrelaxation of the monolayer host system, this trend is mostly pre-served. The decreasing excitation energies are more pronouncedfor singlets than for doublets and are often vanishingly smallfor triplets and quadruplets, which is a direct measure of howmuch the many-body states are controlled by density–density andexchange Coulomb matrix elements, as illustrated for each de-fect below.3.5. Many-Body States3.5.1. CBCN Many-Body StatesThe CBCN carbon dimer impurity has been widely discussed be-fore, see, e.g., Refs. [29, 31, 34]. It can be approximately mod-eled as a half-filled Hubbard dimer[33,61] whose many-body eigen-states are solely controlled by the ratio of the local density–densityCoulomb repulsion U and the hopping t between the sites. In ourcase this dimer is formed by localized pz-like orbitals at the two Cpositions with local U0 and interorbital Coulomb interactions U1,see Figure 5a. In this scenario the effective local Coulomb inter-action magnitude is approximately given by U* = U0 − U1.[62,63]Upon increasing the screening from monolayer to bulk, both U0and U1 are significantly but similarly reduced (e.g., ΔU0 ≈ 1.3 eVand ΔU1 ≈ 1.1 eV), such that the resulting U* differ by ΔU* ≈0.2 eV. This corresponds to a relative change of nearly 40%. Asthe electronic hopping between the two pz Wannier functions isaffected by only 3% through the monolayer to bulk transition,we can again state that modifications to the many-body proper-ties are mostly driven by the changes in the dielectric screen-ing. This explains the reduction of the S1 state by about 0.2 eVand is in agreement with previous GW+BSE and TD-DFT basedresults.[31,34] In contrast, the lower triplet states T1 is less affectedby this screening, as it depends more strongly on Hund’s ex-change elements, which are nearly unaffected (see SupportingInformation Table SIV). Further, the data presented in Figure 5dSmall 2023, 19, 2300144 © 2023 The Authors. Small published by Wiley-VCH GmbH2300144 (7 of 13) 16136829, 2023, 41, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202300144 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.small-journal.comFigure 5. a–c) DFT band structures of each considered impurity complexes embedded into bulk hBN with highlighted in red impurity states, whichwere parametrized using Wannier functions. Black dots sketch the initial occupations and their modification upon optical excitations. d–f) Many-bodyimpurity states embedded into bulk, constrained monolayer, and free-standing monolayer of hBN. Arrows demonstrate the possible excitations, whichproperties are given in Table 5.shows that atomic relaxations of the defect between bulk andmonolayer do not play a significant role here (c.f. constrainedmonolayer and monolayer).3.5.2. CBVN Many-Body StatesFor CBVN the spinless DFT Kohn-Sham (KS) ground state isapproximately given by two electrons mostly residing in a C-centered px-like state (see side panel of Figure 5b) with a dis-tortion of the C atom position out of the plane.[55] The fully in-teracting many-body ground state of this impurity can be wellapproximated as a single Slater determinant with two electronsof opposite spin in the lowest KS state forming a singlet groundstate.[28,30,57] The many-body singlet–singlet transitions approxi-mately promote one of these electrons either into a C-centeredpz-like state (S0 → S1, which lets the C atom relax back to thehBN plane)[28,57] or into a delocalized state with two pz-like wave-functions centered at the neighboring B atoms (S0 → S2 withreduced out-of-plane C position distortion). The bulk screeningreduces the corresponding transition energies by 0.2 eV and by0.24 eV for S1 and S2, respectively (c.f. constrained-monolayer tobulk transitions in Figure 5e). A similar approximate U* analysis(using orbital averaged U0 and U1 given in the Supporting In-formation) yields an estimate of ΔU* ≈ 0.2 eV (ca. 20%). As thesingle-particle energies only change by less than 5% and sincethe Hund’s exchange elements are nearly unaffected between thebulk and constrained monolayer hosts, the overall trend in themany-body levels can again be explained based on modificationsto the density–density Coulomb matrix elements. Full relaxationstoward the free-standing monolayer enhance the out-of-plane Cposition distortion, which mainly affects the single-particle en-ergies, while screened density–density and Hund’s exchange el-ements are nearly unaffected (see Table SV, Supporting Infor-mation). The single-particle KS energies change by about 0.1 to0.2 eV in the freestanding monolayer with respect to the con-strained monolayer, which is the same order of magnitude as themany-body excitation energies change between the two systems.3.5.3. CBCNVN Many-Body StatesThis defect complex can be interpreted as a combination of CBCNand CBVN. The KS ground state corresponds approximately toa fully occupied pz-like state centered on the C atom farthestfrom the vacancy and a half occupied px-like state centered at theSmall 2023, 19, 2300144 © 2023 The Authors. Small published by Wiley-VCH GmbH2300144 (8 of 13) 16136829, 2023, 41, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202300144 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.small-journal.comTable 5. The absolute change of configuration coordinates ΔQ (in√amu Å), Franck-Condon energy EFC (in eV) and Huang-Rhys factor SHR for impuritycomplexes embedded into free-standing monolayer and bulk hBN.CBCN CBVN CBCNVNmonolayer bulk monolayer bulk monolayer bulk# ΔQ EFC SHR ΔQ EFC SHR ΔQ EFC SHR ΔQ EFC SHR ΔQ EFC SHR ΔQ EFC SHR1 0.24 0.06 0.65 0.26 0.07 0.75 3.37 0.80 33.0 2.00 0.62 17.20 2.95 0.59 24.80 1.68 0.44 12.192 1.51 0.23 8.0 0.78 0.18 3.67 1.79 0.65 15.78 1.46 0.64 12.813 2.81 0.13 11.00 1.12 0.08 3.38C adjacent to the vacancy. This impurity thus hosts three elec-trons. Due to the spin degree of freedom of the unpaired elec-tron, the many-body ground state is two-fold degenerate forminga doublet state, whose wave functions can again be well approxi-mated by a single Slater determinant in the band basis (see TableSX, Supporting Information). The ground state structure showsan out-of-plane distortion, mostly in the C position. Doublet–doublet excitations approximately promote the unpaired electronfrom the px orbital to delocalized pz-like orbitals at the neighbor-ing B (D0 → D1 letting the C atom relax back to the plane), one ofthe electrons from the C pz to the C px (D0 → D2 with enhancedout-of-plane distortions) and the unpaired electron from the C pxto C pz (D0 → D3 also letting the C position relaxing back to theplane). Here, the many-body energies are less influenced by mod-ifications to the dielectric environment upon changing from bulkto monolayer hBN hosts. D1, D2, and D3 change by about −0.03,−0.13, and −0.08 eV respectively, although, as before the effec-tive density-density Coulomb matrix element is reduced by aboutΔU* ≈ 0.15 eV, while the single particle energies are affected byless than 5% when changing from the constrained-monolayer tothe bulk hBN host. Thus only the D2 transition approximatelyfollows ΔU*, while the other excitations are less affected. We at-tribute this different behavior to the difference in the nature ofthe excitations: D1 and D3 are single electron excitations, while D2takes place among three electrons and involves significant modi-fications to the impurity charge density. Under further relaxationtoward the free-standing monolayer hBN, which enhances theout-of-plane distortion, the main difference is that the many-bodystates D1 and D2 become nearly degenerate. This is driven bymodifications to the single-particle Kohn-Sham energies, whilethe screened Coulomb interactions are approximately the same(see monolayer to constrained-monolayer transition in Table SV,Supporting Information).3.5.4. Huang-Rhys FactorsWe now turn our focus to the Huang-Rhys factors SHR. In Table 5we summarize all ΔQ, EFC, and SHR for both the free-standingmonolayer and bulk hBN hosts. For both cases, we find thatSHR corresponding to the excited states in CBCN are considerablysmaller than in CBVN and CBCNVN (note that our CBCN SHR isin line with Refs. [29, 34]). Quantitatively this is driven by the de-cisive out-of-plane distortion of the CBVN/CBCNVN ground state,which is reduced in their excited states, yielding large ΔQ. Thisis related to the decisively modified impurity charge densitiesin CBVN/CBCNVN upon excitation. In the ground states, we findan in-plane charge density with px symmetry localized partiallyon carbon and partially on the nearest boron sites in presenceof the N vacancy. Hartree terms (i.e., bare Coulomb repulsions)between these two partial densities destabilize the planar crys-tal structure, shifting the carbon up in z-direction in the groundstate. Further full relaxation toward the free-standing monolayerlimit enhances this out-of-plane distortion, which consequentlyincreases SHR via enhancing ΔQ. However, we note that in actualexperiments these additional distortions are likely suppressed bythe presence of a substrate or capping layer. In addition to ΔQ,there is also a slight but consistent trend of EFC being less inbulk than in monolayer (CBCN is again the exception, though thechange is negligible).4. DiscussionIn Section 2, we experimentally showed via PL measurementsthat the environment of carbon-based defects affected their prop-erties, such as the energy of optical resonances and Huang-Rhysfactors SHR. In Section 3, we explored theoretically the possi-ble mechanisms for these changes by comparing monolayer andbulk hBN hosts. In the following we summarize our findings, fo-cusing mostly on defect complexes with technologically relevantintradefect transitions.4.1. Mechanisms for Environmental Effects on Defect PropertiesFirst, we consider differences in defect structure between mono-layer and bulk hBN. These changes are relatively minor whenthe distortions related to the defect are purely in the hBN plane,as is the case for the single vacancies, single C impurities, andCBCN (see, e.g., Table 5 and Supporting Information). For de-fects with ground-state out-of-plane corrugation (e.g., CBVN), thedisplacements are larger in monolayer than in bulk hBN hostsas there are no other hBN layers to constrain them (see Sup-porting Information). The effect of the increased distortions onthe many-body energy differences and Frank-Condon relaxationenergies is relatively modest. However, SHR can be significantlylarger in freestanding monolayer hBN hosts than in bulk onesfor excitations accompanied by out-of-plane to in-plane geometryrelaxations, which significantly increase ΔQ for the correspond-ing transitions.We furthermore show that changing the hBN host environ-ment mostly affects screened density–density Coulomb matrixSmall 2023, 19, 2300144 © 2023 The Authors. Small published by Wiley-VCH GmbH2300144 (9 of 13) 16136829, 2023, 41, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202300144 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.small-journal.comFigure 6. Configuration coordinate diagram for defect complexes embedded to bulk and monolayer sample.elements within the impurity states, which can be reduced in thebulk by up to 1.3 eV, while single particle energies are modi-fied by only up to 0.2 eV, and Hund’s exchange Coulomb ma-trix elements barely change. Both the interorbital and intraor-bital density–density interactions are reduced significantly inbulk hBN hosts by up to ≈ 50% compared to the monolayer case.Quantifying this change with an effective dielectric constant give𝜖bulk ≈ 3 to 4, while 𝜖ML ≈ 2 to 2.5 depending on the defect.However, the extent to which these changes affect the intrade-fect many-body energies depends on how much the many-bodylevels depend on the interplay and compensation of the (partiallyvarious) density–density interactions and Hund’s exchange pa-rameter J. Overall, the resulting trend is that the energy sep-aration of intradefect levels decreases in bulk hBN hosts com-pared to monolayer ones, which is, in general, most prominentin charge excitations.4.2. Implications for Defect IdentificationOur findings have significant implications for the identificationof defects in hBN and 2DvdW layered hosts in general. First, wepoint out that renormalization of optical transitions from inter-actions and the correct treatment of spin symmetry is importantfor accurate comparison between experiment and theory, as isclear from comparing the KS single-particle states in Figure 5a–cwith the many-body states (d–e). Here, this was achieved by ourembedding approach, which can often also be achieved, e.g., viaΔSCF calculations utilizing hybrid functionals and possibly cor-rections for spin contamination[64] or higher-level multireferencequantum chemistry approaches. A representative example of thisobservation is CBVN, wherein the single particle picture two tran-sitions of about 2 eV are expected, while in the many-body picturethe singlet–singlet transitions appear at energies above 3 eV, andtriplet–triplet transition occur at about 1 eV (see Figure 5b,e).Moreover, we have shown that intradefect transitions within2DvdW hosts are controlled by additional degrees of freedom,i.e., the surrounding environment of the defects. In our opti-cal measurements, this was tuned by the host-material thick-ness, while in theoretical calculations, we considered defectsin bulk and monolayer hBN. The general trend of increas-ing intradefect energies and increasing SHR in monolayer com-pared to bulk hosts are summarized in Figure 6: the host mate-rial can change both, ΔQ and EFC. Thus, for experimental de-fect identifications within 2DvdW hosts, shifts in many-bodyenergies and changes in SHR should be taken into accountacross samples. On the other hand, such trends can be usedto aid defect identification and characterization if a signal canSmall 2023, 19, 2300144 © 2023 The Authors. Small published by Wiley-VCH GmbH2300144 (10 of 13) 16136829, 2023, 41, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202300144 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.small-journal.combe correlated between samples of different thicknesses, or ondifferent substrates.In light of these results, we can reexamine the defects observedin the PL measurements. In the considered energy range, we ob-served four types of defects with varying emission energies andHuang-Rhys factors SHR, which behave differently upon variationof the hBN film thickness. The defects A2 and A4 have PL res-onances in the range of the doublet–doublet transitions relatedto VN and VB containing defects, respectively. However, SHR forthese defects is much larger than observed experimentally.[65] A3likely involves a defect similar to the carbon dimer, which mightbe coupled to neighboring vacancies, as it shows a relatively smallSHR and strong sensitivity of the PL resonance energy to the hBNfilm thickness. Defect A1 is characterized by the largest SHR andlowest emission energy (within our series). Assuming that A1 is asimple defect, we could attribute it to the triplet–triplet transitionwithin CBVN. However, it is also plausible that this defect doesnot belong to the defect space considered here. Interestingly, theinvestigated defects exhibit vanishing small Zeeman splitting,when the magnetic field is applied perpendicularly to the hBNplane (see Figure S5, Supporting Information). This observationmay indicate that the optically active transitions are dominatedby singlet–singlet transitions, however, the transitions betweenthe parallel branches of the spin-split doublets and triples cannotbe excluded.Finally, we point out that we have not explicitly considered theeffect of a substrate in this work. Depending on the substrate,the impact on the defect, properties may change significantly. Onone hand, we would expect that the substrate could confine out-of-plane lattice distortions caused by the defect similar to the ad-ditional layer within bulk hBN hosts. On the other hand, the di-electric screening from certain substrates might be significantlylarger compared to the moderate enhancement from monolayerto bulk hosts. Substrates with significant lattice contributions tothe dielectric susceptibility may result in effective relative dielec-tric constants of 20 and above.[66,67]4.3. Defects for Detecting Local Dielectric SusceptibilityWe can use the sensitivity of defects in 2DvdW materials asprobes of the local dielectric environment of, e.g., other surfaces,adsorbed molecules, or confined liquids.[68] An ideal defect forsuch applications should exhibit an optically accessible intra de-fect transition that has a strong dependence on the dielectric en-vironment. Also, the defect should be characterized by a spec-trally narrow optical signal providing good sensitivity and resolu-tion for determining the properties of the dielectric environment.Thus, a small SHR is desired. Finally, it would be ideal if the de-fect properties were relatively insensitive to other environmentalstimuli such as external magnetic or electric fields.Experimentally, defect A3 appears as the best candidate for de-tecting the local dielectric constant. The PL emission resonanceshifts by ΔE = 60.1 meV between the 50 nm thick hBN film andten-layer-thick hBN film. We can estimate the upper bound of themodification of the relative dielectric constant sensed by the de-fect within the two films based on our calculation to be Δ𝜖 < 2.Therefore, we can conclude that our sensitivity toward the localdielectric constant is better than Δ𝜖/ΔE < 0.03 meV−1. The res-olution is limited by the linewidth of the PL resonance 𝛿, whichalso depends on the layer thickness. In the bulk form, the lineexhibits its largest broadening, 𝛿 = 7.4 meV, which correspondsto the change of the relative dielectric constant Δ𝜖 < 0.22. In themonolayer form, 𝛿 = 1.5 meV, which yields Δ𝜖 < 0.05.From our theoretical point of view, the singlet–singlet tran-sition in CBCN is a promising candidate. Although it hasrather high energy, its ZPL has been detected and identifiedpreviously.[20,22,29,58] Here, we have demonstrated that it is highlysensitive to the defect environment. Also, it has a Huang-Rhysfactor that does not change with the dielectric environment, so itsZPL will remain sharp. Finally, it is a singlet–singlet transition,such that it likely will not be affected by stray magnetic fields. Itmay however, shift under the application of electric fields sincethe defect has a relatively low symmetry (point group C2v).Therefore, carbon defect centers in hBN can be used as sensi-tive detectors of local dielectric constants, addressing the need ofmeasuring the environmental effects on a nanoscale.5. ConclusionUsing a combination of experiment and theory, we have eluci-dated the effect of the dielectric environment and the local struc-ture on defect properties in hBN. We show via PL measurementsof few-layer and bulk-like samples, that carbon-based defects ex-hibit shifts in ZPL energies, as well as changes in phonon side-bands and ZPL lineshapes, which can be quantified via Huang-Rhys factors. Using first-principles theory and embedding meth-ods applied to monolayer and bulk hBN, we show that the keyeffect of the environment arises from modifications in the effec-tive local dielectric screening acting on the correlated impurities.This alters the inter- and intra-orbital impurity Coulomb interac-tions and plays a role in reducing optical emission energies forintradefect transitions in bulk versus mono/few-layer hBN hosts.These effects must be taken into account when performing defectidentification via a comparison between experiment and theory.A plausible application of our findings is a detector for local di-electric constants, easily integrateble with solid, soft, and liquidmatter systems in pristine and/or functionalized form.6. Experimental SectionCrystal Growth: The pristine ultra-pure hBN crystals had been grownvia the high-pressure temperature-gradient method. A part of the crystalsfrom a single growth batch was annealed in a graphite furnace at a tem-perature of 2000°C for an hour.Sample Fabrication: The hBN:C films were isolated through mechani-cal exfoliation of bulk crystals onto 300 nm thick Si/SiO2 substrates. In thetransfer process, the substrates were heated to 50°C to increase the yieldof thin hBN:C flakes, which exhibit homogeneous and reproducible opticalemission. The thickness of the films, typically ranging between a few layersto a few tens of nanometers, was determined by optical force microscopy.For STM studies, hBN:C crystals were mechanically exfoliated onto 90nm-thick Si/SiO2 substrates. one up to three layer-thick films, identifiedvia optical contrast and AFM techniques, were subsequently lifted with aPDMS/PC stamp and transferred onto a large graphite flake that was par-tially covered by a Cr/Au film, providing a conductive surface for the STMmeasurements. The samples were cleaned with DCM, ACE, and IPA, andfinally annealed in an ultra-high vacuum in the STM chamber to removeany polymer residues from the transfer process.Small 2023, 19, 2300144 © 2023 The Authors. Small published by Wiley-VCH GmbH2300144 (11 of 13) 16136829, 2023, 41, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202300144 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.small-journal.comOptical Characterization: The PL spectra were measured in a back-scattering geometry under continuous-wave 514 nm excitation. The sam-ple was cooled down via exchange gas in a closed-loop cryostat or via coldfinger in a cryostat cooled with liquid helium. The laser was focused on thesurface of the sample to a spot of about 1 𝜇m via an objective. The samplewas positioned under the objective using x-y-z piezo-scanner system. ThePL signal was resolved by a spectrometer and detected by a charge-coupleddevice camera. The second-order photon correlations were measured inthe Hanbury-Brown and Twiss configuration with avalanche photodiodesacting as photon detectors.Scanning Tunneling Microscopy: The scanning tunneling microscopywas done in a low-temperature Createc system with base pressure below10−10 mbar. In these samples, the tunnelling occurred between the tung-sten tip and graphite substrate through a three-layer-thick carbon-dopedhBN barrier. The dI/dV tunnelling spectra were measured at a modulatedvoltage between 3 and 10 meV at the frequency of 700–900 Hz. The tip wascalibrated for spectroscopy against the surface state of gold in 111 orien-tation.Theoretical Methods: Density functional (DFT) electronic structurecalculations were performed within the Vienna ab initio simulation pack-age (vasp)[69,70] utilizing the projector-augmented wave (PAW)[71] formal-ism with PBE generalized-gradient approximation (GGA)[72] of exchange-correlation functional. For details on of the mapping to the minimal mod-els and their solutions see Supporting Information.Supporting InformationSupporting Information is available from the Wiley Online Library or fromthe author.AcknowledgementsThis project was supported by the Ministry of Education (Singapore)through the Research Centre of Excellence program (grant EDUN C-33-18-279-V12, I-FIM), AcRF Tier 3 (MOE2018-T3-1-005). This material wasbased upon work supported by the Air Force Office of Scientific Researchand the Office of Naval Research Global under award number FA8655-21-1-7026. This research was supported by the Ministry of Education, Singa-pore, under its Academic Research Fund Tier 2 (MOE-T2EP50122-0012).J. Lu acknowledges the support from Agency for Science, Technology andResearch (A*STAR) under its AME IRG Grant (Project no. M21K2c0113).K.W. and T.T. acknowledge support from JSPS KAKENHI (Grant Numbers19H05790, 20H00354, and 21H05233). P.H. acknowledges the supports ofthe National Key Research and Development Program (2021YFB3802400)and the National Natural Science Foundation (52161037) of China. M.Packnowledges the support from EU Graphene Flagship and FNP-Poland(IRA - MAB/2018/9 grant, SG 0P program of the EU). C.R.F acknowledgesthe European Union’s Horizon 2020 research and innovation programmeunder the Marie Skłodowska-Curie grant agreement N° 895369. The com-putational work for this article was performed on resources at the NationalSupercomputing Centre, Singapore. C.E.D. acknowledges support fromthe National Science Foundation under Grant no. DMR-2237674. D.I.B.was supported by the European Research Council (ERC) under the Euro-pean Union’s Horizon 2020 research and innovation programme, grantagreement 854843-FASTCORR.Conflict of InterestThe authors declare no conflict of interest.Data Availability StatementThe data that support the findings of this study are available from the cor-responding author upon reasonable request.Keywordscarbon centers in hexagonal boron nitride, dielectric environment, embed-ded impurities, screening effects to impuritiesReceived: January 5, 2023Revised: June 1, 2023Published online: June 17, 2023[1] J. R. Weber, W. F. Koehl, J. B. Varley, A. Janotti, B. B. Buckley, C. G. Vande Walle, D. D. Awschalom, P. Natl. Acad. Sci. 2010, 107, 8513.[2] B. E. Kane, Nature 1998, 393, 133.[3] J. J. Pla, K. Y. Tan, J. P. Dehollain, W. H. Lim, J. J. L. Morton, D. N.Jamieson, A. S. Dzurak, A. Morello, Nature 2012, 489, 541.[4] Y. Wu, Y. Wang, X. Qin, X. Rong, J. Du, npj Quantum Inf. 2019, 5, 9.[5] I. 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