Introduction

Excitons are energetically favorable bound states of excited electrons and holes with an energy less than the single-particle (electronic) band gap. This energy difference between the exciton energy and the electronic gap is the exciton binding energy. Since excitons are bound by the Coulomb interaction, the binding energy can primarily be enhanced by weak dielectric screening and charge localization1,2,3,4,5,6,7,8. Strong interactions, manifested by large exciton binding energies, can stabilize excitons up to room temperature9 and enhance the associated emergent phenomena, such as exciton condensates10 and charged excitons11. In addition to stabilization, large binding energies lead to enhanced absorption, sharp spectral emission, and electrical tunability12, all of which are promising properties for next-generation optoelectronic applications, including photovoltaics13 and single photon emitters14. These large exciton binding energies are most prevalent in monolayer materials, namely monolayer transition metal dichalcogenides (TMDs) where binding energies exceed 500 meV, because of reduced dielectric screening and geometrical confinement1,4,7,8,9. Bulk materials, with significantly stronger screening environments, typically host excitons with binding energies orders of magnitude lower4 yet would provide a platform on which to explore the role of dimensionality in many-body interactions. In theory, bulk materials with strong Coulomb interactions encouraged by anisotropy could overcome this limitation15,16. Particularly, van der Waals magnets, in which anisotropy is necessary to stabilize magnetic order down to the 2D limit17 and which host exotic quasiparticle interactions18, could thus provide promising materials in which to detect tightly bound bulk excitons and explore their rich many-body physics.

Among 2D magnets, CrSBr stands out as a highly promising candidate due to its strong optical response and pronounced anisotropy. CrSBr is a van der Waals antiferromagnet that crystallizes in the orthorhombic space group Pmmn (space group number 59) with lattice parameters a = 3.50 Å, b = 4.75 Å, and c = 7.94 Å near T = 200 K19,20. Its structure is comprised of van der Waals layers in the ab plane with each layer consisting of two buckled CrS sheets capped by Br (Fig. 1a). The corresponding Brillouin zone is shown in Fig. 1b. Below the bulk Néel temperature TN = 132 K, CrSBr develops A-type antiferromagnetic (AFM) order where the Cr magnetic moments couple ferromagnetically within the ab plane but antiferromagnetically in the stacking direction with the b-axis being the easy axis19,21. The monolayer has also been proven to remain stable in air and exhibits ferromagnetic ordering within the plane below 146 K 20,22. Moreover, an extraordinary phase transition, which is indicative of weak interlayer coupling, has been revealed in bulk CrSBr where the surface transitions into the AFM state at 140 K while the bulk transitions at 132 K23. Importantly, CrSBr exhibits a large optical response arising from Wannier excitons with binding energies predicted to be between 500–900 meV for the monolayer24,25. Correspondingly, a large exciton-photon coupling strength of  ~20 meV per monolayer was measured in 2D CrSBr polaritons26. Coupling between charge, spin, and lattice further demonstrates the presence of rich quasi-particle interactions24,25,26,27,28. Alongside structural and magnetic anisotropy, calculations and electrical transport measurements suggest that the electronic structure exhibits strong electronic anisotropy in the form of a quasi-one-dimensional (quasi-1D) conduction band25,29. This is also well captured by our self-consistent GW (scGW) calculation on monolayer CrSBr shown in Fig. 1c. From these optical and computational results, it has been suggested that CrSBr hosts tightly-bound quasi-1D exctions24,25,30. However, direct experimental observation of these remarkable excitonic properties and their underlying mechanisms has not yet been realized.

Fig. 1: Structural, electronic, and optical properties of CrSBr.
figure 1

a Three-dimensional representation of the real-space crystal structure of CrSBr, created with the VESTA software package69. The pink rectangular prism represents the unit cell. b The corresponding Brillouin zone of CrSBr with high symmetry points labeled. c Self-consistent GW calculations of the electronic band structure of monolayer CrSBr along high symmetry directions in the ΓXSY plane. d Reflection contrast and photoluminescence spectra of bulk CrSBr at 40K. Features in the two spectra are in excellent agreement, revealing an optical gap at Γ of 1.362 eV.

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In this study, we report the direct measurement of an exceptionally large exciton binding energy in single crystals of CrSBr through angle-resolved photoemission spectroscopy (ARPES) and optical measurements. We attribute the large binding energy to strong charge localization due to electronic and structural anisotropy as well as enhanced Coulomb interactions from diminished dielectric screening, confirming the inferences from optical and computational results. To our knowledge, this is the first direct measurement of a large exciton binding energy in a 2D magnet. We also demonstrate that the introduction of surface dopants can reduce the band gap and thus tune the electronic and optical properties. Through our study, we shed light on the role of anisotropy in the stabilization and tunability of many-body interactions in bulk excitonic materials.

Results

To evaluate the exciton binding energy, we must compare the electronic gap (Eg, the energy necessary to separately create an electron and a hole) with the optical gap (Eo, the energy necessary to create a bound electron-hole pair). We first investigated the optical gap through reflection and photoluminescence (PL) measurements. At 40K, both reflection and PL data show strong, narrow features and reveal the optical gap to be 1.362 eV (Fig. 1d and Supplementary Note 1). The good agreement between the PL and reflection data suggests that we are probing the direct transition at Γ.

We next explored the electronic gap by performing ARPES measurements on single crystals of CrSBr. Figure 2a, b show the band dispersion of pristine CrSBr at T = 192K along the high symmetry directions Γ − X and Γ − Y, respectively. Note that high symmetry points are designated based on photon energy scans and iso-energy plots at high binding energies (Supplementary Note 2). The relatively high temperature was chosen to avoid charging effects seen below TN although low-temperature measurements confirm that the key features do exist at lower temperatures as well (Supplementary Note 3). The shape of the band structure along the high symmetry directions is in strong agreement with previously published ARPES data31 and our scGW calculations (Fig. 2e, f). All of these results corroborate the insulating nature of the material, as evidenced by the lack of intensity at the Fermi level EF. It should be noted that while our calculations predict a valence band maximum (VBM) at Γ, we do not immediately observe this in the ARPES data due to matrix element effects in the first Brillouin zone31. However, the maximum at Γ can be observed in the second Brillouin zone and we thus determine the VBM to be at an energy of 1.84 eV below EF (Supplementary Note 4). Since we do not observe the conduction band in our pristine samples, this implies that the electronic gap is greater than 1.84 eV. The minimum gap size estimate of 1.84 eV is derived from high-temperature data, which does not exhibit the charging issues present at lower temperatures. Importantly, however, low-temperature measurements are able to demonstrate that this lower bound (i.e. minimum possible value) on the electronic band gap does not decrease at either 97 K or 40 K (Supplementary Note 3) and that changes to the band structure are minor, in agreement with theory25. This establishes the electronic gap to exceed 1.84 eV at 40K. This measure of the lower bound on the electronic gap is larger than the gap of 1.5 ± 0.2 eV found with scanning tunneling spectroscopy (STS)21,25 and contains significantly less uncertainty. Additionally, previous bulk ARPES measurements found the VBM to be at 1.51 eV below EF, also with no observation of the conduction band, and thus established 1.51 eV as a lower bound of the electronic gap31. Yet, due to significant charging at low temperatures, the authors speculated that the gap is likely much larger than 1.51 eV31. Our ARPES measurements are able to establish a larger lower bound on the electronic gap because our sample is heavily n-doped, as evidenced through comparison to our calculations. This eliminates (or significantly reduces) charging effects in our samples (Supplementary Note 5). This n-doping arises from our growth procedure where the high vapor pressure helps reduce halide vacancies.

Fig. 2: Direct visualization of the quasi-1D conduction band and electronic gap.
figure 2

Experimental band dispersion along the Γ − X and Γ − Y high symmetry lines for (a, b) pristine and (c, d) highly dosed (0.35 electrons per unit cell) bulk CrSBr at T = 192 K (paramagnetic regime without charging effects). The emergence of the conduction band and upward energy shift of the valence band upon dosing are evident. scGW calculations of monolayer CrSBr along the (e) Γ − X and (f) Γ − Y. g Energy dispersion curves (EDCs) as a function of K dosing detail the gradual emergence of conduction band as a function of K dosing, highlighted by the arrows. The numbers above each spectrum corresponds to the dosing level in units of electrons per unit cell. The downward shift in energy of the conduction band along with the upward shift of the valence bands results in a reduction of the band gap as a function of K dosing (h). Error bars indicate estimated error in determining peak positions from the EDCs.

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Because our sample is heavily n-doped, we expect the conduction band to be just above EF. Therefore, it is likely that the conduction band can be detected through in-situ alkali metal surface doping (dosing), a procedure expected to donate electrons to the system, thus raising the Fermi level and causing a rigid downward energy shift of the bands. Using potassium (K) as our dopant, we performed five rounds of dosing to reach a surface dopant density of 0.35 electrons per unit cell (Supplementary Notes 6 and 7). The successful dosing of K is confirmed through the growth of the K 3p core peak at  ~19 eV below EF, as shown by the vertical dashed line in Fig. 2g. More importantly, a feature arises at EF upon dosing, highlighted by the black arrows in Fig. 2g. To explore this feature more closely, we present the high symmetry band dispersion along Γ − X and Γ − Y for fully dosed (0.35 electrons per unit cell) bulk CrSBr in Fig. 2c, d. In the dosed samples we observe a band near EF that is nearly flat along kx but highly dispersive along ky. By comparing the dispersion to our scGW calculations, we conclude that this feature is the conduction band and not from other sources such as degenerate defect states. We observe that the experimental band gap is indirect, as the conduction band minimum is located at the X point, ~50 meV lower than at Γ (Supplementary Note 8). Note that, despite the indirect gap, the dominant optical response is due to the transition at Γ24,32. Additionally, we observe that, besides broadening, the shapes of the valence bands remain qualitatively similar upon K dosing but that the energy of the top valence band has increased while the energies of the lower valence bands remain roughly unchanged. As the conduction band shifts downward while the valence band shifts upward, we necessarily observe a reduction in the band gap as a function of K dosing (Fig. 2h). This evolution diverges from the expected rigid downward band movement and fixed band gap.

Discussion

Thus, our ARPES experiments have unveiled three notable findings: (1) a substantial band gap exceeding 1.84 eV; (2) a highly anisotropic conduction band; and (3) an anomalous evolution in the band gap upon K dosing. We now explore the implications of each of these observations. First, it is evident that CrSBr hosts tightly bound excitons. Comparing the values of Eo to the lower bound of Eg extracted from our ARPES data, we establish the exciton binding energy (Eb = Eg − Eo) in single crystal CrSBr to be greater than 478 meV. An exciton binding energy of this magnitude is significantly larger than other bulk inorganic semiconductors1,4,8,33,34,35,36,37,38,39,40,41,42,43,44, including bulk TMDs (Fig. 3). Rather, this lower bound on the exciton binding energy roughly follows the universal trend of monolayer semiconductors where \({E}_{{{{\rm{b}}}}}=\frac{1}{4}{E}_{{{{\rm{g}}}}}\)45, suggestive of the strong van der Waals character of bulk CrSBr.

Fig. 3: Large exciton binding energy of CrSBr in comparison to other inorganic bulk semiconductors.
figure 3

Plot of the exciton binding energy versus the electronic band gap for various monolayer and bulk semiconductors 1,4,8,33,34,35,36,37,38,39,40,41,42,43,44. For the bulk TMDs, the gap at the K point (which is the relevant transition for the excitons) is plotted instead of the indirect gap. For all the other materials, the direct band gaps are used. For CrSBr, the gap plotted is the lower bound on the direct electronic gap found through our pristine ARPES measurements. The rainbow background represents the spectrum of the exciton energy (optical gap) for a given electronic gap and exciton binding energy. Notably, bulk materials have an exciton binding energy below 250 meV except for bulk CrSBr, which has a binding energy similar to those for monolayer TMDs. The gray dashed-line plots the \({E}_{{{{\rm{b}}}}}=\frac{1}{4}{E}_{{{{\rm{g}}}}}\) trend for monolayer semiconductors45 which bulk CrSBr roughly obeys. All data points are from experimental works. The measurements for anisotropic black phosophorous (BP) were performed on a bilayer, hence the asterisk.

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To explore the physical mechanisms for this exceptionally large bulk exciton binding energy, we examine the evolution of the electronic structure upon K dosing. First, we analyze the significant anisotropy of the conduction band to understand charge localization. Figure 4a presents the iso-energy plot in the ΓXSY plane at an energy corresponding to the conduction band at Γ (0.29 eV below EF). The presence of a single stripe-like feature that is extended along kx and truncated along ky directly confirms the in-plane anisotropic nature of the conduction band. This feature is in excellent agreement with our scGW calculation (Fig. 4b). Additionally, we observe a stripe-like feature extended along kz and truncated along ky, indicating a relatively flat conduction band along the kz direction (Supplementary Note 7). From these observations along with Fig. 2c, d, it is evident that the conduction band exhibits quasi-1D characteristics. While such quasi-1D nature has been theoretically predicted and inferred through electrical transport and optical measurements24,25,29, this presents the first direct, momentum- and energy-resolved confirmation of the quasi-1D electronic structure in bulk CrSBr.

Fig. 4: Charge localization arising from the quasi-1D conduction band.
figure 4

a Experimental iso-energy plot in the ΓXSY plane at the conduction band energy at Γ taken at T = 192K (Paramagnetic regime). b Calculated iso-energy plots in the ΓXSY plane at the conduction band minimum. The stripe-like feature is indicative of a quasi-1D electronic structure. c Experimental (solid markers and lines) and calculated (empty markers and dashed lines) energy dispersion of the CB along Γ − X and Γ − Y lines. The markers are band positions found from peak positions of the EDCs. The lines are quadratic fits to the peak positions. The calculated curves have been offset in energy to match the position of the experimental data. scGW calculations of the spatial extent of the dominant orbitals of the bottom conduction band and top valence band viewed along the (d) c- and (e) b-axes. The electrons are localized in 1D chains along the b-axis.

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We quantify this observed anisotropy by fitting the peaks of the energy dispersion curves (EDCs) along Γ − X and Γ − Y to extract the effective mass of the electrons in the vicinity of Γ (Fig. 4c). We report experimental electron effective masses of \({m}_{x}^{*}=12.26\,{m}_{{{{\rm{e}}}}}\) (3.58 me, calculated) and \({m}_{y}^{*}=0.48\,{m}_{{{{\rm{e}}}}}\) (0.22 me, calculated), where me is the free electron mass. These values correspond to an effective mass ratio of \(\frac{{m}_{x}^{*}}{{m}_{y}^{*}}=25.63\) (16.43, calculated). The discrepancy between experimental and calculated values is likely due to strong correlation effects outside the scope of our calculations. This effective mass anisotropy is large, even among other known quasi-1D materials46. Notably, the mass anisotropy is significantly greater than previous reports of \(\frac{{m}_{x}^{*}}{{m}_{y}^{*}}=6.50\) in exfoliated CrSBr47, likely due to substrate effects in exfoliated CrSBr. Furthermore, calculations reveal significant anisotropy in the valence band as well with hole effective masses of \({m}_{x}^{*}=3.75\,{m}_{{{{\rm{e}}}}}\) and \({m}_{y}^{*}=0.17\,{m}_{{{{\rm{e}}}}}\) which correspond to a ratio of \(\frac{{m}_{x}^{*}}{{m}_{y}^{*}}=22.32\).

This effective mass anisotropy in both the conduction and valence bands is indicative of localization of the charge carriers. To confirm this charge localization, we calculated the real-space densities of the dominant orbitals comprising the lowest conduction and highest valence band (Fig. 4d, e). We observe a strong charge localization along the crystallographic b-axis, consistent with previous calculations25 and our observations in the electronic structure. We thus confirm the robust confinement of charge carriers along 1D channels in bulk CrSBr. Crucially, spatial confinement, typically arising from geometric origins, is known to enhance exciton binding energy in van der Waals semiconductors1,4,5,7. Therefore, we argue that the charge localization arising from the quasi-1D electronic structure contributes to the tightly bound excitons observed here in CrSBr. Notably, this localization arises from electronic anisotropy, potentially offering insight into the mechanisms enabling the existence of tightly bound excitons in a bulk material.

We next examine the anomalous evolution of the electronic band structure upon K dosing. From Fig. 2a–d, it is clear that in the K-dosed dispersion the conduction band has shifted downward in energy while the valence band has shifted upward, relative to the pristine bands. By measuring the conduction and valence band positions as a function of K dosing, we track this evolution more closely (Supplementary Note 7). We observe that the conduction band suddenly appears after the first dosing round (0.07 electrons per unit cell). It then proceeds to shift only slightly downward in energy upon further dosing. In contrast, the valence band initially shifts downward before gradually moving upward. Importantly, we observe a significant reduction in the direct gap at Γ from greater than 1.84 eV down to 1.65 eV at a dosing level of 0.07 electrons per unit cell before a roughly linear reduction down to 1.39 eV at 0.35 electrons per unit cell (Fig. 2h). This tunability of the band gap with K dosing is consistent within multiple samples.

We now discuss the physical explanations for this anomalous behavior. We first rule out the possibility of negative electronic compressibility (NEC), a decrease of the Fermi level upon an increase in carrier density, as we do not observe an upward energy shift in neither the deep valence bands nor the core levels48 (Fig. 2a–d, g and the traces for Sample 1 in Supplementary Fig. 6). Additionally, the evolution cannot be explained by the intercalation of K atoms as the K 3p core peak maintains its sharp line shape throughout the entire dosing process49 (Supplementary Note 7). Instead, we propose that the anomalous band evolution and reduction of the gap are most likely due to carrier-induced band gap renormalization (BGR) and the surface Stark effect (SSE).

First, BGR is a well-documented effect where the introduction of free carriers alters the screening environment and causes a sharp decrease in the electronic gap50,51. Notably, this effect is theorized to be most pronounced for low doping levels and for a high density of states of the free carriers51. Indeed, our data demonstrate a large band gap reduction upon light K dosing and the quasi-1D nature of the conduction band supports a large electronic density of states. Furthermore, previous ARPES studies on low-dimensional CrSBr exfoliated on Au and Ag report band gaps of only 1.14 and 1.18 eV47, respectively, which, we argue, is suggestive of substrate-induced BGR. Similar to the addition of free carriers, a substrate can drastically alter the screening environment of exfoliated flakes, reducing electron-electron interactions and causing a sharp drop in the electronic gap. The observation of this substrate-induced effect suggests that the analogous carrier-induced BGR may also be observed in CrSBr. Additionally, a similar evolution of the valence and conduction bands has been observed in K-dosed WS2/h-BN where the increase in screening from the filling of the conduction band is found to be responsible for the band renormalization52. This is consistent with our BGR description of the band normalization we observe in CrSBr. Thus, we ascribe the sudden emergence of the conduction band and immediate decrease in gap size to carrier-induced BGR.

The linear upward shift in the valence band and the resultant reduction in band gap for dosing levels greater than 0.07 electrons per unit cell can then be explained by the SSE (Fig. 2h and Supplementary Note 7). The SSE is a phenomenon whereby a vertical electric field localizes the conduction and valence electrons at different potentials in real space which results in a renormalization of the band gap53,54. Notably, this effect has been throughly demonstrated in alkali metal-dosed van der Waals materials44,55 where the ionized alkali metal atoms on the surface are responsible for the vertical electric field. Furthermore, previous theoretical work has demonstrated that an external vertical electric field applied to CrSBr is capable of creating the strong real-space separation of the conduction and valence electrons necessary to produce the SSE56. Additionally, the band gap reduction from the SSE is expected to be linear as a function of electric field for large fields53,54, in agreement with our ARPES data (Fig. 2h). We also note that competition between the chemical potential shift and the SSE has been previously reported and can explain the initial downward movement of the valence band55.

Importantly, large contributions from both carrier-induced BGR and SSE are indicative of weak dielectric screening and strong 2D character44,50,51,52,57. This strong 2D character and related weak screening observed in bulk samples of CrSBr can be understood through the highly anisotropic electronic structure. In tandem with the formation of in-plane quasi-1D chains of electron density, weak interlayer coupling driven by the orbital contents of the lowest energy conduction band and highest energy valence band contributes to out-of-plane localization. Specifically, these bands have minimal contribution from Br orbitals (<10% Br any orbital,  <5% Br pz) throughout the Brillouin zone, as deduced from our scGW calculations (Supplementary Note 9) and visible in Fig. 4d, e. Since Br atoms form capping layers on each van der Waals sheet, the lack of Br pz orbital contribution results in weakened interlayer coupling and, consequently, out-of-plane localization and reduced screening25. Compared to MoS2, a TMD with weak contributions from S pz orbitals near K ( <6%) but significant contributions near Γ (28%)58, CrSBr demonstrates even weaker interlayer coupling and thus the electronic properties are less affected by the inclusion of additional layers, enabling weak screening even in the bulk. Crucially, weak dielectric screening strengthens Coulomb interactions and has been shown to contribute significantly to large exciton binding energies in 2D materials2,3,4,5,6,7,8. Thus, the large observed exciton binding energy in bulk CrSBr can be further explained by weak dielectric screening arising from the anisotropic electronic structure. Additional experiments exploring the effects of doping and externally-applied electric fields can further establish the role of BGR and SSE in CrSBr.

We therefore determine that bulk CrSBr hosts tightly bound excitons due to quasi-1D charge localization and weak dielectric screening. The key role of the anisotropy, both structural and electronic, provides optimism that large exciton binding energies can be found in other highly anisotropic bulk van der Waals systems. Particularly, other van der Waals magnets, where predictions of extraordinarily large exciton binding energies in CrBr3, CrI3, and MnPS3 have already been made6,59, could provide additional systems in which to study the dimensionality of excitons. The band gap tunability upon surface doping also presents a method to easily adjust the electronic and optical properties of CrSBr, an effect that opens possibilities for both the further study of many-body physics and the development of semiconductor devices.

Methods

Crystal growth

CrSBr single crystals were synthesized via the direct solid-vapor method following the one we described before20. Chromium powder (Cr, Alfa Aesar, 99.97%), sulfur powder (S, Alfa Aesar, 99.5%), and solidified Br2 were loaded into a clean quartz ampoule in an atomic molar ratio of 1:1.1:1.2, then sealed under vacuum using a liquid nitrogen trap. The sealed ampoule was gradually heated to 930 °C, maintained at this temperature for 20 hours, and then slowly cooled to 750 °C at a rate of 1 °C per hour followed by the furnace cooling to room temperature. Large CrSBr single crystals formed naturally at the bottom of the ampoule, while a small amount of CrBr3 appeared at the top of the ampoule and was easily separated from the CrSBr crystals.

Optical measurements

Reflection contrast (RC) and photoluminescence (PL) spectroscopies were conducted by real-space imaging of the sample. The sample was kept in a Montana Fusion system for temperature control. An objective lens with a numerical aperture (NA) of 0.42 was used for both focusing and collection. A supercontinuum white light laser (NKT Photonics, SuperK) with a beam size of  ~2 μm in diameter was used as the white-light source for RC measurements. A continuous-wave solid-state laser at 532 nm with a power of 100 μW and a beam size of  ~2 μm in diameter was used as a pump for PL measurements. The collected signals were detected by a Princeton Instruments spectrometer with a cooled charge-coupled camera.

ARPES measurements

All ARPES measurements were performed at Beamline 7.0.2 (MAESTRO) of the Advanced Light Source. The beamline is equipped with a R4000 spectrometer with deflectors that enable data collection across the full Brillouin zone without moving the sample. Bulk CrSBr crystals were mounted on Cu pucks with Epotek H20E silver epoxy and cleaved in situ at vacuum better than 5 × 10−11 mbar. All measurements with fixed photon energy were performed with 83-84eV photons with linear horizontal polarization. The beam spot size was 15 μm x 15 μm. Paramagnetic measurements were performed at temperatures of 192K, 195K, and 195K for Sample 1, Sample 2, and Sample 3, respectively while antiferromagnetic measurements (Supplementary Note 3) were performed on Sample 1 at a temperature of 97K and on a fourth sample at roughly 40K (Supplementary Note 3). Potassium dosing experiments were performed by evaporating potassium onto the cleaved CrSBr surface in situ from a SAES getter source such that the sample was not moved from the measurement position. The level of potassium adsorption was estimated using Luttinger’s theorem (Supplementary Note 6).

Self-consistent GW calculations

Self-consistent GW calculations were performed on the Matsubara axis using a finite-temperature Gaussian-orbital based self-consistent Green’s function solver60,61,62. Calculations used the gth-szv-molopt-sr basis set63 with the gth-pbe pseudopotential64.

Self-consistent GW iterations were initialized with a density functional calculation using the PBE functional65, using the pyscf66 software package, and iterated to self-consistency.

Results were converged on the Matsubara axis using convergence acceleration67. Results were then analytically continued to the real frequency axis using Nevanlinna analytical continuation68.

The results shown here are obtained in a monolayer geometry using a periodic k-space mesh of size 6 × 8 × 1. Supporting Materials present additional details regarding the symmetrized atomic orbital decomposition, charge density calculation, and 2-D density of state iso-energy surface.