domain: phys phys

text:
Etienne Carré

Lorenzo Sponza

Alain Lusson

Ingrid Stenger

Sébastien Roux

Victor Zatko

Bruno Dlubak

Pierre Seneor

Etienne Gaufrès

Annick Loiseau

Julien Barjon
email: julien.barjon@uvsq.fr

Quantum well confinement and competitive radiative pathways in the luminescence of black phosphorus layers Authors

Black phosphorus (BP) stands out from other 2D materials by the wide amplitude of the band-gap energy (Eg) that sweeps an optical window from Visible (VIS) to Infrared (IR) wavelengths, depending on the layer thickness. This singularity made the optical and excitonic properties of BP difficult to map. Specifically, the literature lacks in presenting experimental and theoretical data on the optical properties of BP on an extended thickness range. Here we report the study of an ensemble of photoluminescence spectra from 79 passivated BP flakes recorded at 4 K with thicknesses ranging from 4 nm to 700 nm, obtained by mechanical exfoliation. We observe that the exfoliation steps induce additional defects states that compete the radiative recombination from bound excitons observed in the crystal. We also show that the evolution of the photoluminescence energy versus thickness follows a quantum well confinement model appreciable from a thickness predicted and probed at 25 nm. The BP slabs placed in different 2D heterostructures show that the emission energy is not significantly modulated by the dielectric environment.

Introduction

Confinement effects in 2D materials have attracted considerable interest because of their layered and exfoliable structures that allow angstrom scale increment of the thickness, from monolayer to bulk. Hence, ultrathin layers of 2D semiconductors, such as the MoS2 and other TDMCs, have demonstrated stunning physical properties, of high significance for optoelectronics and spintronics 1,2 . In this context, black phosphorus (BP), firstly exfoliated in 2014 3,4 , stands out from other 2D materials by the peculiar and strong confinement effects on its physical properties. One emblematic feature is the wide amplitude of the band-gap energy (Eg) that sweeps an optical window from Visible (VIS) to Infrared (IR) wavelengths, depending on the number of layers. Indeed, the optical activities in bulk and monolayer BP are centered at 0.3 eV and 2 eV 5 , respectively, corresponding to an amplitude energy Eg-BP =1. 7 eV considerably wider compared with that observed in the MoS2 family, roughly Eg-MoS2 = 0.3 eV 6 . Moreover, this variation of Eg-BP spreads over a large layer number and, for instance, a significant photoluminescence (PL) energy modulation is still observed in the range of 20-30 BP layers. 7 These singularities make the optical and excitonic properties of BP difficult to map. Specifically, the literature lacks in presenting experimental and theoretical data on the optical properties of BP on an extended thickness range. The difficulty to probe BP samples is, furthermore, exacerbated by the poor sensitivity of the detectors in the IR optical range and by the fast and intrinsic photodegradation of BP when placed in ambient conditions. 8,9 The intercrossing of dielectric, geometrical, defects and mechanical effects with quantum confinement effects also need to be clarified in the ultrathin, thin and bulk-like exfoliated BP flakes. Theoretical 5,10 and experimental [11][12][13][14][15] studies have been focusing on the behavior of few layers BP emitting in the visible / NIR range. In particular, few works analyzed the strong and anisotropic absorption/emission bands associated to the direct band gap in BP in the 1-4 layers regime 11,12,16 . BP samples were also investigated in the semi-bulk and bulk-like thickness ranges, i.e. arbitrary 4 nm to 40 nm and over 400 nm. 17,[START_REF] Ye | Highly polarization sensitive infrared photodetector based on black phosphorus-on-WSe 2 photogate vertical heterostructure[END_REF] The study of infrared luminescence of thicker layers of BP has accelerated, partly driven by the emergence of infrared optical applications such as BP based photodetectors and lasing. 7,17,[START_REF] Chen | Widely tunable mid-infrared light emission in thin-film black phosphorus[END_REF][START_REF] Wang | Mid-infrared Polarized Emission from Black Phosphorus Light-Emitting Diodes[END_REF] More recently, the fingerprints of bulk BP PL spectra at 2K were also reported, showing the existence of free and bound excitons energy around a refined value of the bandgap of 0.287 eV at cryogenic temperature [START_REF] Carré | Excitons in bulk black phosphorus evidenced by photoluminescence at low temperature[END_REF] .

Here we report on an unprecedented ensemble of low temperature PL spectra from 79 passivated BP flakes with thicknesses ranging from 4 nm to 700 nm, obtained by mechanical exfoliation. We observed that the exfoliation steps induce additional defects states that compete the radiative recombination from bound exciton observed in the as-grown crystals. We described our data set with a quantum well model applied to a thickness dependent BP slab and we identified a correlation between the optical band gap energy variation and the footprint of the electron-hole wave function within a BP bulk crystal. Finally, we evidenced that the dielectric environment does not modulate the photoluminescence energy from BP layers sandwiched in different 2D heterostructures.

Main text

Defect-related emission from exfoliated crystals.

The BP crystals were purchased from HQ Graphene and were mechanically exfoliated by an usual repeatable stamping process in a glovebox under argon atmosphere (<0.5 ppm O2, <1 ppm H2O) to prevent BP degradation. The thickness of the flakes was characterized by combining optical and Atomic Force Microscopy (AFM), both installed in the glovebox. A custom-built transfer box was used to transfer the samples into an Atomic Layer Deposition (ALD) chamber to perform a homogeneous deposition of a 10 nm Al2O3 passivation layer. Further details on the sample preparation and PL characterization can be found in the supplementary information file [SupMat1]).

In Figure 1 we compare representative photoluminescence spectra recorded at 4K from: an asreceived BP crystal (1a); a crystal resulting from a single-stamping exfoliation deposited on a Si/SiO2 substrate (1b); and a thick BP flake (700 nm) obtained from a multiple-stamping exfoliation deposited on substrate (1c). The corresponding photography and optical images are shown in Figure 1d. As described in [START_REF] Carré | Excitons in bulk black phosphorus evidenced by photoluminescence at low temperature[END_REF] , the PL of the as-received crystal consists of two structures. A fine emission signal is formed mainly of free and bound excitons (I°X) lines, and is highlighted by the pale-blue strip in the Figure 1a-c. A second contribution at about 20 meV lower energy is related to defect-assisted recombination. Because size effects are negligible at this thickness, as we will demonstrate below, the PL fingerprints are expected to look similar in all three samples. Surprisingly, even if the overall PL emission bands fall in the same tight energy interval (50 meV), the line shape is significantly modified. The excitonic fine structure of BP crystal (sharp peak) clearly disappeared leaving a broadened emission line for the sample resulting from multiple-stamping exfoliation (Figure 1c). The characteristics of this emission peak (line widths ranging from 15 meV to 45 meV) are consistent with the PL spectra reported in the literature for different thick flakes obtained via exfoliation 7,17,[START_REF] Chen | Widely tunable mid-infrared light emission in thin-film black phosphorus[END_REF][START_REF] Wang | Mid-infrared Polarized Emission from Black Phosphorus Light-Emitting Diodes[END_REF] , but its origin has not been elucidated yet. To investigate it, we operated a single-stamping exfoliation of a BP crystal, i.e. we stamped the crystal with PDMS only once, hence obtaining a BP chip of micron scale thickness, corresponding to an intermediate thickness sample of a measure between the size of the BP crystal and the ultrathin samples commonly reported in literature. Low temperature micro-PL spectra recorded at different selected areas of this BP chip (areas labeled 1 to 4 in the image of the sample in Figure 1d) interestingly exhibit different PL line shapes (Figure 1b). On one hand, a bright and narrow band dominates the emission fingerprint for areas 1 and 2. The emission energy and linewidth present strong similarities with those of the I°X bound exciton found in the pristine crystal [START_REF] Carré | Excitons in bulk black phosphorus evidenced by photoluminescence at low temperature[END_REF] . Furthermore, micro-Raman measurements performed in the same region indicate a good structural quality with a narrow linewidth for all BP Raman modes, (see Figure the radiative pathway is driven by bound excitons. On the other hand, the area 4 displays a broad emission band centered at lower energy, similar to the low-energy emission band of the BP crystal reported by Carré et al. [START_REF] Carré | Excitons in bulk black phosphorus evidenced by photoluminescence at low temperature[END_REF] . We observed that this area is characterized by an overall broadening of the Raman modes (Figure 1e), suggesting the presence of an additional defect density caused by the exfoliation process. An intermediate situation with the coexistence of the two bands is observed in some regions of the chip, alike area 3. They reproduce the overall macro PL spectrum recorded on the crystal. Our micro-PL and micro-Raman analysis of BP samples resulting from single-and multiple-stamping exfoliations clearly indicate that the emergence of this band is concomitant with the formation of defects upon exfoliation.

In the light of these observations, we conclude that the exfoliation based on PDMS stamping, commonly used for producing BP samples, induces deep changes in the nature of the luminescent process in BP layers, resulting in a broad emission band that may rely to the lowenergy band observed in the pristine crystal in macro PL. The anisotropic structure of BP and the low value of the Young modulus, compared to other 2D materials, could explain the high sensitivity of BP to the mechanical exfoliation processing.

Quantum-confinement on a large thickness range

To further investigate the characteristics of this emission peak, we probed the photoluminescence of 79 BP exfoliated flakes at 4K, with a thickness ranging from 4 nm to 700 nm (see SupMat1 in the supplementary information file for details on PL measurements procedure). The AFM image of a representative thin sample is displayed in Figure 2a whereas a set of optical and AFM images of other flakes are presented in SupMat2. Figure 2b presents the normalized PL spectra at cryogenic temperature of typical BP flakes. The grey line indicates the part of the spectra disturbed by the CO2 absorption band coming from the free-space part of the collection beam in our apparatus. In all exfoliated samples we probed, the luminescence spectra consist in a single and broadened emission band. This observation is consistent with the PL spectra reported in the literature for different thicknesses 7,17,[START_REF] Chen | Widely tunable mid-infrared light emission in thin-film black phosphorus[END_REF][START_REF] Wang | Mid-infrared Polarized Emission from Black Phosphorus Light-Emitting Diodes[END_REF] with a line width ranging from 15 meV to 45 meV. Note that the PL band from the thickest flakes coincides with the broadened band reported in Figure 1c. For each flake, we associate the energy of the emission to the maximum of the PL spectrum. When it falls in the CO2 absorption range (gray rectangle in Figure 2b), for example for the 92 nm film, the PL signal is modeled with a Gaussian peak and the energy at maximum is extrapolated.

The evolution of the PL peak energy over the wide thickness interval considered, from about 10 to about 1000 layers, follows two distinct regimes with a transition thickness (Lb) around 25 nm (about 50 layers), as highlighted by a black dashed line in Figure 2c.

At thicknesses higher than Lb, the PL energy evolves moderately, from 0.27 eV to 0.32 eV, and remains close to that of the bound exciton (0.275 eV at 2 K, green line in Figure 2c). These values are, in average, slightly lower than those presented in the literature for the same thickness range (0.32 eV [START_REF] Wang | Mid-infrared Polarized Emission from Black Phosphorus Light-Emitting Diodes[END_REF] and 0.31 eV 17 for 70 nm and 220 nm-thick BP layers probed at 80K, respectively) but this can be explained by the fact that the gap of black phosphorus blueshifts with temperature [START_REF] Villegas | Anomalous temperature dependence of the band-gap in Black Phosphorus[END_REF][START_REF] Baba | Photoconduction of Black Phosphorus in the Infrared Region[END_REF] . Measures have been taken at two different excitation fluences: black bullets and empty circles correspond to laser powers of 3x10 4 and 3x10 3 W•cm -2 , respectively.

No degradation was observed under exposure, i.e. no trace of the laser spot under the microscope and no decrease of the luminescence signal over time. We only notice a weak dependence of the peak energy on the excitation fluence, which could be ascribed to the Burstein Moss effect [START_REF] Burstein | Anomalous Optical Absorption Limit in InSb[END_REF][START_REF] Moss | The Interpretation of the Properties of Indium Antimonide[END_REF] (more details in supmat3).

At thicknesses lower than Lb, the peak energy increases significantly, reaching 0.7 eV for the 4 nm-thick BP flake, which is much higher than what is reported in literature 7 because of the different experimental conditions. The evolution of the experimental peak energies as a function of the thickness is well described by the law 𝐸 = ħ²𝜋² 2𝑚 * 𝐿² + 𝐸 0 , where E0 is an offset energy and L is the thickness of the slab. This is the energy behavior of two independent particles of opposite sign moving freely with an effective reduced mass m* inside an infinitebarrier quantum-well (see red fit in Figure 2c andSupMat6). Fitting the law with parameters 𝐸 0 = 0.279 eV (Average energy of PL measurements on the 700 nm thick flake) and the measured thicknesses, we extract an effective mass of m*=0.049m0 (m0: mass of the electron in vacuum). Intriguingly, the effective mass extracted from our measurements is comparable to the exciton effective mass (mw) predicted for the bulk by previous calculations [START_REF] Carré | Excitons in bulk black phosphorus evidenced by photoluminescence at low temperature[END_REF][START_REF] Asahina | Electronic Structure of Black Phosphorus in Self-Consistent Pseudopotential Approach[END_REF][START_REF] Takao | Electronic structure of black phosphorus: Tight binding approach[END_REF] including the anisotropic Wannier model introduced by some of us [START_REF] Carré | Excitons in bulk black phosphorus evidenced by photoluminescence at low temperature[END_REF] . We explored further this similarity by reverting to our bulk Wannier model and calculating the corresponding probability P(z) of finding an electron-hole pair of vertical extension z and the distribution of the average electron-hole pair vertical distance |z|P(z). A plot of these two quantities is reported in Figure 3a while the details of the calculation are discussed in the SupMat5. The result is that both distributions start being negligible at around Lw = 25 nm which indicates that, in the bulk, excitons extend in the stacking direction for Lw = 25 nm at most. In a BP slab, the same exciton would start "feeling'' the presence of the surfaces for thicknesses L < Lw , thus entering into a quantum well regime. After the similarity of the effective masses, it is remarkable that also the threshold thickness for the quantum well regime measured experimentally (Lb) is so close to a corresponding quantity predicted for the bulk exciton (Lw).

Finally, we notice that in our bulk excitonic model, the kinetic contribution to the energy was much more sensitive to changes of the parameters than the potential contribution. If this is still the case in the BP slabs, then the PL signature should not be affected by changes of the dielectric environment. Experimentally, we tested this hypothesis by comparing the evolution of the photoluminescence energy of some hBN / BP / hBN heterostructures (hBN has a dielectric constant of about 3 [START_REF] Laturia | Dielectric properties of hexagonal boron nitride and transition metal dichalcogenides: from monolayer to bulk[END_REF] ) with the SiO2/BP/Al2O3 samples studied above (the dielectric constant of alumina is around 9 [START_REF] Vila | Dielectric spectroscopy of alumina ceramics over a wide frequency range[END_REF] ). More details on the heterostructures fabrication and their luminescence are given in the SupMat4. Results are reported in Figure 3b in red triangles (hBN encapsulated) and grey bullets (alumina-passivated samples of Figure 2). The superposition of the PL energy as a function of thickness demonstrate that, indeed, the quantum confined emission is not significantly affected by changes of the dielectric environment, at least in this range of thicknesses. This can be explained by the fact that, at these thicknesses, the wavefunctions of the particles are much more strongly affected by the confinement than the screening of their interaction. We notice also that this feature justifies "a posteriori'' the independent-particle approximation implicit in the two-particle quantum well model, which nicely describes the observed emission behavior.

The characteristics of this quantum well regime depict a quite unique situation among the 2D materials. Indeed it is found to start below a threshold thickness Lb of about 25 nm, corresponding to about 50 layers and the increase of the peak energy is quite slow upon reducing thickness as attested by the very low effective mass of particle pair (m*=0.049m0) involved in the model. For a comparison, in MoS2 the PL energy enters the quantum well regime at a thickness of about 5 layers (Lb=2-3 nm) and increases steeply (see Figure 3c). This slow change of the emission energy may be an advantage for possible applications because it makes the modulation of the emission easier to control. Furthermore, we observed that these characteristics are robust against changes of the substrate or the passivation layer, which may be another advantage as it simplifies the integration of BP slabs in actual devices.

Finally, we noticed that these two quantities (the threshold thickness and the pair effective mass) together with the low sensitivity to changes in the dielectric medium are common also to the bulk excitonic model we introduced in a previous study. These similarities suggest that excitons are somehow involved in the formation and the evolution of the broadened emission band we studied.

Conclusions

To conclude, our work reports the evolution of luminescence at cryogenic temperature of many exfoliated slabs of black phosphorus over a wide and so far unexplored range of thicknesses.

Without being able to identify clearly the origin of the broad luminescence emission peak from exfoliated BP samples, our measures and analysis allows us to formulate the hypothesis that the robust quantum well emission we observed comes from excitons whose recombination pathways are altered by some structural disorder or defects introduced by the exfoliation procedure itself. This peak coincides with the low-energy band of the bulk crystal spectrum at very large thickness. As a consequence, this emission has properties that are unique to black phosphorus slabs. They are related to the high softness of the material, which explains the impact of the exfoliation on the photoluminescence properties, and to the low effective mass of paired particles that explains the exceptionally wide range of thicknesses over which the quantum well emission can be observed, contrasting with other semiconducting 2D materials.

Quantum well confinement and competitive radiative pathways in the luminescence of black phosphorus layers Authors

Etienne Carré1,2 , Lorenzo Sponza 1 , Alain Lusson 2 , Ingrid Stenger 2 , Sébastien Roux 1,2 , Victor Zatko3 , Bruno Dlubak 3 , Pierre Seneor 3 , Etienne Gaufrès 4 , Annick Loiseau 1 and Julien Barjon 2 *

Experimental methods

Exfoliation and passivation of BP thin layers:

Thin layers of black phosphorus are exfoliated from a high quality crystal (HQ Graphene -99.995% purity) in glove box under argon atmosphere (<0.5 ppm O2, <1 ppm H2O) to prevent its photo-oxidation under ambient 1 . Conventional PDMS (1:10 ; cross-linked 1h at 80° C) buffers are used to reduce BP single crystals to thin layers, which are then deposited on a SiO2(300 nm)/Si substrate. For each sample, the area of interest is identified with an optical microscope and the thickness is measured with an Atomic Force Microscope (Bruker Innova AFM). These measurements are done within the glove box. A hermetically argon filled suitcase was used to transport and introduce BP samples in an atomic layer deposition (ALD) growth chamber (BENEQ TFS 200) for passivation purposes. A 10 nm continuous Al2O3 has been deposited on BP samples from the reaction of tri-methyl-aluminum and H2O, which provides an efficient barrier against air exposure 2,3 . From the storage of the crystal to the passivation of the thin layers, the BP is never in contact with ambient air and oxidizing agents, which results in good quality samples.

Luminescence experiments:

Micro photoluminescence measurements are performed at 4 K with a He cold finger cryostat.

The excitation source is a frequency-doubled Nd:YAG Laser (532 nm). It is focused on the BP flakes with a laser spot of about 2 µm diameter thanks to a reflective objective lens (Cassegrain Thorlabs, x25, NA 0.4). The PL signal is collected back through the objective and detected with a Fourier Transform Infra-Red (FTIR) spectrometer (BOMEM DA8, CaF2 or Quartz beamsplitters, spectral resolution 0.5 meV) equipped with an InSb detector cooled at 80 K. A chopper and a lock-in amplifier are needed to eliminate the undesirable blackbody radiations in the infrared spectral range.

Macro photoluminescence measurements are quite similar. The set-up differs in two ways: the cryostat used is a helium bath and the laser is no longer focused by an objective but by a lens allowing it to reach a diameter of about 100 µm.

RAMAN experiments:

The Raman spectrum was recorded in backscattering configuration using a high-resolution Raman spectrometer setup (Labram HR800 from HORIBA Jobin-Yvon). A He-Ne laser operating at 633 nm is focused on the sample through a 100x objective with a numerical aperture of 0.80 to for an illumination spot diameter of about 1 μm. To avoid any form of degradation, the laser power is set below 2 mW and the measurements are performed under argon flow to avoid BP oxidation. Figure SM4a shows the power dependence of a typical thick BP flake PL spectrum at 4 K. As mentioned before, we fit the PL signal with a Gaussian to extract the energy (Figure SM4b) as well as the integrated intensity (Figure SM4c) of the peak, partially in the area of the CO2 absorption. It is worth noting that the PL spectrum does not widen with power and remains stable around 30 meV, which attests that the heating due to the beam seems to be negligible.

Raman measurement

AFM images

The PL spectrum shows a clear blueshift of few tens of meV over two decades of excitation power. This behavior is due to the filling of the conduction band by the high number of carriers generated by the high LASER power. This energy difference is commonly known as the Burstein-Moss shift (BMS) 4,5 . We can estimate the density of state in the conduction band of the BP at a given temperature (1) :

𝑁 𝐶 = 2. ( 2𝜋𝑚 𝑒 * 𝑘𝑇 ℎ 2 ) 3 2 (1)
With m*e = 0.413m0, the effective electron mass averaged over the 3 crystallographic axes 6 , we find a density of state in the conduction band at 4 K of NC(4 K) = 1.024.10 16 cm -3 . A similar calculation allows us to obtain the density of state in the valence band of the BP : NV(4 K) = 7.482.10 15 cm -3 .

This value must be compared with the charge carrier density created in photoluminescence by a 532 nm laser. We can estimate the diameter of the beam dbeam (2) as well as the penetration depth pdepth (3) from the complex refractive index N measured in the literature 7 . The excitation being circularly polarized, we take the average of the absorption coefficients following armchair and zigzag.

𝑑 𝑏𝑒𝑎𝑚 = 1.22 𝜆 𝑁𝐴 = 1.6 µ𝑚 (2)
𝑝 𝑑𝑒𝑝𝑡ℎ = 1 𝛼 = 𝜆 4𝜋.𝐼𝑚(𝑁) ~ 180 𝑛𝑚 (3)
We then find an excitation volume of V = 3.7.10 -13 cm 3 . Assuming a beam reflection of 0 and a quantum efficiency of 1 we can then calculate the density of injected charge carriers neh for a 30 mW laser power (4). The lifetime of the charge carriers does not seem to be known for the BP so we choose to take an arbitrary value of τ = 0.1 ns which is the same order of magnitude as the lifetime in other classical semiconductors 8 .

( We then find that neh >> NC(4 K), NV(4 K), therefore, the strong photo excitation regime forces us to consider the filling effects of the conduction band (CB). As the excitation power increases, more electrons are sent to the CB and thermalize at the band edge. The newly generated electrons cannot access the band edge states and are then forced to occupy higher energy states and increase the optical gap, this is what is observed in figure 2b. BMS effect is also observed on BP samples by several experimental measurements requiring similar excitation powers [9][10][11] .

In Figure SM4c, the integrated intensity of the PL I is fitted with a classical power model 𝐼 = 𝑃 𝑘 , with P the incident laser power and k a factor that depends on the nature of the radiative transition 12,13 . The over linear behavior of the PL peak intensity (k = 1.18) suggests an excitonic nature (radiative de-excitation of a bound or free exciton) rather than a defect-related emission, generally characterized by a sublinear k-factor. However, this analysis should be completed, in particular by extending the power range on which this study is made. Figure SM5a displays the different steps necessary to the fabrication of a hBN/BP/hBN heterostructure. A high pressure / high temperature hexagonal Boron Nitride (hBN) crystal 14 is classically exfoliated using PDMS and deposited on a SiO2(300 nm)/Si substrate (1-). Using the "all dry" method described by Castellanos Gomez et al. 15 , an exfoliated BP flake is spotted on PDMS by optical contrast (2-) and then slowly deposited on a large flat hBN flake (3-).

Heterostructures

Before encapsulation, the thickness of the BP flake is measured by AFM (4-). Similarly, another large flat hBN flake is spotted by optical contrast on the PDMS (5-) and deposited on the top of the pre-existing BP/hBN stack (6-) to build a hBN/BP/hBN heterostructure (7-). The entire process is carried out in a glove box under inert atmosphere to avoid BP oxidation. About 15 similar stacks were made with different black phosphorus thicknesses as shown in Figure 3a. ) , with N a normalization constant.

The associated kinetic energy, also referred as confinement energy, is The fit reported in Figure 2 gives an effective mass of the same order of magnitude of the the experimental values used to predict the excitonic properties in the bulk model [START_REF] Ye | Highly polarization sensitive infrared photodetector based on black phosphorus-on-WSe 2 photogate vertical heterostructure[END_REF] . We also stress that this is an independent-particle model because there is no particle-particle interaction term.

< 𝐾 > = -
The adoption of such assumption to describe our measurements is justified a posteriori by the observation that the emission is insensitive to changes of the dielectric environment.

Figure 1 .

 1 Figure 1. Photoluminescence spectra recorded a 4 K from (a) an as-received BP crystal, (b) from a thick BP chip obtained after a single-stamping exfoliation step, and (c) from a 700 nm thick flake obtained after multiple-stamping exfoliation steps. The excitation wavelength is 532 nm and the fluence is 2.10 2 W.cm -2 for the macro-PL (a) and 3.10 3 W.cm -2 for the micro-PL (b,c) experiments. (d) Photography and optical images corresponding to the three samples analysed in (a,b,c) from top to bottom. Labels 1 to 4 refer to the areas where spectra in (b) are recorded. (e) Raman spectra recorded from areas 1 and 4 of the sample (b) with a 633 nm excitation laser.

Figure 2

 2 Figure 2 (a) Representative AFM image of exfoliated BP and deposited on Si/SiO2. (b) Selection of photoluminescence spectra of BP flakes at different thicknesses probed at 4 K and excited at a wavelength of 532 nm and at a fluence of 3.10 4 W.cm -2 . The grey line indicates the part of the spectra disturbed by the CO2 absorption band. (c) Evolution of the PL peak energy as a function of the thickness (black bullets and circles) and its fit based on a quantum well (QW) model. (red dashed line). The bound exciton energy of the BP crystal at 4 K is marked as a horizontal blue line. A black dashed line indicates the threshold thickness at 25 nm. The blue star corresponds to the energy of the PL peak shown in Figure 1c.

Figure 3 .

 3 Figure 3. (a) Probability densities P(z) (black solid line) and z P(z) (blue dashed line). Inset: cartoon representing the electron-hole z-distance probability zP(z) inside a bulk-BP slab. (b) Variation of the photoluminescence peak energy with the thickness. hBN / BP / hBN (red triangles) and SiO2/BP/Al203 (gray dots) heterostructures. c Comparison of the emission band energy variation in BP and MoS2 6,28,29 . Red arrows indicate the limit beyond which the energy band gap varies by less than 3% of the total band gap modulation from the monolayer to the bulk.

Figure SM2 -

 SM2 Figure SM2 -Raman spectra recorded from areas 1 to 4 of Figure 1b sample with a 633 nm excitation laser.

Figure SM3 -Figure SM4 -

 SM3SM4 Figure SM3 -Optical and AFM images of the seven flakes from which the spectra shown in Figure 2b are recorded. AFM profiles in insets of the images are extracted along the grey lines marked in the AFM images.

Figure SM5a -

 SM5a Figure SM5a -Optical images and associated diagrams describing the manufacturing process of a hBN/BP/hBN heterostructure: 1-Crystal of hBN on SiO2/Si. 2-Crystal of BP on PDMS. 3-BP transfer on hBN, the wavefront corresponds to the PDMS -SiO2/Si contact line. 4-BP/hBN heterostructure on SiO2/Si. 5-hBN crystal on PDMS. 6-Transfer of hBN on BP/hBN. 7-hBN/BP/hBN heterostructure.

Figure SM5b -

 SM5b Figure SM5b -PL spectra of a 5 nm thick alumina-passivated black phosphorus flake (gray) and a hBN / BP (4.6 nm thick) / hBN (red) heterostructure at 4 K. PL intensities are normalized (normalization factors are x1.5 and x1 respectively as indicated in the figure).

Figure 6 .

 6 FigureSM5bshows typical spectra of two BP flakes of comparable thickness, one encapsulated in hBN flakes (red) and the other one passivated with alumina (gray) at 4 K. The PL spectrum of BP encapsulated in hBN does not differ drastically from that of alumina passivated BP. In particular the intensities and full widths at half maximum of the peaks (29 meV for BP encapsulated in hBN ; 37 meV for passivated BP) are of the same order of magnitude contrary to what could be observed for the atomic layers of TMDC encapsulated in boron nitride16 . The energy difference between the two peaks (0.53 eV for BP encapsulated in hBN ; 0.49 meV for passivated BP) can be explained by the slight difference in BP thickness. The variation of the dielectric environment does not seem to influence the luminescence energy at these thicknesses as it had been shown for the atomic layers of BP17 .

  ) 𝑛 𝑒ℎ =

	𝑃.𝜏.𝜆 ℎ.𝑐	.	1 𝑉	= 2.2. 10 19 𝑐𝑚 -3

  where mz are the effective masses of the single charges and 𝑚 * =

	ℏ 2 2	∫ ∫ 𝜙 𝐿 {	1 𝑚 𝑒	𝜕 2 𝜙 𝐿 𝜕 𝑧 𝑒 2 +	1 𝑚 𝑧	𝜕 2 𝜙 𝐿 𝜕 𝑧 ℎ 2 } 𝑑𝑧 𝑒 𝑑𝑧 ℎ =	ħ²𝜋² 2𝑚 * 𝐿²
						𝑚 𝑒 𝑚 ℎ 𝑚 𝑒 +𝑚 ℎ	is the reduced effective
	mass of the pair.

Laboratoire d'Étude des (LEM), UMR 104 CNRS-Onera, Université Paris Saclay, 92322 Châtillon, France.

Groupe d'Etude de la Matière Condensée (GEMaC)), CNRS-UVSQ, Université Paris Saclay, 45 Avenue des Etats-Unis 78035 Versailles, France

Unité Mixte de Physique, CNRS, Thales, Université Paris-Saclay, 91767 Palaiseau, France.

Laboratoire Photonique Numérique et Nanosciences, CNRS -Institut d'Optique -Univ. Bordeaux, 33400 Talence, France

Acknowledgments.

The authors acknowledge funding from the French national research agency (ANR) under the grant agreement No. ANR-17-CE24-0023-01 (EPOS-BP). This project has also received funding from the European Union's Horizon 2020 research and innovation program under grand agreement N° 881603 (Graphene Flagship core 3).


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Lorenzo Sponza

Impact of edge morphology and chemistry on nanoribbons' gapwidth

In this work, we scrutinise theoretically how the gap of C and BN armchair nanoribbons changes upon variations of the bond length between edge atoms and their distance from passivating species. Our DFT calculations indicate that the gap of C-based nanoribbons is more sensitive to the relaxation of the bonding length between edge atoms (morphology) whereas in BN-nanorribons it is more sensitive to the distance between edge atoms and passivating hydrogens (chemical environment). To understand the origin of these two different behaviours, we solved a tight-binding ladder model numerically and at the first-order perturbation theory, demonstrating that the different dependence is due to the interference of the wavefunctions of the top valence and the bottom conduction states.

I. INTRODUCTION

In recent decades, graphene and hexagonal boron nitride (BN) have attracted a great deal of interest because of their remarkable transport and optical properties [1][2][3][4][5] . A much explored way to modulate them is by adding extra confinement (as in 2D quantum dots, nanoribbons or nanotubes). The presence of confining edges endows them with novel sizedependent features dominated by the characteristics of the edge itself. This is why graphene and BN nanoribbons are often classified according to their edge shape, which can be zig-zag, armchair, fall in an intermediate chiral angle, or present structures that require a more general nomenclature 6 . In zigzag nanoribbons, well localised edge-state are formed which confer antiferromagnetic properties to C-based zig-zag nanoribbons [6][7][8][9][10][11][12] . Instead, BN-based zig-zag nanoribbons have an indirect gap and display an intrinsic dipole moment 9,[13][14][15][16][17][18][19] . At variance, both graphene 6,[8][9][10][11][12][20][21][22][START_REF] Nishad | Proceedings of the IEEE Conference on Nanotechnology[END_REF][START_REF] Prabhakar | [END_REF][25][26][27] and BN 9,14-18 armchair nanoribbons (AGNR and ABNNR), have no magnetic states and display a direct size-dependent gapwidth To take full advantage of this richness of properties, several methods have been explored including the application of external electromagnetic fields 9,10,14,18,27 , strain 17,[START_REF] Prabhakar | [END_REF]28 and edge engineering 17,19,[21][22][START_REF] Nishad | Proceedings of the IEEE Conference on Nanotechnology[END_REF][START_REF] Prabhakar | [END_REF][25][26]29 .

As a matter of fact, the edge characteristics are crucial for the performances of nanoribbonbased devices such as transistors, interconnects and logical devices [START_REF] Nishad | Proceedings of the IEEE Conference on Nanotechnology[END_REF][29][30][31][32][33] , photovoltaic applications 33,34 , or chemical sensing 33,35 .

Experimentally, edge engineering 34,36,37 , chemical treatment 38 or selective passivation 29 have been demonstrated to have a significant impact on the device quality, precisely because of their action on the edges.

Alterations of the electronic structure due to edge modifications can be divided into morphology effects (variation of the bondlengths) and chemistry effects (variation of the passivating species and their distance from the edges) 6,26 . The sensitivity of AGNR and ABNNR gap to the passivation has been investigated by many authors 6,17,19,[21][22][START_REF] Nishad | Proceedings of the IEEE Conference on Nanotechnology[END_REF][START_REF] Prabhakar | [END_REF][25][26]29 who showed that its effect depends on the type of atoms involved, and/or on the number and position of the passivated sites. Most of these first-principle studies 17,19,[21][22][START_REF] Nishad | Proceedings of the IEEE Conference on Nanotechnology[END_REF][START_REF] Prabhakar | [END_REF][25] discuss the role of passivation on fully relaxed structures, so morphology and chemistry effects are actually treated on the same footing. At best of our knowledge, only two studies 6,26 have been conducted in frameworks that separate the two effects, but in both publications the focus is put on other aspects than the dependence of the gapwidth on morphology and chemistry modifications.

On the other hand, rare are the studies on genuine morphological effects 6,12,26 and they are done only on AGNRs. Actually, in 39,40 , a thorough study of edge reconstruction and edge stress in graphene and BN nanoribbons is actually carried out, but the investiga-tion stops at a stability level and the relation to the gapwidth is not explored. However, both effects seem to be decisive in determining the gap of nanoribbons and we deemed that the subject deserved a more focused study.

In this article, we employ density functional theory (DFT) to study the evolution of the gap, the top valence (TV) and the bottom conduction (BC) states of AGNRs and ABNRs as a function of the nanoribbon size upon variations of the distance between edge atoms and between these and the passivating species. Our objective is to compare the effect of morphological and chemical variations on the gapwidth and understand which of them is dominant and in which situation. We demonstrate that the response of the gapwidth to changes of the distance between edge atoms (morphology) or between edge atoms and passivating atoms (chemical environment) is opposite in the two materials and we rationalise this different behaviour by means of a tight-binding model which we solved both numerically and perturbatively.

II. STRUCTURAL AND COMPUTATIONAL DETAILS

All nanoribbons studied in this article have armchair edges passivated with H atoms. They form an infinite periodic structure in the y direction and are confined along x. The extension of the periodic cell along y is the cell parameter a, while the width is expressed by the number N a which indicates the number of dimers aligned along y inside the unitary cell (number of rows). To indicate a specific structure we will attach the index N a after the label of the material, as in Figure 1, so for instance AGNR5 designates an armchair graphene nanoribbon of size N a = 5.

Density functional theory calculations were carried out within the generalized gradient approximation using the PBE 41 exchange correlation potential as implemented in the Quantum ESPRESSO 42 simulation package. Long-range van der Waals corrections were included via the DFT-D2 method 43 . To avoid interactions between consecutive cells, we included 15 Å and 20 Å of empty space in the z and x directions respectively. In electron density calculations and relaxation runs, the periodic axis was sampled with 20 k-points centered in Γ (corresponding to 11 irreducible k-points). This mesh was dense enough to converge total energies in the smallest nanoribbons. For density of states (DOS) calculations, a five times denser sampling was adopted for all systems and the resulting spectra have been broadened with a Gaussian distribution with a width of 0.02 eV.

We used norm-conserving pseudopotentials 44 and set the kinetic energy cutoff at 80 Ry in both materials. It is worth stressing that using a large vertical empty space and a high energy cutoff is essential even in the relaxation runs in order to prevent nearly free-electron states from hanging below the p z states hence jeopardizing the gap description. In fact, as already well known for free-standing layers [45][46][47][START_REF] Paleari | First-principles approaches to the description of indirect absorption and luminescence spectroscopy: exciton-phonon coupling in hexagonal boron nitride[END_REF][START_REF] Latil | Structural clas-sification of boron nitride twisted bilayers and ab initio investigation of their stacking-dependent electronic structure[END_REF] and nanotubes [START_REF] Blase | [END_REF][51][52] in BN nanomaterials there is a competition at the bottom conduction between 2p z and 3s states, whose right alignment requires a dedicated convergence study. If sometimes one can overlook this issue in BN layers, because the two competing states originate direct and indirect band gaps, this is not the case in ABNNRs where both states give rise to a direct gap at Γ.

In non-relaxed structures, all atoms occupy the sites of a regular honeycomb lattice with an inter-atomic distance of 1.42 Å. Structural relaxation runs have been performed with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm for all systems with the stopping criterion of all forces being lower than 5 × 10 -5 eV/Å. We allowed variations of the cell parameter a and all atomic positions. As clarified in the following, we also run some calculations letting only specific atoms to move. In Figure 1 we report the relaxed structures of AGNR and ABNNR at N a = 5 for sake of example, and we introduce some notable structural parameters. In the AGNRs, the main modifications with respect to nonrelaxed structures are a contraction of the distance between edge atoms d E and between C and H d HC .

In ABNNR, we observe a similar contraction of the B-N distance on the edges d E , and different contractions of the distances between H-B and H-N (d HB = d HN ). We observed also that these modifications are basically independent on the size of the nanoribbon both qualitatively and quantitatively, so the structural parameters undergo minimal variations when comparing nanoribbons of different size.

III. GAP EDGE STATES

A. AGNRs

The electronic structure of AGNRs has been already studied in the past 6,[8][9][10][11][20][21][22][START_REF] Nishad | Proceedings of the IEEE Conference on Nanotechnology[END_REF][START_REF] Prabhakar | [END_REF][25][26]33,34 . Both non-relaxed and relaxed ribbons display a band gap at Γ of gapwidth ∆ Na . Because of the 1D confinement, the gapwidth falls in one of the three families N a = 3m -1, 3m or 3m + 1 (with m ∈ N * ). Each family follows a different trend which asymptotically tends to zero for growing nanoribbon sizes and follows the general rule ∆ 3m-1 < ∆ 3m < ∆ 3m+1 . This is depicted in Figure 2 where we plot the gapwidth of AGNRs versus N a for both non-relaxed and relaxed structures (red dashed and solid blue curves). The effect of relaxation is to open the gap by about 0.1 eV in families N a = 3m + 1 and 3m -1, while in the N a = 3m the opening is observed only in small nanoribbons, while the gap closes in larger ones. Our results are in quantitative agreement with previous works both for relaxed 11,12,21,25 , and unrelaxed simulations 26 .

To characterise better the gap states, we analyzed in more detail the nature of the TV and the BC states at Γ in the relaxed structures. In panels a) and b) of Figure 3, we report the band structure and the density of states (DOS) of the AGNR8, chosen as a representative example. For sake of comparison, in panel b) we They form a π bonding with nodes on the xy plane, as expected. Instead, the first empty σ state is found at 3 eV above the BC. To go deeper in the analysis of the gap-edge states, we look at the site-projected DOS. We integrated the bare data inside an interval of 0.1 eV encompassing the TV and BC (shaded bands in the instet of Figure 3b). The outcome of this analysis is summarised in Figure 3c), where the site-projected DOS of gap-edge states is reported as a function of the row index (note that the curves are plotted on the same y axis). At variance from what observed in zigzag nanoribbons 7 , the gap states are not concentrated on the edge atoms, but rather delocalized throughout the full nanoribbon and present a modulation that nicely displays the characteristics of a static wave. This observation is confirmed by the wave-like modulation of the charge probability |ψ(r)| 2 associated with the TV and BC states, reported aside panel c). The wavefunction plot shows also that there is no spill-out on the passivating hydrogens and that, with respect to the edge-bbondings d E , TV and BC states display respectively a bonding and an antibbonding character.

B. ABNNRs

The gapwidth of ABNNRs fall in the same three families with the same hierarchy 17,18,28 . This similarity with the graphene ribbons is actually quite general and can be understood from a simple tight-binding model (see section IV B). The evolution of the ABN- NRs gapwidth for sizes going from N a =5 to 19 in the relaxed and non-relaxed configurations is presented in Figure 4 by the solid blue and the red dashed lines. The non-relaxed structures present a gap that monotonically tends to the limit N a → ∞ in a way that is similar to non-passivated calculations 17 . We estimate N a → ∞ = 3.885 eV from the weighted average of the curves extrapolated at 1/N a = 0 (cfr. inset of the Figure ). This value is about 0.8 eV lower than the gapwidth of the isolated BN sheet (4.69 eV in PBE). All these aspects are consistent because, as it will become clearer later, in non-relaxed calculation, H atoms are too far to saturate efficiently the dangling bonds located at the edges of the ribbbon. As a consequence, these form edge states inside the gap that lower the gapwidth similarly to what happens in non-passivated (bare) ribbons.

As a result of the structural optimisation, the gapwidth of all families opens and tends to an asymptotic limit that is still about 0.1 eV lower than in the isolated monolayer, in agreement with similar calculations 14,17 . This discrepancy is ascribed to a nonnegligible edge contribution to the BC state, obviously absent in the isolated monolayer (cfr. the rowprojected DOS analysis here below, and 14 ). Finally, we note that the first empty σ state, i.e. the near free-electron state, is only 0.5 eV above the BC.

Similarly to what done before, in Figure 5 we report the band structure, the projected DOS and the row-resolved DOS of the TV and BC states of the representative ABNNR8 system. We verify that the TV and the BC states are formed essentially of Ncentered and B-centered p z orbitals respectively. The row-projected DOS of both TV and BC, reported in panel c), shows again a very nice static-wave-like modulation with nodes in rows 3 and 6, but at vairance with the AGNR8 case, here the TV and BC states localize differently: while the TV states are delocalised on the entire nanoribbon as in the previous case, the BC states are clearly peaked at the edges. The visualization of the associated charge density confirms that the TV state is characterised by a wavefunction equally delocalised on all the N atoms except those on rows 3 and 6. Instead, the BC state presents a wavefunction more concentrated on the edge B atoms with non negligible tails touching passivating H and edge nitrogens, in contrast to the isolated monolayer.

The compared study of the TV and BC states of AGNRs and ABNNRs suggests that the gap of the two materials responds differently to modifications of the morphology and the passivation of the edges. To test this intuition, we have performed a detailed analysis by separating the two effects.

IV. MORPHOLOGY VS CHEMISTRY OF THE EDGES

A. Distinguishing the effects through selective relaxation in DFT Several investigations can be found in literature on the effects of edge reconstruction on the gapwidth of AGNR and ABNNR 6,12,17,19,[21][22][START_REF] Nishad | Proceedings of the IEEE Conference on Nanotechnology[END_REF][START_REF] Prabhakar | [END_REF][25][26] . However, a study that systematically compares the effects of passivation and edge morphology is absent. Here we monitor the gapwidth in the family N a = 3m -1 by relaxing separately the H-X distances d HX (X = C, B or N) and the C-C or B-N distance on the edges d E . We did calculate data from the other two families but we do not report them because they have qualitatively the same behaviour.

In Figure 6, a variation of d HX is represented by a change in the line's type (color and dash), while a variation of d E is represented by a change in the symbols (colour filled or empty). Let us examine first the case of AGNRs in panel a). We can start from a non-relaxed configuration where all atoms are equidistant d HC =d E =1.42 Å (empty bullets, red dashed line), then we reduce d HC to its relaxed value 1.08 Å (empty bullets, blue solid line). We observe that there is basically no variation on the AGNRs' gapwidth. Instead, contracting the edge bonds from d E =1. 42 that in AGNRs, the variations of the gapwidth induced by the relaxation and reported in Figure 2 are essentially due to changes of bond length d E between C atoms at the edge. Interestingly, this gap opening is approximately independent on the width of the ribbon.

Passing now to the study of ABNNRs (bottom panel), we observe an opposite behaviour. The gapwidth undergoes very small changes upon relaxation of d E whereas the passage from the unrelaxed H-B and H-N distance (1.42 Å) to the relaxed values clearly opens the gap by about 0.8 eV. To be more precise, by changing separately the two distances d HB and d HN (not shown), we found that it is the bonding between H and B that plays a major role in the opening of the gapwidth, indicating a dominant contribution from conduction states consistently with the observations we drew from Figure 5. According to this analysis, the gapwidth of ABNNRs is more sensitive to the passivation than to the very morphology of the edge. Once again we notice that the gap opening is basically independent on N a . This clarifies why our non-relaxed DFT gapwidth look very similar to the non-passivated results of Topsakal and coworkers 17 .

B. Unperturbed tight-binding model

To investigate further the reasons of this different behaviour, we generalise a ladder tight-binding model introduced initially for AGNRs to the heteroatomic case. Changes in the edge passivation and morphology will be successively introduced through variations of the on-site and hopping parameters of the model, as suggested in 6,12 , and the modified Hamiltonian solved both numerically and perturbatively 12 .

Following references [6][7][8]10,12,16,20 , the gap of an armchair nanoribbon whose TV and BC states are formed of p z orbitals can be described with a ladder tightbinding model as the one reported in Figure 7. The Hamiltonian of the model reads:

H 0 = j,µ   µj |Φ µj + j ,µ t µµ jj |Φ µ j   Φ µj | .
(1) The index j ∈ [1, N a ] labels the position of a dimer in the x coordinate (row coordinate), while µ = 1, 2 indicates the atomic site within the dimer (C 1 or C 2 in AGNRs and B or N in ABNNRs). The basis function r|Φ µj = Φ µ (r -r j ) is the p z orbital of the atom µ of the dimer placed at r j = x(j -1)a. For µ = 1, Φ µ (r -r j ) is centered on the bottom rung if j is odd and in the upper rung if j is even, and the opposite for µ = 2.

At the unperturbed level, µj does not depend on the row-index j and is equal to for µ = 1 andfor µ = 2, with ≥ 0. In the first-neighbour approximation, the hopping term t µµ jj = t ∈ R if µ = µ and j -1 ≤ j ≤ j + 1 and vanishes otherwise. The unperturbed solutions of this model are:

E 0 n± = ± 2 + τ 2 n = ±E n , (2)
where

τ n = t [1 + 2 cos (θ n )],
the discrete index n comes from the confinement in the x direction and θ n = nπ/(N a + 1). The eigenfunction associated to these states read

Ψ n± = Na j=1 µ=1,2 sin (jθ n ) D n± µ Φ µ (r -r j ) (3)
with

D n± 1 = E n ± (N a + 1)E n D n± 2 = ±sgn (τ n ) E n ∓ (N a + 1)E n (4)
where the function sgn (x) = 1 if x ≥ 0 and -1 if x < 0. At this point, it is worth stressing two aspects. First, if one poses τ n = 0, then the Hamiltonian becomes diagonal and equivalent to that of a The first neighbours distance is a, the index j defines the position of a dimer. Atoms µ = 1 are placed above µ = 2 if j is even, below if j is odd.

t t a a ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 j=1 2 3 4 5 6 7 N a =8
non-interacting system. Consistently, the coefficients D n± µ become those of a pure system:

D n+ 1 = -D n- 2 = 2/(N a + 1) and D n- 1 = D n+ 2 = 0.
If instead one takes the homoatomic limit, i.e. → 0, then the coefficients become a bonding and antibonding pair, with

D n± 1 = 1/ √ N a + 1 and D n± n = ±sgn (τ n ) / √ N a + 1.
The last occupied state (TV) |ñ, -and the first empty state (BC) |ñ, + are found at the integer quantum number ñ that minimizes the quantity E n , i.e. that minimize |τ n |. If N a = 3m or 3m+1 with m ∈ N * , then ñ = 2m+1. Note that the interacting term τ 2m+1 changes sign in passing from one family to the other. Instead if N a = 3m -1, then the integer ñ = 2m and τ n = 0. These considerations leads to the unperturbed gap of a heteroatomic system ( > 0):

∆ 0 Na = 2 for N a = 3m -1 2E 2m+1 for the other values of N a (5)
and the eigenstates of the TV and BC of the N a = 3m -1 family are pure states. The gap of a hosystem ( = 0) reads:

∆ 0 Na = 0 for N a = 3m -1 2|τ 2m+1 | for the other values of N a (6)
and the eigenstates of the TV and BC of the N a = 3m -1 family are the bonding and antibonding combinations of C 1 and C 2 .

C. Distinguishing the effects through perturbation theory

As in 6,12 , we now add to H 0 a perturbation Hamiltonian δH which consists in adding δt to the hopping term connecting the atoms of the edge rows (j = 1, N a ) and in changing their on-side energy by δ µ . The hopping perturbation δt accounts for changes in d E , so it is more strongly related to the edge morphology, while the on-site one δ takes into account variations of d HX and of the passivating species. The perturbative correction to the energy of the generic state |n± reads

n, ±|δH|n, ± = 2 sin 2 (θ n )× × (D n± 1 ) 2 δ 1 + (D n± 2 ) 2 δ 2 + 2D n± 1 D n± 2 δt (7)
In the heteroatomic case > 0, the perturbative correction to the gap is always δ∆ = ñ, +|δH|ñ, +ñ, -|δH|ñ, -. Using (7), the coefficients (4) or their appropriate limit, and remembering that ∆ 0 Na = 2E ñ, then the gap correction for the case > 0 reads,

δ∆ = (δ 1 -δ 2 ) /m (8)
for N a = 3m -1; and

δ∆ = 8 sin 2 (θ 2m+1 ) (N a + 1)∆ 0 × × (δ 1 -δ 2 ) + 2τ 2m+1 δt (9)
for N a = 3m and N a = 3m + 1. Notice that, by construction, τ 2m+1 is the closest to zero among the accessible values, so the term 2τ 2m+1 δt is always negligible. The result shows that in ABNNRs the variations of the gap are mostly due to the chemical environment of the edge atoms. This dependence comes ultimately from an interference between the TV and the BC wavefunctions. These two states are very close to pure states, so the mixed products D + 1 D + 2 and D - 1 D - 2 of equation ( 7) are systematically negligible, and they do actually vanish in the family N a = 3m -1 where the two states are perfectly pure.

In the homoatomic case ( = 0) the corrected gap can be obtained following the same approach as before, and taking the appropriate limits of the coefficients (4). However, more attention must be paid in studying the family N a = 3m -1. In fact this case corresponds to the double limit → 0 and τ n → 0. Even though the final eigenvalues do not depend on the order with which the two limits are taken, the eigenstates do, therefore also the perturbative corrections depend on this choice. In DFT calculation and experiments, the system itself is well defined at the very first place, because one works either with ABN-NRs or with AGNRs. So, for comparisons with DFT to make sense, the right order with wich the limits must be taken is: first → 0, followed by τ n → 0. Finally, one has to pay attention to another point: in the N a = 3m-1 family, the TV and the BC states are degenerate and the unperturbed gap is 0. So there is no reason to define δ∆ = ñ, +|δH|ñ, + -ñ, -|δH|ñ,rather than its opposite. However, the correction must be positive, so the correction must be defined as the modulus of the difference above. Putting all these things together, one gets for the homoatomic ( = 0) case

δ∆ = 2 m |δt| for N a = 3m -1 sgn (τ 2m+1 ) 8 sin 2 (θ2m+1) (Na+1)
δt otherwise (10) This result shows that in AGNRs most of the variations of the gap is accounted by δt, so by morphological changes of the bonding between edge atoms, and not by changes of their chemical environment. Once again this result can be understood from the symmetries of the TV and BC wavefunctions. In fact, when = 0, the TV and BC states are perfect bonding and antibonding combinations at any N a , so their difference causes the terms in (D ñ± µ ) 2 of equation ( 7) to always cancel out. This result, although in perfect agreement with 12 , seems to be in blatant contradiction with results from 2H-passivated AGNRs 26 , where the gap is found independend on the C-C edge distance. Actually, these systems present a hybridisation of the sp3 type and their gapwidth can not be described by this model.

D. Validitation of the perturbative approach

Besides the perturbative approach, we also solved the perturbed Hamiltonian H = H 0 + δH numerically. For the unperturbed problem, we parametrized the model with values that fit the band structure of the isolated graphene and hBN monolayers. Instead, the perturbation parameters δ and δt have been adjusted to recover as best as possible the DFT curves reported in Figures 2 and4. The best parameters are reported in Table I. Successively we explored how the gap changes upon variations of the perturbative parameters δt and δ µ in the range -1 eV, +1 eV in the nanoribbons of width N a =11, 12 and 13, i.e. one representative nanoribbon per family. Guided by physical intuitions we took δ 1 = δ 2 = δ in the case of AGNRs, and δ 1 = -δ 2 = δ in the case of ABNNRs.

Globally, the numerical and the perturbative gapwidth are in very good agreement for both ABNNRs and AGNRs in the range explored, confirming our conclusions. In all cases, the numerical solution displays a quadratic trend with respect to δ which adds on top of the invariance (AGNR) or the linear (ABNNR) dependence predicted by the perturbative approach. The deviations between the two approaches are larger 2 and4.

for this parameter than for δt, with the larger deviations of the order of 0.2 eV in the N a = 3m and N a = 3m + 1 families of ABNNRs. Instead, the deviations for the parameter δt are in general very small and never larger than 0.1 eV. Note however that for extreme values of δt, the numerical solution may undergo a band crossing in the top valence and the bottom conduction which would lead to a sudden closing of the gap, as it is the case at δt = -0.9 in AGNR13 and δt = 0.9 in AGNR12. This physics is not accessible in our first order expansion and clearly sets the limit of applicability of the perturbative approach.

V. CONCLUSION

We have calculated with DFT the gapwidth of graphene and boron nitride armchair nanoribbons (AGNRs and ABNNRs) for ribbon sizes going from N a = 5 rows to N a = 19 rows both for relaxed and unrelaxed structures. We have relaxed selectively specific interatomic distances and reported how the gapwidth changes upon variations of the bondlength with passivating atoms (chemistry-driven changes) and between edge atoms (morphology-driven changes). Thanks to this selective relaxation, we showed that the variations of the gapwidth in AG-NRs are morphology-driven, while in ABNNRs are chemistry-driven. To understand why, we adopted and extended the tight-binding approach introduced by Son and coworkers 12 and we demonstrated that the interference between the wavefunctions of the top valence and the bottom conduction are at the origin of these two distinct responses.

In the AGNR case, these states are basically a bonding and antibonding pair. As the two states are equally distributed on the atoms, the difference between BC and TV leads to a mutual cancellation of on-site changes, and only hopping terms survive. This explains the stronger dependence of the gapwidth on interatomic distances and hence on the morphology of the edges rather than the chemical environment. At variance, in ABNNR case, the TV and the BC states are basically pure states and the effective Hamiltonian is quasi non-interacting. As a result, the two states are mostly insensitives to variations in the hopping term and are instead strongly affected by on-site variations (chemical environment).

Our results can help pushing further the research on nanoribbon-based devices, as they clarify the role played by edge-engineering, and selective passivation and provide the tools to investigate more complex scenarios.

Figure 1 .

 1 Figure 1. Scheme of relaxed AGNR5 and ABNNR5 structures. Unitary cells are reported as black dashed rectangles and some relevant structural parameters are labelled in red. In blue, the row index j and the edge or inner character of atoms are also reported.

Figure 2 .

 2 Figure 2. Energy gap of graphene nanoribbons as a function of the width Na. Relaxed calculations (blue solid line), unrelaxed (red dashed line) and tight-binding numerical solution (black dotted) with parameters indicated in Table I. The three families are reported with different symbols. A blue arrow at Na = 8 indicate the nanoribbon chosen for the analysis presented in Figure 3.

Figure 3 .

 3 Figure 3. Electronic structure of the relaxed AGNR8. a) Band structure. b) and Inset: Total density of states (thick black) and projected on pz orbital character (red bullets) compared with the DOS of the graphene sheet (dashed blue). c) Row-projected DOS from the integration of the total DOS around the band-edge states (shaded areas of panel b) and charge density associated with the TV and BC states at Γ.

Figure 4 .

 4 Figure 4. Energy gap of BN nanoribbons as a function of the size Na. Relaxed DFT (blue solid line), unrelaxed (red dashed line) and the numerical tight-binding solution (TableI). The three families are reported with different symbols. Horizontal dashed lines indicate the gapwith of the DFT hBN sheet (4.69 eV) and the asymptotic Na = ∞ limit (∼3.885 eV). The blue arrow pointing at the ABNNR8 indicates the system analysed in Figure4. Inset: extrapolation of non-relaxed calculations at 1/Na = 0. The red arrow in the inset indicates the Na = ∞ limit as the weighted average of the extrapolation of the three families.

Figure 5 .

 5 Figure 5. Electronic structure of the relaxed ABNNR8. a) band structure; b) total density of states (thick black) and projected on pz orbital character (red and green dotted for B and N states) compared to the hBN sheet DOS (dashed blue). c) Row-projected DOS integrated around the band-edge states (shaded areas of panel b). Insets: charge density of the TV and BC states at Γ.

Figure 6 .

 6 Figure 6. Gapwidth of the Na = 3m -1 family of a) AGNRs and b) ABNNRs. Full (empty) symbols stand for relaxed (non-relaxed) edge-atom bondings. Blue solid (red dashed) lines for relaxed (non-relaxed) passivatingto-edge-atoms bondings.

Figure 7 .

 7 Figure 7. Scheme of the ladder model of width Na = 8.The first neighbours distance is a, the index j defines the position of a dimer. Atoms µ = 1 are placed above µ = 2 if j is even, below if j is odd.

Figure 8 .

 8 Figure 8. Deviation from ∆ 0 Na as a function of δt (squares) or δ (circles) in AGNRs (top panels) and ABNNRs (bottom panels). From left to right: Na = 11, 12, 13. Colored symbols result from equations (8 -10); empty symbols from the numerical solution.

Table I .

 I Parameters of H = H 0 + δH in eV used to plot curves in Figures

	AGNR		ABNNR
	C tCC δt	B	N	tBN δ
	0.0 -2.6 -0.4	2.32 -2.32 -2.46 -0.3


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Multi-technique analysis, correlation between ultrasound and x-ray diffraction on commercial cells

Li-ion batteries currently dominate the electrochemical systems market but still suffer from ageing problems. Being able to determine the State of Health (SOH) of batteries is a critical need 1 to ensure user safety. Many studies on the ageing of battery materials have already been carried out, using multiple techniques, in particular using large scale facilities. These analyses allow operando monitoring of degradation mechanisms, but they are very long and difficult to set up 2-3 and unfortunately cannot be used for use to monitor the day-to-day real-life conditions.

Ultrasound technology, which has been already widely used in many industrial sectors such as nuclear 4 and medical [START_REF] Juengert | AIP Conference Proceedings 1949[END_REF] , has many advantages such as simple to implement, inexpensive, and enable nondestructive & operando measurements, allowing "in vivo" monitoring of the battery. However, it is difficult to know what an ultrasonic sensor can detect in a battery, whether it can determine the state of ageing of a battery and, above all, if the results obtained can be linked to any structural or morphological changes.

The objective of the study is therefore to correlate the analysis of ultrasound signals with X-ray diffraction measurements at the synchrotron (BM02 line at ESRF) in operando mode as a function of electrochemistry. For this purpose, a commercial cell composed of multiple electrodes of LiNi0.6Mn0.2Co0.2O2 (NMC 622) versus graphite (Z-stacking) was paired with an ultrasound sensor. X-ray diffractograms were acquired at several cycling rate and also during relaxation periods. The aim was to correlate the structural evolutions as a function of the charge/discharge rate and as a function of the ultrasound signal signature.

After checking if there is no beam damage on the sample, we were able to follow the evolution of the position of the two main reflections of the materials: the NMC (003) and the graphite (002) as a function of the lithiation rate (SOC) (Figure 1a). As can be seen in Figure 1b, we can correlate the electrochemical signal (derivative curve) to the change in the lattice parameters of graphite and NMC and to the time of flight (ToF) of the ultrasound showing that ultrasound can be an excellent non-destructive sensor to track structural changes within battery during cycling.

Through this talk, we will be presenting the methodology employed to correlate the results between XRD and ultrasound, as well as the additional techniques used to determine ageing parameters (impedance, X-ray tomography, dilatometer measurement, etc…).

Figure 1 .

 1 Figure 1. a) X-ray diffractogram acquired on the complete cell; b) Correlation between the electrochemical signal (derived from the galvanostatic curve) and the ultrasound signal

  Benjamin Mercier-Guyon a , Corentin Renais a , Maxime Servajon a , Morian Sonnet b , David Wasylowski b ,

	Sandrine Lyonnard c & Claire Villevieille a
	a-Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, Grenoble INP, LEPMI, 38000 Grenoble,
	France
	b-Chair for Electrochemical Energy Conversion and Storage Systems, Institute for Power Electronics
	and Electrical Drives (ISEA), RWTH Aachen University, Jägerstrasse 17-19, 52066 Aachen, Germany
	c-Université Grenoble Alpes, CEA, CNRS, IRIG, SyMMES, Grenoble, France
	Benjamin.mercier-guyon@grenoble-inp.fr

Benjamin Mercier-Guyon, Corentin Renais, Maxime Servajon, Morian Sonnet, David Wasylowski, Sandrine Lyonnard, Claire Villevieille To cite this version: Benjamin Mercier-Guyon, Corentin Renais, Maxime Servajon, Morian Sonnet, David Wasylowski, et al.. Multi-technique analysis, correlation between ultrasound and x-ray diffraction on commercial cells. International Society of Electrochemistry (ISE) -2023, Sep 2023, Lyon, France. hal-04290284


text:
Sylvain Latil

Binh Van

Elisa Serrano Vu

Hakim Richaud

Lorenzo Sponza Amara

Sponza

Van Binh Vu

Elisa Serrano Richaud

Hakim Amara

Lorenzo Sponza

Electronic localization in high-angle twisted boron nitride bilayer

The wide spectrum of physical properties exhibited by twisted bilayers has led to significant advances giving rise to the promising field of twistronics [START_REF] Carr | Electronic-structure methods for twisted moiré layers[END_REF][START_REF] Andrei | The marvels of moiré materials[END_REF][START_REF] Inbar | The quantum twisting microscope[END_REF][START_REF] Du | Moiré photonics and optoelectronics[END_REF][START_REF] Liu | Higher-Order Band Topology in Twisted Moiré Superlattice[END_REF]. By stacking 2D atomic layers to form van der Waals heterostructures, a geometric moiré superlattice emerges as a result of a lattice mismatch or a rotational twist [START_REF] Gratias | Crystallography of homophase twisted bilayers: coincidence, union lattices and space groups[END_REF]. The resulting pattern modulates the potential at the supercell scale and hence changes the electronic band structure typically through the formation of low dispersing bands which possibly lead to peculiar transport characteristics, as unconventional superconductivity [START_REF] Yankowitz | Tuning superconductivity in twisted bilayer graphene[END_REF][START_REF] Balents | Superconductivity and strong correlations moiré flat bands[END_REF] or insulating behavior driven by correlations [START_REF] Cao | Correlated insulator behaviour at half-filling in magic-angle graphene superlattices[END_REF]. Typical examples of moiré composites include the pioneering twisted bilayer graphene [START_REF] Bistritzer | Moiré bands in twisted double-layer graphene[END_REF], twisted hexagonal boron nitride (hBN) [START_REF] Liu | Higher-Order Band Topology in Twisted Moiré Superlattice[END_REF][START_REF] Xian | Multiflat Bands and Strong Correlations in Twisted Bilayer Boron Nitride: Doping-Induced Correlated Insulator and Superconductor[END_REF][START_REF] Kim | Electrostatic moiré potential from twisted hexagonal boron nitride layers[END_REF], hetero-and homobilayers of transition metal dichalcogenides (TMDs) [START_REF] Scuri | Electrically tunable valley dynamics in twisted wse2/wse2 bilayers[END_REF] as well as graphene on lattice-mismatched hBN substrates [START_REF] Wang | New generation of moiré superlattices in doubly aligned hbn/graphene/hbn heterostructures[END_REF].

In this context, semiconducting twisted systems, like hBN or TMDs, are peculiar. In these bilayers, the width of band edge states decreases continuously with the angle of twist (no magic angle) and the presence of different atomic species generates several stacking possibilities with specific electronic properties, hence providing an additional degree of freedom with respect to graphene bilayers [START_REF] Zhao | Formation of bloch flat bands in polar twisted bilayers without magic angles[END_REF][START_REF] Walet | Flat bands, strains, and charge distribution in twisted bilayer h-BN[END_REF][START_REF] Latil | Structural classification of boron nitride twisted bilayers and ab initio investigation of their stacking-dependent electronic structure[END_REF]. More specifically, hBN having the widest band gap in the monolayer (larger than 7 eV [START_REF] Blase | Quasiparticle band structure of bulk hexagonal boron nitride and related systems[END_REF][START_REF] Wirtz | Excitons in boron nitride nanotubes: Dimensionality effects[END_REF][START_REF] Ribeiro | Stability of boron nitride bilayers: Ground-state energies, interlayer distances, and tight-binding description[END_REF][START_REF] Galvani | Excitons in boron nitride single layer[END_REF]) presents outstanding optical properties [START_REF] Sponza | Direct and indirect excitons in boron nitride polymorphs: a story of atomic configuration and electronic correlation[END_REF][START_REF] Schué | Bright luminescence from indirect and strongly bound excitons in h-bn[END_REF] with many possible applications [START_REF] Knobloch | The performance limits of hexagonal boron nitride as an insulator for scaled cmos devices based on two-dimensional materials[END_REF].

Since the early stages of the research on twist-angle physics, the scientific community has been investigating the small twist angle limit. In fact, there are no exhaustive theoretical studies in the literature about largeangle twisted bilayers. At best of our knowledge, only 30 • twisted hBN bilayers [START_REF] Chernozatonskii | BN Diamanelike Quasicrystal Based on 30°Twisted H-BN Bilayers and Its Approximants: Features of the Atomic Structure and Electronic Properties[END_REF] and graphene [START_REF] Yu | Dodecagonal bilayer graphene quasicrystal and its approximants[END_REF] have been considered. Even thought density functional the-ory (DFT) calculations suggest that this BN material is a new wide-gap 2D quasicrystal [START_REF] Chernozatonskii | BN Diamanelike Quasicrystal Based on 30°Twisted H-BN Bilayers and Its Approximants: Features of the Atomic Structure and Electronic Properties[END_REF], its electronic and optical properties have never been addressed specifically. Very recently it has been highlighted in AlN and GaN binary materials [START_REF] Kang | Multiple extremely flat bands in twisted bilayer binary materials at large twist angles induced by atomic reconstruction[END_REF] the emergence of localized states responsible of unique features. A similar physics is expected to occur also in large-angle twisted hBN bilayers. Most importantly, a 30 • rotation produces quasicrystals without translational symmetry in all hexagonal bilayers, encompassing all most popular 2D materials like hBN, graphene, TMDs and their heterostructures, so all these systems must share some specific characteristics when twisted at large angles.

In this Letter, we determine the evolution of the band structure and optical response as a function of the twist angle in the vicinity of 30 • for different stacking sequences of twisted hBN bilayers. By introducing a tight-binding (TB) model parametrized on DFT calculations, we show two surprising features. The first is that as the twist approaches 30 • (the asymptotic quasicrystal limit), the electronic properties of all stackings tend to the same band structure. This can be understood by the fact that the system slowly approaches the asymptotic limit where it loses all translational symmetries. In other words, the system tends to a limit where all local configurations coexist and the very notion of stacking sequence has no meaning. The second is that this common electronic structure is characterised by a strong electron localization resulting from the emergence of a bundle of lowdispersing states in the middle of the conduction band. We also show that, despite being in the middle of the conduction, the bundle is entirely responsible of a very intense and robust absorption onset which may give rise to spectacular excitonic properties.

To identify univoquely the structures we investigate, we use the definitions and nomenclatures that has been introduced in [START_REF] Latil | Structural classification of boron nitride twisted bilayers and ab initio investigation of their stacking-dependent electronic structure[END_REF] by some of us. Let us recall the most important results for the current work. Every BN bilayer with hexagonal symmetry can be identified univoquely by a stacking label and a couple of integers (q, p) defining the moiré supercell. There are only five possible stacking sequences for each (q, p)-pair. The five stacking labels take their name from the pair of atoms placed exactly on top of each other on the three high-symmetry points of the hexagonal supercell that are the origin, the point (1/3 1/3) and the point (2/3 2/3). The five stackings divide into single-coincidence stackings (labelled BB, BN and NN) and double coincidence stackings (labelled BBNN and BNNB). The (q, p) indeces define two matrices that, once applied to monolayer unitary vectors, generate the supercell of the lower layer and the twisted supercell of the top layer. If we take without loss of generality p > q, then the rotation of the top layer with respect to the bottom layer is given either by an angle θ (in the single-coincidence systems) or -θ ′ (in the doublecoincidence case) with θ ′ = π/3θ (cfr. Figure 1. Both twist angles can be derived from p and q with formulae derived in [START_REF] Latil | Structural classification of boron nitride twisted bilayers and ab initio investigation of their stacking-dependent electronic structure[END_REF].

In order to study in detail low-energy excitations in these systems, we developed a tight-binding model purposely designed to describe accurately the last occupied and the first empty states. We took inspiration from a previous model developed for graphene moiré bilayers [START_REF] Trambly De Laissardière | Localization of dirac electrons in rotated graphene bilayers[END_REF][START_REF] Trambly De Laissardière | Numerical studies of confined states in rotated bilayers of graphene[END_REF] and extended to twisted bilayer MoS 2 [START_REF] Venkateswarlu | Electronic localization in twisted bilayer mos2 with small rotation angle[END_REF]. The basis is constituted of the p z orbitals of B and N and the Hamiltoinan relies on few parameters: the onsite energy difference ∆ = 4.90 eV between B and N species and the first-neighbour in-plane hopping t ∥ = -2.65 eV. The interlayer hopping is included only for the σ components and has a distance-dependent form t XY ⊥ (r) = γ XY e -q XY (a-r)/a with a being the interlayer distance 3.22 Å [START_REF] Latil | Structural classification of boron nitride twisted bilayers and ab initio investigation of their stacking-dependent electronic structure[END_REF], r being the distance between two sites belonging to different layers and XY labelling the pairings BN , BB and N N . The values of the γ XY and q XY parameters are reported in Table I. The quality of the model can be appreciated in Figure 2 where we report the top valence and bottom conduction states of the BB and the BN stackings at twist angle θ = 27.80 • . These two stackings have been chosen as paradigmatic because of some specific characteristic of their band structure that is discussed in more detail below, and because the corresponding bilayers belong to two different layer symmetry groups. These have been computed with our TB model (black solid lines) and the DFT free simulation package Quantum ESPRESSO [START_REF] Giannozzi | Quantum espresso: a modular and open-source software project for quantum simulations of materials[END_REF][START_REF] Giannozzi | Advanced capabilities for materials modelling with quantum espresso[END_REF] (red dashed lines).

Our TB model is parametrized to reproduce well some specific features of the band structure like the general dispersion of the electronic states and the gapwidth. Particular attention has been paid on the description of the two bands forming the bottom of the conduction. Indeed, the shape and the flatness of the bottom conduction bands are correctly reproduced by the model as well as the ordering. In fact, the DFT predicts the formation of a pretty flat dispersion in the M-K region in both systems. Actually the two bands avoid each other in the BB(1,3) electronic structure, even though the splitting is extremely small, whereas they cross at K in the BN [START_REF] Carr | Electronic-structure methods for twisted moiré layers[END_REF][START_REF] Inbar | The quantum twisting microscope[END_REF], consistenlty with what simulated at smaller angles [START_REF] Latil | Structural classification of boron nitride twisted bilayers and ab initio investigation of their stacking-dependent electronic structure[END_REF]. Our model catches very well these features although the splitting in the BB stacking is somewhat overestimated.

The imaginary part of the independent-particle transverse dielectric function ε(ω) at first order in the coupling with the vector potential A(r, t) is obtained through the formula [START_REF] Grosso | Chapter 12 -Optical Properties of Semiconductors and Insulators[END_REF]:

ε(ω) = e 2 π m 2 0 ε 0 ω 2 k,m,µ |v kµm (ê)| 2 δ(E kµ -E km -ℏω) , ( 1
)
where e is the absolute value of the electron charge, m 0 is the electron mass and ε 0 is the vacuum permittivity. This expression describes the absorption of a photon with energy ℏω and polarization vector ê through the promotion of an electron from the valence state |m, k⟩ of energy E km to the conduction state |µ, k⟩ of energy E kµ . The velocity matrix element v kµm = ⟨µ, k| ê • v |m, k⟩ is obtained from the eigenstates of the operator ℏv = i[ Ĥ, r]. We evaluated ε(ω) in the same systems as before using eigenvalues and eigenfunctions coming from our DFT calculations, by means of the Yambo [START_REF] Marini | yambo: An ab initio tool for excited state calculations[END_REF] code, and from our TB model. Results are reported in Figure 3. Both methods predict a well-detached peak at 4.5 eV, corresponding to transitions towards the bottom conduction states. As expected from the electronic structure, differences between the two stackings are essentially negligible, indicating that not only the band structure but also the corresponding wavefunctions are remarkably similar.

Now that we have validated the TB model, we can extend our investigation to twist angles closer to 30 • and systems that can hardly be addressed with DFT. In Figure 4 we report the bottom conduction states of the five stackings in the supercells [START_REF] Inbar | The quantum twisting microscope[END_REF][START_REF] Balents | Superconductivity and strong correlations moiré flat bands[END_REF], [START_REF] Du | Moiré photonics and optoelectronics[END_REF][START_REF] Xian | Multiflat Bands and Strong Correlations in Twisted Bilayer Boron Nitride: Doping-Induced Correlated Insulator and Superconductor[END_REF] and [START_REF] Xian | Multiflat Bands and Strong Correlations in Twisted Bilayer Boron Nitride: Doping-Induced Correlated Insulator and Superconductor[END_REF][START_REF] Venkateswarlu | Electronic localization in twisted bilayer mos2 with small rotation angle[END_REF] corresponding to twist angles ranging from 29.41 • to 29.96 • in the single coincidence stackings (BB, BN and NN) and from 30.59 • to 30.04 • in the double coincidence ones (BBNN and BNNB). The tendency observed already in the (1,3) supercells is here confirmed and strengthen: all stackings present basically the same band structure at fixed supercell.

More interestingly, many low-dispersing states forming a bundle of flat bands are present above the bottom conduction and concentrated in an interval ∆E centered around 4.40 eV and pretty narrow (about 100 meV). Quite interestingly, this interval is at about the same energy where the M-K flat region was encountered in the (1,3) stackings (cfr. Figure 2). Such behaviour is in contrast to small-angles twisted hBN bilayers where one or more single states are formed directly in the gap and clearly separated in energy by about 0.1 eV [START_REF] Liu | Higher-Order Band Topology in Twisted Moiré Superlattice[END_REF][START_REF] Xian | Multiflat Bands and Strong Correlations in Twisted Bilayer Boron Nitride: Doping-Induced Correlated Insulator and Superconductor[END_REF][START_REF] Zhao | Formation of bloch flat bands in polar twisted bilayers without magic angles[END_REF]. Furthermore, BN being recognized as a cardinal compound in 2D material research for its exceptional optical features, it is therefore particularly worthwhile to understand the impact of these emerging states on absorption properties.

To this aim we computed with TB the independentparticle absorption spectra of the BB [START_REF] Inbar | The quantum twisting microscope[END_REF][START_REF] Balents | Superconductivity and strong correlations moiré flat bands[END_REF], the BB(4,11), the BN [START_REF] Inbar | The quantum twisting microscope[END_REF][START_REF] Balents | Superconductivity and strong correlations moiré flat bands[END_REF] and the BN(4,11) bilayers and we report them in Figure 5 as red solid and black dashed lines. As in the case of the (1,3) stackings, the spectral onset is dominated by a very intense and well detached peak at an energy compatible with the bundle states. As a confirmation of its origin, we computed the spectra including only transitions toward the bundle states (shaded patterns). We recover the same peak at essentially the same intensity indicating that, despite being in the middle of the conduction band, the bundle states are responsible of the absorption onset, the bottom conduction contributing only at higher energies.

To conclude, we have investigated the electronic and optical properties of hBN bilayers at twist angles close to 30 • . To this aim, we have developed a simple and versatile tight-binding model which accurately describes low-energy excitations in all stacking sequences and twist angles. Its parametrisation has been validated against ab initio calculations of band structure and optical absorption. With our model, we have demonstrated that at twist angles close to 30 • all hBN bilayers develop the same electronic properties, irrespective to the stacking sequence. This common electronic structure is characterised by the emergence of a bundle of low-dispersing states right above the bottom of the conduction band which are responsible of an intense and robust peak at the onset of the absorption spectrum. Our results suggest that 30 • -twisted BN bilayers may host extremely strong excitonic phenomena independent on the stacking sequence. More generally, the tendency of developing the same electronic structure is a consequence of the fact that any hBN bilayer becomes a quasicrystal in the 30 • limit, i.e. a structure with no translation symmetry where all the possible local configurations co-exist. Since this very characteristic is actually common to the majority of homobilayers, we predict that a similar trend can be found in the 30 • limit in all most common 2D materials including graphene, transition metal dichalcogenides, buckled graphene structures (antimonene, silicene), transition metal monochalcogenides, and all homostructures formed of hexagonal single-layers.

FIG. 1 .

 1 FIG.1. The five hexagonal stackings of BN bilayers clased into the single and double sublattice coincidence groups. Their stacking sequence takes the name from the atoms perfectly stacked at high symmetry points (red circles). The rotation angles are defined as the angles spanned by the B-N bonding at the coincidence point moving from the bottom layer to the upper layer.

FIG. 2 .FIG. 3 .

 23 FIG. 2. DFT and TB bandstructure of the BB and BN stackings in the (1,3) supercell, corresponding to a twist angle of θ = 21.79 • .

FIG. 4 .

 4 FIG.[START_REF] Du | Moiré photonics and optoelectronics[END_REF]. From top to bottom: Bundle states in the conduction bands of the[START_REF] Inbar | The quantum twisting microscope[END_REF][START_REF] Balents | Superconductivity and strong correlations moiré flat bands[END_REF], the[START_REF] Du | Moiré photonics and optoelectronics[END_REF][START_REF] Xian | Multiflat Bands and Strong Correlations in Twisted Bilayer Boron Nitride: Doping-Induced Correlated Insulator and Superconductor[END_REF] and the[START_REF] Xian | Multiflat Bands and Strong Correlations in Twisted Bilayer Boron Nitride: Doping-Induced Correlated Insulator and Superconductor[END_REF][START_REF] Venkateswarlu | Electronic localization in twisted bilayer mos2 with small rotation angle[END_REF] supercells of all the five stacking sequences (in different colors).

FIG. 5 .

 5 FIG.5. Normalised independent-particle absorption spectra at the onset. BB stacking in the left panel, BN stacking in the right panel. Supercells[START_REF] Inbar | The quantum twisting microscope[END_REF][START_REF] Balents | Superconductivity and strong correlations moiré flat bands[END_REF] and[START_REF] Du | Moiré photonics and optoelectronics[END_REF][START_REF] Xian | Multiflat Bands and Strong Correlations in Twisted Bilayer Boron Nitride: Doping-Induced Correlated Insulator and Superconductor[END_REF] in black dashed and solid red respectively. Patterned areas stand for spectra obtained including allowing µ of equation (1) to run only on the bundle state index.

TABLE I .

 I Set of parameters for the interlayer hopping.

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 Γ M K Γ

(4,11) (4,11) (4,11) (4,11) 4.25 4.30 4.35 4.40 4.45 4.50 Γ M K Γ (11,30) Energy (eV) Γ M K Γ (11,30)Γ M K Γ[START_REF] Xian | Multiflat Bands and Strong Correlations in Twisted Bilayer Boron Nitride: Doping-Induced Correlated Insulator and Superconductor[END_REF][START_REF] Venkateswarlu | Electronic localization in twisted bilayer mos2 with small rotation angle[END_REF]