Pressure Dependence of Electronic, Vibrational and Optical Properties of wurtzite-Boron Nitride

Wurtzite Boron Nitride ($w$BN) is a wide band gap BN polymorph with peculiar mechanical properties (hardness and stiffness). After its first synthesis in 1963 as a transformation of hexagonal BN ($h$BN) under high temperature and pressure conditions, a lot of progress have been made in order to stabilize wurtzite phase at atmospheric pressure. Today the crystallization of good quality samples is finally possible. This fact motivates our first principles study of the electronic, vibrational and light absorption and emission properties of $w$BN over a wide range of pressures. Our findings are important in view of the potential use of $w$BN as a dielectric for integration in BN-based technologies in optoelectronics and harsh environment applications.


I. INTRODUCTION
Thanks to the recent advances in high-pressure physics a new palette of materials is being synthesized that would previously have been impossible at ambient pressure.1 Pressure can drastically changes elastic, electronic, magnetic, structural, and chemical properties, and transform materials from simple insulators to superconductors, amorphous to crystalline solids, ionic to covalent compounds. Some of these transformations can be irreversible, as the transition between graphite and diamond, allowing new stable materials to be created at ambient pressure. Among the different materials that are created by means of pressure there are the wide-band gap insulators. 2 The term wide-bandgap material in general refers to any semiconductor with an energy bandgap much greater than conventional semiconductors such as silicon (Si) and gallium arsenide (GaAs). The interest revolving around this class of materials fits into the context of the permanent quest for smaller, faster, more reliable, and more efficient electronic components than their silicon (Si)-based counterparts and it has promising applications for future generations of high power electronics, as well as deep-UV optoelectronics, quantum electronics, and harsh-environment (high temperature and high pressure conditions) applications.
In optoelectronic applications, wide-bandgap materials are the viable choices for efficient green and blue light emitting diodes (LEDs). Gallium nitride (GaN), for example, is used to make bright blue LEDs. 3 In the shorter wavelength regime, they find applications such as UV-light germicide sources for water sanitation applications and advanced solar-blind optical communication systems. Only recently hexagonal Boron Nitride, a wide band gap material, has attracted a lot of attention because it is a very efficient UV light emitter despite the fact of being an indirect band gap material. 4 and 5 This finding has initiated a series of theoretical and experimental studies focusing on its unexpected properties. [6][7][8][9][10][11][12][13] However the wide-bandgap materials applications are relatively immature compared to current commercialized semiconductor technologies based on silicon, gallium arsenide (GaAs), and even indium phosphide (InP). Due to the tremendous commercialization opportunities for wide bandgap semiconductors in next-generation microelectronics and optoelectronics, they are currently under substantial research and development efforts and in some case much remains to be done to get to an accurate electronic and optical characterization. In this context wurtzite Boron Nitride (wBN), isostructural to other group III nitrides with optoelectronic applications such as GaN, AlN and InN stands out because of its outstanding mechanical (hardness and elastic stiffness)14 and 15 and electronic properties (high thermal conductivity and large spontaneous polarization)16 and 17. Since its first synthesis in 1963 18 obtained as transformation from hBN under high temperatures and pressures conditions a lot of efforts have been devoted to the fabrication of good quality samples. The prevailing synthesis method for a while has been the shock compression technique of hBN, which unfortunately produces only wBN powders with µm grains size. However in 2019 by using wBN powders as a starting point the pure single-phase polycrystalline bulk material has been successful synthetized 19 with a well controlled process under ultra-high pressure (200 kBar) and high temperature (1150 • C) conditions. The major technical obstacle in the hBN → wBN transformation, is represented by the stabilization of the latter at atmospheric pressure, it tends to transform back to the ambient phase, because of its low kinetic barrier. Several efforts moved in this direction testifying the possibility to stabilize wurtzite phase at atmospheric pressure, and finally wBN has been recently obtained from hBN in an upstroke cycle from ambient pressure conditions up to 200 kBar. While measuring optical properties of wBN in the IR-visible spectral energy range, it has been shown that the transition is completed at 130 kBar20 and the optical features of wBN structure are retained in a pres- sure downstroke cycle up to ambient conditions. Furthermore Chen et al. 21 successfully fabricated millimeter size wBN crystals via the hexagonal-to-wurtzite phase transformation at high pressure and high temperature, and demonstrated that the stability at atmospheric pressure is guaranteed by the presence of a 3D high density network of planar defects. This stabilization mechanism defies the common knowledge that the crystal defects in materials facilitate the occurrence of phase transformations, and sheds light on the possibility of devising advanced materials under extreme conditions. Motivated by the experimental evidence of getting good quality crystalline samples of wBN and the pressure induced phase transition from hBN to wBN, we present here the first theoretical study of the optical properties (absorption and emission) of wBN in the UV energy range over a wide range of pressures from ambient pressure to 200 kBar. The electronic and vibrational properties are also investigated as a function of pressure, and they will serve as a support in the discussion. The aim of this study is to enrich the panorama of the known physical properties of wBN with an investigation of its the light absorption and emission properties, and as a consequence to enable and boost the realization of Boron Nitride based optoelectronic technology.

II. COMPUTATIONAL METHODS
The wurtzite phase has hexagonal symmetry. The crystal structure can be seen as two interpenetrating hexagonal close-packed (hcp) sub-lattices, each of them formed by one atom type. The two individual atom types are displaced along the c-axis of amount z in a ABAB... stacking sequence. Boron atoms sit in the ( 1 /3, 2 /3, 0), ( 2 /3, 1 /3, 1 /2) crystallographic positions, while nitrogen atoms in ( 1 /3, 2 /3, z), ( 2 /3, 1 /3, 1 /2 + z) ones. The unit cell is shown in Fig. 1. We used the lattice parameters a = 2.55Å, c = 4.22Å taken from Ref. 22 as a starting point parameters to relax the structure at the different pressures. The internal parameter z is defined as the length of the BN bond along the z-axis (in the present case 1.56Å) in units of the c lattice parameter.
We sampled the pressure range 0-200 kBar at a pace of 50 kBar. The geometry relaxation and the ground state electronic properties have been calculated within Density Functional Theory (DFT) framework using the Quantum ESPRESSO package23. In the DFT calculations we used PBEsol exchange-correlation functional 24 and DOJO pseudopotentials. 25 We first relaxed the lattice parameters and the atomic positions at each pressure by sampling the Brillouin zone with a 12×12×8 k-grid and a value of 70 Ry was required for the kinetic energy cutoff of the plane wave expansion. The forces acting on the cell and on the atoms were converged to be lower than 10 −5 a.u. We used the final coordinates and lattice parameters as starting points for investigating the vibrational, electronic and optical properties of wBN, that we are going to discuss in the following. [see Supplemental Material (SM) Ref. 26 for further details.] The excited electronic properties and the optical ones have been obtained within Many Body Perturbation Theory (MBPT) framework, implemented in the Yambo code. 27 Vibrational Properties Starting from the atomic geometries obtained for each pressure chosen in the range 0-200 kBar, the vibrational properties (phonon frequencies, eigenvectors, dielectric constants and Born effective charges) of wBN as a function of pressure were calculated within Density Functional Perturbation Theory framework (DFPT) as implemented in the Quantum ESPRESSO package.23 and 28 The dynamical matrices have been sampled on a regular 9×9×6 q-grid. Then a Fourier interpolation is used to obtain the entire phonon dispersion along a q-path connecting high symmetry points of the Brillouin zone. The phonon dispersion at 0 kBar is represented in Fig. 2. The four atoms in the unit cell give rise to twelve phonon branches for a given q-point. According to the analysis of the wBN symmetry group C 6v , phonon modes can be decomposed into N ( nents because of the macroscopic electric field associated with the motion of the longitudinal-optical phonon vibrations and the consequent dependence on the q-point of the Born effective charges.29 Phonon modes are identified in Fig. 3 according to notation used in Ref. 30. We are going to examine now the evolution of phonon modes as a function of pressure. As hydrostatic pressure increases from 0 to 200 kBar, with a consequent compression of the entire crystal structure, phonon frequencies in general increase as well as expected. The general tendency is captured by the phonon density of state represented in Fig.4 (top panel), obtained by sampling the Brillouin zone on a uniform grid 30×30×20 q-points. We observe that low energy optical phonons (in the frequency range 400−800 cm −1 ) are slightly affected while the high energy optical phonons shift significantly.
This point is highlighted in Fig. 4 (bottom panel) where we report the phonon energies at the Γ point as a function of pressure (with the exception of the first three acoustic). Note that, at variance with all the other modes which have a similar slope, the two degenerate E 1 2 mode is less susceptible to changes with pressure. This is consistent with the fact that this mode is associated to the phase transition to the zincblende phase (cBN). The E 2 mode is in fact the result of folding of the cubic phase transverse acoustic phonon branch onto Γ in the wurtzite phase. In Tab. I we report the calculated phonon frequencies at Γ point, which are in agreement with previous results 2 and 31. We observe that the E 1 LO-TO splitting amounts to 217.2 cm −1 at 0 kBar and decreases up to 213.7 cm −1 at 200 kBar, accordingly to the decrease of the Born effective charges and the dielectric tensor matrix elements (see SM). This finding goes in the same direction as what happens for wurtzite phase GaN 32 and other III-V compounds 33. As pressure increases such splitting is always more pronounced than the E 1 -A 1 splitting (which amounts to 11.7 cm −1 at 0 kBar and 9.7 cm −1 at 200 kBar, (see SM). Therefore wBN be-  Electronic Properties Here we focus on the evolution of electronic properties as a function of pressure. By sampling the Brillouin zone with a 12 × 12 × 8 k-point grid we found that at 0 kBar and at DFT level the lowest energy gap occurs for a transition between Γ and K points of the Brillouin  Fig. 6 the QP corrected band structure at the extreme of the considered pressure range are shown. It is visible that along Γ-M line (where the direct transition occurs) the lowest conduction band shifts up, while at K it is slightly shifted down. The first valence bands are essentially unaffected.
As pressure increases the k-space positions (X DF T and X GW ) of the direct gap do not change and in the 0-200 kBar range the QP correction is almost constant. On the other hand indirect and direct band gaps undergo a different behavior (see Fig. 7). We observe that while the first tends to decrease the latter does the opposite. This is in contrast of what observed for direct and indirect band gaps in hBN under pressure. 2 We ascribed this behaviour to the different character of the band edges, that we deduced by projecting the Kohn-Sham states on selected atomic orbitals of B, N atoms (see SM). The states taken into consideration are 2s and 2p for each of the four atoms in the primitive cell. The electrons contributing to conduction bands are mostly from the B atoms while the electrons contributing to the valence ones come from the N atoms, this is in particular true for k-points very close Γ points. However at the K point the contribution comes from a mixture of the orbitals of two atoms, making the behaviour of the indirect gap different from the direct one.
Optical Properties Optical properties of wBN have been investigated by solving the Bethe-Salpeter equation for the two-body correlation function that takes into account the screened interaction between excited electrons and the leftover holes when electrons are excited by an external field.37 It can be shown that the Bethe-Salpeter equation can be recast in a two-particle Hamiltonian38 in the form: where nk ,f nk are the quasi-particle energies and occupations respectively, V is the Coulomb potential derived from the variation of the Hartree term and W the screened electron-hole interaction derived from the screened exchange. 37 We used a 12 × 12 × 8 k-points grid and included 3 valence and 3 conduction bands. After diagonalization of Eq. 1 we obtain the eigenvectors A λ nn k and eigenvalues E λ that are used to construct the macroscopic dielectric function M as: (2) In Fig. 8 we report the calculated imaginary part of the dielectric function. We observe that the absorption spectrum shifts to higher energies as pressure increases reflecting the behaviour of the direct band gap.
We found a lowest energy exciton which is dark, a twice degenerate excitons at 9.59 eV with a small dipole, while the first bright exciton energy determining the optical gap at 0 kBar amounts to 9.64 eV and it is twice degenerate. Its energy evolution as a function of pressure is resumed in Fig. 9, it increases linearly from 0 kBar to 200 kBar, almost following the linear behaviour of the quasi particle corrected direct band gap. As a consequence the exciton binding energy (E bind = E gap − E exc ) is almost constant with pressure, assuming a value of 0.38 eV at 0 kBar. Such behaviour can be explained by considering how the screening acts on the band gap and the electron-hole pair. While the states contributing to the bands determining the band gap are localized inside each cell, the exciton is highly delocalized well beyond the boundaries of the unit cell. Therefore the screening acting on the electron-hole interaction is less affected by the change of the system geometry.
Luminescence Since wBN is an indirect gap material, luminescence is possible only if assisted by phonon modes as it happens or hBN 5. It means that the excitonphonon coupling must be taken into account. This can be done by following two strategies: the finite differences approach 11 and 12 and by a direct calculation of the exciton-phonon dipole matrix elements 39. Here we calculated the phonon assisted luminescence following the first strategy. As discussed previously the lowest indirect gap at independent and quasi-particle levels occurs between points Γ and K. The momentum responsible for the indirect emission would be then q = K − Γ = K = ( 1 /3, 1 /3, 0).
We first diagonalized the Bethe Salpeter equation at transferred momentum q = K by including or not the quasi particle corrections. In this way we estimated a scissor correction that reproduce exactly the position of the exciton at K. In Table II we report the position of the excitons, with and without GW corrections, and the obtained energy shift. q = K Exc. Energy (@ IP) Exc. Energy (@ GW) QP correction (eV) (eV) (eV) 4.73 6.14 1.41 We built a supercell which maps this q-point at Γ (see Fig. 10). We calculated the dielectric constant that enter BSE with the same parameter of the primitive cell. Then we solved the Bethe Salpeter equation on the supercell at transferred momentum q = 0 applying the scissor op-  Table II and we found the excitons already present in the unit cell and new lower 4 times degenerate dark excitons, corresponding to the indirect excitons mapped at Γ. Being dark, the indirect excitons do not contribute to the optical response without the coupling with phonons.
On the basis of N=12 phonon modes calculated at the momentum q = K (see SM) we generated N supercells one for each phonon mode, with atoms displaced along the phonon eigenmodes times 0.1 Bohr. For each displaced supercell we diagonalized the Bethe Salpeter equation, where single particle energies have been corrected using a scissor operator to reproduce the exact exciton energy at momentum K in the primitive cell. 40 Then we calculated the second order derivatives of the excitonic dipole matrix elements ∂ 2 |T S | 2 ∂R 2 λq by finite differences, where S is the exciton index and λ, q the phonon mode ? . Luminescence then can be calculated with a generalization of the Roosbroeck-Shockley relation 41 to the excitonic case derived in Refs. 12 and 42: where n r (ω) is the refractive index, B E S , T exc is the Boltzmann occupation of the lowest excitons and Ω λq are the phonon frequencies. In Eq. 3 we used an excitonic temperature of 75 K and a lattice one of 55 K in analogy with the measurements on hBN.12 Moreover the sum over q points reduces to q=K. We calculated the phonon assisted luminescence at 0 kBar, and spatially averaged it on the three Cartesian directions. The result is showed in Fig. II. One observes two main peaks, the most intense at about 6.02 eV and another one at 6.05 eV, each of them split in a doublet. The contribution of each phonon mode at K point of the Brillouin Zone is also highlighted, allowing to identify each peak as the result of the coupling of the indirect exciton iK and a precise phonon mode. We found that the highest-energy optical modes are those that couple most closely with K excitons to give the luminescence signal. On the basis of the evolution as a function of pressure of the QP direct band gap and of the direct exciton energy (see Figs. 5-9), we deduce that the indirect exciton energy will decrease as pressure increases. Beside that the phonon frequencies at K point (see SM) tend to increase with pressure even if with a relatively different rates if optic and acoustic modes are compared. Putting all this information together we will expect a slight red shift with pressure of the luminescence spectrum accompanied by a small deviation of the two main peaks, since the originally located peak at 6.02 eV will red shift more than the other at 6.05 eV, because of the contribution coming from different phonon modes.

III. CONCLUSIONS
In this work we studied wBN for a wide pressure range. We found a large gap indirect material. We predict a direct gap close to Γ point that increases with pressure while the indirect gap, between Γ → K, decreases. Then we investigated the optical properties and from the BSE we found the first optical active excitations at 9.64 eV, while due to the large difference between direct and indirect gap the light emission mediated by phonons, is expected at about 6 eV. This work, together with new wBN synthesis techniques, opens the way for future applications of wBN in optoelectronics. 20-CE47-0009-01-NOTISPERF. Centre de Calcul Intensif d'Aix-Marseille is acknoledged for granting access to its high performance computing resources. This work was granted access to the HPC/AI resources of TGCC under the allocation 2022-AD010913493 made by GENCI. The research leading to these results has received funding from the European Union Seventh Framework Program under grant agreement no. 785219 Graphene Core2. This publication is based upon work from COST Action TU-MIEE CA17126, supported by COST (European Cooperation in Science and Technology). The authors acknowledge A. Saul and K. Boukari for the management of the computer cluster Rosa.