We study the electroacoustic properties of aluminum scandium nitride crystals Al$_{1-x}$Sc$_x$N with the metastable wurtzite structure by means of first-principles calculations based on density functional theory. We extract the material property data relevant for electroacoustic device design, namely the full tensors of elastic and piezoelectric constants. Atomistic models were constructed and analyzed for a variety of Sc concentrations $0\le x\le50$ %. The functional dependence of the material properties on the scandium concentration was extracted by fitting the data obtained from an averaging procedure for different disordered atomic configurations. We give an explanation of the observed elastic softening and the extraordinary increase in piezoelectric response as function of Sc content in terms of an element specific analysis of bond lengths and bond angles.
We present results of a density functional theory study of MnO in the wurtzite structure. Our motivation is provided by recent experiments reporting ferromagnetism in Mn-doped wurtzite structure ZnO. We find that wurtzite MnO a) is not strongly energetically disfavored as compared with the ground state rocksalt MnO, b) shows strong magnetostructural coupling and c) has a piezoelectric response that is larger than that of ZnO. These predictions augur well for the creation of ferromagnetic piezoelectric semiconductor based on Mn-doped ZnO.
First principles calculations based on density functional theory are performed to investigate the structural, electronic and magnetic properties of wurtzite ZnO/NiO (0001) interface. By using DFT+U method we discover that the half-metallic behavior of wurtzite NiO (w-NiO) retains in the ZnO/NiO (0001) interface. Through analyses of density of state, charge population and magnetic moments, we find the half-metallicity is weakened around the interface but interface effect is quite localized. More over the interface system keeps a ferromagnetic ground state as bulk w-NiO does. Based on the simula…
Gallium nitride (GaN) is a key semiconductor for solid-state lighting, but its radiative processes are not fully understood. Here we show a first-principles approach to accurately compute the radiative lifetimes in bulk uniaxial crystals, focusing on wurtzite GaN. Our computed radiative lifetimes are in very good agreement with experiment up to 100 K. We show that taking into account excitons (through the Bethe-Salpeter equation) and spin-orbit coupling to include the exciton fine structure is essential for computing accurate radiative lifetimes. A model for exciton dissociation into free carr…
Density-functional calculations are used to study various plausible structures of the wurtzite InN $(0001)$ and $(000{\bar 1})$ surfaces. These structures include the unreconstructed surfaces, surfaces with monolayers of In or N, several possible coverages and locations of In or N adatoms and vacancies. The stable structure of the $(0001)$ surface under N-rich conditions is the unreconstructed, In-terminated, surface, while under In-rich conditions the stable surface has a 3/4 monolayer of In atoms. The stable structure of the InN $(000{\bar 1})$ surface corresponds to a full monolayer of In a…
Employing first-principles calculations we study the structural, electronic and magnetic properties of the [Sc$_{1-x}$V$_x$]C and [Sc$_{1-x}$V$_x$]Si alloys. In their equilibrium rocksalt structure all alloys are non-magnetic. The zincblende and wurtzite structures are degenerated with respect to the total energy. For all concentrations the alloys in these lattice structures are half-metallic with the gap located in the spin-down band. The total spin moment follows the Slater-Pauling behavior varying linearly between the -1 $μ_B$ of the perfect ScC and ScSi alloys and the +1 $μ_B$ of the perfe…
Magnesium Selenide (MgSe) is a wide bandgap semiconductor with applications in optoelectronics and energy conversion technologies. Understanding thermal conductivity (k) of MgSe is critical for optimum design of thermal transport in these applications. In this work, we report the temperature and length dependence lattice thermal conductivity of magnesium selenide (MgSe) with different crystallographic phases; zincblende, rocksalt, wurtzite and nickel arsenic, using first principles computations. Computations reveal significant differences in thermal conductivity (k) of MgSe for different phase…
The spontaneous and piezoelectric polarizations of III-nitrides considerably affect the operation of various III-nitride-based devices. We report an ab initio study of the spontaneous polarization (SP) and piezoelectric (PZ) constants of the III-nitride binary and ternary alloys with the hexagonal reference structure. These calculated polarization properties offer us a profound principle for polarization engineering of nitride semiconductor devices, based on which we propose a few heterojunctions which have nearly-zero polarization effect at the junctions that can potentially enhance optical a…
First-principles calculations of $e$-ph interactions are becoming a pillar of electronic structure theory. However, the current approach is incomplete. The piezoelectric (PE) $e$-ph interaction, a long-range scattering mechanism due to acoustic phonons in non-centrosymmetric polar materials, is not accurately described at present. Current calculations include short-range $e$-ph interactions (obtained by interpolation) and the dipole-like Fröhlich long-range coupling in polar materials, but lack important quadrupole effects for acoustic modes and PE materials. Here we derive and compute the lon…
The thermodynamics properties of the wurtzite and zinc-blende \InGaN alloys are calculated using first-principles density-functional calculations. Special quasi-random structures are used to describe the disordered alloys, for $x= 1/4, 1/2$, and 3/4. The effect of lattice vibrations on the phase diagram, commonly omitted from semiconductor alloy phase diagram calculations, are included through first-principles calculations of phonon spectra. Inclusion of lattice vibrations leads to a large reduction in the order-disorder critical temperature ($\sim 29$% and $\sim 26$% for the wurtzite and zinc…
We employ first-principles techniques tailored to properly describe semiconductors (modified Becke-Johnson potential added to the exchange-correlation functional), to obtain the electronic band structures of both the zinc-blende and wurtzite phases of GaAs, GaSb, InAs, and InSb. We extract the spin-orbit fields for the relevant valence and conduction bands at zone center, by fitting the spin-splittings resulting from the lack of space inversion symmetry of these bulk crystal structures, to known functional forms---third-order polynomials. We also determine the orientations of the spin-orbit ve…
A first-principles study has been performed to evaluate the electronic and optical properties of wurtzite Zn1-xCdxO up to x=0.25. We have employed the Perdew-Burke-Ernzerhof (PBE) form of generalized gradient approximation within the framework of density functional theory (DFT). Calculations have been carried out in different configurations. With the increasing Cd concentrations, the band gap of Zn1-xCdxO is decreased due to the increase of s states in conduction band. The results of imaginary part of dielectric function indicate that the optical transition between O 2p states in the highest v…
In this paper some specific issues related to point defects in GaAs nanowires are addressed with the help of density functional theory calculations. These issues mainly arise from the growth of nanowires under conditions different from those used for thin films or bulk GaAs, such as the co-existence of zincblende and wurtzite polytypes, the use of gold particles as catalyst, and the arsenic-limited growth regime. Hence, we carry out density-functional calculations for As vacancies, Ga_As antisites, and Au impurities in ZB and WZ GaAs crystals. Our results show that As vacancies can diffuse wit…
Systematic comparison of approximations and functionals in first-principle calculations of aluminum-based III-V ferroelectric nitrides
University of Arkansas（阿肯色大学）；Guangdong Technion – Israel Institute of Technology（广东以色列理工学院）；Colorado School of Mines（科罗拉多矿业学院）；National Laboratory of the Rockies（落基山国家实验室）；Tel Aviv University（特拉维夫大学）；Université Paris-Saclay（巴黎萨克雷大学）；CentraleSupélec（中央苏佩雷勒高等电信学院）
First-principles full-potential linearized augmented plane wave (FLAPW) calculations are performed to clarify the role of the interface geometry on piezoelectric fields and on potential line-ups at the [0001]-wurtzite and [111]-zincblende GaN/Al junctions. The electric fields (polarity and magnitude) are found to be strongly affected by atomic relaxations in the interface region. A procedure is tested to evaluate the Schottky barrier in the presence of electric fields and used to show that their effect is quite small (a few tenths of an eV). These calculations assess the rectifying behaviour o…
An ab initio pseudopotential method based on density functional theory, generalized gradient corrections to exchange and correlation, and projector-augmented waves is used to investigate structural, energetical, electronic and optical properties of MgO, ZnO, and CdO in rocksalt, cesium chloride, zinc blende, and wurtzite structure. In the case of MgO we also examine the nickel arsenide structure and a graphitic phase. The stability of the ground-state phases rocksalt (MgO, CdO) and wurtzite (ZnO) against hydrostatic pressure and biaxial strain is studied. We also present the band structures of…
Interaction between point defects and vertical inversion domain walls in wurtzite AlN
Université de Toulouse, CNRS, CEMES（图卢兹大学，法国国家科学研究中心，材料科学与化学研究所）；Smart Ferroic Materials Center, Physics Department and Institute for Nanoscience and Engineering, University of Arkansas（阿肯色大学智能铁电材料中心、物理系与纳米科学与工程研究所）；Université Paris-Saclay, CentraleSupélec, CNRS, Laboratoire SPMS（巴黎萨克雷大学，中央理工-高等电力学院，法国国家科学研究中心，SPMS实验室）；Department of Materials Science and Engineering, Tel Aviv University（特拉维夫大学材料科学与工程系）；CEMES-CNRS, Université de Toulouse（材料科学与化学研究所-法国国家科学研究中心，图卢兹大学）
In this work we study the structural stability and electronic properties of the Beryllium sulphide nanowires (NWs) in both zinc blende (ZB) and wurtzite (WZ) phases with triangle and hexagonal cross section, using first principle calculations within plane-wave pseudopotential method. A phenomenological model is used to explain the role of dangling bonds in the stability of the NWs. In contrast to the bulk phase, ZB-NWs with diameter less than 133.3 (angstrom) are found to be less favorable over WZ-NWs, in which the surface dangling bonds (DBs) on the NW facets play an important role to stabili…
First-principles calculations are made for the primary pyroelectric coefficients of wurtzite GaN and ZnO. The pyroelectricity is attributed to the quasiharmonic thermal shifts of internal strains (internal displacements of cations and anions carrying their Born effective charges). The primary (zero-external-strain) pyroelectricity dominates at low temperatures, while the secondary pyroelectricity (the correction from external thermal strains) becomes comparable with the primary pyroelectricity at high temperatures. Contributions from the acoustic and the optical phonon modes to the primary pyr…
We report ab initio investigations of hexagon-shaped, [111]/[0001] oriented III-V semiconductor nanowires with varying crystal structure, surface passivation, surface orientation, and diameter. Their stability is dominated by the free surface energies of the corresponding facets which differ only weakly from those of free surfaces. We observe a phase transition between local zinc-blende and wurtzite geometry versus preparation conditions of the surfaces, which is accompanied by a change in the facet orientation. The influence of the actual III-V compound remains small. The atomic relaxation of…
In this Letter we present first principle calculation of surface energies of rock-salt (B1), zinc-blende (B3), and wurtzite (B4) AlN allotropes. Out of several low-index facets, the highest energies are obtained for mono-atomic surfaces (i.e. only by either Al or N atoms): $γ_{\{111\}}^{\rm B1}=410\uu{meV/Å}^2$, $γ_{\{100\}}^{\rm B3}=346\uu{meV/Å}^2$, $γ_{\{111\}}^{\rm B3}=360\uu{meV/Å}^2$, and $γ_{\{0001\}}^{\rm B4}=365\uu{meV/Å}^2$. The difference between Al- and N-terminated surfaces in these cases is less then $20\uu{meV/Å}^2$. The stoichiometric facets have energies lower by $100\uu{meV/Å…
Using the linear response theory, the vibrational and dielectric properties are calculated for c-BN, w-BN and h-BN. Calculations of the zone-center optical-mode frequencies (including LO-TO splittings) are reported. All optic modes are identified and agreement of theory with experiment is excellent. The static dielectric tensor is decomposed into contributions arising from individual infrared-active phonon modes. It is found that all of the structures have a smaller lattice dielectric constant than that of electronic contribution. Finally, the infrared reflectance spectrums are presented. Our …
Nitride semiconductors are ubiquitous in optoelectronic devices such as LEDs and Blu-Ray optical disks. A major limitation for further adoption of GaN in power electronics is its low hole mobility. In order to address this challenge, here we investigate the phonon-limited mobility of wurtzite GaN using the ab initio Boltzmann transport formalism, including all electron-phonon scattering processes, spin-orbit coupling, and many-body quasiparticle band structures. We demonstrate that the mobility is dominated by acoustic deformation-potential scattering, and we predict that the hole mobility can…
We report a theoretical study of the structural, electronic and optical properties of hBN-AlN superlattice heterostructures (SL) using a first-principles approach based on standard and hybrid Density Functional Theory. We consider short-period ($L<10$ nm) SL and find that their properties depend strongly on the AlN layer thickness $L_{AlN}$. For $L_{AlN}\lesssim1$ nm, AlN stabilizes into the hexagonal phase and SL display insulating behavior with type II interface band alignment and optical gaps as small as $5.2$ eV. The wurtzite phase forms for thicker AlN layers. In these cases built-in elec…
In this work, we report thermal conductivity(k) of magnesium telluride (MgTe) with various crystallographic phases such as rocksalt, zincblende, wurtzite and nickel arsenic (NiAs) using density functional theory and Boltzmann transport equation. Our first principles calculations results show the low thermal conductivity of MgTe with kNiAs < krocksalt < kwurtzite < kzincblende. We systematically investigated the phonon group velocity, phonon scattering rate and mode contributed thermal conductivity of transverse acoustic (TA), longitudinal acoustic (LA) and optical phonons. Our first principles…
High spin states of cation vacancies in GaP, GaN, AlN, BN, ZnO and BeO were analyzed by first principles calculations. The spin-polarized vacancy-induced level is located in the band gap in GaP, ZnO and BeO. In the nitrides, the stronger exchange coupling forces the vacancy states to be resonant with valence bands, forbids formation of positively charged vacancies in GaN and BN, and allows Al vacancy in p-AlN to assume the highest possible S=2 spin state. The shape of the spin density, isotropic in the zinc blende structure, has a pronounced directional character in the wurtzite structure. Sta…
We systematically study the pristine and doped chromium arsenide (CrAs) in six different crystal structures to investigate the structural, magnetic, and optical properties for real applications by first-principles calculations. First, we found that the ground-state structure is an orthorhombic MnP-type structure with antiferromagnetic spin order. The rocksalt structure is an low-energy metastable phase and a ferromagnetic metal with high spin polarization at the Fermi level. Secondly, the NiAs structure and MnP structure have a higher absorption coefficient than other structures in the infrare…
Surface doping of ZnO allows for tailoring the surface chemistry of the material while preserving the superb electronic structure of the bulk. Apart from obvious changes in adsorption energies and activation energies for catalysis, surface doping can alter the workfunction of the material and allow it to be tuned for specific photocatalytic and optoelectronic applications. We present first-principles electronic structure calculations for surface doping of Mn on the ZnO (0001) surface. Various dopant concentrations have been considered at the out-most (surface) layer of Zn atoms, while the inte…
A number of diverse bulk properties of the zincblende and wurtzite III-V nitrides AlN, GaN, and InN, are predicted from first principles within density functional theory using the plane-wave ultrasoft pseudopotential method, within both the LDA (local density) and GGA (generalized gradient) approximations to the exchange-correlation functional. Besides structure and cohesion, we study formation enthalpies (a key ingredient in predicting defect solubilities and surface stability), spontaneous polarizations and piezoelectric constants (central parameters for nanostructure modeling), and elastic …
Stable partial dislocation complexes in GaN as charge carrier lifetime modifiers for terahertz device applications by molecular dynamics and first-principle simulations
V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine（乌克兰国家科学院V.拉什卡廖夫半导体物理研究所）；Educational Scientific Institute of High Technologies, Taras Shevchenko National University of Kyiv（基辅塔拉斯舍甫琴科国立大学高等技术教育科学研究所）；National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”（乌克兰国立伊戈尔·西科尔斯基基辅理工学院）；University of Rome “Tor Vergata”（罗马托维尔加塔大学）
The structural phase transitions and the fundamental band gaps of Mg(x)Zn(1-x)O alloys are investigated by detailed first-principles calculations in the entire range of Mg concentrations x, applying a multiple-scattering theoretical approach (Korringa-Kohn-Rostoker method). Disordered alloys are treated within the coherent potential approximation (CPA). The calculations for various crystal phases have given rise to a phase diagram in good agreement with experiments and other theoretical approaches. The phase transition from the wurtzite to the rock-salt structure is predicted at the Mg concent…
We report Auger recombination rates for wurtzite InGaN calculated from first principles density-functional and many-body-perturbation theory. Two different mechanisms are examined -- inter- and intra-band recombination -- that affect different parts of the emission spectrum. In the blue to green spectral region and at room temperature the Auger coefficient can be as large as 2x10^-30cm^6s^-1; in the infrared even larger. Since Auger recombination scales with the cubic power of the free-carrier concentration it becomes an important non-radiative loss mechanism at high current densities. Our res…
The challenge of finding an absolute reference energy from first-principles simulations to realigning semiconductor's valence band-top and conduction band-bottom, a theoretical methodology is proposed based on plane-wave calculations as implemented within state-of-art density functional theory. We have studied some of inorganic binary semiconductors, including both oxides and non-oxides, as for example rutile- and anatase TiO2, wurtzite ZnO, rutile SnO2, blende phase of GaP, GaAs, InP, ZnTe, CdS, CdSe, and SiC, those are well known and qualitatively important for photoelectrochemical, optoelec…
The non-equilibrium boron nitride (BN) phase of zinc oxide (ZnO) has been reported for thin films and nanostructures, however, its properties are not well understood due to a persistent controversy that prevents reconciling experimental and first-principles results for its atomic coordinates. We use first-principles theoretical spectroscopy to accurately compute electronic and optical properties, including single-quasiparticle and excitonic effects: Band structures and densities of states are computed using density functional theory, hybrid functionals, and the $GW$ approximation. Accurate opt…
First-principles density-functional calculations have been performed for zinc monochalcogenides with zinc-blende- and wurtzite-type structures. It is shown that the local-density approximation underestimates the band gap, misplaces the energy levels of the Zn-3d states, and overestimates the crystal-field splitting energy. Without spinorbit coupling, the order of the states at the top of VB is found to be normal for all the ZnX phases considered. Upon inclusion of the spinorbit coupling in calculations, ZnO in zinc-blende- and wurtzite-type phases become anomalous. It is shown that the Zn-3d e…
In the framework of density functional theory (DFT), the structural and magnetic properties of FeN mono nitride have been investigated using the all electrons augmented spherical wave method (ASW) with a generalized gradient GGA functional for treating the effects of exchange and correlation. Calculation of the energy versus volume in hypothetic rocksalt (RS), zinc-blende (ZB) and wurtzite (W) types structures shows that the RS-type structure is more stable than the others. Spin polarized calculation results at equilibrium volume indicate that the ground state of RS-FeN is ferromagnetic with a…
We determine atomic and electronic structure, formation energy, stability and magnetic properties of native point defects, such as Gallium (Ga) and Nitrogen (N) vacancies in bulk and at non-polar (10$\overline{1}$0) surface of wurtzite Gallium Nitride (\textit w-GaN) using, first-principles calculations based on Density Functional Theory (DFT). Under both Ga-rich and N-rich conditions, formation energy of N-vacancies is significantly lower than that of Ga-vacancies in bulk and at (10$\overline{1}$0) surface. Experimental evidence of the presence of N-vacancies was noted from electron energy lo…
A density functional theory study of substitutional carbon impurities in ZnO has been performed, using both the generalized gradient approximation (GGA) and a hybrid functional (HSE06) as exchange-correlation functional. It is found that the non-spinpolarized C$_\mathrm{Zn}$ impurity is under almost all conditions thermodynamically more stable than the C$_\mathrm{O}$ impurity which has a magnetic moment of $2μ_{\mathrm{B}}$, with the exception of very O-poor and C-rich conditions. This explains the experimental difficulties in sample preparation in order to realize $d^{0}$-ferromagnetism in C-…
We performed a first principles investigation on the structural, electronic, and optical properties of crystals made of chemically functionalized adamantane molecules. Several molecular building blocks, formed by boron and nitrogen substitutional functionalizations, were considered to build zincblende and wurtzite crystals, and the resulting structures presented large bulk moduli and cohesive energies, wide and direct bandgaps, and low dielectric constants (low-$κ$ materials). Those properties provide stability for such structures up to room temperature, superior to those of typical molecular …
We present a first-principles study of the atomic, electronic, and magnetic properties of two-dimensional (2D), single and bilayer ZnO in honeycomb structure and its armchair and zigzag nanoribbons. In order to reveal the dimensionality effects, our study includes also bulk ZnO in wurtzite, zincblende, and hexagonal structures. The stability of 2D ZnO, its nanoribbons and flakes are analyzed by phonon frequency, as well as by finite temperature ab initio molecular-dynamics calculations. 2D ZnO in honeycomb structure and its armchair nanoribbons are nonmagnetic semiconductors but acquire net ma…
We report first principles density functional calculations of the spontaneous polarizations, piezoelectric constants and elastic constants for the II-VI wurtzite structure ZnO, MgO and CdO. Using our self interaction corrected implementation of density functional theory we obtain polarization values of we obtain polarization values of -0.05, -0.17 and -0.10 C/m$^2$, and piezoelectric constants, $e_{33}$ ($e_{31}$) of 1.34 (-0.57), 2.26 (-0.38) and 1.67 (-0.48) C/m$^2$ for structurally relaxed ZnO, MgO and CdO respectively. These properties are consistently larger in magnitude than the correspo…
Understanding the luminescence of GaN doped with erbium (Er) requires a detailed knowledge of the interaction between the rare-earth dopant and the nitride host, including intrinsic defects and other impurities that may be present in the host material. We address this problem through a first-principles hybrid density functional study of the structure, energetics, and transition levels of the Er impurity and its complexes with N and Ga vacancies, substitutional C and O impurities, and H interstitials in wurtzite GaN. We find that, in the interior of the material, Er$_{\rm Ga}$ is the dominant E…
In this paper, we study the elastic properties of the entropy-stabilized oxide (Mg, Co, Ni, Cu, Zn)O, using experimental and first principles techniques. Our measurements of the indentation modulus on grains with a wide range of crystallographic orientations of the entropy-stabilized oxide revealed a high degree of elastic isotropy at ambient conditions. First principles calculations predict mild elastic anisotropy for the paramagnetic structure, which decreases when the system is considered to be non-magnetic. When the antiferromagnetic state of CoO, CuO and NiO is accounted for in the calcul…
In this paper we present a detailed analysis of the atomic and electronic structure of GaAs nanowires using first-principles pseudopotential calculations. We consider six different types of nanowires with different diameters all grown along [111] direction and reveal interesting trends between cohesive energy and nanowire type with varying diameters. Generally, the average cohesive energy of nanowires with wurtzite stacking is higher than those with zinc blende stacking for small diameters. We found that most of the bare nanowires considered here are semiconducting and continue to be semicondu…
Relying on generalized-gradient and hybrid first-principles simulations, this work provides a complete characterization of the electronic properties of ZnO ultra-thin films, cut along the Body-Centered-Tetragonal(010), Cubane(100), h-BN(0001), Zinc-Blende(110), Wurtzite(10$\bar{1}$0) and (0001) orientations. The characteristics of the local densities of states are analyzed in terms of the reduction of the Madelung potential on under-coordinated atoms and surface states/resonances appearing at the top of the VB and bottom of the CB. The gap width in the films is found to be larger than in the c…
The electronic structure and chemical bonding of wurtzite-GaN investigated by N 1s soft x-ray absorption spectroscopy and N K, Ga M1, and Ga M2,3 emission spectroscopy is compared to that of pure Ga. The measurements are interpreted by calculated spectra using first-principles density-functional theory (DFT) including dipole transition matrix elements and additional on-site Coulomb interaction (WC-GGA+U). The Ga 4p - N 2p and Ga 4s - N 2p hybridization and chemical bond regions are identified at the top of the valence band between -1.0 and -2.0 and further down between -5.5 and -6.5 eV, respec…
Based on the first-principles calculations, we reveal that TlN, a simple binary compound with Wurtzite structure, is a three-dimensional (3D) topological insulator (TI) with effectively negative spin-orbit coupling $λ_{eff} < 0$, which makes it distinguished from other TIs by showing opposite spin-momentum locking effect in its surface states. The sign of $λ_{eff}$ depends on the hybridization between N-$2p$ and Tl-$5d$ states, and can be tuned from negative to postive by lattice strain or chemical substitution, which drive the system into a Dirac semimetal with 3D Dirac cones in its bulk stat…
Recent studies suggest that piezoelectric polarization can play an important role in determining the electronic and optical properties of nanoscale nitride heterostructures. Among a few models available, recent first-principles calculations performed by Prodhomme et al. provide a simple yet accurate description of linear and nonlinear piezoelectric coefficients in reduced dimensionality structures having wurtzite crystal symmetry. In this paper, first, within a fully atomistic VFF-sp3s* tight-binding framework, we employ the model proposed by Prodhomme et al. to evaluate the importance of nonl…
