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Abstract (English):
Indium selenide (InSe) is one of the representatives of family of layered semiconductors A3B6 with the anisotropic physical properties finding application in the field of nonlinear optics and optoelectronics. The present paper provides the results of calculations of the major structural parameters (lattice constants, lengths of interatomic bonds, layer thickness and interlayer spacing) and the energy of interlayer coupling Eb in bulk InSe, and also the electronic spectra of the bulk crystal and isolated monolayer performed with the use of computational tools of the density functional theory (DFT). A comparative assessment of accuracy of various approximations of DFT allowing to judge their productivity during the studies of physical characteristics of the A3B6 compounds has been provided. It has been shown that the use of van der Waals functionals of the vdW-DF family gives an opportunity to increase significantly the accuracy of determination of values of the structural parameters of InSe and results in Eb from -50 to -67 meV/atom which is comparable to the energy of interlayer interaction in graphite and a number of related compounds. The modeling of structure of a separate monolayer shows a negligible deviation from the characteristics of layers in a bulk crystal. The calculated electronic spectra provide a conclusion about an essential growth of width of the forbidden energy band of indium selenide upon the transition from bulk material to a monolayer

Layered semiconductors, indium selenide, electronic structure, first-principles calculations, van der Waals density functional
INTRODUCTIONThe crystal compounds of the family A3B6 (GaSe, InSe, GaS, GaTe) have pronounced anisotropic properties and are of interest for use in the eld of nonlinear optics, electronics and optoelectronics, in particular, for the generation of monochromatic [1] and broadband terahertz radiation [2-4], the creation of sources for near-eld IR-nanoscopy systems [5, 6], photodetectors and detectors of ionizing radiation and solid-state batteries [7]. The layered structure of these materials and low density of dangling bonds on the surface allows to apply bulk crystals A3B6 as the substrates for growth of molecular and metal nanostructures and also to form heterosystems on the basis of semiconductors with various symmetries of crystal lattice [8-10].The development of nanotechnology in the last several years has provided the expansion of “classical” elds of use of group-III metal chalcogenides with an essentially new trend - obtaining quasi-two-dimensional (2D) materials with perspective properties. The steady interest for 2D systems emerged right after the publications of the rst works on graphene in 2004-2006, and it became very quickly clear that graphene is only one of the representatives of the new class of materials, “the top of the iceberg” as it is guratively noticed in [11]. At the present time the ultrathin layers of transition metal dichalcogenides (MoS2, MoTe2, WSe2, etc.), and also Bi2Se3, Bi2Te3, h-BN, Bi2SrTa2O9, etc., the simplest method of forming of which is the method of micromechanical exfoliation of single crystals, are being actively studied. The general feature of the majority of 2D systems considered in literature is the layered nature of the corresponding three-dimensional (3D) crystals. For example, the crystals of dichalcogenides of transition metals consist of the layers each of which is a set of three atomic planes in the sequence X-M-X (M=Mo, W; X=S, Se). The layers M2X3 (M=Bi, Sb; X=Se, Te) are about 1 nanometer thick and include ve atomic planes where the atoms of V and VI groups successively alternate. The group III metal chalcogenides (GaSe, InSe, GaS, GaTe) are intermediate in the complexity of structure of layers in comparison with two above-stated classes of materials.The sharp growth of practical interest to 2D materials on the basis of A3B6 systems can be judged by an increase in the number of publications for the last 3-4 years. The ultrathin layers of GaSe and GaS, quite big in their lateral sizes, were obtained for the rst time using the method of mechanical exfoliation and characterized by means of atomic force microscopy and Raman spectroscopy in [12, 13]. The measurements of photoresponsivity Si showed superiority of GaSe in this parameter clearly already in the rst works (Si = 30 mA/W for the layers deposited on mica [14], and 2.8 A/W for the samples obtained usingScience Evolution, 2017, vol. 2, no. 112the method of micromechanical exfoliation [13]) over pure graphene (Si ~ 1-6 mA/W). On the basis of the nanolayers of GaS and GaSe obtained using the method of mechanical exfoliation and deposition on the substrates Si/SiO2, the eld-effect transistors have been successfully manufactured. Higher performance has been reached in the devices on the basis of GaSe: the charge-carrier mobility of 0.6 сm2V-1s-1 and the relation of currents in the opened and closed states 105 compared to 0.1 cm2V-1s-1and 105 in case of use of GaS respectively [15].The thin plates of InSe obtained from an ingot of rhombohedral γ-polytype of InSe, which is a typical polymorphic modication of crystalline indium selenide, have been for the rst time studied in [16]. The most interesting results of this work are: 1) the conrmation of optical activity of InSe nanolayers at the room temperature in a technologically important near IR-range, 2) the observation of blanking out of a luminescence signal upon the transition to the plates of L < 6 nm thick (including about 7 monolayers of InSe) caused by a change in the band gap type, 3) a considerable blue shift of photoluminescence with the reduction of thickness of layers. The papers [17-19] testify to a high appeal of the properties of InSe suitable to create new ultrathin and exible optoelectronic devices. Thus, it has been established in [17] that photoresponsivity of InSe samples in the range of 450-785 nm takes the values 3.9-12.3 A/W depending on the choice of substrate. These values are higher than the values Si both of graphene and MoS2, and GaSe which determines the advantages of the matrix image sensors on the basis of indium selenide provided in [18]. Apparently, even the provided values Si can be considerably increased by the use of the samples with a higher quality of structure. Thus, it is stated in [19] about obtaining ultrahigh values of photoresponsivity at a level of 104A/W in the wide range (UV - near IR), for detectors on the basis of InSe-phototransistors with the response time ~ 5 ms. The characteristics of InSe-phototransistors obtained in [19] considerably surpass the parameters of the silicon photodetectors used at the present time and also the devices discussed in literature on the basis of other 2D materials which sets a task about a check and conrmation of results of this work.The theoretical studies of electronic and mechanical properties of bulk InSe were performed within the density functional theory (DFT) in [20-22] and earlier works with the use of both pseudopotential approximation and full-potential methods. As a rule, the early calculations of structural and electronic characteristics of InSe were performed with the use of classical approximations of local density or the generalized gradient. The common fault of these approximations is the inability to describe the dispersive forces playing an important role in layered crystals. At the same time, the computing approaches and new exchange and correlation functionals offered in recent years give an opportunity to increase the accuracy of calculations of electronic properties for the crystals with van der Waals bonds which stimulated this work. In the presented paper the calculations of structural, energy and electronic parameters of bulk and monolayer indium selenide allowing to estimate the modication of fundamental characteristics of this material upon the transition to ultrathin layers have been performed with use of new DFT methods.MATERIALS AND METHODSThe majority of the published theoretical studies of physical characteristics of the compounds А3В6 were performed within the local density approximation (LDA) or the generalized gradient approximation (GGA) which give quite satisfactory results when calculating the lattice parameters and binding energies of the semiconductors, for example, of A3B5 family crystallizing in the zinc-blende and wurtzite structures. However the LDA and GGA are not capable to consider the van der Waals interaction playing an important role in layered compounds and molecular crystals which provides errors of the corresponding quantum chemical calculations for indium selenide and similar materials. To obtain more exact results the recently developed methods DFT-D2 [23] and vdW-DF [24, 25] were used in the present paper.The main requirement for any scheme of calculation on the basis of DFT is the provision of asymptotic behavior of energy of interaction of remote particles under the law -1/r6 where r is the distance between the particles. The simplest approach for the achievement of this purpose is the introduction of the additional term into the total energy expression which takes into account all the absent long-range interactions: Etot = EDFT + Edisp . (1)where EDFT is the DFT energy calculated with use of the corresponding exchange and correlation functional and Edisp is the term for the dispersion interaction: ./,6,,6disp∑-=BABABArCE (2)The dispersion coefcient C6A,B depends on the sort of particles A and B. Owing to the divergence (2) with r → 0, the expression (2) in practice is employed together with a damping function ƒ(rA,B, A, B) which quickly decreases at small distances: ./),,(,6,,6disp∑-=BABABAA ,BrCBArfE (3)In the computing scheme DFT-D2 of Grimme [23] the calculation of dispersion coefcients is performed on the basis of the formula including the ionization potentials and static polarizabilities of the isolated atoms. Owing to the simplicity of realization and availability of data for all the elements up to Xe, at the present time DFT-D2 is the most widely used method for the account of dispersion effects. Note that in [26] an approach has been also provided which relies on the atomic polarizabilities and atomic coefcients C6 by means of which the corresponding pair coefcients are found for the denition of Edisp.The methods described above demand the input parameters determined in advance to calculate a dispersion interaction no matter if coefcients C6 are Science Evolution, 2017, vol. 2, no. 113found directly or from atomic polarizabilities. The approaches are of great interest which do not rely on the external input parameters but express the energy of dispersive interaction directly through electronic density. A use of the recently developed non-local correlation functionals which are written as follows is the basis for these methods: Ecnl = ∫∫drdr′n(r)φ(r,r′)n(r′), (4)where φ(r,r′) is the function that depends on the difference of coordinates |r - r′|, the electronic density n(r) and its gradient. In the work of Dion et al. [24] an equation for φ(r,r′) was offered which is suitable for the systems of any geometry and provides the required asymptotic performance O(-1/|r - r′|6). Within this approach the total exchange-correlation energy is dened as: EXC = ExGGA + EcLDA + Ecnl , (5)where the terms in the right-hand side are, respectively, the exchange energy in the revPBE approximation, the LDA correlation energy and a non-local amendment for correlation energy.The functionals constructed according to (5) are called van der Waals density functionals and are now actively applied when studying adsorption processes and the energy characteristics of molecular and crystal systems. After the emergence of the original scheme vdW-DF a number of the works devoted to its careful analysis and increase in efciency has followed. As various variants of GGA are described in literature, the choice of expression for the rst term in (5) is ambiguous. In the present paper a number of new functionals the brief description of which is given below was used to calculate the structural parameters of the compound InSe and to assess the productivity of various approximations. The optimization of structure was performed in the package Quantum ESPRESSO [27] with use of the projector-augmented wave (PAW) method to treat the electron-ion interactions which provides a high accuracy of calculations of total energy and interatomic forces. The calculations of electronic spectra were performed with use of norm-conserving In and Se pseudopotentials obtained by us within the Troullier-Martins scheme and which were earlier successfully applied when studying the properties of the ternary compound LiInSe2 [28]. The computing parameters (the plane-wave energy cutoff and the density of k-mesh for integration over the Brillouin zone) were selected in a way to provide a high level of convergence of the calculated values.RESULTS AND DISCUSSIONThe layered crystals of the family A3B6 show the variety of crystal modications caused by a possibility of various arrangement of layers relative to each other. At the same time all the layers have the same internal structure, are characterized by hexagonal symmetry, and consist of four parallel atomic planes arranged in the sequence VI-III-III-VI, i.e. two internal planes of atoms of the III group are concluded between the planes of atoms of the VI group. The strong covalent bonding in the layers VI-III-III-VI and the weak van der Waals interlayer bond with a low ion-covalent contribution results in the pronounced anisotropy of properties of InSe, GaSe, GaS - optical characteristics, electrical conductivity, heat conductivity, etc. which signicantly differ in the directions along the crystallographic axis c (perpendicular to layer planes) and along the layers.The most general crystal modications of A3B6 are ε-, β-, γ- and δ-polytypes with the spatial groups P6¯m2, P63/mmc, R3m and P63mc respectively described in literature. The polytypes ε, β, and δ have a hexagonal lattice with the number of atoms in an elementary cell from 8 (ε, β) to 16 (δ) while the γ-phase is characterized by a rhombohedral lattice (4 atoms in a cell). The monocrystals of GaSe grown up using the Bridgman method have, as a rule, the structure of ε-polytype with a small impurity of γ-phase. Stable modications of InSe are γ-and β-phases. The corresponding elementary cells are presented in Fig. 1. For the γ-polytype we provide a hexagonal cell which is more convenient for comparison and includes three layers (12 atoms).Fig. 1. Successively from left to right the elementary cells of the polytypes γ-InSe, β-InSe and the structure of a separate layer. Green spheres are indium atoms and yellow spheres are selenium atoms.

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