Ab. Initio Study of the Structural, Elastic, Electronic and Optical Properties of Cu3N

Department of physics, Faculty of science, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran Nano research group, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran Thin film laboratory, Faculty of science, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran Nano research centre, Department of physics, Malek Astar University of technology, Shahinshahr, Isfahan, Iran Condensed matter lab, Department of physics, faculty of science, Shahid Chamran University, Ahvaz, Iran


Introduction
Many interests were concentrated on copper nitride for several reasons; copper nitride can be used as an optical engraving medium for optoelectronic devices 1 , a barrier material in spin tunnel junction 2 , a high density optical storage media 3,4 , micrometric conductive dots and lines 5 and good candidate for hybrid inorganic solar cells 6 .
Copper nitride has several phases among which cubic anti-ReO 3 structure is more stable. In this phase Copper nitride has simple cubic structure and is crystallized in space group pm-3 m (221) with atomic positions Cu (0, 0, 0) and N (1/2, 1/2, 1/2). Unit cell of Cu3N is illustrated in Figure 1.
There are always noticeable divergences between values of theoretically calculated band gap energy and experimental ones. According to the theoretical works, Cu 3 N is semiconductor with narrow band gap in 0.23 to 0.9 eV regions 7,8 whereas, experimental results change between 0.8 to 1.9 eV 6,[9][10][11] .
In this study, we have studied structural, elastic, optical, and electronic properties of yttrium oxide compound in cubic phase using the full-potential augmented plane waves (FP-LAPW) within density functional theory (DFT) framework. Moreover, we have used new approximation for exchange and correlation terms (MBJ-GGA) for band calculation which greatly improves the band gap. Our results agree very well with experimental studies. Some experimental results were incorporated to validate theoretical results.

Computational Method
The full-potential linear augmented plane wave (FP-LAPW) method implemented in WIEN2K (2011) computer package 12 . In this method the space is divided into an interstitial region (IR) and non-overlapping (MT) spheres centered at the atomic sites. the basis set consists of plane waves, in the IR region. In the MT spheres, the basis sets is described by radial solutions of the one particle Schrodinger equation (at fixed energy) and their energy derivatives multiplied by spherical harmonics. To achieve energy eigenvalues convergence, the wave functions in the interstitial region were expanded in plane waves with a cut-off, K max = 14/R mt where R mt denotes the smallest atomic sphere radius and K max gives the magnitude of the largest K vector in the plane wave expansion. For Cu 3 N, the radius values R mt are taken to be 1.7 and 1.5 atomic units (a.u.) for Cu and N respectively. The full-potential methods the calculation is dependent on the choice of sphere radii in the full-potential methods. The valence wave functions inside the spheres are expanded up to l max = 10, while the charge density was Fourier expanded up to G max = 14 (a.u.) -1 . The exchange-correlation (XC) effect was treated by the generalized gradient approximation (GGA), GGA+U, mBJ and mBj+U. The self-consistent calculations are considered to be converged when the total energy of the system is stable within 10 -4 Ry. The integrals over the irreducible Brillouin zone (IBZ) are performed up to 35 k-points, using the Monkhorst-Pack special k-points approach 13 . To validate the results, the calculated reflectance spectra, refractive index and absorption coefficient were compared with their corresponding experimental results. Figure 2 shows the calculated total energy of Cu 3 N structure by GGA, LDA and GGA+SP approaches. To obtain the ground state properties, the total energies are fitted to Murnaghan 14 equation of state

Structural properties
where P is the pressure, V is the volume at pressure P, V 0 is the volume at ambient pressure, B 0 is the bulk modulus at ambient pressure and B' is the pressure derivative of bulk modulus B 0 . The equilibrium lattice constant, bulk modulus and pressure derivative of the bulk modulus are given in Table 1, together with some theoretical results and the available experimental data. It is clearly seen that the LDA

Electronic properties
In order to investigate electronic properties, the band structure and density of states (DOS) were calculated. Figure 3 depicts the calculated band structure and total DOS for Cu 3 N by GGA. There are three regions. First; a sharp peak under -15 eV (related to Nitrogen 2s orbitals), second; wide valance band about 8 eV, third; conduction band. From partial DOS (Figure 4), there is strong hybridization between N and Cu states mostly related to N 2p and Cu 3d orbitals. Moreover, it is understood that Cu 3 N compound is a semiconductor with indirect band gap. The maximum valance band is at R point and the minimum conduction band is at M point.
There are noticeable differences between values of theoretically calculated band gap energy and experimental ones. According to the theoretical works, Cu 3 N is semiconductor with narrow band gap in 0.23 to 0.9 eV regions 6,7 whereas, experimental results change between 0.8 to 1.9 eV [6,[9][10][11] . The calculated bandgap energy by different approaches was listed in Table 2. By applying GGA-PBE (PBE0) and MBJ-GGA approaches, the direct band gap were estimated to be 0.85 and 1.04 eV that improved in comparison to GGA and predicted in the range of experimental results.
To coincide the calculated bandgap energy with the experimental one, the Hubbard parameter U was introduced to guarantee spatial distribution of the partially filled d orbitals 15 . The U values were calculated to be 5.2eV for Cu 3 N using Wien2K package. By applying GGA+U and MBJ+U approaches, the band gap was calculated 0.95 and 1.12 eV, respectively. This confirms the latter approach was suitable to calculate band gap in semiconductors as reported by Tran et al. 16 . Using MBJ, the band gap was obtained 1.04 eV that improve our previous results and other's results. Then effective potential (U) was added to MBJ and the band gap is calculated 1.12 eV (see Table 2). In Figure 4, the band structure, the total and atomic densities of states of Cu 3 N by GGA+U and MBJ+U are depicted to analogy with other approaches.

Optical properties
The optical properties of matter can be described by the complex dielectric function ε(ω), which represents the linear response of a system to an external electromagnetic field. It can be expressed as ε(ω) = ε 1 (ω) + iε 2 (ω), where ε 1 (ω) and ε 2 (ω) are the real and imaginary parts of the dielectric function, respectively. Generally, there are two contributions to ε(ω), namely, intra-band and inter-band transitions. The contribution from intraband transitions is crucial only for metals. The interband transitions can further be split into direct and indirect transitions. The indirect inter-band transitions which involved scattering of phonon was ignored. The imaginary part ε 2 (ω) is directly related to the electronic band structure and it can be computed by summing up all possible transitions from the occupied to the unoccupied states, taking into account the appropriate transition dipole matrix elements. A full detail description of the calculation of these matrix elements is given by Ambrosch-Draxl et al. 17 . The real part ε 1 (ω) can be derived from the imaginary part using the familiar Kramers-Kronig transformation. The knowledge of both real and imaginary parts of the dielectric function allows the calculation of various optical constants, such as the spectral reflectivity, the refractive index, and the electron energy loss function using standard expressions 18 .
Absorption coefficient is given according to Tauc-Pankove relation 19 where E is the photon energy, and B is a factor depends on the transition probability and can be assumed to be constant within the optical frequency range, and the index p is related to the distribution of the density of states. The index p has discrete values like 1/2, 3/2, 2, more depending on whether the transition is direct or indirect and allowed or forbidden.  The optical energy gap was deduced from the intercept of the extrapolated liner part of the plot of (aE) 1/p versus the photon energy E with abscissa. The band energy Cu 3 N is calculated according to (aE) 1/p -E using Figure 5ain the frame of GGA, GGA+U and MBJ approaches. the optical band gap from these approaches were estimated 0.9, 1.3 and 1.5 eV, respectively.
The refractive index dispersion was calculated using GGA, GGA+U and MBJ approaches (Figure 5b). The phonon contributions are not included to the dielectric screening. ε 1 (ω) corresponds to the static optical dielectric constant. The static refractive index was 3.3, 2.9 and 2.7 for GGA, GGA+U and MBJ, respectively.
Transmittance and reflectance spectra were measured for as-deposited Cu 3 N thin films by reactive magnetron sputtering at different substrate temperature 9 . Figure 6a depicts the plot of Tauc-pankove relation versus photon energy. It is seen the theoretical and experimental bandgap energy are in good agreement. Figure 6b illustrates the refractive index dispersion of the Cu 3 N thin films. Cauchy dispersion relation 20 n(λ) = n 0 + n 1 E 2 + n 2 E 4 (3) was fitted with refractive index extracted from experimental transmittance and reflectance data and the n 0 , static refractive index, was obtained as 2.66-3.04 for Cu 3 N thin films (see the corresponding theoretical value).

Electronic charge distribution
where c A and c B is the electronegativity of the involving atoms. c Cu = 1.90 and c N = 3.04. The iconicity of the chemical Cu-N bound is around 0.63. Hence, Cu 3 N is mostly ionic. It confirms small deformation in electronic distribution between Cu-N bound.

Elastic constants and mechanical properties
The elastic constants C ij are the proportionality coefficients relating the applied strain (ε i ) to the stress (σ i ), σ i = C ij ε i . So, C ij determines the response of the crystal to external forces 22 . Ab initio calculation of the elastic constants is based on induced changes in the total energy due to the changes in the strain [23][24][25] .
The symmetry of cubic crystal lattice reduces the 21 elastic constants to three independent elastic constants namely C 11 , C 12 and C 44 [26] . The present values of elastic constants of Cu 3 N using GGA were given in Table 3. The elastic stability criteria for a cubic crystal 27 at ambient condition (P=1bar and room temperature) are simultaneously C 11 +2C 12 >0, C 44 >0 and C 11 -C 12 >0. The elastic constants were calculated using the tetrahedral and rhombohedral distortions on the cubic structure. The systems are fully relaxed after each distortion in order to reach the equilibrium state 28 . This optimization procedure was important for rhombohedral case, in which the accurate results could be obtained for C 44 . The elastic anisotropy ratio is another important physical quantity which tells about the structural stability and it is highly correlated with the possibility of inducing micro-cracks in the materials. This is defined as A=2C 44 /(C 11 -C 12 ). For a completely isotropic system, A is unity and the deviation from unity measures the degree of elastic anisotropy 28 . This factor was also calculated. The anisotropy factor values of Cu 3 N do not deviate much from the unity, hence Cu 3 N was not described by a profound anisotropy.
Voigt-Reuss 29-30 approach, the actual effective modulus for crystals could be approximated by the arithmetic mean of the two well-known bounds for mono crystals. Then the main mechanical parameters for cubic structures, i.e. shear modulus G, bulk modulus B, Young's modulus E, Poisson's ratio σ and Lame's coefficients m and λ were calculated from the elastic constants of the single crystals using the following relations: G = (C 11 -C 12 + 3C 44 )/5 (5a)   (5b) The value of the Poisson ratio σ for covalent materials is small as σ = 0.1, whereas for ionic materials a typical value of σ is 0.25 [31] . The value of σ was calculated to be about 0.28, i.e. a ionic contribution in intra-atomic bonding for Cu 3 N was confirmed (see section 3.4).
The bulk modulus B as calculated from elastic constant (116.342GPa) is in good agreement with the corresponding data in Table 1. According to the empirical Pugh formula 32 , the critical value of the ratio B/G separates the ductile and brittle behavior of materials is around 1.75; i.e. if B/G > 1.75, the material behaves in a ductile manner; otherwise the material behaves in a brittle manner. Therefore, Cu 3 N was classified as brittle material.
Using the Young's modulus E, bulk modulus B, and shear modulus G, mean velocity (v m ) and Debye temperature (θ D ) could been calculated. θ D is an important fundamental parameter closely related to many physical properties such as elastic constants, specific heat and melting temperature. θ D can be estimated from the elastic constants data, by the following classical relations 33 : where v m is the mean sound velocity, h is Plank's constant, k B is Boltzmann's constant and V a is the atomic volume. The mean sound velocity in the polycrystalline material is given by 34 : where v l and v t are the longitudinal and transverse elastic wave velocities, respectively, in an isotropic material, which can be obtained using the shear modulus G and the bulk modulus B from Navier's equation 35 : The calculated sound velocity and Debye temperature as well as the density for Cu 3 N are given in Table 4. Unfortunately, as far as the authors know, there are no data available related to these properties in the literature for this compound. Future experimental work will testify the calculated results.

The cohesive energy
First principles calculations based on the DFT is a useful tool to determine the cohesive energy of the solids. The cohesive energy is defined as the difference between the energy of an isolated atom and the energy of the same atom in the solid. In order to calculate cohesive energy, the energy of an isolated atom by considering it to be in a large unit cell containing just one atom, with FCC crystal structure irrespective of crystal structure of the corresponding solid. This does not introduce any errors in the results, due to the fact that a very large unit cell has been considered in the calculations. The size of the unit cell was chosen sufficiently large so that the energy convergence with respect to the size of the cell was less than 0.0001 Ry: a large cubic cell with dimensions of 25 a.u. was used for Cu and N. The energy of isolated Cu and N atoms are E tot (Cu)= -3309.77129537 Ry and E tot (N)= -108.91055867 Ry, respectively. The energy of the same atom in the solid as Cu 3 N is E tot (Cu 3 N)= -10039.61508342Ry.The cohesive energy of Cu 3 N is calculated as ∆

Conclusion
In summary, ab-initio calculations of structural, electronic, optical and elastic properties of Cu 3 N using full-potential linearized augmented plane-wave method. The calculations were done using different approaches for exchange-correlation term including GGA, LDA, PBE, and as well as new approximation MBJ-GGA. The addition of Hubbard potential parameter improved the predicted bandgap energy. The optimized structural parameters were in good agreement with the experimental ones. The elastic constants and derived bulk modulus, shear modulus, Young's modulus, Poisson's ratio, and sound velocities for longitudinal and shear waves were calculated.
The calculated elastic constants satisfied the stability criteria. The Debye temperature was calculated. The results showed that Cu 3 N is brittle. Moreover, the results of electronegativity concept, electronic charge density and Poisson's ratio revealed that the bonding nature between Cu-N was governed by ionic identity. The direct band gap energy (about 1.12 eV) which obtained by MBJ+U, are well in agreement with experimental result. The static refractive index was 2.7-3.3 that extracted from dispersion of refractive index. The optical band gap energy obtained using Tauc -Ponkove relation is greater than that of predicted by band structure.
The static equilibrium lattice structure, elastic constants bulk modulus and Debye temperature of the cubic anti-ReO 3 structural copper nitride (Cu 3 N) were calculated using a firstprinciples full-potential linearized augmented plane wave (FP-LAPW). For the exchange-correlation terms, GGA, GGA + U, PBE0, MBJ and MBJ + U approximations have been used. The bandgap energy of Cu 3 N has been calculated using effective potential U estimated by ab-initio study and improved with respect to the other theoretical works. The refractive index and optical gap have been calculated and compared with other studies. Moreover, mechanical stability of Cu 3 N was confirmed using the calculated mechanical and elastic constants.