First-principles study of the electronic and optical properties of a new metallic MoAlB

The structural, elastic, electronic and optical properties of MoAlB were investigated by first-principles calculations. The hardness of MoAlB is 12.71 GPa, which is relatively softer and easily machinable compared to the other borides. The analysis of the band structure and density (DOS) of states indicates that MoAlB has a metallic nature. The analysis of the electron localization function (ELF) shows that the Mo-B bond is a polar covalent bond with a short distance, which may increase the stability of the compound. The calculation of the phonon frequencies confirms the dynamical stability of MoAlB. Optical properties of MoAlB are investigated. In the energy range up to ~19 eV, MoAlB possesses high reflectivity and has the strongest absorption in the energy range of 0–23.0 eV. In addition, the plasma frequency of MoAlB is 20.4 eV and MoAlB can change from a metallic to a dielectric response if the incident light has a frequency greater than 20.4 eV.

The bulk modulus B and shear modulus G are two important parameters in mechanics. The bulk modulus is a physical quantity that allows one to estimate rigidity, while the shear modulus is used to determine the ductility of a material. Using the CASTEP program 18 , all the elastic stiffness constants C ij for the MoAlB crystal are calculated. According to the Voigt-Reuss-Hill approximation 19 , the bulk modulus B and shear modulus G can be obtained by taking the average of Voigt's and Reuss's schemes, which represent upper and lower bounds:  13 23 In which S ij represents the elastic compliance constants consisting of the compliance tensor. Young's modulus E and Poisson's ratio ν can be obtained by the following equations: In addition, an empirical model 20 correlating the elastic moduli and Vickers hardness was used to calculate Vickers hardness: The obtained bulk modulus B, shear modulus G, Vickers hardness H ν , Young's modulus E and Poisson's ratio ν are listed in Table 1. The available experimental values are also included. Ade et al. 21 synthesized MoAlB crystals and obtained a hardness of 13.6 GPa. They found that the hardness of different crystal planes ranged from 11.4 to 13.6 GPa. Sankalp et al. 17 measured the hardness of the MoAlB crystal to be 10.3 GPa. In this paper, the obtained hardness is 12.71 GPa, which is closer to the experimental results 21,17 . In addition, compared to the other borides such as MoB (H v = 23 GB) 22 , MoB 2 (H v = 21-27 GPa) 23 , MoAlB is relatively softer, so it is most readily machinable and it is damage tolerant. Compared with MoB, the major difference in behavior for MoAlB is due to the mobile dislocations in the crystal structure. The bulk modulus of MoAlB is 210.99 GPa, which is smaller than that of MoB 2 (304 GPa) 23 , but much bigger than those of TaS 2 (155.74 GPa) and TaSe 2 (148.39 GPa) 24 .
In this paper, we performed the minimization of the total energy for different lattice volumes at a given energy cutoff value (850 eV). Then, fitting the data of total energy and the lattice volume to the Birch-Murnaghan equation of state 25 3 , which shows that our calculation is reliable. Figure 2 presents a plot of the energy versus lattice volume for MoAlB.
Electronic properties can be used to determine the metallic, semiconducting or insulating character of a compound. An energy gap between the valence and conduction bands can be deduced from the density of state (DOS) and band structure, which are investigated and shown in Fig. 3. The band structure calculations for the MoAlB crystal are performed along high symmetry directions of the Brillouin zone. The calculated energy band structures in Fig. 3(a) indicates clearly that the energy band curves pass through the Fermi energy, which shows that MoAlB is a metal at ambient pressure. The bands which cross E F originate primarily from Mo-d states, with some contributions from B-p states. The DOS of materials can provide more information about the electronic structure. From Fig. 3(b), the relatively low DOS at the Fermi energy also reveals the metallic character of the MoAlB  crystal. In addition, it also indicates that MoAlB should be stable within the perspective of electronic structure according to the free-electron model 26 .
The energy of MoAlB in the DOS ranges from − 37 to 5 eV, with a dominant structure having a split peak appearing at about − 36 eV. The narrow subband at about − 36 eV is completely attributed to Mo-p orbitals and is well separated from other orbitals. In order to present the DOS more clearly, in Fig In order to obtain the detailed bonding character of MoAlB, the electronic localization function (ELF) was calculated. Based on the Hartree-Fock pair probability of parallel spin electrons, ELF can be used to describe and visualize chemical bonding in molecules and solids 29 . ELF ranges from 0 to 1. ELF = 1 means the perfect localization characteristic of covalent bonds or lone pairs, while ELF = 0.5 the electron-gas like pair probability (i.e., a metallic bond), ELF = 0 corresponds to no localization (or delocalized electrons). Figure 4 shows the obtained ELF of MoAlB at ELF = 0.5 and 0.8, respectively. Clearly, when ELF is equal to 0.8, ELF at the Mo sites is negligible, while it has the local maximum values at the B sites. This shows that the bond between B and Mo atoms is partially Mo-B covalent and has an ionicity with B withdrawing charge from Mo. The polar covalent bonds and the short distance (about 2.37 Å) for Mo-B bond may increase the stability of the compound. The maximum ELFs between Mo and Al atoms are 0.5. So the bonds between Mo and Al atoms are metallic.
In order to determine the dynamical stabilities of MoAlB, the phonon dispersion curve is presented in Fig. 5. The absence of negative frequencies (imaginary frequencies) in the whole Brillouin zone confirms the dynamical stability of the Cmcm-MoAlB crystal at standard pressure.
Optical properties are some of the most important properties for a material, indicating a material's response to electromagnetic radiation and, in particular, to visible light. The frequency-dependent dielectric function is an important optical parameter, ε ω ε ω ε ω = +i ( ) ( ) ( ) 1 2 , and has a close relationship with the electronic structure. The imaginary part ε 2 (ω) of the dielectric function can be expressed as: where ω is the frequency of light, e is the electronic charge, û is the vector defining the polarization of the incident electric field, and ψ K C and ψ K V are the conduction and valence band wave functions at k, respectively. The real part ε 1 (ω) of the dielectric function can be derived from the imaginary part by the Kramers-Kronig relations. The other optical properties, such as the refractive index, absorption spectrum, loss-function, reflectivity and photoconductivity are derived from ε 1 (ω) and ε 2 (ω) 30 . In metal and metal-like systems there are intraband contributions from the conduction electrons mainly in the low-energy infrared part of the spectra. So an empirical Drude term with plasma frequency 3 eV and damping 0.05 eV is used for the dielectric function in our calculation. From the analysis of the band structure and DOS, MoAlB behaves as a metallic compound, so 0.5 eV Gaussian smearing is used in all calculations. Figure 6 presents the dielectric function, refractive index for incident photon energies up to 45 eV.
The dielectric function is the most general property of a material and can characterize how a material responds to the incident electromagnetic wave of light. Figure 6(a) presents the real and imaginary parts of the dielectric function. The peak of ε 2 (ω) is related to the electron excitation. The large negative value of ε 1 (ω) shows that the MoAlB crystal has a Drude-like behavior. In Fig. 6(a), ε 2 (ω) approaches zero from above about 18 eV, which is an additional indication of metallic conductivity. In the high energy region (ultraviolet region), ε 1 (ω) is close to unity and ε 2 (ω) reaches nearly zero, which indicates that MoAlB becomes almost transparent with very little  The refractive index is another important property of optical materials and its real part n indicates the phase velocity, while its imaginary part k is often called the extinction coefficient and indicates the amount of absorption loss when the electromagnetic wave propagates through the material. For MoAlB, the static refractive index n(0) is about 20 from Fig. 6(b). n decreases drastically and then increases to its highest peak at around 35 eV. The extinction coefficients k first increases drastically to 3.2 at about 5.5 eV and then decreases to the minimum value at about 20 eV. In the photon energy range from 4.8 eV to 20.4 eV, the extinction coefficient k is larger than the refractive index n, which means that light cannot propagate in this region.  Figure 7 presents the conductivity, absorption, energy-loss function and reflectivity for incident photon energies up to 45 eV. Optical conductivity is a good gauge of photoconductivity that could shed light on the electrical conductivity of the material 35 . Since MoAlB is metallic with no band gap, Fig. 7(a) presents the photoconductivity starting with zero photon energy. The optical conductivity σ shows a sharp increase to reach the maximum value of ~11.50 in the energy range from 4.3 to 5.0 eV in the ultraviolet region and then decreases to the minimum, then increases to reach the second peak from 37.2 to 37.9 eV. Obviously, MoAlB should be more conductive when the incident photon energy ranges from 4.3 eV to 5.0 eV.
The reflectivity is the ratio of the energy of a wave reflected from a surface to the energy of the wave incident on the surface. Figure 7(b) presented the reflectivity spectra as a function of photon energy. The reflectivity of MoAlB starts from about 0.9, increases to reach the maximum value of about 0.95 at the photon energy of about 19 eV in the ultra-violet region, then decreases drastically to reach the minimum. This indicates that MoAlB possesses high reflectivity in the energy range up to ~19 eV, and the reflectivity decreases to a very low value (high transparency) for short wavelengths.
The absorption coefficient defines the extent to which a material absorbs energy. Figure 7(c) presents the absorption spectrum of the title compound. It is noted that the absorption spectrum begins at zero photon energy  The energy loss function can be used to describe the optical spectra and the excitations produced by swift charges in a solid can be obtained from the imaginary part of the reciprocal of the complex dielectric function. Figure 7(d) shows the energy loss spectrum of MoAlB. The highest peak of the energy loss function appears at a particular incident light frequency known as the plasma frequency ω p 36 of the material. The plasma frequency of MoAlB is 20.4 eV and corresponds to the rapid decrease of reflectivity in Fig. 7(b). This shows that MoAlB will change from a metallic to a dielectric response if the incident light has a frequency greater than 20.4 eV. It is noted that the energy loss spectrum does not exhibit any distinct maxima in the range from 0 to 20 eV because of the larger ε 2 37 .

Methods
The space group for MoAlB is Cmcm space group 10 . Density functional calculation implemented in the Vienna ab-initio simulation package (VASP) code 38 were performed for the energy and electronic structure of MoAlB crystal. The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) functional 39 for exchange and correlation is employed. The adopted method is all-electron projector augmented wave (PAW) method 40 for Mo, Al, and B atoms with valence electrons of 4s 1 3d 5 , 3s 2 3p 1 and 2s 2 2p 1 , respectively.
During geometry optimization, no symmetry was used and no constraints were applied for the unit cell and the atomic positions, and a plane-wave cutoff energy of 850 eV was used. The k-points sampling in the Brillouin zone was 9 × 9 × 9 based on the Monkhorst-Pack method in order to ensure the energy convergence with energy differences of less than 1 meV per atom. The band structure and electron localization function (ELF) were also calculated. To obtain the electronic density of state (DOS), the tetrahedron method with Bloch corrections was used for the Brillouin-zone integration and 11 × 11 × 11 k-points sampling was used. The phonon structures were determined by using a supercell approach implemented in the PHONOPY code 41 .
Elastic constants were calculated by using CASTEP (Cambridge Serial Total Energy Package) program 18 . The bulk modulus B and shear modulus G were obtained from the calculated elastic constants C ij . And the Vickers hardenss (H ν ) was estimated by using the empirical mode correlating the elastic moduli and Vickers hardness.