Electronic And Nonlinear Optical Features of Inorganic Ga12N12 Nanocage Decorated With Alkali Metals (Li, Na and K)

The effect of alkali metals (Li, Na and K) interaction on the nonlinear optical response (NLO) of Ga 12 N 12 nanocage has been performed using density functional theory (DFT) calculations. The results show that the exo-M@Ga 12 N 12 structures are energetically favorable with negative interaction energies in the range of ‒ 1.50 to ‒2.28 eV. The electronic properties of decorated structures are strongly sensitive to interaction with the alkali metals. The HOMO-LUMO gap of Ga 12 N 12 is reduced by about 70% due to the decoration with alkali metals. It is obtained that the adsorption of alkali metals over the tetragonal ring of Ga 12 N 12 nanocage remarkably enhances the first hyperpolarizability up to 6.5×10 4 au. The results display that decorating Ga 12 N 12 nanocage with alkali metals can be introduced it as a novel inorganic nanomaterial with significant NLO properties. decorated structures are ‒ 1.62 (Na@fN) > ‒ 1.60 (Na@R4) > ‒ 1.50 (Na@R6) > ‒ 0.72 (Na@inside) as well as ‒ 1.691 (K@R4) > ‒ 1.679 (K@R6) > +0.281 eV (K@inside). The results show that the most stable complexes of lighter alkali metals (Li, Na) are formed by the interaction of their adsorption with the electronegative N atom of the Ga 12 N 12 nanocage.


Introduction
Since the discovery of C60 fullerene in 1985 by Kroto et al. [1], many theoretical and experimental researches have been done on fullerene carbon systems. Carbon fullerenes Cn (20 < n < 60) have been synthesized experimentally and have attracted much attention. Detailed studies of the carbon clusters are important in many applied fields, such as astrophysics, interstellar chemistry, electronics, and combustion processes [2][3][4][5]. The discovery of interesting properties and applications of carbon fullerenes led to extensive research in the design and synthesis of inorganic fullerenes Therefore, a variety of different inorganic-based fullerene-like nanocages have been reported.
The design and synthesis of novel materials with excellent nonlinear optical (NLO) properties has attracted great interest in experimental and theoretical fields over the past several decades due to their potential application in optical, electro-optical devices, optical switching and other laser devices [22][23][24][25][26][27][28][29]. Among many strategies for enhancing the NLO response of materials, introducing the diffuse excess electron, such as alkali metals, proposed an efficient approach to improve the NLO properties of different systems. The excess electron is a kind of special anion with dispersivity, which plays an important role to improve the NLO properties of different systems and the first hyperpolarizability (β0) [30,31]. In addition to their various other applications, they are potential candidates for materials with large nonlinear optical response [32][33][34][35][36][37][38][39][40][41].
In this paper, we study the decorated Ga12N12 structures with alkali metals both exohedrally and endohedrally. The effect of decorating in different sites of the nanocage on NLO properties is investigated in detail.

Computational Details
All calculations are performed using Gaussian 09 quantum chemistry code [42] with default convergence criteria; the SCF convergence criteria are set to 10 -8 Hartree on the density (SCF=Tight) as well as the convergence of geometric optimizations are adjusted to maximum force and root-mean-square (RMS) force of 4.5×10 -4 and 3.0×10 -4 Hartree.Bohr -1 , respectively, and maximum and RMS displacements of 1.8×10 -3 and 1.2×10 -3 Bohr, respectively. The geometries of all considered structures are fully optimized at B3LYP/6-31+G(d) level of theory and the nature of the stationary points are checked by frequency analysis at the same computational level. The spinunrestricted approach is applied to describe the geometry optimization, electronic structure and NLO properties of M@Ga12N12 (M=Li, Na, K); whereas the restricted approach is used for the isolated clusters. The corresponding 2 S values for spin-unrestricted approach are in the range of 0.752−0.754 for these mentioned structures, which are very close to the value 0.750 for the pure doublet state, indicating that the spin contamination is negligible and the computational results are reliable. The first static hyperpolarizability is evaluated using an analytical density functional Coulomb-attenuated hybrid exchange-correlation functional CAM-B3LYP approach and 6-31+G (d) basis set. The magnitude of the applied electric field is chosen as 0.001 au for calculation of the hyperpolarizability.
The interaction energy (Eint) between the alkali metal and nanocage is computed as: EM@nanocage is the total energy of the M@nanocage as well as Enanocage and EM are the energies of the isolated nanocage and alkali atoms, respectively. Energies of the frontier molecular orbitals The energy of a system in the weak and homogeneous electric field can be defined as [43,44]: where E 0 is the molecular total energy without the electric field and Fα is the electric field component along α direction. The, μα, ααβ and βαβγ denote dipole, polarizability, and the first hyperpolarizability, respectively. The dipole moment (μ), main polarizability (α), anisotropy of polarizability (Δα) and first hyperpolarizability (β0) are noted as: The polarizability (α) is a second rank tensor or a 3 × 3 matrix with 9 elements. The diagonal elements describe the response parallel to the applied electric field and their eigenvalues αii (i = x, y, and z) are used to calculate the mean polarizability α (isotropic of polarizability Eq. (6)). Some materials also become polarized in directions perpendicular to the applied electric field. Anisotropy of polarizability (Δα) is calculated from diagonal and off-diagonal elements of polarizability tensor according to the following equation: The first hyperpolarizability (β0) is a third rank tensor or a 3 × 3 × 3 matrix with 27 components and known as nonlinear optical response (NLO) coefficient. Time-dependent density functional theory (TD-DFT) calculations were performed at the CAM-B3LYP/6-31+G(d) method to obtain the crucial excited states, and the differences of dipole moments between the ground state and crucial excited state.

Optimized structures
The optimized geometry of inorganic Ga12N12 nanocage at B3LYP Å) [19]. The bond lengths of b64 bonds are larger than b66 ones, it indicates that more p orbital participation is responsible for such increasing of b64-bonds than b66-ones. The distance between the nitrogen atoms (lN-N) and the gallium atoms (lGa-Ga), that is, the diameters of the four-membered rings in Ga12N12, is 2.755 and 2.621 Å, indicating that the four rings are rhombic, not square. The HOMO and LUMO distribution pictures of the pristine Ga12N12 nanocage are also shown in Fig. 1.
It is clear that HOMO orbitals are on nitrogen atoms and LUMO orbitals are on gallium atoms. In other words, nitrogen atoms are electron donors due to having non-bonded electron pairs and gallium atoms are electron acceptors due to having empty orbitals.
The present computational study is performed to understand the influence of alkali metals interaction with Ga12N12 nanocage on its electronic and nonlinear optical properties. The most important atomic distance (l) and bond lengths of all structures are listed in The interaction energies (Eint) of alkali metals decorated Ga12N12 nanocage are computed as Eq. (1).
The obtained interaction energy values are listed in Table 1

Electronic properties
The obtained frontier molecular orbital energies (εL and εH), energy gap (Eg), percentage of variation of Eg (%ΔEg) and dipole moment of all studied structures are listed in Table 1 Therefore, it can be concluded that the interaction of alkali metals with Ga12N12 leads to the formation of a higher energy level as the location of the new HOMO level between the original HOMO and LUMO of the pristine Ga12N12, which is responsible for significant narrowing of energy gap. The picture of FMOs is displayed in Fig. 4. In these pictures, the presence of the LUMOs on the alkali metal is observed and it shows that the charge transfer (CT) from the alkali metal atoms to the nanocage has taken place.

Linear and nonlinear optical properties
The calculated dipole moment (μ), polarizability (α), the first hyperpolarizability (β0) and its components (βx, βy, and βz), anisotropy of polarizability (Δα) and the Bader charge of metal atom of considered structures are listed in Table 3.  Table 3 and Fig. 5 The diffuse excess electron release from alkali metals into the Ga12N12 causes a large NLO response. And this diffuse of electrons from outside the nanocage is much more effective in the position of the R4 ring and front of nitrogen atom. It is noteworthy that the interaction in these two positions causes the highest amounts of negative interaction energy (Eint) and increased the first hyperpolarizability. In other words, complexes with significant NLO response have the highest polarizability (α) and the most interaction energy among the studied complexes. Therefore, it can be confirmed that a more stable system can show remarkable first hyperpolarizability.
The ionization potential energy (IPE) of alkali metal atoms (Li, Na, and K), the interaction distance and the position of them with the Ga12N12 nanocage are important factors in the diffuse excess electron to the nanocage. Heavier alkali metals have easier electron diffusion into the nanocage and cause a large NLO response. And also, the interaction distance between the alkali atom and the nanocage, plays a very important role in the amount of electron transfer, and the shorter the interaction distance leads to a greater NLO response. These factors challenge with each other. The interaction distances of the structures considered in Table 1. Among studied structures, some of them lead to insignificant NLO response (M@inside). The results of this table indicate that the shortest interaction distance is belonged to Li metal, but the greatest values of β0 are related to Na@R4, K@R4, Na@fN and Li@fN structures. Therefore, the ionization potential, interaction distance and the position of alkali metals with Ga12N12 play an essential role in changing the NLO response of structures.

TD-DFT calculations
To investigate the decorating effect of an alkali atom and its position on β0 value of Ga12N12, the time-dependent density functional theory (TD-DFT) computations were performed to obtain the crucial excited states of the all systems and the two levels of expression can be used [45][46][47]: where ΔE, f0, and Δμ are the crucial transition energy, the largest oscillator strength, and the difference of dipole moment between the ground state and the crucial excited state (the excited state with the largest oscillator strength), respectively. Our computed results are listed in Table 4. The maximum absorption wavelength (λmax) and dominant transitions of all studied systems are also given in Table 4. Pristine Ga12N12 nanocage has an electron excitation with 3.18 eV which its wavelength appear at 390.1 nm in near-UV region as displayed in Fig. 6. The transition energies (ΔE) of decorated nanocage are in the range of 1.13 to 1.75 eV and are much smaller than the value of the Ga12N12 structure. According to eq. 5, the β0 value is inversely proportional to the third power of transition energy, so the β0 value increases with decreasing transition energy. Also, the β0 value is proportional to the values of f0 and Δμ. The data in Table 4 show that the f0 and Δμ for all decorated structures are larger than the pristine Ga12N12. The UV-Visible spectrum of all structures is plotted in Fig. 6. Therefore the obtained results show that the decoration of Ga12N12 nanocage with alkali metals could be introduced as an effective strategy to induce remarkable first hyperpolarizability and it could be considered as promising innovative nonlinear optical inorganic-based nanomaterial.

Compare with previous reports
In the study of the effect of alkali metals on the nonlinear optical properties of nitride of other elements of this group (B12N12 and Al12N12), several references can be mentioned [15][16][17]. In 2014, Niu et al. investigated the effect of alkali metals on NLO properties of inorganic Al12N12 nanocage [15], and they reported that the excess electron play a key role in enhancing the static first hyperpolarizability of Al12N12 nanocage. In 2016, Hou et al. [16] and Shakerzadeh et al. [17] separately studied the interaction of alkali metals with B12N12 nanocage on NLO properties of Mdoped structures. For comparison, significant values of β0 from the three references are listed in Table 5. Based on the data in Table 5, the interaction of alkali metals with the group (III) nitride nanocages (B12N12, Al12N12 and Ga12N12) has significantly increased the nonlinear optical response.
Depending on the type of alkali atom, nanocages and the location of the alkali atom, in some structures the β0 value has increased by about 10 4 -10 5 au. Decoration of the inorganic nanocages with alkali metal atoms has improved the NLO properties of the decorated inorganic nanocages.
This increase is due to the charge transfer and excess electron from the alkali metal atom to the nanocages.

Conclusion
The present DFT study on interaction of alkali metals with Ga12N12 nanocage shows that the electro-optical properties of Ga12N12 are remarkably sensitive to interaction with alkali metals. It is found these decorations narrow the HOMO-LUMO gaps of Ga12N12 nanocage. The results display that the interaction process are energetically favorable with negative interaction energies in the range of -1.50 to -2.28 eV. Indeed, it is found that the adsorption of alkali metals over the tetragonal ring of Ga12N12 nanocage remarkably enhances their first hyperpolarizability.
Exohedrally adsorption of alkali atoms on Ga12N12 nanocage leads to significant increases in dipole moment, polarizability and remarkable NLO response. In particular, the adsorption of Na atom on nanocage (Na@R4) with Eint = -1.62 eV leads to outstanding NLO response of 64723.4 au. causes a dramatic response to NLO. It is indicated that the ionization potential of alkali metal atoms, interaction distance and the position of them with Ga12N12 nanocage are important factors in NLO response. It seems that it can be concluded that decorated Ga12N12 nanocage with alkali metals can be promising for the design and synthesis of novel NLO nanomaterial.   Table 2. The interaction energy (Eint), HOMO and LUMO energies (ε H and ε L ) and energy gap (E g ) in eV, percentage of variation of E g (%ΔEg) and dipole moment in Debye of optimized structures Table 3. The calculated dipole moment (μ), polarizability (α), the first hyperpolarizability (β 0 ) and the Bader charge of metal atom in the considered structures Table 4. The calculated transition energy (ΔE), the difference of dipole moment (Δµ) between the ground state and the crucial excited state, the largest oscillator strength (f 0 ), maximum absorption wavelengths (λmax) and the dominated transition of all studied structures The optimized structure and HOMO-LUMO distribution of Ga12N12 nanocage The stable decorated structures of M@Ga12N12 (M=Li, Na, K)

Figure 3
The total density of states (TDOS) spectrum of all studied nanocages Figure 4 The picture of FMOs of obtained Na@Ga12N12 and K@Ga12N12 structures The obtained values for μ, α, Δα and β0 of all decorated structures in terms the position and type of alkali atom Figure 6 The UV-Visible spectrum of all studied structures The maximum wavelength (λmax) of all studied structures in terms the position and type of alkali atom