Theoretical study of the non linear optical properties of alkali metal (Li, Na, K) doped aluminum nitride nanocages

Abstract

The effect of alkali metal (Li, Na, and K) doping in aluminum nitride (Al₁₂N₁₂) nanocages is studied through density functional theory (DFT) methods. Six new stable compounds of M@Al₁₂N₁₁ and M@Al₁₁N₁₂ are designed theoretically where alkali metal replaces an atom (Al/N) of a nanocage. The stability of the doped nano-cages is evaluated through binding energy calculations. Doping alkali atom M (M = Li, Na, K) into a nanocage significantly reduces the band gap (HOMO–LUMO gap). Polarizability and first hyperpolarizability are calculated using long range separated methods to evaluate the non-linear optical (NLO) properties of these doped systems. The hyperpolarizability of MAl₁₂N₁₁ nanocages is much higher than that of M@Al₁₁N₁₂ nanocages. The higher hyperpolarizability of M@Al₁₂N₁₁ nanocages is believed to arise from participation of excess diffuse electrons, revealed from PDOS.

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Introduction

Over last few decades, significant efforts have been made to design novel materials exhibiting a large non-linear optical (NLO) response due to their versatile potential applications in photonics and opto-electronic devices.^1–10 In view of designing high performance materials, a number of effective strategies have been proposed that can cause dramatic improvement in the NLO properties of a system (both organic and inorganic).^11–22

For organic materials the concept of extended conjugation combined with an enhanced electron push–pull mechanism has been a successful method of enhancing NLO response. In this approach, an electron donor and an acceptor fragment are carefully chosen and connected through π conjugation in a molecule.^11,14,15 Different high performance NLO materials were designed using this strategy such as donor–acceptor-π bridge frameworks.^13,16 Another approach for increasing NLO response of a system is to design metal–ligand frameworks in which transition metal is incorporated in organic molecules.^23–25 Transitions caused by charge transfer from metal to ligand were exploited in this strategy. There are several other materials in which the same charge transfer transitions are exploited such as “octupolar materials”^20,26 and “X-type chiral π-conjugated oligomers”.²¹

Recently, introduction of diffuse excess electrons in a system (organic/inorganic) has emerged as a proficient strategy for enhancement of NLO properties.^27–33 Systems that contain diffuse excess electrons have considerably larger β₀ (hyperpolarizability) value. Alkali metal atoms are excellent candidate for delivering diffuse excess electrons. Excess electrons from alkali metals can lower the excitation energy and hence play a decisive role for increasing β₀ and enhanced NLO response.^{30,32,34–36} Different molecules such as calix[4]pyrrole,³⁰ B₄H₁₀ basket,³³ HCN trimer,²⁸ π-conjugated aromatic rings like benzene, thiophene²⁷ and most recently, AlN nanocages have been studied in this context.³⁷ Interaction of these systems was studied with alkali metals (Li, Na, K), and it was found that valence electrons of alkali can be pushed out to generate diffuse excess electrons due to which excitation energy decreases which result in remarkably enhanced β₀.

For last few years, nano structures such as fullerene hollow nanocages of elements other than carbon have gained much attraction due to their unique optical/electronic properties.^38–43 Group III–V nitrides are among most promising nanocages.^44,45 From group III nitrides, AlN nanocage has versatile applications in nanoscience due to its thermodynamic stability, low electron affinity and high thermal conductivity ^42,46. A study by Wu et al., on the stability of aluminum nitride nanocages (n = 2–41) revealed that (AlN)₁₂ nanocage is energetically the most stable one, and can be considered as an ideal nanocage. sp² hybridization of group III metal atom and nitrogen is responsible for this stability.⁴⁷ Al₁₂N₁₂ nanocages has been studied for its potential applications in hydrogen storage and in gas sensing. According to Wang et al., each aluminum atom in cage can adsorb one hydrogen molecule but when the nanocage is decorated with nickel, this adsorption capacity increases several times. Ni incorporated AlN nanocage can adsorb up to 20 molecules of hydrogen outside and inside the nanocage.⁴⁸ NO adsorption on aluminum nitride nanocage was also studied and it was revealed that the nanocage in presence of CO can selectively adsorb/detect NO molecules.⁴² Beheshtian showed that fluorine atom can be adsorbed on aluminum site of nanocage, and this fluorination can significantly alter the electronic properties of nanocage to p-type semiconductor.⁴⁹ Al₁₂N₁₂ has been studied for NLO response when doped with alkali metal. Alkali metal incorporation can be done in many ways, such as placing alkali metal atom at the surface of the nanocage, encapsulation of alkali metal inside nanocage and replacement of any atom of nanocage by alkali atom. Surface doping on Al₁₂N₁₂ nanocage enhances the first hyperpolarizability by several orders of magnitude.³⁷ Second approach is recently used by Zeinab et al., for group III nitrides where alkali metal was encapsulated in the nanocage that results in larger first hyperpolarizability of the system.⁴⁵ Nevertheless, a third approach of doping is yet to be explored where alkali atom replaces any atom of the nanocage.

In current study, we investigate the NLO properties of inorganic nano clusters by doping them with alkali metal atom. This work mainly aims to investigate the impact of alkali metals (Li, Na, K) doping on NLO properties of (AlN)₁₂ nanocage. Here, in this work, the replacement of aluminum or nitrogen atom of nanocage with alkali atom gives rise to two types of structures, M@Al₁₂N₁₁ (where nitrogen is replaced with alkali metal) and M@Al₁₁N₁₂ (where aluminum is replaced). Main objective of our work is to investigate thoroughly the influence of alkali metal doping at different sites on its optical and electronic properties. The results are quite useful for designing of new materials for their potential applications in electronics and high-performance NLO materials.

Methodology

Gaussian 09 suit of programs is used for all calculations.⁵⁰ All the geometric structures are optimized at B3LYP/6-31G(d, p) level of theory. No imaginary frequency is found for these structures which confirms them as true minima. Stability of undoped and doped systems was evaluated by binding energies and cohesive energy:

(Cohesive)

Undoped system

E_coh = E_AlN − (12E_Al + 12E_N)

N-site doped system

E_coh = E_AlNM − (12E_Al + 11E_N + 1E_M)

Al-site doped system

E_coh = E_AlNM − (11E_Al + 12E_N + 1E_M)

(Adsorption)

For M@Al₁₂N₁₁

E_bind = E_AlNM − (E_Al12N11 + E_M)

For M@Al₁₁N₁₂

E_bind = E_AlNM − (E_Al11N12 + E_M)

where, E_AlN is energy of undoped aluminum nitride nanocage, E_Al and E_N is energy of aluminum and nitrogen atom respectively, E_AlNM is energy of alkali doped aluminum nitride nanocages and E_M is energy of alkali that is doped on the system. Zero point corrected vibrational energies (ZPVE) are considered for estimation of binding energies. Natural bond orbital (NBO) calculations were also performed at B3LYP/6-31G(d, p) for charge analysis and also for finding % S and % P character of alkali–atom (Al/N) bond.

Polarizability and first hyperpolarizability calculations are also carried out on long range correlated methods such as CAM-B3LYP/6-311+g(d) and LC-BLYP/6-311+G(d) level of theory.^51–55 CAM-B3LYP/6-311+g(d) and LC-BLYP/6-311+G(d) are the best reported level of theory for the calculation of first hyperpolarizability.⁴⁵ However, a recent report shows that CAM-B3LYP slightly overestimate hyperpolarizabilities.^36,56,57 Our results (vide infra) are in accordance with statement.

First hyperpolarizability and mean polarizability is denoted as

Polarizability

α = 1/3(α_xx, α_yy, α_zz)

First hyperpolarizability

β₀ = [(β_xxx, β_xyy, β_zzz)² + (β_yyy, β_yzz, β_yxx)² + (β_zzz, β_zxx, β_zyy)²]^1/2

Moreover, for calculation of crucial excited states of all the structures TD-DFT (time dependent DFT) calculations are executed on CAM-B3LYP/6-311+g(d) level of theory. Same basis set is employed for the calculation of dipole moment of excited states.

Density of states of all the structures are plotted by using GaussSum and partial density of states (PDOS) are generated through multiwfn software.⁵⁸

Results and discussion

All alkali doped aluminum nitride nano cages (M@Al₁₂N₁₂) geometries were optimized without any symmetry constraints through DFT method. Two sets of geometries were optimized namely, M@Al₁₂N₁₁ and M@Al₁₁N₁₂ (M is alkali metal). These two series differ according to doping site in nanocage. In M@Al₁₂N₁₁ series, nitrogen is replaced with alkali metal whereas, aluminum is replaced with alkali atom in M@Al₁₁N₁₂ (Fig. 1). For the assessment of stability of both series, binding and cohesive energies were calculated and compared with pure Al₁₂N₁₂ nanocage. Pure Al₁₂N₁₂ nanocage (E_coh of 3136 kcal mol⁻¹) is more stable than metal doped Al₁₂N₁₂ nanocage. It was observed that stability of doped nanocages decreased slightly with increase in atomic number of alkali metal. In M@Al₁₂N₁₁ series, Li@Al₁₂N₁₁ has highest cohesive energy value of 2926 kcal mol⁻¹ which decreases very slightly for Na@Al₁₂N₁₁ (2914 kcal mol⁻¹) and K@Al₁₂N₁₁ (2912 kcal mol⁻¹). Similarly M@Al₁₁N₁₂ series also showed a very little difference in cohesive energy values among different doped nanocages. Li@Al₁₁N₁₂ has cohesive energy of 3020 kcal mol⁻¹ which is higher than 2998 kcal mol⁻¹ for Na@Al₁₁N₁₂ and 2995 kcal mol⁻¹ for K@Al₁₁N₁₂. M@Al₁₁N₁₂ series showed higher cohesive energy than M@Al₁₂N₁₁ series.

Fig. 1 Optimized structures of undoped and alkali doped aluminum nitride nanocages (all the bond lengths are in angstrom (Å)).

Furthermore, binding energy (E_bind) values are also coherent with cohesive energy values where, Li@Al₁₂N₁₁ has the highest adsorption energy value of −45 kcal mol⁻¹ among M@Al₁₂N₁₁. Na@Al₁₂N₁₁ has adsorption energy of −33 kcal mol⁻¹ which decreases to −31 kcal mol⁻¹ for K@Al₁₂N₁₁. Similarly, Li@Al₁₁N₁₂ has the highest value of E_bind (−89 kcal mol⁻¹) in M@Al₁₁N₁₂ series. Na@Al₁₁N₁₂ has binding energy value of −67 kcal mol⁻¹, and it decrease further for K@Al₁₁N₁₂ (−64 kcal mol⁻¹). The high binding energy of M@Al₁₁N₁₂ can be attributed to the strong binding of nitrogen with alkali metal in doped nanocages. Interaction of nitrogen with alkali atom results in strong binding interactions. While in M@Al₁₂N₁₁ series, bonds are formed between alkali metal and aluminum atom which have weak binding interactions than M@Al₁₁N₁₂.

Bond length of M–Al and M–N (M = alkali metal atom) in pure Al₁₂N₁₂ is 1.84 Å, which shows a constant increase with increase in atomic number of doped alkali metal. M–Al bond in Li@Al₁₂N₁₁ is 2.5 Å which is less than (2.85 Å) for Na@Al₁₂N₁₁ whereas, K@Al₁₂N₁₁ has highest bond length of 4.0 Å. Likewise, for M@Al₁₁N₁₂ series, M–N bond also showed significant increase in bond length for heavier alkali metals. K@Al₁₁N₁₂ showed bond length of about 2.8 Å which is greater than 2.4 Å for Na@Al₁₁N₁₂. Li@Al₁₁N₁₂ has lowest bond length of 2.07 Å. M@Al₁₂N₁₁ series displayed larger increment in bond length than M@Al₁₁N₁₂ series (Table 1).

Table 1 Obtained properties doped aluminum nitride nanocages

Properties	Pure Al12N12	MAl12N11			MAl11N12
		Li	Na	K	Li	Na	K
Symmetry	Th	C1	C1	C1	C1	C1	C1
Bond length (Å)	1.85	1.98	2.85	3.98	2	2.4	2.8
Charge q		0.517	0.561	0.761	0.744	0.496	0.788
% P		54	57	61	72	—	—
% S		45	41	37	27	—	—
Ecoh (kcal mol^−1)	3136	2926	2914	2912	3020	2998	2995
Ebind (kcal mol^−1)	—	−45	−33	−31	−89	−67	−64
Eg (eV)	3.9	1.39	1.78	1.74	2.9	2.8	2.7

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NBO analysis was carried not only to find out % S and % P character in all M–Al and M–N bonds but also to analyze charge transfer in nanocage. In all doped nanocages considered here, alkali metal was found to be more electropositive than remaining nanocage. Charge is transferred from alkali atom to the nanocage causing the later more electron rich. Hence, an assumption was made that s valence electrons were transferred from alkali atom to the nanocage forming alkali metal more electropositive. NBO analysis showed that each bond (vide infra) (connecting alkali to nanocage atom) has more % P character than % S character which is assumed to be responsible for bond lengthening of M–Al and M–N than all other bonds in nanocage. % P character in orbitals was seen to contribute more in M@Al₁₂N₁₁ series than in M@Al₁₁N₁₂ series which leads to increase in bond length in the former series due to very weak bonding (long bond) between alkali metal and nanocage. For M@Al₁₂N₁₁ series, % P character increases for aluminum atom with increase in atomic number of bonded alkali atom. K@Al₁₂N₁₁ has highest % P character of 61% for aluminum atom (Table 1). The % P and % S character could not be found for Na@Al₁₁N₁₂ and K@Al₁₁N₁₂ nanocages. This is probably due to very weak interactions of alkali metal with the nanocage.

Electronic properties of all considered doped and undoped nanocages were examined to explore the effect of doping. For this purpose, we have studied frontier molecular orbitals, band gap and density of states. Energies of HOMO and LUMO as well as band gap (E_g) were calculated for both series. Pure Al₁₂N₁₂ has a large band gap of 3.9 eV which is a barrier in the way of its applications in electronic devices. It was observed that electronic properties for doped nanocages were much different than undoped nanocages since a significant decrease was noticed in HOMO–LUMO gap for all doped Al₁₂N₁₂ nanocages (Table 1).

Band gap is lowered down to 1.39 eV for Li@Al₁₂N₁₁ which slightly increases to 1.78 eV for Na@Al₁₂N₁₁. E_g for K@Al₁₂N₁₁ is 1.74 eV which is slightly different from band gap of Na@Al₁₂N₁₁. On the other hand, M@Al₁₁N₁₂ series showed decreasing trend of band gap with increase in atomic number of alkali metal hence lowest value was found for K@Al₁₁N₁₂ (2.7 eV). Na@Al₁₁N₁₂ and Li@Al₁₁N₁₂ showed band gap of 2.8 eV and 2.9 eV, respectively. M@Al₁₂N₁ series showed much decrease in band gap than M@Al₁₁N₁₂ series (Fig. 2, Table 1). Comparison of these E_g values with original band gap value of pure AlN (3.9 eV) suggests larger first hyperpolarizability of doped nanocages.

Fig. 2 Graphical representation of band gap.

Decrease in band gap in alkali doped AlN nano cages is assumed to arise from the formation of a new high energy level above the original HOMO of pure nanocage. This new HOMO is located between old HOMO and LUMO which minimizes the gap between them. In case of pure AlN nano cage, LUMO appeared at −2.54 eV and HOMO was present at −6.46 eV. HOMO in Li@Al₁₂N₁₁ were shifted at −5.09 eV from −5.97 eV (Fig. 3). For Na@Al₁₂N₁₁ and for K@Al₁₂N₁₁ energy levels were shifted to −4.47 eV and −4.16 eV from original value of −5.77 eV and −5.53 eV, respectively. Likewise, M@Al₁₁N₁₂ series have fermi levels changed from-5.66 eV to −5.45 eV for Li@Al₁₁N₁₂. Fermi levels of Na@Al₁₁N₁₂ and K@Al₁₁N₁₂ were changed to −5.20 eV and −4.96 eV from −5.34 eV and −5.01 eV, respectively (Fig. 3).

Fig. 3 TDOS showing energy levels of doped aluminum nitride nanocage.

To better understand the electronic properties, total density of states (TDOS) were plotted for both pure and doped AlN nanocages. For M@Al₁₂N₁₁ series, TDOS indicated that doping the nanocage with alkali metal from Li to K results in formation of new levels. These new energy levels resides at higher energy than old HOMO hence, constitute the new HOMO. Diffuse excess electrons from alkali metal atom were transferred to nanocage due to which new level was generated and therefore, band gap is lowered in doped AlN nanocages. Same is the case for M@Al₁₁N₁₂ series, where TDOS clearly point to the formation of new HOMO. Old HOMO serves as HOMO-1 in doped nano cages.

In all doped nanocages, LUMO was present at almost same energy value as of original LUMO (−2.5 eV) however, in case of Li@Al₁₂N₁₁, the new LUMO also appeared at much lower energy value (−3.70 eV). Hence, Li@Al₁₂N₁₁ has lowest band gap in all doped AlN nano cages. It is obvious from the results that presence of diffuse excess electrons in a system can make a great change in its conducting properties through formation of new energy levels and as a result lowering the band gap. To confirm the involvement of alkali's valence electrons in formation of new HOMO, PDOS were plotted. In Fig. 4, PDOS of alkali atom doped structures are present which confirms the contribution of alkali metal in generating new HOMO level.

Fig. 4 PDOS of doped structures (dashed line corresponds to HOMO).

Earlier studies confirmed that the presence of diffuse excess electrons can cause large NLO response of a system.^1,5,14,37,45 Presence of these electrons can considerably change polarizability and first hyperpolarizability of a system.

In this work we computed polarizability and first hyperpolarizability of pure and alkali doped AlN nanocages. Polarizability (α) of pure Al₁₂N₁₂ was found to be 287 au, which significantly increased for all considered nano cages. In M@Al₁₂N₁₁ series, increasing trend of polarizability with increase in atomic number of alkali was seen hence, K@Al₁₂N₁₁ showed highest value of 397 au. Li@Al₁₂N₁₁ has the lowest α of 350 au while, Na@Al₁₂N₁₁ has α value of 374 au which is larger than Li@Al₁₂N₁₁ and smaller than K@Al₁₂N₁₁. On the other hand, M@Al₁₁N₁₂ series showed increasing trend of polarizability however, very slight increase in α value of pure nano cage was observed. Li@Al₁₁N₁₂ has α of 288 au which increases for Na@Al₁₁N₁₂ (294 au) and further increases to 301 au for K@Al₁₁N₁₂. Here, we noticed that M@Al₁₂N₁₁ series showed much increase in α than M@Al₁₁N₁₂ series.

Much attention is given to first hyperpolarizability (β₀) due to interesting change observed in its value for alkali doped AlN nanocages. First Hyperpolarizability of pure Al₁₂N₁₂ was negligible however, β₀ is considerably increased for doped nanocages. Two different long range corrected methods were used to calculate hyperpolarizability (CAM-B3LYP and LC-BLYP).

CAM-B3LYP

With CAM-B3LYP method, for M@Al₁₂N₁₁ series, β₀ was found to be 2.5 × 10³ au (2561.6 au) for Li@Al₁₂N₁₁ which increases to 4.8 × 10³ au (4834.6 au) for Na@Al₁₂N₁₁ and 9.1 × 10⁴ (9134.7 au) for K@Al₁₂N₁₁. A relatively different trend is observed for M@Al₁₁N₁₂ series, where highest β₀ is calculated for Na@Al₁₁N₁₂ that is 7.7 × 10² au (779.4 au). β₀ for lighter alkali Li@Al₁₁N₁₂ and heavier alkali K@Al₁₁N₁₂ are 4.2 × 10² au (428.8 au) and 7.1 × 10² au (716.2 au), respectively. M@Al₁₂N₁₁ series, have larger first hyperpolarizability than M@Al₁₁N₁₂ series.

LC-BLYP

Hyperpolarizability at LC-BLYP for M@Al₁₂N₁₁ and M@Al₁₁N₁₂ follows the same trend as for CAM-B3LYP. However, the calculated values at LC-BLYP are slightly lower than CAM-B3LYP. This is consistent with the literature that CAM-b3lyp overestimate the hyperpolarizability value.^56,57 In M@Al₁₂N₁₁ series, Li@Al₁₂N₁₁ has lowest β₀ of 2.1 × 10³ au (2100.3 au), which increases to 4.6 × 10³ au (4603.5 au) for Na@Al₁₂N₁₁. K@Al₁₂N₁₁ has largest value of 6.6 × 10³ au (6615.8 au) in the series. For M@Al₁₁N₁₂ series, Na@Al₁₁N₁₂ has largest β₀ of 4.8 × 10² au (487.6 au) whereas, Li@Al₁₁N₁₂ has lowest value of 2.6 × 10² au (263.2 au). K@Al₁₁N₁₂ was found to have β₀ of 3.9 × 10² au (394.1 au). It is assumed that in former series bonding interactions were relatively weak due to larger distance present among alkali and its bonded atom. This greater distance may create larger dipole moment hence, give rise to larger β₀ for M@Al₁₂N₁₁. It is depicted here that in M@Al₁₂N₁₁ series, ionization energy plays its role due to which monotonic behavior is seen.

Both the method showed significant increase and trends in hyperpolarizability of considered compounds however, values obtained from CAM-B3LYP method were greater than values obtained from LC-BLYP method (Fig. 5).

Fig. 5 Hyperpolarizability comparison on two different methods.

For deep understanding of this tremendous increase in first hyperpolarizability due to doping of alkali metal in AlN nano cage, “two level model” was considered. According to two level model

β₀ ≈ Δμ × f₀/ΔE³

where, Δμ is difference of dipole moment between ground state and crucial excited transition state (energy state having largest oscillator strength), f₀ is largest oscillator strength and ΔE is crucial transition energy. The expression clearly shows that β₀ is inversely proportional to third power of ΔE thus, crucial transition energy plays significant role in estimation of β₀. Time-dependent DFT (TD-DFT) calculations were performed to obtain crucial transitions of all considered systems. It was seen that valence electrons from alkali metal were expelled out as diffuse excess electrons due to which forces of interaction become very weak among valence and core electrons of the metal. These electrons then formed a new HOMO in system from where the electrons for crucial transition state were excited. ΔE values obtained for M@Al₁₂N₁₁ series showed an irregular trend with increase in atomic number of alkali thus, Na@Al₁₂N₁₁ has largest value of 5.5 eV. Li@Al₁₂N₁₁ has lowest ΔE value of 4.6 eV whereas, K@Al₁₂N₁₁ has ΔE of 5.3 eV. Similar is case with M@Al₁₁N₁₂ series where, Li@Al₁₁N₁₂ showed largest ΔE of 6.7 eV which decreased to 5.4 eV for Na@Al₁₁N₁₂, and it further dropped to 5.2 eV for K@Al₁₁N₁₂.

Monotonic dependency

Earlier work shows that heavier alkali atom have larger effect on NLO response of system.^27,33,56,59 Ionization potential of alkali decreases with increase in atomic number of alkali therefore, valence electrons have weak interactions with metal center for larger atomic number and are easy to push out.

Hyperpolarizability of a system can be decided by several factors. For example, ionization potential can affect the hyperpolarizability if diffuse excess electrons are present as the only factor subsequently a monotonic behavior is predicted. This is attributed to the fact that diffuse excess electrons are related to ionization potential and with increase in the atomic number of alkali ionization potential also increases. In our results M@Al₁₂N₁₁ have shown monotonic increase in hyperpolarizability values which is in agreement with the trend discussed above.

There are also reports in literature which supports our results for monotonic dependency of hyperpolarizability on atomic number.^56,60 PDOS are also plotted for sustenance of the results.

Another factor that may play its role in deciding hyperpolarizability is the interaction distance (distance between alkali atom and the system). Shorter distance leads to high hyperpolarizability since the interaction of alkali atom and the system is stronger whereas, longer distance reverse the trend of hyperpolarizability. Thus, if interaction distance is only factor for deciding hyperpolarizability then a decrease in hyperpolarizability is estimated with increase in atomic number of alkali. There are also reports in literature which shows this type of trend.^37,57

Nevertheless, the behavior observed for our calculated hyperpolarizability values of M@Al₁₁N₁₂ is unmonotonic, which proposes that not a single factor from the above discussed factors is playing its part in the series. Hence, it is anticipated that both the factors together are affecting this trend. An increase in hyperpolarizability of Na@Al₁₁N₁₂ over Li@Al₁₁N₁₂ may be due to ionization potential and decrease in hyperpolarizability of K@Al₁₁N₁₂ than Na@Al₁₁N₁₂ is probably due to interaction distance.

Moreover, β₀ was also calculated from above mentioned ‘two level model’ expression and the results are given in Table 2.

Table 2 Comparison of different values of doped vs. undoped structures

Properties	Pure Al12N12	MAl12N11			MAl11N12
		Li	Na	K	Li	Na	K
α (au)	284	350	374	397	282	294	301
β (CAM-B3LYP)		2.5 × 10^3	4.8 × 10^3	9.1 × 10^3	4.2 × 10^2	7.7 × 10^2	7.1 × 10^2
β (LC-BLYP)		2.1 × 10^3	4.6 × 10^3	6.6 × 10^3	2.6 × 10^2	4.8 × 10^2	3.9 × 10^2
f0	0	0.0596	0.3099	0.3784	0.0169	0.0169	0.0155
ΔE (eV)		4.6	5.5	5.3	6.7	5.4	5.2
Δμ (eV)		4.8	0.5	5.6	0.5	1.10	1.4
Δμ × f0/ΔE^3		3261	1033	15 795	31	138	171
CT		H → L + 1	H → L + 4	H → L + 6	H − 5 → L	H − 1 → L	H → L
			H → L + 3	H → L + 10	H − 3 → L	H − 1 → L + 3	H − 1 → L

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We analyzed all components of β₀ in two level model to realize which component has the highest contribution. ΔE, does not yield the trend as observed with calculated β₀. For example, Na doped nanocage (Na@Al₁₂N₁₁) showed higher ΔE than other alkali metals (Li, K). Then, f₀ (largest oscillator strength) is analyzed for M@Al₁₂N₁₁. f₀ shows monotonic behavior where value for K@Al₁₂N₁₁ is higher (0.3784) than both Na@Al₁₂N₁₁ (0.3099) and Li@Al₁₂N₁₁ (0.0596). In M@Al₁₂N₁₁, f₀ shows a trend consistent with β₀ which suggests that oscillator strength is playing a key role in deciding the trend for hyperpolarizability.

For M@Al₁₁N₁₂ series, ΔE and f₀ are also analyzed which showed unmonotonic trend of both factors. Both ΔE and f₀ decrease with increase in atomic number. ΔE for Li@Al₁₁N₁₂ (6.7 eV) is higher than Na@Al₁₁N₁₂ (5.4 eV) and K@Al₁₁N₁₂ (5.2 eV). Likewise, Li@Al₁₁N₁₂ and Na@Al₁₁N₁₂ have almost same value of f₀ 0.0169 which decreased for K@Al₁₁N₁₂ (0.0155). These results again suggests that f₀ is the key factor in deciding the trends of hyperpolarizability.

Above discussion clearly depicted that f₀ can be conclusive factors for explaining the particular trend of first hyperpolarizability for M@Al₁₂N₁₁ series. Consequently, dipole moments (Δμ) of ground state and excited state of crucial transitions were calculated (Table 2).

Conclusion

In the present study six new alkali doped aluminum nitride nanocages are designed and studied through DFT where, alkali metal replaces either aluminum or nitrogen atom (M@Al₁₂N₁₁ and M@Al₁₁N₁₂). % S and % P character for alkali metal–atom (M–Al/M–N) bond was calculated and it was found that contribution of % P is more than % S character which results in lengthening of the bond. Band gap (E_g) is narrowed significantly from 3.9 eV for pure Al₁₂N₁₂ nanocage to 1.39 eV for Li@Al₁₂N₁₁. For other nanocages band gap lie in the range of 1.74 eV to 2.9 eV. Valence electrons from alkali atom are pushed out to form new energy levels near HOMO. This new HOMO results in lowering the gap between HOMO and LUMO.

Moreover, introduction of these diffuse excess electrons in M@Al₁₂N₁₁ and M@Al₁₁N₁₂ are responsible for increasing hyperpolarizability of system. Hyperpolarizability was calculated on long range corrected methods such as CAM-B3LP and LC-BLYP. β₀ for our newly designed structures are increased several times where, K@Al₁₂N₁₁ has largest value of 9.1 × 10³ au. M@Al₁₂N₁₁ series is found to have larger hyperpolarizability values than M@Al₁₁N₁₂. Two state model is used for estimation of first hyperpolarizability of optimized structures and results indicate that oscillator strength is a key factor in deciding the hyperpolarizability of the newly designed nanocages.

Acknowledgements

Author acknowledge financial support for the research of the manuscript from Higher Education Commission (HEC) Grant number 1899, 2469, 2981. The author acknowledge full support from Higher Education Commission, COMSATS Institute of Information Technology, Abbottabad, and University of Agriculture, Faisalabad.

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