Nonlinear optical ASnX (A = Na, H; X = N, P) nanosheets with divalent tin lone electron pair effect by first-principles design†

Shengzi Zhang ^ab, Lei Kang *^a and Zheshuai Lin *^ac
aTechnical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190, China. E-mail: kanglei@mail.ipc.ac.cn; zslin@mail.ipc.ac.cn
^bUniversity of Chinese Academy of Sciences, Beijing 100049, China
^cCenter of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China

First published on 3rd July 2020

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1. Introduction

Two-dimensional (2D) materials have received more and more attention due to the important optoelectronic capability in the micro-nano devices.^1,2 As the development of optoelectronic 2D materials, the typical nonlinear optical (NLO) properties have become an emerging hotspot with extensive experimental and theoretical research.^3,4 For example, the second-harmonic generation (SHG) effect of MoS₂ monolayer (ML) was reported in 2013, exhibiting comparable SHG signal to GaAs material.⁵ Further, its SHG effect can be significantly enhanced by the edge states.⁶ Subsequently, the NLO properties of several series of 2D materials such as metal chalcogenides and black phosphorus were reported, with bandgaps spanning from 0.3 to 2.5 eV.⁷ In addition, 2D NLO materials provide a research platform for condensed matter physics. In 2015, the SHG process of MoS₂ ML was considered to rely significantly on the selection of energy band valley, so SHG can be applied in valleytronics for information coding and operation.⁸ Moreover, the SHG response in 2D materials is extremely sensitive to the lattice orientation, defect states, and edge states, and is easily affected by the external field, which can introduce more control methods and switch mechanism for potential NLO technology. In 2016, the MoS₂ bilayer (BL) was converted into an NLO material by the electrical manipulation, thereby achieving the valley Hall effect, indicating that 2D materials can promote the development of electronic storage and computing devices.⁹ Note that the development of optoelectronic technology is closely related to the advance of 2D NLO materials. Therefore, exploring new 2D materials with potential SHG capabilities is of great significance for enriching the structural chemistry of 2D NLO materials and expanding the field of optoelectronic applications.

In this work, different from the 2D structural characteristics of metal chalcogenides, we attempt to construct 2D NLO layered structures in nitride and phosphide systems by introducing the polar lone electron pairs (LEP) into the covalent framework. This design proposal is mainly based on the following three reasons: (i) Nitride and phosphide-family semiconductors can exhibit wide bandgap range and strong SHG effects, such as GaN (≈3 eV, 5 pm V⁻¹), ZnGeP₂ (≈2 eV, 70 pm V⁻¹) and GaAs (≈1 eV, 110 pm V⁻¹);^10–12 (ii) Polar effect of LEPs could enlarge the second-order NLO response such as the SHG effect (e.g., in BiB₃O₆ with Bi³⁺ LEP) and shift-current effect (e.g., in CsPbI₃ with Pb²⁺ LEP);^13,14 and (iii) LEP might enhance the layered structural stability to some extent due to the stable dangling bonds between layers.

By screening nitride and phosphide compounds from the inorganic crystal structure database (ICSD), we have focused on a series of non-centrosymmetric layered structures with divalent tin (i.e., Sn²⁺) LEP effect, including ASnX (A = Na, K; X = N, P, As, Sb).^15–18 We selected two representative structures including NaSnN and NaSnP as the initial crystal models for the first-principles design. Accordingly, we have systematically investigated the SHG properties and structural stabilities of their bulk and 2D materials. Detailed analysis of the structure–property correlations provides important clues for us to understand the origin of SHG effects. Our study can play a positive role in promoting the development of 2D NLO materials in the nitride and phosphide systems.

2. Computational methods

Thanks to the development of high-performance computing, material theorists can now predict the NLO properties and design new NLO structures in an ab initio way. In the past decade, we have developed a complete set of computational tools from first principles, which have effectively and efficiently discovered a series of new NLO materials, and guided the follow-up experimental explorations.¹⁹ In this study, CASTEP is used to perform the first-principles calculations based on the plane wave pseudopotential method and density functional theory.^20,21 The norm-conserving pseudopotentials are adopted for all constituent elements. Energy cutoff of 770 eV and Monkhorst–Pack k-point meshes of 9 × 9 × 4 for bulk and 9 × 9 × 1 for 2D structures in the Brillouin zone are chosen. BFGS scheme and dispersion correction are adopted for the structural geometry optimization and van der Waals (vdW) interactions.²² The linear response method is employed to obtain the phonon dispersion of crystals.²³ The dispersion separation of 0.01 per ų is adopted to make sure the computational accuracy.

Based on the optimized structure, the electronic energy bands, partial density of states (PDOS), atomic charge and orbitals are obtained. Further, the linear (e.g., refractive indices n and birefringence Δn) and NLO (e.g., static SHG coefficients d_ijk = χ_ijk/2) properties are calculated. The SHG coefficients herein are calculated using the following expression:²⁴

χαβγ = χαβγ(VE) + χαβγ(VH) + χαβγ(TB)

where χ^αβγ(VE), χ^αβγ(VH) and χ^αβγ(TB) denote the contributions from virtual-electron (VE) processes, virtual-hole (VH) processes and two-band (TB) processes, respectively. The formulae for calculating χ^αβγ(VE), χ^αβγ(VH) and χ^αβγ(TB) are given as follows:

Here, α, β and γ are Cartesian components, v and v′ denote valence bands, and c and c′ denote conduction bands. P(αβγ) denotes full permutation and explicitly shows the Kleinman symmetry of the SHG coefficients. The band energy difference and momentum matrix elements are denotes as ℏω_ij and p^α_ij, respectively, and they are all implicitly k dependent.

The screened exchange local density approximation (sX-LDA) method is employed to accurately (in particular for the narrow-gap semiconductors) predict the energy bandgaps E_g,²⁵ and LDA is used to calculate the optical properties corrected by the scissor operator, which is set as the difference between LDA and sX-LDA bandgaps.^26,27 This self-consistent ab initio approach has been proven to be effective to investigate the optical properties in many types of NLO materials without introducing any experimental parameter.¹⁹ Note that the SHG coefficients of 2D materials are related to the thickness (unit is Å pm V⁻¹). For comparison, herein we use relative SHG effect, defined as the SHG effect per unit thickness (unit is pm V⁻¹), to evaluate the NLO performance of ML structures.

Based on the proposed method, we have calculated the energy bandgaps and SHG properties for the important NLO systems of ZnGeP₂ and MoS₂ as the benchmark for bulk and ML, respectively. It is demonstrated that the calculated results as listed in Table 1 are in good agreement with the experimental measurements (relative error < 10%).

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3. Results and discussion

We will discuss the layered structural characteristics and the LEP-induced SHG effects according to the evolution from NaSnX (X = N, P) bulks to nanosheets, including the MLs and surfaces of NaSnX (X = N, P) and novel layered structures of HSnX (X = N, P) by surface hydrogen modification. We will further investigate the NLO origin and possible control means such as strain to adjust the material properties of energy bands and SHG effects.

3.1. NaSnN bulk material

NaSnN is a novel ternary tin(II) nitride crystal with the non-centrosymmetric symmetry of P6₃mc.¹⁵ Its unit cell structure is shown in Fig. 1a, with the lattice constant a = b = 3.176 Å and c = 10.774 Å. The calculated data as listed in Table S1 of the ESI† is in good agreement with the experimental results, demonstrating the accuracy of LDA scheme for the present structure prediction. The bulk NaSnN structure is formed by stacking layered anionic framework (SnN)⁻ and interlayer cation Na⁺ (see Fig. 1b). This buckled (SnN)⁻ ML structure is like covalent silicene, but its tin(II) (Sn²⁺) exhibits a relatively strong LEP effect, which can be clearly seen from the calculated electron localization function (ELF) analysis as plotted in Fig. S1 of the ESI.†

An interesting structural feature is that the interlayer cations Na⁺ are attracted to the adjacent N³⁻ from the lower layer but are repelled by the adjacent Sn²⁺ from the upper layer, which results in the weak connection between adjacent NaSnN layers. First-principles calculations for Mulliken bond population (MBP) as listed in Table 2 show that there are anti-bonds (MBP < 0) between Na⁺ and Sn²⁺ while weak covalent bonds (MBP ≈ 0.17, bond length L ≈ 2.47 Å) between Na⁺ and N³⁻.²⁸

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The band structure and PDOS of bulk NaSnN are plotted in Fig. 2a, showing that it is an indirect-gap semiconductor. The ionic Na–N coupling and covalent Sn–N hybridization lead to the bandgap between valence bands (VB) and conduction bands (CB). The other orbitals in the top of VB and the bottom of CB are composed by the s orbitals of Sn, which are mainly non-bonding states contributing to the LEP effect.

Table 1 lists the bandgap E_g, SHG coefficients d_ijk, refractive indices n_o, n_e and birefringence Δn of bulk NaSnN. Its bandgap calculated by sX-LDA is 1.34 eV, within the suitable bandgap range of promising photovoltaic materials. The SHG coefficient d₁₁₃ (or d₂₂₃) is relatively small (≈2.7 pm V⁻¹), since it mainly originates from the in-plane polarization of the layered orbitals but the polarization directions for the upper and lower layers are opposite. As a comparison, the d₃₃₃ caused by the polar LEP effect along the c-axis is much larger (≈27 pm V⁻¹), which can be confirmed from the analysis of SHG-weighted density as shown in Fig. 3a. Based on the SHG-weighted charge density analysis tool developed by our group, the SHG coefficient is “decomposed” onto the respective orbital or band according to a “band-resolved” scheme, and then the SHG-weighted bands are used to sum the charge densities of all occupied (OCC) or unoccupied (UNOCC) states.²⁹ As a result, the electronic states irrelevant to SHG are not shown in the OCC or UNOCC SHG density, and the orbitals vital to SHG are highlighted in the real space. It is clearly illustrated from Fig. 3a (for details see Table S2 in the ESI†) that the SHG density is mainly located in the Sn and N atoms from the VE process, indicating that the dominant contribution of the SnN backbone with the LEP effect to the SHG effect. Moreover, the calculated birefringence is much larger (Δn ≈ 0.63) than most of NLO materials such as β-BaB₂O₄ (Δn ≈ 0.12), LiIO₃ (Δn ≈ 0.17) and BiB₃O₆ (Δn ≈ 0.18), demonstrating that it has larger structural anisotropy and sufficient phase-matching ability for the effective NLO conversion.

3.2. NaSnN ML and nanosheets

It should be emphasized that NaSnN itself is a novel NLO structure. However, since the direction of light-induced polarization is reversed, the polar effects in the a–b plane are cancelled, so the total NLO response for both layers becomes weaker. If the bulk structure can be transformed into an ML structure, the intralayer SHG effect will be exhibited. Considering the weak connection between adjacent NaSnN layers in the bulk material, we can theoretically strip out the NaSnN ML with similar Sn²⁺ LEP effect. MBP analysis as listed in Table 2 shows that there is still some covalent bonding (MBP ≈ 0.07, L ≈ 2.38 Å) between Na⁺ and N³⁻.

Unfortunately, the bandgap of NaSnN ML is closed as shown from the band structure in Fig. 2b, demonstrating that it has become a semimetal from an insulator. The PDOS analysis further shows that the s orbitals of Sn and hybrid Sn–N p orbitals mainly contribute to the VB maxima, while the Na states make up the CB minima. This is mainly due to the relatively free Na⁺ that are not bound by the covalent layers. Detailed Mulliken atomic population (MAP) as listed in Table 2 shows that the charge transfer (≈0.38) from Na to SnN layer in the NaSnN ML is less than that in the NaSnN bulk (≈0.63), which results in the great decrease of bandgap in ML.

Note that the proper band gap is very important for NLO applications, especially for frequency conversion in transparent regions. Therefore, in order to open a bandgap so as to meet the requirement of NLO transparency, we have further designed NaSnN BL, trilayer (TL) and multilayer (e.g., 5L) as shown in Fig. S2a of the ESI† to enlarge the charge transfer between Na and SnN layer by increasing the number of layers. However, their bandgaps are still closed as shown from the band structures in Fig. S2b of the ESI,† because there is still little charge transfer (e.g., 0.27 for BL, 0.24 for TL, 0.21 for 5L from the calculations) between exposed Na and adjacent SnN, which causes the Na states still located around the Sn–N coupling gap from the PDOS analysis in Fig. S2c of the ESI.†

3.3. 2D design from NaSnN to HSnN

From the calculation point of view, NaSnN (001) surface has similar structural and electronic properties as the 5L structure. Fig. 1c displays the structure of NaSnN surface with Na⁺ exposed. Its band structure and PDOS in Fig. 2c show that the free Na orbitals are like the NaSnN surface states located in the bulk bands, which can result in the zero bandgap as similar as that of NaSnN nanosheets. However, in some cases, these surface states may be saturated due to the possible passivation. For the NaSnN surface, we assume that the Na ions on the surface are totally freed, and then the exposed N atoms are passivated by hydrogens, i.e., NaSnN layer on the surface becomes HSnN layer by hydrogen passivation as shown in Fig. 1d. Unfortunately, the bandgap is still close from the band structure as shown in Fig. 2d. Combined the analysis of PDOS with MAP (see Table 2), it is still mainly due to the insufficient charge transfer (≈0.29) between H and SnN layer despite it has been improved compared with that (≈0.20) on the clean NaSnN surface.

Note that eliminating the influence of substrate may be effective to improve the charge transfer of HSnN on the surface. Considering the weak interlayer connection between substrate NaSnN and covered HSnN, we theoretically peel off the covalent HSnN from the NaSnN substrate as a single layer as depicted in Fig. 1e. Analysis of MBP and MAP shows that the framework of HSnN ML is more covalent than that of NaSnN ML, and the charge transfer (≈0.33) between H and SnN for HSnN ML are more than that (≈0.29) of HSnN on the NaSnN substrate. Consequently, HSnN ML exhibits an opened bandgap (≈1.08 eV) by the sX-LDA calculation. Fig. 2e further shows that it is an indirect-gap semiconductor, and the states around the bandgap are mainly from the s orbitals of Sn and the hybrid p orbitals between Sn and N. The H orbitals are far from the bandgap, different from those states of free Na in the NaSnN nanosheets.

Table 1 lists the SHG coefficients of HSnN ML, in which the largest coefficient d₁₁₁ (or d₁₂₂) is in-plane and about 147 pm V⁻¹, as large as that of MoS₂ (cal. d₁₁₁ ≈ 149 pm V⁻¹, exp. d₁₁₁ ≈ 158 pm V⁻¹).⁵ Further from the analysis of SHG-weighted density as shown in Fig. 3b, the effect of d₁₁₁ is mainly originated from the Sn LEP states in the VE process. In addition, the d₁₁₃ (or d₂₂₃ ≈ 46 pm V⁻¹) becomes much larger than that of NaSnN bulk although the d₃₃₃ becomes smaller (≈11 pm V⁻¹). These SHG coefficients are also from the polar Sn²⁺ LEP along the c-axis, which MoS₂ does not have. Moreover, the phonon vibration spectrum is calculated to confirm the structural stability of designed HSnN ML. The results are plotted in Fig. S3a of the ESI,† indicating that it is dynamically stable without any virtual frequency. Therefore, HSnN ML may be used as a possible 2D NLO structure that can exhibit strong SHG effect comparable to that of MoS₂ ML.

3.4. Strain regulation from NaSnN ML to NaSnP ML

In fact, there is another way, i.e., employing strain to NaSnN ML, to regulate the charge transfer effect. As such, we have theoretically obtained the band evolution with strain varying from 0 to 15% and the relation curve of bandgap with respect to the strain as plotted in Fig. 4a and b. It can be found that the bandgap is gradually opened from 0 (strain ≈ 8%) to 0.18 eV (strain ≈ 10%) and 0.66 eV (strain ≈ 15%) by LDA. This is mainly due to the stronger charge transfer (0.44) than those of ML (0.38) or surface (0.20). Meanwhile, the corresponding SHG effects are gradually decreased as the increasement of bandgap, but still located in the strong range (e.g., d₁₁₁ ≈ 300–500 pm V⁻¹) as plotted in Fig. 4c, which are greatly enhanced compared with NaSnN bulk and HSnN ML, and much larger than that of MoS₂ ML (for details see Table 1). The SHG density in Fig. 3c shows that its SHG effect is caused by the synergic effect of the Sn LEP states and Sn–N hybrid orbitals in the VE process (also see Fig. 2f). It notes that more than 5% of strain is too difficult to achieve in the practical applications. Fortunately, there is a phosphide analogue system, i.e., NaSnP with the lattice constant a = b = 3.880 Å and c = 11.667 Å, which exhibits a larger lattice constant than NaSnN (equivalent to 16% strain).¹⁶ Therefore, it is possible to realize a ML structure in this NaSnP system that may exhibit not only an opened bandgap but also a significant SHG effect.

First-principles results from Table 1 demonstrate that NaSnP bulk exhibits a smaller indirect bandgap (≈1.1 eV) and SHG effect (d₃₃₃ ≈ 16 pm V⁻¹) than those (≈1.3 eV and 27 pm V⁻¹) of NaSnN bulk. Its band structure and PDOS are shown in Fig. 2g, which is basically similar to the energy bands of NaSnN bulk with slightly different dispersion. Interesting, when stripped into the NaSnP ML, it exhibits an opened bandgap (≈0.87 eV) and huge SHG effects in d₁₁₁ (≈203 pm V⁻¹), d₁₁₃ (≈618 pm V⁻¹) and d₃₃₃ (≈533 pm V⁻¹). The SHG effects are much larger than those of HSnN ML (≈147 pm V⁻¹) and MoS₂ ML (cal. ≈149 pm V⁻¹), and comparable to the strained NaSnN ML (≈300–500 pm V⁻¹). This is mainly because that the covalent bonding between Na and (SnP) layer is relatively weaker than that between Sn and P. The further analysis demonstrates that the charge transfer (0.28) from P to Sn in NaSnP ML is more than that (0.19) in NaSnP bulk, which can enlarge the polar effect of Sn LEP and further enhance the SHG effect. It can be confirmed from Fig. 3d and e that, compared with the SHG density of NaSnP bulk, there is much stronger SHG density for d₁₁₃ of NaSnP ML on the Sn LEP and P site in the main VE process. Meantime, different from those for NaSnN bulk, there is some density around Na atoms that also contributes to SHG effect in the VH process (see Table S2 in the ESI†). It can be also found from PDOS that the Na orbitals are relatively isolated at the bottom of CB. Moreover, these bands around the bandgap are flatter than those in bulk, exhibiting a quasi-direct gap, which is not only beneficial to the enhancement of SHG but also favorable to the optoelectronic applications.

In order to confirm the structural stability of strained NaSnN ML and NaSnP ML, the phonon spectra are obtained from first principles as shown in Fig. S3b and S3c of the ESI.† The results show that there exists a virtual frequency in the 15%-strained NaSnN ML, indicating that it may be a metastable structure. For NaSnP ML, the dispersion of the lowest vibration frequencies is normal without any virtual frequency. Thus, the NaSnP ML structure could exhibit the dynamical stability, and needs to be verified by the experiment.

Similar as HSnN ML, we can also design the HSnP ML structure. Unfortunately, its SHG effect (≈14 pm V⁻¹) is much weaker than those of NaSnP systems although its bandgap (≈2.18 eV) is abnormally larger. This is because the SHG effect is mainly originated from the covalent SnP framework (see PDOS of Fig. 2i and SHG density of Fig. 3f), but the small anionic charge (−0.41 for P) and cationic charge (0.37 for Sn) as listed in Table 2 cause the electronic polarization to be weak and the SHG effect to be decreased. Moreover, it is metastable with a virtual frequency as shown in Fig. S3d of the ESI.† Thus, it cannot exhibit usable SHG effect like HSnN ML.

In addition to the dynamical stability, we also need to provide more clues to evaluate the structural rationality of HSnN and NaSnP ML materials. For such, the interlayer interaction energies of the two structures are calculated with respect to the interlayer distance as well as those of MoS₂ as plotted in Fig. S4 of the ESI.† It is found that the interlayer binding energy for NaSnP (275 meV per atom) is larger than those of HSnN (115 meV per atom) and MoS₂ (114 meV per atom), indicating that its nanosheet is more difficult to be cleaved than HSnN and MoS₂. Meanwhile, the thermodynamic stability can be approximated by the formation energy, i.e., the difference in energy and the lowest value for any mixture of bulk materials. The calculations show that NaSnP has indeed higher formation energy (428 meV) than those of HSnN (172 meV) and MoS₂ (133 meV).³⁰ The cohesive energy, usually used to discuss the energetic stability of the structure, is also obtained by the difference between the energy of the dispersed gas of its constituent atoms (or molecules) and the energy of the crystal at absolute zero of temperature.^31,32 The calculated cohesive energy decreases from MoS₂ (7.8 eV per atom) to HSnN (6.0 eV per atom) and NaSnP (4.2 eV per atom), indicating that the energetic stability decreases from MoS₂ to HSnN and NaSnP. The above results demonstrate that the NaSnP and HSnN nanosheets maybe more difficult to be obtained than MoS₂.

4. Conclusions

In this work, we focused on a series of ASnX (A = Na, H; X = N, P) layered structures with divalent tin LEP effect. We investigated their electronic structures and SHG properties systematically from the first-principles calculations. We illustrated in detail the PDOS, charge transfer and SHG density for the bulk and 2D nanostructures. All the results demonstrated that the polar divalent tin effect could produce large birefringence in bulk materials, which can also greatly enhance the SHG effect in the corresponding 2D systems by hydrogen passivation or strain engineering. As two remarkable 2D structures, (1) HSnN ML exhibits a moderate bandgap (≈1.1 eV) and comparable SHG effect (d₁₁₁ ≈ 147 pm V⁻¹) as MoS₂ ML (d₁₁₁ ≈ 158 pm V⁻¹); (2) NaSnP ML exhibits a direct optical bandgap (≈0.87 eV) and much stronger SHG effect (d₁₁₁ ≈ 203 pm V⁻¹, d₁₁₃ ≈ 618 pm V⁻¹, d₃₃₃ ≈ 533 pm V⁻¹) than MoS₂ ML. Moreover, their 2D monolayered structures are both dynamically stable. Once experimentally confirmed, they will facilitate the NLO research of 2D materials in the nitride and phosphide systems.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We thank Dr Shunhong Zhang (University of Science and Technology of China) and Dr Fei Liang (Shandong University) for fruitful discussions on the structural stability and optical properties of 2D materials. This work was supported by the National Natural Science Foundation of China Grants No. 11704023 and 51890864. Z. Lin acknowledges support from outstanding member in Youth Innovation Promotion Association at CAS.

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Footnote

† Electronic supplementary information (ESI) available: Tables S1, S2 and Fig. S1–S4. See DOI: 10.1039/d0nr03778h

This journal is © The Royal Society of Chemistry 2020