Theoretical investigation on the balance between large band gap and strong SHG response in BMO₄ (M = P and As) crystals

Abstract

Nonlinear optical (NLO) materials with outstanding performances are particularly important in laser science. To design an efficient NLO material with an optimal balance between the optical band gap and the NLO coefficient is still a huge challenge. In this paper, the electronic structures and optical properties of two isostructural non-centrosymmetric BMO₄ (M = P and As) crystals exclusively containing tetrahedral groups have been studied by the first-principles method. Our calculated results show the compounds exhibit excellent properties with a suitable balance of large band gap and large NLO coefficient for BPO₄ (10.4 eV and 0.64 pm V⁻¹) and BAsO₄ (7.8 eV and 1.58 pm V⁻¹). The unanticipated large second-harmonic generation (SHG) response can be understood by the relatively strong s–p hybridization in BMO₄ (M = P and As). It is found that substitution of PO₄ groups for AsO₄ groups can increase the SHG effect and the birefringence but decrease the band gap. Whether the redshift of bandgap will be responsible for the change of SHG effect between BMO₄ (M = P and As) is explored by energy correction method. The increased polarizability anisotropy of tetrahedral groups explained the enhanced birefringence.

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Introduction

Nonlinear optical (NLO) crystals have attracted much attention because they can be used to extend the working wavelength range of a solid state laser through frequency conversion.^1,2 For ultraviolet (UV) or deep-UV NLO applications, ideal NLO materials require a strict combination of properties that need to be satisfied simultaneously in a non-centrosymmetric crystal: a wide transparency window down to the deep-UV region, relatively strong effective second-harmonic generation (SHG) response, and suitable birefringence to realize phase matching. Borate and phosphate crystals have attracted much interest for their excellent performance as NLO crystals, such as, KH₂PO₄ (KDP),³ CsB₃O₅,⁴ LiB₃O₅ (LBO),⁵ SrB₄O₇,⁶ LiCs₂PO₄,⁷ Ba₃P₃O₁₀Cl,⁸ Ba₅P₆O₂₀,⁹ and RbBa₂(PO₃)₅.¹⁰ The tetrahedral BO₄ or PO₄ groups existing in the above mentioned crystals enlarge the band gaps of the crystals shifting the cutoff edge to the deep-UV region. However, the crystals with BO₄ or PO₄ groups as building blocks are usually labeled by small NLO coefficients and/or small birefringence. The combination of the π-conjugated planar BO₃ groups into the structures is away to improve the birefringence and optical nonlinearities of these crystals.^11–16 Especially, KBe₂BO₃F₂ (KBBF) has a short deep-UV cutoff wavelength about 150 nm, a relatively large SHG tensor d₁₁ about 0.47 pm V⁻¹, and a moderate birefringence of 0.077. And it is the sole NLO crystal, which can generate coherent light of wavelength below 200 nm by the direct SHG at room temperature.¹⁷ However, KBBF crystal is very difficult to grow in thickness due to its strong layering growth habit, which severely hinders its practical applications.¹⁸

In 1934, the cristobalite like forms of the ternary silica analogues BPO₄ were first discovered by solid-state synthesis at high temperature.¹⁹ The theoretical or experimental researches of BPO₄ on its growth and optical properties have been in progress for many years.^20–24 They measured BPO₄ has a NLO coefficient d₃₆ = 0.76 pm V⁻¹, which is larger than that of KBBF (d₁₁ = 0.47 pm V⁻¹). BPO₄ also has a short deep-UV cutoff wavelength ∼134 nm, 16 nm shorter than that of KBBF.²⁴ BPO₄ should have a smaller NLO coefficient resulting from the larger band gap because the possibility of quantum transition between valence and conduction bands is ratio to the reciprocal of diverse energy at each momentum point. Until 2016, Li et al. explained the mechanism of its large NLO coefficient.²⁵ Unsatisfactorily, BPO₄ has a smaller birefringence (approximately 0.007) than KBBF, which makes it non-phase matched in ultraviolet SHG progress, limiting its application in ultraviolet output. Therefore, the issue of how to improve the birefringence of the available frequency conversion material should be focused on. Previous studies have promised strain engineering and ion-doping methods could modulate or even enhance the birefringence of relevant materials both theoretically and experimentally.^26–30 Following this idea, we “look away” to arsenic, which is similar to phosphorus can coordinate with four oxygen atoms to form the tetrahedral groups. Curiously, does the combination of the BO₄ and AsO₄ groups result in an optimal balance between large band gap and strong SHG response? Does the value of birefringence also increase compared to BPO₄? To the best of our knowledge, theoretical or experimental reports about the optical properties of BAsO₄ are still absent after it was discovered due to the difficulty in growth.^31–33

In this paper, in order to explore the influence on optical properties by the substitution of NLO-active PO₄ groups for AsO₄ groups, BMO₄ (M = P and As) crystals exclusively containing tetrahedral groups are studied. We first introduced the optical properties obtained by ab initio method. Then, the gap-determination factor was discussed to explain the mechanism of large band gaps in BMO₄ (M = P and As) crystals. BAsO₄ has a stronger SHG response with 2.5 times than that of BPO₄. The origin and enhancement of the large SHG effect of BAsO₄ was explored by combining the band structures and the energy correction analysis. Finally, we systemically discussed the polarizability anisotropy of tetrahedral groups to explain why the birefringence of BAsO₄ is 2.6 times than that of BPO₄.

Calculation methods

The first-principles calculations for electronic structure employed the planewave pseudopotential method in the CASTEP.^34–36 The calculations was performed using the crystallographic information files BPO₄: ICSD-4132434; BAsO₄: ICSD-413438. During the calculation, geometry optimization was performed using the BFGS minimization technique. The converged criterions for the geometry optimization are that the residual forces on the atoms, the displacements of atoms and the energy change were less than, 5.0 × 10⁻⁶ eV per atom, 0.01 eV Å⁻¹ and 5 × 10⁻⁴ Å, respectively. Two exchange and correlation functionals, local density approximation/Ceperley–Alder–Perdew–Zunger (LDA/CA–PZ)^37,38 and generalized gradient approximation/Perdew–Burke–Ernzerhof (GGA/PBE)³⁹ were adopted to calculate the electronic structures and NLO coefficients. After a series of successful tests, the LDA/CA–PZ functional with ultrasoft pseudopotentials (USP)⁴⁰ was finally selected, giving the best agreement with the experimental data of BPO₄. The valence electrons were B-2s²2p¹, O-2s²2p⁴, P-3s²3p³ and As-4s²4p³, respectively. The number of plane waves was determined by a cutoff energy of 380 eV and the k-point sampling was set as 7 × 7 × 8.

Based on the electronic structures, the imaginary part of the dielectric constant ε₂ (ω) can be obtained by

In which Ω is the volume of the elemental cell, v, and c represent the valence and conduction bands, respectively, and u is the vector defining the polarization of the electric field of the incident light. The real part of the dielectric constant, and then the refractive index n and birefringence Δn = n_max − n_min can be obtained using the Kramers–Kronig transform.⁴¹ It is well known that although the LDA functional always underestimates the bandgap, after the scissors corrections were employed, reliable refractive indices and birefringence can still be got from the LDA calculations.^42–45 To get suitable scissors corrections of these two compounds, the modified Becke–Johnson (mBJ) exchange potential^46,47 and implemented in WIEN2k⁴⁸ was used to calculate the optical band gap because the mBJ method can get similar accuracy comparison with very expensive GW calculations^49,50 and experimental values.²² Besides above methods, PBE0 hybridization functional has also been utilized to estimate the bandgaps in this work.

In addition, Becke three-parameter Lee–Yang–Parr (B3LYP) exchange correlation functional with the Lee–Yang–Parr correlation functional at the 6-31G basis set in Gaussian 09W program⁵¹ were employed to calculate the polarizability of the tetrahedra groups MO₄ (M = B, P and As). The anisotropic polarizabilities (Δα) are defined by the following equation:

Δα = (1/2)^1/2[(α_xx − α_yy)² + (α_xx − α_zz)² + (α_zz − α_yy)²]^1/2 (2)

SHG coefficients were calculated at a zero frequency limit by the length-gauge formalism derived by Aversa and Sipe,⁵² and later developed by Zhang et al.²⁷

χ⁽²⁾_αβγ = χ⁽²⁾_αβγ(VE) + χ⁽²⁾_αβγ(VH) (3)

where,here, α, β, γ are Cartesian components, v and v′ denote valence bands, c and c′ denote conduction bands, and P(αβγ) denotes full permutation. The band energy difference and momentum matrix elements are denoted as ω_ij and p_ij^α, respectively.

Results and discussion

Structure and optical properties

BPO₄ and BAsO₄ crystallize in the tetragonal space group I4̄ (no. 82), with a = b = 4.3401 Å, c = 6.6470 Å, Z = 2 for BPO₄, a = b = 4.4658 Å, c = 6.8139 Å, Z = 2 for BAsO₄, respectively. As sketched in Fig. 1, a continuous 3D framework of BMO₄ (M = P and As) crystals is built by sharing the corner of the BO₄ and MO₄ (M = P, As) tetrahedra.

Fig. 1 The crystal structure of BPO₄ (a) and BAsO₄ (b).

The band gaps, SHG coefficients in static limit and the birefringences for BMO₄ (M = P and As) were obtained (shown in Table 1). As compared with the experimental results, the PBE0 functional gives reliable values that are estimated of 10.4 eV and 7.8 eV, respectively. It is well known that although the LDA and GGA functional always underestimates the bandgap, after the scissors corrections were employed, reliable SHG coefficient and birefringence can still be got from the LDA calculations. It is clearly shown that the largest calculated NLO coefficient d₃₆ = 1.58 pm V⁻¹ of BAsO₄ also is larger than that d₃₆ = 0.64 pm V⁻¹ of BPO₄, which is very closed to the experimentally measured value of about two times of KDP.²² And the calculated birefringences are 0.011 for BPO₄, and 0.029 for BAsO₄ at 589 nm. The calculated birefringence about 0.011 is consistent well with the experimental value 0.007 at 589 nm.²⁴ It should be noticed that the band gap and NLO coefficient of BAsO₄ are comparable to these of the commercial NLO crystals LBO (7.8 eV and 1.17 pm V⁻¹), β-BaB₂O₄ (6.6 eV and 2.26 pm V⁻¹) and KTiOPO₄ (3.2 eV and 3.24 pm V⁻¹).⁵³ A subtle balance is obtained between large band gap and strong SHG response in BAsO₄ crystal. It indicates that BAsO₄ may have a promising potential to be a NLO material in UV or even DUV region.

Table 1 The optical properties of BMO₄ (M = P and As)

	Gap (eV)	SHG (pm V^−1)	Δn
BPO4	10.4	0.64	0.011
BAsO4	7.8	1.58	0.029

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Role of AsO₄ in band gap

LDA and GGA functionals are performed to calculate the band gap of BMO₄ (M = P and As), as shown in Table 2. The LDA and GGA results are smaller than the experimental data 9.25 eV (corresponding to the measured optical cutoff edge of 134 nm). It is not strange at all because the Kohn–Sham schemes based on LDA and GGA usually underestimate the band gap of materials primarily owning to the discontinuity of exchange–correlation energy. For comparison, the experimental optical bandgaps, the PBE0 and mBJ methods are also employed. It is found that PBE0 result is close to the experimental one. The results clearly indicate the validity of the method used here, so the optical bandgaps of BMO₄ (M = P and As) are very large (10.4 eV and 7.8 eV), comparable to the crystals aimed in deep-UV region such as Ba₃P₃O₁₀Cl (6.9 eV),⁸ Li₄Sr(BO₃)₂ (6.5 eV)¹⁴ and KBBF (8.3 eV),¹⁷ NaSr₃Be₃B₃O₉F₄ (7.2 eV)⁵⁴ and can be used in the UV region as NLO materials.

Table 2 Band gaps (eV) based on various exchange and correlation functionals

	BPO4	BAsO4
LDA	7.62	5.32
GGA	7.22	4.75
PBE0	10.35	7.77
mBJ	10.62	7.55
Exp.	9.25 (ref. 22)	—

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To explore the reason for the large band gaps, the partial density of states (PDOS) for BMO₄ (M = P and As) are plotted in Fig. 2a and b. For BPO₄, the upper of the valence band (VB) is mainly occupied by the non-bonding 2p orbitals of O; the bottom of the conduction band (CB) mainly consists of the hybridization orbitals of B–O and P–O. Similar to BPO₄, the contributions of O 2p orbitals primarily dominate the upper VB of BAsO₄. However, the bottom of CB of BAsO₄ mainly consists of As-s, B-p, O-p. To gain a deeper understanding of the electric structure features and check the band-gap determiner, we plotted slices of orbital density of CB minimum and VB maximum as shown in Fig. 2c–f. From Fig. 2c and e, it concludes that the VB maximum in BMO₄ (M = P and As) is mainly occupied by the nonbonding oxygen orbitals. While CB minimum exhibits totally different characters for the title compounds. It is clearly shown that the orbitals from P atoms, O atoms and B atoms dominate VB maximum of BPO₄ (see Fig. 2d). However, for BAsO₄, the VB maximum is primarily only controlled by orbitals from As and O atoms without B atoms as shown in Fig. 2f. Therefore, one can make a conclusion that the band gap determiner is nonbonding 2p of O under Fermi level and hybridization orbitals of B/P–O in CB for BPO₄ crystal, and As–O in CB for BAsO₄ crystal. In addition, the orbital density also reveals a strong hybridization between As-s and O-p in Fig. 2e. Such strong sp hybridization in BAsO₄ leads to a slight red-shift of the deep-UV cutoff edge compared with BPO₄.

Fig. 2 The partial density of states (PDOS) for BPO₄ (a) and BAsO₄ (b). The energy range of the specific orbitals of the BPO₄ crystals (c), (d) and the BAsO₄ crystals (e), (f).

In the light of the frontier molecular orbitals theory, the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of MO₄ (M = B, P and As) and BO₃ groups were also obtained (Table 3). Evidently, the band gaps ΔE_g of the MO₄ (M = B, P and As) tetrahedral groups and BO₃ groups obey the following relative orders: ΔE_g (BO₄) > ΔE_g (PO₄) > ΔE_g (AsO₄) ≈ ΔE_g (BO₃). It should pay attention to that the band gaps of AsO₄ groups is comparable with that of BO₃ groups revealing AsO₄ to be as a NLO active unit which is also beneficial to the large band gap.

Table 3 HOMO and LUMO of MO₄ (M = B, P and As) groups and BO₃ groups

Units	HOMO (a.u.)	LUMO (a.u.)	Gap (a.u.)
BO4	1.064	1.459	0.395
PO4	0.443	0.775	0.332
AsO4	0.424	0.729	0.305
BO3	0.578	0.898	0.311

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Mechanism for strong and enhanced SHG response

SHG coefficients d_ij were estimated by the sum-over-states under the static limit within the length gauge.²⁷ BMO₄ (M = P and As) both belong to I4̄ space group, owing to the restriction of Kleinman's symmetry, there are two independent elements: d₃₆ and d₁₅. The calculated results are listed in Table 4. It is clearly shown that the largest SHG tensor for BPO₄ is d₃₆ = 0.64 pm V⁻¹ which is comparable to the reported value d₃₆ = 0.78 pm V⁻¹.²² As for BAsO₄, the SHG coefficients d₃₆ = 1.58 pm V⁻¹ and d₁₅ = 1.04 pm V⁻¹ are obtained, respectively.

Table 4 The SHG response of two SHG tensors (pm V⁻¹) of BMO₄ (M = P and As)

Compound	d36	d15
BPO4	0.64	−0.07
BAsO4	−1.58	1.04

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As mentioned above, BMO₄ (M = P and As) should have a weaker NLO response resulting from the large band gap. Curiously, SHG response is 2 × KDP for BPO₄ and 4 × KDP for BAsO4, which is larger than many crystals with equivalent bang gap like KBBF (1.2 × KDP, 8.3 eV). To illuminate this question, we carefully check the band structures of BMO₄ (M = P and As) (shown in the Fig. 3). We mainly focused on the electric structure around the Fermi energy level because the optical properties of the crystal are particularly determined by the states close to the Fermi level. Evidently, the lowest conducting band is only dispersive around the G point (dyed by blue), and valence bands are clearly dispersive. It implies a relatively strong hybridization between the O 2p orbitals and the 4s orbitals from As which can enhance the interband Berry connection, and contribute a relatively large SHG effect in BAsO₄. The same situation (see the Fig. 3a) also occurs in the BPO₄ crystal.²⁵

Fig. 3 The band structures of BPO₄ (a) and BAsO₄ (b).

To identify the contribution of each atom in the SHG processes in real space, the SHG-density method was analyzed for both compounds. The SHG process contains the two virtual transition processes, namely virtual electron (VE) and virtual hole (VH) processes. Here we will just show the SHG-density of the largest SHG tensors d₃₆ to discuss the origin of the SHG effect. The obtained SHG-densities of the two compounds are shown in Fig. S1† and 4. More attention will be paid on that the SHG-density of BMO₄ (M = P and As) mainly distributes around the BO₄ and MO₄ groups (especially on O²⁻ anions) for unoccupied states and occupied states, illustrating their dominant contributions to the SHG.

Fig. 4 SHG-density of BAsO₄: (a) VE occupied states; (b) VE unoccupied states; (c) VH occupied states; (d) VH unoccupied states.

It is worth to notice that BAsO₄ has SHG response about 2.5 × BPO4. Curiously what is the origin of the largely enhanced SHG coefficients of BAsO₄? It is well known that band gap has a significant influence on SHG effect that the small gap can change the SHG response dramatically. To check the influence of the decrease of band gap in BAsO₄, we use the scissor operator corrections to the experimental band gaps of BPO₄. It turns out that the band-gap-corrected SHG coefficient d₃₆ is 0.79 pm V⁻¹ for BAsO₄, which is close to that of BPO₄ (d₃₆ = 0.64 pm V⁻¹). Compared with 1.58 pm V⁻¹, we found that the contribution induced by band gap reduction is around 85%. And we know from the electronic structure that the band gap reduction or the red-shift of UV-cutoff from BPO₄ to BAsO₄ attributes to the strong hybridization interaction between As and O. Therefore, the stronger covalent interaction of As–O is a significant factor to the enhanced SHG response in BAsO₄ compared that of BPO₄.

Origin of enhanced birefringence

The obtained birefringence of BAsO₄ is about 2.6 × BPO₄. In order to figure out the enhancement, we calculate the polarizability anisotropy of BO₄, PO₄ and AsO₄ groups in BMO₄ (M = P and As) using the DFT implemented via the Gaussian09 package at the 6-31 G level. The polarizability anisotropy Δα of BO₄, PO₄ and AsO₄ groups are shown in the Fig. 5. It is clearly the Δα of tetrahedral groups of BAsO₄ are larger than those of BPO₄. The sequence of the calculated polarizability anisotropy is in good agreement with the sequence of birefringence.

Fig. 5 Calculated polarizability anisotropy Δα of tetrahedral groups in BMO₄ (M = P and As).

Conclusion

In this paper, the electronic structures and optical properties of BMO₄ (M = P and As) were calculated by DFT method. The band gap of BAsO₄ is still large after a substitution of NLO active PO₄ groups for AsO₄ groups. The band gaps ΔE_g of the groups MO₄ (M = B, P and As) and BO₃ groups obey the following relative orders: ΔE_g (BO₄) > ΔE_g (PO₄) > ΔE_g (AsO₄) ≈ ΔE_g (BO₃). It implies AsO₄ groups to be as a NLO active unit which is also in favour of the large band gap. We also found BMO₄ (M = P, As) have large SHG responses. A relatively strong hybridization between the O 2p orbitals and the orbitals from B/P or As contributes a relatively large SHG effect in BMO₄ (M = P, As). The strong s–p hybridization influence between As and O gives the band gap reduction and the 85% enhancement of SHG response from BPO₄ to BAsO₄. The calculated birefringence of BAsO₄ is about 2.6 × BPO₄ due to the stronger polarizability anisotropy of tetrahedral groups in BAsO₄ than those of BPO₄. The results show the replacing PO₄ tetrahedra by AsO₄ groups do not only achieve an optimal balance between large band gap and strong SHG response but also are helpful to increase birefringence. And one may be able to design new compounds with the adjustment effect of tetrahedral groups.

Acknowledgements

This work is supported by the National Key Basic Research Program of China (Grant No. 2014CB648400), the “National Natural Science Foundation of China” (Grant No. 51425206, 11474353), the Xinjiang International Science & Technology Cooperation Program (20146001).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra26568e

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