Theoretical perspective of the lone pair activity influence on band gap and SHG response of lead borates

Abstract

Metal lone pairs play an important role in determining the SHG enhancement and bandgap red shift. In this work, the electronic structures and optical properties of a class of lead borates Pb₂B₅O₉Cl, BaPb[B₅O₉(OH)]·H₂O, and Ba₂Pb(B₃O₆)₂ have been investigated to uncover the influence of the lead atom on the band gap and SHG response. It is found that BaPb[B₅O₉(OH)]·H₂O and Ba₂Pb(B₃O₆)₂ have little bandgap redshift due to their weak distortion and stereochemical activity. The intensity of the lead lone pair stereochemical activity plays an important role in determining the reduction of the band-gap.

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Introduction

The prerequisite for good ultraviolet (UV) or deep-ultraviolet (deep-UV) nonlinear optical (NLO) materials follows at least the following four criteria: wide band gap, high second-harmonic generation (SHG) response, appropriate large birefringence to meet the phase matching condition, good chemical stability and mechanical properties.^1,2 Due to the rich chemical structures and good performances, borates have long been prominent sources of nonlinear optical (NLO) materials, such as β-BaB₂O₄ (BBO),³ LiB₃O₅ (LBO),⁴ KBe₂BO₃F₂ (KBBF),⁵ CsLiB₆O₁₀ (CLBO),⁶ K₃B₆O₁₀Cl (KBOC),⁷ Li₄Sr(BO₃)₂.⁸ In order to obtain borates with large SHG intensity, a general strategy is to introduce asymmetric structural units such as the d⁰ metal centered distorted polyhedra or stereochemically active lone-pair anions.^9–15 Inspired by this idea, lead borates have long been attracting increasing attentions and becoming one kind of important NLO materials with large SHG responses. However, it is found that the cost of good SHG performance is the narrow band gap, which limits the transmittance in the UV region. Typical examples possessing remarkable strong SHG responses are Pb₂B₅O₉I (3.1 eV, 13.5 KDP),¹¹ Pb₄O(BO₃)₂ (3.3 eV, 3.0 KDP),^16,17 Pb₂(BO₃)(NO₃) (3.6 eV, 9 KDP),¹⁸ CsPbCO₃F (4.12 eV, 13.4 KDP).¹⁹ Therefore, the key point to obtain lead borates used in UV region is getting the subtle balance between the SHG response and bandgap.

In this paper, in order to clarify how to keep balance on the seesaw of the energy band gap and SHG coefficients, a class of lead borates Pb₂B₅O₉Cl,^20–22 Ba₂Pb(B₃O₆)₂ (ref. 23) and BaPb[B₅O₉(OH)]·H₂O²⁴ are studied. For comparison, the electronic structures of Ba₂B₅O₉Cl,²⁵ the isostructural virtual compound Ba₂[B₅O₉(OH)]·H₂O and Ba₃(B₃O₆)₂ were also investigated. It is found that two of these lead borates, Ba₂Pb(B₃O₆)₂ and BaPb[B₅O₉(OH)]·H₂O have relative large band gaps and relative small band gap shift. The influence of the lead lone pair on the bandgap and SHG response will be discussed.

Numerical calculation details

Geometry optimization, electronic structures and optical properties were carried out by the density functional theory (DFT) method within the generalized gradient approximation (GGA) in the scheme of Perdew–Burke–Ernzerhof (PBE) functional,²⁶ which was implemented in the CASTEP package.^27–30 And the norm-conserving pseudopotential (NCP) was adopted,²⁷ the following orbital electrons were treated as valence electrons: Pb 5s²5p⁶5d¹⁰6s²6p², Cl 3s²3p⁵, Ba 5s²5p⁶6s², B 2s²2p¹, O 2s²2p⁴, H 1s¹. The number of plane waves included in the basis was determined by a cutoff energy of 910 eV (Pb₂B₅O₉Cl, BaPb[B₅O₉(OH)]·H₂O, Ba₂B₅O₉Cl), and 820 eV (Ba₂Pb(B₃O₆)₂), throughout the calculations. Meanwhile, the Monk-horst–Pack k-point scheme of Pb₂B₅O₉Cl, BaPb[B₅O₉(OH)]·H₂O and Ba₂Pb(B₃O₆)₂ were set as 4 × 4 × 2, 4 × 4 × 2, 4 × 4 × 4 in the irreducible Brillouin zone. The optimized parameters are given in the ESI.†

The length-gauge formalism derived by Aversa and Sipe was used to calculated the SHG coefficients at a zero frequency limit.³¹ The static second order coefficients χ⁽²⁾_αβγ can be written as:³²

χ⁽²⁾_αβγ = χ⁽²⁾_αβγ(VE) + χ⁽²⁾_αβγ(VH) + χ⁽²⁾_αβγ(two-bands) (1)

In this sum-over-states type formalism, the total SHG coefficient χ⁽²⁾ is divided into virtual-electron (VE) process, virtual-hole (VH) process and two-bands. And it should also be noted that the contribution of two-band process has been proved strictly to be zero.³³ Hence, the new formalisms are represented as:

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

where α, β, γ are Cartesian components, v, v′ refer to valence bands, c and c′ refer to conduction bands, and the P(αβγ) denotes full permutation. To calculate the optical properties accurately a gap correction is needed.

To show the distribution of the quantum states relevant to SHG, the SHG-density technique was implemented by using the effective SHG of each band (occupied and unoccupied) as weighting coefficient (after normalized with total VE or VH χ⁽²⁾ value) to sum the probability densities of all occupied or unoccupied states. The effective SHG of each band was obtained by partially summing over two out of the three band indices for χ⁽²⁾_αβγ (VE) or χ⁽²⁾_αβγ (VH), and two kinds of summing sequences induce a consequent decomposition of the SHG strength into occupied and unoccupied band representations of orbital contributions. Therefore, the resulting distribution of SHG density represents a highlight of the origin of SHG optical nonlinearity in real space.

Results and discussion

Electronic structure and band gap

The crystal data of Pb₂B₅O₉Cl, Ba₂Pb(B₃O₆)₂, BaPb[B₅O₉(OH)]·H₂O and Ba₂B₅O₉Cl are shown in Table S1 in the ESI.† Pb₂B₅O₉Cl, Ba₂B₅O₉Cl and BaPb[B₅O₉(OH)]·H₂O crystallize in the acentric space group Pnn2, Pnn2 and Cc respectively. Pb₂B₅O₉Cl and its isostructural Ba₂B₅O₉Cl exhibit three-dimensional borate frameworks. The boron–oxygen groups of BaPb[B₅O₉(OH)]·H₂O form a 2-D layer with 9 member ring. The Ba₂Pb(B₃O₆)₂ crystallizes into a trigonal system with centric space group R3̄.

Using the method described above, the band structures of Pb₂B₅O₉Cl, BaPb[B₅O₉(OH)]·H₂O and Ba₂Pb(B₃O₆)₂ are obtained, and the calculated band gaps are 3.359 eV, 4.749 eV, 4.573 eV respectively. The calculated result shows that Ba₂Pb(B₃O₆)₂ and BaPb[B₅O₉(OH)]·H₂O have relatively large band gaps. The tendency of calculated bandgaps is consistent well with the experimental values. The experimental values of Ba₂Pb(B₃O₆)₂ and BaPb[B₅O₉(OH)]·H₂O are 5.17 eV and 6.20 eV, respectively. It is well known that introduction of the lead into alkaline-earth borates usually leads to a red-shift of the UV cutoff edge. Curiously, relative large bandgap was found in BaPb[B₅O₉(OH)]·H₂O. Why?

In order to investigate the influence of the lead cations on the bandgap, the electronic structures of their isostructural compound Ba₂B₅O₉Cl, Ba₂[B₅O₉(OH)]·H₂O and Ba₃(B₃O₆)₂ are studied for comparison. The virtual structures are obtained by substituting Pb atoms with Ba atoms and the electronic structure of these virtual structures are investigated after the geometry optimization. The convergence tolerance of Ba₂[B₅O₉(OH)]·H₂O and Ba₃(B₃O₆)₂ are shown in ESI.† The calculated band gaps of Ba₂B₅O₉Cl, Ba₂[B₅O₉(OH)]·H₂O and Ba₃(B₃O₆)₂ are 5.315 eV, 4.958 eV and 4.549 eV, respectively. It is found that the gap difference between Pb₂B₅O₉Cl and Ba₂B₅O₉Cl is 1.956 eV. While the calculated band gaps of compounds Ba₃(B₃O₆)₂ and Ba₂[B₅O₉(OH)]·H₂O are close to that of Ba₂Pb(B₃O₆)₂ and BaPb[B₅O₉(OH)]·H₂O, respectively. Furthermore the bandgap of Ga₂[B₅O₉(OH)]·H₂O,³⁴ the isostructure of BaPb[B₅O₉(OH)]·H₂O, with an experimental band gap about 6.2 eV also can indirectly show the little redshift of BaPb[B₅O₉(OH)]·H₂O. This phenomenon indicates that introducing of lead in borates will not always reduce the band gap like most of lead borates do. To visually show the redshift of the cutoff edge, a factor Δλ = λ(I) − λ(II) is defined to clarify the redshift of the wavelength in which λ(I), and λ(II) are the wavelength cutoff edge of the one containing lead atoms (compound II) and its isostructural alkaline-earth borates (compound I). The results show that the shift Δλ from BaPb[B₅O₉(OH)]·H₂O to Ba₂[B₅O₉(OH)]·H₂O is 12.36 nm (GGA), and the redshift from Ba₂Pb(B₃O₆)₂ to Ba₃(B₃O₆)₂ is almost zero. While, for Pb₂B₅O₉Cl (3.359 eV), and its isostructural Ba₂B₅O₉Cl (5.315 eV),^25,35 the Δλ is as large as 135.79 nm. It is well known that the GGA-PBE in the standard DFT always underestimates the band gap.³⁶ To get reliable results, the bandgaps of BaPb[B₅O₉(OH)]·H₂O and the virtual compound Ba₂[B₅O₉(OH)]·H₂O were recalculated using the hybrid XC functional PBE0 again. And the bandgaps of BaPb[B₅O₉(OH)]·H₂O and Ba₂[B₅O₉(OH)]·H₂O calculated by PBE0 are 6.896 eV, 7.353 eV, respectively, and Δλ from BaPb[B₅O₉(OH)]·H₂O to Ba₂[B₅O₉(OH)]·H₂O is 11.17 nm, which has a similar tendency in band gap changes with GGA calculations. The calculated band gap for BaPb[B₅O₉(OH)]·H₂O by PBE0 is in accord with the experimental value.²⁴ The values of Δλ indicate that BaPb[B₅O₉(OH)]·H₂O and Ba₂Pb(B₃O₆)₂ have less red-shift than Pb₂B₅O₉Cl when barium is replaced by lead atoms in the compounds.

As BaPb[B₅O₉(OH)]·H₂O owns an acentric structure and large band gap, more attention will be paid on it and its isostructural Ba₂[B₅O₉(OH)]·H₂O. In order to further observe the influence of lead cation on the band gap, the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) (shown in Fig. 1 and 2) are analyzed. Fig. 1 shows that the oxygen atoms surround the lead atom are crucial for the HOMO of BaPb[B₅O₉(OH)]·H₂O, whereas, the contribution of the LUMO mainly comes from the hybrid orbitals of Pb, O9 and O11. Noting that O9 is terminal oxygen of BO₃ group, and O11 belongs to water molecules. Comparatively, in Ba₂[B₅O₉(OH)]·H₂O, the oxygen atoms at the LUMO are not only the O9, O11 but also O10, and the metallic cation barium is not the decisive element for LUMO. As for HOMO, compared with BaPb[B₅O₉(OH)]·H₂O, when the lead atom was substituted by the barium, it is observed that there is no contribution from the oxygen atoms O2, O7, O11 in Ba₂[B₅O₉(OH)]·H₂O. Thus, one can find that the Pb atoms affect the band gap, but such influence does not make the band gap of BaPb[B₅O₉(OH)]·H₂O red-shift too much. So what does this phenomenon happen? To address this issue, the structure and density of states (DOS) of three lead containing compounds are analyzed.

Fig. 1 The band structures (a) and the HOMO (b) and LUMO (c) orbitals of BaPb[B₅O₉(OH)]·H₂O.

Fig. 2 The band structures (a) and the HOMO (b) and LUMO (c) orbitals of Ba₂[B₅O₉(OH)]·H₂O.

Generally speaking, microscopic structures influence the macroscopic properties. To better understand the emergence of a large band gap, the coordination environment of the lead atom is firstly considered. Fig. 3 shows the coordination environment of Pb atom with the bond length less than 3.1 Å. The Pb₁ and Pb₂ of compound Pb₂B₅O₉Cl are both surrounded by seven oxygen atoms, which distributes similarly as a plane. And five of the oxygen atoms around Pb₁ show obvious directivity by one side. The PbO₈ polyhedron of BaPb[B₅O₉(OH)]·H₂O is falling with the sphere around lead. And the hexa-coordinated lead of compound Ba₂Pb(B₃O₆)₂ exhibits a symmetric distribution with no stereochemical active. It is known that stereochemical active cause distortion, to quantitatively analyze, the distortion of lead–oxygen polyhedron in compounds Pb₂B₅O₉Cl, Ba₂Pb(B₃O₆)₂, and BaPb[B₅O₉(OH)]·H₂O are calculated. The distortion value of two different lead atoms Pb₁ and Pb₂ of compound Pb₂B₅O₉Cl are 4.023 and 2.683. The distortion value of Ba₂Pb(B₃O₆)₂, BaPb[B₅O₉(OH)]·H₂O are 0.002, 0.982 respectively. The results show that the distortion of Ba₂Pb(B₃O₆)₂, BaPb[B₅O₉(OH)]·H₂O are much smaller than Pb₂B₅O₉Cl. The degree of the distortion of the lead–oxygen polyhedral in these lead borates is shown in Fig. 4, along with the redshift of the bandgaps of these compounds. As shown in Fig. 4, similar tendency can be found for the distortion of lead–oxygen polyhedron and the redshift of bandgap, indicating the more distortion value the lead–oxygen polyhedral are, the larger redshift the lead borate owns.

Fig. 3 The coordination of lead atom (the bond length of Pb–O is in the range less than 3.1 Å are drawn). (a) Pb₁ and (b) Pb₂ in Pb₂B₅O₉Cl, (c) BaPb[B₅O₉(OH)]·H₂O, (d) Ba₂Pb(B₃O₆)₂.

Fig. 4 The relationship between the distortion of Pb–O polyhedron and the redshift of band gaps of lead borates.

To better understand the internal mechanism of how the more largely distorted Pb²⁺ cause smaller gaps for lead borates, the electronic structures of these lead borates were then calculated. The calculated project density of states (PDOS) of Pb₂B₅O₉Cl, Ba₂Pb(B₃O₆)₂, and BaPb[B₅O₉(OH)]·H₂O are shown in Fig. 5. It is clearly shown that the bands spanning from about −9 eV to the Fermi level of these compounds are mainly derived from O 2p with small B 2p, B s components, and there are also some contributions from Pb s, Pb p states at the top of the valance band (VB) near the Fermi level. The states at the bottom of the conduction band (CB) are mainly the 2p states of lead. An overlap between Pb p and O 2p states was found at the energy region close to the Fermi level. It is clearly shown that the Pb atom plays a decisive role in determining the band gap of these three compounds.

Fig. 5 The PDOS of (a) BaPb[B₅O₉(OH)]·H₂O, (b) PbBa₂(B₃O₆)₂ and (c) Pb₂B₅O₉Cl.

On the other hand, according to the lone pair model,^37,38 due to the interaction of Pb and O, an mixture between Pb 6s and O 2p states exists, producing a filled antibonding state with some Pb 6s character at the top of the VB. Further mixing between the antibonding state and empty Pb 6p states is responsible for the distortion of Pb–O group. Therefore, the asymmetric electron density for a distorted Pb–O group arises predominantly from a mix of O 2p and Pb 6p orbitals at the top of the VB. At the top of the VB of BaPb[B₅O₉(OH)]·H₂O there is a small mixing between O 2p and Pb 6p orbitals which indicates a weak interaction between lead and oxygen. The PDOS of Ba₂Pb(B₃O₆)₂ at the top of VB has a similar tendency with BaPb[B₅O₉(OH)]·H₂O, there is also no obvious Pb 6p at the top of the VB in Fig. 5(b). In contrast, it shows that there is an apparent Pb p state with Pb s state at the top VB of Pb₂B₅O₉Cl (Fig. 5(c)). The strength of stereochemical lone-pair activity was then calculated by the contribution of Pb p state. The contribution ratio³⁹ of BaPb[B₅O₉(OH)]·H₂O, Ba₂Pb(B₃O₆)₂ and Pb₂B₅O₉Cl are 0.0399, 0.0277 and 0.1348 respectively, which indicates that compound Pb₂B₅O₉Cl has a stronger stereochemical activity. In addition, the electron localization function (ELF) of BaPb[B₅O₉(OH)]·H₂O are also calculated, Fig. 6 shows that the electron localization are non-spherical distributed around Pb but no asymmetric lobes on it, which proves that there is no obvious stereochemical lone-pair activity of BaPb[B₅O₉(OH)]·H₂O. Comparatively, compound Pb₂B₅O₉Cl has a larger lead–oxygen polyhedron distortion and distribution of O 2p state, but a smaller band gap. Comparing with the project density of states and distortion values, one can deduce that the larger the distortion of lead–oxygen polyhedral owns, the stronger interaction between lead and oxygen has. And because the bandgap is determined by the lead–oxygen states, the relatively strong interaction makes lead borates own smaller gaps. In a word the intensity of stereochemical activity is one reason leading to reduction of the band-gap.

Fig. 6 Electron localization function of BaPb[B₅O₉(OH)]·H₂O.

SHG effect

As BaPb[B₅O₉(OH)]·H₂O has a little redshift of UV cutoff edge due to the weak stereochemical activity of lead lone pair, the influence of lead to SHG response is investigated. Here the SHG coefficients are calculated firstly, the result shows 3.2 × KDP for the largest component, which is almost in accord with experimental value of 2.2 × KDP. To observe the SHG coefficient difference between BaPb[B₅O₉(OH)]·H₂O and Ba₂[B₅O₉(OH)]·H₂O, the SHG coefficient tensors of Ba₂[B₅O₉(OH)]·H₂O are also calculated, the maximal tensor element is 1.7 × KDP. It is also found that all the SHG coefficients are decreased when the lead atom is substituted by the barium atom (Table 1).

Table 1 The SHG coefficients (pm/V) of BaPb[B₅O₉(OH)]·H₂O(I) and Ba₂[B₅O₉(OH)]·H₂O(II)

	d11	d15	d12	d13	d24	d33
I	−1.26	−0.76	0.96	−0.66	−0.52	0.86
II	−0.34	−0.45	0.56	−0.01	−0.38	0.67

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The contribution of total SHG coefficients can be divided into two processes, virtual-electron and virtual-hole. The virtual-electron contribution to the total SHG coefficients of BaPb[B₅O₉(OH)]·H₂O are 76.91% (d₁₁), 73.38% (d₁₅), 84.87% (d₁₂), 58.13% (d₁₃), 84.51% (d₂₄), 114.8% (d₃₃). Here we will just show the SHG-density for virtual-electron process of the largest SHG tensors d₁₁ to discuss the original of the SHG. Fig. 7 illustrate that the oxygen atoms have a dominant contribution to the SHG coefficients for both virtual-electron occupied states and unoccupied states. And the Pb atom takes an important role in the virtual-electron contribution for unoccupied states. In Fig. 7(a), it is observed that the SHG density of the oxygen atom on occupied states concentrate on about 9.151 × 10⁻³, and the density of unoccupied states (Fig. 7(b)) shows that the density of Pb cations is approximate 2.231 × 10⁻³. Hence, the Pb cations contribute to SHG response but not the dominator.

Fig. 7 The SHG-density of the virtual-electron of the largest SHG tensors of BaPb[B₅O₉(OH)]·H₂O. (a) Occupied process (b) unoccupied process.

Based on the experimental values, it is suggested that the BaPb[B₅O₉(OH)]·H₂O has a relatively large band gap that can reach UV region. The SHG response of BaPb[B₅O₉(OH)]·H₂O also can meet the need of the UV application. Through theoretical calculation and simulation, the introduction of lead atom makes a little redshift of the cutoff edge of BaPb[B₅O₉(OH)]·H₂O but indirectly increases to the intensity of SHG response. This balance of band gap and SHG is related to the little stereochemical activity of the lead atom.

Conclusions

In conclusion, BaPb[B₅O₉(OH)]·H₂O, Pb₂B₅O₉Cl, Ba₂Pb(B₃O₆)₂, and their isostructual compounds “Ba₂[B₅O₉(OH)]·H₂O”, Ba₂B₅O₉Cl and “Ba₃(B₃O₆)₂” are studied mainly via DFT calculation. Analyzing the structure and electronic structure of these compounds, we found that the lead containing borates with little red-shift also has a small distortion and weak stereochemical activity. Futhermore, the SHG-density of BaPb[B₅O₉(OH)]·H₂O also shows that the lead atom has a contribution to its SHG response. Therefore, by controlling the intensity of stereochemical activity, one may obtain lead borates with large SHG response and wide transmittance or lead-free borates with moderate SHG response and wide transmittance. This balance of the band gap and SHG response can offer guidance to material design and experimental synthesis.

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. 11104344, 51425206, U1129301), the Western Light Joint Scholar Foundation Program of Chinese Academy of Sciences (Grant No. LHXZ201101), and the High-level Professional and Technical Personnel of Autonomous region.

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Footnote

† Electronic supplementary information (ESI) available: The convergence tolerance, calculated band structure, lattice parameters and optimized parameter. See DOI: 10.1039/c5ra13647d

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