Metal-free perovskites for non linear optical materials

Abstract

We identify the existence of non linear optical (NLO) activity in a number of novel ABX₃-type metal-free perovskites, where A is a highly tuneable organic cation, B is a NH₄ cation and X is a halide anion. Through systematic first-principles calculations, we identify important trends to chart the second-harmonic generation of this class of materials. We study three perovskites MDABCO–NH₄I₃, CNDABCO–NH₄I₃ and ODABCO–NH₄I₃ for use as deep-UV second-harmonic generation materials. We identify the role of the dipole moment imparted by the organic group on the A cation as an important parameter to tune the NLO properties of these materials. We apply this knowledge functionalising the organic group DABCO with the highly polar cyanide CN⁻ group, and we demonstrate a significant improvement of the NLO response in this family of materials. These findings can accelerate the application of metalfree perovskites as inexpensive, non-toxic, earth-abundant materials for the next generation of optical communication applications.

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1 Introduction

Light has been utilized as a communication device for many centuries. Recently, non-linear optics (NLO) and second harmonic generation (SHG) have been at the heart of several technological revolutions. With the advent of the internet, conveying information and data by means of fiber-optics and lasers have transformed telecommunications. The development of fiber-optic devices with increased performance has fuelled a surge of interest in the development of materials with ever-increasing data-transfer capabilities.^1–10 Materials, such as LiNbO₃ and LiTaO₃ with NLO and SHG are at the core of high-speed electro-optic modulator devices, significantly boosting the transmission capacities of the telecommunication infrastructure (∼10 Gbit s⁻¹).^1,11 In parallel NLO materials for deep-UV lasers are being used in semiconductor manufacturing, photolithography, laser systems, and advanced instrument development.

The demand for apparatus with increased performance requires the development of novel inexpensive NLO materials and every year the electronic and telecommunication industries demands the production of ∼40 000 tons of LiNbO₃. The soaring costs of lithium and niobium,^12,13 and thus LiNbO₃ requires the development of the new generation of NLO materials relying on more earth-abundant elements.^14–16 This is further aggravated by the fact that LiNbO₃ melts incongruently and the manufacturing of congruently lithium niobate single crystals requires cooling of Li-poor nonstoichiometric melts of LiNbO₃,¹⁷ adding additional production costs.

Meanwhile, for deep-UV applications Tran et al.⁹ have shown that only a handful of materials, such as KBe₂BO₃F₂, RbBe₂BO₃F₂ and CsBe₂BO₃F₂, sharing similar structural features currently fulfil the desired requirements. However, these materials contain toxic Be, whose usage is prohibited in many countries. Therefore, materials with better NLO characteristics are required to supplant deep-UV lasers.¹⁰ As identified by Halasyamani and Rondinelli,¹⁰ NLO materials for application in deep-UV lasers must fulfil a minimal number of requirements: (i) the atoms in the material should present a noncentrosymmetric arrangement, (ii) the material absorbs light in the deep-UV spectrum (i.e., absorption wavelengths ≥ 175 nm), and (iii) the material should respect the phase-matching criteria discussed in detail within the manuscript.

One promising route towards sustainable NLO materials lies in organic materials, which have shown promising SHG properties.^3,18–28 However, to date organic NLO suffer from thermal instability and difficult fabrication. In contrast to typical organic-based NLO, metal-free perovskites have been shown to be structurally stable beyond 200 °C.²⁹

In this study we present an in-depth analysis of the NLO properties of a new class of materials with ferroelectric response termed metal-free perovskites developed by Ye and co-workers.²⁹ We use first-principles calculations, based on density functional theory (DFT) to chart the optical and NLO properties of a these novel metal-free perovskites. In contrast to typical inorganic perovskite A²⁺B⁴⁺O₃²⁻ where A and B are metal cations (e.g., CaTiO₃), in metal-free perovskites the A and B cations are replaced by organic units. The rich choice of organic units introduces the possibility of tailoring the functional properties of such metal-free perovskites, while being easy-to-synthesise, affordable and non-toxic. Note that NLO materials with inorganic perovskite structures exist (e.g., K₃B₆O₁₀Cl) which are more closely related to the metal-free perovskites than organic-based NLO materials.³⁰

We investigate such metal-free perovskites as NLO materials for SHG applications, and verify whether these perovskites are suitable as deep-UV NLO materials. Our findings reveal that metal-free perovskites, based on the organic moieties N-methyl-N-diazabicyclo[2.2.2]octonium (MDABCO²⁺) and N-hydroxy-N-diazabicyclo[2.2.2]octonium (ODABCO²⁺), posses SHG response, with magnitudes similar to some inorganic contenders, such as KBe₂BO₃F₂, RbBe₂BO₃F₂ and CsBe₂BO₃F₂. We demonstrate that organic groups presenting intrinsic dipole moments can contribute positively to the ferroelectric response of the material, and provide an increased NLO response.

On the basis of our predictions, we provide guidelines to improve the SHG response of the metal-free perovskites, by tailoring the structural features of the organic cations. Following this principle, we extend the computational search to new structures. For example, we show the case where the hydroxy-group in ODABCO²⁺ (A-cation) is effectively replaced by a cyanide CN⁻ group with increased polarity, which provides remarkable NLO response. While preliminary observations²⁹ showed experimentally SHG response in MDABCO–NH₄I₃, we also identify ODABCO–NH₄I₃ and CNDABCO–NH₄I₃ as superior SHG materials, and their properties should be carefully verified experimentally.

2 Results

2.1 Structure of ferroelectric metal-free perovskites

The first point when considering the suitability of a material for NLO applications is the crystal structure. All materials reported herein and shown in Fig. 1 feature the typical perovskite ABX₃ structure (where A and B are cations and X are anions), with BX₃ corner-sharing octahedra and charge-balancing A cations in the cavities of the framework. The structures investigated have NH₄⁺ as B sites and halide X⁻ sites, as seen in Fig. 1. A series of different A site divalent molecular cations are considered (i) N-methyl-N-diazabicyclo[2.2.2]octonium (MDABCO²⁺), (ii) M-hydroxy-N-diazabicyclo[2.2.2]octonium (ODABCO²⁺), (iii) R-3-ammonioquinuclidinium (R-3AQ²⁺), and (iv) S-3-ammoniopyrrolidinium (S-3AP²⁺). Starting from the ODABCO–NH₄X structures, we also replaced the hydroxy group with a more polar cyanide CN⁻ group forming a new metal-free perovskite termed CNDABCO–NH₄X₃. In studying the halide chemical space of these metal-free perovskites, we proposed three new structures, including MDABCO–NH₄Cl₃, ODABCO–NH₄I₃ and S-3AP–NH₄I₃ that were not identified by Ye et al.²⁹ In the case of ODABCO–NH₄I₃ and CNDABCO–NH₄X₃ we assumed a R3 space group. However, we also computed the iodine-based polymorph with Pca2₁ space group (as in ODABCO–NH₄Cl₃), which converted to the R3 polymorph.

Fig. 1 Polyhedra representation of a metal-free ABX₃-type perovskite, where A can be one of the four organic cations (also shown by the red ball), B is NH₄⁺ (grey ball) and X the halide ion, i.e. Cl⁻, Br⁻ and I⁻ (purple balls). The nature of the A cation is also shown.

To be NLO active a material must be non-centrosymmetric. As shown in Tables S1 and S2 of the ESI,† the materials considered here fall into three space groups, R3, Pca2₁ and P2₁, all of which are non-centrosymmetric. For those metal-free perovskites that have been synthesized and characterized previously the non-centrosymmetric space groups persisting up to 448 K.²⁹

In addition, metal-free perovskites, and inorganic materials, such as LiNbO₃ posses an intrinsic polarization, which is typical of ferroelectric materials. LiNbO₃ charts among the most popular of NLO materials with a polarization of ∼70–75 μC cm⁻².³¹ The presence of dipolar molecules and the off-centring of the B-site cation in the metal-free perovskites indicates the existence of an intrinsic polarization, which for MDABCO–NH₄I₃ has been measured to be ∼19 μC cm⁻².²⁹

2.2 Birefringence activity of metal-free perovskites

Having established the crystallographic criteria for NLO materials, we now consider the optical properties that must be satisfied for SHG. The birefringence is the maximum difference refractive indices (n) which depends on the propagation directions of light in the crystal at a fixed frequency (ω) (as in eqn (1)).

Δn = n_max(ω) − n_min(ω) (1)

SHG occurs efficiently whenever the refractive index n of the 2^nd harmonic is equal (or close) to that of the generating wave at half-frequency, i.e., n(2ω) = n(ω); this condition is termed phase-matching. Phase-matching requires a suitable range of frequencies with refractive indexes, n_max(ω) − n_min(2ω) > 0, with n_min and n_max the lowest and the largest refractive index in the spread.³²

If Δn of a material is too small the phase-matching condition for SHG will not occur. If Δn is too large the material will exhibit spatial beam walk-off, where the intensity distribution of the wave drifts away from the direction of propagation resulting in reduced SHG intensity. In general a moderate birefringence (Δn ≈ 0.07) is desired.

In uniaxial systems, such as LiNbO₃ and the MDABCO–NH₄X₃ systems the direction of ordinary and extraordinary rays occurs along the optical axis, which lays along the highest symmetry axis.

In contrast, biaxial systems have multiple optic axes, whose directions of propagation depend on the diffraction index and measured by the 2V acute angle (see below).³² Therefore, the 2V angle provides information about the nature of the birefringence in biaxial crystals. In general, orthorhombic, monoclinic and triclinic systems are biaxial crystals.

Fig. 2 plots Δn (obtained from the dielectric constants in the static regime) and the 2V acute angle. From Fig. 2, we observe that all the systems considered here have Δn ranging between values of −0.03 and +0.05, with CNDABCO, ODABCO, S-3AP and R-3AQ-based perovskites charting among the highest in magnitude. The R3 materials, which are uniaxial NLO systems, all exhibit positive birefringence. In contrast, the Pca2₁ and P2₁ systems, which are biaxial NLO materials, display both positive and negative birefringence. A closer look at Fig. 2 shows that with the exception of MDABCO-based perovskites and S-3AP–NH₄I₃ (whose 2V ∼ 0), the remaining materials all show a rather complex biaxial response.

Fig. 2 Computed birefringence Δn and 2V angle of metal-free perovskites at their experimental volume and lattice constants. Δn is plotted on the radial axis and using the colour-bar, with red indicating positive Δn. Values of n are given in Table S3 of the ESI.† The 2V angles provide information about the nature of the birefringence in biaxial crystals.

2.3 NLO response in metal-free perovskites

We now discuss the NLO response of the metal-free perovskites. Among the prerequisites for SHG the materials under investigation should display a non-negligible value of the second order dielectric tensor, χ⁽²⁾.

χ ⁽²⁾ _ijk = 2d_ijk is a third-rank tensor as defined in eqn (6), and therefore can be difficult to analyse intuitively. The i, j and k components of the d tensor identify the directions of the applied electric fields (i.e., j and k) of the incident radiation and the polarization of the generated second harmonic (i.e., i), respectively, (see eqn (6)). In SHG experiments, the directions of the electric fields of incident radiations have frequencies ω₁ and ω₂ (with ω₁ = ω₂ in SHG) and the second harmonic wave with frequency ω₃ = ω₁ + ω₂. A complete discussion of the χ⁽²⁾ and d tensors and their dependence can be found in ref. 33.

A useful proxy for presenting the ijk component of the static χ⁽²⁾ tensor is the norm of the d_ijk components of Table 1 and eqn (2):

d_norm has been utilized previously^16,34 and is more effective of a simple geometric mean of the d_ijk components, which can assume values approaching numerical zeros given the oscillations in sign of the d_ijk values (see Table 1). We note that the values d_norm are by definition larger in magnitude of any component d_ijk, but here d_norm serves us simply to compare each materials investigated on the same footing.

Table 1 Computed static d_ijk components of the χ⁽²⁾ tensor and their norm d_norm (in pm V⁻¹) of the metal-free perovskites at the experimental volumes. Spg. is the space group

X^−	Spg.	d xxx	d xxy	d xxz	d xyy	d xyz	d xzz	d yyy	d yyz	d yzz	d zzz	‖dnorm‖
MDABCO–NH 4 X 3
Cl	R3	0.140	−0.410	0.048	−0.140	0.000	0.000	0.410	0.048	0.000	−0.490	0.79
Br	R3	0.230	−0.620	−0.058	−0.230	0.000	0.000	0.620	−0.058	0.000	−0.580	1.10
I	R3	−0.170	1.200	0.430	0.170	0.000	0.000	−1.200	0.430	0.000	0.021	1.80
ODABCO–NH 4 X 3
Cl	Pca21	0.000	0.000	0.230	0.000	0.000	0.000	0.000	−0.076	0.000	−0.076	0.25
Br	R3	0.019	0.510	−0.320	0.150	−0.290	−0.069	0.099	−0.100	−0.280	−0.230	0.79
I	R3	0.270	0.620	−0.670	−0.190	−0.400	0.100	−0.840	−0.490	−0.280	0.007	1.50
CNDABCO–NH 4 X 3
Cl	Pca21	0.000	0.000	0.650	0.000	0.000	0.000	0.000	−0.160	0.000	0.075	0.68
Br	R3	−0.450	0.850	0.260	0.230	0.120	0.092	−0.880	0.350	0.071	−0.590	1.50
I	R3	−0.450	1.310	0.760	0.450	−0.001	0.0006	−1.310	0.760	0.004	−0.014	2.20
R-3AQ–NH 4 X 3
Cl	P21	0.000	0.091	0.000	0.000	0.068	0.000	0.0121	0.000	−0.160	0.000	0.19
Br	P21	0.000	−0.075	0.000	0.000	0.004	0.000	−0.320	0.000	0.400	0.000	0.52
S-3AP–NH 4 X 3
Cl	P21	0.000	−0.025	0.000	0.000	−0.041	0.000	−0.160	0.000	−0.150	0.000	0.22
Br	P21	0.000	−0.029	0.000	0.000	−0.055	0.000	−0.120	0.000	−0.048	0.000	0.14
I	P21	0.000	−0.210	0.000	0.000	−0.075	0.000	−0.140	0.000	0.029	0.000	0.26
LiNbO 3
—	R3c	0.170	0.000	5.000	0.170	0.000	0.000	0.000	5.000	0.000	16.450	17.91

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Fig. 3 presents the static dielectric constant ε (y-axis) as a function of the band gap (x-axis) for the materials studied, the data points are coloured according to the magnitude of d_norm, which are obtained at the experimental volumes at room temperature. The values of band gap, ε, χ⁽¹⁾, χ⁽²⁾_ijk and d_norm of these materials are found in Tables S4 and S5.†

Fig. 3 Computed static d_norm (colour-bar) as function of the band gap (x-axis, eV) and average dielectric constant (y-axis) of the metal-free perovskites at the experimental volume. The ε values (in the static limit) computed for the metal-free perovskites are normalized against the LiNbO₃ (∼4.39). Similarly, the values of d_norm are rescaled to that of LiNbO₃ (∼17.91 pm V⁻¹). The space group of each structure is also reported.

In general, volumes obtained after relaxation in hybrid systems containing a number of van der Waals interactions, such as these metal-free perovskites, tend to be highly distorted. Therefore, the main text concentrates on results obtained on structures where all the atomic coordinates are relaxed at the experimental volumes and lattice constants, whereas results on fully relaxed structures are available in the ESI.† For clarity and ease of comparison, the dielectric constant and the d_norm are scaled by that of LiNbO₃, calculated at the same level of theory (Tables S4 and S5†).

The y-axis of Fig. 3 presents the average dielectric constant (in the static limit) of the metal-free perovskites rescaled to that of LiNbO₃ ∼ 4.39, which compare well with experimental data (∼4.87).³¹ The computed d_zzz values of 16.45 pm V⁻¹ (at the experimental volume of LiNbO₃) slightly underestimated the experimental value of (∼22.0 pm V⁻¹ at λ ∼ 407.2 nm) by Magel et al.³⁵ DFT has been known to systematically underestimate values of d in NLO materials.³⁶ The complete dielectric and d tensors of each metal-free perovskites are reported in the ESI.†

The computed band gaps of Fig. 3 are characteristic of high-gap insulators ranging between ∼5.5 and 8 eV. The metal-free perovskites absorb light between 155 and 220 nm, i.e. the deep-UV portion of the electromagnetic spectrum, meaning that these materials fulfil another of the criteria for deep-UV SHG applications. All computed d_norm values in metal-free perovskites with the same organic A-site cation follow the order I > Br > Cl. The order of increasing d_norm moving down the halide group follows the trend in band gaps. We find that the computed d_norm of all the metal-free perovskites is a fraction (ranging between 1% and 10%) of the d_norm of LiNbO₃ (Fig. 3), which is used as a reference.

From these data, we identify two iodine-based structures, MDABCO–NH₄I₃ and ODABCO–NH₄I₃ as the best NLO materials across the metal-free perovskites investigated. When the OH⁻ group in ODABCO–NH₄X₃ is replaced by a polar CN⁻ group (see Fig. 3), forming CNDABCO–NH₄X₃, we find improved NLO properties. In CNDABCO–NH₄X₃, the computed d_norm of ∼2.23 pm V⁻¹ outperforms all other metal-free perovskites, suggesting that CN-substituted A cations offer a promising strategy to improve the optoelectronic properties of these materials.

Fig. 3 shows that both R-3AQ–NH₄X₃ and S-3AP–NH₄X₃ posses small values of d_norm, covering a narrow range between ∼0.14 and ∼0.52 pm V⁻¹, with the highest value set by R-3AQ–NH₄Br₃ and the lowest by S-3AP–NH₄Br₃. Hence, both R-3AQ–NH₄X₃ and S-3AP–NH₄X₃ are expected to show low SHG behaviour and therefore, will not be considered further in this analysis.

3 Discussion

We have identified that MDABCO–NH₄I₃, ODABCO–NH₄I₃ and CNDABCO–NH₄I₃ show the largest NLO response. Our results show the largest components of the d tensor in MDABCO–NH₄I₃ are d_xxz = d_yyz ∼ 0.43 pm V⁻¹ and d_xxy ∼ 1.20 pm V⁻¹, respectively, d_yyy ∼ 0.84 pm V⁻¹ in ODABCO–NH₄I₃ and d_yyy ∼ 1.31 pm V⁻¹ in CNDABCO–NH₄I₃, which represent three promising NLO materials with good SHG. Therefore, the discussion will only focus on the subset of metal-free perovskites showing significant SHG activity, including CNDABCO–NH₄I₃, ODABCO–NH₄X₃ and MDABCO–NH₄X₃.

The values for SHG are similar and in most cases even larger than the prescribed minimal conditions for deep-UV applications, i.e. d_xyz ∼ 0.39 pm V⁻¹ (with λ ∼ 1.064 μm) KH₂PO₄.^9,37 The calculated band gaps indicate that while the materials absorb in the 2ω range required for SHG, they do not absorb in the region ω, a critical consideration, as absorption at ω would lead to material damage in operation. It is important to note that in a operation scenario the light absorbed will enhance the susceptibility but will also reduce the emission, thus the calculate values of d_xyz serve to demonstrate the very promising properties of these materials for NLO and SHG applications, but should be considered as guides, rather than definitive numbers of real device operation. Most importantly, the composition–structure–property relationships driving NLO response in these test systems can provide rules for designing even more SHG active materials.

As seen in Fig. 3, the primary factor influencing the size of d_norm is the magnitude of the band gap. This is a well established trend in semiconductor materials in general and is related to lower carrier concentrations in wider gap materials at finite temperatures. A correlation between band gap and the halide species exists, we find a similar correlation in the values of the static dielectric constants vs. the anion species, following the trend ε(I⁻) > ε(Br⁻) > ε(Cl⁻). This observation is consistent with the density of states of Fig. 4 of ODABCO–NH₄Br₃ (top), MDABCO–NH₄Br₃ (middle) and MDABCO–NH₄I₃ (bottom), respectively, which shows the valence band dominated by halide species and the conduction band populated by the A organic cation, i.e., ODABCO²⁺ or MDABCO²⁺. As the frontier valence orbitals are primarily halide in character (Fig. 4), it is not surprising that the largest change in band gap is controlled by halide exchange. These findings are also supported by the DOS presented in ref. 38 and are consistent with observations in hybrid halide perovskites.³⁹

Fig. 4 (a) Total and projected density of states of ODABCO–NH₄Br₃, MDABCO–NH₄Br₃ and MDABCO–NH₄I₃ obtained with PBESol0-D3. The red line (dash) is the Fermi energy, which is set at the top of the valence band of each material. (b) Variance of dipole moment for DABCO cations substituted with a number of organic groups (functions).

There is a secondary contribution to the d_norm arising from the nature of the organic cation at the A-site. As seen in Fig. 3, d_norm follows the order d_norm(CNDABCO²⁺) ≫ d_norm(MDABCO²⁺) > d_norm(ODABCO²⁺). However, this trend becomes d_norm(CNDABCO²⁺) ≫ d_norm(ODABCO²⁺) > d_norm(MDABCO²⁺) when the structures are fully relaxed (see Table S5†). This is despite the fact that both CNDABCO²⁺ and ODABCO²⁺ systems have larger dielectric constants and smaller band gaps than the MDABCO²⁺ counterparts. As an example, while the ODABCO²⁺ cation is structurally very similar to the group MDABCO²⁺ (see Fig. 1), the apical methyl group in MDABCO²⁺ is substituted by a polar OH⁻ in ODABCO²⁺. The presence of polar moieties in the organic cations, such as the CN⁻ and OH⁻ groups in CNDABCO²⁺ and ODABCO²⁺, can result in a greater intrinsic electric dipole moments, whose polarization induces an increase of the dielectric constant. The intrinsic electric dipole moment introduced by the CN⁻ or the OH⁻ groups increase the overall dipole of CNDABCO²⁺ from ∼3.61 Debye to ∼7.64 Debye, compared to MDABCO²⁺ ∼ 1.64 Debye, which is reflected by the values of dielectric constants of Fig. 3.

Thus, the highly modular nature of the A site cation, in particular the DABCO group, afforded by a metal-free scaffold can be exploited to fine-tune the NLO response of these perovskites. For example, the substitution of the CH₃ group on MDABCO with highly polar organic groups, such as –SH (thiol), –NH₂ (amine), –NO (nitroso) and –NO₂ (nitro), whose computed electric dipole moments are shown in Fig. 4b, represents a viable strategy to increase the NLO response of these metal-free perovskites. Indeed, in Fig. 2 we have demonstrated that replacing the CH₃ (or OH) with the CN moiety, increases substantially the values of d_norm, thus providing a clear design rule to design novel metal-free perovskites.

The properties and tunability demonstrated in metal-free perovskites are comparable to organic-based NLO materials. Organic NLOs can supersede the performance in terms of descriptors, χ⁽²⁾ (see eqn (6)), even by 1–2 orders of magnitude of the inorganic NLO materials discussed so far.^3,18–28 Some of these materials are even commercial.^3,40,41 For example, the 4-N,N-dimethylamino-4′-N′-methyl-stilbazolium tosylate (DAST) forms organic crystals with a superior non-linear susceptibility d ∼ 580 pm V⁻¹ (with d = 0.5χ⁽²⁾).^21,25 However, organic-based NLOs display at least three major limitations curbing their use in optical devices:^23,27 (i) difficult to obtain sufficiently large-size crystals, (ii) limited thermal stability above 200 °C and (iii) low mechanical strength. For example, while a good NLO organic material, such as the 3-methyl-4-methoxy-4′-nitrostilbene has a 1^st hyperpolarizability β (see eqn (6)) approximately 300 times larger than the inorganic KH₂PO₄, it melts at 109 °C.²³ Metal-free perovskites also offer the potential to overcome these challenges.

All of the above trends are also dependent on crystal structure and therefore a further understanding of how composition and structure are related is required to achieve truly targeted synthesis. In systems like these metal-free perovskites an interplay of weak forces, such as dispersion, hydrogen bonding and entropy will be important for driving structure and phase transitions, as exemplified in the hybrid halide perovskites.^42,43

4 Conclusions

Using first-principles calculations, we have demonstrated the existence of NLO activity in a number of novel metal-free perovskites. In the past, density functional theory has been shown to systematically underestimate the values of χ⁽²⁾_ijk. While we believe the superior quality of hybrid DFT/PBESol0-based simulations may still underestimate χ⁽²⁾, our data show that MDABCO–NH₄I₃ and ODABCO–NH₄I₃ both fulfil the criteria to useful as deep-UV second-harmonic generation materials. We identify the role of the dipole moment imparted by the organic group on the A cation as an important parameter to tune the NLO properties of these materials. We apply this knowledge introducing the cyanide CN⁻ group on the DABCO–NH₄X₃ structure, improving significantly the NLO properties of metal-free perovskites. Furthermore our systematic calculations allows us to extract useful trends to chart the second-harmonic generation properties of this novel class of materials. We have shown that the material band gap, and hence the d_norm can be tuned by altering the halide anion. We also show the selection of the A-site cation provides an extra degree of tuning the χ⁽²⁾. By combining engineering of both the A- and X-sites we believe that these findings provide a blueprint for how to achieve high non-linear optics and second-harmonic generation activity in the desired part of the electromagnetic spectrum. We hope that these findings will help to accelerate the application of metal-free perovskites as non-toxic, earth-abundant materials for the next generation of optical communication applications.

5 Linear and non-linear optic effects

Materials respond differently to incident electromagnetic waves, whose induced electric polarization P follows:³³where E is the electromagnetic field, ε₀ is the vacuum permittivity, and χ⁽¹⁾, χ⁽²⁾ and χ⁽ⁿ⁾ are the 1^st, 2^nd and n^th order electric susceptibilities. The dependence of the polarization on the 2^nd order term (P⁽²⁾) and successive electric susceptibilities suggests that the response of a permanent dipole in non-linear optical materials assumes an anharmonic behaviour once the electric dipole is perturbed by an incident electromagnetic radiation.³³

Here, we assess the linear and NLO properties up to χ⁽²⁾ in the static regime, and accurately derived from first-principles calculations using the coupled perturbed Kohn–Sham theory (CPKS).^44,45 We compute the n^th-order derivatives of the total energy E_Tot. with respect to derivatives of the electric field components in [small epsilon, Greek, vector]_ijk (i, j and k are the Cartesian direction of the electric field), which are cast in the form of order-n + 1 tensors. Such derivatives link to important optical descriptors, such as the electric dipole moment μ_i and the polarisability, α_ij,

From α_ij, the components of the dielectric tensor ε are derived as with ε₀ the vacuum permittivity and V the unit-cell volume. We note that ε = n² = 1 + χ⁽¹⁾, with n the refractive index. n and Δn (of eqn (1)) are computed directly from the dielectric tensor.

The third-order rank tensor χ⁽²⁾_ijk relates to 1^st hyperpolarizability β_ijk.

where j and k are the directions of the incident waves and i the direction of the SHG wave. From eqn (3), χ⁽²⁾_ijk links directly to the electric field that in materials is subjected to the spatially uniform electric fields of the incident radiations, and reads as:where χ⁽²⁾_ijk is a component of the χ⁽²⁾ tensor proportional to the polarization generated along the i-axis (e.g., x), from the j and k (e.g., y and z) components of the electric fields of the incident radiations oscillating at frequencies ω₁ and ω₂, respectively. In SHG materials ω₁ = ω₂.

Typically, χ⁽²⁾_ijk is reported as d_ijk = 0.5χ⁽²⁾_ijk. Note that for hybrid functional calculations—the choice in this study—CRYSTAL17 computes only β_ijk, χ⁽²⁾_ijk and d_ijk values in the static limit.

6 First-principles calculations

DFT is used to investigate the structural and optical properties of the metal-free perovskites. Structure relaxations of the experimental coordinates of the metal-free perovskites was performed using the PBESol⁴⁶ functional as in VASP.^47–49 The Grimme D3 (ref. 50) correction was added to capture dispersive forces. The wave functions of the valence electrons were expanded with plane-waves with a cutoff of 520 eV, and core electrons were treated with the projector augmented wave theory.⁵¹ The Brillouin-zone was integrated on a mesh with reciprocal density of 64 k-points per Å⁻¹. The total energy was converged within 10⁻⁵ eV and forces to 10⁻² eV Å⁻¹.

An accurate treatment of the electronic structure and in particular the assessment of the optical band gaps is crucial for the description of NLO properties.^8,10,52 Starting from the PBEsol-D3 structures with VASP, we re-optimised the coordinates (or volume and coordinates) with the hybrid functional PBE0Sol-D3,^46,53 including 25% of exact exchange as available in CRYSTAL17.^54,55 We exploited the all-electron linear combination of atomic orbitals of CRYSTAL17 expanded by consistent triple-ζ plus polarization basis-sets, see ref. 56 and 57. Given the electronic configuration of I ([Kr]4d¹⁰5s²5p⁵) a fully-relativistic pseudo-potential is used.⁵⁸ The total energy was converged within ∼3 × 10⁻⁹ eV and integrated over a well converged and symmetrized 4 × 4 × 4 k-point mesh (i.e., the shrinking factor is set to 4). The truncation of the (infinite) Coulomb and exchange series was set by the tolerances (TOLINTEG): 10⁻⁷, 10⁻⁷, 10⁻⁷, 10⁻⁷ and 10⁻³⁰. Tables S1 and S2 of ESI† show the PBE0Sol+D3 geometries. We computed the non-linear-optical properties with these settings. The iterative solution of the CPKS equations is reached for values below 10⁻⁴. Table S6† shows a comparison of the performance of a number of DFT functionals.

The dipole moments of the DABCO units substituted by the organic groups were computed with Gaussian16 (ref. 59) using the PBE0+D3 (ref. 53) functional and the same basis-set used in the periodic calculations with CRYSTAL17.

Author contribution

P. C. and K. T. B. conceived the manuscript, performed the data analysis and wrote the first draft. T. W. K., A. M. M., C. H. H. and K. T. B. performed the initial VASP calculations. P. C., K. T. B. and Z. D. performed the CRYSTAL and Gaussian calculations. All authors contributed to the revision of the manuscript.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

P. C. acknowledges support from the Singapore Ministry of Education Academic Fund Tier 1 (R-284-000-186-133). The computational work for this article was partially performed on resources of the National Supercomputing Centre, Singapore (https://www.nscc.sg). This work benefited from access to the University of Oregon high performance computer, Talapas. P. C. acknowledges Dr Y. Wu and Prof. A. K. Cheetham for useful discussion.

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Footnotes

† Electronic supplementary information (ESI) available: Structural properties, band gaps, dielectric constants, refractive indexes values of χ⁽¹⁾, values of χ⁽²⁾ and d_norm, respectively. See DOI: 10.1039/c9sc03378e

‡ These authors contributed equally.

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