Bi₃TeBO₉: electronic structure, optical properties and photoinduced phenomena

BTBO powders were synthesized by means of conventional solid-state reaction route, similarly to the procedure described by Daub et al. [6]. The molar ratio of used precursor compounds: Bi₂O₃ (SIAL, 99.95%), TeO₂ (MERCK, 99.99%) and B₂O₃ (SIAL, 99.999%), was 3:2:1.1, respectively. The 5% excess of boron oxide was used to prevent the non-stoichiometry due to the presence of water and possible evaporation of B₂O₃ at elevated temperatures. The precursor compounds were mixed and thoroughly ground in an agate mortar till the homogeneous mixture was obtained. It was initially preheated at 450 °C for 12 h to evaporate the residuals of adsorbed water. In the following steps, the mixture was ground and calcined three times at 690 °C for 12 h each time. After the final grounding, the pure, white powders of BTBO were obtained. Their phase uniformity was confirmed using the XRD analysis. The diffraction pattern of synthesized BTBO powder was collected using the BRUKER D8 Discover, equipped with standard CuK _α radiator (λ _Kα1  = 1.54056 Å, λ _Kα2  = 1.54439 Å, Siemens KFL CU 2 K, 40 kV voltage and 40 mA current in operating mode). The radiator was equipped with Göbel FGM2 mirror. The Bragg–Brentano diffraction geometry was applied. The diffraction angle 2θ _B range was set to 10–70° with a step of 0.015° and acquisition time 2 s per step. The measurements were done at 25 °C in the temperature-stabilized Anton Paar HTK 1200N chamber. For the data processing, the DIFFRAC.SUITE EVA application was used. The λ _Kα2 signal component was deconvoluted and removed, the background was subtracted and the data were smoothed using fast Fourier transform (FFT). The phase analysis was performed with support of Crystallography Open Database (COD). For crystallographic calculations, the MAUD ver. 2.32 program was used, allowing to perform the Rietveld refinement and calculate the precise unit cell parameters. The reference CIF file used for this analysis was received from the supplementary data given in Ref. [7]. The obtained BTBO powder diffraction pattern is shown in Fig. 1.

Calculations of the electronic, optical and elastic properties of BTBO crystal were performed within the framework of density functional theory (DFT) implemented in CASTEP (Cambridge Serial Total Energy Package) program—package of Biovia Materials Studio 8.0 [8]. The initial BTBO crystal structure was taken from Ref. [7]. We applied the plane-wave (PW) basis set. Eigen function and the eigenvalues were estimated from self-consistent solving of the one-electron Kohn–Sham equation. Exchange and correlation effects were taken into account by using local-density approximation (LDA) with Ceperley–Alder–Perdew–Zunger parameterization [9, 10] and generalized gradient approximation (GGA) together with Perdew–Burke–Ernzerhof parameterization [11]. The cutoff energy of plane-wave basis was set to be equal to E _cut = 830 eV. The Monkhorst–Pack k-points grid sampling was set at 2 × 2 × 2 points for the Brillouin zone used for electronic charge density calculations and geometry optimizations, respectively. Both the cell parameters and atomic positions were optimized (relaxed) using Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm [12], until the force on each ion was smaller than 0.01 eV/Å. The resulting structures were then used as input to the electronic structures and related properties calculations.

The core electron orbitals were calculated using the norm-conserving pseudopotential (NCPP). The following electron configurations were used: B—2s ² 2p ¹, O—2s ² 2p ⁴, Te—5s ² 5p ⁴, Bi—5d ¹⁰ 6s ² 6p ³. The convergence parameters were as follows: total energy tolerance 5 × 10⁻⁶ eV/atom, maximum force tolerance 1 × 10⁻² eV/Å, maximum stress component 0.02 GPa and maximum displacement 5.0 × 10⁻⁴ Å. The band structure was calculated at the following points and along directions of high symmetry of BZ: Г → A→ H → K →Г → M → L → H (see Fig. 2).

The crystal structure of BTBO crystal is presented in Fig. 3. BTBO belongs to hexagonal symmetry (space group P6₃, no. 173) with Z = 2. The experimental and theoretically calculated principal band structure parameters, using DFT-LDA and DFT-GGA exchange-correlation functionals, together with NCPP, after the optimization of lattice geometry are presented in Table 1. Following Table 1, the calculated lattice constants are in good agreement with the experimental ones. Lattice constants calculated using the LDA functional are a bit smaller than the experimental ones, while in the case of GGA functional, these parameters are a bit higher. It can be explained by the renormalization of principal electronic parameters within the LDA functional, which leads to underestimation of basic structural parameters.

For calculations of electronic structure and other properties, the optimized crystal structure was used. Calculated band structure of BTBO crystal using LDA and GGA functionals for exchange–correlation effects description is shown in Fig. 4. Band structure is plotted at following BZ line directions: Г → A → H → K → Г → M → L → H. One can see that the band structure has a large dispersion. The lowest dispersion is observed for A → H and L → H BZ directions. Valence band maxima (VBM) and conduction band minima (CBM) are in the segment between the Г and A points (Fig. 4) and are shifted in relation to each other. Thus, band gap of the titled crystal is of the indirect type. The smallest band gap value is equal to 2.96 eV for LDA and 3.21 eV for GGA, respectively (Table 2), while as previously reported by Xia et al. [7], the experimental and calculated values were 2.70 and 3.23 eV, respectively. So, using both LDA and GGA together with NCPP, one can perfectly describe the model of BTBO crystal, since they give good agreement with the experimental values of the band gap and lattice constants. Density of states (DOS) and partial density of states (PDOS) charts give a quick qualitative picture of the crystal electronic structure. The calculated total and partial density of states for BTBO crystal are depicted in Fig. 5.

In the spectral energy range within −15 to 10 eV, one can see two distinctive groups of peaks corresponding to principal electronic states of BTBO crystal. The first group corresponds to the energy states within the range of −10.6 to 0 eV. The 6s and 6p electrons of bismuth form bands in the range from −10.6 to −7 eV and from −6.5 to 0 eV. Also, a minor peak of 6s electrons is observed near 0 eV. Tellurium electrons form several peaks for 5s electrons, which are situated nearby −10 eV and at −4 eV, while the 5p electrons form peak at −8 and −1 eV, respectively. In the case of boron, the 2s peak at −6 eV is spectrally shifted toward relatively higher energies when compared to the peak formed by tellurium. Wide band originating from −6.3 to −1 eV is formed by 2p electrons of boron. The electrons of oxygen atoms, originating from bands that are situated in the range from −10.3 to 0 eV, correspond to 2p subshell. As shown in Fig. 5, the top of the valence band originates from oxygen 2p states, bismuth 6s and tellurium 5p states. The bottom of the conduction band is formed by p states of oxygen and s states of tellurium. It can be assumed that the optical transitions in absorption spectra are likely formed by oxygen 2p to bismuth 7s or 7p states.

Additional important information can be obtained from atomic charge bond population. In contrast to plane-wave basis, a linear combination of atomic orbitals (LCAO) basis set provides a natural way of specific quantities such as atomic charge, bond population and charge transfer. Population analysis in CASTEP is performed using a projection of the PW states onto a localized basis using a technique described by Sanchez-Portal et al. [13]. Population analysis of the resulting projected states is then performed using the Mulliken formalism [14]. This technique is widely used for the analysis of electronic structure calculations performed within a framework of LCAO basis sets. Calculated atomic populations, in units of the proton charge for BTBO crystal are presented in Table 3. One can see that the obtained charges calculated using LDA functional are close to the calculated using GGA.

The shortest bond lengths between constituent atoms in BTBO crystal, calculated using LDA and GGA functional together with overlap populations, are collected in Table 4. As can be seen, the calculated bond lengths and population values are in good agreement between one to another. All bonds of the titled crystal are of covalent type. It is easy to see that B–O bonds possess the highest degree of covalency, while Bi–O—the lowest one. Based on these results, we can conclude that BTBO crystal possesses covalent type of chemical bonding.

The optical properties of solids are described by their dielectric function ε = ε ₁ + iε ₂, which real part ε ₁ corresponds to the refractive indices of the material, while its imaginary part ε ₂ describes the material absorption coefficients, which are directly related to the absorption spectra. The imaginary part of complex dielectric function ε ₂, which corresponds to the absorption, can be obtained from the inter-band transition by numerical integration in k-space of elements of dipole matrix operator, between the filled VB states and empty CB levels (Eq. 1):where E—energy of probing optical beam; u—incident photon polarization vector; \( \psi_{k}^{c} \) and \( \psi_{k}^{v} \)—wave functions of the conduction and valence bands in k-space, respectively; Ω—unit cell volume; e—electron charge; \( \varepsilon_{0} \)—dielectric permittivity of vacuum; r—operator of electron position; and ω—frequency of incident radiation.

Using Kramers–Kronig relation (Eq. 2) [15] from the imaginary part of dielectric function, its real part can be estimated as follows:

Figure 6 depicts the dispersion of real ε ₁ and imaginary ε ₂ parts of dielectric functions calculated using GGA functional. The crystals of hexagonal symmetry have two principal directions that correspond to the direction along polar z-axis and direction perpendicular to z. In Fig. 6, we present the calculated dielectric function, which corresponds to photon polarization E || z and \( E\; \bot \;z \). The dielectric function shows insignificant anisotropy for BTBO crystal. The real part of dielectric function in the region from 7.5 to 18 eV has inversion of sign that can correspond to metallic reflection.

In terms of crystal anisotropy studies, elastic properties of the crystal can be of special interest. The optimized crystal structure of investigated compound as input for the elastic constant simulations was used. The elastic constants are defined by means of the Taylor expansion of the total system energy E(V, δ), with respect to a small lattice strain δ of the primitive unit cell with volume V. The energy of a strained system is expressed as follows [16]:where E(V ₀, 0) is the energy of the unstrained system with equilibrium volume V ₀, \( \tau_{0} \) is an element in the stress tensor and \( \xi_{i} \) is a factor to take care of Voigt index.

For the hexagonal crystal, its elastic properties can be described by asymmetric 6 × 6 matrix of elastic coefficients. In case of hexagonal symmetry, we get the following relation between elastic constants: C ₁₁ = C ₂₂; C ₁₃ = C ₂₃; C ₄₄ = C ₅₅; C ₆₆ = 1/2(C ₁₁−C ₁₂); it is known that for practical application it is important to obtain a stable crystal structure. Knowing of the elastic constants and Young’s modulus is important for technological aims, in the manufacturing process and crystal preparation. For crystal structure stability of hexagonal symmetry, the obtained elastic constants should satisfy the requirements of mechanical stability criteria [17], which are: C ₃₃ > 0; C ₄₄ > 0; C ₁₂ > 0; C ₁₁ > |C ₁₂|; (C ₁₁ + 2C ₁₂)C ₃₃ > 2C ₁₃ ² ).

Following Table 5, the calculated elastic constants completely satisfy the mechanical stability criteria. Unfortunately, according to already published data concerning the BTBO crystal, the experimental results of its mechanical properties are not yet known. The values of C ₁₁ and C ₃₃ differ one from another, indicating the anisotropy of elastic properties. The universal anisotropy index for BTBO crystal is equal to A ^U  = 0.412/0.254 from the calculation using GGA/LDA functionals. Calculated elastic compliance matrix S _ij for BTBO crystal is presented in Table 6.

The Young’s modulus components \( E_{x} \), \( E_{y} \) and \( E_{z} \) can be estimated from elastic compliance S _ij using following relations: \( E_{x} = S_{11}^{ - 1} \), \( E_{y} = S_{22}^{ - 1} \), \( E_{z} = S_{33}^{ - 1} \). The axial compressibilities k _a , k _b and k _c for hexagonal BTBO crystal are also calculated in the following way [18]: \( k_{a} = S_{11} + S_{12} + S_{13} \), \( k_{b} = S_{12} + S_{22} + S_{23} \), \( k_{c} = S_{13} + S_{23} + S_{33} \). Both Young’s modulus E and compressibility k, calculated using GGA/LDA functionals, are collected in Table 7.

Additional useful method to show mechanical anisotropy can be given by visualization of a three-dimensional dependence of the bulk modulus and Young’s modulus as a function of direction. For hexagonal crystal class, the directional dependencies of bulk modulus and Young’s modulus are given by the following relations [19]: where B is the bulk modulus and E is the value of Young’s modulus, in the direction determined by \( l_{1} ,l_{2} \) and \( l_{3} \)—the direction cosines of [uvw] directions and \( S_{ij} \) are the elastic compliance constants, which are inverse to the matrix of elastic constants C _ij calculated for BTBO crystal. The surfaces of directional dependence of bulk modulus B and Young’s modulus E for BTBO crystal, plotted in spherical coordinates, are depicted in Fig. 7.

The surfaces of directional dependence of bulk and Young’s modulus (Fig. 7) show that the BTBO crystal possesses low anisotropy. Both surfaces have the similar shape. In order to receive additional information and a better understanding of bulk modulus and Young’s modulus relation along different directions, the projections of the moduli at ab and ac crystal plane are plotted (see Fig. 8).

It is easy to see that at ab plane both bulk modulus and Young’s modulus are of a spherical shape, which means in this plane the BTBO crystal is almost isotropic. In the case of ac plane, some anisotropy of mechanical properties is shown.

Following the performed band-structure calculations and the elastic features, one can expect that the titled crystal may be operated by an external laser light of wavelength near the fundamental absorption edge. For this reason, we used two coherent nitrogen 371 nm 9 ns laser pulses with frequency repetition about 10 Hz in order to study the possible changes of the effective second-order nonlinear optical susceptibilities. As a probing beam, an Er:glass 20 ns laser (at λ = 1540 nm) was used. The detection was performed using interferometric filter at 770 nm and the photomultiplier connected to the oscilloscope. In order to monitor the possible photoluminescence near 371 nm laser treatment, the additional control of the spectral lines and their time dependencies were performed in the range of 330–905 nm by spectrometer. There was any wider spectral noticeable photoluminescence in this spectral range. Additionally, the time shift between the excitation and the induced nonlinear signals did not exceed few nanoseconds, while for the fluorescence it should be of an order higher. The use of two coherent beams interacting at different angles should stimulate the occurrence of additional birefringence [20], which also changes the nonlinear optical susceptibilities. Moreover, during the photoinduced changes in the common borate crystals, the contributions of anharmonic phonons begin to play an additional role [21, 22]. They may be responsible for the photoinduced second-order nonlinear optical susceptibility changes versus the cationic content, as shown in Fig. 9.

It is crucial that, contrary to the well-studied α-BIBO crystal, for the titled BTBO there exists a huge opportunity to operate the output SHG using external photoinducing UV coherent nanosecond laser pulses. The arrow indicates an excitation of the prevailing number of anharmonic phonons by the nitrogen pulsed laser beams. Because they are anharmonic, their symmetry is the same as for the second-order nonlinear optical susceptibility. As a consequence, it is manifested in Fig. 9 and the optical functions are changed significantly. The effect disappears immediately after the switching off the photoinducing beam treatment.