Density Functional Study of the Electronic, Elastic and Optical Properties of Bi₂O₂Te

2.1 Computational Settings

The Vienna Ab initio Simulation Package (VASP) code [17] is used to optimise the structure and calculate the properties of Bi₂O₂Te. Projector-augmented-wave pseudo-potentials [18] are employed to describe the interaction between ions and electrons, where the electronic configuration of Bi is 5d¹⁰6s²6p³, that of O is 2s²2p⁴ and that of Te is 5s²5p⁴. The Perdew–Burke–Ernzerhof functional [19] within the generalised gradient approximation (GGA) is adopted to calculate the exchange–correlation energy of the electrons. The atomic wave functions are expanded by a group of plane waves with a truncated energy of 520 eV. Brillouin zone integral is performed on a discrete 7 × 7 × 7 mesh [20] in lattice relaxation, which increases to 15 × 15 × 15 when calculating the electronic and optical properties. The convergence criterion of energy is set to 1 × 10⁻⁶ eV/atom, and the maximum force between ions is set to 0.01 eV/Å. The tests verify that these above parameters can ensure the convergences of the studied properties. Considering that pure density function calculations often underestimate the band gaps of materials, we use the modified Becke–Johnson (mBJ) potential [21] when studying the electronic and optical properties to make the results reasonable.

2.2 Calculation of Elastic Properties

The elastic constants of Bi₂O₂Te are calculated by the density functional perturbation method implemented in VASP. Tetragonal crystals have six independent elastic constants: C₁₁, C₃₃, C₄₄, C₆₆, C₁₂ and C₁₃. For polycrystals, comprehensive physical quantities, such as bulk modulus (B) and shear modulus (G), are usually used to describe their elastic properties [22]. The methods proposed by Voigt and Reuss are commonly used in the literature to derive the bulk modulus and shear modulus of a polycrystal. Voigt’s formulas for calculating the bulk modulus and shear modulus of a tetragonal crystal are as follows [22]:

Reuss’ formulas for calculating these two moduli are as follows:

where M=C11+C12+2C33−4C13, C2=(C11+C12)C33−2C132. In practice, the average values of the above variables are taken as the final values, i.e.

Poisson’s ratio of a polycrystal can be obtained using the following formula:

The shear wave velocity (V_t), longitudinal wave velocity (V_l) and average wave velocity (V_m) propagating in a polycrystal can be deduced from the following formulas, respectively [23]:

The Debye temperature Θ, related to the thermal properties of materials, can be obtained via the following expression:

where h is the Planck constant, k is the Boltzmann constant, n is the number of atoms contained in one molecular formula, N_A is the Avogadro number, ρ is the density of the material and M is its molecular weight.

3.1 Crystal Structure

Bi₂O₂Te is a tetragonal crystal and its space group is I4/mmm (no. 139), as shown in Figure 1. The reported lattice constants are a = 3.980 Å, c = 12.704 Å, and the fractional coordinates of atoms are Bi (0, 0, 0.3472), O (0.5, 0, 0.25) and Te (0, 0, 0) [14]. One unit cell contains four Bi atoms, four O atoms and two Te atoms. The lattice constants after structural relaxation are a = 4.019 Å, b = 12.885 Å. The fractional coordinates of Bi are (0, 0, 0.3467) and those of other two atoms are kept unchanged due to their special high site symmetries. These three relaxed parameters are close to the experimental values with relative errors of 1.0 %, 1.4 % and −0.14 %, respectively, which could be acceptable in common density functional calculations.

Figure 1:

Crystal structure (left) and primitive cell and Brillouin zone (right) of Bi₂O₂Te. The large, medium and small balls represent the Bi, Te and O atoms, respectively.

3.2 Electronic Properties

The calculated band structure along the high symmetry directions in the Brillouin zone of Bi₂O₂Te is shown in Figure 2a. Because the bands near the Fermi level have the greatest influence on the electronic properties of crystals, we mainly presented the bands between −8.0 and 6.0 eV. The results show that the top of the valence band is at the N point while the bottom of the conduction band is at the X point, so Bi₂O₂Te is an indirect crystal. The band gap calculated by the GGA functional is almost equal to zero, far smaller than the experimental value of 0.23 eV [14]. It increases to 0.20 eV when the mBJ potential is used with the GGA functional.

Figure 2:

Calculated (a) band structure and (b) partial DOS of Bi₂O₂Te. The green dashed lines represent the Fermi level.

The band structure of Bi₂O₂Te seems dispersive. To find their origins, we calculated its density of states (DOS), as illustrated in Figure 2b. It reveals that the bands located in the energy range from −6.5 to −2.5 eV come mainly from the O 2p and the Bi 6p orbitals; the bands located between −2.5 eV and the Fermi level mainly originate from the Te 5p orbital, with a little contribution from O 2p orbital and Bi 6s and 6p orbitals; and the bands from 0.5 to 6.0 eV are mainly from the Bi 6p orbital, together with some contribution from the O 2p and the Te 5p orbitals. It can also be seen that the band gap of Bi₂O₂Te is formed by the Te 5p orbital located at the top of the valence band and the Bi 6p orbital located at the bottom of the conduction band.

From the calculated partial DOS, one can find that the overlap between the s and p orbitals within each type of atom is rather limited, which indicates that the hybridisation in the atom itself is very weak. However, the p orbitals of the O and Bi atoms overlap seriously and have high densities in the energy range from −6.5 to −2.5 eV, which implies that there exists certain hybridisation between the p orbitals of O and Bi. The p orbital of Te is relatively localised, having high peak values in the energy range from −2.5 eV to the Fermi level. It has a weak hybridisation with the Bi 5s and 5p orbitals, indicating a very weak covalent interaction between the Te atoms and the Bi atoms. The hybridisation between the p orbitals of the constituent atoms makes the bands of Bi₂O₂Te fluctuated.

Bi₂O₂Te seems like a layered crystal, so it is meaningful to study its bonding properties. For this end, we calculated its charge density difference and displayed it in Figure 3. It is shown that the O atoms obtain electrons (blue colour) while the Bi atoms lose electrons (red colour) when forming the crystal, so there also exists an ionic interaction between the Bi and O atoms. There are no electrons accumulating between the Te and Bi atoms, suggesting that the interaction between them is ionic. The distance between the Te and Bi atoms is 3.461 Å and that between the Te and O atoms is 3.797 Å, which is much longer than the value of 2.364 Å between the Bi and O atoms, so the interaction between the Te and Bi (O) atoms is weaker than that between the Bi and O atoms. Bader analysis [24] indicates that each O atom obtains 1.16 electrons, each Te atom obtains 0.82 electrons, while each Bi atom loses 1.57 electrons. From the above analyses, we can infer that the Bi and O atoms form the Bi–O layers by ion-covalent interaction, and then these layers form the crystal mainly through the ionic interaction between the Bi and Te atoms.

Figure 3:

Charge density difference (left) and that in the (100) plane (right) of Bi₂O₂Te. The blue (negative) and red (positive) colours represent electron enhancement and depletion, respectively.

3.3 Elastic Properties

The elastic constants of materials can be used to describe their responses to external stresses within their elastic limits, which are closely related to the mechanical stability of materials. The calculated elastic constants of Bi₂O₂Te are C₁₁ = 140.3 GPa, C₃₃ = 98.9 GPa, C₄₄ = 24.3 GPa, C₆₆ = 51.8 GPa, C₁₂ = 63.1 GPa and C₁₃ = 41.5 GPa. For tetragonal crystals, the criteria of mechanical stability are [22] C₁₁ > 0, C₃₃ > 0, C₄₄ > 0, C₆₆ > 0, C₁₁ − C₁₂ > 0, C₁₁ + C₃₃ − 2C₁₃ > 0, 2(C₁₁ + C₁₂) + C₃₃ + 4C₁₃ > 0. The calculated elastic constants satisfy the above criteria, so Bi₂O₂Te is mechanically stable.

The elastic constant C₁₁ is larger than C₃₃, which indicates that the stiffness of Bi₂O₂Te along the a/b axis is greater than that along the c axis. In other words, the strain of the a axis is smaller than that of the c axis when a given normal stress is applied along the a axis and thec axis, respectively. C₁₂ is larger than C₁₃, which implies that the strain of the c axis is larger than that of the b axis when a given normal stress is applied along the a axis. C₄₄ and C₆₆ are elastic constants related to shear strain. C₆₆ is larger than C₄₄, which means that the latter will produce a greater strain when the same shear stress is applied to the [0 ± 10](±100) and the [0 ± 10](00 ± 1) shear systems, respectively. The elastic constants C₁₃ and C₄₄ are both related to the bonding properties along the c axis of the crystal. C₁₃ and C₄₄ are the two smallest elastic constants among all, which means that the bonds along the c axis are the weakest and the crystal is easier to fracture along the c axis under shear stress.

Bulk modulus describes the relative volume change of a material under hydrostatic pressure, while shear modulus reflects the resistance of a material to shear deformation. The calculated bulk modulus of Bi₂O₂Te is 72.8 GPa and the shear modulus is 34.0 GPa. The former is much larger than the latter, which implies that the shear deformation is the main factor determining the mechanical stability of the crystal. Pugh [25] proposed that the brittleness and ductility of a material can be discriminated by the value of B/G. A value >1.75 means that the material is ductile, or else it is brittle. The calculated B/G ratio of Bi₂O₂Te is 2.14, which suggests that it is ductile. Haines et al. [26] pointed out that Poisson’s ratio can be used to determine the bonding properties of materials: for pure covalent crystals, the value is around 0.1 and for ion-covalent crystals, the value is between 0.2 and 0.3. The calculated Poisson’s ratio of Bi₂O₂Te is 0.3, which suggests that it is an ion-covalent crystal. This result is consistent with the conclusion drawn previously. Chen et al. [27] put forward that the hardness of a material could be estimated from its bulk modulus and shear modulus. They gave a formula for calculating Vickers hardness: H_v = 2(G³/B²)^0.585−3, from which we deduced that the hardness of Bi₂O₂Te is about 3.5 GPa. The Debye temperature is closely related to the thermal properties of materials. The values of the shear, longitudinal and average wave velocities of Bi₂O₂Te calculated from the elastic constants are 1921.3, 3579.9 and 2145.4 m/s, respectively. The Debye temperature is finally predicted to be about 232.2 K. Using the empirical formula T_m = 607 + 9.3 × B (bulk modulus) [28], we obtained a value of 1284.0 K for the melting point of Bi₂O₂Te.

Previous studies have revealed that a material is prone to cracking if its elastic anisotropy is large [29]. Therefore, it is important to study the elastic anisotropy of materials. Here, we studied the elastic anisotropy of Bi₂O₂Te using the directional bulk modulus and Young’s modulus. The former characterises the linear compressibility of a material in all directions under a hydrostatic pressure, while the latter characterises the compressibility of a material along the direction of an applied uniaxial stress. The formulas for calculating these two quantities of tetragonal crystals are as follows [30]:

where (l₁l₂l₃) denotes the directional cosine and S_ij is the element in the elastic yield tensor matrix, which is the inverse of the elastic tensor matrix.

Figure 4 shows the directional bulk modulus and Young’s modulus. Both graphs are not spherical, indicating that the elastic properties of Bi₂O₂Te are anisotropic. The maximum bulk modulus is 290.4 GPa in the (001) plane and the minimum is 138.4 GPa in the [001] direction, which means that the c axis of Bi₂O₂Te is easiest to contract under a hydrostatic pressure. The maximum Young’s modulus is 128.3 GPa in the diagonal directions of the (001) plane and the minimum is 68.6 GPa in the diagonal directions of the (100) and (010) planes, so the former directions are difficult to be compressed and stretched.

Figure 4:

Directional (a) bulk modulus and (b) Young’s modulus of Bi₂O₂Te (unit: GPa).

3.4 Optical Properties

The optical properties of a crystal, such as its refraction, absorption and reflection functions, can be deduced from its dielectric function ε(ω). The imaginary part of the dielectric function ε₂(ω) of a crystal can be obtained by calculating the allowable transition of electrons in Brillouin zone from the occupied orbitals to the unoccupied orbitals under an incident light [31]:

where q is the Bloch vector of the light; ω is the frequency of the light; ck and vk denote the empty and occupied orbitals at the k point of the Brillouin zone, respectively; u_ck and u_vk are the cell periodic part of the orbitals at the k point; and E_ck and E_vk are the corresponding energies. The real part of the dielectric function ε₁(ω) can be obtained using the Kramers–Kronig relation:

where P means the principal integral and η is the smearing factor with a value of 0.1.

Bi₂O₂Te is a tetragonal crystal, so it is a uniaxial crystal with the c axis as its optical axis. When a beam of light having a polarisation direction perpendicular to the c axis propagates in it, it will have the same optical response regardless of the propagating direction. However, its optical response will be different if the transiting light has a polarisation direction parallel to the c axis. In the following sections, we will use the signs “⊥” and “//” to represent the polarisation directions of lights perpendicular and parallel to the c axis, respectively. Obviously, the former relates to the case of the ordinary light while the latter connects with the case of the extraordinary light.

The calculated dielectric curves of Bi₂O₂Te are illustrated in Figure 5. For the convenience of discussion, the total DOS of Bi₂O₂Te is also given in the figure. The dielectric curves are different in the two polarisation directions. There is one apparent peak and a broad shoulder in the curves of ε₂(ω) in both cases. The peak is located at about 2.6 eV and the shoulder extends to 4.8 eV for the perpendicular polarisation case. The shoulder starts at about 2.7 eV and ends at the peak located at about 4.2 eV for the parallel polarisation case. From the calculated DOS and bond lengths, one can know that these peaks and shoulders should mainly be induced by the transition of electrons from the p orbitals of Te and O to the 6p orbitals of Bi. The curves of the parallel polarisation case shift right and meanwhile lower their values, which should be caused by the difference of the transition details of electrons under the perturbation of light. The calculated curves of ε₁(ω) reveal that the zero-frequency dielectric constants along the two directions are ε_⊥ = 12.985 and ε_// = 9.547.

Figure 5:

Dielectric curves in the directions perpendicular and parallel to the c axis of Bi₂O₂Te, together with the total DOS.

The refraction function n(ω) determines the propagation speed of light in materials and the extinction function k(ω) reflects the attenuation of light in materials. They can be derived from the following formulas, respectively [32]:

The calculated refraction and extinction curves of Bi₂O₂Te are given in Figure 6. Because the extinction function mainly reflects the absorption of light by medium when light propagates in it, its profile is very similar to that of ε₂(ω), which can also be explained by the electron transition as done previously. There are two broad peaks in the extinction curves: one is located at the energy range from 1.0 to 11.5 eV, which should be mainly caused by the electron transiting from the orbitals located between −6.0 and 0 eV to the permitted orbitals above the Fermi level. The second is at the energy range of 11.5–15.0 eV, which may be induced by the electrons transiting from the s orbitals at the energy range from −12 to −9 eV to the allowable orbitals above the Fermi level. From the calculated dielectric constants at zero frequency, one can obtain that n_o = n_⊥ = 3.60 and n_e = n_// = 3.09. Thus, Bi₂O₂Te has a birefringence of Δn = n_e − n_o = −0.51 < 0, which means that it is a negative uniaxial crystal. Its large birefringence can compare to those of some liquid crystals [33], implying that it can be used as tuning material in optical fields like those liquid crystals.

Figure 6:

Refraction curves n(ω) and extinction curves k(ω) in the directions perpendicular and parallel to the c axis of Bi₂O₂Te.

The absorption function denotes the losing energy of light when it transmits through a material by one unit thickness, which has a direct connection with the Joule heat produced in that material [34]. The reflection function weights the percentage of energy of light reflected from the surface of a material. The energy loss function measures the energy loss of a fast electron when it traverses in a homogeneous dielectric material [35]. The functions of absorption α(ω), reflection R(ω) and energy loss L(ω) can be derived from the following expressions, respectively [32]:

where c is the speed of light.

The computed absorption, reflection and energy loss curves of Bi₂O₂Te are illustrated in Figure 7. Because the absorption function is proportional to the extinction function, their figures seem alike. Bi₂O₂Te has non-zero absorbance >1.0 eV, which covers the energy scope of the visual light, meaning that Bi₂O₂Te is not transparent in this range. Reflection curves reveal that it has non-zero reflectivity in the energy range studied. In the intermediate energy range, classically 1–50 eV, the energy loss is mainly induced by single electron excitation and collective excitation [36]. Peaks in the energy loss curves reflect the plasma oscillation and the corresponding frequencies are called plasma frequencies, which can be obtained by letting ε₁(ω) = 0 while restricting ε₂(ω) < 1. A material is dielectric [ε₁(ω) > 0] just above these frequencies, whereas it behaves like a metal [ε₁(ω) < 0] just below them [34]. There are two sharp peaks located at about 11 and 15 eV in the energy loss curves of Bi₂O₂Te, which correspond to plasma frequencies of about 2.6 × 10¹⁵ and 3.6 × 10¹⁵ Hz, respectively. The origins of these peaks in energy loss curves cannot be explained by the DOS because they are induced by the collective excitation of the electrons within the crystal.

Figure 7:

(a) Absorption (b) reflection and (c) energy loss L(ω) curves in the directions perpendicular and parallel to the c axis of Bi₂O₂Te.