† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No. 51171156).
The hydrostatic-pressure-dependent mechanical stability and optoelectronic behavior of Zn4B6O13 (ZBO) are calculated using the exchange-correlation functional Perdew–Burke–Ernzerhof generalized gradient approximation and the hybrid functional PBE0 based on density functional theory. The calculated and experimental unit cell volumes and Vickers hardness of ZBO at zero pressure agree well. ZBO is mechanically stable under the critical pressure of 52.98 GPa according to the generalized stability criteria. Furthermore, Youngʼs modulus and Vickers hardness decrease with increasing hydrostatic pressure. The strength and type of ZBO bonds are investigated by population and electron density difference. The electronic structure at zero pressure reveals that ZBO is an indirect band gap semiconductor, and the calculated 5.62-eV bandgap coincides well with the 5.73-eV experimental value, highlighting the success of the hybrid functional PBE0 calculations of electronic properties. The band gap almost increases as a second-order polynomial of pressure, and the indirect nature does not change with the applied external pressure. The optical reflectivity and absorption coefficient show that ZBO is an excellent ultraviolet photodetector. Our calculation results suggest that the elastic and optical properties of ZBO are highly stable over a wide pressure range.
Inorganic borates have attracted considerable attention over the past few decades because of their rich structures and distinctive physical/chemical properties.[1–5] In the crystal structures of inorganic borates, generally, a flat triangular structure can be formed with one boron atom and three oxygen atoms (BO3), or a tetrahedral-shaped structure can be composed of one boron atom and four oxygen atoms (BO4), i.e., a boron atom can be sp2-hybridized with three oxygen atoms or sp3-hybridized with four oxygen atoms. The common oxygen-atom-linked BO3 or BO4 groups can be composed of isolated rings and cages or polymerize into infinite chains, sheets, and networks to form rich borate structures.[1] Plentiful borates have been found, and some have been synthesized and industrialized. For example, iron borate (FeBO3) is a magneto-optical crystal for magnetic storage materials;[2] magnesium borate (MgB4O7) is a thermosensitive material;[3] lithium tetraborate (Li2B4O7) is a new type of piezoelectric crystal used in mobile communication engineering; and NYAB (NdAl3(BO3)4) is an important laser crystal material in the research field of solid-state lasers.[4,5] Rare earth borates are significant host materials for fluorescence because of their high photoluminescence performance and chemical stability. In addition, borate materials like LiB3O5,[6] Sr2Be2B2O7,[7] CsLiB6O10,[8] and CsB3O5 [9] are perfect linear and non-linear optical materials.[10]
Zinc borates have been widely used in many fields, e.g., as polymer additives of flame-retardant synergists, anti-arcing agents, preservatives in wood composites and additives for smoke and afterglow inhibition. In the binary system of ZnO-B2O3, at least three compounds have been proposed, including zinc tetraborate (ZnB4O7), zinc orthoborate (Zn3B2O6) and zinc metaborate (ZBO).[11] The ZBO crystal is special because it is one of the few anhydrous metaborates with all the boron atoms in a fourfold coordination.[12] The qualitative measurements of the photoelastic properties of ZBO performed by Bohaty showed high unexpected effects that suggest an application for acousto-optical deflection devices.[13] Over the past decade, many researchers have shown great interest in UV-irradiation-induced defects in ZBO and found that ZBO is a valuable substrate for afterglow luminescence materials because of its multiple defect centers.[14,15] Recently, Jiang[16] prepared ZBO single crystal via solid-state reaction. The as-received ZBO single crystal was a transparent semiconductor with an optical band gap of 5.73 eV and an important three-dimensional near-zero expansion character. As a result, ZBO has promise for applications in ultraprecise optical instruments often used in the UV region.
The scientific community has no detailed study on the hydrostatic-pressure-dependent mechanical stability and optoelectronic behavior of ZBO. This work aims to conduct a complementary investigation of ZBO under conditions of different hydrostatic pressures with first-principle methods and provides much information regarding the mechanical and optoelectronic behavior of this important material below a critical pressure.
ZBO crystallizes in a cubic structure within an I-43m space group forming a B24O48 sodalite cage in which the O1 atom at the center of the unit is surrounded by four ZnO4 tetrahedra, three BO4 groups are bonded by one Zn atom via relatively strong Zn–O2 bonds, and 24 BO4 tetrahedra are corner-shared by O2 atoms as shown in Fig.
The calculations presented in the current work were performed using the Cambridge Serial Total Energy Package code which uses pseudopotentials to depict electron–ion interactions and represents electronic wave functions by using a plane-wave basis set.[17–19] In the calculations of ZBO geometric and mechanical properties and population and electron density differences, the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation was used for the electron exchange correlation, the ultra-soft pseudopotential was adopted to describe the interactions between the valence electrons and ionic core, and the kinetic energy cutoff of the plane-wave basis set was sufficiently set at 500 eV. Note that, in the current work, the band structures, density of states and optical properties of ZBO were calculated by adopting the hybrid functional PBE0 and the norm-conserving pseudopotential with a kinetic energy cutoff of 750 eV. The hybrid functional PBE0 overcomes the shortcomings of previous band structure calculations of solids performed within generalized gradient approximation and LDA approaches which often significantly underestimate the band gaps in comparison with experimental results.[20] In all calculations, the electrons Zn3d104s2, B2s22p1, and O2s22p4 were treated as valence states. The Monkhorst–Pack[21] scheme k-points grid sampling was set at 8 × 8 × 8 points for the first Brillouin zone. All atoms were fully relaxed to their equilibrium positions when the maximum force on each atom was less than 0.01 eV/Å, the total energy change converged to 5.0 × 10−7 eV/atom, the maximum stress on each atom was below 0.02 GPa, and the displacement change of each atom was less than 5.0 × 10−4 Å.
Within the framework of density functional theory calculations, the study of the ZBO crystal structure provides an initial passage to investigate its other physical properties. After geometrical optimization, the ZBO lattice parameters and unit cell volumes at different external hydrostatic pressures less than 50 GPa are shown in Table
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![]() | Table 1.
ZBO pressure-dependent lattice constants and unit cell volume. . |
ZBO has only three independent elastic constants, namely, C11, C12 and C44 because of its cubic structure with space group I-43m. The elastic constant C11 represents the resistances to linear compression in the x, y and z directions, and a longitudinal strain induces a change in C11. The elastic constants C12 and C44 representing the elasticity in shape are shear constants. The calculated elastic constants Cij of ZBO at different hydrostatic pressures are shown in Table
![]() | Table 2.
Pressure-dependent elastic constants Cij (in GPa). . |
To ensure that ZBO is stable at a given hydrostatic pressure, the three independent elastic constants must match the generalized elastic stability criteria[23] listed below
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![]() | Table 3.
Pressure-dependent Mi (in GPa) for ZBO. . |
Based on elastic constants, Voigt and Reuss approximations[24] are widely used to calculate the bulk modulus B and the shear modulus G of solids. The bulk modulus B reflects the ability of solids to defend compression, while the shear modulus G stands for the ability of solids to defend shear deformation. For a cubic system, the Voigt and Reuss approximations of B and G can be obtained by
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Hill[25] confirmed that the Voigt and Reuss models represent the extreme upper and lower bounds, respectively, and the arithmetic average value VRH (Voigt–Reuss–Hill) is close to the experimental results. The arithmetic average values of B and G can be written as
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![](cpb_27_5_057101/cpb_27_5_057101_ieqn1.gif)
![](cpb_27_5_057101/cpb_27_5_057101_ieqn2.gif)
![](cpb_27_5_057101/cpb_27_5_057101_ieqn3.gif)
![](cpb_27_5_057101/cpb_27_5_057101_ieqn4.gif)
![](cpb_27_5_057101/cpb_27_5_057101_ieqn5.gif)
![]() | Table 4. B (in GPa), G (in GPa), E (in GPa), γ, Hv (in GPa), B/G and AU of ZBO. . |
Based on the calculated B and G above, Pughʼs ratio B/G, Poissonʼs ratio γ, Vickers hardness Hv, Youngʼs modulus E and anisotropic index AU can be obtained[27–30]
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![](cpb_27_5_057101/cpb_27_5_057101_ieqn6.gif)
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The Vickers hardness Hv often described by an empirical formula is a very important property used to measure the hardness of solids.[29] The obtained theoretical Vickers hardness of ZBO at zero pressure was 11.96 GPa, which is close to the experimental value of 1304 kg/mm2 (namely, 12.37 GPa),[16] confirming that ZBO has high mechanical strength. However, as shown in Table
The Youngʼs modulus E represents the ratio of a simple tensile stress to a corresponding tensile strain and is calculated by Eq. (
The directional dependence of Youngʼs modulus of ZBO is illustrated in Fig.
![]() | Fig. 5. (color online) Directional dependence of the Youngʼs modulus of ZBO: (a) 0 GPa, (b) 5 GPa, (c) 10 GPa, (d) 20 GPa, (e) 30 GPa, (f) 40 GPa, and (g) 50 GPa. |
The universal elastic anisotropy index AU for all crystal phases introduced by Ranganathan[30] is often used to quantitatively study monocrystalline anisotropy. For a completely isotropic material, the elastic anisotropy factor AU is equal to 0, while any value larger than 0 indicates anisotropy. The higher the value, the higher the degree of anisotropy. The anisotropy values shown in Table
To investigate the stress effects on the bonds in ZBO, we present only the bond lengths, bond populations and electron density differences for 0 and 50 GPa in Table
![]() | Fig. 6. (color online) ZBO electron density differences: (a) {110} plane and (b) {111} plane for 0 GPa and (c) {110} and (d) {111} plane for 50 GPa. |
![]() | Table 5.
ZBO bond lengths and bond populations. . |
From the electron density differences of ZBO in Figs.
After confirming the mechanical stability of ZBO under hydrostatic pressures less than 52.98 GPa, we used the hybrid functional PBE0 to investigate its density of states and band structures within the same pressure range. According to the total and partial density of states of ZBO at zero pressure displayed in Fig.
Figure
Considering that energy band gaps of solids can be described as a function of pressure, this variation can be described as
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![](cpb_27_5_057101/cpb_27_5_057101_ieqn8.gif)
![](cpb_27_5_057101/cpb_27_5_057101_ieqn9.gif)
![](cpb_27_5_057101/cpb_27_5_057101_ieqn10.gif)
![](cpb_27_5_057101/cpb_27_5_057101_ieqn11.gif)
During the ZBO optical property calculations, 216 empty bands (three times the number of occupied bands) were adopted to consider as many electron transitions as possible.
The study of radiation–matter interaction causing electronic transitions is essential to investigate the optical behavior of ZBO. This interaction can be described by the well-known complex dielectric function
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The imaginary part
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![](cpb_27_5_057101/cpb_27_5_057101_ieqn14.gif)
![](cpb_27_5_057101/cpb_27_5_057101_ieqn15.gif)
![](cpb_27_5_057101/cpb_27_5_057101_ieqn16.gif)
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From the imaginary part
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Figure
The absorption coefficient α(ω), refractive index n(ω), loss function L(ω) and conductivity σ(ω) of ZBO at different pressures are plotted in Fig.
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![](cpb_27_5_057101/cpb_27_5_057101_ieqn27.gif)
![]() | Fig. 12. (color online) Pressure-dependent ZBO optical properties: (a) absorption coefficient α(ω), (b) refractive index n(ω), (c) loss function L(ω), and (d) conductivity σ(ω). |
The current work provides a theoretical prediction of the effect of hydrostatic pressure on the mechanical properties and optoelectronic behavior of ZBO crystal in the mechanical stability range 0–52.98 GPa. The obtained structural parameters and Vickers hardness were consistent with the observed experimental results. ZBO has high mechanical strength, high Youngʼs modulus and weak anisotropy, and all of these mechanical properties are slightly dependent on external pressure. The B–O2 bonds are strongest in ZBO while the Zn–O1 and Zn–O2 bonds are covalent with ionic characteristics. The top valence bands of ZBO are dominated by O-2p and Zn-3d states, while the conduction bands consist of Zn-4s states. The band gap of ZBO increases as a function of pressure (i.e.,
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