^{1}

^{1}

Self-consistent ab initio calculations are performed on the structural, electronic and optical properties of wurtzite ZnO. The Full Potential Linearized Augmented Plane Wave (FP-LAPW) method is applied to solve the Kohn-Sham equations. Results are obtained by using the PBE-GGA and mBJLDA exchange correlation potentials. The energy and charge convergence have been examined to study the ground state properties. The band structure and Density of States (DOS) diagrams are plotted from the calculated equilibrium lattice parameters. The general profiles of the optical spectra and the optical properties, including the real and imaginary part of dielectric function, reflectivity, refractive index, absorption co-efficient, electron energy loss function and optical conductivity of wurtzite ZnO under ambient conditions are discussed. The optical anisotropy is studied through the calculated optical constants, namely dielectric function and refractive index along three different crystallographic axes.

Zinc oxide (ZnO) is the most promising candidate of II-VI semiconductor family due to its vital applications in various fields. It has attracted much interest of the research community for its electronic properties such as a wide band gap, ~3.34 eV and a large exciton binding energy, 60 meV [

ZnO crystallizes in three different structures such as hexagonal wurtzite (B4), cubic zincblende (B2), and cubic rocksalt (B1). Hexagonal wurtzite structure of ZnO is the most stable structure under ambient conditions, which belongs to the space group P6_{3}mc. Each zinc atom is surrounded by four oxygen atoms, which are located at the corner of a regular tetrahedron and vice versa [

The theoretical interpretations of optical properties are very important, because the electronic structure has a large impression on optical and energy loss properties [

First principle calculations are performed on structural, electronic and optical properties of wurtzite ZnO. To obtain reliable results, a highly accurate Full Potential Linearized Augmented Plane Wave (FP-LAPW) method is applied, as implemented in WIEN2k code based on Density Functional Theory (DFT) [

In FP-LAPW method, the basis set is obtained by dividing the unit cell into non-overlapping spheres surrounding each atom and creating an interstitial region between them. The potential and the charge density are expanded by spherical harmonics inside the muffin-tin sphere and by plane wave basis set in the interstitial region of the unit cell. The equilibrium volume V_{0}, bulk modulus B_{0}, pressure derivative of bulk modulus _{0} are determined by fitting the total energy versus the reduced and extended volume of the unit cell into third-order Birch-Murnaghan’s equation of state (EOS) [

respectively. In these expressions, E_{0} is the total energy; V_{0} is the equilibrium volume; B_{0} is the bulk modulus at pressure P = 0; and

Geometry minimization is also done for obtaining the optimized positions. To achieve the energy convergence of the eigenvalues, the wave function in the interstitial regions is expanded in plane waves with a k cut-off, k_{max} = 7.0/R_{MT}, where R_{MT} denotes the smallest atomic sphere radius (muffin-tin radius) and k_{max} denotes the magnitude of the largest k-vector in the plane wave expansion. The valence wave functions inside the muffin-tin sphere are expanded up to l_{max} = 10, while the charge density is Fourier expanded up to G_{max} = 12 (Ryd)^{1}^{/2}. The R_{MT} values are 1.75 and 1.53 a.u. for Zn and O, respectively. A dense mesh of 1000 k points is used in the irreducible wedge of the brillouin zone. The self-consistent calculations are iterated till the total energy converges below 10^{−4} Ry and force converges below 1 mRy/a.u. Zn 3d, 4s and O 2s, 2p orbitals are considered as valence states and all lower-lying states are treated as core. The atomic positions are (1/3, 2/3, 0) (2/3, 1/3, 0.5) for Zn and (1/3, 2/3, 0.38) (2/3, 1/3, 0.88) for O, respectively. The volume optimization curve is shown in

The ground state structural properties such as, equilibrium lattice parameters (a_{0} and c_{0}), anion position parameter u (which governs the positions of oxygen ions), equilibrium volume (V_{0}), bulk modulus (B_{0}) and its pressure derivative (_{0}) and its pressure derivative (

The electronic properties of ZnO are discussed using the band structure, total density of states (TDOS) and partial density of states (PDOS) calculated with optimized values. Spin polarized and non-spin polarized calculations are performed using both mBJLDA and PBE-GGA potentials. The calculated band structure along the higher symmetry points Г-M-K-Г-A and higher symmetry directions Σ, Δ, Λ in the brillouin zone using mBJLDA

a_{0} (Å) | c_{0} (Å) | c/a | u | V_{0} (Å/f. u.) | B_{0} (GPa) | ||
---|---|---|---|---|---|---|---|

PW | 3.286 | 5.269 | 1.603 | 0.380 | 24.61 | 128.72 | 4.38 |

Exp. | 3.249^{[a]} | 5.204^{[a]} | 1.601^{[a]} | 0.381^{[a]} | 23.79^{[a]} | 183^{[a]} | 4.0^{[a]} |

Cal. | 3.286^{[b]} 3.270^{[c] } 3.283^{[f]} | 5.241^{[b] } 5.268^{[c] } ^{ } 5.309^{[f]} | 1.595^{[b] } 1.611^{[c] } 1.604^{[e] } 1.617^{[f]} | 0.383^{[b] } 0.378^{[c]} 0.385^{[e] } 0.378^{[f]} | 24.93^{[e]} | 154.46^{[d] } ^{ } 129.73^{[e] } 131.5^{[f]} | 4.2^{[d] } ^{ } 4.68^{[e] } 4.2^{[f]} |

PW-present work; Exp.-experiment; Cal.-Calculations. [a] Exp. [

and PBE-GGA approach are shown in

The valence band maximum and the conduction band minimum are located at Г point, resulting in a direct band gap. The E_{g} using PBE-GGA and mBJLDA are 0.814 eV and 2.683 eV respectively. F. Tran and P. Blaha have reported the E_{g} of 0.75 eV and 2.68 eV using the LDA and mBJLDA potentials, respectively [_{g} using mBJLDA shows better agreement with experimental values, which confirms that mBJLDA potential, is best suited for w-ZnO, compared to GGA and LDA. So, in the present study the further calculations such as electronic and optical properties of w-ZnO are done using mBJLDA and presented in the subsequent sections.

Total and partial densities of states of w-ZnO are shown in Figures 3(a)-(c) for the energy range −8 eV to +10 eV. The first valence band is located between 0 to −3.5 eV and it comes from the admixture of O “p” state, Zn “p” state and a small amount of Zn “d” states. This mainly comes from the p-d hybridization between O “p” and Zn “d” states. The second valence band is located between −3.4 eV to −4.0 eV. This is predominantly from “d” states of Zn. The third valence band below −4 eV to −5.5 eV is the admixture of “d” and “p” states of Zn and “p” state of O. Figures 4(a)-(e) represent the partial density of states (PDOS) of s, p, d of Zn and s, p of O, respectively, using mBJLDA. This clearly shows the hybridization discussed above and nature of bonding.

We have investigated the electronic charge density contour of w-ZnO in (110) plane, to analyse nature of chemical bond between Zn and O atoms, as shown in

Optical properties play an active role in the understanding of the nature of material and provide a clear picture for the usage of a material in opto electronic devices. It is generally known that the interaction of a photon with the electrons inthe system can be described in terms of time-dependent perturbations of the ground-state electronic states. Transitions between occupied and unoccupied states are originated by the electric field of the photon. The spectra resulting from these excitations can be described as a joint density of states between the valence and conduction bands. The optical response of a material to the electromagnetic field at all energy levels, can be described by means of complex dielectric function ɛ(ω) as,

where, ɛ_{1}(ω) and ɛ_{2}(ω) are real and imaginary part of the dielectric function. Real part of the dielectric function ɛ_{1}(ω), means the dispersion of the incident photons by the material, while the imaginary part of the dielectric function ɛ_{2}(ω), corresponds to the energy absorbed by the material. There are two contributions to complex dielectric function ɛ(ω), namely intraband and interband transitions. The contribution from intraband transitions is influential only for metals. The interband transitions can be further divided into direct and indirect transitions [

The imaginary part ɛ_{2}(ω) of the dielectric function is calculated from the contribution of the direct interband transitions from the occupied to unoccupied states and the calculation is associated with the energy eigenvalue and energy wave functions, which are the direct output of band structure calculation. ɛ_{2}(ω) can be calculated using the following expression [

where, M is the dipole matrix; i and j are initial and final states respectively; f_{i} is the Fermi distribution function for the i^{th} state; E_{i}_{ }is the energy of electron in the i^{th} state and ω is the frequency of the incident photon. Real part ɛ_{1}(ω) of the dielectric function can be found from its corresponding ɛ_{2}(ω) by Kramers-Kronig transformation in the form [

where, P stands for the principle value of the integral.

_{2}(ω) and real ɛ_{1}(ω) part of the dielectric function, respectively, for the electric field vector parallel and perpendicular to the crystallographic c-axis. Since w-ZnO has hexagonal symmetry, we need to calculate two different independent principle components for ɛ(ω) such as, ɛ_{zz}(ω) and ɛ_{xx}(ω) corresponding to light polarized parallel and perpendicular to c-axis. Due to this reason, all optical constants are compared together in two directions.

In _{1}(ω)_{xx} and ɛ_{1}(ω)_{zz} spectra, which falls to a first critical point at 2.68 eV. This is again showing the direct band gap nature of ZnO as discussed earlier in section 3.2.1 and as shown in _{2}(ω) is 2.68 eV, which is close to the calculated band gap and also known as the fundamental absorption edge. This confirms the direct optical transitions, between the valence band maxima and the conduction band minima at Г. The prominent large peak in the spectra is situated at 11.7 and 13.3 eV for ɛ_{2}(ω)_{xx} and ɛ_{2}(ω)_{zz} respectively. There are two small humps in ɛ_{2}(ω)_{xx} spectrum, situated at 8.0 and 13.1 eV and one shoulder at 12.3 eV whereas in ɛ_{1}(ω)_{zz} spectrum there are three small humps located at 8.2, 11 and 12 eV. The transition between Zn 4s and O 2p orbitals may drive to the peak at around 8.0 eV. The peak at 11.0 eV is mainly from the transition between O 2p and Zn 3d orbitals. The peaks around 13 eV come from Zn 3d and O 2s states. These results are consistent with the other reported results [

The static dielectric constant value (the value of the dielectric constant at zero energy) of ɛ_{1}(0) is 2.8. A higher value of energy gap produces a smaller ɛ_{1}(0), which can be explained on the basis of the Penn model [_{1}(ω) and ɛ_{2}(ω), all the other optical properties, such as, reflectivity R(ω), refractive index n(ω), extinction coefficient κ(ω), energy loss function L(ω), absorption coefficient α(ω) and optical conductivity σ(ω) can be calculated [

where, n is the real part of the complex refractive index (refractive index) and κ is the imaginary part of the refractive index (extinction co-efficient).

enhanced after 13 eV. From the reflectivity spectra we observed, the anisotropy behaviour of w-ZnO is small up to the photon energy 10 eV. The reflectivity data of the present calculation are compared with other experimental data. The line shape of our calculated reflectivity spectra is in reasonable agreement with the previously measured reflectivity [

L(ω) is an important factor describing the energy loss of a fast moving electron in a material. The peaks in L(ω) spectra represent the characteristic combined with the plasma resonance and the corresponding frequency is the so-called plasma frequency (ω_{pl}), above which the material shows the dielectric behaviour [ɛ_{1}(ω) > 0], while below which the material exhibits the metallic property [ɛ_{1}(ω) < 0]. The peaks in L(ω) spectra reveal that the point of transition from the metallic property to dielectric property for a material [

_{1}(ω) due to relation

Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent. The birefringence is quantified as the difference between the extraordinary and ordinary refractive indices, Δn(ω) = n_{e}(ω) - n_{o}(ω), where n_{e}(ω) is the index of refraction for an electric field oriented along the c-axis and n_{o}(ω) is the index of refraction for an electric field perpendicular to the c-axis [_{e}(ω) is 1.663 and n_{o}(ω) is 1.648. We find that the positive birefringence Δn(0) is equal to 0.015 for w-ZnO.

The extinction co-efficient k(ω) indicates strongest absorption at the edge and above 8 eV (in the UV region). The line shape of k(ω) spectra in

The calculated optical conductivity and absorption coefficient as a function of photon energy for pure w-ZnO with mBJLDA are shown in _{1}(ω) and ɛ_{2}(ω) are given in Equations (10) and (11).

Optical conductivity starts at 2.68 eV in both σ(ω)_{xx} and σ(ω)_{zz} spectra, which confirms that ZnO is a semiconductor. The highest optical peak is obtained at 11.7 eV in σ(ω)_{xx} and at 13.2 eV in σ(ω)_{zz}. The line shape of σ(ω)_{xx} and σ(ω)_{zz} are similar as ɛ_{2}(ω) spectra. Absorption co-efficient is another important factor to evaluate the optical properties ofa material. The peaks and valleys in the absorption curve are related to the possible transition between states in the energy bands. The absorption spectra in

The calculated real and imaginary parts of optical conductivity as a function of photon energy for pure w-ZnO with mBJLDA are shown in _{2}(ω) in all frequencies. The peaks in Re σ(ω) are mainly from the interband transitions between the occupied and unoccupied states.

We have analysed the structural, electronic and optical properties of wurtzite ZnO using Full Potential Linearized Augmented Plane Wave (FP-LAPW) method. Exchange and correlation effects are treated by PBE- GGA and mBJLDA potentials. The structural parameters show good agreement with experimental values. The

band structure calculations are done using both the exchange correlation potentials. Since mBJLDA gives better band gap than PBE-GGA, further studies are carried out with the former potential. Total and partial densities of states of ZnO are also performed to understand the relative energetic positions of electrons and to know about the hybridization and nature of bonding. From the investigation of electronic charge density, it is found that ZnO has iono-covalent bonding nature. The optical properties, such as real and imaginary parts of dielectric function, reflectivity R(ω), refractive index n(ω), extinction co-efficient k(ω), absorption co-efficient α(ω), electron energy loss function L(ω) and optical conductivity σ(ω) are calculated. Our optical properties reasonably agree with other reported experimental and theoretical results.

One of the authors (S.P) acknowledges the financial support from Department of Science and Technology-

Promotion of University Research and Scientific Excellence-phase II Fellowship. Mr. T. Samuel, application programmer of the Department of Theoretical Physics is acknowledged for his timely support during this work.

Rita John,S. Padmavathi, (2016) Ab Initio Calculations on Structural, Electronic and Optical Properties of ZnO in Wurtzite Phase. Crystal Structure Theory and Applications,05,24-41. doi: 10.4236/csta.2016.52003