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 P63mc. 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 V0, bulk modulus B0, pressure derivative of bulk modulus
respectively. In these expressions, E0 is the total energy; V0 is the equilibrium volume; B0 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, kmax = 7.0/RMT, where RMT denotes the smallest atomic sphere radius (muffin-tin radius) and kmax 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 lmax = 10, while the charge density is Fourier expanded up to Gmax = 12 (Ryd)1/2. The RMT 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 (a0 and c0), anion position parameter u (which governs the positions of oxygen ions), equilibrium volume (V0), bulk modulus (B0) 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
a0 (Å) | c0 (Å) | c/a | u | V0 (Å/f. u.) | B0 (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 Eg using PBE-GGA and mBJLDA are 0.814 eV and 2.683 eV respectively. F. Tran and P. Blaha have reported the Eg of 0.75 eV and 2.68 eV using the LDA and mBJLDA potentials, respectively [
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, ɛ1(ω) and ɛ2(ω) are real and imaginary part of the dielectric function. Real part of the dielectric function ɛ1(ω), means the dispersion of the incident photons by the material, while the imaginary part of the dielectric function ɛ2(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 ɛ2(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. ɛ2(ω) can be calculated using the following expression [
where, M is the dipole matrix; i and j are initial and final states respectively; fi is the Fermi distribution function for the ith state; Ei is the energy of electron in the ith state and ω is the frequency of the incident photon. Real part ɛ1(ω) of the dielectric function can be found from its corresponding ɛ2(ω) by Kramers-Kronig transformation in the form [
where, P stands for the principle value of the integral.
In
The static dielectric constant value (the value of the dielectric constant at zero energy) of ɛ1(0) is 2.8. A higher value of energy gap produces a smaller ɛ1(0), which can be explained on the basis of the Penn model [
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 (ωpl), above which the material shows the dielectric behaviour [ɛ1(ω) > 0], while below which the material exhibits the metallic property [ɛ1(ω) < 0]. The peaks in L(ω) spectra reveal that the point of transition from the metallic property to dielectric property for a material [
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) = ne(ω) - no(ω), where ne(ω) is the index of refraction for an electric field oriented along the c-axis and no(ω) is the index of refraction for an electric field perpendicular to the c-axis [
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
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 ɛ2(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
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