Abstract
Using first principles calculations based on density functional theory and a coupled finitefields/finitedifferences approach, we study the dielectric properties, phonon dispersions and Raman spectra of ZnO, a material whose internal polarization fields require special treatment to correctly reproduce the ground state electronic structure and the coupling with external fields. Our results are in excellent agreement with existing experimental measurements and provide an essential reference for the characterization of crystallinity, composition, piezo and thermoelectricity of the plethora of ZnOderived nanostructured materials used in optoelectronics and sensor devices.
Introduction
ZnO is a wideband gap, transparent, polar semiconductor with unparalleled optolectronic, piezoelectric, thermal and transport properties, which make it the material of choice for a wide range of applications such as blue/UV optoelectronics, energy conversion, transparent electronics, spintronic, plasmonic and sensor devices^{1}. Understanding the infrared (IR) properties of the bulk crystal is crucial for the intrinsic characterization of the material and is a prerequisite for the viable application of all ZnOderived systems (e.g. nanostructures, interfaces, alloys) that are now the leading edge of research in many optoelectronic fields. In fact, at IR wavelengths, the vibrational polar modes affect also the dielectric response of the material (i.e. the optical properties) and thus its final performance. This justifies the uninterrupted sequence of experimental reports on vibrational properties of wurtzite ZnO crystal, which started in the sixties and continues nowadays^{2,3,4,5,6,7,8}.
Raman spectroscopy is one of the most common investigation techniques exploited for this purpose because it provides information on the structural and chemical properties such as orientation, crystallinity and existence of defects and impurities. However, despite the profuse research efforts pursued in the last decades, there are still open questions on the unambiguous identification of highfrequency phonon modes or unusual linewidth variations^{4} and the interpretation of angulardependent Raman spectra of ZnO: The uniaxial crystal structure along with the pronounced mass difference and the strong bond polarity imparts a large LOTO splitting that mixes phonon bands and makes the interaction with external radiation largely anisotropic. Clearly, a unified theoretical approach able to treat at the same level of accuracy both the phonon dispersion and the Raman spectra is highly desirable. Although semiempirical approaches could provide a simple and computationally inexpensive way to treat this problem^{9,10}, the parameterization of the force fields strongly reduces the transferability of the interatomic potentials, which must be carefully tested for each specific system and morphology especially in the presence of doping, defects and impurities. On the other hand, while a few attempts to reproduce ZnO phonon bands through ab initio techniques exist^{7}, no simulations of Raman spectra have appeared so far. The challenge resides in the proper treatment of the response of a highly polar material like ZnO to external electric fields.
In principle, Density Functional Perturbation Theory (DFPT)^{11,12} is an elegant and accurate approach for simulation of both phonon and Raman spectra^{14}. In practice, the required computational effort and the difficulty in improving the description of the electronic structure beyond the common functional approximations (LDA, PBE, etc.) prevent the application of this techniques for a large class of realistic structures and materials. ZnO is a prototypical example: a standard Density Functional Theory (DFT) description of the band structure of bulk ZnO severely underestimates repetition the band gap ( vs ). This is not simply due to the lack of manybody corrections typical of DFT, but rather to an unphysical enhancement of the covalent character of the ZnO bonds^{15}. In turn, the wrong description of the electronic structure affects the electrostatic properties of the system and thus the phonon distribution and the coupling with the external fields. This problem may be easily overcome by including Hubbardlike corrections^{16} or using hybrid functionals^{17}. However, these features are not readily accessible in DFPT approaches for their large computational cost^{18}.
In this paper we have solved the problem of simulating with comparable accuracy both the electronic and vibrational spectrum of ZnO using a unified finitefields/finitedifferences approach that is independent of the choice of the functional. This allows us to simulate the phonons, the high and lowfrequency dielectric constant and the nonresonant Raman spectra of any material through a finite set of electronic structure and force calculations using the finite displacement method^{19}.
Results
Unified finitefields/finitedifferences theory
The interatomic Force Constants (IFCs) K and the electronic susceptibility χ are obtained as finite differences of forces and polarizations with respect to atomic displacements or external electric fields^{20}.
IFCs are the second derivatives at equilibrium of the total energy versus the displacements of the ions u(R), or alternatively, the derivative of the atomic forces F versus the ionic displacements:
where R and R′ are Bravais lattice vectors, I, J are the Ith and Jth atom of the unit cell and α and β represent the Cartesian components. Phonon frequencies (ω_{ph}) and normal modes () are routinely obtained from the resolution of the eigenvalue equation: , where the dynamical matrix D_{Iα}_{,Jβ}(q) is the Fourier transform of the IFC. The acoustic sum rule (ASR) in real space is defined as in Ref. 12, while longwavelength dipoledipole effects are included by means of specific nonanalytical correction to the IFCs, as proposed by Yi Wang et al.^{21}.
The proper treatment of the long wavelength macroscopic polarization field requires also the evaluation of the dielectric tensor and of the Born effective charges. For this purpose we consider the electronic susceptibility:
where is the total energy in the presence of electric field and P^{el} is the electronic polarization along the direction of . Here, and P^{el} are evaluated following the method proposed by Umari and Pasquarello^{20}, where the introduction of a non local energy functional allows electronic structure calculations for periodic systems under finite homogeneous electric fields. E^{0} is the ground state total energy in the absence of external electric fields; P^{ion} is the usual ionic polarization and P^{el} is given as a Berry phase of the manifold of the occupied bands^{22}.
The highfrequency dielectric tensor is then computed as and the Born effective charge tensor is defined as the induced polarization along the direction i by a unitary displacement of the Ith atom in the j direction, or alternatively in terms of atomic forces in the presence of the electric field :
Finally, the contribution of the polar optical modes to the static dielectric constant is computed using the LyddaneSachsTeller relation^{12,13}: where ω(LO) (ω(TO)) are the LO (TO) phonon frequencies and s denotes the electric field polarizations parallel () and perpendicular (⊥) to the polar axis. This alternative method is implemented and fully integrated in the Quantum ESPRESSO suite of codes^{23} (www.quantumespresso.org; the finitefields/finitedifferences approach is implemented in the package FD in PHonon.).
The calculation of the Raman spectra proceeds along similar lines. The nonresonant Raman intensity for a Stokes process is described by the Placzek's expression^{24}:
where e_{i} (e_{s}) are the polarization of the incident (scattered) radiation, ω_{m} is the frequency of the generated optical phonon, n_{m} is the BoseEinstein distribution and is the Raman tensor, defined as:
Within the finitefield approach, the third rank tensor is evaluated in terms of finite differences of atomic forces in the presence of two electric fields^{25}:
In practice, the tensor is obtained from a set 19 calculations, which combine the finite electric fields along the cartesian coordinates. is then symmetrized to recover the full C_{6v} symmetry of the wurtzite structure.
Dielectric and vibrational properties
The calculated high and low frequency dielectric constants, summarized in Table I, are in agreement with experimental spectroscopic ellipsometry data^{3}, as well as HartreeFock calculations^{26}. Born effective charges (Table I) are also in agreement with values extracted from transverse IR reflectance spectra^{27} and may be easily associated to piezolectric properties of ZnO and its modifications under pressure^{28}.
PBE and PBE + U phonon dispersions are reported in Fig. 1 and phonon frequencies at the Brillouin Zone center (Γ) are summarized in Table II, where we also display the LDA values for comparison. Experimental Inelastic Neutron Scattering (red diamond and blue circles) and Raman (green squares) data are extracted from Ref. 7 and superimposed in Fig. 1 to the first principles data. While LDA and PBE results clearly underestimate the experimental frequencies, mostly in the optical branches, with PBE + U the agreement is very good. Here only the highestfrequency LO bands are slightly overestimated. Notably, these phonon branches suffer of anharmonic renormalization effects, which are not included in the present approach. Furthermore, the error bar on the experimental Raman results is higher for these bands, due to the low crosssection of the corresponding phononphoton scattering processes (see below).
Following the irreducible representation of the wurtzite symmetry group C_{6v}, the phonon modes of ZnO can be classified as Γ = 2A_{1} + 2B_{1} + 2E_{1} + 2E_{2}. One low energy A_{1} and one double E_{1} modes correspond to the transverse and longitudinal acoustic branches, the others are optical modes. B_{1} modes are IR and Raman inactive. The two E_{2} modes are non polar IR inactive but Raman active. In A_{1} and E_{1} polar modes, atoms move parallel and perpendicular to the caxis respectively and are the ones responsible for the LOTO splitting. The labeling in Fig. 1 and Table II is based on the direct analysis of the phonon eigenmodes and their underlying symmetry. Note that PBE calculations give an incorrect ordering of the B_{1} and A_{1}(LO) modes that is instead correct in PBE + U (see Table II).
Raman spectroscopy
We calculated the Raman spectra for different propagation (k) and polarization (e_{i}_{,s}) directions of the incoming and scattering light, as shown in Fig. 2. We simulated the Raman spectra for different rotation angles (θ) of the polarization vectors in the planes parallel (aplane) and perpendicular (cplane) to the caxis. Following the standard experimental setups, for cplane spectra we considered only parallel polarization of the incoming and scattered light . We used the Porto notation: symbols inside the parentheses indicate the direction of e_{i} and e_{s}, while symbols outside the parentheses indicate the directions of propagation of light (k). In the aplane case, two different orientations of e_{i}_{,s} can be experimentally detected: and .
Results for room temperature boson distribution are summarized in Fig. 2. Panel (a) shows the aplane spectra for parallel polarization vector configurations, which reproduces the angular modification of experimental spectra^{3,8} and the reciprocal intensity ratio of the main peaks. The spectra are, in fact, dominated by Stokes processes due to the A_{1}(TO) and phonon modes at 384 and 436 cm^{−1}, respectively, only 2 cm^{−1} from the experimental values. The intensity of the A_{1}(TO) peak is independent from the orientation of incident polarization, while the peak reaches a maximum at 90° and minima at 0° and 180°. For 20°, 40°, 120° and 140° a small contribution from E_{1}(TO) also appears at 410 cm^{−1}. The phonon mode at 106 cm^{−1} is not visible in this energy scale, the LO modes are not allowed by symmetry for this incident light direction.
Similar arguments apply for the aplane spectra for e_{i} ⊥ e_{s} polarizations, as displayed in panel (b). Here the main contributions stem from the E_{1}(TO) and modes, whose reciprocal intensity depends on the θ angle. The two peaks are in counter phase: the former has maxima at 0°, 90° and 180°, where the latter reaches the minimum and vice versa. Besides these two peaks, experimental results for crossed polarization vectors exhibit small, angulardependent features associated to A_{1}(TO) and E_{1}(LO) that should be symmetry forbidden in this geometry. This is indeed a consequence of the deviation from the ideal geometry setup and/or a deviation from perfect hexagonal crystallinity of the sample. Following Ref. 8 we considered nonorthogonal polarization vectors,
through the inclusion of an arbitrary constant p. The angular distribution and intensity of the peaks depend on the choice of the p factor, i.e. on the characteristic crystallinity of the sample. Results for p = 0.5 are reported in the inset of panel (b) where the A_{1}(TO) peak and a small contribution from E_{1}(LO) appear.
From the spectra of Fig. 2 we can also extract the relative intensity ratio between the main Stokes peaks, which are univocally related to the nonzero matrix elements of the Raman tensor for the C_{6v} point group (see for instance Ref. 29). Calculated ratios between A_{1} (E_{1}) and , taken as reference, are 0.54 and 0.23, respectively; the corresponding experimental values are 0.6 and 0.4^{8}. The agreement is quite satisfactory for A_{1}, while it is slightly worse for E_{1}. However, in the latter case the experimental value is extracted from e_{i} ⊥ e_{s} polarizations spectra, which may suffer of nonideal hexagonal symmetry, as discussed above (Fig. 2b).
Discussion
Our results demonstrate that the coupled finitefields/finitedifferences DFT approach is essential to accurately evaluate the dielectric properties, phonon dispersions and Raman spectra of ZnO. By reducing high order derivatives of the total energy (i.e. of the charge density) to lower order differences of polarizations and forces obtained from a limited set of selfconsistent DFT simulations, it favors the inclusion of advanced corrections to the groundstate electronic structure that are essential for the proper description of the system. In particular, the ground state electronic structure and the coupling with external fields in ZnO require the use of ad hoc corrections to the position of the dbands of Zn that can be achieved with a DFT + U treatment, as summarized in Fig. 3. In the case of standard LDA and PBE calculations we observe a severe underestimation of the bandgap (E_{gap}) and a wrong energy position of the dbands of the Zn atoms, that in turn give an inadequate description of the phonon dispersions (see Fig. 1(b) and discussion therein). The choice of functional influences directly the electrostatics and thus also the characteristics of the vibrational response of the system. In fact, the IFCs are the sum of two contributions:
where are the second derivatives of Ewald sums corresponding to the ionion repulsion potential, while the electronic contributions are the second derivatives of the electronelectron and electronion terms in the ground state energy. It is the last term that reflects directly in the vibrational spectrum the wrong position of the dstates and the dramatic underestimation of the gap. On the other hand, LDA, with its overestimation of the bonding (shorter lattice parameter and stiffer IFCs) accidentally gives results that are marginally closer to the experimental data thanks to a fortuitous cancellation of errors. The inclusion of the Hubbard U, by correcting the dZn pO hybridization, properly describes not only the electronic bands but also the vibrational and dielectric properties in agreement with the experimental results^{15}. Notably, the values of the dielectric constants obtained with LDA (, ) and PBE (, ) are much higher than in experiments, a clear indication of a lesser polarity of the system in these approximations.
Finally, the finitefields/finitedifferences framework allows straightforward developments of the theory to include 3rd order terms in the IFCs (i.e. anharmonic contribution to phonons) or susceptibility χ^{(2)} for the simulation of nonlinear optical properties such as hyperRaman or electrooptic effects. One example of the latter is the Raman signal along the parallel cplane , shown in Fig. 4, where the A_{1}(LO) and modes dominate the spectrum for all angles (0–180°) (black solid line). However, although allowed by symmetry, A_{1}(LO) is invisible or hardly visible in most experimental data collections^{2,3,6,8}. This effect is due to a competitive interaction with the Fröhlich term^{2}, which couples with the macroscopic field generated by the A_{1}(LO) mode and cancels most of the deformation potential contributions. This effect may be evaluated including a nonlinear correction to the tensor for the longitudinal q_{l} mode^{30}
where the third order tensor may be straightforwardly calculated extending the finite field approach to the case of three electric fields, or alternatively:
Indeed, the inclusion of the Fröhlich term suppresses the A_{1}(LO) (red dashed line) in agreement with experiments.
In conclusion, we have demonstrated that our unified finitefields/finitedifferences technique provides a general framework for the calculation of the vibrational, dielectric and Raman properties of materials, without any limitation in the choice of the DFT framework. We have applied this method to the study of ZnO, a material that for its complex electronic structure requires a special treatment in order to obtain a faithful representation of its properties. Our results are in excellent agreement with available experimental data.
Methods
Computational details
We calculated vibrational and Raman spectra of wurtzite ZnO bulk using PBEGGA^{31} ultrasoft pseudopotentials^{32} for all the atomic species, a 28 (280) Ry energy cut off in the plane wave expansion of wavefunctions (charge) and a (12 × 12 × 8) regular kpoint mesh. LDA calculations have been done with normconserving TroullierMartins pseudopotentials^{33}, with a kinetic energy cutoff of 100 Ry. 3d electrons of Zn have been explicitly included in the valence for both cases. For PBE + U, a Hubbardlike potential with U = 12.0 eV and U = 6.5 eV is applied on the 3d orbitals of zinc and on the 2p orbitals of oxygen, respectively^{16}. Note that the Hubbard U values included in the calculations do not have to be considered as physical onsite electronelectron screened potentials, in the sense of the manybody Hubbard Hamiltonian, but as empirical parameters introduced to correct the gap. This is an efficient and computationally inexpensive way to correct for the severe underestimation of the bandgap and the wrong energy position of the dbands of the Zn atoms^{15,34}. This procedure has also been applied to other similar systems^{35,36,37,38}. P^{el} and α^{m} are obtained applying a.u along the cartesian axes. All calculations have been run with lattice parameters optimized for each functional^{16}.
For the calculation of phonon modes we considered a (8 × 8 × 2) hexagonal supercell (512 atoms) and we calculated forces (i.e. IFC) displacing only 2 atoms in primitive ZnO cell along the two inequivalent directions. This corresponds to a set of 4 DFT calculations. The Raman tensor has been calculated on the primitive wurtzite cell.
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Acknowledgements
We would like to thank Michele Lazzeri, Paolo Umari and Marco Fornari for support and useful discussions and Rosa Di Felice for a critical reading of the manuscript. Rita Stacchezzini provided valuable graphical aid. This work was supported, in part, by SRC through Task ID P14924 in the Center for Electronic Materials Processing and Integration at the University of North Texas and ONRMURI N000141310635. Computational resources have been provided by the UNT HPC Initiative and the Texas Advanced Computing Center (TACC) of the University of Texas, Austin.
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A.C. and M.B.N. designed the research, performed the calculations, analyzed the data and cowrote the paper. M.B.N. wrote the finitedifference codes for phonon calculations. A.C. wrote the code for the finitefield calculation of the Raman tensor.
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Calzolari, A., Nardelli, M. Dielectric properties and Raman spectra of ZnO from a first principles finitedifferences/finitefields approach. Sci Rep 3, 2999 (2013). https://doi.org/10.1038/srep02999
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DOI: https://doi.org/10.1038/srep02999
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