Detailed check of the LDA + U and GGA + U corrected method for defect calculations in wurtzite ZnO
Introduction
Zinc oxide (ZnO), with wide band gap (3.44 eV) and large exciton binding energy (60 meV), has obtained a large interest for potential applications in optical and optoelectronic devices [1]. The behaviors of intrinsic point defects are fundamentally important. Theoretical studies of defects are useful for understanding these issues. The pivotal quantity of theoretical studies is the defect formation energy ΔH, from which one can calculate the defect concentrations and the transition levels of the defects. To date, a large number of density functional theory (DFT) calculations, which are based on local density (LDA) or generalized-gradient approximation (GGA), have been performed to elucidate the behavior of intrinsic and extrinsic point defects. However, these methods suffer from the band-gap problem (calculated energy differences suffer from the band-gap problem, see Ref. [4], Section III(A)). To overcome these problems, some authors propose to use LDA + U or GGA + U method to correct the formation energy and transition energy level [2], [3], [4]. The LDA + U or GGA + U method was originally developed to improve LDA or GGA description of Mott insulators by introducing Hubbard-type interactions in LDA or GGA via adjustable Hubbard parameters U and J, but it is also noticed that it can also improve the description of wurtzite ZnO with proper Hubbard parameter [2], [3], [4]. (In the somewhat simplified, yet rotationally invariant method of Dudarev et al. [5], these two parameters are combined into a single parameter .) Though these methods seem to be successful in some sense, wide spread of predicted transition levels are obtained by different versions of correction method, and some researchers doubt the validity of these correction methods. It is very necessary to elucidate and check the various properties of this class of methods.
In this work, we follow the methods of Janotti and Van de Walle [2], and check the dependence of various quantities (lattice parameters, electrostatic potential, valence band maximum, transition energy level and total energy) on the effective U parameter () for both zinc vacancy and oxygen vacancy. Based on the calculated results, we obtained some interesting and maybe important findings of the correction methods based on LDA + U or GGA + U.
Section snippets
Calculational method and models
The density functional calculations were carried out using the plane-wave based Vienna ab initio simulation package (VASP) [6], [7], based on the local density approximation (LDA) [8] and generalized-gradient approximation (GGA) with the functional of Perdew, Burke, and Ernzerhof (PBE) [9]. The electron wave functions were described using the projector augmented wave (PAW) method of Blöchl [10] in the implementation of Kresse and Joubert [11]. Plane waves have been included up to a cutoff
Results and discussion
Fig. 1 displays the dependence of the lattice parameters a and c on in the range 0.0–8.0 eV. The experimental values of a and c are also plotted in Fig. 1 as reference, where a is 3.248–3.250 Å, c is 5.207–5.210 Å [12]. We can see that when increases, both a and c decrease. LDA + U worsens the agreement with experiments. For GGA + U, when is taken as 4.0–5.0 eV, the calculated lattice parameters are in good agreement with the experiment values.
Second, we check the dependence of
Conclusion
We have performed a detailed check of the LDA + U and GGA + U corrected method. We found that the transition energy levels depend almost linearly on the effective U parameter. If the linear extrapolate formula is used to obtain the transition energy levels [2], the results are not sensitive to the specific value of the effective U parameter in principle. Combine all of results, GGA + U seems to be better than LDA + U, with effective U parameter of about 5.0 eV. However, though the results between LDA
Acknowledgement
The work is supported by “973 Project” (Ministry of Science and Technology of China, Grant No. 2006CB605102).
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