Detailed check of the LDA + U and GGA + U corrected method for defect calculations in wurtzite ZnO

doi:10.1016/j.cpc.2012.03.017

Computer Physics Communications

Volume 183, Issue 8, August 2012, Pages 1749-1752

https://doi.org/10.1016/j.cpc.2012.03.017 Get rights and content

Abstract

Based on a detailed check of the LDA + U and GGA + U corrected methods, we found that the transition energy levels depend almost linearly on the effective U parameter. GGA + U seems to be better than LDA + U, with effective U parameter of about 5.0 eV. However, though the results between LDA and GGA are very different before correction, the corrected transition energy levels spread less than 0.3 eV. These more or less consistent results indicate the necessity and validity of LDA + U and GGA + U correction.

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 $U_{eff} = U – J$ .) 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 ( $U_{eff}$ ) 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 $U_{eff}$ 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 $U_{eff}$ increases, both a and c decrease. LDA + U worsens the agreement with experiments. For GGA + U, when $U_{eff}$ 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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Citation Excerpt :
Therefore for the correct estimation of the band gap in case of highly correlated systems, an orbital-dependent correction term (Hubbard U) [44] is added to the exchange and correlation potential. But, the band gap is still underestimated for certain transition metal oxides like Cu2O, ZnO, TiO2, etc even after adding the U term to the 3d orbital [5,45,46]. On the other hand, the addition of Hubbard U parameter on the p-orbital of oxygen in addition to the d-orbital of the corresponding cations has resulted in the correct estimation of the band gap [43,47].
The present work deals with the first principle calculations under the framework of density functional theory to realize n-type conductivity in Cu₂O by doping it with practically feasible dopants: In and Al. From the electronic density of states and band structure calculations, it was found that both In and Al create deep level donor states at 0.84 eV and 1.32 eV respectively above the valence band maximum. The optical band gap was found to be widened from 2.04 eV for pure Cu₂O to 2.66 eV and 3.05 eV for In and Al doping respectively. The optical absorption coefficient was found to be ∼10⁶ m⁻¹ for pure Cu₂O and ∼10⁷ m⁻¹ for the doped systems. The charged defect formation energy calculation shows that both these cationic substitutional defects are thermodynamically more favourable under Cu rich conditions.
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Detailed check of the LDA + U and GGA + U corrected method for defect calculations in wurtzite ZnO

Abstract

Introduction

Section snippets

Calculational method and models

Results and discussion

Conclusion

Acknowledgement

Physica B

J. Appl. Phys.

Phys. Rev. B

Phys. Rev. B

Phys. Rev. B

Phys. Rev. B

Phys. Rev. B