Correlation Effects on Lattice Relaxation and Electronic Structure of

Article pubs.acs.org/JPCC

Correlation Effects on Lattice Relaxation and Electronic Structure of ZnO within the GGA+U Formalism Xinguo Ma,*,†,‡ Ying Wu,§ Yanhui Lv,‡ and Yongfa Zhu*,‡ †

School of Science, Hubei University of Technology, Wuhan 430068, China Department of Chemistry, Tsinghua University, Beijing 100084, China § Department of Physics, University of Bath, Bath, BA2 7AY, United Kingdom ‡

ABSTRACT: The electronic structures of ZnO were calculated using density functional theory, in which the electronic interactions are described within the GGA+U (GGA = generalized gradient approximation) formalism, where on-site Coulomb corrections are applied on the Zn 3d orbitals (Ud) and O 2p orbitals (Up). The relaxed GGA+U calculation can correct completely the band gap, the position of Zn 3d states, the transition levels of O vacancy in band gap, and so on, which is different from other GGA+U (equivalent LDA+U) calculations partially correcting the energy band structure for fixed lattice constants. By comparing with experimental data, the pair of Ud = 10 and Up = 7 eV was identified as an optimum choice for the energy band structure of W-ZnO. Then, the proper pair of Ud and Up parameters was taken to predict the energy band structure of ZB- and RS- ZnO, of which the former is in good agreement with experimental values, and the latter is in dispute, relating to the decrease of the octahedral symmetry. Subsequently, we pay special attention to the possible causes of the decrease of lattice constants deriving from the +U correction. Further, the formation energies and transition levels of O vacancy in W-ZnO were calculated using three different schemes to address possible routes to presenting the defect states in band gap. Our results provide some guidance for improving electronic structure of ZnO using the GGA+U approach.

1. INTRODUCTION ZnO is probably one of the most studied compounds in materials science, due to its versatility and large range of applications from optoelectronic devices1−3 to photocatalysts.4,5 At ambient conditions, the thermodynamically stable phase is wurtzite (W-ZnO), with an experimental band gap of 3.4 eV. The zinc blende structure ZnO (ZB-ZnO) can be stabilized only by growth on cubic substrates, and the rocksalt structure ZnO (RS-ZnO) may be obtained at a relatively high pressure of about 9.5 GPa.6,7 It is well-known that the electronic structure of a given semiconductor is pivotal in determining its potential utility. Consequently, an accurate knowledge of the band structure is critical if the semiconductor in question is to be incorporated in the family of materials considered for device applications or photocatalysis. Even now, the band gap value of the RS-ZnO is still in dispute between Segura’s experimental studies8,9 and other theoretical calculations.10−12 The W-ZnO is the most studied, and its defect physics has already received considerable experimental and theoretical attention. However, the electronic structure of ZnO with oxygen vacancy is not yet firmly established.6,13 The differences of these results are mainly of two respects: the value of defect formation energy and the position of transition levels in band gap.14−23 Many researchers had calculated the formation energy of oxygen vacancy in ZnO; however, their results are inconsistent due to different methods used for © 2013 American Chemical Society

density functional theory (DFT) calculations. Especially, part of the previous calculations for the oxygen vacancy in ZnO presents a negative-U behavior,6,13−16 which means that the system has lower energy in the neutral state than in the 1+ states, due to a much more efficient structural relaxation in the neutral state. In addition, some researchers think that the (2+/ 0) transition level lies in the upper half of the band gap;13−17 however, others propose that it lies in the lower half of the band gap.18−22 In the past decade, many of theoretical investigations employed the local density approximation (LDA)24 and the generalized gradient approximation (GGA)25 to calculate the electronic structures of wide band gap transition metal oxides. However, they fail to give even reasonably accurate results for their band characteristics and especially band gap values.26 These approaches underestimate the binding energy of the semicore d states and consequently overestimate their hybridization with the anion p-derived valence states, enhancing the effects of the p−d coupling. The artificially large p−d coupling pushes up the valence band (VB) edge, leading to a reduction of band gap. The GGA+U (or LDA+U) approach aims to correct for this by adding an orbital-dependent term to the Received: July 23, 2013 Revised: November 12, 2013 Published: November 21, 2013 26029

dx.doi.org/10.1021/jp407281x | J. Phys. Chem. C 2013, 117, 26029−26039

The Journal of Physical Chemistry C

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difficulties of some previous schemes and may provide some guidance for improving electronic structure of ZnO using the GGA+U approach.

exchange and correlation potential. This is the true for the highly correlated systems exhibiting strong effective on-site Coulomb interactions (U) between localized elections, such as TiO2, MnO2, CoO, ZnO, and so forth. Previous +U studies did point to the fact that the band gap of transition metal oxide strongly depends on the Hubbard U parameters.27 When the correction is only applied on the d orbitals of transition metal, however, the band gap is still underestimated compared with the experimental one, even at a high U value.28−30 Recently, a few theoretical studies on transition metal oxides discussed the effect of U parameter on the p orbitals (Up) of oxygen in addition to the d orbitals (Ud) of the transition metal.29−32 Besides the +U approach, the recent theoretical developments of exchange and correction functionals,33−38 such as the Heyd− Scuseria−Ernzerh (HSE) of screened hybrid density functional33−35 and GW implementations,11,36,38 have shown a correction of energy band structure of ZnO for some of severe shortcomings of the LDA and GGA. However, the underestimation of band gap and the under-binding of Zn 3d still prevails. In addition, these approaches spend a lot of time calculating electronic structure of dozens or hundreds of atom systems. By comparing them, it can been identified that the +U approach is a good choice for quick calculations of some relative large systems.28−38 A previous work has recommended a variety of U values for the Zn 3d orbitals based on a number of experimental data published regarding the electronic states of W-ZnO. Depending on the experimental quantity fitted, the U values for the Zn 3d orbitals were in a wide range from 4 to 12 eV.29−32 There is no consensus regarding the procedure by which U is chosen or the best value for U. To determine U, most of the previous work proposed that the energies of some of the Kohn−Sham orbitals match the excitation energies measured by “one-electron spectroscopy”, such as the band gap of ZnO, the defect states in band gap of reduced ZnO, or the Zn 3d levels in VB from the XPS spectrum. Paudel and Lambrecht18 and Lany and Zunger19 discussed that the LSDA+U approach applied to both of Zn 3d and 4s orbitals. However, one should first realize that a rather large and seemingly unphysical Us (43.54 eV) is required. In addition, the shift of band edge induced by the Us occurs mainly at the Γ-point only. Instead of a rigid shift, the gaps at other k points, such as the K- and M-points, are not raised as much, which leads to an overall wrong curvature of the lowest conduction band (CB), with an overestimated effective mass (see Figure 1 of ref 18). In particular, since the VO defect levels containing contributions from several host VB states at different k-points are deep, they are not sufficiently raised along with the band gap correction due to much less sensitive to computational model than with respect to the band-edges.21,39 Here, we report the calculated results of electronic properties of ZnO using first-principles DFT incorporating the GGA+U formalism, where U parameters are applied on the Zn 3d orbitals and O 2p orbitals. First, we calculate the energy band structure of W-ZnO using the GGA+U methods under the fixed or relaxed lattice constants. Here, the band gap and the absolute energy positions of Zn 3d states are compiled for ZnO based on experimental data. We analyzed the mechanism about the decrease of lattice constants deriving from the +U correction. Then, the U parameters which match the electronic structure of W-ZnO are taken to test the energy structure of other two phases: ZB- and RS-ZnO. Further, the formation energies and the transition levels of O vacancy of W-ZnO in band gap were discussed. Our strategies can overcome the

2. CALCULATION METHODS AND MODELS All calculations were performed using the projector augmented wave (PAW) pseudopotentials40 with the exchange and correlation in the Perdew−Burke−Ernzerhof (PBE)25 formalism of DFT as implemented in the Vienna ab initio simulation package (VASP). In addition, the Ceperley−Alder (CA)41 formalism was also used to obtain the structure parameters and bond overlap population of ZnO. The valence atomic configurations are 4s23d10 for Zn and 2s22p4 for O, respectively. Geometry optimizations were done before single point energy calculations, and the self-consistent convergence accuracy was set at 1 × 10−6 eV/atom. The convergence criterion for the maximal force on atoms is 0.02 eV/Å. The maximum displacement is 5 × 10−4 Å, and the stress is less than 0.02 GPa. For a perfect crystal of ZnO, a cutoff energy is 500 eV, and the Monkhorst−Pack k-meshes of 14 × 14 × 10, 12 × 12 × 12, and 12 × 12 × 12 are used for W-, ZB-, and RS-ZnO, respectively. The unit cell structures of three phases are shown in Figure 1.

Figure 1. Unit cell structure of W-ZnO (a), ZB-ZnO (b), and RS-ZnO (c). The big gray ball denotes a Zn atom, and the small red one is an O atom.

The GGA+U method was taken for the description of the exchange and correlation energy of electrons.27 As is wellknown, the Hubbard-U correction was first introduced by Anisimov et al.42 to deal with open shell narrow d or f bands. One can also apply it to a full shell band such as the Zn 3d band in ZnO, the Cu 3d band in Cu2O, and so on. The spherically averaged approach to the GGA+U, proposed by Dudarev et al.,27 can be understood as adding an additional term in the total energy functional given by EGGA + U = EGGA +

(U − J ) 2

∑ [(∑ ρjjσ̂ ) − (∑ ρjiσ̂ ρiĵσ )] σ

j

j,i

(1)

ρσji

where is the density matrix of the orbitals treated by GGA +U. U and J are the spherically averaged matrix elements of the screened Coulomb electron−electron interaction. The additional term forces the on-site occupancy density matrix in the direction of idempotency, that is, ρ̂σ = ρ̂σ ρ̂σ. The matrix of the one-electron potential is given by the derivative of eq 1 with respect to ρ̂σji, V jiσ =

δEGGA ⎡1 ̂ ⎤ δji − ρjiσ̂ ⎥ σ + (U − J )⎢ ⎣ ⎦ 2 δρij

(2)

where = njδ̂ji and the corresponding elements j are adjusted by the occupation numbers nj. For ZnO, J = 0 is set here, due to ρ̂σji

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a closed shell of atomic orbitals and adjust U only. So the oneelectron potential obtains a downward shift by −U/2. Equation 1 bridges the orbital-dependent formulation by Anisimov et al.42 with the rotationally invariant functional proposed by Liechtenstein et al.,43 retaining the simplicity of the first and the covariant character of the second. To obtain ideal electronic structure, an on-site Coulomb correction has been applied on the O 2p orbitals, besides on the Zn 3d orbitals. Using these methods, satisfactory results have been obtained in our previous studies of Ag3PO4.30 Here, we applied various Ud and Up to investigate their effect on the energy band structure of ZnO and find the optimum Ud and Up parameters. Note that our values for Hubbard U correspond to the differences of U−J, where U and J denote the intrasite Coulomb and exchange interactions, respectively. To introduce an isolated O vacancy, an interior O atom is removed from the supercell consisting of a 3 × 3 × 2 periodic repetition of the primitive unit cell, so that it includes 36 Zn and 35 O atoms. A larger supercell of 4 × 4 × 3 containing 96 Zn and 95 O atoms was used to check the finite-size effects, and the difference of defect formation energies between two supercell systems is only 0.08 eV. In fact, for the 3 × 3 × 2 supercell system, the distance of the nearest defects is more than 9 Å, indicating that the model is large enough to calculate the electronic structure of the defect system.17,19 A cutoff energy of 400 eV and the Monkhorst−Pack k-mesh of 4 × 4 × 4 are used in the calculations of the defect supercell system.

Table 1. Structure Parameters and Bond Overlap Population of ZnOa type

method

c (Å)

ε3d (eV)

Eg (eV)

W-ZnO

PAW-PBE

3.259

5.218

−5.2

0.78

(P63mc)

PAW-CA

3.187

5.108

−4.7

0.62

GGA+Ud+Up

3.052

4.912

−8.5

3.40

3.242 3.286 3.148 3.196 3.257

5.188 5.241 5.075 5.149 5.223

−8.2 −7.5

4.587

−6.5 −7.2 −5.9 −6.7 −5.4

3.30 3.44 0.38 1.51 1.81 3.18 3.20 1.22

ZB-ZnO

exp. exp. LCAO LDA+Ub PBE+Uc PBE0 scGWd PAW-PBE

(F4-3m)

PAW-CA

4.483

−5.0

1.02

GGA+Ud+Up

4.276

−9.2

3.25

e

exp.

3. RESULTS AND DISCUSSION 3.1. Evaluation of U. By minimizing the crystal total energy, the equilibrium lattice parameters of three phases of ZnO have been calculated using LDA and GGA approaches, and the results are given in Table 1 with the experimental values.44−54 The former approximation, LDA, underestimates the lattice constant and overestimates cohesive energies relative to the experimental values, while the opposite is true for GGA. Our calculated equilibrium structural parameters are in good agreement with experimental values within deviations of 2% in the lattice constants. A little difference of the lattice constants among experimental data in Table 1 is attributed to a different modification condition and characterization technique.50 Based on the equilibrium lattice parameters of W-ZnO, first, we calculated the energy band structure and density of states (DOS) spectra and found that minimum direct band gaps were about 0.78 and 0.62 eV at G for GGA-PBE and LDA-CA, respectively, which were significantly smaller than the experimental values reported previously due to well-known limitations in DFT. Especially, these results do not present correctly the actual position of the Zn 3d levels. Figure 2a shows that the VB consists of two separated regions, the widths of 2.6 and 4.1 eV, respectively. Combining the DOS, the lower part is mainly from the Zn 3d states and a little from hybridized O 2p states, while the upper part is mainly from O 2p states and partially from Zn 3d states. The lower part of CB is dominated by the Zn 4s states. Although the GGA calculations improve the discrepancy in band gap and Zn 3d levels, the energy for the transition metal d states is too high, which results in their overhybridization with the p(O) states from the top of VB. The use of GGA+U approach would decrease both hybridization and the energy of Zn 3d levels. In most GGA+U (or LDA+U) studies for metal oxides, the Ud values were fitted in an empirical way, according to the band gap, the position of special atom orbitals, and/or defect states

a (Å)

RS-ZnO

exp. exp. PAW-PW91 GW PAW-PBEf

4.463, 4.37, 4.47 4.580 4.400 4.627 4.530 4.248

(Fm3m)

PAW-CAf

3.27

ref this work this work this work 44 45 46 47 48 34 49 this work this work this work

50

−5.5

3.10 3.17 0.64 2.53 0.84

4.160

−5.3

0.52

GGA+Ud+Upf

4.087

−9.4

4.05

exp. PBE PAW-PW91 GW GW

4.237 4.334 4.280 4.250

2.45 1.29 0.75 4.51 4.27

51 52 53 11 this work this work this work 8, 9 54 53 10 11

a

Eg denotes the band gap energy. The data in this work are gained using the PAW method with PBE and CA formalism. The Ud and Up are taken as 10 and 7 eV in GGA+Ud+Up calculations. bUeff = 4.7 eV. c Ueff = 7.5 eV. dIncluding the electron−hole interaction. eMeasured by using RHEED, XRD, and TEM. fUnder 10 GPa in present work.

in band gap. To investigate the effect of U for Zn 3d and O 2p orbitals, we employed several calculations, in which a range of Ud and Up values was chosen, as shown in Figures 3 and 4. There are two strategies: fixed or relaxed lattice constants in GGA+U calculations. When the lattice constants are fixed, a = 3.259 and c = 5.218 Å in GGA+U calculation, if the U is incorporated only for the Zn 3d orbitals, the band gap energies increase with the increasing Ud values from 0 to 14 eV, from a value of 0.78 eV for the conventional GGA to 2.27 eV for GGA +Ud, as shown in Figure 3a; however, the band gaps are still underestimated compared with the experimental value of 3.4 eV. When the Ud value is more than 14 eV, the self-consistent convergence cannot be completed at any accurate sets of our DFT calculations. Figure 3a also shows a great fluctuation of total energies with the increasing Ud values, which indicates that the GGA+Ud approach may not be adequate to describe the electronic structure of ZnO. A corrected band gap energy can be obtained, however, if in addition to the U values for d orbitals the U value is included 26031



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relation between the band gap energies and U values for GGA +Ud+Up calculations, where Ud is chosen for 10 eV, the Up ranges from 0 to 12 eV, and the above-mentioned lattice constants are also fixed. Although the band gap energies increase with the increasing Up values, they are still significantly smaller than the desired experimental values. No fluctuation of total energies with the increasing Up values is present in Figure 3b, which indicates that the addition of Up can improve the total energies of systems. How can we increase the band gap to the desired experimental values? In a succeeding work, we attempt to relax freely the geometry crystal structure of ZnO in GGA+U calculations. Figure 4a shows that, with the increase of Ud value, the band gap energy increases steeply, and the lattice constants decrease a lot. We note that the GGA+U approach systematically produces smaller lattice constants than GGA, which is due to the fact that the application of U causes the d states become more localized, and thus results in small lattice constants. We found that the Ud = 12.5 eV is proper to gain a corrected band gap, where the lattice constants are a = 3.020 and c = 4.812 Å, smaller than general experimental values. But we found that the highest peak of Zn 3d states is at about −9.5 eV, which is lower than experimental results. Göpel et al.45 and Powell et al.55 had carried out angle-integrated and UV photoemission measurements on W-ZnO cleaved in vacuum, respectively, which both placed the Zn 3d core level at about 7.5 eV below the VB maximum (VBM), is slightly different with the X-ray photoemission results reported by Ruckh et al.44 (8.2 eV), Vesely et al.56 (8.5 eV), and Ley et al.57 (8.81 eV). So another alternative scheme must be taken. Subsequently, we performed the calculation of energy band structure of ZnO for relaxed configuration using GGA+Ud+Up. Our results show that the band gap energies increase with the increasing Up values, where the optimum Up is 12.5 or 7 eV if Ud value is 8 or 10 eV, respectively. We note that, to describe consistently electronic structures in Al doped SiO2 and Li doped MgO,58,59 a Up value of 7 eV is sufficient. This suggests that the description of O 2p states is independent of oxide materials. More importantly, there exist experimental data in which the Up−p for O 2p states has been determined. Auger spectroscopy studies of different oxides demonstrated that the on-site Coulomb interaction energy for a hole in an O 2p orbital is 5−7 eV, entirely consistent with the present work.60,61 Therefore, we can now suggest that, for oxide materials, GGA+U with a Up value of 7 eV will be suitable for first-principles calculations, even where experimental data are lacking, providing a predictive capability for GGA+U approach. The effect of U parameters on Zn 3d states is discussed in detail in a future work. If we fixed lattice constants, the highest peak of Zn 3d states moves down evidently with the increasing Ud values, while the O 2p states only have a little downward shift, due to the p−d coupling become weaker and the localization of Zn 3d states become stronger, as shown in Figure 5a. Figure 5b presents that the effect of Up on the Zn 3d states is not obvious. However, the upper part of VB from the hybridized O 2p and Zn 3d states shifts downward a little, and the CB minimum (CBM) is almost pinned, resulting in an increase of band gap. If we relax freely the geometry structure of ZnO in GGA+Ud calculations, the addition of Ud leads to the downward shift of the Zn 3d and hybridized O 2p states, and the upward shift of Zn 4s states, which is consistent with Figure 4a of ref 47, as shown in Figure 6a. Although the band gap energy is about 3.4 eV when the Ud is 12.5 eV, but the Zn 3d

Figure 2. Energy band structure of W-ZnO calculated from GGA (black solid lines), GGA+Ud (red dashed lines), and GGA+Ud+Up (blue dashed lines) approach. The right one (b) is an enlarge energy band structure, in the energy range from −1.5 to 7.5 eV. The Fermi energy is not normalized to zero. The inset shows the distribution of the four oxygen atoms for the studied point group (C6v) and the stacking sequence of closed-packed diatomic plane, (0001) planes for W-ZnO.

Figure 3. Band gap energies and free energy of W-ZnO as a function Ud (a) and Up (b). For latter, the Ud is fixed as 10 eV. In this figure, the lattice constants are fixed as a = 3.259 and c = 5.218 Å. E0 in a and b represent the total energy per two ZnO units of −17.872 eV for the GGA and −17.851 eV for the GGA+U (Ud = 10, Up = 0 eV), respectively.

Figure 4. Band gap energies and lattice constant a of W-ZnO as a function Ud (a) and Up (b) after geometry relaxation. For latter, the Ud is fixed as 10 eV. The dashed arrows specify the U parameters and lattice constants corresponding to the experimental band gap.

for p orbitals as well. This raises an interesting question regarding the predictive ability of GGA+U approach used in this work, in which the U value is required to be determined empirically through fitting to experimental data. In fact, different ways of determining U or the form of localized orbitals can impact on the results of a GGA+U calculation. With this in view, we performed extensive tests to determine the appropriate U parameters for Zn 3d and O 2p orbitals to reproduce the corrected energy based on correct band gap energy and Zn 3d DOS peak in ZnO. Figure 3b shows the 26032



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be predicted accurately. Here, the calculations of ZnO at relatively modest external hydrostatic pressures were performed by applying different hydrostatic pressures and minimizing the enthalpy: H = U + PV with respect to all structural parameters. Transition pressures were taken from the curve crossings, that is, those pressures where the enthalpies of two phases coincide, and no phase has a lower enthalpy. From the enthalpy− pressure behavior now shown for each crystal phase in Figure 7,

Figure 5. TDOS and PDOS of W-ZnO obtained by (a) GGA+Ud and (b) GGA+Ud+Up for fixed lattice constants. In case of b, Ud is fixed as 10 eV. The Fermi energy is not normalized to zero. Due to complete symmetry between spin-up and spin-down states, we only showed spin-up DOS.

Figure 7. Enthalpy versus pressure data obtained by the GGA (red lines) and GGA+Ud+Up (Ud = 10, Up = 7 eV) (blue lines) approaches for the high-pressure forms ZB- and RS-ZnO are shown relative to that of W-ZnO. Thus, the relative enthalpy of W-ZnO is set to zero for the GGA and GGA+Ud+Up approaches, respectively.

we can deduce the transition pressures of the crystal phase, which depend on the calculated methods. The calculated results show that the enthalpy of ZB-ZnO is almost tantamount to that of W-ZnO at each pressure, indicating the ZB phase is relatively stable. Because of the tetrahedral coordination of wurtzite and zinc-blende structures, the four nearest neighbors and twelve next-nearest neighbors have the same bond distance in both structures. The main difference between these two structures lies in the stacking sequence of closed-packed diatomic planes. The experimental results show that the ZB-ZnO structure is metastable and can be stabilized by heteroepitaxial growth on cubic substrates, such as ZnS62 and GaAs/ZnS,50 reflecting topological compatibility to overcome the intrinsic tendency of forming a wurtzite phase. The present calculations predict that the wurtzite and zincblende to rocksalt transformations all occur at about 10.7 GPa in GGA calculations and 9.2 GPa in GGA+Ud+Up calculations. The results are in reasonable agreement with experimental observations.6,7 The structural parameters of three phases at stable conditions are shown in Table 1. We note that, compared with GGA, the GGA+Ud+Up approach induces the decrease of the lattice constants from 4.587 to 4.262 Å for ZBZnO and from 4.298 to 4.021 Å for RS-ZnO, respectively, which are about 5% smaller than the experimental values. Of course, the discrepancy in our calculated values is larger than the experimental measured one. By comparing the energy band structure of Figures 2 and 8 for all three phases of ZnO calculated using the GGA functional, we note that the position of the Zn 3d levels and the overlap of Zn 3d and O 2p levels is similar for W- and ZBZnO. For ZB-ZnO, a minimum direct band gap of about 0.66 eV was found at G in the GGA calculation, which is significantly smaller than of the experimental value (3.10− 3.27 eV),50−52 as shown in Figure 8a. For RS-ZnO, a minimum direct band gap of about 0.95 eV was found at G. However, the energy difference of VBM at L and G points is 0.09 eV. In other

Figure 6. TDOS and PDOS of W-ZnO obtained by GGA+Ud (a) and GGA+Ud+Up (b) after geometry relaxation. In case of b, Ud is fixed as 10 eV. The Fermi energy is not normalized. Due to complete symmetry between spin-up and spin-down states, we only showed spin-up DOS.

states move too much, to −9.5 eV, comparing with experimental data (average value 8.27 eV) and thus produce an unphysical description of the electronic interactions (a oversize split of Zn 3d orbitals, see the bottom of Figure 6a). The addition of Up only has a little effect on the s states due to the weak hybridization of O 2p and Zn 4s orbitals, which leads to a tiny downward shift of O 2p states and a small upward shift of Zn 4s states. We note that first that the pair of Ud = 10 and Up = 7 eV is an optimum choice to gain the desired band gap and the position of Zn 3d levels, as shown in Figures 4b and 6b. Furthermore, comparing the energy band structure near the top of VB and the bottom of CB calculated using GGA, GGA+Ud, and GGA+Ud+Up, we find that the three curves at VBM are almost overlapped, and the three curves at CBM have a similar shape (the same dispersion with k), indicating that the addition of U parameter does not change the effective mass of holes in the top of VB and electrons in the bottom of CB, as shown in Figure 2b. It can be concluded that, for the W-ZnO, the optimum Ud and Up values are about 10.0 and 7 eV, respectively. Is the Ud−Up pair proper to predict correctly the energy band structure of ZB-ZnO and RS-ZnO? 3.2. Electronic Structures of ZB- and RS-ZnO. In spite of several studies on the phase relations of ZnO, the intricate phase transition kinetics has prevented a clear-cut definition of phase boundaries at low pressures. Thus, if we want to know the structural parameters of three ZnO phases at their stable conditions, first, their transition pressures of crystal phase need 26033



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ZnO, because at the G point the p and d orbitals belong to different representations of the underlying point group (Oh) the mixing of p and d states is almost forbidden. Figure 8b also presents that, besides the vertical shifting of atom orbitals, the effect of U parameters on the energy band structure of RS-ZnO includes the subdued change of the curves in VB at highsymmetry points. We especially observe an octahedral distortion, decreasing the octahedral point group symmetry from Oh to D4h. It is noted that two apical Zn−O bonds becomes shorter than four equatorial Zn−O bonds after using GGA+U approach. The distribution of six oxygen atoms for Oh and D4h on an octahedron centered by a Zn ion is also present in the inset of Figure 8b. 3.3. Lattice Relaxation Mechanism. For ZnO, the +U correction leads to a decrease of lattice constants under relaxing freely the geometry crystal structure, which is different from that of TiO2 due to their different electronic structure.28 In rutile TiO2, the CB mainly consists of unoccupied Ti 3d states (divided into two sets of t2g and eg); therefore, the only Ud correction has almost no effect on the electronic wave function, and the increase of the lattice constants is due to the increased energy separation between Ti 3d and O 2p bands, thus leading to the higher charge transfer energy and more ionic character.28 We think that there is the competition of two contributions of +U correction, which have an effect on the lattice constants of ZnO. The first factor is that the +U correction increases the localization of Zn 3d and/or O 2p orbitals and thus decreases their hybridization, which results in a decrease of lattice constants.47 For this, we systematically calculated the electronic population function corresponding to each individual atom using the Bader charge analysis, as shown in Table 2. It is interesting to note that Bader volumes for Zn and O ions in ZnO crystal decrease with the increase of Ud (or Ud and Up).

Figure 8. Energy band structure of ZB-ZnO (a) and RS-ZnO (b) obtained by the GGA (black solid lines) and GGA+Ud+Up (Ud = 10, Up = 7 eV) (blue dashed lines) approaches after geometry optimization. The Fermi energy is not normalized. The inset of part a shows the distribution of four oxygen atoms for the studied point group Td (left) and the stacking sequence of closed-packed diatomic plane, (111) planes for ZB-ZnO (right). The inset of part b shows the distribution of six oxygen atoms for Oh (left) and D4h (right) on an octahedron centered by a Zn ion in RS-ZnO.

words, the PAW pseudopotential calculations yield an indirect transition from VB at L to CB at G as the minimum band gap, as shown in Figure 8b. To improve the situation, we calculated the energy band structure of ZB- and RS-ZnO using the GGA+Ud+Up approach, where the Ud and Up values are taken as 10 eV and 7 eV, respectively. In this case, the band gap energies of ZB- and RSZnO are 3.25 eV and 4.05 eV, respectively, as shown in Figure 8. The former is in good agreement with experimental values, and the latter is similar with part of other theoretical results using the empirical pseudopotential method12 or the GW approximation,10,11 however, seriously conflicting with Segura’ experimental result (2.45 eV).8,9 Chen et al.63 identified that, even above the phase-transition pressure of 11 GPa, the ZnO nanosheets had at least a direct band gap of more than 3.5 eV at Γ point, but they did not present how much was a indirect band gap. No other experimental results can further identify them. Simultaneously, we also see that, using GGA+U d+Up approach, the Zn 3d levels of ZB- and RS-ZnO shift downward from −5.2 to −9.2 eV and from −5.3 to −9.4 eV, respectively, which are slightly larger than that of W-ZnO. The difference of energy band structures of three phases of ZnO should be related to the coordination structure of Zn or O atoms in ZnO (four-, four-, and six-coordination for W-, ZB-, and RS-ZnO, respectively). The W- and ZB-ZnO lattices have no inversion symmetry, where their point groups are C6v and Td, respectively. The main difference between two structures lies in the stacking sequence of closed-packed diatomic plane, (0001) planes for W-ZnO (see inset of Figure 2b) and (111) planes for ZB-ZnO (see inset of Figure 8a). Compared with the wurtzite structure, the zinc blende structure along the [111] direction exhibits a 60° rotation, which is very well illustrated in Figures 2b and 8a. The pair of Ud = 10 and Up = 7 eV can also successfully predict the energy band structure of ZB-ZnO, due to the similar local symmetry of tetrahedral coordination for every atom. By comparing Figure 8a and b, the main difference of the band dispersion of ZB- and RS-ZnO is at the G point, which implies that the hybridization of p and d orbitals of ZB-ZnO is significantly stronger than that of RS-ZnO, so the addition of U parameters will weaken the hybridization of p and d orbitals of ZB-ZnO, thus increase the more localization of VB, while the opposite is true for the hybridization of p and d orbitals of RS-

Table 2. Bader Charge, Minimum Distance, and Atomic Volume of W-ZnO with Various Values of Ud and Upa U value Ud = 0 Ud = 6 Ud = 10 Ud = 12 Ud = 14 Ud = 10, Up = 4 Ud = 10, Up = 8 Ud = 10, Up = 12

atom

charge/e

min. dist. (Å)

atomic vol. (Å3)

Zn O Zn O Zn O Zn O Zn O Zn O Zn O Zn O

10.745 7.255 10.753 7.247 10.755 7.245 10.689 7.310 10.717 7.283 10.675 7.325 10.656 7.344 10.641 7.359

0.764 0.867 0.757 0.852 0.742 0.835 0.710 0.812 0.704 0.780 0.708 0.810 0.703 0.804 0.702 0.803

10.849 13.153 10.244 12.584 9.597 11.857 8.644 11.496 7.852 10.282 8.568 11.410 8.371 11.164 8.315 11.090

a

The number of normal valence electrons on Zn and O atom is 12 and 6, respectively.

Another factor is the effect of +U correction on the occupied Zn 3d orbitals. Here, Zn 3d orbitals are completely occupied and have almost no contribution to CB; after the +U correction, the occupied Zn 3d orbitals have a significantly downward shift, which seriously brings the Zn atom potential down, since the correction is dependent on the occupancy of 26034



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that, if we apply it to partial filled d orbitals (nd < 10), the lattice volume will expand with the increase of the Ud parameter, and the change of one-electron potential is complex. It is inferred from above-mentioned viewpoints that the second factor is preferable. Thus, a detailed theoretical calculation is underway to assess this specific point. 3.4. Electronic Structure of the Oxygen Vacancy of WZnO. Nowadays, the oxygen vacancy is the most mentioned defect in the ZnO literature; it is therefore worthwhile devoting special attention to this defect. It has most frequently been invoked as the source of unintentional n-type conductivity, but recent research indicates that this assignment is not correct,64 which perhaps relates to the different calculation approaches, such as the approximation to exchange correlation, models and boundary conditions in simulations of defects, and the suggested postcorrection schemes. In succedent work, as a case, the U parameters mentioned above will be used to predict the electronic properties of intrinsic defect systems and subsequently compare with other theoretical and experimental results. The formation energy is one of the significant quantities in defect physics and determines the equilibrium defect concentration. In a future work, the formation energy of O vacancy in W-ZnO is calculated. In thermodynamic equilibrium the concentration c of point defects is given by the expression c = NsiteNconfig exp(−Ef/kT).65 Here, Ef is the defect formation energy, and a lower Ef corresponds to higher defect concentration c, which directly determines the physical and chemical behavior of ZnO. In other words, defects with lower formation energies are more likely to form. The formation energy of a defect with charge state is defined as30,65−67

the orbitals. Thus increased energy separation between Zn 3d and O 2p bands results in higher charge-transfer energy. We note that Bader charges on Zn ions in ZnO crystal first increase and then decrease with an increase of Ud, which show a reverse trend to that on O ions, indicating that the effect of +U parameter on Bader charges is complicated and sensitive. By applying both Ud = 10 and Up = 8 eV, the fully converged electron population on each Zn atom is reduced by 0.09 e while that on each O atom is raised by same number, indicating that the on-site Coulomb correction predicts a more ionic bonding character and the elongation of Zn−O bonds. For ZnO, the first factor achieves complete dominance in the effect of U parameter on lattice constants, which is in contrast to that of TiO2. To unravel more information about the change of charge density distribution after GGA+U calculation, we present the charge density difference of W-ZnO between GGA and GGA +Ud+Up (Ud = 10, Up = 7 eV) approaches. Here, we defined the charge density difference as δρ = ρGGA+U − ρGGA, where ρGGA and ρGGA+U are the charge density of W-ZnO using GGA and GGA+U approaches, respectively. To address them, the lattice structure of W-ZnO with GGA+U approach is taken to be the same as that with GGA approach. The inset of Figure 9 shows

Ef (Dq) = E T(Dq) − E T(perfect) + nOμO q + q(E F + E VBM )

Figure 9. Volume difference ratio as a function Ud of transition metal ions. All atoms in metal oxides are freely relaxed in the GGA+U calculation. Here, the U value is not included for the p orbitals of O atoms. The inset is the charge density difference (isosurface value = 0.02 e/Å3) of W-ZnO between GGA and GGA+Ud+Up (Ud = 10, Up = 7 eV) approaches, in which yellow (i) and green (ii) indicate the positive and negative charge values, respectively.

(3)

where ET(Dq) is the total energy of a defective supercell with charge state q and ET(perfect) is the total energy of the perfect supercell. Here nO is the number of O atoms removed from the perfect supercell to introduce O vacancies. To succeed, a certain number of difficult problems had to be solved. First, EqVBM value of a defective supercell is obtained from that of the perfect supercell and a difference ΔV in average potentials (Vav) between the perfect supercell and defective supercells as follows:30,67

that the addition of U parameter leads to the charge density redistribution of ZnO, especially around the Zn atom. It is obvious that the electron density around Zn atom decreases a lot; however, the electron density around O atom hardly changes. Here, the positive charge value in the inset I of Figure 9 corresponds to the increase of the localization. These indicate that the effect of U parameter on the localization of atomic orbitals mainly focuses on Zn atoms. Subsequently, we calculated the volume difference ratio as a function Ud of transition metal ions for several common oxides, as shown in Figure 9. Now, it is perhaps surprising that the change of lattice constants depends on the occupation number of d orbitals of transition metal. It is clear that, if we apply it to the filled d orbitals (nd = 10) in oxides (e.g., Cu2O, ZnO, Ag2O, CdO), the lattice volume will shrink with the increase of the Ud parameter, and the one-electron potential will obtain a downward shift by −U/2, and thus the total energy decreases with the increase of the Ud parameter. In these systems, the first factor mentioned above is preferable. A similar phenomenon is also found in GaN and InN by Janotti et al.47 It is also notable

q perfect E VBM = E VBM + ΔV

(4)

The first term of the right-hand side of eq 4 can be contained by perfect E VBM = E T(perfect; +1) − E T(perfect; 0)

(5)

where ET(perfect;0) is the total energy of the neutral perfect supercell; ET(perfect;+1) is the total energy of the +1 charged perfect supercell corresponding to the situation that one electron is removed from the top of VB of the neutral perfect supercell. The thermodynamic transition level of a defect between charge states q1 and q2, ε(q1/q2), corresponds to the Fermi level at which formation energies for charge states q1 and q2 equalize. When measured from VBM, the thermodynamic transition level is given as15,66 26035


The Journal of Physical Chemistry C ε(q1/q2) =

Article

Ef (q1) − Ef (q2) q2 − q1

(6)

where Ef(q1) denotes the defect formation energy for charge state q1. Errors in the defect formation energies due to the spurious electrostatic interactions in the finite-sized cells were analyzed. The defect−defect interactions in neighboring supercells and those between defect and the jellium background converge slowly with the cell size.17−19 According the corrections from eqs 4 and 5, the 72-atom cell employed in GGA and GGA+U calculations produces small errors, no more than 0.05 eV. The formation enthalpies calculated using GGA and GGA+Ud+Up approaches are −2.90 and −3.83 eV, respectively. These values indicate that the defect formation energies can in principle vary over in a wide range, corresponding to the magnitude of the formation enthalpy of ZnO. The former, GGA, overestimates the lattice constant and underestimates cohesive energies relative to the experimental values, so the formation enthalpies are underestimated, while the opposite is true for GGA+Ud+Up. The experimental value of the formation enthalpy of ZnO is −3.63 eV, which is larger than the results reported by Oba et al.34,48 using the HSE (a = 0.375) hybrid functional (−3.13 eV) and Lany et al.20 using GGA (−2.93 eV). These results indicate that the formation enthalpies of ZnO and defect formation energies strongly depend on the calculated methods. Calculations that are based purely on GGA functionals carry a large uncertainty in the formation energies and transition levels due to the underestimation of band gap of ZnO by ∼77%. If an interior O atom is removed from the supercell, four Zn dangling bonds are involved and thus produce defect state (a1 state) located in band gap, which is related to the local lattice relaxation around the oxygen vacancy.15 When the defect state is occupied by electron, the formation energy for the relevant charge state will be underestimated as well. In this case, transition levels related to defects that induce defect state in band gap will be underestimated. To overcome this problem, we checked three strategies to screen the effect of the difference of the band gap on the formation energy. A simple correction scheme is to shift the CB edge upward so that the band gap agrees with an experimental value. If the defect-induced electronic states have host conduction band-like orbital characteristics, they will be considered to follow this shifting upward when the CB shifts upward. The formation energy of a defect with such an electronic state is corrected by mΔEg, where m is the number of electrons at the defect state and ΔEg denotes the difference between experimental and exp 67 calculated band gaps, that is, ΔEg = ΔEcal g − ΔEg . Thus, the formation energies increase 5.24 and 2.62 eV for neutral and 1+ charge states, respectively. There is no any transition level seen in band gap of ZnO; perhaps the correction is not appropriate. The second correction scheme is based on the difference of the positions of the defect states in band gap between the GGA and GGA+Ud+Up calculations. Thus, before the correction, we need ensure the actual positions of the defect states in the GGA and GGA+Ud+Up calculations. Oba et al.34 and Paudel et al.18 calculated the energy band structure of ZnO with an O vacancy using the HSE hybrid functional and the LDA+Ud+Us method, respectively. For VO0, their results show that the localized occupied state is at 0.9 eV above VBM, similar to our results of Figure 10d. For VO1+, there is two spin-splitting states, in which spin-up state is occupied, another is empty, in agreement with the results of mentioned-above references, as shown in Figure

Figure 10. TDOS of W-ZnO with an O vacancy in the different charge states obtained by the GGA (blue lines) and GGA+Ud+Up (Ud = 10, Up = 7 eV) (red lines) approaches after geometry optimization. The Fermi energy is normalized, so Fermi level is set to the zero of energy (dashed lines).

10e. But for VO2+, there is a little difference of defect states between their two methods, no localized states near CBM reported by Oba et al.34 and localized states at the middle of band gap reported by Paudel et al.18 Our results is in agreement with the latter. By comparing the Figure 10a and d, for VO0, the defect state occupied by two electrons shifts 1.1 eV upward above VBM, thus the formation energy will enhance significantly about 2.2 eV. Similarly, for VO1+, the partial occupied defect state shifts 1.2 eV upward, the formation energy will also enhance about 1.2 eV (see Figure 10b and e). For VO2+, the defect state has a shift relative VBM. It is empty and thus does not change the formation energy after +U calculation. The results are shown in Table 3. The third strategy is that the total energies of perfect and defect systems calculated using GGA+Ud+Up approach were substituted into eq 3 to gain directly the defect formation energies. Here, we directly adjust the band gap to experimental value and thus avoid the demand of the correction ΔEg mentioned above in the prediction of formation energies and transition levels. The results are shown in Table 3. Here, the U parameters for Zn 3d and O 2p orbitals considered have only influence on the total energy of systems, of which Zn 3d and O 2p orbitals are occupied by electrons. For three charge states: 0, 1+, 2+, the occupied electron numbers of Zn 3d and O 2p orbitals in the VB are the same, and only the occupied electron numbers of the defect states in band gap are different. So the effect of the U parameters on the total energy of system is also same for three different charge states. Our approach is perhaps more optimal than that using the U parameters on Zn 3d and Zn 4s orbitals. It should be mentioned that the total energies of the O2 molecule and the hexagonal crystal of element Zn were calculated using Up on O 2p orbitals and Ud on Zn 3d orbitals in our GGA+Ud+Up calculations, which is completely different with that of Paudel and Lambrecht.18 For VO0, the formation energy of O vacancy is about 1.0 eV near VBM (EF = 0 eV) under extreme O-poor conditions, which is equal to that of our GGA calculation. But for VO2+, the difference of formation energies of O vacancy in ZnO between GGA and GGA+Ud+Up approaches is large due to significantly difference of formation enthalpy of ZnO. In predicting the transition levels, other LDA +U (or GGA+U) calculations are significantly different with our calculations. First, the other calculations only included a Ud 26036



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Table 3. Formation Energies of O Vacancy with Three Charge States in W-ZnO under Extreme Zn-Rich and O-Rich Conditions, in eV Zn-rich

O-rich

method

0

1+

2+

0

1+

2+

GGA GGA+correct GGA+Ud+Up

0.994 3.195 0.996

0.275 1.475 −0.843

−1.145 −1.145 −3.370

3.892 6.092 4.827

3.172 4.372 2.988

1.752 1.752 0.461

hybrid density functional.16,34 To improve the electronic structure of ZnO, LDA+Ud+Us scheme have also been taken by Paudel and Lambrecht18 (ε(2+/0) = EVBM + 0.80 eV), Boonchun and Lambrecht21 (ε(2+/0) = EVBM + 0.64 eV), and Lany and Zunger19 (ε(2+/0) = EVBM + 0.60 eV), respectively. It is obvious that the LDA+Ud+Us scheme will produce a 2+/0 transition level lying in the lower half of the band gap. The first group found a very small negative-U (−0.05 eV); however, the second one presents even a positive-U (0.15 eV). More research groups unanimously agree that VO has a deep donor, and the ε(2+/0) transition level is located at ∼1 eV below CBM; namely, VO is stable for the neutral charge state and has a negative-U behavior in n-type ZnO, which is identified by both of our proposed two schemes. Qualitatively speaking, however, it is concluded that the O vacancy has a deep donor level.

which only partially corrects the band gap. Second, Erhart et al.17 extrapolated the defect states shift in proportion to the difference of band gap between calculated and experimental values, while Lany and Zunger19 simply looked at how much Ud shifts the one-electron levels and then correspondingly an a posteriori shift of the transition levels. Table 3 presents the formation energies of O vacancy in various charge states at VBM (EF = 0 eV) under extreme O-rich conditions (μO = μO[O2]) and extreme Zn-rich conditions (μZn = μZn[bulk]) using GGA, GGA+Ud, and GGA+Up+Ud approaches. Here, we only discussed the defect formation energies of O vacancy under Zn-rich conditions and corresponding results with the neutral, 1+, and 2+, as shown in Figure 11. As defined

4. CONCLUSIONS An extensive study of the electronic structures of ZnO was performed using the GGA+U approach, and we tested the Ud and Up parameters comparing with other experimental and theoretical data. We found that a relaxed GGA+U calculation can completely correct the energy band structure of W-ZnO, which is significantly different from the partially correction of band gap under fixed lattice constants in other on-site Coulomb correction calculations. The decrease of lattice constants is related to the competition of two contribution of +U correction, the increased localization of Zn atomic orbitals, and the elongation of Zn−O bonds originating from more ionic bonding character. By comparing the energy band structure using different U parameters, the pair of Ud = 10 and Up = 7 eV is an optimum choice for W-ZnO to overcome the limit of GGA in describing the energy band structure and the energy of system. Taking the same U parameters, the band gaps of ZBand RS- ZnO are 3.25 and 4.05 eV, respectively. The former is in good agreement with experimental data. The latter is in agreement with other GW calculations, but both are larger than Segura’s experimental result. Further, we found that the formation energies and transition levels of O vacancy in WZnO are sensitive to the calculated schemes. Our GGA+U formalism leads to the 2+/0 transition level lying at 2.2 eV above the VBM, close to our a posteriori correction based on the shift of the Kohn−Sham one-electron levels in band gap and the recent other hybrid functional calculations. A negativeU behavior is obtained with ΔU′ = 0.4 eV, which is significantly different from the results using LSDA+Ud+Us reported by Paudel el al. and Boonchun et al. It is demonstrated theoretically that the GGA+Ud+Up formalism can be a candidate for considerable improvement of the distribution of energy band structure of ZnO.

Figure 11. Formation energies of O vacancies in W-ZnO as a function of the Fermi level under Zn-rich conditions. The zero energy of EF corresponds to the valence band maximum, while the 3.4 eV indicates the conduction band minimum using the experimental band gap value.

by eq 3, the formation energy depends on the Fermi level, with the slope corresponding to the charge state. All of our three methods show that, when the Fermi level is low, the 2+ charge state is energetically the most preferable for the O vacancy. As the Fermi level rises, the formation energy reaches that of the neutral state, above the Fermi level, the neutral state becomes more favorable. The defect has a negative-U behavior, that is, U′ = ε(+/0) − ε(2+/+) < 0, indicating the instability of VO1+. Figure 11 shows the formation energies of the O vacancy in ZnO as a function of the Fermi level EF under the O-poor growth conditions. Filled circles denote the position of thermodynamic transition levels such as ε(2+/0) is defined as the Fermi-level position where charge states 0 and 2+ have equal energy. The thermodynamic transition levels are usually called the thermal ionization energy or the acceptor ionization energy EA.65 As the name implies, the level will be observed in experiments where the final charge state can fully relax to its equilibrium configuration after the transition. Figure 11a clearly exhibits a 2+/0 transition level at 1.1 eV above VBM predicted using GGA. After a posteriori correction of the position of the defect state, the 2+/0 transition level moves significantly up to 2.1 eV above the VBM, as shown in Figure 11b. Based on GGA +Ud+Up approach, a 2+/0 transition level is located at 2.2 eV above VBM, as shown in Figure 11c. The latter results are in good agreement with Ud-extrapolation scheme13 and the HSE

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Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (20925725, 50972070, and 51102150), the National Postdoctoral Science Foundation of China (20100480254 and 201104085), Chutian Scholar Program. The authors gratefully acknowledge Professor M. A. Van Hove and R. Q. Zhang (City University of Hong Kong) for valuable discussion on this topic.

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