First-principles characterization of native-defect-related optical transitions in ZnO

The role of native defects in influencing the electrical and optical properties of ZnO has been studied and debated for decades. Native donors such as the oxygen vacancy (V_O) and the zinc interstitial (Zn_i) have been touted as causes of the observed unintentional n-type conductivity in as-grown material.^1–4 Native acceptors such as zinc vacancies (V_Zn) and oxygen interstitials (O_i) have been proposed to give rise to luminescence bands⁵ and to act as sources of electrical compensation.⁶ Numerous theoretical and experimental studies have investigated the properties of these species (for an overview, see Ref. 7 and references therein), as well as their interaction with common impurities such as hydrogen.⁸ One important result that most experts in the field now agree on, after the first report in Ref. 9, is that V_O is a deep donor that cannot cause n-type conductivity in ZnO.^10–14 Still, despite the large body of work associated with native defects in ZnO, many questions remain regarding the optical and electronic properties of these centers.

An ongoing issue in ZnO research is achieving full control over electrical conductivity. While recent work has cast doubt on whether a p-type material can be achieved,^15–17 ZnO is still highly promising for applications that require only unipolar doping, such as field-effect transistors.¹⁸ A full understanding and control over electrical conductivity are essential for these applications. While V_O cannot contribute to n-type conductivity, native acceptors can influence electrical conductivity through compensation.

Studying optical properties of native defects is also of high interest, not only because they may affect optoelectronic devices, but also because optical techniques such as absorption, photoluminescence (PL) and photoluminescence excitation (PLE) are commonly used to characterize materials. Point defects in ZnO may also be utilized as single-photon sources,^19–21 an application that would benefit from the detailed knowledge of defect optical properties. The association of optical signals with specific defect centers thus remains an open topic in ZnO. A notable controversy involves a possible native-defect source of green luminescence (GL),^8,22–26 which has been linked with both V_O (Refs. 22 and 24) and V_Zn.²³ This GL signal, which is relatively featureless, is distinct from the GL associated with Cu_Zn,²⁷ which features distinct phonon replicas and a zero-phonon line (ZPL). In the present work, we will focus on native defects and not discuss impurities, but we note that it is entirely possible that there would be more than one source of GL. In a more recent work, V_Zn has been associated with lower-energy photoluminescence peaks; for instance, in Refs. 28 and 29, PL peaks of 1.6 eV were linked with isolated V_Zn. With regard to oxygen vacancies, detection of V_O-related luminescence has been challenging.²⁸ however, electron paramagnetic resonance (EPR) studies on irradiated³⁰ and as-grown²⁸ ZnO have found that the EPR signal of V_O can be excited with energies peaking between 2.5 and 3.0 eV.

An important focus of the present work is the comparison with experiments. Direct measurements of formation energies of defects are scarce.^6,31 In a addition, comparison with calculated values is complicated by the fact that (as we will describe in detail in Sec. II) formation energies depend on the growth conditions through chemical potentials, which are difficult to quantitatively assess. Qualitative comparisons are more feasible, based on the fact that only defects with low enough formation energies will occur in observable concentrations in as-grown material. In principle, defect formation energies can also be determined from activation energies for diffusion if migration barriers are known;³² but again the lack of formation about chemical potentials makes direct comparisons with calculations challenging. In the present work, we will, therefore, focus on optical studies, which are more readily available, and where direct comparisons can be made between calculated and measured transition levels.

Motivated by the open questions concerning the behavior of native defects in ZnO, we have conducted a systematic study using the most advanced technique that is currently available for comprehensive studies of point defects, namely, density functional theory with a hybrid functional.³³ We compare the stabilities of native vacancies and interstitials under representative growth conditions. Throughout this paper, our implicit assumption (unless otherwise stated) is that we are considering n-type material; this is indeed the most relevant situation, in light of the absence of reliable p-type doping, and given that most not-intentionally doped samples are n-type.

We then investigate defect-related optical transitions, beginning with a case study of N_O, for which experimental results are available.¹⁶ Turning to native defects, we examine those that are most likely to be present in n-type conditions, considering exchange of carriers with both the valence and conduction bands. V_O is found to have a high formation energy in n-type material, indicating that it is unlikely to be present in high enough concentrations to affect the properties of as-grown samples. Our calculated absorption peak for V_O (2.67 eV) is similar to the peak excitation energies experimentally observed in irradiated samples.^28,30

For Zn vacancies, we find that they are the most stable defect in n-type ZnO. We also find that these vacancies are further stabilized by the formation of vacancy-hydrogen complexes. Hydrogenation of V_Zn strongly modifies the properties of V_Zn, affecting its possible charge states, lowering its ionization energies, and shifting its luminescence signals to higher energies. The presence of such vacancy-hydrogen complexes also leads to characteristic vibrational frequencies, providing a connection to experimental studies of such modes.^34–36 

Finally, we examine the electrical and optical transition levels of oxygen and zinc dangling bonds (DB) in ZnO. Such centers would be expected to occur near grain boundaries, dislocations, and in irradiated or implanted samples. They could also be sources of single-photon emissions that have been detected in ZnO nanoparticles.^19,20

We begin by discussing in Sec. II the methodology used in this work. We then report our results for isolated native defects in Sec. III, examining their electronic properties, stability, and optical transitions. In Sec. IV, we investigate the interaction between hydrogen (a common donor impurity) and zinc vacancies (which are likely to be the most common acceptor defect). Our results for dangling bonds in ZnO are discussed in Sec. V, and we conclude by summarizing our key results in Sec. VI.

Our calculations are based on generalized Kohn-Sham theory^37,38 with the screened hybrid functional of Heyd-Scuseria-Ernzerhof (HSE)³⁹ and the projector-augmented wave method,^40,41 as implemented in the VASP code.⁴² The parameter that determines the fraction of Hartree-Fock exchange is set to 0.36, which yields a band structure, band gap (3.35 eV), and wurtzite lattice parameters (a = 3.23 Å, c = 5.24 Å) that are in excellent agreement with experimental values.¹⁵ 

Formation energies and defect transition levels are calculated using a well established formalism.³³ For example, the formation energy of V_Zn in charge state q is given by

where $Etot(VZnq)$ is the total energy of a supercell containing one $VZnq$ in charge state q, and E_tot (ZnO) is the total energy of a perfect ZnO crystal in the same supercell. Any electrons added to or removed from the supercell are exchanged with the electron reservoir in the semiconductor host, i.e., the Fermi level (E_F), which is referenced to the valence-band maximum (VBM) (ε_v) of ZnO (i.e., the Kohn-Sham eigenvalue of the highest occupied state at the Γ point). The final term (Δ^q) represents the correction for the finite size of charged supercells, calculated using the procedure of Refs. 43 and 44.

We employ 400 eV plane-wave cutoffs and 2 × 2 × 2 special k-points for all calculations. For all defects except V_O, we used a 96-atom wurtzite supercell. Due to the more extended nature of the defect-related states of the oxygen vacancy, 192-atom supercells were necessary to ensure converged levels for V_O. Spin polarization is explicitly taken into account for defects with unpaired electrons.

The Zn atom that is removed from the crystal is placed in a reservoir of energy μ_Zn, referenced to the energy per atom of bulk Zn. The range of μ_Zn is limited by the enthalpy of formation of ZnO (–3.63 eV, Ref. 45), meaning that it can theoretically take on values between 0 eV (Zn-rich conditions) and –3.63 eV (Zn-poor limit). Similar considerations apply to μ_O, which is referenced to the energy of O₂ at T = 0. As will be discussed below, not all of these conditions are likely to be accessed during growth or processing, and we will focus on chemical-potential conditions that are relevant for actual ZnO samples.

Figure 1 displays our results for formation energies of the prevailing native point defects in ZnO. The energies are plotted for a specific value of μ_O, namely –1.50 eV, which is roughly halfway between Zn- and O-rich conditions. While μ_O can, in principle, vary between 0 eV and –3.63 eV (corresponding to μ_Zn = 0 eV), the extreme values are usually not achieved under realistic growth conditions. μ_O = –1.50 eV corresponds⁴⁶ to conditions that might occur during pulsed laser evaporation or vapor-phase epitaxy growth of ZnO,⁴⁷ at temperatures near 800 °C and pressures ∼10⁻³atm. Our choice of μ_O for the purposes of plotting is not limiting in any way; formation energies under other conditions can always be obtained by referring back to equations such as Eq. (1).

Formation energies provide information about the likelihood that defects will form in the crystal; this aspect will be discussed in Sec. III B. However, the Fermi-level positions at which the formation-energy curves exhibit kinks (i.e., at which the most stable charge state switches from one value to another) correspond to the thermodynamic charge-state transition levels of the defect,³³ which determine its electronic properties. These properties are discussed first, in Sec. III A. Note that Fig. 1 does not contain information about antisite defects. Those tend to be high in energy,⁴⁸ and therefore, we judge them to be unlikely to occur and we have not pursued them any further in the present study.

Our present results for the electronic properties of native defects in ZnO are in general qualitative agreement with the notions that have emerged from the large body of previous theoretical work (reviewed in Ref. 7), but provide a quantitative accuracy that reflects the use of the state-of-the-art techniques employed here. As shown in Fig. 1, we find the zinc interstitial (Zn_i) to be a shallow donor,^11,12,48,49 stable in the 2+ charge state across the band gap of ZnO. This defect has a high formation energy in n-type ZnO and is thus unlikely to be present in large concentrations; i.e., it cannot be responsible for unintentional doping. Zn_i have also been shown to be highly mobile, and hence unlikely to ever be present as a stable isolated species⁴⁸ (except when introduced at very low temperatures⁵⁰).

The oxygen interstitial (O_i) is a deep acceptor when occupying the octahedral site in the ZnO wurtzite lattice, with a (–/2–) transition level 2.90 eV above the VBM of ZnO, and a (0/–) level 1.85 eV above the VBM (Fig. 1). O_i can also assume a split-interstitial configuration, in which case it is stable in the neutral charge state across the band gap of ZnO. The qualitative behavior of the split and octahedral O_i is consistent with the results reported in Ref. 48; the most notable difference is the position of the acceptor transition levels of O_i(oct), which we find to be closer to the conduction-band minimum (CBM). This behavior likely originates from the use of the hybrid functional in the present work.

Regarding the oxygen vacancy, we find it to be a deep donor in ZnO, and hence it cannot cause n-type conductivity. This behavior has been reported by a number of research groups using a variety of computational methods.^9–14 Also in agreement with previous work, our results show that V_O is a negative-U center, with 2+ and 0 being the only charge states stable within the band gap of ZnO, and the + state never being thermodynamically stable. We find that the (2+/0) transition level is at 1.93 eV above the VBM, as shown in Fig. 1. The (+/0) transition level is at 1.67 eV, and the (2+/+) transition at 2.18 eV above the VBM of ZnO (not shown in Fig. 1). These values are similar to the results reported for other hybrid functional studies^11–13,51 and to an extrapolation based on the local density approximation (LDA) and LDA + U.¹⁰ As noted in Sec. II, we employed 192-atom supercells for V_O calculations and found that formation energies and transition levels could vary by as much as 0.3 eV if only 96-atom cells were used.

The ultimate conclusion that all methods agree on is that V_O cannot lead to n-type conductivity. Although there are some quantitative differences in the exact position of the V_O transition levels, it has been found¹³ that results from different methods agree very well after taking into account shifts in valence-band edges by aligning to a common reference. We note that through the use of the hybrid functional we are able to make quantitative predictions for defect properties (and compare with experiment via optical transitions; see below) without having to rely on extrapolation or ad hoc band-edge shifts.

For the zinc vacancy (V_Zn), we find that the defect states are localized mainly on nearest-neighbor O atoms, as is shown in Figs. 2(a)–2(c). In the 2–charge state, the corresponding Kohn-Sham states in the band gap are all filled. When an electron is taken away from $VZn2−$⁠, the resultant hole is localized almost exclusively onto a single nearest-neighbor O atom, as is shown in Fig. 2(c). This localization behavior is consistent with EPR measurements^28,52,53 that also indicate that $VZn−$ traps a hole on a nearest-neighbor O atom. This tendency for localization, which would be missed if traditional functionals such as LDA or the generalized gradient approximation (GGA) were used, has also been reported by other theoretical groups using LDA + U (Ref. 54) or the so-called screened-exchange (sX) functional,¹² as well as HSE.⁵¹ Further removal of electrons results in additional localized holes: for $VZn0$ two holes are localized on distinct O neighbors, and the vacancy can even be placed in positive charge states $VZn+$ (with three holes localized on distinct O atoms) and $VZn2+$ (with four holes localized on distinct O atoms). In Fig. 2, we plot spin densities of $VZn+, VZn0$⁠, and $VZn−$ that illustrate this hole localization. We find the (–/2–) transition level to occur at 1.84 eV, the (0/–) level at 1.39 eV, the (+/0) level at 0.81 eV, and the (2+/+) level at 0.23 eV above the VBM. V_Zn thus acts as a compensating center in n-type ZnO. These values are within 0.2 eV of other recent HSE calculations.⁵¹ There are discrepancies with other hybrid calculations,^11,49 for instance, Ref. 49 reported a (0/–) of 0.62 eV (but a similar (–/2–) level of 1.80 eV), while Ref. 11 reported a (0/–) level near 0.8 eV, and a (–/2–) level near 2.5 eV. We speculate that this discrepancy may be caused by reporting a higher-energy spin state for $VZn0$⁠, and the use of different charge-state correction schemes and a Γ-only k-point mesh in the case of Ref. 11.

We now examine how likely native defects are to form in ZnO under typical growth conditions. As argued above, μ_O = –1.50 eV is representative of conditions that might occur during growth of ZnO. Regarding Fermi level, unless acceptor doping is explicitly pursued, as-grown samples of ZnO typically exhibit n-type conductivity even in the absence of intentional doping.^7,8

Figure 1 shows that V_O and V_Zn are the most stable native defects in ZnO when μ_O = –1.50 eV. For E_F < 3.0 eV, $VO0$ is the most stable species, while for E_F > 3.0 eV, $VZn2−$ is the lowest energy species. For as-grown ZnO samples that are (unintentionally or intentionally) n-type, E_F is near the conduction-band minimum (CBM), and V_Zn will be the most important native defect, acting as a compensating center.

The formation energy of V_Zn under these conditions (1.70 eV when E_F is at the CBM) would indicate a small concentration of zinc vacancies at equilibrium. However, this energy is strongly dependent on both μ_Zn and E_F; less Zn-rich conditions, or Fermi levels above the CBM (which occur in the case of degenerate doping) would lead to an increase in the concentration of V_Zn. Indeed, ZnO samples which were highly n-type doped were reported to contain large concentrations of compensating zinc vacancies or donor-V_Zn complexes.⁶ 

Achieving maximum conductivity requires minimizing the concentration of compensating acceptors. One approach would be to manipulate growth conditions to be more Zn-rich, thereby raising the formation energy of V_Zn. Another approach is through passivation by complex formation with donor species; this will be discussed in Sec. IV for the example of hydrogen donors.

In this section, we examine the optical transitions associated with defects in ZnO, with the aid of configuration-coordinate (CC) diagrams (which are explained in detail in Sec. II E of Ref. 33). We illustrate the need for exploring all possible processes. We present results for the case study of the deep N_O acceptor, followed by a discussion of the optical properties of V_O and V_Zn. V_Zn-H complexes are also capable of generating optical signals, but these will be discussed in Sec. IV.

We have calculated the properties of N_O using the approach outlined in Sec. II. We find a (0/–) acceptor level that is 2.04 eV above the VBM. This value is somewhat deeper than that reported in Ref. 15 due to two factors: distortions associated with the neutral state (which stabilize $NO0$ by 0.45 eV and thus increase the ionization energy) and implementation of more accurate charge-state corrections^43,44 (which raise the energy of $NO−$ by 0.25 eV, also increasing the ionization energy). The value for the (0/–) value reported here is more consistent with other hybrid functional studies^17,55,56 which also found symmetry-breaking distortions for $NO0$⁠. Consistent with Ref. 56 we also find a (+/0) donor level associated with N_O that is 0.87 eV above the VBM.

The optical transitions associated with N_O in ZnO have been put forward as a means of experimentally verifying¹⁶ the theoretical prediction of its deep acceptor character.¹⁵ In addition to studying the N_O thermodynamic transition level, we have also examined the optical transitions associated with this acceptor in a CC diagram (Fig. 3). For the situation illustrated in Fig. 3(a), the excited state is the $NO0$ plus an electron at the CBM. The transition is to the ground state, which is the acceptor in the negative charge state (⁠$NO−$⁠). The two curves are displaced along the horizontal axis because generally they have different atomic configurations. In the case of N_O, the generalized coordinate corresponds to the relaxation of the nearest-neighbor Zn atoms. For $NO0$⁠, one N-Zn bond length increases by 7% of the ZnO bond length, while for $NO−$⁠, all N-Zn bond lengths decrease by 3%. The minima of the curves are displaced by the energy of the zero-phonon line (ZPL), which corresponds to the energy difference between the (0/–) charge state transition level and the CBM. In a classical approximation, the radiative transitions are vertical, with a transition energy corresponding to the peak of the absorption or emission spectrum. The emission process leaves the electronic ground state (⁠$NO−$⁠) in a vibrationally excited state (corresponding to the atomic configuration of $NO0$⁠), which will decay to the equilibrium state by emitting phonons and losing the relaxation energy (0.67 eV). Full calculations of the optical spectra are feasible,⁵⁷ but for the purposes of the present paper information about peak energies, as well as peak widths, which can be estimated from the magnitude of the relaxation energy, will suffice.

Transitions involving an electron recombining with $NO0$ had originally been thought to give rise to PL peaking near ∼1.7 eV,¹⁵ in agreement with the experimental PL peak.¹⁶ Here we find that such transitions actually peak near 0.64 eV, with a zero-phonon line ZPL of 1.31 eV, and absorption peaking near 1.85 eV, as shown in Fig. 3(a). Such transitions therefore cannot be the source of the PL observed in Ref. 16. Instead, transitions involving recombination of a hole with $NO−$ are a more likely source of such PL peaks. As shown in Fig. 3(b), we calculate that this process has a ZPL of 2.04 eV, with absorption peaking at 2.71 eV and PL peaking near 1.50 eV. In fact, recombination of holes with the negatively charged $NO−$ center can be expected to produce much stronger recombination than electrons recombining with a neutral center.

These results indicate that the hole capture process of N_O, which had not been considered previously, is more likely to be responsible for the ∼1.7 eV PL observed in N-doped ZnO. This case study also draws attention to the importance of considering all possibilities for optical transitions, including not just recombination with electrons in the conduction band but also with holes in the valence band.

In Fig. 4(a) we plot an optical transition associated with V_O. Absorption of an electron from the occupied defect states of $VO0$ into the CBM leads to absorption peaking at 2.67 eV, with a zero-phonon line of 1.68 eV and a relaxation energy of 0.99 eV. For the subsequent emission process, whereby an electron from the CBM recombines radiatively with $VO+$⁠, we predict a luminescence peak of 0.62 eV and a relaxation energy of 1.06 eV.

Due to the nearly mid-gap position of the V_O (+/0) transition level (at 1.67 eV), and to the similar relaxation energies between the two charge states, the calculated transitions shown in Fig. 4(a) are almost identical to transitions which would involve hole capture by V_O. For such transitions, we predict an absorption peak of 2.73 eV, a zero-phonon line of 1.67 eV, and a luminescence peak of 0.68 eV.

In principle, optical transitions associated with the (2+/+) transition level of V_O are also possible. Due to the proximity of this level to the CBM (the ZPL is at 1.17 eV), we predict a PL peak of only 0.16 eV for electron capture by $VO2+$⁠. Emission at such low energies is much more likely to be a nonradiative than a radiative process; indeed, for transitions at these energies, typical nonradiative capture coefficients^58,59 are much larger than radiative coefficients.⁶⁰ Another possible process is hole capture by $VO+$⁠; we predict that this would occur with a ZPL at 2.18 eV and a PL peak of 0.91 eV. However, since this process would involve hole capture by a positively charged defect, it is not expected to be efficient.

ZnO often exhibits a broad luminescence near 2.5 eV,^8,22,23 commonly referred to as the GL, which is distinct from the Cu-related GL that features distinct phonon replicas. The luminescence peaks we predict here for the possible V_O-related transitions occur at much lower energies; so we can exclude oxygen vacancies as the source of this band. Our predicted PL peaks are also quite different from the ∼2 eV signal (with an estimated 2.48 eV ZPL) that has previously been linked to V_O based on optically detected EPR experiments (ODEPR).⁵³ Although our calculated ZPL for hole capture by $VO+$ is at 2.18 eV, due to a large relaxation energy (1.27 eV) our predicted PL peak (0.91 eV) is far from the signals observed in Ref. 53.

In fact, the V_O luminescence peaks we predict, at less than 1 eV, are outside the range of energies that are usually scanned in PL studies of ZnO. This may explain why they have not been reported; in addition, as already mentioned at these low energies nonradiative recombination is likely to be stronger than radiative recombination.^58,60

The situation is different for optical absorption. Various absorption signals have been attributed to oxygen vacancies in the experimental literature, in the context of EPR experiments. Observation of EPR signals requires a paramagnetic state, which for V_O occurs only in the positive charge state. Due to the negative-U character, this charge state is not stable in equilibrium, but it can be generated by illumination. In Ref. 30, the absorption associated with the optically induced V_O EPR signal was measured to have an onset near 1.9 eV and a peak near 2.5 eV [although the authors noted that the exact position of the peak may have been affected by the generation of $(VZn−−H)0$ centers]. Similar results were reported in Ref. 28, where the V_O-associated EPR signal was excited with an onset below 2 eV and peaking near 2.65 eV. Both of these reports are in good agreement with our calculated excitation energies for $VO+$⁠, presented in Fig. 4(a): onset (ZPL) at 1.68 eV, peak at 2.67 eV.

We present selected optical transition levels of isolated V_Zn in Figs. 4(b) and 4(c). We focus on transitions leading to PL peaks above 1.3 eV, as signals in this energy region are the most frequently reported in the experimental literature.^{28–30,53,61} At energies below 1.3 eV, nonradiative recombination becomes more likely than radiative recombination.^58–60 

We first consider transitions in which $VZn−$ is the ground state, as depicted in Fig. 4(b). The $VZn−$ charge state can occur due to direct photoexcitation of $VZn2−$ (a process which would feature an absorption peak of 2.31 eV). In Fig. 4(b) we consider the process whereby $VZn2−$ captures a photogenerated hole. The generalized coordinate in Fig. 4(b) is the displacement of a specific nearest-neighbor O atom (the one to which the hole is bound, see Sec. III A) measured with respect to the nominal V_Zn site. We find the absorption peak for this process occurs at 2.57 eV, with a zero-phonon line at 1.84 eV. The calculated emission energy is 1.37 eV, lower than PL signals usually reported for ZnO. While electron capture into (–/2–) is in principle possible, we calculate a PL peak at 0.78 eV, which is unlikely to be observed, both due to the low energy and due to the fact that electron capture by a negatively charged defect is expected to be relatively inefficient.

Optical transitions involving the (0/–) level are, in principle, also possible, but again the calculated PL peaks are all low in energy. For instance, electron capture by $VZn0$ gives rise to a ZPL of 1.96 eV and a PL peak of 1.29 eV. Hole capture by $VZn−$ gives rise to even lower energies: a ZPL at 1.39 eV and a PL peak of 0.86 eV.

Finally, we consider recombination of an electron into the (+/0) level of V_Zn, which is depicted in Fig. 4(c) (hole capture into this level would give rise to emission less than 0.7 eV and is thus not discussed). For this process, we calculate a ZPL of 2.54 eV and a PL peak of 1.90 eV. (Hole capture by this level is also possible, but it leads to a ZPL of only 0.81 eV and a predicted PL peak of 0.26 eV.) Electron-capture transitions involving the (+/0) level are in closest agreement with the ∼1.6 eV signals that have previously been attributed to V_Zn from PL²⁹ and ODEPR studies.^28,50 Yet it is unclear whether $VZn+$⁠, at which 3 holes are trapped, would give rise to an EPR signal similar to that of $VZn−$ (at which only a single hole is trapped).

Obtaining high concentrations of the $VZn+$ charge state in n-type material may also be difficult, even at high optical excitation intensities. This means that electron recombination into the (2+/+) level (for which we predict a ZPL of 3.12 eV, a PL peak of 2.40 eV, and an absorption peak of 3.76 eV), will be even less likely to occur.⁵¹ However, the discussion in Sec. IV will make clear that V_Zn-related complexes can also play a role; a discussion of their optical signals will be presented in Sec. IV B.

Hydrogen is a common impurity in ZnO. It is present in almost all growth or processing environments and easily incorporates.^7,61,62 It is therefore, important to consider the interaction between hydrogen and point defects. We have already noted that vacancies are the most likely native defects to occur in ZnO, and we will focus on interactions between H and V_Zn. Interactions between hydrogen and oxygen vacancies have been studied previously;⁶³ the resulting complex is equivalent to a substitutional H on an oxygen site (in a multicenter bond) and acts as a shallow donor.

We investigate the interaction between hydrogen and V_Zn in ZnO by calculating the formation energies of V_Zn-H and V_Zn–2H complexes. To assess the stability of these complexes, we compare with the formation energies of isolated V_Zn and of isolated interstitial hydrogen (H_i); all of these energies are included in Fig. 5(a). For the purposes of this figure, the same chemical potential for μ_O (–1.50 eV) as in Fig. 1 is used, while for H we chose a chemical potential that corresponds to H₂O as a solubility-limiting phase. Based on the tabulated enthalpy of formation of H₂O at 0 K [–3.06 eV (Ref. 45)], this leads to μ_H = –0.78 eV.

$Hi+$ is stable exclusively in the positive charge state across the band gap of ZnO.⁶² The Zn vacancy can bind one or two hydrogen atoms through the formation of O-H bonds. Hydrogen forms strong bonds with the nearest-neighbor O atoms, with bond lengths of 0.99 Å, quite similar to the 0.96 Å O-H bond length in the water molecule.⁶⁴ The most favorable configuration of the (V_Zn-H)^– complex has the H bonded to an oxygen atom in the basal plane, with the bond oriented at 109° with respect to the c-axis, in excellent agreement with experimental observations.⁶⁵ The calculated energy difference between the basal and axial configurations is small, with the basal O-H configuration more favorable by only 40 meV. The process of binding a hydrogen atom effectively passivates one of the O dangling bonds that give rise to the defect states of V_Zn.

The V_Zn-H complex is a single acceptor; attaching the hydrogen reduces the charge state under n-type conditions from 2– in V_Zn to – in V_Zn-H. The (0/–) transition level is calculated to occur at 1.26 eV above the VBM. Removing a second electron from this center stabilizes the + charge state; the (+/0) transition level occurs at 0.65 eV above the VBM. Removing a third electron from this center stabilizes the 2+ charge state; the (2+/+) transition level occurs at 0.22 eV above the VBM. We can calculate a removal energy for the V_Zn-H complex, defined⁶⁶ as the energy cost to separate (V_Zn-H)^– into $VZn2−$ and $Hi+$⁠; the resulting value is 3.11 eV. This value is similar to values obtained for hydrogenated cation vacancies in other semiconducting oxides.⁶⁷ The magnitude of this value also indicates that, once the complex is formed, it is unlikely to dissociate.

For V_Zn–2H two configurations are possible, due to the inequivalency of the oxygen atoms neighboring the vacancy. One possibility is to have one H bonded to the axial O neighbor and the other to a planar O. We label this center (V_Zn–2H)a, and find that the axial H is aligned at 11° off the c-axis, while the other O-H bond is oriented almost perpendicular (97°) to the c-axis. This atomic geometry is in excellent agreement with the experimentally determined structure reported in Ref. 34, where the bond angles with respect to the c axis were found to be 100° for the basal O-H and 10° for the axial O-H.

The other possibility is to have both H bonded to planar H; we label this center (V_Zn–2H)b. In our calculations, we find (V_Zn–2H)b to be slightly less stable than (V_Zn–2H)a, although the energy difference is exceedingly small (10 meV). A metastable (V_Zn–2H) center with both H bonded within the basal plane has also been experimentally identified.³⁶ 

Due to passivation of the second oxygen dangling bond, V_Zn–2H is neutral across most of the band gap of ZnO [Fig. 5(a)], but not completely passivated; it still exhibits two transition levels above the VBM, a (+/0) level at 0.60 eV and a (2+/+) level at 0.09 eV above the VBM. This is due to the fact that V_Zn–2H can still trap holes onto the nearest-neighbor oxygens that are not bonded to hydrogen. The removal energy of the neutral V_Zn–2H complex [i.e., the energy cost to separate into (V_Zn-H)^– and $Hi+$] is 2.17 eV.

Although hydrogenation reduces the charge state of V_Zn under n-type conditions, other electrical properties are preserved; we observed similar behavior for hydrogenated cation vacancies in GaN.⁶⁸ We illustrate this by plotting the transition levels of V_Zn and of the two hydrogenated complexes in Fig. 5(b). Addition of H (moving to the right in the diagram) removes the highest thermodynamic transition level. The remaining levels are more or less aligned, i.e., they fall within an energy range 0.2 eV, as indicated by the narrow range of the shaded energy regions.

These results indicate that H_i can passivate V_Zn, offering a strategy for reducing compensating acceptors in ZnO. The complex with one hydrogen atom is still electrically active but is only a single acceptor (as opposed to the double acceptor V_Zn). The complex with two H atoms, while still potentially optically active, is electrically neutral in n-type ZnO, and is therefore no longer a source of compensation. Considering that hydrogen is likely to be present unintentionally in ZnO, and given the large removal energies associated with the hydrogenated zinc vacancies, we conclude that these complexes are very likely to spontaneously form during growth or subsequent processing of ZnO. Once formed, they are unlikely to dissociate.

Since both hydrogenated vacancies have thermodynamic transition levels within the band gap of ZnO, optical transitions involving these centers are possible. CC diagrams for V_Zn-H and V_Zn–2H are shown in Fig. 6. For these defects, we only present results for electron capture, as the hole capture processes give rise to low PL peak energies. For instance, hole capture into the (0/–) level of V_Zn-H gives rise to a ZPL of 1.26 eV and a PL peak of 0.73 eV. Other hole capture processes produce even lower PL peaks. As noted before, radiative emission at such low energies is unlikely to be observed.

For electron capture into the (0/–) level of V_Zn-H, we predict a broad emission signal peaking at 1.31 eV and a ZPL at 2.09 eV, with the absorption peak occurring at 2.62 eV, as shown in Fig. 6(a). Electron capture into the (+/0) level of V_Zn-H, Fig. 6(b), is also possible. For this process, we predict a ZPL of 2.70 eV, emission peaking at 1.96 eV and absorption peaking at 3.31 eV. These optical transitions are quite similar to those predicted for electron capture into the corresponding (+/0) level of isolated V_Zn [Fig. 4(c)].

If V_Zn-H can capture three holes, electron capture into the (2+/+) level is also possible. We predict a ZPL of 3.13 eV, emission peaking at 2.35 eV and absorption peaking at 3.82 eV. We again note that such transitions may not be observed experimentally due to the unlikelihood of V_Zn-H capturing three holes.

Addition of a second hydrogen to V_Zn causes a further blueshift of the predicted emission signals. For electron capture into the (+/0) level, we find optical signals very similar to those predicted for V_Zn-H, as evidenced by the similarity in the CC diagrams for V_Zn-H in Fig. 6(b) and for V_Zn–2H in Fig. 6(c). Starting from the neutral charge state, we predict an absorption peak at 3.20 eV and a ZPL of 2.75 eV. For the emission process, which corresponds to the capture of a free electron by (V_Zn–2H)⁺, we calculate a signal peaking at 2.08 eV. Electron capture into the (2+/+) level of V_Zn–2H, shown in Fig. 6(d), leads to a ZPL of 3.26 eV and absorption peaking at 3.78 eV. For the emission process, we predict a PL peak of 2.30 eV.

Optical transitions attributed to V_Zn have been widely reported. Based on ODEPR experiments in Ref. 50, a 1.65 eV PL signal with an excitation onset near 2.4 eV was observed in irradiated material and proposed to be related to V_Zn, either as isolated V_Zn or as a V_Zn-donor complex. A similar signal was subsequently reported in a study of as-grown ZnO:²⁸ a signal at ∼1.6 eV was detected, whose EPR signature was found to be very similar to the signal observed in Ref. 50.

Our predicted optical emission peak for V_Zn is at 1.37 eV, which is lower in energy than the PL signals peaking near 1.6 eV that have been attributed to V_Zn in the experimental literature.^28,50 However, for the V_Zn-H complex we find PL peaks at 1.31 eV and 1.96 eV, which are also close to the observed vacancy-related PL peaks. We propose here that the PL/EPR signals previously linked to $VZn−$ may also be due to (V_Zn-H)⁰, which is also paramagnetic and hence observable in EPR. Since these two defects exhibit similar spin densities (both featuring a hole localized onto an O neighbor), it would be difficult to distinguish between the two using EPR techniques.

Optical studies of single-photon emitters in ZnO have linked PL emissions in the 1.56–1.88 eV region with isolated V_Zn,¹⁹ while other studies have observed single-photon emission at 1.88–2.11 eV from an unknown defect.^20,21 Depending on which charge states are involved, and whether V_Zn is complexed with H, we find that PL peaks associated with V_Zn can vary from 1.31 to 2.30 eV. Thus V_Zn (or V_Zn-H complexes) may be consistent with the signals observed for these single-photon emitters.

Vibrational spectroscopy has also produced very useful information about hydrogen-related defects and complexes in ZnO treated with H plasma.^34,36,65 To aid in the identification of the observed signals we have calculated the vibrational properties of hydrogenated V_Zn complexes. We use the formalism outlined in Refs. 67 and 69, fully, including anharmonic contributions (which are sizeable for hydrogen due to its light mass). We find all frequencies to be slightly overestimated with respect to experiment, which is typical of hybrid functional calculations⁷⁰ and has also been observed for hydrogenated cation vacancies in other oxides.⁶⁷ Relative differences between vibrational modes are much more reliable for comparison with the experiment. To compare more directly to the experiment, we adopt a systematic correction of –67 cm⁻¹, which shifts the calculated axial O-H frequency of the (V_Zn–2H)⁰a complex to the observed local vibrational mode (LVM) at 3312 cm⁻¹, since the assignment of this mode has been supported by both experiment and theory.³⁴ 

The calculated vibrational frequencies are listed in Table I. Experimentally, (V_Zn–2H)a exhibits LVMs at 3312.2 cm⁻¹ for $O−H∥c$ and 3349.6 cm⁻¹ for $O−H⊥c$⁠.³⁴ Consistent with the experiment, we find the magnitude of the LVM for the planar H to be larger than that of the axial H, although the splitting is slightly overestimated.

The LVM signal in ZnO at 3326 cm⁻¹ has been attributed to a (V_Zn-H)^– complex with a planar O-H bond.³⁵ Photoexcitation experiments determined that this defect also exhibits a slightly higher LVM of 3358 cm⁻¹ when in the excited neutral charge state.³⁵ In terms of the LVMs, we find the (V_Zn-H)^– with the basal O-H has a calculated frequency of 3408 cm⁻¹, while (V_Zn-H)^– with an axial O-H has a slightly lower frequency of 3373 cm⁻¹. In the neutral charge state, we observe higher frequencies, consistent with the observed increase in the frequency of the LVM associated with the 3326 cm⁻¹ signal under photoexcitation.³⁵ Based on the observation in Ref. 35 that the vibrational modes of this center could not be excited with sub-band-gap light lower than 1.96 eV, (V_Zn-H)⁰ was deduced to have a transition level roughly 2.0 eV below the CBM of ZnO. This result agrees well with our calculated (0/–) transition level for (V_Zn-H), which we find to occur 2.1 eV below the ZnO CBM.

To model isolated dangling bonds (DB) in ZnO, we employ the methodology described in Refs. 71 and 72. We create the Zn (O) DB by initially removing an O (Zn) atom to create a primary dangling bond, followed by the removal of that atom's nearest neighbors, creating nine secondary dangling bonds. We passivate these secondary dangling bonds using H atoms with fractional charges, set to balance and passivate the dangling bonds of the nearest neighbors (0.5 for Zn, 1.5 for O). The structures are illustrated in Figs. 7(b) and 7(c). The partially charged H atoms are relaxed to optimize the O-H (Zn-H) distances and then fixed for subsequent calculations.

These dangling-bond calculations were performed in 64-atom zinc-blende supercells. Although zinc blende is not the lowest-energy structure for ZnO, the local tetrahedral bonding environment is very similar to that in wurtzite. As the phases differ only in the positions of third-nearest neighbors, localized defects are not expected to exhibit significant differences between these two phases.⁷³ Our intent with the dangling-bond studies is not to exactly mimic a specific structure, but to provide insight into the generic behavior of a dangling bond that would occur in a non-crystalline or defected region of the crystal. We thus expect the results obtained for the zinc-blende phase to be representative of the behavior of dangling bonds in ZnO regardless of the crystal phase.

We probe the electrical properties of the isolated DBs by occupying them with zero, one, or two electrons. Thermodynamic transition levels between these occupancies can be calculated using the same equations to calculate standard point defects (details can be found in Refs. 71 and 72). In Fig. 7(a) we show the transition levels for both Zn and O DBs in ZnO. For each DB, only one transition level occurs within the band gap. In the case of the O DB, the transition between one-electron occupation (1e) and two-electron occupation (2e) occurs at 1.45 eV above the VBM. The transition between 1e and the 0e (unoccupied) state [not shown in Fig. 7(a)] occurs at 0.04 eV below the VBM. The O DB has the character of an O 2p orbital, as shown by the spin density in Fig. 7(b), quite similar to the localized holes which arise due to V_Zn. The O atom that carries the O DB occupied with 1e moves 0.48 Å off the original lattice site, away from the absent Zn neighbor. This displacement is consistent with a tendency to move towards sp² hybridization when an electron is removed. When the DB is occupied with 2e, the O atom is displaced by only 0.14 Å, indicating it stays closer to the original sp³ hybridization.

One can consider the O DB to be the building block for the states of the zinc vacancy. In a tetrahedral environment, four O DBs combine into a fully symmetric a₁ state and three t₂ states (the latter would be split further if the symmetry is lowered). These states can be occupied with 4, 5, 6, 7, or 8 electrons (corresponding to the 2+, +, 0, –, and 2– charge states of V_Zn, see Fig. 1), i.e., the a₁ states are always filled (indeed, a bonding combination of oxygen orbitals is expected to lead to a state well below the VBM), and the t₂ states can have varying number of electrons. The transition levels of V_Zn span a range of 1.6 eV, from 0.23 eV to 1.84 eV; the energy differences between them reflect Coulomb repulsion energies associated with adding electrons, as well as exchange splittings.

For the Zn DB, the (0e/1e) transition level occurs at 3.25 eV above the VBM of ZnO, i.e., very close to the CBM, while the (1e/2e) level [not shown in Fig. 7(a)] occurs at 3.75 eV above the VBM (i.e., 0.40 eV above the CBM). As shown in Fig. 7(c), the Zn DB has Zn 4s character. The Zn atom that carries the Zn DB with 1e is displaced 0.27 Å off the lattice site towards the site of the absent O neighbor. When occupied by 0e, the Zn atom moves away from the absent O neighbor by 0.30 Å.

We can consider the Zn DB to be the building block for electronic states in an oxygen vacancy; we expect the interactions between these DBs to be stronger than those between O DBs, since the Zn states are less localized in nature, and hence the splitting between a₁ and t₂ states is expected to be larger. In fact, due to the magnitude of this splitting, and the fact that the Zn DB state itself occurs very close to the CBM, the t₂ states are all well above the CBM, and the 2+, +, and 0 charge states of V_O reflect occupation of the a₁ state with 0, 1, or 2 electrons,¹⁰ with transition levels clustered around 1.4 eV below the CBM.

We analyze the optical properties of the ZnO DBs by constructing CC diagrams using the same procedure as used in Figs. 4 and 6. Transitions involving electron capture by an oxygen DB containing one electron are shown in Fig. 8(a). For this transition, we obtain a ZPL of 1.90 eV, an absorption peak of 2.27 eV and emission peaking at 1.49 eV. Hole capture by a doubly occupied O DB (not shown) would involve a ZPL of 1.45 eV, absorption peaking at 1.86 eV and an emission peaking at 1.08 eV.

The calculated CC diagram for hole capture by the Zn DB is shown in Fig. 8(b). Due to the position of the Zn DB transition levels, this is the only transition likely to be relevant. For this process we predict a ZPL of 3.25 eV, an absorption peaking at 3.38 eV, and emission peaking near 2.41 eV.

In Sec. III C 2 we discussed the GL, peaking around 2.5 eV, that is often observed in ZnO and has frequently been attributed to V_O.^22,24–26 We noted that our calculated properties for V_O were not consistent with a PL peak at this energy. For the Zn DB, however, we find a PL peak at 2.41 eV. The Zn DB might then offer an explanation for the unstructured, intrinsic (i.e., not related to Cu impurities) GL in ZnO. We note that this assignment would be consistent with the observations of GL in powdered,²² porous,²⁴ sputter-deposited²⁵ and highly defective²⁶ ZnO material, all of which can be expected to contain high concentrations of dangling bonds.

Using hybrid density functional calculations, we have examined native vacancies, interstitials, and dangling bonds in ZnO. V_O is found to be a neutral defect with high formation energy under n-type conditions; it is unlikely to influence electrical conductivity. The highest-energy PL peak associated with V_O is predicted to occur at 0.62 eV. We predict V_O-related absorption to peak at 2.67 eV, in good agreement with ODEPR studies. V_Zn is the most important native defect in ZnO; it acts as an acceptor which can compensate n-type conductivity. V_Zn gives rise to multiple transition levels above the VBM of ZnO and emission between 1 and 2 eV.

Hydrogen forms highly stable complexes with V_Zn, lowering its charge states and moving its acceptor levels closer to the VBM. Calculated vibrational frequencies of hydrogenated V_Zn agree well with recent experimental reports, supporting the identification of singly and doubly hydrogenated V_Zn species as the source of these signals. Hydrogenated V_Zn leads to similar optical transitions as isolated V_Zn, and our calculated (0/–) level for V_Zn-H is consistent with what was reported in a vibrational study that employed sub-band-gap illumination.³⁵ 

Isolated O DBs have deep transition levels and emission peaking at 1.49 eV, and Zn DBs have transition levels near the CBM. We find that Zn-DB-related emission peaks at 2.41 eV, which may be an intrinsic source of GL in ZnO.

Fruitful discussions with N. Jungwirth and G. Fuchs are acknowledged. This work was supported by the Army Research Office (W911NF-16-1-0538) and by the National Science Foundation MRSEC program (DMR-1121053). Computational resources were provided by the Center for Scientific Computing at the CNSI and MRL (an NSF MRSEC, DMR-1121053) (NSF CNS-0960316), and by the Extreme Science and Engineering Discovery Environment (XSEDE), supported by NSF (ACI-1053575). Work at NRL was supported by the Office of Naval Research through the Naval Research Laboratory's Basic Research Program, and had computational support from the DoD Major Shared Resource Center at AFRL.