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Keywords:

  • nitrogen molecule;
  • oxygen vacancy;
  • p-type doping

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The oxygen vacancy
  5. 3 Nitrogen doping
  6. 4 Conclusions
  7. Acknowledgments
  8. References
  9. Biographies

Our recent calculations of the oxygen vacancy in ZnO based on the local density approximation with Hubbard U corrections (LDA inline image U) are compared with experimental deep level transient spectroscopy (DLTS) results and found to give excellent agreement for the inline image and inline image transition levels. While N O gives a deep acceptor level in ZnO, we show that N2 on the Zn-site can give a shallow acceptor. Results of hybrid functional and generalized gradient approximation are compared. Calculations for the g-factor and analysis of the wave function character also show that an electron paramagnetic resonance (EPR) center for N2 in ZnO also corresponds to the Zn-site rather than O-site.The observation of a donor–acceptor pair recombination photoluminescence center at 3.235 eV along with the EPR center suggests that the latter and its associated shallow acceptor level at 165 meV correspond both to the N2 on Zn. Finally, we discuss how N2 may be preferentially incorporated on the Zn site on Zn-polar surfaces and under Zn-poor, O-rich conditions.

1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The oxygen vacancy
  5. 3 Nitrogen doping
  6. 4 Conclusions
  7. Acknowledgments
  8. References
  9. Biographies

Native defects and doping in ZnO have already been extensively discussed in the literature. Here we focus on two aspects we have studied in our recent computational work: (i) the oxygen vacancy, (ii) the prospects for p-type doping with nitrogen. The oxygen vacancy has been the subject of large controversies in the literature about the position of the inline image transition level. This discussion was already summarized in two of our previous papers [1, 2]. Here we briefly remind the reader what the discussion is about and discuss the comparison with experiment using experimental data we were not aware of when writing our previous papers.

In the main part of the paper we turn to nitrogen and its relation to p-type doping. Substitutional nitrogen on the oxygen site, inline image is well-known by now to give an extremely deep level, unsuitable for p-type doping [3-6]. On the other hand, it is known that a N-related shallow acceptor exists in ZnO. In samples with high N concentration a donor–acceptor pair (DAP) recombination band in the photoluminescence (PL) at 3.235 eV is observed and an acceptor level of 165 meV was extracted from these data by Zeuner et al. ([7]). The question is what defect complex is this level related to?

Recently, Lautenschlaeger et al. ([8]) proposed that inline image pairs coupled with a hydrogen atom could be responsible for this shallow acceptor. We have studied this proposal in Ref. ([9]) and found it to be invalid. On the other hand, Liu et al. ([10]) proposed that a inline imageinline image complex would be a shallow acceptor. They also studied the incorporation of N on Zn-polar and O-polar surfaces. It is known experimentally that N is preferentially incorporated with higher concentration and with the characteristic DAP mentioned above at Zn-polar surfaces ([11]). Liu et al. ([10]) discussed computationally why N gets incorporated preferentially on the Zn-polar surface under high O concentration and then preferentially occupies a Zn-site. However, they found that there is then also a tendency to incorporate inline image next to the NZn. They proposed that the N could switch to the O site and leaves a Zn-vacancy behind, thereby forming the shallow defect complex. This however requires overcoming a rather large energy barrier and would thus not be a very efficient process.

We have recently pursued alternative scenarios involving N2 molecules in ZnO. We found that N2 on the O-site would behave as a donor whereas N2 on the Zn site would be a double acceptor ([12]). This proposal is based essentially on examining the electronic structure of the N2 molecule and the line-up of its energy levels with the energy bands of ZnO. Here we present more detailed evidence for the shallow nature of this defect using both hybrid functional Heyd–Scuseria–Ernzerhof (HSE) [13, 14] and generalized gradient approximation (GGA) calculations. In a recently submitted paper, we also showed that when this shallow inline image derived level is singly occupied, it corresponds to a inline image radical and the g-tensor of such a center was shown to agree well with the observed g-tensor for an electron paramagnetic resonance (EPR) center observed by Garces et al. ([15]) and already assigned to N2 based on the characteristic hyperfine splitting by two inline image nuclear spins. Our proposal for the Zn location also better explains the magnitude of the isotropic hyperfine splitting constant than models with N2 on the oxygen site. Since the DAP of the shallow acceptor is also observed in the same samples as the EPR center, it is likely that they correspond to the same defect. Extending Liu et al.’s ([10]) discussion, we argue that N2 could also be preferentially incorporated on the Zn-polar surfaces. We conclude by discussing recommendations for p-type doping based on these models.

2 The oxygen vacancy

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The oxygen vacancy
  5. 3 Nitrogen doping
  6. 4 Conclusions
  7. Acknowledgments
  8. References
  9. Biographies

The oxygen vacancy is one of the most studied point defects in ZnO. Originally thought to be one of the main sources of background n-type conductivity, it was later shown by various calculations to have a rather deep energy level. In fact, it was proposed to be a so-called negative-U center with a inline image transition. The position of this transition level in the gap has been the subject of controversy, with some authors [1, 16-22] claiming it lies in the lower half of the band gap and others [2, 23-25] it lies in the upper half of the band gap. In view of the band gap of ZnO being as large as 3.44 eV, this means an unreasonable uncertainty of more than 1 eV on this defect level position. What is the origin of this large discrepancy between seemingly similar calculations?

Table 1. Transition levels of inline image in ZnO
 BLaJVbvdWcLZdCRLZeLZ(GGA + GW)fLZ(HSE+GW)gObahPLiPjExpt.k
  1. a

    inline image Boonchun and Lambrecht ([2]): LDA inline imageU with U on Zn and O, image corrections.

  2. b

    Janotti and Van de Walle ([24]): GGAinline imageU extrapolated, no image corrections.

  3. c

    Van de Walle ([23]): LDA no Zn-3d, no image corrections.

  4. d

    Lany and Zunger ([16]): LDAinline imageU with only inline image on Zn, image corrections.

  5. e

    Clark et al. ([22]): screened exchange, image corrections.

  6. f

    Lany and Zunger ([18]): GGAinline imageGW, image corrections.

  7. g

    Lany and Zunger ([18]): HSEinline imageGW, image corrections.

  8. h

    Oba et al. ([25]): HSE, image corrections.

  9. i

    Paudel and Lambrecht ([1]): LDAinline imageU with U for Zn-s and d, no image corrections.

  10. j

    Patterson ([19]): B3LYP, image corrections.

  11. k

    Hoffmann et al. ([26]): DLTS.

2+/02.822.422.701.602.201.361.662.200.800.402.91
2+/+3.342.903.302.24 1.46  0.850.643.30

The reason for the discussion is that the local density approximation (LDA) to density functional theory (DFT) severely underestimates the band gap of ZnO and hence different a posteriori corrections were proposed to treat how this affects the defect level position. In Ref. ([1]), we proposed to address this problem by constructing an explicit Hamiltonian which gives the correct band gap and then apply it to the defect problem. We did this by extending the so-called LDAinline imageU method beyond its usual domain of application. LDAinline imageU is mainly used to deal with strongly correlated electrons in open shell systems. In ZnO it had been applied before to the Zn-3d bands [16, 24]. These narrow (but completely filled) bands are shifted down in energy when an additional Coulomb energy U is added for these states compared to the orbital independent interactions already incorporated in the LDA. The method essentially allows one to deal with orbital dependent Coulomb interactions. This in turn affects the valence band maximum (VBM) because of the hybridization of the O-2p electrons with Zn-3d. It shifts the VBM down and hence slightly increases the gap. However, not sufficiently to agree with the experimental gaps. The main origin of the controversy was then how this affects the defect level. According to one camp ([16]), the remainder of the gap correction should not further affect the defect level. According to the other camp ([24]), one should extrapolate the correction to the case of the full gap correction, even if the remainder of the gap correction is not due to the d-band shift. This led to respectively levels staying in the lower half of the gap or moving up above the middle of the gap. Our first extension of the method added U-shifts to the Zn-s levels, thereby shifting empty states up and hence the conduction band at the inline image-point could be adjusted to the experimental value. The defect level however stayed below the middle of the band gap. We noted that the main effect of the LDA underestimate of the gap was that in the inline image and inline image charge states the Kohn–Sham level of the defect state (the Zn-dangling bond combination of inline image symmetry) lies above the conduction band minimum (CBM). This means that in the LDA (or GGA) calculations the occupation of the inline image charge state is completely erroneous because the electron would be placed in the CBM instead of the defect level. Hence considerable uncertainty exists on the position of the inline image total energy and hence the corresponding transition levels inline image and inline image.

We noted however that this band gap correction is insufficient as it only corrects the minimum gap, whereas in reality the conduction band should shift up more or less rigidly at all k-points. Since the defect is rather deep, one expects, in a conceptual expansion of the defect state in bulk host states, that it should involve states from the entire Brillouin zone, not just from near the inline image-point. Furthermore, the important question is also how the band edges shift individually with respect to the average electrostatic potential, when corrections beyond LDA are incorporated, not just how the gap changes. We then constructed a new LDAinline imageU model in Ref. ([2]) including O-p, O-s as well as Zn-s, Zn-p, and Zn-d shifts. The shifts in this model were adjusted to the quasiparticle self-consistent QSGW ([27]) band structure, on an absolute scale, i.e., relative to the average electrostatic potential.

Although there were other differences between the two studies, pertaining to how to treat the supercell size corrections, the essential difference was that using this procedure, a more accurate Hamiltonian was used for the host band structure incorporating the shifts of the individual band edges throughout the Brillouin zone. This model finally gave a inline image level rather high in the gap, at 2.82 eV. In fact this level is even higher than obtained by any of the previous calculations and in particular than in the hybrid functional calculations by Oba et al. ([25]), which gave 2.2 eV or the screened exchange calculations by Clark et al. ([22]). These two methods are by many regarded as superior since they do not involve empirically determined parameters and correct the band gap of the semiconductor pretty accurately in many cases. The GW calculations by Lany and Zunger ([18]) remarkably still give fairly low position of the level, but this is in part because these authors added an additional size correction of the one-electron levels.

At the time of publication we were not aware of the experimental results by Hoffmann et al. ([26]) which determined the inline image level directly using a deep level transient spectroscopy (DLTS) signal E4 which was shown to be correlated with a deep PL band at 2.45 eV already known to be related to the oxygen vacancy by its relation to the EPR inline image signal from an excited state of the inline image ([28]). The correlation between these defects was established by studying the concentrations of these experimental signatures under varying annealing conditions in vacuum and oxygen atmospheres. Among the various calculated levels, our inline image appears to agree the best with the DLTS results of 2.91 eV. This uses a gap of 3.44 eV and their value of inline imageeV. Furthermore, by using a special fast filling pulse technique these authors were able to temporarily create the meta-stable inline image charge state and an additional DLTS signal E4R corresponding to the inline image transition could be determined by them to be at 3.30 eV. Our calculations placed this level at 3.34 eV. A summary of the transition levels obtained by different methods and authors is given in Table 1. Ironically, the oldest calculation by Van de Walle ([23]) comes also close to the experimental and our values ([2]) even though it used arguably the least accurate method, namely pseudo-potentials of ZnO, which did not include the Zn-3d levels as bands and LDA only without corrections for supercell size effects.

Other experimental data pertaining to the oxygen vacancy level, such as the optical activation and deactivation energies of the inline image state EPR center were argued in our previous paper ([2]) to be closer related to the Kohn–Sham one electron levels than to the transition levels and our calculations for those were also found to be consistent with the data. Evans et al. ([29]) determined the optical activation of the inline image EPR active state by the reaction inline image with the electron going to the conduction band to be about 2 eV. This can be viewed as saying that the one-electron level of the initial neutral charge state lies 2 eV below the CBM, or at 1.4 eV. Our calculations place this level at 1.5 eV.

On the other hand, the deactivation of the EPR center is related to the ODEPR (optically detected) EPR study of Vlasenko and Watkins ([30]) which corresponds to the reaction inline image. The optically detected signal of this process, however, corresponds to the subsequent recombination of the defect electron with a hole from the valence band (VB), not the direct transition from the effective mass-like donor EM to the inline image. Thus it measures in some sense the position of the empty level Kohn–Sham level in the short lived inline image meta-stable state. Our calculation gave 1.86 eV for this, whereas the experiment gave inline imageeV.

The position of the empty Kohn–Sham level in the inline image state is also important for the observation of persistent photo-conductivity. Lany and Zunger ([16]) claim that the persistent conductivity is related to the meta-stable conducting state of the inline image state which relies on this level lying above the CBM. We do not find this, but persistent conductivity can be caused by other defects than point defects. Relatedly, Lany and Zunger ([16]) explain the green emission in ZnO by means of the Franck–Condon shift of the absorption and emission from the inline image to the inline image state. On the other hand, Leiter et al. ([28]) have shown that the green emission at 2.45 eV is related to an inline image triplet excited state of the inline image state, which lies above the CBM as evidenced by a Fano resonance line shape ([31]).

3 Nitrogen doping

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The oxygen vacancy
  5. 3 Nitrogen doping
  6. 4 Conclusions
  7. Acknowledgments
  8. References
  9. Biographies

3.1 Substitutional N on oxygen-site

The most successful attempts at p-type doping ZnO have been with nitrogen ([32]). Nonetheless significant problems remain to achieve reliable and stable p-type doping. In part the problem is to avoid compensating native defects relating to n-type doping. Although the oxygen vacancy is now believed to be relatively deep, even in the estimates that place the inline image closest to the conduction band, hydrogen is a pervasive n-type dopant. The problem with nitrogen substitutional on the oxygen site inline image is that it has an extremely deep acceptor level. While LDA calculations originally placed this level at about 0.4 eV ([33]), later hybrid functional ([3]) calculations found it to be as high as 1.3 eV.

The reason for this is the self-interaction error in LDA. In an LDA or GGA calculation, one finds a Kohn–Sham level or defect band close to the VBM resulting from the slightly higher N-p states than O-p states. The majority spin and 2/3 of the minority spin levels are filled, because there is just one less electron. In hybrid functional calculations, the energy levels depend on the individual orbital's filling. The empty inline image orbital is thereby allowed to move much deeper in the gap. This “orbital polarization” effect is accompanied by a structural relaxation and is hence often called a “polaronic” effect. This more localized solution is inhibited in the LDA because of the remaining spurious self-interaction error.

Exactly how deep the level lies depends on how one corrects for the self-interaction. Lany and Zunger ([4]) use an explicit additional so-called hole potential which is chosen in strength so that the generalized Koopman's theorem (GKT) is fulfilled and found the level at 1.6 eV. On the other hand, if one chooses the Hartree–Fock contribution (inline image) in HSE so as to fix the band gap, then one may overestimate the self-interaction correction. We found in ([9]) that if one uses inline image) the gap is fixed at 3.38 eV but the level inline image level moves to 1.8 eV. This suggests that a better approach is to keep inline image at the standard 0.25 level or to adjust it so as to fulfill the GKT. Other effects play a role, namely the size of the supercell and whether or not one includes image charge corrections. These tend to push the level deeper in the gap. With inline image and a 96 atom cell, we find 1.23 eV, without, and 1.48 eV, with image charge corrections, whereas Lyons et al. ([3]) found 1.3 eV but without image charge corrections and using inline image and 72 atom cell. We also used slightly different lattice constants, optimized for the particular inline image in HSE. Experimentally, the level was found to be 1.3 eV by studying the optical deactivation of the EPR signal related to the singly occupied inline image state by absorbing light which releases the electron from the defect to the CBM. This happens at 2.1 eV, indicating a level at 1.3 eV [6, 34]. Since for a deep level localized system, image charge corrections are important, we think our results indicates that even inline image still somewhat overestimates the self-interaction correction.

3.2 Models for a shallow N-related level

So, the simple inline image defects can be ruled out as a source of p-type doping. On the other hand, a shallow acceptor has been identified to be related to N by analyzing the DAP at 3.235 eV. Zeuner et al. ([7]) showed that this corresponds to a shallow level at 165 inline image40 meV by measuring the residual donor concentration involved in the DAP using lifetime measurements. Several proposals have been made for the nature of this shallow acceptor complex.

Lautenschlaeger et al. ([8]) proposed that inline image pairs could be stabilized by hydrogen and that the DAP interaction between H and the inline image could push the level closer to the VBM. They analyzed statistically the concentration of N required to have a sufficient number of N-pairs. We tested this model in a previous paper ([9]) and found that this model does not work. In fact, H instead of weakly interacting with the two nitrogens would strongly bind to one of them and thereby remove that level completely from the gap and into the VB. The remaining inline image however would be essentially unchanged and stay a deep level in the gap.

Recently, we examined related possibilities, namely, one of the inline image could bind to the other N and form an N2 molecule on the oxygen site leaving behind a vacancy. A similar process was recently found to happen for C in ZnO ([35]). The inline image on the O-site however was found to have donor rather than acceptor behavior and so does the inline image as discussed above. Placing a H next to the inline image perpendicular to the bond axis at the mid-bond altitude, does not help either because a H-s level cannot interact (by symmetry) with the inline image state of the molecule that constitutes the defect level in this case.

On the other hand, examining the basic electronic structure of the N2 molecule it becomes clear that a much more promising location for a shallow acceptor level is to place the molecule on the Zn-site. The reason is as follows. In the N2 molecule the highest occupied molecular orbital (HOMO) is a inline image state, corresponding to inline image-bonding between the inline image orbitals pointing along the axis of the molecule. This level lies in fact slightly higher than the inline image bonding state, inline image because it is anti-bonding to the lower N-s states. The lowest unoccupied molecular orbital (LUMO) on the other hand is the inline image state. Here the subscripts g and u mean even or odd under the inversion center of the molecule and the superscript + means it is an anti-bonding state between p and s-states. The high strength of the N2 bond in fact results from the fact that all bonding combinations of orbitals are filled while all anti-bonding orbitals are empty. This situation, however changes when we place the molecule in the ZnO on an O or Zn site. Assuming that the molecular levels stay more or less intact, if it resides on an O site, it needs to capture two additional electrons, while on the Zn-site it needs to release two of its electrons. This is because as a substitutional defect, it needs to play either the anion (on O site) or cation (on Zn-site) role. However, this means that on the O site, the inline image level will become half filled whereas on the Zn site, the inline image level is emptied. Clearly, the Zn-site is much more likely to behave as an acceptor than the O site, simply because the relevant orbital (inline image) is lying lower in energy. N2 has also been studied in various interstitial sites ([36]). In that case, in principle it should show no active levels in the gap since the inline image stays occupied and the inline image stay empty but in some configurations, the molecule was found to distort the surrounding bonds and this could give rise to active levels in the gap. However, these did not lead to acceptor like levels.

3.3 Qualitative features obtained from FP-LMTO

To test our hypothesis that N2 on the Zn-site would be a shallow acceptor, we first carried out full-potential linearized muffin-tin orbital (FP-LMTO) calculations [37, 38] in the LDA to determine the relative position of the molecular levels relative to the ZnO band structure. Supercells with 128 atoms were used and the structure was relaxed before calculating the partial densities of states (PDOS). By projecting on the N-p and N-s levels, we can identify the inline image and the inline image states. These calculations ([12]), showed that indeed the molecular states were still clearly visible and split by about the same amounts as in the free molecule. For N2 on the Zn-site, it was found that the inline image state formed a resonance seemingly just below the VBM. Note that compared to N O where the N-p orbitals lie above the O-p states in the VB, the bonding between N-p orbitals in the molecule places these levels deeper even when the molecule sits on the Zn-site and hence feels repulsive Coulomb interactions from the surrounding O-ions. N-p and s character is found in the PDOS throughout the VB but gives rise to a resonance just below the VBM. Counting the electrons, the VB is just short of two electrons. In contrast, for inline image on the O site, the Fermi level is lying near the ZnO CBM, which is of course too low in LDA but close to the inline image molecular level. This establishes the basic nature of the two defects as being acceptor-like for the Zn site, and donor-like for the O site. Now, we need to try to determine more carefully if the acceptor state is indeed shallow.

The first indication is that no clearly separated state occurs in the gap, rather a slight resonance near the VBM. For a truly shallow level, supercell calculations often actually give a resonance below the VBM rather than a shallow level just above it because the long-range Coulomb potential which is responsible for the shallow level in a hydrogenic model is not properly included in the supercell approach. The short-range part of the potential of the defect that can be included in the limited size supercell is simply not strong enough to bind a state. The difficulty is that in supercell calculations it is not trivial to actually determine the precise location of the band edges, as they themselves are perturbed by the defect and the finite size interaction effects of the defect. We can examine the defect wave function by integrating the charge density over the narrow energy range of the inline image like peak. This showed a fairly strongly delocalized state with very little weight on the N itself but rather spread first, over the nearest neighbor O atoms and with its tail spread out significantly further. This is a second good indication of a shallow level.

In summary, the results so far indicate that N2 on a Zn site could be a shallow acceptor. This can be understood in a simple manner by counting the electrons. An N2 molecule has ten electrons, whereas a Zn atom has 12 valence electrons, if we include the deep Zn-3d electrons. In fact, due to the molecular bonding in N2 several of the valence electrons are also placed in rather deep levels, even the inline image state lies near the bottom of the O-2p VB in ZnO. In this sense, the lower levels of N2 play the role of the Zn-3d electrons but the molecule ends up with essentially two less shallow valence electrons than a Zn atom. The molecule has a total nuclear charge of inline image but four 1s core electrons reduce the total ionic charge to inline image, matching the ten valence electrons. It is thus a double acceptor and to some extent the defect can be simply interpreted as a hydrogenic effective mass like shallow acceptor defect.

3.4 GGA and HSE calculations of the level position

image

Figure 1. Partial densities of states for three charge states for N2 on the Zn site. Black solid line: total inline image1/100, blue dotted line: unperturbed host, red dashed line: N-s, green dash-dotted line: N-p, vertical long dashed orange lines: VBM and CBM. The relevant molecular states and the band edges are indicated. One may see the defect acceptor level becoming deeper with charge state. These calculations are done in VASP using GGA, a 300 eV cut-off and using a inline image k-point mesh.

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Next, we try to determine the defect levels more quantitatively. Although this could in principle also be done with FP-LMTO we switched to the VASP program ([39]) because it gives us more flexibility in comparing different exchange correlation functionals. We performed GGA calculations in the Perdew–Burke–Ernzerhof (PBE) ([40]) approximation. Supercells with 96 atoms and a cut-off energy up to 400 eV were used. The Brillouin zone integrals were performed using inline image-point sampling. Similar results for the structure and the PDOS were obtained but with a slightly more separated defect peaks just above the VBM. This is shown here in Fig. 1.

image

Figure 2. Band gap with Kohn–Sham energy levels in different charge states for N2 on Zn-site in GGA and determined from the levels at the inline image-point. The open and closed circles indicate the filling with electrons.

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By calculating the core level at some atom far away from the defect and separately determining in a bulk calculation where the VBM of ZnO lies with respect to this core level, we attempt to determine more precisely where the host VBM lies in the PDOS figures. We also examine separately the levels at the inline image-point. Combining these methods we can locate the Kohn–Sham level relative to the VBM. We also carried out calculations for different charge states, inline image, inline image, and inline image. The KS levels determined in this way for each charge state are shown in Fig. 2. They give the empty KS level just 0.11 eV above the VBM in the neutral charge state. This can be identified as the empty acceptor level involved in the donor–acceptor pair recombination and is within 0.05 eV from the experimental value of 165inline imagemeV determined by Zeuner et al. ([7]). Definitely it is compatible with a shallow level. In the inline image state, the majority spin and minority spin level are different and the occupied spin level lies at about 0.19 eV, while the empty minority level lies at 0.24 eV. It moves further in the gap (to 0.51 eV) as we add a second electron in it because of the mutual Coulomb repulsion of the levels.

Another way to determine the defect levels is to calculate the transition levels. We calculate the energies of formation as

  • display math(1)

where inline image is the total energy of the supercell containing the defect, inline image is the total energy of the same size supercell for the host without the defect, inline image is the chemical potential of Zn which is removed and inline image is the chemical potential of the N2 molecule, q is the charge state, inline image is the VBM of the host, determined by means of a core level as explained above, and inline image is the Fermi level measured from the VBM. The Zn chemical potential inline image was calculated separately from metallic Zn in the hexagonal closed packed structure, corresponding to the Zn-rich limit. In the oxygen rich limit, on the other hand, inline image which inline image its value in the O2 molecule and inline image. Here inline image is the enthalpy of formation of ZnO from its elements in their equilibrium phases at room temperature and standard pressure. The inline image is determined in the N2 molecule. The negative values of the energy of formation in Fig. 3 result from the fact that we look at the O-rich limit in which the Zn-vacancy forms relatively easily and so we can also easily put an N2 molecule in this vacancy. The binding in this N2 molecule is only slightly weakened when it enters the solid in this configuration and the weak bonding of the N2 with the surrounding O will somewhat heal the Zn-vacancy so be even less costly than forming an actual vacancy. In the Zn-rich limit, all energies would go up by 1.60 eV and make the energy of formation positive. The inline image transition level is the Fermi level position where the ground state switches from q to inline image. We find the transition level inline image and inline image at 0.22 and 0.37 eV, respectively as shown in Fig. 3. Since the transition levels are based on differences between relaxed total energies for each charge state, we may view these levels as the thermal activation energy for an electron from the neutral state to the single occupied and doubly occupied charge states. These are slightly deeper but still consistent with a shallow level. These are the actual activation energies relevant for ionization of the acceptor and p-type doping.

image

Figure 3. Energy of formation as function of Fermi level indicating the transition levels for the N2 on Zn-site defect. The energy of formation corresponds to the O-rich limit and is calculated in PBE–GGA.

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Now, one may worry about the accuracy of the GGA just as for the inline image case. Therefore we have also carried out hybrid functional HSE calculations. It should be noted from the start, however, that it is not a priori clear that the HSE would be more accurate in this case. In fact, the nature of the defect states is clearly different. For the inline image case, the defect level is located clearly on the N atom and in HSE on a single orbital of the N atom. In such a case, self-interaction effects are expected to be important. In the present case, our calculations already showed that the defect wave function is more spread out and in fact rather located on the nearby oxygens than on the N atoms of the N2 molecule. This is because the inline image level of the molecule would actually lie below the VBM and oxygen would donate electrons to it to keep it filled but then we end up with electrons missing from some of the surrounding oxygens, in other words with a O-dangling bond related state similar to the inline image in which we have placed the N2 molecule. The LUMO of the inline image state in the GGA calculation is shown in Fig. 4 for both spins. The delocalized nature of these states means that self-interaction effects are expected to play a minor role. They also justify a posteriori why we do not include image charge corrections in our calculations. The latter are based on a multipole expansion and require a sufficiently localized defect density. For a delocalized defect electron density, they tend to become negligibly small.

image

Figure 4. Wave function modulo squared of the empty defect state in the gap for N2 on Zn-site in ZnO, calculated in GGA from inline image-point level summed over both spins in neutral charge state. Small red spheres are O, large gray sphere are Zn. The N2 molecule is seen in the middle of the cell as two small gray spheres bonded to each other but not to the rest of the system. The yellow colored iso-surface of the partial charge density corresponds to 0.015 e Å−3.

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image

Figure 5. Band gap and Kohn–Sham levels for N2 on Zn site in various charge states in HSE.

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Nevertheless we carried out HSE calculations to see what happens. The results for the KS levels are shown in Fig. 5. We find that the occupied levels are similar to the GGA–PBE case for each state, but the unoccupied levels moved up significantly. This is related to the fact that in Hartree–Fock, empty states feel a different potential than occupied ones and is responsible among other for the large overestimate of the gap in Hartree–Fock. While this effect is reduced by the mixing factor of only 0.25 of Hartree–Fock in HSE, it still shows that the empty and filled levels behave quite different in HSE. Calculating the transition levels in HSE we find inline image at 0.30 eV and inline image at 1.07 eV using a 300 eV cut-off. These are deeper than in the GGA–PBE results. However, the most relevant inline image level is still relatively shallow. In contrast to the N O case, where the level moved from 0.4 to 1.3 eV, it here only moves from 0.22 to 0.30 eV. There is clearly no “orbital polarization” effect at work in this case. Examining the wave functions still shows a stronger contribution on the neighboring O atoms than on the N although the states are somewhat more localized than in the GGA–PBE.

This indicates that in fact, HSE is less accurate than GGA to describe this defect. This can further be proven by examining whether the Generalized Koopmans Theorem (GKT) is fulfilled. According to this theorem, for the exact theory the vertical transition energy inline image, i.e., the transition energy between the −and 0 charge state calculated at the frozen geometry of the neutral charge state, should be equal to the one-electron Kohn–Sham level inline image. This is indeed nearly the case for PBE–GGA, where the deviation inline image equals only 0.01 eV, whereas in HSE, the deviation is inline imageeV. The signs are as expected for GGA and HSE, respectively and correspond to the convex (concave) curvature of the total energy versus occupation in GGA and Hartree–Fock, while the exact theory should correspond to a straight line. The point is that for delocalized states, as occur in this defect, there is no self-interaction error to begin with, and hence HSE which tries to correct this error, overcompensates.

3.5 Relation to EPR center

In Ref. ([12]), we have shown that this defect can also explain the observed EPR center. When the defect level is singly occupied, i.e., in the inline image state, there is a single unpaired spin and the defect is thus EPR active. An EPR signal clearly related to an N2 molecular species has indeed been observed in ZnO by Garces et al. ([15]). The identification with N2 is based on the characteristic number and relative intensities of hyperfine lines (1:2:3:2:1) for interaction with the nuclear spins inline image of the almost 100 % abundant14 N nuclei. The authors of that paper identified their center with an inline image radical and speculated it would correspond to the O site even though they mention that the defect level should be in a inline image-like bonding state. However, with our present understanding of the molecular levels, it is clear that at the O site, the electron would be in a inline image state. This could in fact occur by ionizing the donor inline image related level removing one of the two electrons with which it should be occupied in the neutral defect state.

There are two problems with this scenario. First, the neutral state with two electrons would likely be in a inline image state (according to Hund's rules) but such a state has not been observed in EPR. Second, in a inline image-like state, the electrons should have no s-contribution on N. This means that there should be no contact hyperfine term because only s-electrons have a finite probability to be near the nucleus. This means that the isotropic part of the hyperfine splitting should strictly speaking be zero and only dipole anisotropic terms can contribute to the hyperfine splitting. However, the hyperfine splitting in the Garces et al. ([15]) center does have a rather sizable isotropic part, in fact larger than that observed for N2 in MgO on O site or on anion sites in alkali halides This already points to the Zn site as a more plausible candidate because in that case our analysis showed that the defect state is related to the inline image state of the molecule which does have a significant s-contribution on the N atoms.

Now, one can argue that in the crystal, the molecular symmetries are not perfectly observed any longer and thus the absolute argument that the isotropic hyperfine part of the hyperfine tensor should be zero for a inline image state is not strictly valid any more, but still it is significant, that the isotropic hyperfine term for Garces center (16.7 MHz) is about four times stronger than for the similar N2 on O site in MgO (4.5 MHz). Instead of interpreting the center as a inline image center, we therefore think it should be interpreted as a inline image radical, corresponding to the Zn site location. The hyperfine splittings for a inline image radical were calculated by Bruna and Grein ([41]) and its isotropic part is 88 MHz. This is larger than the 16.7 MHz observed by Garces, but this is in fact consistent with a shallow nature of the defect if its wave function is about 20% localized on the N-s states. Thus we argue that the observed hyperfine splitting is in much better agreement with the Zn location of an inline image radical than with the O location and of an inline image radical.

Further evidence comes from the g-factor. The g-factor determines the splitting of the singly occupied state in a magnetic field and is the fingerprint of each EPR center. For the inline image radical the deviation from the free-electron g-value is larger for the direction along the axis of the molecule, whereas for the inline image radical it is the other way around. This is again related to the different nature of the states. The deviation of the g-factor results from the orbital contribution to the response to the magnetic field and can be evaluated in second order perturbation theory with one of the perturbations being the spin–orbit coupling and the other the orbital Zeeman term. In the case of the inline image radical, the singly occupied molecular orbital (SOMO) is the inline image state, and the only term contributing to the perturbation theory is the higher lying inline image state. This gives a contribution only for the p-orbitals perpendicular to the molecular axis and hence affects inline image.

For the inline image radical in MgO, on the other hand, has a inline image state as SOMO, split by a crystal field splitting and spin–orbit coupling and leads to a quite different g-tensor with the inline image component along the axis of the molecule having the largest (negative) deviation from the free electron value. The experimentally observed center by Garces et al. ([15]) also observed and discussed by Gallino et al. ([42]) has the largest negative g-shift perpendicular to the direction of the molecular axis and thus agrees with the inline image radical and not with the inline image radical. Again, this indicates the Zn-position rather than the O-position. In Ref. ([12]), further details are given of the g-factors and we also presented a calculation of the g-factor for the inline image case based on a tight-binding model for the molecule, which can be fully solved analytically with parameters adjusted to the DFT calculations.

The study of the EPR center as function of orientation of the magnetic field led Garces et al. ([15]) to the conclusion that the center is axially oriented along the c-axis of the crystal. However, in the calculations we found the N2 molecule to be tilted away from the c-axis and very little interacting with the lattice. In fact, within normal N–O and N–Zn bond lengths there are no bonds between the N2 and the ZnO surrounding lattice at all. This suggests the molecular can rapidly rotate between various equivalent energy minima around the c-axis. These rotations would however happen at a much faster time scale (THz) than the EPR microwave absorption (GHz). This means that effectively the EPR center would exhibit axial symmetry even if the molecule is tilted.

Thus, we arrive at the conclusion that inline image on Zn site would be a shallow acceptor, compatible with the 3.235 DAP signal in PL and when singly occupied provides the explanation for the EPR signal. Significantly the 3.235 eV DAP PL was also observed by Garces et al. (actually at 3.232 eV) in the same samples which show the EPR signal although they rather assigned this level to the N O related EPR signal. This however is incompatible with the now accepted deep acceptor character of the latter. The fact that the 3.235 eV DAP and EPR signal appear in the same samples suggests they arise from the same defect. This reinforces our argument that both are related to the N2 on Zn site.

In order to explain the existence of the N2 on the Zn-site in the single negative charge state, which is EPR active, we must assume that the Fermi level stays below the inline image transition level of 0.37 eV. This means that already there must be a low concentration of compensating donors, which would otherwise pin the Fermi level much higher in the gap. It also means that illumination with fairly low energy photons, could pump electrons from the VB to the defect and make them doubly occupied, thereby quenching the EPR signal. Garces et al. ([15]) show that exposure to 442 nm light almost completely quenches the EPR signal of the N2 center while at the same time activating the N O EPR center. This corresponds to 2.8 eV and they interpreted this as moving the electron from the N2 defect to the conduction band and hence locating the level deep in the gap. We interpret it as quenching it by double filling the acceptor level. It would be interesting to study if much lower energy photons, e.g., of energy 0.4–0.5 eV in the IR already have the same effect.

3.6 The NOinline image complex

The only known alternative candidate for the shallow level is the inline imageinline image complex suggested by Liu et al. ([10]). The inline image is a native acceptor level but with a rather deep inline image level at 0.95 eV ([43]). The N O also has a deep level. It is thus somewhat surprising that their combination would produce a shallow level as low as 0.11 eV (in GGA) or 0.16 eV in HSE as claimed by Liu et al. ([10]). We recalculated this defect in GGA and found 0.21 eV, slightly higher but still compatible with a shallow level.

We offer the following tentative explanation. This defect complex differs from the inline image in that one of the four surrounding O is replaced by a N and there is one less electron. In the inline image there are six electrons in the dangling bonds so, here only five. Two go in a deep inline image level of the inline image tetrahedral group, leaving 4 (and 3) in the inline image state for the V Zn (and inline imageinline image). Now, let's add the distortion to inline image which is present in any case because of the wurtzite structure and here reinforced by the N being one of the corners of the tetrahedron. Then the inline image splits in an e and an inline image level. If the e-level is the lower one, it is exactly filled for the inline image case, and the first hole state is the higher inline image level, but here there is still a hole in the e-level. Further distortion may occur to split the e-level but it appears that the lowest empty orbital for the inline image state may be lower in energy than for the inline image, essentially because there is one less electron in the complex. So, this defect complex appears to be a viable candidate for a shallow level. Further study of this defect is in progress. However, to produce this defect, these authors have to assume a conversion from the inline imageinline image complex, which they find preferentially to be formed considering surface incorporation processes. While they indeed find the inline imageinline image to have a lower energy, they also find that the barrier required to make the transition from one complex to the other by the switching of the N atom is of order 1 eV.

3.7 Discussion of N-incorporation on different polar surfaces

On the other hand, the paper by Liu et al. ([10]) provides useful insights about the incorporation of atoms on the surfaces. They find indeed that on the O-polar surface N is not easily incorporated. It would make only a single bond to a surface O and the resulting NO-molecule could easily leave the surface by desorption. On the other hand on a bare Zn-polar surface, it is also not strongly bonding because of the weak Zn–N bond. However, if we consider an O covered Zn-polar surface, N would form three bonds with O if it lands in a Zn position. N which is in competition with Zn in this case has thus a better chance at surviving at the Zn-site. Upon the next growth step, O would avoid the position on top of N as it could again desorb as NO and then leave a inline imageinline image divacancy but rather go on top of the available Zn sites. They propose this as an explanation for how inline imageinline image complexes would preferentially form and be a temporary step in the subsequent formation of a shallow level inline imageinline image by letting the N switch from the Zn to the O vacancy position. We think that their arguments to N incorporation would also hold for N2 molecules, or once N is incorporated, further N would probably bond stronger to the N already there then to the Zn sites because of the strong bonding in N2. These discussions suggest a natural reason why N2 could be preferentially incorporated on Zn-sites on the Zn-polar surface. This in turn could explain the results of Lautenschlaeger et al. ([11]) that N incorporation is preferential on the Zn-polar surface and leads to the characteristic broad DAP recombination at high enough concentration that is associated with the shallow level ZPL DAP found by Zeuner et al. ([7]) at lower concentrations of N. The site preference energy for Zn versus the O site for an N2 molecule depends strongly on the chemical potential conditions of the growth. In Zn-poor conditions the Zn-site is obviously preferred as is explained in more detail in Ref. ([12]).

4 Conclusions

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The oxygen vacancy
  5. 3 Nitrogen doping
  6. 4 Conclusions
  7. Acknowledgments
  8. References
  9. Biographies

In summary, we have reviewed our previous results on the oxygen vacancy and shown that they agree better with experimental results than any of the other models. They confirm that the oxygen vacancy is a deep donor but not as deep as sometimes claimed in the literature. They also show that the use of LDAinline imageU approach viewed as a way to modify the host band structure with U-parameters chosen to mimic the GW band structure corrections, is a promising approach to defect calculations.

Our studies of N and N2 in ZnO, have shown that N2 on the Zn site behaves as a shallow acceptor whereas N2 on the O-site behaves as a donor and single N O behaves as a deep acceptor. When in a singly occupied state, the g-factor and isotropic hyperfine structure of inline image agree qualitatively with an EPR center observed ([15]) for N2 in ZnO and provide strong additional evidence for the Zn-site. The fact that the EPR center and the DAP recombination at 3.235 eV occur in the same samples strongly suggest that both are related to the same defect and hence we assign both to the N2 on Zn. Finally, extending some previous results of Liu et al. ([10]) on surface incorporation, the preferential incorporation of N2 on Zn site on the Zn-polar surface (in agreement with experiments) suggests natural way for this defect to occur. We cannot exclude the presence of inline imageinline image suggested by Liu et al. ([10]). It could provide a shallow level close to that of the N2 on Zn but a characteristic EPR signal for this complex has not been identified and a more complex process is required to produce it.

Our identification of the shallow level with N2 on Zn, results in several new recommendations for achieving p-type doping: (i) work in Zn-poor, O-rich conditions, (ii) use N2 rather than atomic or excited or other N species such as NO, NH3 to incorporate nitrogen, (iii) as already established experimentally, use the Zn-polar surface. The recommendation (i) has the added benefit of avoiding oxygen vacancies and the recommendation (ii) has the advantage of avoiding H incorporation as a compensating n-type donor. These recommendations present a change in direction from several previously held beliefs, according to which high N incorporation would require more reactive species than N2 and placing it on O site would require O-poor conditions and making the levels shallower might have required H.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The oxygen vacancy
  5. 3 Nitrogen doping
  6. 4 Conclusions
  7. Acknowledgments
  8. References
  9. Biographies

We thank S. Limpijumnong for useful discussions in the early stages of our work on N-pairs in ZnO. We thank B. K. Meyer for telling us about N2 in ZnO and encouraging us to study the g-factors. W. L. thanks the Fulbright foundation for a scholarship allowing his sabbatical at the Forschungszentrum Jülich (FZJ) Peter Grünberg Institute (PGI), where most of the recent work on N2 was carried out, as well as the hospitality of the Stefan Blügel group at the FZJ-PGI. The work at CWRU was funded by NSF-DMR-1104595 and DOE-BES-0008933. Calculations were performed at the Ohio Supercomputer Center and at Tsukuba NIMS.

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The oxygen vacancy
  5. 3 Nitrogen doping
  6. 4 Conclusions
  7. Acknowledgments
  8. References
  9. Biographies

Biographies

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The oxygen vacancy
  5. 3 Nitrogen doping
  6. 4 Conclusions
  7. Acknowledgments
  8. References
  9. Biographies
  • Image of creator

    Adisak Boonchun received his B.Sc. degree from Kasetsart University in Thailand and his PhD in physics in 2011 from Case Western Reserve University in Cleveland, Ohio. Since 2012, he holds a postdoctoral position at the National Institute for Materials Science (NIMS) in Tsukuba, Japan. He is interested in the study of properties of materials using density functional calculations on a variety of materials. Since his PhD, Adisak Boonchun has worked on electronic structure of semiconductor photocatalysis and has continued working on atomistic modeling of defects in ZnO that may lead to p-type doping.

  • Image of creator

    Walter R. L. Lambrecht obtained his Dr. Sc. at the University of Gent, Belgium in 1980. He is professor of physics and materials science at Case Western Reserve University in Cleveland, Ohio. He is a Fellow of the American Physical Society. He has worked on electronic structure, phonons, defects, and optical properties of a wide variety of materials, in particular on wide band gap semiconductors, such as ZnO, SiC, and group-III-nitrides, dilute magnetic semiconductors, chalcopyrites, and rare-earth compounds. The present work was facilitated by a sabbatical stay at the Forschungszentrum Jülich supported by a Fulbright award in 2012.