Introduction
Recently, ZnO has attracted attention not only as a suitable closely lattice-matched substrate for GaN but also as a potentially useful active optoelectronic material in its own right [Reference Reynolds, Look and Jogai1, Reference Reynolds, Look, Jogai and Morkoç2, Reference Look, Reynolds, Sizelove, Jones, Litton, Cantwell and Harsh3, Reference Reynolds, Look, Jogai and Collins4, Reference Reynolds, Look, Jogai, Litton and Collins5] This raises new questions not only about the basic properties of ZnO but also about those of related materials which can be combined with ZnO in heteroepitaxial device structures. In fact, one would need a material with a higher band gap to use as barrier in order to make ZnO quantum wells. Ohtomo et al. [Reference Ohtomo, Kawasaki, Koida, Masubichi, Koinuma, Sakurai, Yoshida, Yasuda and Segawa6] showed that it is possible to grow Mg x Zn1−x O alloy films with the wurtzite structure over a certain range of concentrations. This possibility was far from obvious because MgO has the rocksalt rather than the wurtzite structure.
In this paper, we first re-examine the band structure of ZnO. Although this material has been studied extensively in the past, [Reference Rössler7, Reference Girard, Tjernberg, Chiaia, Söderholm, Karsson, Wigren, Nylén and Lindau8, Reference Schröer, Krüger and Pllmann9, Reference Massida, resta, Posternak and Baldereschi10] there are some remaining open questions about its band structure. Most notably, there has been a long-standing controversy over the nature of the valence band ordering. The question is: is it the usual Γ9 above Γ7 as proposed by Park et al. [Reference Park, Litton, Collins and Reynolds13] or the inverse ordering as suggested by Thomas [Reference Thomas12]. Recently, new results supporting Park et al.’s point of view has been obtained by Reynolds et al. [Reference Reynolds, Look, Jogai, Litton, catnwell and Harsh14]. We show that this ordering depends crucially on the position of the Zn3d bands, which is not correctly obtained in the local density approximation (LDA). We also present results of our total energy results for ZnO in both wurtzite and zincblende structure.
Next, we present results of first-principles calculations of the relative stability of the wurtzite, zincblende and rocksalt forms of MgO and obtain an estimate for the expected gap in wurtzitic MgO. We also present preliminary results for an ordered ZnMgO2 structure, which allows us to obtain a first idea of the expected band gap dependence in this system. Using the dielectric midgap energy charge neutrality point model, [Reference Cardona and Christensen11] we obtain an estimate of the band-offset between ZnO and MgO.
Method of Calculation
The computational method used in this work is the density functional theory in the local density approximation (LDA). The Kohn-Sham equations are solved using the full-potential linear muffin-tin orbital (LMTO) method developed by Methfessel [Reference Methfessel15]. Details of the computational procedure (sphere radii choice, angular momentum cut-off paramters, size of basis sets) are similar to those in recent work on GaN and related materials and were carefully checked for convergence [Reference Kim, Lambrecht, Segall and van Schilfgaarde17].
For the study of the alloys, we used zincblende derived structures as a model system. The gaps in wurtzite are usually slightly larger than those in zincblende forms. The differences between the gaps in the two structures typically vary linearly across the alloy system as we found in previous
work on InxGa1−xN alloys [Reference Kim18]. As a model 50 % compound, we used the chalcopyrite structure. While this is usually a low energy modification and does not quite display the properties of a random alloy, it does nevertheless give some idea of the band-gap bowing. More extensive studies of the disorder effects on band gap bowing are required and are in progress.
The valence band ordering depends crucially on the nature of the spin-orbit coupling. We have studied these within the atomic sphere approximation [Reference Andersen, Jepsen, Šob and Yussouff16] to the LMTO method. Prior checks show that the band-structure without spin-orbit coupling in this approximation is close to that obtained in the full-potential method.
Results
ZnO total energy and band structure
Our calculated total energy and related parameters are summarized in Table I and are seen to be in good agreement with experimental data.
Table I: Properties of ZnO
| wurtzite | zincblende | ||
| expt | theory | theory | |
| a (Å) | 3.25 | 3.21 | |
| c/a (Å) | 1.602 | 1.608 | |
| u | 0.382 | 0.380 | |
| acubic | 4.49 | ||
| volume per molecule (Å3) | 23.8 | 23.4 | 22.6 |
| Ecoh (eV/molecule) | 8.9 | 8.9 | |
| bulk modulus B (GPa) | 157 | 159 | |
In Table II, we compare some of the band structure energies with photoemission data by Girard et al. [Reference Girard, Tjernberg, Chiaia, Söderholm, Karsson, Wigren, Nylén and Lindau8]. One may see that the deeper eigenvalues are slightly too high in LDA (except for A 1,3) which is a result of the increasing self-energy corrections. In particular, one may notice the underestimate of the Zn 3d bands. The latter is of special importance because its hybridization with the O2p band strongly affects the details of the valence band maximum near the gap.
Table II: Selected eigenvalues (in eV) from the top of the valence band.0393
| present FP-LMTO | Pseudopotential a | GW b | expt c | |
|---|---|---|---|---|
| Γ5v d | −0.7 | −0.85 | −0.8 | −0.85 |
| Γ3v | −3.8 | −4.0 | −5.0 | −4.3 |
| A1,3 | −2.1 | −2.37 | −2.48 | −2.0 |
| −A5,6 | −0.4 | −0.43 | −0.43 | −0.45 |
| Zn 3d average | −5.1 | −5.21 | −6.4 | −6.95 |
a Schröer et al. [Reference Schröer, Krüger and Pllmann9]
b Massida et al. [Reference Massida, resta, Posternak and Baldereschi10]
c Girard et al. [Reference Girard, Tjernberg, Chiaia, Söderholm, Karsson, Wigren, Nylén and Lindau8]
d Note that we use Rashba’s [Reference Rashba19] notation for the symmetrylabeling and thus our Γ5 and Γ6 are interchanged from those in Girard et al. [Reference Girard, Tjernberg, Chiaia, Söderholm, Karsson, Wigren, Nylén and Lindau8]
Table III gives our results for the eigenvalues near the band gap. These were obtained in the ASA because we only have spin-orbit coupling included in that version of our LMTO codes. However, the ASA band structure without spin-orbit coupling is in good agreement with that obtained from the FP-LMTO method, except for having a slightly larger band gap. Since the gap needs to be corrected anyway, this is not a serious problem.
Table III: Band gaps and valence band splittings in ZnO.
| theory | Expt a | ||
|---|---|---|---|
| without | with | ||
| Zn 3d shift | |||
| minimum gap at Γ | 1.4 | 1.8 | 3.437 |
| E A − E B (meV) | 37 | 9.5 | 9.5 |
| E B − E c (meV) | 80 | 340 | 39.7 |
| Zn 3d band position (at Γ) (eV) | −5.05 | −6.25 | −6.95 b |
We now discuss the effects of the Zn3d band position on the valence band maximum. According to the Hopfield model,[Reference Hopfield and Phys20] the Γ9 − Γ7 splittings are given by
The Δ s denotes the spin-orbit splitting that would be obtained in the cubic zincblende structure and is 3 times the value of the Δ2 = Δ3 parameters of the Rashba-Shika-Pikus model Hamiltonian.[Reference Kim, Lambrecht, Segall and van Schilfgaarde17]
The experimental values for the A-B, and B-C splittings, where the A, B, and C valence band states are given in order of decreasing energy, are given in the last column of Table III. Now, one can interpret these with A having Γ9 and B and C having Γ7 symmetry, in which case one obtains as crystal field and spin-orbit coupling parameters Δ c = 42.0 meV and Δ s = 16.7 meV, or with B having Γ9 and A and C having Γ7 character, in which case one obtains Δ c =43.3 meV and Δ s = −13.1 meV. The possibility of a negative spin-orbit coupling arises from the hybridization with the Zn3d band. Indeed, a lower lying d band contributes a negative component to the spin-orbit splitting.[Reference Shindo, Morita and Kamimura21, Reference Cardona22]
Without spin-orbit coupling and in a pure LDA calculation, we obtain a band gap of 1.3 eV and a crystal field splitting of the valence band maximum Γ6 − Γ1 of 95 meV. The results with spin-orbit included also indicate too large a spin-orbit and crystal field splitting. When we empirically shift the Zn3d band position in our Hamiltonian, we find that the gap, the crystal field splitting and the spin-orbit splitting depend sensitively on the Zn3d band position. However, if we shift the Zn 3d band all the way down to its experimental position [Reference Girard, Tjernberg, Chiaia, Söderholm, Karsson, Wigren, Nylén and Lindau8] of −6.95 eV, we find that the crystal field splitting becomes as low as 16 meV. The best fit is obtained for a shift of about 1.2 eV and is included in Table III. In order to see whether this case corresponds to a negative or positive spin-orbit splitting, we perform an analogous calculation for zincblende ZnO. In that case, the spin-orbit coupling splits the Γ15 into a quadruplet Γ8 and a doublet Γ7 and we can unambigously determine the sign of the spin-orbit splitting without the need for a tedious inspection of the eigenvectors. When we shift the Zn3d band in zincblende to the same position as in wurtzite, we find a spin-orbit splitting of −12 meV. This is consistent with the value extracted from the wurtzite case using Hopfield’s model. Hence, our present results seem to support a negative spin-orbit splitting. Our calculated values extracted from column 2 of Table III are Δ s = −12.9 meV and Δ c = 37.4 meV. However, we caution that the results are quite sensitive to the d-band position. A slightly larger d-band shift results in a positive spin-orbit splitting. However, we then would face the problem of how to explain the serious underestimate of Δ c . We note that our LDA calculaton should tend to overestimate the Δ c because the underestimate of the gap increases the repulsive coupling between the Γ1 valence and Γ1 conduction band states. Further work is required to fully interpret the data of Ref. [Reference Reynolds, Look, Jogai, Litton, catnwell and Harsh14].
The question now arises as to why the required shift of the Zn3d band appears to be less than that indicated by the photoemission data. This is a rather subtle question. The origins of the discrepancies between theory and experiment are the difference between Kohn-Sham eigenvalues of density functional theory and quasiparticle energies. Because of the energy dependence of the self-energy operator, it may be impossible to obtain both valence band maximum and Zn3d bands consistently from one energy independent Hamiltonian. Clearly, part of the shift must arise from a better treatment of exchange, one of the main errors of LDA being the incomplete cancellation of the self-interaction by the averaged exchange [Reference Städele, Majewski, Vogl and Görling23, Reference Vogel, Krüger and Pollmann24]. This can be handled by an improved energy independent Hamiltonian, (here obtained empirically but in possibly justifiable by means of improved density functionals). However, there still remains an energy dependent part in the true quasiparticle shift, in particular for deep localized narrow bands such as Zn 3d bands.
MgO wurtzite/rocksalt stability
As a starting point for investigating the rocksalt versus wurtzite relative stability in the alloy system, we here examind the end compound MgO. In Table IV we give our total energy results for four structures of MgO, the equilibrium structure (rocksalt), the relaxed (and ideal) wurtzite structure, and the zincblende structure. The ideal wurtzite has bond lengths and cohesive energy very close to that of zincblende. When the structural parameters for MgO in a wurtzitic lattice (c/a and u) are relaxed, however, we find that u = 1/2. This implies that the Mg and O atoms lie in the same plane. In other words the buckled hexagonal rings of the wurtzitic basal plane become flat. It also means that the Mg is at equal distance from the O below and above it along the c-axis. The coordination is thus either 3-fold if one only counts the atoms in the plane or rather 5-fold when one includes the atoms above and below, for which the bond lengths are slightly larger than the in-plane ones. (See Table IV). The five-fold coordination is consistent with the tendency for ionic compounds to prefer a high coordination. The energy of this hypothetical structure is 0.10 eV/molecule above that of the rocksalt structure and lower than zincblende (or ideal wurtzite) by 0.2 eV/molecule. We found that a further continuous distortion can turn this “degenerate” wurtzitic structure into the rocksalt structure. Calculations with the structure confined to such a transformation path indicate that there is a barrier which would mean that the wurtzitic structure is indeed metastable. Details of this will be presented elsewhere. The c/a and u parameters of MgO are thus very different from those in ZnO, which are closer to ideal wurtzite and are given in the previous section.
Table IV: Total energy results for MgO.
| theory | expt. a | ||||
| relaxed wurtzite | ideal wurtzite | zinc-blende | rock-salt | rock-salt | |
| a (Å) | 3.413 | 3.199 | |||
| c/a | 1.204 | 1.633 | |||
| u | 0.5 | 0.375 | |||
| bond length (Å) | 1.971 ⊥ c | 1.959 | 1.959 | 2.071 | 2.1075 |
| 2.055 ∥ c | |||||
| acubic | 4.524 | 4.141 | 4.215 | ||
| volume per molecule (Å3) | 20.7 | 17.8 | 17.8 | 17.8 | 18.7 |
| Bulk modulus B (GPa) | 148 | 129 | 178 | 153−162 | |
| Ecoh (eV/molecule) | 11.56 | 11.39 | 11.36 | 11.67 | 10.41 |
a From Ref. [Reference Timmer and Borstel25] and refs. therein.
Alloys
In view of the above noted complication of MgO and ZnO having very different types of c/a and u parameters, it is far from obvious what will happen to the structure at intermediate compositions. In order to bypass this problem, we decided to first investigate the alloy band gap bowing behavior in zinblende alloys first.
Our LDA band gaps for zincblende ZnO, MgO and chalcopyrite ZnMgO2 are given in Table V along with those of some of the other structural modifications. From the zincblende results we obtain a slight bowing toward lower gaps with bowing coefficient b of 0.56 eV, defined in the usual manner by
where A=ZnO, B=MgO and x is the relative amount of MgO in the alloy.
Table V: LDA band gaps.
| ZnO | MgO | ZnMgO2 | |
| zincblende | 0.75 | 3.76 | 2.115 |
| wurtzite | 1.27 | 3.69 | |
| rocksalt | 5.24 |
The experimental gap of MgO in rocksalt is 7.8 eV, which requires a gap correction of 2.56 eV. With this correction applied to wurtzitic MgO, we obtain an estimated gap of 6.25 eV. This is consistent with the experimental results of Ohtomo et al. [Reference Ohtomo, Kawasaki, Koida, Masubichi, Koinuma, Sakurai, Yoshida, Yasuda and Segawa6] who obtained Mg x Zn1−x O alloys up to x = 0.4. Beyond 0.3, MgO phase-separation started to occur. These results were obtained by a laser ablation growth technique. The band gaps obtained from the shift of the optical transmission measurements varied from 3.3 eV in ZnO to 4.1 eV at about x = 0.36. These are all room temperature gaps which differ from our theory estimates by about 0.1 eV. Using our estimated value of 6.25 for MgO and a gap of 3.4 eV for ZnO, and our bowing coefficient of 0.56 eV predict a gap of 4.2 eV at 33 % MgO, consistent with their results.
Band-offsets
Within the dielectric midgap energy model, the band-offsets are obtained by aligning a suitably defined midgap point which plays the role of a charge neutrality point. In other words, when this level is aligned, negligible charge transfer from one material to the other is assumed to occur. This defines the self-consistent line-up of the two band structures. So, one only needs to calculate where the valence band maximum lies with respect to this midgap point. Our midgap energy points for relaxed wurtzitic MgO and ZnO are 3.03 eV and 1.55 eV above their valence band maxima respectively. This would result in a valence band offset between pure MgO and ZnO of 1.48 eV. With the above estimated gap difference of 2.85 eV, this gives a conduction band offset of 1.37 eV. Taking into account the various uncertainties, the conduction band to valence band ratio is predicted to be about 1. The alignment is of type-I.
Conclusions
In summary, we have obtained total energy and band structure results for ZnO and MgO in various relevant crystal structures. Our results of LDA calculations for ZnO total energy properties and band structure are in good agreement with experimental data, except for the details of the valence band splitting at the meV scale and the usual LDA shortcomings regarding gaps and deep semicore like band positions. We showed that the latter play a crucial role in determining the sign of the spin-orbit coupling and while we could not definitively solve the controversy about the valence band ordering, our present results seem to support a negative spin-orbit splitting. We found that MgO confined to the wurtzite structure has an unusual behavior with a structure which is really 5-fold rather than 4-fold coordinated. We found a small gap bowing and an overall gap dependence consistent with the data by Ohtomo et al. [Reference Ohtomo, Kawasaki, Koida, Masubichi, Koinuma, Sakurai, Yoshida, Yasuda and Segawa6]. We find that the gap of wurtzitic MgO is significantly lower than that of rocksalt MgO. We predict type-I band offsets between ZnO and Mg x Zn1−x O alloys with an approximately 1/1 valence band to conduction offset ratio.
Supported by the Office of Naval Research.
