Electronic structure of (ZnO)1−x(InN)x alloys calculated by interacting quasi-band theory

We calculated the electronic structure of (ZnO)1−x(InN)x, (=ZION), which belongs to a novel category of hybrid (II–VI)1−x(III–V)x alloys, by the interacting quasi-band theory aided by the sp3 tight-binding model of the wurzite structure. The tight-binding parameters of the irregular bonds (Zn–N and In–O) were estimated by iterating the relevant normal bonds and the absolute atomic levels were corrected by the electron affinities of ZnO and InN. We thus obtained the quasi-band structure of ZION at various concentrations. Across the entire range of concentrations, ZION exhibited a direct energy gap at Γ, and the band-gap energy continuously changes from 0.7 to 3.3 eV with a large band-gap bowing. A particularly, large shift was observed around x = 0.5. The obtained theoretical results imply that ZION (x = 0.1–0.3) is a suitable material for visible-light devices.


Introduction
Semiconducting materials have usually been developed under the octet rule, which stipulates that the number of valence electrons in a unit cell is a multiple of eight. 1) Initial investigations of Si and Ge were followed by investigations of III-V and II-VI compounds with multiple combinations of elements. Another important technique in semiconductor technology is alloying, in which the alloy composition dictates the physical properties of alloyed material. Most of the investigated alloy systems fall into the III-V or II-VI categories. Among the well-known combinations are three component alloys such as (II 1−x II′ x )-VI, (III 1−x III′ x )-V formed by cation substitution and II-(VI 1−x VI′ x ), III-(V 1−x V′ x ) of anion substitution. 2) These alloy compounds have provided the foundation of the so-called band gap engineering. [3][4][5] Extension to four-component alloys would extend the variety in combination of available materials. However, as the combination of elements within the octet rule is limited, we must pursue new combinations.
In 2005, Ref. 6 successfully fabricated (ZnO) 1−x (GaN) x , which belongs to a novel category of (II-VI) 1−x (III-V) x alloys. This alloy, composed of ZnO (E g = 3.3 eV) and GaN (E g = 3.4 eV), has been investigated as a new type of solar cell and photocatalyst material. Quite recently, Ref. 7 have synthesized another semiconductor alloy comprising ZnO (E g = 3.3 eV) and InN (E g = 0.7 eV). This material has been reported as a pseudo-binary alloy, (ZnO) 1−x (InN) x , with a wurzite structure. It can be synthesized by coherent growth using RF-magnetron sputtering, and its electronic structure can be modified by altering the alloy composition. Although this material is potentially applicable to optoelectronic devices, there has been no theoretical attempt to clarify its electronic property. Such novel hybrid alloy systems are expected to contain "irregular" II-V and III-VI bonds that violate the abovementioned octet rule (Fig. 1). How these irregular bonds affect the electronic structures of the systems is a challenging problem.
Bloch's theorem, the most important key theorem in the crystalline materials, is inapplicable to semiconductor alloys because of the randomness in the atomic occupations. The most primitive treatment for random systems is the virtual crystal approximation (VCA), which replaces the random potentials by the average potential, recovering the translational symmetry that guarantees the Bloch theorem. In the VCA, the band structures in the alloys linearly depend on the concentration, but the theory is too simple to describe the intricate electronic structures in real alloy systems. Green's function methods, such as the coherent potential approximation (CPA) has been proposed in the late 1960s, have proven to be the best approach among the single-band tight-binding modeling schemes. 8,9) However, CPA is not easily extendible to multiple bands in compound semiconductor alloys with zincblende or wurtzite structures because the self-consistent equations for multiple coherent potentials are difficult to solve. Alloys can be modelled by another powerful method called supercell firstprinciples calculation, 10) however, the typically used supercells are insufficiently large to realistically represent the randomness in alloys.
Recently, we have developed a new method called the interacting quasi-band (IQB) theory. 11) This method is applicable to various alloys with diagonal and off-diagonal randomness, at any concentration. By extending the IQB theory and using the sp 3 s* (sp 3 ) empirical tight-binding model, we then calculated the electronic structures of typical III-V and II-VI semiconductor alloys with both the zincblende 12) and wurtzite structures 13) and successfully described the general behaviour of the concentration dependence of the top of the valence and the bottom of the conduction bands. When calculating the electronic structure of a cation-substituted alloy, for example III 1−x III′ x -V, the only necessaries are the sp 3 s* crystal parameters in the constituent III-V and III′-V semiconductors. The calculation procedures are detailed elsewhere. [12][13][14] In this study, we further extend the IQB theory to investigate the electronic structure of (ZnO) 1−x (InN) x , a typical novel hybrid of the alloy systems (II-VI) 1−x (III-V) x . As stated above, we must properly account for the randomness due to the statistical occupation of atoms. We must also determine the tight-binding parameters of the irregular bonds and the correct off-sets of the potential energies.

Theory and calculation model
According to Ref. 7 an appropriate mixture of ZnO and InN generates a pseudo-binary system (ZnO) 1−x (InN) x with neutral average charge. In other words, there may be a charge mismatch in some unit cells, but statistically the whole alloy satisfies the octet rule. Based on the experimental results, we assume a wurzite structure with a regular lattice as shown in Fig. 2. The cation sites 1 and 2 are occupied by a cation (C=Zn or In) atom with respective probabilities 1−x and x. Meanwhile, the anion sites 3 and 4 are occupied either by an anion (A=O or N) atom with respective probabilities 1−x and x.
Reference 15 applied the tight-binding model with the sp 3 model to crystal compound semiconductors with wurzite structure. They successfully reproduced the band structures near the valence and conduction bands in several III-V and II-VI crystals. Using the same notations for physical parameters in Ref. 15, we apply the sp 3 model to (ZnO) 1−x (InN) x alloy as follows.
We first construct the quasi-Bloch wave function for a wave vector k k k k , , )denotes the atomic orbital σ (=s, p z , p x , p y ) at the ith site at the inner position τ i (i = 1, 2, 3, 4) in a unit cell located at R, and N is the number of unit cells. The atomic orbitals are assumed to be orthogonal with each other: It should be reminded that because of randomness the wave vector k in alloys is no more a good quantum number, but remains a suitable index in statistical sense. The local amplitude i b s is assumed as a variational parameter dependent on the occupation at the ith site. In the case of (ZnO) 1−x (InN) x , there are following 32 parameters, each corresponding to the atomic orbital : We first consider the expectation value of the Hamiltonian for the wave function (1). We then obtain, for example, the factor for a vertical bond between sites 2 and 3. Here U , ZnO ss¢ for example, denotes the transfer energy between Zn-σ orbital and O-s¢ orbital. It should be noted that the above obtained factor contains U ZnN ss¢ and U InO ss¢ for irregular bonds in addition to U ZnO ss¢ and U InN ss¢ for normal bonds. We next take the ensemble average over the statistical occupation of each site, neglecting the correlation among different sites. The result is then extremized with respect to , i * b s under the constraint condition, We thus obtain a secular equation for i b s 's with 32 × 32 non-Hermitian matrix.
The matrix in Eq. (4) contains the tight-binding parameters of pure ZnO and InN crystals (Table I), the concentration x, and the structural factors for the k-vectors in the Brillouin zone of a wurzite structure (Fig. 3). It would be informative to show here the relation between the VCA and IQB theory. If we apply the following constraint to the amplitude β i σ 's in the trial wave function (1) , , ,a n d , 7  a and c (Fig. 2). As ZnO and InN have different lattice constants, even U ZnO ss¢ in alloy (ZnO) 1−x (InN) x , for example, should differ from that in pure ZnO. It is shown to be satisfactory that the transfer energy depends on the inverse-square of the bond length. 1) Therefore, we corrected them by the following equations.    both c and a, the lattice constant along the a-axis. In this correction, we have assumed that the lattice constants in alloys satisfy Vegard's rule. Second, we must evaluate the transfer energies of irregular bonds, i.e. Zn-N and In-O bonds, in ZION, since these values are absent in Table I. They should be carefully verified since they directly affect the band gap energy. According to Shiba,17) the transfer energy U FG ss¢ between F and G atoms can be factorized in general as, Here the quantity F a s ( G a s¢ ) is specific to atom F (G), and u depends on the spatial geometry of the F-G bond. Another choice of the transfer energies of the irregular bonds will be discussed later.
Third, we usually adjust the site energy in the sp 3 model so that the resulting valence band maximum is to be located at 0 eV (=Fermi level) in each semiconductor. The relative values , , , e e e e are then meaningful within the same material, but not among two different materials. In the IQB theory, the absolute positions of the site energies are crucial. In our previous study on alloys, such as InGaN, [12][13][14] the N atom was common to both InN and GaN. In this scenario, the site energies , s p N N e e so can be shifted to coincide in GaN and InN. However, the present novel hybrid alloy (ZnO) 1−x (InN) x has no common element, so the site energies are corrected by considering the electron affinities that is, the difference between the vacuum level and the bottom of the conduction band. 18,19) Figure 4 schematizes the present correction for ZION, which sets the absolute origin of the energy level to be the vacuum level. Here, we use the electron affinities listed in Table I. Note that the Fermi level is not 0 eV after this correction.  Table I shows the tight-binding parameters used in this study, based on the values in Refs. 15 and 16 with relations given in Table 3 in Ref. 15. A set of these parameters quantitatively reproduces the electronic band structure of pure ZnO and InN crystals.

Results and discussion
In the present study, the origin of the energy has been chosen as the vacuum level = 0 eV, we illustrate here those energy levels <0 eV, since those above the vacuum level are inaccurate in the tight-binding model. In these plots, the present results computed by IQB theory (blue solid lines) are compared with those of VCA (green dashed lines) using Eqs. (8) and (12). In both sets of results, ZION exhibits a direct band gap at the Γ point over the entire range of composition ratios. This suggests that ZION is applicable to optical devices.
As stated in our previous papers, 13,14) the electronic structures obtained by IQB theory are governed by mixing between the quasi-extend states which are approximately reproduced in VCA, and the quasi-localized state (QLS) which represents the effective fluctuation. Thus, the IQB theory gives quasi-Bloch states modulated by randomness in alloys. For (ZnO) 1   As stated before, the transfer energy of the irregular bonds influences the band gap. We have tried several choices of the transfer energy for the irregular bonds, and compared with the experiments on x = 0.3 (ZnO-rich) samples, where most accurate and reproducible experimental results are obtained. 7) Figure 6 shows the calculated results assuming that U U and ( ) are (a) geometrical mean of the regular bond as shown in Eq. (12), and (b) arithmetic mean, which is larger than a geometrical mean, and (c) zero. It is readily seen that the geometrical mean gives satisfactory results with experimental values. Case (b) does not show high-energy shift of the valence band resulting a similar result in VCA. Case (c) shows simple superpositions of reduced pure ZnO bands and reduced pure InN bands, which do not mix with each other. Thus, we have concluded that the choice of the geometrical mean for the irregular bonds, which is consistent with theoretical investigation Eq. (12), is appropriate. Figure 7 shows the concentration dependence of the bandgap energy at the Γ point, determined by IQB theory and in previous experiments. 7) The VCA results are also shown to guide the eye, by dotted line. The present result shows a continuous change of the band gap energy from 0.7 to 3.3 eV, with a large band-gap bowing and a particularly large shift around x = 0.5. This bowing may be derived from randomness in the ZION alloy. The bowing tendency is consistent with the experimental results in ZnO-rich regions (x = 0.1-0.3). In InN-rich regions, the present results largely differ from the experimental values. This discrepancy may arise from the reported difficulty of forming InN films by sputtering methods. 7) In particular, an InN film tends to incorporate O as an impurity. 20) Reference 7 reported that in ZION samples with InN-rich composition, peaks of In 2 O 3 and Zn 3 N 4 appear in XRD measurements. These presence might affect the band gap energy. While we assume the wurzite structure in calculation with all composition. This would be a reason for the discrepancy appeared in InN-rich region. From the present results, we conclude that the visiblelight region of the band gap energy occurs around x = 0.1-0.3, narrower than those estimated by VCA.
In the present study, we used the lattice constant of ZION computed by Eq. (10) and adjusted the tight-binding transfer energies in each alloy composition. Let us now discuss how the deviation of the lattice constant from these ideal values influences the electronic structure. Itagaki's group fabricated ZION by coherent growth on a ZnO substrate, 21) thereby inducing a shift in the lattice constant (see Table II). When the film is thin, the ZION lattice matches that of the ZnO substrate. On a thick film, the lattice relaxes to stabilize the lattice distortion. The electronic structure differences among ZnO, (ZnO) 0.7 (InN) 0.3 (coherent growth), and (ZnO) 0.7 (InN) 0.3 (after relaxation) are shown in Fig. 8. Although relaxation does not remarkably change the conduction band, it removes the degeneracy in the valence band at the Γ point, thereby reducing the band gap.