Electronic structure and band gap of zinc spinel oxides beyond LDA: ZnAl2O4, ZnGa2O4 and ZnIn2O4

We examine the electronic structure of the family of ternary zinc spinel oxides ZnX2O4 (X=Al, Ga and In). The band gap of ZnAl2O4 calculated using density functional theory (DFT) is 4.25 eV and is overestimated compared with the experimental value of 3.8–3.9 eV. The DFT band gap of ZnGa2O4 is 2.82 eV and is underestimated compared with the experimental value of 4.4–5.0 eV. Since DFT typically underestimates the band gap in the oxide system, the experimental measurements for ZnAl2O4 probably require a correction. We use two first-principles techniques capable of describing accurately the excited states of semiconductors, namely the GW approximation and the modified Becke–Johnson (MBJ) potential approximation, to calculate the band gap of ZnX2O4. The GW and MBJ band gaps are in good agreement with each other. In the case of ZnAl2O4, the predicted band gap values are >6 eV, i.e. ∼2 eV larger than the only reported experimental value. We expect future experimental work to confirm our results. Our calculations of the electron effective masses and the second band gap indicate that these compounds are very good candidates to act as transparent conducting host materials.


Introduction
Zinc alluminate (ZnAl 2 O 4 ) and zinc gallate (ZnGa 2 O 4 ) are wide-band-gap semiconductors with the reported band gaps of 3.8-3.9 and 4.4-5.0 eV, respectively [1]. These wide-bandgap structures are useful in photoelectronic and optical applications and are being studied as candidate materials for reflective optical coatings in aerospace applications [2,3]. Because of their wide band gap, they have attracted much interest as possible transparent conducting oxide (TCO) materials [4,5]. For effective material design for this purpose, a sound knowledge of the electronic properties of these materials is essential. The structural properties and electronic structure of these materials have been studied previously [5][6][7] within the framework of standard density functional theory (DFT) [8,9]. But these studies were hampered by the well-known problem that within that framework the band gap of semiconductors and insulators is severely underestimated [10]. Indeed, although the structural parameters obtained within DFT are in fairly good agreement with experiment, the band gaps are not so. For instance, the calculated DFT band gap of ZnGa 2 O 4 is 2.79 eV [6], an underestimation of 42% with respect to the experimental value. Interestingly, in the case of ZnAl 2 O 4 , the DFT band gap is found to be 4.11 eV [6], which is roughly 5% higher than the experimental value. This is in stark contrast to the common trend and has led Sampath [6] to indicate that since the band gaps in [1] were derived from reflectance measurements of powder samples, a correction due to the particle-size dependence of light scattering may be necessary. Thus, the exact band gap value of ZnAl 2 O 4 is at present an open question.
Fortunately, at present there are first-principles techniques that have been demonstrated to be able to describe accurately the electronic structure of semiconductors and insulators. As examples first we mention the GW approximation [10] and thereafter the recently proposed modified Becke-Johnson (MBJ) potential [11,12] approximation. Here we apply these methods to study systematically the series ZnX 2 O 4 , where X = Al, Ga and In are successively heavier elements from group III of the periodic table. We focus not only on predicting the real value of the fundamental band gap in these materials, but also on other key properties in TCOs, such as the second band gap (between the two lowest conduction bands) and the electron effective mass.

3
The GW method is a Green's function technique that involves the ejection or injection of electrons. It links the N -particle system with the (N ± 1)-particle system. In this way, the GW approximation offers a strong physical basis to correlate the band energies obtained using Green's function with the experimental band gap measured using photoemission spectroscopy. Band gaps calculated by GW are observed to be much closer to the experimental values than are DFT band gaps [13]. The MBJ exchange correlation potential proposed recently by Tran and Blaha [11,12] is a parameterized functional that recovers the local density approximation (LDA) as a limiting case. The number of parameters that were tuned by applying this method to a test set is small (only two). MBJ calculations require barely more computation time than do regular LDA calculations, and provide band gaps that are observed to be very similar to GW band gaps [11]. The GW band gaps are calculated on top of the DFT band structure at the point, using a pseudopotential (PP) and a plane wave basis set. Note that we use the nonself-consistent or 'single-shot' approximation [10]. The MBJ calculations are performed with an all-electron method using an augmented plane wave + local orbital (APW + lo) basis set.
Transition metal oxides can be particularly challenging for first-principles calculations and this is the case for the GW method as well. Indeed, while there is ample evidence that the nonself-consistent GW approximation works well in combination with PPs and a plane wave basis set within DFT-LDA [14], it has been observed that it can underestimate band gaps in transition metal oxides if no special care is taken. The exchange part of the self-energy operator within the GW approximation is inadequately treated if only cation d-states are included as valence states [15]. Therefore, a 'standard' PP with only semi-core d-states is not suitable for calculating a GW band gap in transition metal oxides. For ZnO, we have found before that the 20-electron cation PP is essential for an adequate treatment of the exchange part of the self-energy within the GW approximation [16]. In this work, we also address the question of whether the complete n = 3(4) shell must be included in the Ga(In) PP to obtain accurate GW results. Thus, in this paper we present the non-self-consistent GW band gap calculated with two sets of PPs. Firstly, with the 'standard' PP containing the semi-core states (3d 10 , 4s 2 for Zn, 3d 10 , 4s 2 , 4p 1 for Ga and 4d 10 , 5s 2 , 5p 1 for In) and then with the entire n = 3(4) shell treated as valence (3s 2 , 3p 6 , 3d 10 , 4s 2 for Zn; 3s 2 , 3p 6 , 3d 10 , 4s 2 , 4p 1 for Ga and 4s 2 , 4p 6 , 4d 10 , 5s 2 , 5p 1 for In). We discuss how these different PPs affect the structural properties as well as the GW band gap. The accuracy of these PPs is examined by comparison with the all-electron calculations with LDA and the MBJ potential.

Pseudopotentials (PPs)
We use two sets of ab initio norm-conserving PPs for Zn, Ga and In as defined below.
(a) The 'standard' Zn 12+ , Ga 13+ and In 13+ PPs in which the semi-core 3d(4d) state is treated as valence. The inclusion of the wide d-orbital is necessary for a correct description of the structural properties by DFT for group-IIB and -IIIA elements. Hereafter this set of PPs will be referred to as PP1.
(b) The Zn 20+ , Ga 21+ and In 21+ PPs generated with the entire n = 3(4) shell as valence. Since the exchange energy contribution to the self-energy operator depends on the spatial overlap of atomic orbitals, the 's' and 'p' states are also included in the valence for an adequate treatment of the self-energy. It should be noted that we do not construct our Zn 20+ , Ga 21+ and In 21+ PPs for 4 the neutral zinc and gallium (indium) atoms, but rather for the ion with the 4s(5s) and 4p(5p) states unoccupied. The cut-off radius for the Zn and Ga atoms is chosen to be 0.43 Å for 3s, 3p and 3d orbitals and 1.38 Å for 4s and 4p orbitals. For the In atom, we choose a cut-off radius of 0.52 Å for 4s, 4p and 4d orbitals and 1.2 Å for 4s and 4p orbitals. These values for the cut-off radius show the smallest transferability error for ionic configurations of Zn/Ga/In (neutral, +1 and +2), at the cost of an increased plane wave cut-off. We have used 90 Ha as the cut-off energy for plane waves, when Zn 20+ /Ga 21+ /In 21+ PP is used. The PP becomes harder with the inclusion of localized core orbitals in the valence. These PPs are generated with the OPIUM code (http://opium.sourceforge.net/index.html) according to the Troullier-Martins method [17] with Perdew-Zunger LDA [18]. Hereafter this set of PPs will be referred to as PP2.

Density-functional theory (DFT), GW and modified Becke-Johnson (MBJ)
The electronic structure and the quasiparticle (GW) correction to the band gap at the point have been calculated using the plane wave PP code abinit [19][20][21]. For the electronic structure the plane wave cut-off is chosen using the total energy convergence criterion of 2 × 10 −2 eV. The atomic positions and structural parameters have been optimized by calculating the Hellmann-Feynman forces. The stresses are minimized with the criterion of 2 × 10 −5 eV Å −3 . We choose a 4 × 4 × 4 Monkhorst-Pack [22] k-point mesh, which yields 10 k-points in the irreducible Brillouin zone.
The parameters used within abinit to calculate the self-energy are optimized with a convergence criterion of 0.01 eV for the band gap at . We have found that for both the screening and the self-energy calculation, 600 bands are sufficient to converge the GW band gap. The dielectric matrix is calculated with the plasmon-pole model [10] and is used to calculate the screening.
All-electron calculations with the APW + lo method were performed using the wien2k code [25,26]. In this method, the wave functions are expanded in spherical harmonics inside non-overlapping atomic spheres of radius R MT and in plane waves in the remaining space of the unit cell (the interstitial region). The radii for the muffin tin spheres were taken as large as possible without overlap between the spheres: The maximum for the expansion of the wave function in spherical harmonics inside the spheres was taken to be max = 10. The charge density was Fourier expanded up to G max = 16 Ry. Atomic positions were relaxed until the forces were below 0.5 mRy au −1 . The plane wave expansion of the wave function in the interstitial region was truncated at K max = 4.7. A converged k-mesh of 16 k-points in the irreducible part of the Brillouin zone was used.

Structural properties and electronic band structure using DFT
ZnX 2 O 4 (X = Al, Ga and In) adopt the normal spinel structure (space group Fd-3m). They are characterized by the lattice parameter a and an internal parameter u. The Zn atoms are located at the Wyckoff positions 8a (1/8, 1/8, 1/8) tetrahedral sites, whereas Al, Ga or In atoms are located at the 16d (1/2, 1/2, 1/2) octahedral sites and the O atoms at 32e (u, u, u) of the facecentered cubic structure. It has been shown by experiment [23] as well as theory [24] that for these compounds (ZnAl 2 O 4 and ZnGa 2 O 4 ), the normal spinel structure is more favorable than  It is well known that DFT typically underestimates the band gap, as mentioned above, but it does so even more in the case of p-d hybridized systems. Thus the apparent band gap overestimation by DFT-LDA in the case of ZnAl 2 O 4 is anomalous. Previous theoretical calculations on LDA level [5][6][7] found results similar to ours and have suggested that the experimental results require revision. In the case of ZnIn 2 O 4 , the DFT-LDA band gap is found to be 1.71 eV. No experimental information is available for comparison, as this material has not been synthesized experimentally.
The electron effective mass is listed in table 3 for ZnX 2 O 4 . The effective mass is calculated along the [111] direction, and it compares well to the known TCO materials such as ZnO (0.23 m 0 ) and In 2 O 3 :Sn (0.30 m 0 ). The effective mass with MBJ is larger than LDA, as also observed by Kim et al [27]. Another key property of a good TCO is a large second band gap between the two lowest conduction bands. The larger value of the second band gap lowers the   To predict the band gap values accurately and to describe the conduction bands, we performed a calculation of the excited states using the GW approximation as well as the MBJ potential. Our findings are reported in the following sections.

GW and MBJ band gaps
We first calculate the quasiparticle correction to the band gap using the GW approximation. The Kohn-Sham(KS)-DFT band structure calculated with a norm-conserving PP serves as a starting point for the ab initio excited state calculation. The self-energy operator is calculated as = iGW , where G is the one-particle Green function and W is the screened Coulomb interaction. The quasiparticle equation is then solved to obtain the quasiparticle energies qp i and the wave functions i . In the above expression, T is the kinetic energy operator and V ext and V H are the external potential and the Hartree potential, respectively. In practice, both the G and W operators are constructed within the quasiparticle approximation by using the KS wave functions i and energies i obtained by DFT calculations. In this work, the self-energy is calculated with the now well-known non-selfconsistent G 0 W 0 approximation [10], where G 0 is the electron Green function corresponding to the DFT eigenvalues and eigenfunctions G 0 (r, r ; ) = lim and W 0 is the dynamically screened Coulomb interaction Here E f is the Fermi energy, ν is the bare Coulomb interaction and ε −1 is the inverse dielectric matrix.
In the following, we present the quasiparticle band gaps obtained with the two sets of PPs (PP1 and PP2) for Zn, Ga and In (table 2). We discuss ZnGa 2 O 4 first. The band gap with plain LDA is about 2.6 eV, with minor influences due to the type of PP or the use of an allelectron method. This gap is 2 eV below the experimental value. When the GW method is used for the standard PP (PP1), the resulting band gap is 1.2 eV larger. This is considerably closer to experiment, but still almost 1 eV too small. If, however, we use the Zn 20+ and Ga 21+ PP (PP2), we obtain a band gap of 4.57 eV, which agrees nicely with the experimental value. Hence, we confirm that similar to ZnO [16], the Zn 20+ and Ga 21+ PP (PP2) is essential for an To elucidate the contribution of the Ga 21+ PP to the self-energy, we now provide an interesting comparison. We have also calculated the GW band gap of ZnGa 2 O 4 using a combination of Zn 20+ and Ga 13+ PP. The calculated GW band gap is 4.39 eV and the quasiparticle correction to the band gap is 1.54 eV, which is lower than 1.74 eV when the Zn 20+ and Ga 21+ PP is used. This confirms that PP2 should be used for both the cations, Zn and Ga. For ZnAl 2 O 4 , the situation is somewhat different. As mentioned in section 3.1, the LDA band gap is larger than the experimental value, which is an anomalous situation. LDA is known to provide band gaps that are considerably too small for oxides. Indeed, applying GW with the standard PP (PP1) gives a 1.6 eV increase in the band gap. Using the Zn 20+ and Al 3+ PP (PP2) increases the band gap further by yet another 0.7 eV. The final value of 6.55 eV is somewhat larger than the MBJ value of 6.18 eV, and either of both is more than 2 eV larger than the reported experimental value. This strongly suggests that the experimental value is indeed incorrect, and a re-measurement is suggested.
ZnIn 2 O 4 shows qualitatively similar behavior: using GW with the standard PP increases the band gap, whereas using PP2 rather than PP1 gives an additional increase. In contrast to the previous two compounds, the second step introduces the larger change. This is consistent with the observation that even at the LDA level the introduction of PP2 increased the band gap by 0.5 eV. The MBJ band gap is again similar to the GW+PP2 value.
As table 2 shows, the MBJ band gaps fall within a range of at most 7% from the GW band gaps (PP2). This is fair agreement considering the large difference with the plain LDA band gaps. Possible reasons for the GW-MBJ differences are the following: (a) the fact that both band gaps are determined at the equilibrium lattice parameter as predicted by the corresponding code (abinit/wien2k) at LDA level (table 1)  unfavorable. This appears to be correlated with the crystal geometry as explained below. The building block of the spinel structure is a distorted oxygen octahedron that contains one atom of element X. Table 5 shows how the average O-O distance and O-X distance for this octahedron depend on X: the bond lengths are almost identical for X = Al and X = Ga and increase by more than 10% for X = In. A similar octahedron appears in the corundum structure. The corresponding average bond lengths are given as well: there is a 5% increase when Al is replaced by Ga and a more than 10% increase when Al is replaced by In. The latter agrees with the spinel case. The difference between corundum and spinel lies in the small bond lengths for the ZnGa 2 O 4 spinel. Apparently such small bond lengths cannot be maintained for ZnIn 2 O 4 , which renders this crystal unstable.

Conclusions
We calculate the quasiparticle band gap with two sets of PPs for Zn, Ga and In.