Electronic structure and bandgap of γ-Al2O3 compound using mBJ exchange potential
© Yazdanmehr et al.; licensee Springer. 2012
Received: 28 May 2012
Accepted: 4 August 2012
Published: 31 August 2012
γ-Al2O3 is a porous metal oxide and described as a defective spinel with some cationic vacancies. In this work, we calculate the electronic density of states and band structure for the bulk of this material. The calculations are performed within the density functional theory using the full potential augmented plan waves plus local orbital method, as embodied in the WIEN2k code. We show that the modified Becke-Johnson exchange potential, as a semi-local method, can predict the bandgap in better agreement with the experiment even compared to the accurate but much more expensive green function method. Moreover, our electronic structure analysis indicates that the character of the valence band maximum mainly originates from the p orbital of those oxygen atoms that are close to the vacancy. The charge density results show that the polarization of the oxygen electron cloud is directed toward aluminum cations, which cause Al and O atoms to be tightly connected by a strong dipole bond.
KeywordsBandgap mBJ exchange potential Density functional theory
γ-Al2O3 is an important material in nanotechnology because of its porous structure, high surface area, and high catalytic surface activity. This material is widely used as a catalyst, an adsorbent, and a support for industrial catalysts in the oxidation of organics and the catalytic reduction of automotive pollutants such as nitric oxide . This oxide material is a metastable ‘transition’ phase of aluminum oxide or alumina which can be formed via dehydrating boehmite γ-AlOOH at rather low temperatures (350°C to 1,000°C)  and is able to keep its crystalline structure unchanged up to about 1,200°C . Recently, this material is considered as a suitable alternative to silicon for producing semiconductor nonvolatile random access memories for future applications . Thus, investigation of the atomic and electronic structures of γ-Al2O3 has also received much attention over the last decade [5, 6].
In this work, we have calculated the electronic structure of γ-Al2O3 compound using the mBJ exchange potential . Our result shows that the semi-local mBJ method predicts the γ-Al2O3 bandgap better than the LDA and GGA when compared to the experimental data. The results which are obtained by the semi-local mBJ method are in good agreement with those of the experimental data. In fact, the precision of theoretical predictions in the mBJ scheme is comparable with the accurate but much more expensive Green function (GW) method. In the mBJ exchange potential, there is a correction (c)-factor which can be used as an adjustable parameter for improving the bandgap prediction. Therefore, the mBJ method has the capability to overcome the well-known shortcoming of the DFT-based methods in predicting the bandgaps. In this paper, we have utilized this capability to yield the bandgap of γ-Al2O3 by determining and fixing the adjustable c parameter of the mBJ potential. The calculated bandgap within our fixed mBJ calculations is found to be in better agreement with that of the experiment than that of the GW method.
RMT-KMAX(bohr × Ry1/2)
Separation energy (Ry)
Muffin-tin radii (bohr)
8 × 8 × 3
The perfect crystalline structure of γ-Al2O3 is described as a defective spinel, denoted as O32, (□ = Al vacancy) [5, 11]. To understand this formula, we should consider a perfect spinel, such as the MgAl2O4 structure. However, due to stoichiometry, γ-Al2O3 is not a perfect spinel because its cation/anion ratio is less than a stoichiometric spinel structure (i.e., 24/32). Hence, aluminum vacancies are formed in each cell. Aluminum vacancies can present both in tetrahedral and octahedral sites, as the cations are divided in tetrahedral and octahedral positions in a stoichiometric spinel. However, the lowest energy configuration of γ-Al2O3 occurs when all of the aluminum vacancies are in octahedral sites with the largest possible inter-distances. The space symmetry group of a perfect spinel, such as the MgAl2O4 structure, is . In this symmetry group, the primitive cell is a triclinic cell of 14 atoms, and hence, it is denoted as Mg2Al4O8. An ideal spinel unit is therefore constructed with the lattice constant 7.911 Å , and all of the Mg atoms are replaced by aluminum. The cubic unit cell contains three Al6O8 primitive cells on ‘top’ of each other, and as a result, the unit cell contains 18 Al and 24 O atoms. To get the correct stoichiometry, we subsequently remove two Al atoms from the octahedral sites which have the largest inter-distances. In this way, a 40-atom triclinic cell containing eight Al2O3 formula units and two Al vacancies is obtained. In order to prepare a hexagonal cell similar to the structure already reported by Pinto and co-workers , we change the basis vectors as where .
Density of states
Partial p-orbital DoSs (partial p-DOSs) are calculated for each of the nonequivalent oxygen and aluminum atoms using mBJ in the γ-Al2O3 compound. The p-DOSs of the nonequivalent O atoms are added to each other to obtain total p-O-DOS, as shown in Figure 2c. Similarly, total p-DOS for the Al atoms is the sum over the partial p-DOSs of the nonequivalent Al atoms (Figure 2d). The total p-DOSs for O and Al atoms show that contribution of the oxygen p-states dominates valence states in the vicinity of the Fermi level. Therefore, most of the occupied states which are touching the Fermi level are constituted by the p-O-DOS. This implies that the oxygen atoms and, in particular, their p-sates are more important for tuning the band gap of γ-Al2O3. The latter result is deduced from the total p-DOS, obtained by adding p-DOSs of the nonequivalent O atoms. Hence, it is not still clear which of the oxygen atoms play a more important role. To clarify which of the oxygen atoms in the γ-Al2O3 compound are more appropriate for engineering the bandgap, we perform more elaborations on the electronic structure of the system by classifying the O atoms in two different groups. The first group contains the O atoms that are far from the vacancies. In the second group, the O atoms are selected to be close to the vacancies. The partial p-DOSs of the first and second groups are shown in Figure 2e,f, respectively. The results elucidate that the oxygen atoms in the second group play a more important role in the engineering of the bandgap, as a contribution of the valence p-DOS of those of oxygen atoms which are closer to the vacancies predominates in touching the Fermi level.
Electronic structure and bandgap
The bandgap of the system is improved by performing the non-regular mBJ calculation. However, it is mandatory to examine whether the internal structure of bands is destroyed by increasing the strength of the repulsion mBJ potential. To this end, one needs to compare Figure 3a and Figure 3b. The comparison authenticates that the band structure is not internally affected by the stronger repulsion potential. Indeed, the conduction bands are altogether shifted up by the same value after increasing the c parameter, keeping their previous structures almost the same as before. This is consistent with our discussion presented earlier concerning the behaviors of the regular mBJ- and GGA-DOSs, as can be rechecked by comparing Figure 2a and Figure 2b.
Physical backbone of increasing the c-factor for the defective spinel γ-Al2O3
where the two 1.023 and −0.012 (dimensionless) values were fitted by Tran and Blaha  in order to reproduce the experimental bandgaps of several compounds. From Equation 1, it is clear that if c = 1, then we have, which is consistent with that of . The c-factor in many cases can be well reproduced within a self-consistent calculation by the self-consistently converged charge density, ρ σ . However, for the γ-Al2O3 compound, the self-consistently reproduced c-factor is not calculated to be so large to be used for predicting the bandgap close to the experimental value. The c-factor may not be large enough for our case because the electron charge density for the defective spinel O32 around the Al vacancy, as indicated by the empty box (□), is very small. Therefore, due to the vacancy, the c-factor which linearly depends on the square root of the average of is self-consistently calculated by Equation 2 to be smaller than an actual and necessary c value for reproducing the bandgap of the γ-Al2O3. Hence, we have increased the c-factor to overcome the low electron charge density that originated from the vacancy for our case. It is worth to note that, in practice, the c-factor cannot be unlimitedly increased. Thus, the experimental bandgap cannot be always exactly reproduced by increasing the c-factor in the current version of the mBJ method for every case without considering the electron charge density variation and bonding nature of the case. Indeed, there can be a critical value for the c-factor, as reported in . For larger c values than the critical c value, the bandgap may be drastically decreased to a meaningless value far from the experiment. This shows that the mBJ potential may result in an incorrect electron charge density, if one uses a very large c-factor. The latter point was perfectly demonstrated  by calculating the electric field gradient, as an extremely sensitive physical quantity to the shape of the valence electron charge density, versus the c-factor for the Cu2O compound. Consequently, the c-factor, as a measure for the electron charge density, should be carefully examined and optimized at least for those cases having low electron charge density. The low charge density may originate from some hollow spaces due to the vacancies, as what existed in the γ-Al2O3 case, or from weak van der Waals bonds, as what existed in the fcc-C60 fullerite. Therefore, from the above discussion, we would, as a corollary result, anticipate that the c-factor may not be also well produced by the regular mBJ method for the case of fcc-C60. Thereby, one needs to first optimize the c-factor for this case as well. The latter anticipation needs more elaboration which is out of the scope of this work.
We have successfully simulated the γ-Al2O3 compound based on the DFT method using the full potential augmented plan waves plus local orbital method, as embodied in the WIEN2k code, and applied the mBJ exchange potential on this system to predict its bandgap more precisely. We found that the Al-O has a highly polar bond in this compound which is consistent with previous reports. We showed that distribution of the valence charge density is not uniform around different Al lattice sites. Besides, the calculated partial DOSs indicated that majority of the valence electronic charges correspond to the p orbitals of oxygen atoms which are consistent with those of previous reports. However, we showed that different oxygen atoms have different contributions in the valence electronic charges. Contributions of oxygen atoms which are closer to the vacancies in γ-Al2O3 dominate. A charge-free region is observed at the aluminum vacancy site which confirms the charge neutrality and insulating behavior of the stoichiometric γ–Al2O3 structure. Our result shows that mBJ can significantly improve the electronic structure of the system if a suitable c-factor is used. A direct bandgap of 8.02 eV, which is very close to the experimentally measured value of 8.7 eV, was obtained at the Γ point by adjusting the c-factor internally to a value of 1.8. This c-factor value can be utilized for the correct estimation of the electronic and optical properties of γ-Al2O3 compound based on the full potential augmented plan waves plus local orbital DFT method using the mBJ exchange potential.
SJA has a PhD in Computational Condensed Matter Physics. AN has a PhD in Experimental Condensed Matter Physics. MY, MG, and MR are MS students.
Saeid Jalali Asadabadi is thankful to Dr. Kourosh Zarringhalam for his valuable comments. This work, as part of Mohsen Yazdanmehr's thesis, is supported by the Office of Graduate Studies, University of Isfahan (UI), Isfahan, Iran.
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