Post-transition-metal oxides, such as In₂O₃ and SnO₂, have been extensively used as transparent conducting oxides (TCOs), a unique class of materials that combine the high electrical conductivity and optical transmission in the visible or near-infrared spectral range, which are needed in various optoelectronic devices, such as solar cells, flat-panel displays, light emitting diodes (LEDs) and transparent transistors^[1-7]. Obviously, one of the most critical properties of these materials is related to their band gaps that essentially determine their transparency. However, the determination of band gaps of these TCOs has not been a trivial work. For instance, early experimental measurements on In₂O₃ revealed a strong optical absorption at 3.75 eV and a much weaker absorption at 2.62 eV^[8]. Initially, the former absorption was attributed as the direct band gap of In₂O₃, while the latter was attributed to the indirect transitions. Although this interpretation seems able to explain some of the experiments, the indirect transitions in In₂O₃ has never been confirmed by first-principles calculations^{[9, 10]}, hindering further understanding of the band structure and optical properties of this material. More recently, by combining the experimental measurements and theoretical calculations, Walsh et al.^[11] identified that In₂O₃ actually possesses a direct fundamental gap of 2.9 eV and a much larger optical gap of 3.7 eV. The large discrepancy between the fundamental and optical gaps was attributed to the inversion symmetry of its bixbyite structure and the resultant dipole-forbidden transitions between the valence band maximum (VBM) and conduction band minimum (CBM) states. This argument, capable of explaining various theoretical and experimental results, has now been commonly accepted.

In contrast to the case of In₂O₃, the band gap problem of SnO₂ is still not fully understood, largely because of the lack of detailed experimental measurements. Analogous to bixbyite In₂O₃, rutile SnO₂ also has the inversion symmetry and the dipole-forbidden transitions between the band edge states. As a result, one may expect different fundamental and optical gaps of SnO₂ as well, which, however, has never been reported in experiments. The single-photon and two-photon absorption experiments showed an optical gap of SnO₂ at ~3.60 eV, which has been widely taken as its “band gap”^[12-14]. Moreover, the recent theoretical calculations, based on the density functional theory (DFT) and GW approximation, identified a fundamental gap of 3.65 eV for SnO₂^{[15, 16]}, which is nearly equal to its optical gap. Sabino et al.^[17] attributed the coincidence between the measured fundamental and optical gaps to the intense illumination used in experiments, so that the very weak absorption in the vicinity of Γ point at the band edges was detected. Based on this argument, they also predicted that a much larger optical band gap of 4.34 eV should be detected, if the low illumination condition is satisfied. However, up to now, no experimental demonstration of this difference between fundamental and optical gaps has been reported.

Despite the absence of direct measurements on its fundamental gap, more insight can be obtained from the recent experiments where SnO₂ was employed as the electron transport layer (ETL) in both dye-sensitized and perovskite solar cells to replace the commonly used TiO₂^{[18, 19]}. The band alignment measurements^[19] showed that the CBM of SnO₂ is about 0.4 eV lower than that of TiO₂, which facilitates the electron transport. Assuming the fundamental gaps of SnO₂ and TiO₂ as 3.65 and 3.03 eV^[20], respectively, we can easily deduce that the VBM of SnO₂ is ~1.02 eV lower than that of TiO₂, as illustrated in Fig. 1(a). This result, however, is quite puzzling and contradicts the small VBM offset one would expect in this system^[21-23], because the two compounds are isovalent semiconductors with the same rutile structure and similar lattice constants, and moreover, the Γ₃⁺ VBM state is almost a pure oxygen p state, which only couples weakly with the cation d state but does not couple to the cation p state. Therefore, this VB offset anomaly is difficult to understand if the fundamental gap of SnO₂ is taken as ~3.65 eV, as previously reported.

To address this issue, in this work, we revisit the band gap problem for the rutile SnO₂ by using first-principles calculations. Different-level computational methods and functionals, such as PBE, PBE + G₀W₀, HSE06 and HSE06 + G₀W₀ are employed and the results are carefully compared. We find that the HSE06 calculations yield a VB offset of 0.38 eV between SnO₂ and TiO₂, generally obeying the common-anion rule. The fundamental gap is calculated to be 2.96 eV, which is much smaller than the previously reported value. The conduction band (CB) offset is then found to be 0.45 eV, in good agreement with experiments^{[18, 19]}. Moreover, similar to In₂O₃, our calculations reveal that the direct optical transition between the band edges of SnO₂ is parity forbidden and the optical gap between the lower-lying VB states and CBM is 3.69 eV, which can well explain its optical absorptions^[12-14]. On the other hand, by employing the HSE06 + G₀W₀ method, the fundamental gap of SnO₂ is found to be 3.76 eV that is close to the value reported previously^{[16, 17, 24]}. However, with the same method, the fundamental gap of TiO₂ is calculated to be 3.68 eV, significantly larger than the experimental result of 3.03 eV^[20]. In fact, we find that the fundamental gap of SnO₂ is always similar to that of TiO₂, independent of the computational methods and functionals employed, which implies that the HSE06 + G₀W₀ calculations may overestimate the fundamental gap of SnO₂, which has been the case also found in other sp semiconductors^[25]. The estimated error of HSE06 + G₀W₀ calculation is about 0.8 eV, which in turn results in a fundamental gap of SnO₂ as ~3.0 eV, in good agreement with the HSE06 calculations.

Our calculations are carried out using the projector augmented wave (PAW) method^[26] and the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional^[27] within the density functional theory (DFT) as implemented in VASP^[28]. For both SnO₂ and TiO₂, a 520 eV cutoff energy is used for the plane wave basis set and a Γ-centered 4 × 4 × 6 k-point sampling of the Brillouin zone is employed. Since the band gaps of these compounds are sensitive to their lattice constants, experimental lattice constants (a = 4.737 Å and c = 3.186 Å for SnO₂^[29], a = 4.594 Å and c = 2.958 Å for TiO₂^[30]) are adopted for the electronic structure calculations. During the structural relaxation, the atomic coordination is fully relaxed until the Hellman–Feynman force on each atom is less than 0.01 eV/Å. To compare the influence of different computational methods and functionals on the band gaps, the Perdew–Burke–Ernzerhof (PBE)^[31], GW^[32] with the PBE (PBE + G₀W₀), and HSE06 + G₀W₀ calculations are also performed.

Fig. 2(a) illustrates the tetragonal rutile structure for both SnO₂ and TiO₂ with the space group . As shown in Fig. 2(a), each cation atom (Sn or Ti) binds with six O atoms, forming a distorted octahedron, while each O atom is trigonally coordinated to three cation atoms. The calculated projected density of states (PDOS) are given in Fig. 2(b). We find that the Γ₁⁺ CBM state of SnO₂ is mainly composed of Sn 5s and O 2s states, while the Γ₃⁺ VBM state is mainly composed of O 2p states. For TiO₂, on the other hand, the CBM has mostly Ti 3d character, while the VBM is still the Γ₃⁺ state dominated by O 2p states with minor Ti 3d character. The above analysis indicates that the well-known common-anion rule^{[21, 22]} should apply to this system, i.e., the VBM offset between these two compounds should be relatively small. To verify this expectation, we construct a SnO₂/TiO₂ heterojunction consisting of six SnO₂(001) and six TiO₂(001) layers, as shown in Fig. 2(c), to calculate the VB offset between SnO₂ and TiO₂ using the method described in Ref. [33], that is

$ \Delta {E_\rm{V}}(\rm{SnO}_\rm{2}/\rm{TiO}_\rm{2}) = {({E_\rm{V}} - {E_\rm{CL}})_{\rm{TiO}_\rm{2}}}} - {({E_\rm{V}} - {E_\rm{CL}})_{\rm{SnO}_\rm{2}}} + \Delta {E_\rm{CL}}. $  (1)

The first two terms on the right-hand side are the core-level to VBM energy separations for pure bulk TiO₂ and SnO₂, respectively, and the third term is the difference of core-level binding energy between SnO₂ and TiO₂ in the SnO₂/TiO₂ superlattice. Since the lattice mismatch between SnO₂(001) and TiO₂(001) layers is very small (< 2%), here we set the lattice constants of the superlattice as the average of those of SnO₂ and TiO₂ and fully relax the atomic coordination in the superlattice with a Γ-centered 4 × 4 × 1 k-mesh until the Hellman–Feynman force on each atom is less than 0.01 eV/Å.

The calculated band alignment for SnO₂ and TiO₂ is shown in Fig. 1(b). We find that the VBM of SnO₂ is 0.38 eV lower in energy than that of TiO₂, which generally follows the common-anion rule. The slightly lower VBM of SnO₂ can be attributed to the more polar Sn⁴⁺ ions, compared to Ti⁴⁺ ions. The calculated fundamental gaps of SnO₂ and TiO₂ are 2.96 and 3.15 eV, respectively, with the latter slightly larger than the experimental value of 3.03 eV^[20]. Using the experimental band gap of TiO₂, one can deduce that the CBM of SnO₂ lies ~0.45 eV below that of TiO₂, which falls in line with the experimental measurements^{[18, 19]}. It is interesting to see that the calculated fundamental gap of SnO₂ is slightly smaller than that of TiO₂, which is quite different from the experimentally measured band (optical) gap of ~3.60 eV^[12-14]. To understand this discrepancy, we carefully investigate the band structure of SnO₂, as shown in Fig. 3(a). Apparently, SnO₂ has a direct fundamental gap located at the Γ point. Inspection of the wavefunction at the band edges indicates that the VBM and CBM states both have the even parity with Γ₃⁺ and Γ₁⁺ symmetries, respectively. Therefore, the electric-dipole transition between these two states is forbidden. Table 1 gives the corresponding transition matrix elements. We can see that the transition probability between the VBM and CBM at the Γ point is exactly zero. As the k-point is slightly off the Γ-point, the transition matrix elements between the VB and CB states are still negligibly small, indicating very weak optical transitions in this region. In contrast, we find that the pronounced optical transitions can occur from the low-lying VB states to the CBM at the Γ point, leading to an optical gap 0.74 eV larger than the fundamental gap. As shown in Fig. 3(a), the low-lying VB states have the Γ₅^– symmetry with oxygen and cation p characters, so the optical transitions from these states to the CBM state are allowed. As shown in Fig. 3(b), the calculated absorption coefficient indicates a very slow increase above 2.96 eV and a sharp increase above 3.70 eV, in good agreement with the band structure analysis. From these calculations, we can conclude that SnO₂ should have a relatively small fundamental gap of ~3.0 eV and a much larger optical gap of ~3.7 eV, with the latter accounting for the “band gap” observed in optical absorption experiments^[12-14].

It should be noted that the fundamental gap of SnO₂ obtained here is quite different from the values reported by the previous calculations^[15-17]. For instance, based on the HSE03 + G₀W₀ calculations, Schleife et al.^[15] reported a fundamental gap of SnO₂ as 3.65 eV that is similar to observed optical gap. They attributed the near coincidence between the fundamental and optical gaps to the dipole-allowed transitions in the vicinity of the Γ point, as depicted by the green arrow in Fig. 3. Sabino et al.^[17] further explained that the detection of these weak transitions can be attributed to the intense illumination used in experiments. To compare with these previous calculations, we also employ the GW formalism combined with the hybrid functional calculations (HSE06 + G₀W₀), together with some other methods, such as PBE and PBE + G₀W₀, with the calculated fundamental gaps of SnO₂ and TiO₂ given in Table 2. For the HSE06 + G₀W₀ calculations, the fundamental gap of SnO₂ is found to be 3.76 eV that is consistent with the previous calculations. However, with the same method, the fundamental gap of TiO₂ is calculated to be 3.68 eV, apparently larger than the experimental results^[20]， which indicates that the HSE06 + G₀W₀ calculations cannot describe the band structure well, at least for TiO₂. More interestingly, we can see from Table 2 that the calculated fundamental band gap of SnO₂ is always similar to that of TiO₂, although different computational methods and/or functionals result in different band gap values. This trend suggests that the HSE06 + G₀W₀ calculations, as employed in the previous work, may also overestimate the fundamental gap of SnO₂, which has also been the case found in other sp semiconductors^[25].

One may argue that the hybrid functional and/or GW approximation may have different impacts on the fundamental gaps of SnO₂ and TiO₂, because of the different component of their CBM states, as shown in Fig. 1(b). To clarify this point, we consider a series of metal oxides, including ZnO, CdO, Ga₂O₃, In₂O₃ and GeO₂, for which the CBM and VBM components are all similar to that of SnO₂. Table 3 gives the calculated fundamental gaps by the HSE06 + G₀W₀ method, together with the available experimental data. We find that although the HSE06 + G₀W₀ calculations can result in reasonable fundamental gaps for IIB oxides ZnO and CdO, which have large d orbital component at the VBM, it significantly overestimates the band gaps of IIIA oxides Ga₂O₃, In₂O₃ and IVA oxides GeO₂, which have similar band edge wavefunction characters as SnO₂. We find the differences between the calculated and experimentally measured fundamental gaps increase monotonically as the occupied d orbital binding energy increases, e.g., from ZnO to Ga₂O₃ and to GeO₂ or from CdO to In₂O₃, as summarized in Fig. 4. From this trend, it is reasonable to expect that the HSE06 + G₀W₀ calculations would also overestimate the fundamental gap of SnO₂. By a simple linear extrapolation, we can roughly estimate the overestimation is ~0.8 eV, which gives an error-corrected fundamental gap of SnO₂ as ~3.0 eV, in accordance with the HSE06 calculations.

Recently, a hard X-ray photoelectron spectroscopy (HAXPES) measurement showed that the sharp increase of the VBM occurs at ~3.6 eV below the Fermi energy for undoped SnO₂^[39]. To confirm that this is consistent with our prediction that the fundamental band is only 2.9 eV, we have plotted the total DOS of SnO₂ with high resolution, as shown in Fig. 5. We see that the sharp increase of the band edge DOS occurs at 0.7 eV below the VBM, corresponding to the A points in the band structure (see Fig. 3), that is, it gives also the optical band gap, not the fundamental band gap. This is consistent with the fact that due to the specific band structure of SnO₂ (Fig. 3), the HAXPES signal near the VBM is low, so can be easily missed by the HAXPES measurement by weak signal.

In conclusion, we have performed a systematic first-principles calculations and symmetry analysis to revisit the long-standing band gap problem for SnO_2. Different-level computational methods and functionals, such as PBE, PBE + G₀W₀, HSE06 and HSE06 + G₀W₀ are employed. We find that in all these calculations the band gap of SnO₂ is similar or slightly smaller than that of TiO₂. The HSE06 calculations yield a fundamental gap of 3.0 eV and a much larger optical gap of 3.6 eV for SnO₂, which is consistent with recent experimental measurement of the band alignment between SnO₂ and TiO₂, but smaller than the previously reported fundamental band gap of 3.65 eV for SnO₂. The discrepancy between the fundamental and optical gaps is attributed to the inversion symmetry of rutile SnO₂ and the resultant dipole-forbidden transitions between the VBM and CBM, similar to what has been found in In₂O₃. On the other hand, we find that the HSE06 + G₀W₀ method, as employed in previous calculations, would overestimate the fundamental gaps of both TiO₂ and SnO₂ as well as a series of other oxides, including Ga₂O₃, In₂O₃ and GeO₂. More experimental tests of our predictions are called for. Because SnO₂ is widely used in optoelectronic devices, our new understanding of the band structure and optical properties of SnO₂ should have large impact on the future design of optoelectronic materials.

We acknowledge the computational support from the Beijing Computational Science Research Center (CSRC). This work was supported by the Science Challenge Project (No. TZ2016003), the National Key Research and Development Program of China (No. 2016YFB0700700), and the Nature Science Foundation of China (No. 11634003, 51672023, U1930402 ).