1.
Introduction
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.
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Figure1.
(Color online) Band gaps and band alignments between rutile SnO2 and TiO2. In (a) the band gaps of SnO2 is taken from Ref. [15] and the CB band alignment is taken from Ref. [19], while in (b) both of them are calculated in this work. The band gap of TiO2 is always set to its experimental value (taken from Ref. [20]).
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.
2.
Theoretical calculation
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 ? and c = 3.186 ? for SnO2[29], a = 4.594 ? and c = 2.958 ? 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/?. 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.
3.
Results and discussion
Fig. 2(a) illustrates the tetragonal rutile structure for both SnO2 and TiO2 with the space group
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Figure2.
(Color online) (a) Primitive unit cells and (b) calculated total and partial density of states for rutile SnO2 and TiO2. The green, blue and purple balls represent Sn, Ti and O respectively. (c) Side view of the atomic configuration for SnO2/TiO2 heterojunction (top panel).
$Delta {E_ m{V}}( m{SnO}_ m{2}/ m{TiO}_ m{2}) = {({E_ m{V}} - {E_ m{CL}})_{ m{TiO}_ m{2}}}} - {({E_ m{V}} - {E_ m{CL}})_{ m{SnO}_ m{2}}} + Delta {E_ m{CL}}.$  | (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
m CL}}$
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].
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Figure3.
(Color online) (a) Band structure of SnO2 along two high symmetry lines M–Γ–Z. (b) Optical absorption coefficient of SnO2. The valence band maximum is set at zero. In (a), the red and green arrows represent the possible optical transitions at Γ-point and away from Γ-point, respectively.
| Transition path | EVB (eV) | ECB (eV) | ΔE (eV) | px, py, pz (a.u.) |
| B–C | 0.00 | 2.96 | 2.96 | (0.0 0.0 0.0) |
| D–E | –0.03 | 3.10 | 3.13 | (0.3 0.3 0.0) |
| A–C | –0.74 | 2.96 | 3.69 | (76.6 69.6 0.0) |
Table1.
The calculated optical transition matrix elements (in arbitrary units) between the valence band states and conduction band states for rutile SnO2. Three transition paths are considered: two are at Γ-point and one is at the vicinity of Γ-point, as shown in Fig. 3. The energy for the valance (EVB) and conduction band states (ECB) are calculated with respect to the VBM and their energy difference is ΔE = ECB – EVB.
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| Transition path | EVB (eV) | ECB (eV) | ΔE (eV) | px, py, pz (a.u.) |
| B–C | 0.00 | 2.96 | 2.96 | (0.0 0.0 0.0) |
| D–E | –0.03 | 3.10 | 3.13 | (0.3 0.3 0.0) |
| A–C | –0.74 | 2.96 | 3.69 | (76.6 69.6 0.0) |
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].
| Method | PBE | HSE06 | PBE + G0W0 | HSE06 + G0W0 |
| TiO2 | 1.75 | 3.15 | 3.33 | 3.68 |
| SnO2 | 1.23 | 2.96 | 2.87 | 3.76 |
Table2.
Calculated fundamental band gaps (eV) of rutile TiO2 and SnO2 using different calculation methods and functionals.
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| Method | PBE | HSE06 | PBE + G0W0 | HSE06 + G0W0 |
| TiO2 | 1.75 | 3.15 | 3.33 | 3.68 |
| SnO2 | 1.23 | 2.96 | 2.87 | 3.76 |
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
m g}} = E_{
m g}^{
m{cal.}} - E_{
m g}^{{
m{exp}}{
m{.}}}$
| Metal oxide | ZnO | CdO | Ga2O3 | In2O3 | GeO2 | SnO2 |
| $E_{ m g}^{ m{cal.}}$ (eV) | 3.33 | 1.18 | 5.30[34] | 3.35[34] | 5.14 | 3.76 |
| $ E_{ m g}^{ m{exp} .}$ (eV) | 3.44[35] | 1.09[36] | 4.90[37] | 2.90[11] | 4.50[38] | – |
| $ Delta {E_{ m g}}$ (eV) | –0.11 | 0.09 | 0.40 | 0.45 | 0.64 | – |
Table3.
The calculated fundamental band gaps by HSE06 + G0W0 method (
m g}^{
m{cal.}}$
m g}^{{
m{exp}}.}$
m g}}$
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| Metal oxide | ZnO | CdO | Ga2O3 | In2O3 | GeO2 | SnO2 |
| $E_{ m g}^{ m{cal.}}$ (eV) | 3.33 | 1.18 | 5.30[34] | 3.35[34] | 5.14 | 3.76 |
| $ E_{ m g}^{ m{exp} .}$ (eV) | 3.44[35] | 1.09[36] | 4.90[37] | 2.90[11] | 4.50[38] | – |
| $ Delta {E_{ m g}}$ (eV) | –0.11 | 0.09 | 0.40 | 0.45 | 0.64 | – |
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Figure4.
(Color online) Differences between the HSE06 + G0W0 calculated and experimentally measured band gaps
m g}}= E_{
m g}^{
m{cal}.} - E_{
m g}^{
m{exp}.}$
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.
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Figure5.
(Color online) Calculated DOS of SnO2 for the valence bands below VBM level. The right-hand side shows the DOS with enlarged energy scale below the VBM at E = 0 eV.
4.
Conclusion
In conclusion, we have performed a systematic first-principles calculations and symmetry analysis to revisit the long-standing band gap problem for SnO2. 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.
Acknowledgment
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 ).
