Tin oxides are the most promising electron transport layers in perovskite solar cells. An ambipolar carrier transport property has been recently demonstrated which enables a simple interconnection structure for all-perovskite tandem solar cells. However, the underlying mechanism for its ambipolar behavior is unclear, which cannot be explained by the intrinsic defects in SnO_{2−x}. Here, by using density functional theory calculations, we unveil the origin of the ambipolar carrier transport of non-stoichiometry SnO_{2−x} with a structure of SnO embedded in the SnO_{2} matrix. The hybridization of O 2p and Sn 5s orbitals of SnO introduces mid-gap states in the bandgap of SnO_{2}, enabling hole transport property for SnO_{2−x} when x is > 0.2. Increasing the percentage of SnO in SnO_{2−x} significantly enhances the hole transport capability of SnO_{2−x} due to the enlarged Sn–O–Sn angles that increase orbital overlapping between O and Sn atoms, thus providing strategies for the further tuning of the carrier transport properties of SnO_{2−x} by compositional and structural designs.

Tin dioxide (SnO_{2}) has attracted great research interest for its application as excellent electron transport layers in record-high-efficiency perovskite solar cells due to its high bulk electron mobility, large bandgap, good stability with perovskites, and versatile fabrication processes.^{1–3} It has a deeper conduction band level compared to other electron transport materials like titanium dioxide and, thus, can easily form good band alignment with perovskites to facilitate electron extraction from perovskites.^{4} SnO_{2} can be fabricated by a variety of methods including solution deposition using pre-synthesized nanoparticles,^{3,5} atomic layer deposition (ALD),^{6–8} chemical bath deposition,^{1,9} electrochemical deposition,^{10} sputtering,^{11–13} and other physical vapor depositions.^{14,15} Like all types of oxides, tin oxides are not stoichiometric due to the presence of large density of O vacancies (V_{O}). The incomplete oxidation of Sn^{2+} to Sn^{4+} is nearly inevitable due to low formation energy of V_{O} and Sn interstitials (Sn_{i}) during the synthesis of SnO_{2}, thus commonly leading to a dominant product of SnO_{2−x} (0 < x < 1).^{16}

A recent breakthrough discovery made by Yu *et al.*^{17} showed that using a simplified fullerene (C_{60})/SnO_{2−x} structure as an interconnection layer in all-perovskite tandem solar cells enabled high efficiency tandem cells, which was mainly fulfilled by the ambipolar carrier transport capability of the nonstoichiometric ALD-grown SnO_{2−x} layer. The ambipolar carrier transport of SnO_{2−x} was realized when a certain amount of Sn^{2+} existed, as schematically shown in Fig. 1. However, it has not been fully understood how the presence of Sn^{2+} in SnO_{2−x} affects the electronic structure of SnO_{2−x} and leads to the ambipolar carrier transport. To date, Sn_{i} and V_{O} in SnO_{2} have been widely considered as the origin of *n*-type conductivity for SnO_{2−x} as revealed by early density functional theory (DFT) studies.^{18} Later, the DFT work focusing on the nonstoichiometric defects in SnO_{2} showed that the only defect which could induce *p*-type doping in SnO_{2} was Sn^{4+} vacancies which, however, had even higher formation energy than V_{O} under oxygen-rich conditions,^{16} thus contradicting experiment results. The discrepancies likely arose from the inappropriate models used that only took point defects in SnO_{2} into consideration for the non-stoichiometry of SnO_{2−x}, whereas in experiment the non-stoichiometry could also come from the alloying of SnO_{2} and SnO in a composition [i.e., (SnO_{2})_{1−x}(SnO)_{x}] when O is significantly deficient.^{13,17} Therefore, there was no theoretical investigation on the alloy structure of SnO_{2} and SnO to uncover the composition-tunable electronic properties and the ambipolar carrier transport characteristics of SnO_{2−x}.

In this work, we calculate the influences of incorporating Sn^{2+} into SnO_{2} on the electronic properties of SnO_{2−x} by constructing a SnO_{2}–SnO alloy structure to mimic the ALD-grown SnO_{2−x} within the framework of DFT calculation. Different from previous simulation approaches in which the SnO_{2−x} was constructed by creating local V_{O} or Sn_{i}, we embedded a complete layer-structured SnO into the matrix of SnO_{2} to uncover the ambipolar carrier transport mechanism of SnO_{2−x}. The hole transport property of SnO_{2−x} is further investigated upon the changing of SnO concentrations, providing future experimental design guidelines to improve the carrier transport properties of Sn-based oxides.

The unit cell of SnO_{2−x} with the alloy structure of SnO_{2} and SnO was constructed by first building a 2 × 3 × 4 (x × y × z) SnO_{2} matrix with the primitive cell of SnO_{2} as shown in Fig. 2(a). Then parts of Sn and O atoms in the matrix were removed and replaced with a layer-structured SnO primitive unit [Fig. 2(b)] with the (001) facet of SnO aligned to the (100) facet of the matrix, as schematically shown in Fig. 2(c). The mixing of SnO_{2} and SnO generates defects in the interconnected areas, which was caused by removing of one O atom and replacement of two Sn^{4+} with Sn^{2+} in the interconnected areas on average. These defects could be equivalently regarded as negatively charged traps of the alloy. By changing the numbers of Sn and O atoms of SnO in the matrix, we obtained SnO_{1.95}, SnO_{1.83}, SnO_{1.75}, SnO_{1.7}, SnO_{1.65}, SnO_{1.5}, and SnO_{1.4}, which correspond to nominal perfect alloy compositions of (SnO_{2})_{0.96}(SnO)_{0.04}, (SnO_{2})_{0.84}(SnO)_{0.16}, (SnO_{2})_{0.81}(SnO)_{0.19}, (SnO_{2})_{0.79}(SnO)_{0.21}, (SnO_{2})_{0.75}(SnO)_{0.25}, (SnO_{2})_{0.625}(SnO)_{0.375}, and (SnO_{2})_{0.5}(SnO)_{0.5}, respectively. We have also considered the variation of V_{O} in different locations including in SnO or at the SnO_{2}/SnO boundary, which as a result did not significantly affect the calculation results within the framework of the alloy structure of SnO_{2} and SnO. In addition, we considered the co-presence of V_{O} and Sn_{i} in the SnO_{2} matrix of SnO_{1.95} and SnO_{1.83} [Fig. 2(c)] to further verify whether these point defects could make SnO_{2−x} *n*-type doped.^{18}

DFT calculations were performed by using the CASTEP package. Perdew–Burke–Ernzerhof (PBE) correlation exchange functional at the generalized gradient approximation (GGA) level and ultrasoft pseudopotentials with a plane wave basis set energy cutoff of 340 eV were adopted. The geometric optimization of each periodic cell was carried out until the change in energy and force in each atom were less than 1 × 10^{−5 }eV/atom and 0.03 eV/Å, respectively. For the density of states (DOS) calculation, pseudo atomic calculations were performed for O 2*s*^{2} 2*p*^{4} and Sn 5*s*^{2} 5*p*^{2} with a *Γ*-centered (5 × 3 × 3) Monkhorst–Pack *k*-point grid sampling for the three-dimensional Brillouin zone. To obtain a more accurate bandgap values that are comparable to experimental results, a hybrid functional B3LYP with norm-conserving pseudopotentials and an energy cutoff of 750 eV were set to calculate the density of states (DOS) with high accuracies for selected models.

We first calculated the formation enthalpies (*H*_{F}) of SnO_{2−x} by using^{19,20}

where *E*(SnO_{2−x}) was the total energy of SnO_{2−x} at the ground state after geometric optimization, *n* and *m* were the total numbers of Sn and O atoms in the unit cell of SnO_{2−x}, respectively, and *μ*_{Sn} and *μ*_{O} were the chemical potentials of Sn and O, respectively. Here, *μ*_{Sn} was fixed at the atomic energy of metal Sn which was −95.451 eV, and the atomic energy of O in an O_{2} molecule which was −434.348 eV was chosen as the zero reference for *μ*_{O}. *μ*_{O} = 0, thus, refers to the most oxygen-rich situation. To estimate the range of *μ*_{O} in H_{2}O during the ALD process, we calculated the total energies of H_{2}O and H_{2} molecules, which were −468.716 and −31.562 eV, respectively. *μ*_{O} in H_{2}O was then estimated by using *μ*_{O} = *μ*_{H2O} − 2*μ*_{H} = −2.806 ± Δ*E* eV (Δ*E* indicates the variation of *μ*_{O} in different chemical environments) with the reference zero *μ*_{O} set to the atomic energy of O in O_{2}. Figure 2(d) plots the dependence of *H*_{F} on the change of *μ*_{O} in the vicinity of −2.8 eV for SnO_{2−x}, simulating the synthetic environment of SnO_{2−x} by ALD, where H_{2}O was used as the precursor for O. In general, O-deficient atmospheres (*μ*_{O} < −2.8 eV) favor the growth of SnO_{2−x} with low O/Sn ratios like SnO_{1.4} as they own lower *H*_{F} in this range. In contrast, O-rich atmospheres (*μ*_{O} > −2.8 eV) lead to the growth of more completely oxidized SnO_{2−x} like SnO_{1.95}. It is interesting that the *H _{F}* of SnO

_{2−x}with medial O/Sn ratios, such as SnO

_{1.83}and SnO

_{1.7}, are always larger than the intersectional minimum

*H*of SnO

_{F}_{1.4}and SnO

_{1.95}[indicated by the black dashed line in Fig. 2(d)], indicating that the formations of SnO

_{1.83}and SnO

_{1.7}are not thermodynamically favored during the ALD growing process. This result implies that the growth of SnO

_{2−x}with medium O/Sn ratios of ∼1.7 by ALD is likely determined by the dynamic process of the chemical reaction, rather than controlled by thermodynamics. Experiment results have shown that once the as-synthesized SnO

_{1.76}was oxidized in dry O

_{2}, it turned from the ambipolar to

*n*-type unipolar charge transport layer accompanied by the increase in the O/Sn ratio, verifying the thermodynamically metastable property of ALD grown SnO

_{2−x}with medium O/Sn ratios.

^{17}

Then we move to investigating the electronic properties of SnO_{2−x}. Figure 3 shows the DOS spectra of SnO_{2−x} with 2−x varying from 2 to 1.5. The DOS spectra of SnO_{2}, SnO_{1.83}, and SnO_{1.50} were first calculated by using both the GGA-PBE and B3LYP functionals to check the accuracy of the norm-conserving pseudopotentials in calculating the electronic states of SnO_{2−x}. It is seen that though the GGA-PBE method underestimated the bandgap of SnO_{2} by ∼2.26 eV, whereas B3LYP yields a bandgap of ∼4.15 eV, which is closer to the experimentally measured value of ∼3.6 eV,^{21–23} the relatively energetic distributions of the DOS with respect to the band edges that obtained from both two methods were basically identical.

For SnO_{1.95}, when there is no V_{O}–Sn_{i} pair in the SnO_{2} matrix, trap states are introduced near the valence band edge. After incorporating the V_{O}–Sn_{i} pair to SnO_{2}, additional donor states occur at the conduction band edge of the system, validating that the *n*-type doping of SnO_{2−x} is realized by the co-formation of V_{O} and Sn_{i} in SnO_{2}. Similar scenario of V_{O}–Sn_{i} induced *n*-type doping appears in SnO_{1.83}. With the decrease in the O/Sn ratio from 1.95 to 1.50, the trap states near the valence band edge gradually increase and broaden toward both the valence band and the middle of the bandgap, forming band tail states next to the valence band edge of SnO_{2−x}. This agrees with experiment results that substantial band tail states were detected in ALD-grown SnO_{1.76} from near-infrared absorption measurement.^{17} These high densities of band tail states would serve as charge traps for charge carriers, leading to the low conductivity of ∼10^{−4} S cm^{−1} for ALD-grown SnO_{2−x}.^{17} In the meantime, an additional mid-gap state appears in the bandgap of SnO_{2−x}, when the O/Sn ratio is lower than 1.83, which gradually merges with the band tail states and the conduction band edge with the further decrease in the O/Sn ratio. This result also agrees with the experimentally observed presence of mid-gap states in ALD-grown SnO_{1.76}.^{17} The DOS spectrum of SnO_{1.83} calculated with B3LYP shows that the mid-gap states locate within the energy range of 0.68–1.56 eV below the conduction band edge, consistent with the measured energy range of 0.8–1.4 eV for the ALD-grown SnO_{1.76} by Yu *et al.*^{17} These results demonstrate that current calculations based on the alloy structure of SnO_{2} and SnO could accurately simulate the experimentally ALD-grown SnO_{2−x}, which has an ambipolar carrier transport property. When the ALD-grown SnO_{2−x} was adopted as an interconnection layer for perovskite tandem cells, the mid-gap states in SnO_{2−x} are the key to realize hole transport through them. Therefore, it is important to understand the origin of the mid-gap states in SnO_{2−x} and their correlation with the carrier transport properties upon changing of the composition.

To figure out the origins of the band tail states and mid-gap states in SnO_{2−x}, we calculated the distributions of the isosurface for the electron wavefunctions of these states in SnO_{1.95}, SnO_{1.83}, and SnO_{1.50}, as shown in Fig. 4. For SnO_{1.95}, there are no mid-gap states. The isosurface for the electron wavefunctions of the tail states is localized at the back-bond of Sn^{2+}, where the equivalently existing negatively charged defects accumulate at the SnO_{2}/SnO boundary. These interfacial defects contribute to the band tail states above the valence band edge of SnO_{1.95} as well as in SnO_{1.83} and SnO_{1.50} [Figs. 4(b) and 4(c)]. Figure 4(a) also shows the isosurface of the electron wavefunctions of the donor states in SnO_{1.95} when a V_{O}–Sn_{i} pair exists in the SnO_{2} matrix. As a result, the electron wavefunctions of these donor states mainly localize around the V_{O}–Sn_{i} pair in SnO_{2}. For the mid-gap states of SnO_{1.83} and SnO_{1.50}, it is found that the isosurface of the electron wavefunctions is mainly distributed around the Sn and O atoms of SnO in the alloy structure. This indicates that the mid-gap states are introduced by the SnO component of SnO_{2−x}, specifically caused by the hybridization of O 2p and spherical Sn 5s orbitals. This is similar to the orbital configuration for the valence bands of SnO, as both are caused by the O 2p and Sn 5s orbitals, which give rise to a much higher valence band maximum (VBM) and smaller bandgap of ∼1.2 eV for SnO compared to SnO_{2}.^{24,25} This hybridization leads to a more dispersed VBM with smaller hole effective mass for SnO, thus enabling efficient hole transport in SnO. In contrast to SnO_{2}, which is commonly an *n*-type oxide, SnO has been extensively reported as a *p*-type oxide owing to its efficient hole transport and unintentional doping by Sn vacancies.^{26–29} Therefore, the SnO-induced mid-gap states in the bandgap of SnO_{2} could facilitate hole transport through these states and make SnO_{2−x} as an ambipolar oxide. Experiments of Yu *et al.* further revealed that once the O/Sn ratio of the ALD-grown SnO_{2−x} increased from 1.76 to 1.91, the SnO_{2−x} changed from ambipolar to *n*-type doping only (Fig. 1).^{17} This again consists with our simulation results that in highly oxide SnO_{2−x} (e.g., SnO_{1.95}), there is no more mid-gap states for the realization of hole transportations.

Eventually, we calculated the band structures of SnO_{2−x} and derived the effective masses of holes (*m _{h}*) with the

*E*–

*k*relationship at the vertex of the defect-induced or SnO-induced mid-gap bands to evaluate the hole transport capabilities of SnO

_{2−x}with different compositions. Figs. 5(a)–5(c) representatively show the band structures of SnO

_{1.95}, SnO

_{1.83}, and SnO

_{1.50}. The energy level of 0 eV corresponds to the Fermi level, which is set to the highest occupied electronic state of the system. With the decrease in the O/Sn ratio, i.e., the increase in the SnO concentration, the mid-gap bands become more dispersed. The

*m*of SnO

_{h}_{1.95}, SnO

_{1.83}, and SnO

_{1.50}are 5.27, 1.39, and 0.58

*m*

_{0}, respectively, implying a tendency of enhancement of hole mobilities with the decrease in the O/Sn ratio in SnO

_{2−x}. To further understand the underlying correlation between

*m*and the O/Sn ratio of SnO

_{h}_{2−x}, we calculated the average Sn–O–Sn angles of SnO in geometrically optimized SnO

_{2−x}as the Sn–O–Sn angle determined the orbital overlapping between the dumb-belled O 2p and spherical Sn 5s orbitals of SnO, which eventually affected the

*m*of SnO.

_{h}^{30}The relationship between

*m*and the average Sn–O–Sn angles of SnO in SnO

_{h}_{2−x}with different O/Sn ratios is shown in Fig. 5(d). A clear trend of decrease in

*m*with the decrease in O/Sn ratio is observed for SnO

_{h}_{2−x}, which is attributed to the increase in the Sn–O–Sn angles in SnO that enhances the orbital coupling between O and Sn atoms. The fitted trend of the dependence of

*m*on the average Sn–O–Sn angle of SnO is similar to the observation of Ha

_{h}*et al.*on different

*p*-type Sn-based oxides.

^{26}It is noteworthy that the average Sn–O–Sn angles of SnO in SnO

_{1.75}to SnO

_{1.4}are even larger than that of pristine SnO, which is 116°, thus they own smaller

*m*and potentially higher hole mobilities than pristine SnO. We speculate that the enlargement of Sn–O–Sn angles is mainly caused by the strain imposed on SnO due to the lattice mismatches between SnO and the SnO

_{h}_{2}matrix. To evaluate the strain at the SnO

_{2}/SnO boundaries, we calculated the lattice mismatches between SnO and SnO

_{2}in the Y–Z plane of the alloy structure using (

*y*

_{SnO2}−

*y*

_{SnO})/

*y*

_{SnO2}× 100% and (

*z*

_{SnO2}−

*z*

_{SnO})/

*z*

_{SnO2}× 100%, as schematically illustrated in Fig. 2(c). Figure 2(e) shows the lattice mismatches between SnO and SnO

_{2}in the Y and Z directions of the alloy structure, which are all positive and vary between 1% and 10%. This indicates that compressive and tensile strains are imposed to SnO

_{2}and SnO, respectively. It is the tensile strain that enlarges the Sn–O–Sn angles in SnO after alloying with SnO

_{2}. This result also provides a strategy for acquiring desired carrier transport properties for Sn-based oxides by compositional and structural designs of SnO

_{2−x}with the alloy structure of SnO

_{2}and SnO, to fulfill their functions as multiple-purpose interlayers in high performance perovskite solar cells.

In summary, we investigated the structural and electronic properties of SnO_{2−x} based on the alloy structure of SnO_{2} and SnO that mimics the ALD-grown SnO_{2−x} in experiment for its application as interconnected layers in all-perovskite tandem cells. The co-formation of V_{O} and Sn_{i} in SnO_{2} makes SnO_{2−x} *n*-type doped, while the incorporation of SnO into the SnO_{2} matrix introduces mid-gap states in the bandgap of SnO_{2} and enables the ambipolar carrier transport capability of SnO_{2−x}. The decrease in the O/Sn ratio of SnO_{2−x} (an increase in the SnO concentration) facilitates the hole transport through the SnO-induced mid-gap states as the effective mass of holes reduces. This highlights an important pathway for the further tuning of the carrier transport properties of SnO_{2−x} by changing the SnO concentration in the alloy structure.

We thank the financial support from the Solar Energy Technologies Office (SETO) within the U.S. Department of Energy under Award No. DE-EE0008749 and the National Science Foundation under Award No. ECCS-2050764.

The authors declare no competing financial interest.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.