Organic-inorganic hybrid lead halide perovskites, such as CH₃NH₃PbI₃^[1⇓-3] have attracted enormous research interests for applications in efficient photovoltaic^[4], light- emitting^[5] and photodetection devices^[6]. As their lead-free counterparts, Sn-based halide perovskites, such as CH₃NH₃SnI₃^[7⇓-9] and CsSnI₃^[10⇓-12], have been proposed for such applications because of their nontoxicity. However, because 2+ state is not the most stable valence state of Sn, the Sn-based perovskites are prone to further oxidation, rendering them even less stable than the Pb-based perovskites, which are already well known to have the stability issue^[13]. It is highly desirable to develop air-stable alternatives. For this purpose, Cs₂SnI₆ is a promising material, in which Sn is already in 4+ valence state and resistant to further oxidation^[14⇓-16]. Meanwhile, it has a suitable band gap and strong optical absorption for photovoltaic applications^[17⇓-19].

The halide perovskite materials are usually reported to exhibit defect tolerance in the bulk^[20-21]. Therefore, the surfaces, interfaces and grain-boundaries are usually the main concerns for optimizing the device performance. The surface properties of Cs₂SnI₆, which are expected to play an important role in the devices, remain poorly understood. Recently, several groups have devoted to the study on the surfaces of Cs₂SnI₆. Kapil et al.^[22] suggested the existence of the surface state in Cs₂SnI₆. Shin et al.^[23] investigated the role of the surface states in the presence of a redox mediator. Xu et al.^[24] reported a general approach to synthesize layered nanoplatelets of Cs₂SnI₆. Zhu et al.^[25] revealed that Cs₂SnI₆ crystals have a preferential growth of (111) surface. Several experiments also proved that Cs₂SnI₆ tends to grow along the <111> direction^[26⇓-28]. However, it is not clear that this preference is thermodynamically driven or just a result of growth condition specific to individual experiments.

In this work, as motivated by the experimental works, we study the surface properties of Cs₂SnI₆ using first- principles calculations. As an initiative work, we attempt to understand the preference to the (111) surface in different crystal growth experiments. Different surface models with different surface orientations and terminations were set up to compare their thermodynamic stability. As some terminations are non-stoichiometric, to evaluate their relative stability it is necessary to calculate the chemical potentials to consider the crystal growth conditions. By analyzing the surface stability, it is expected to provide useful information for future experimental synthesis and device fabrication.

Our first-principles calculations were based on density functional theory and performed using the Vienna Ab initio Simulation Package (VASP)^[29]. Projector augmented wave (PAW) potentials were used to describe the interaction between ion cores and valence electrons^[30]. The strongly constrained and appropriately normed (SCAN) functional in combination with the rVV10 van der Waals (vdW) functional was used for the exchange-correlation functional^[31]. The cutoff energy of planewave basis set was taken to be 340 eV and the Γ-centered 3×3×3 k-point mesh was used for optimizing the 9-atom primitive cell of Cs₂SnI₆. A cutoff energy of 272 eV and a Γ-centered 3×3×1 k-point mesh were used for the surface calculations.

Experimentally, it is reported that bulk Cs₂SnI₆ exhibits a cubic structure with Fm-3m space group symmetry, as shown in Fig. 1(a), and the lattice parameter a₀ is 1.165 nm^[20]. The calculated a₀ using the SCAN+rVV10 functional is 1.156 nm, 0.8% smaller than the experimental value. The SCAN functional without considering the vdW effect yields a₀=1.178 nm, 1.1% larger than the experimental value, suggesting that the vdW effect is significant for Cs₂SnI₆. For comparison, the commonly used PBE and HSE functional is also considered, which yield a₀=1.203 and 1.197 nm, respectively. The reason for this large vdW effect is that the material is rather soft. The calculated bulk modulus using the SCAN+rVV10 method is only 13.1 GPa. Using the other functionals mentioned above would yield even smaller bulk modulus.

The band structure and projected density of states (pDOS) of Cs₂SnI₆ were calculated, as shown in Fig. 1(b). The direct gap at the Γ point is 0.19 eV, significantly smaller than the experimental band gap of ~1.3 eV, suggesting that the meta-generalized gradient approximation is not sufficient for studying band-gap-sensitive properties of this material, for which the HSE functional including spin-orbit coupling will be necessary^[20]. According to the pDOS plot, the top valence bands are mainly contributed by I5p orbitals, while the bottom conduction bands are contributed by both I5p orbitals and Sn5s orbitals. The Sn5s orbitals form a separate band, above which is another band gap and the Sn5p bands.

The surface properties are studied in the next step. We adopt the symmetric slab models for the surfaces, which possess a mirror symmetry through the middle of the slabs. Such models also avoid spurious interaction between periodic slabs due to dipole-dipole interactions. For all calculations, sufficient vacuum region (more than 1-nm-thick) was used to ensure negligible interaction between the slabs. Seven different terminations of Cs₂SnI₆ surface models were considered, as shown in Fig. 2. The non-stoichiometric (001) surfaces were modeled with CsI₂-terminated (or A-termination) and SnI₄-terminated (or B-termination) slabs, whose unit-cell formulae were Cs₁₂Sn₅I₃₂ and Cs₈Sn₅I₂₈, respectively. Similarly, the non-stoichiometric (011) surfaces were modeled with I₄-terminated (A-termination) and Cs₂SnI₂-terminated (B-termination) slabs, whose unit-cell formulae were Cs₁₀Sn₅I₃₄ and Cs₁₀Sn₅I₂₆, respectively. Along the [111] direction, the atomic stacking sequence is -Sn-CsI₃- CsI₃- Sn-. Correspondingly, the non-stoichiometric Sn-terminated (A-termination), non- stoichiometric CsI₃-terminated (B-termination) and stoichiometric CsI₃-terminated surfaces were modeled, whose unit-cell formulae were Cs₈Sn₅I₂₄, Cs₁₂Sn₅I₂₆ and Cs₁₀Sn₅I₃₀, respectively.

The cleavage energy are firstly evaluated, which is the energy required to split a crystal into two complementary non-stoichiometric terminations. It is noted that CsI₂- and SnI₄-terminations for Cs₂SnI₆ (001) surfaces are mutually complementary, and so are I₄- and Cs₂SnI₂-terminated slabs for (011) surfaces, as well as Sn- and CsI₃-terminated slabs for (111) surfaces. As two complementary surfaces (also referred to as A- and B-termination above) are created simultaneously when a crystal is cleaved, the total cleavage energy of two complementary surfaces can be obtained by

where $E(\mathrm{~A})_{\text {slab }}^{\text {unrel }} \text { and } E(\mathrm{~B})_{\text {slab }}^{\text {unrel }}$ are the energies of unrelaxed A- and B-terminated slabs, respectively.E_bulk is the energy per unit cell, S represents the surface area and n is the total number of bulk unit cells in the two slabs. Next, the relaxation energy of A-terminated surfaces with both sides relaxed is calculated according to

where $E(\mathrm{~A})_{\text {slab }}^{\mathrm{rel}}$ is the energy of A-terminated slab after relaxation.E_rel(B) is calculated similarly. Finally, the total surface energy of the two complementary surfaces can be obtained as the sum of the cleavage and relaxation energies:

The calculated results of total cleavage energy, total relaxation energy and total surface energy of the two complementary non-stoichiometric terminations with different surface orientations are shown in Fig. 3. For comparison, the cleavage, relaxation and surface energies of the stoichiometric CsI₃-terminated (111) surface are also shown. It can be seen that the total surface energies of the two complementary non-stoichiometric terminations are relatively high compared with that of stoichiometric CsI₃-terminated (111) surface whose surface energy is only 0.11 J/m², regardless of the surface orientations. However, the contributions to the cleavage energy from the A- and B-terminations are not equal. Further study is needed to determine whether A- or B-termination could individually have surface energy lower than 0.11 J/m². In order to evaluate the relative stability of each surface termination under various experimentally preparation conditions, the consideration of chemical potential μ_Cs, μ_Sn and μ_I is necessary^[21,32].

Through varying the chemical potentials, different experimental conditions can be simulated. The relative values of the chemical potentials of Cs, Sn and I are introduced with respect to their most stable phases, $\Delta \mu_{\mathrm{Cs}}=\mu_{\mathrm{Cs}}-E_{\mathrm{Cs}}^{\text {bulk }}, \Delta \mu_{\mathrm{Sn}}=\mu_{\mathrm{Sn}}-E_{\mathrm{Sn}}^{\text {bulk }}$, and $\Delta \mu_{\mathrm{I}}=\mu_{\mathrm{I}}-\frac{1}{2} E_{\mathrm{I}_{2}}^{\text {bulk }}$. The chemical potentials of Cs, Sn and I are constrained by the calculated enthalpy of formation of the primary phase Cs₂SnI₆^[33-34]

where $\Delta E_{\mathrm{f}}\left(\mathrm{Cs}_{2} \operatorname{SnI}_{6}\right)$ is calculated to be -9.16 eV. The chemical potentials are further subject to specific bounds that are set by the existence of secondary phases:

The enthalpies of formation above were all obtained by SCAN+rVV10 calculations. By considering these constraints, the final allowed region for μ_Sn and μ_I is demarcated by the points A, B, C, D and E, where equilibrium growth of bulk Cs₂SnI₆ is possible, as shown in Fig. 4.

Using the determined chemical potential region, the surface energy for each individual termination can be obtained using^[35-36]

where E_slab is the total energy of relaxed A-termination, N_Cs, N_Sn and N_I are the numbers of Cs, Sn and I atoms in the slab, respectively. Considering the variation of chemical potential with reference phase as mentioned above, the surface energy can be finally rewritten as

Here, Ø(A) is a constant term, independent of the chemical potentials. Considering that surfaces will spontaneously form and the crystal will be destroyed if surface energy is negative, it is necessary to satisfyE_surf>0.

The stability diagram of the Cs₂SnI₆ (001) surface is shown in Fig. 5(a). The blue and orange regions represent the regions where CsI₂- and SnI₄-terminations are stable, respectively. The upper part of the green region is located in the blue region, indicating that the CsI₂-termination is favored under the I-poor condition. There is still a small part of the green region located in the orange region, e.g., at chemical potential points C and D, indicating that the SnI₄-termination is more stable than the CsI₂-termination under I-rich condition.

Similarly, the stability diagram of the Cs₂SnI₆ (011) surface is shown in Fig. 5(b). The blue region represents that the Cs₂SnI₂-termination is thermodynamically more stable, while the orange part refers to the region where the I₄-termination is more stable. It can be seen that the green region is also located in both blue and orange regions, indicating that different terminations are favored when varying the experimental environments. Under I-rich condition (e.g., at chemical potential point A) the Cs₂SnI₂-termination is favored, while under I-poor condition (e.g., the chemical point D) the I₄-termination is favored.

In Fig. 5(c), the stability diagram of Cs₂SnI₆ (111) surface is shown. Different from the stability diagrams of the (001) and (011) surfaces discussed above, the whole green region is located in the orange region for the (111) surface, indicating that the stoichiometric CsI₃-terminated (111) surface is the most energetically favored among the three terminations regardless of the chemical potentials.

Finally, the surface energies of the seven terminations of Cs₂SnI₆ low-index surfaces are compared in Fig. 5(d) as a function of the chemical potentials. It can be seen that the stoichiometric CsI₃-terminated (111) surface consistently has the lowest surface energy, indicating that it is the most thermodynamically favored surface among the seven terminations, in agreement with the recent experimental reports^[25-26].

Based on density-functional theory calculations with the SCAN+rVV10 functional, seven models for the low- index Cs₂SnI₆ surfaces were studied with different surface orientations and terminations to compare their thermodynamic stability. Overall, based on the calculated surface energies, we identified that the stoichiometric CsI₃-termination for (111) surface is consistently the most stable, regardless of the chemical potentials, which is in agreement with the experimental observation that the (111) surface is often the most exposed surface. For the (100) and (110) surfaces, two different terminations were considered for each of them. Their relative stability depends on the chemical potentials. From an experimental point of view, when preparing these two surfaces, different terminations can be obtained by varying the growth condition, e.g., by controlling the I-poor or I-rich conditions.

The authors thank Professor Lian Jie, Dr. Zhu Weiguang and Shen Junhua from Rensselaer Polytechnic Institute for enlightening discussions.