Organic–inorganic hybridized perovskite solar cells have recently attracted tremendous attention^[1–11] because of its high power conversion efficiency (PCE),^[12] low cost,^[4,6,13] and manufacturing feasibility.^[1,14,15] While its PCE reaches 23.7%^[12] only 9 years after the first reported 3.8% in 2009,^[16,17] its thermal stability issue under ultraviolet light illumination,^[18] in moisture environment,^[19] or at high temperatures^[20] is still a challenge to the whole community. Organic–inorganic hybridize perovskite CH₃NH₃PbI₃ (MAPbI₃) tends to dissociate into volatile NH₃ and CH₃I gases^[21] and PbI₂^[20,22] solid under those mentioned conditions. Unfortunately, PbI₂ has a larger bandgap compared with CH₃NH₃PbI₃, which leads to largely reduced PCEs in perovskite solar cells. This instability was believed to be one of the most challenging issues in commercialization of perovskite solar cells.

Efforts from all aspects,such as newly designed cell architectures, substantially optimized morphology, or introduced protecting layers, have been exhaustively made to solve this issue,^[13,24–27] among which large-size inorganic cations^[28,29] appear to play an active role. It was found that mixing MA with larger cations, such as Cs⁺, can improve the photo- and thermal-stabilities of the perovskite solar cells.^[29] As a result, CsPbI_xBr_3−x or CsPbI₃ shows better thermal and moisture stabilities with a reasonably high PCE of ∼10%^[30–32] and a large open-circuit voltage of 1.23 V.^[33] These pioneering attempts suggest another route to build practical solar cells that uses purely inorganic perovskites, like CsPbX₃ (X=I, Br). Another issue, however, arises that the cubic phase of inorganic perovskites, with moderate direct bandgaps of ∼1.7–2.0 eV and thus offering the highest performance of light absorption among all allotropes of CsPbX₃, is more stable at higher temperatures while their bulk form undergoes a transition to an orthorhombic phase (Pnma) at 578 K^[34] for CsPbI₃ and 360 K^[17] for CsPbBr₃ or to a tetragonal phase at 403 K^[17] for CsPbBr₃. The room temperature orthorhombic phase has a much larger indirect bandgap, such as 2.6 eV for CsPbI₃,^[34] compared with that of the cubic phase, which, consequently, reduces the PCE, such as 0.09% for CsPbI₃,^[35] and light absorption coefficient.

However, the stability of the cubic phase could be enhanced by downsizing its size to nanoscale where the surface energy dominates, as illustrated by CsPbI₃ nanocrystals^[36,37] or quantum dots^[33] which maintained the cubic phase for months under ambient conditions. Here, we thus comprehensively investigated the role of surface energy in tuning the cubic to orthorhombic phase transition using first-principles calculations. We considered the differences of surface energies of both the cubic and the orthorhombic phases. It turns out that the cubic phase has a higher total energy for its bulk but the surface energies of its solids are substantially lower than those of the orthorhombic phase. These competitive interactions thus lead to a reduced critical temperature for the cubic to orthorhombic phase transition if the surface ratio is enlarged. We have considered zero-, one-, and two-dimensional nanostructures while the bulk forms of both CsPbI₃ and CsPbBr₃ could be stabilized at room temperature if the crystal size is small enough. Given size and temperature dependent phase diagrams, we found characteristic sizes of 7.8–18.0 nm, 3.0–10.0 nm, 1.3–4.8 nm for 0D (nanocrystal), 1D (nanowire) and 2D (nano-plates) CsPbI₃, respectively, while the corresponding values are 14.0–71.9 nm, 9.7–48.3 nm, and 5.4–24.6 nm for CsPbBr₃. With the confined sizes smaller than these critical values, the cubic phase becomes the most stable phase at room temperature in CsPbX₃ nanostructures. These findings offer a feasible route to stabilize the cubic phase and to utilize these materials in solar cell applications.

Figure 1(a)–1(d) show the atomic models of a CsPbI₃ crystal in the cubic and orthorhombic phases, respectively, while the associated lattice constants are summarized in Table 1. The cubic phase of CsPbI₃ [space group Pm-3m (No. 221)] has higher symmetry in a smaller unit cell with a lattice constant of 6.29 Å. The orthorhombic phase [Pnma (No. 62)] has a larger cell size with a = 10.46 Å, b = 4.79 Å, and c = 17.81 Å. These theoretical values are highly comparable with the experimental values with errors less than 0.5%. Figure 1(e) plots the total energies of these two phases as a function of cell volume, which indicates that the orthorhombic phase, with a smaller equilibrium volume, is 0.29 eV more stable than the cubic phase at the zero temperature limit. The structure of the cubic phase of CsPbBr₃ is similar to that of CsPbI₃ (Fig. 2(a)). The calculated lattice constant is 5.92 Å, only 0.83% larger than the experimental value, while the orthorhombic phase of CsPbBr₃ is substantially different from that of CsPbI₃, as shown in Figs. 2(b)–2(d). The theoretical lattice constants of the orthorhombic phase are a = 8.05 Å, b = 8.42 Å, c = 11.81 Å with an error of less than 2% in comparison with the experimental values. Although both orthorhombic phases of CsPbI₃ and CsPbBr₃ share the same Pnma (D2H-16) space group, their atomic details do differ from each other. For CsPbI₃, Pb²⁺ ions are embedded in distorted iodine octahedra forming (Pb₂I₆)²⁻ chains oriented along the y direction (Fig. 1(b)), while Cs cations are filled in the interstitial regions of these chains. Figure 1(d) clearly shows the cross-section of each (Pb₂I₆)²⁻ chain, in which each Pb atom is octahedrally coordinated by six I atoms and these (Pb₂I₆)²⁻ chains are alternately titled to left and right sides as arranged in a zig-zag pattern.

Fig. 1.

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Fig. 1. Schematic models and thermal stability of bulk CsPbI3 in cubic and orthorhombic phases. (a) Perspective view of the high temperature cubic phase bulk CsPbI3. (b)–(d) Side- and top-views of the low temperature orthorhombic phase bulk CsPbI3. Purple, gray and pink balls represent Cs, Pb and I atoms, respectively. (e) Thermal stability of bulk CsPbI3 in cubic (orange line) and orthorhombic (green) phases.

Fig. 2.

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Fig. 2. Schematic models and thermal stability of bulk CsPbBr in cubic and orthorhombic phases. (a) Perspective view of the high temperature cubic phase bulk CsPbBr3. (b)–(d) Top- and side-views of the low temperature orthorhombic phase bulk CsPbBr3. Purple, gray and brown balls represent Cs, Pb and Br atoms, respectively. (e) Thermal stability of bulk CsPbBr3 in cubic (orange line) and orthorhombic (green) phases.

Table 1.

Table 1.

Table 1. Lattice parameters of bulk CsPbI3 and CsPbBr3 and relative errors comparing with the experimental values. . CsPbI3 CsPbBr3 Expt.[34,44]/Å DFT/Å Error/% Expt.[44,45]/Å DFT/Å Error/% Cubic-a 6.29 6.29 0.00 5.87 5.92 0.85 Orth-a 10.46 10.47 0.10 8.21 8.05 −1.95 Orth-b 4.80 4.79 −0.21 8.26 8.42 1.94 Orth-c 17.78 17.81 0.17 11.76 11.81 0.43

Table 1. Lattice parameters of bulk CsPbI3 and CsPbBr3 and relative errors comparing with the experimental values. .

In terms of CsPbBr₃, each Pb is again octahedrally coordinated by six I atoms, however, no chain is formed and these (PbBr₆)²⁻ octahedra are continuously connected, and are slightly tilting, along all of the lattice directions forming a 3D framework, as shown in Figs. 2(c) and 2(d). Table 1 summarizes the calculated lattice constants of both phases in CsPbI₃ and CsPbBr₃, reasonably consistent with the experimental values. The energetic difference between the cubic and orthorhombic phases of CsPbBr₃, i.e., 0.11 eV (Fig. 2(e)), is smaller than that of CsPbI₃, consistent with the nearly 200 K^[17,34] lowered transition temperature for CsPbBr₃.

We further considered the surface energies of these phases. Energies of the surfaces Cubic(001)-asym (Fig. 3(a)), Orth(010) (Fig. 3(b)), Orth(001)-CsI (Fig. 3(c)), and Orth(100)-3I (Fig. 3(d)) are independent of the chemical potentials of the individual atoms. Among these surfaces, the first two are non-polar and the others are quasi-nonpolar. Their surface energies are 9.6 meV/Ų, 16.0 meV/Ų, 13.1 meV/Ų and 23.3 meV/Ų, respectively (see Table 2). The first three surfaces were believed to be most stable among all surfaces considered, although the Cs rich limit of the Orth(001)-Cs (Fig. 3(e)) surface is a bit more stable than the Orth(001)-CsI surface and the I-rich limit of the Orth(100)-4I (Fig. 3(f)) surface is slightly more stable than the Orth(010) surface. Here, we emphasize that the Cs rich limit is very difficult to reach and the surface energy of the Cs-deficient limit of this surface is over 100 meV/Ų which is rather high compared with other surfaces (see Table 2). Although the I-rich condition is relatively more feasible to reach, its energy is comparable with that of the Orth(010) within 1 meV and the stability rapidly decreases to over 40 meV under I-deficient conditions. In light of this, we exclude both Orth(001)-Cs and Orth(100)-4I in the following discussion.

Apparently, surfaces of the cubic phase are more stable than those of the orthorhombic phase, which, again, indicates that reduced dimensions down to nanoscale could stabilize the cubic phase at room temperature. There are two terminations of surface Cubic(001), namely Cubic(001)-PbI and Cubic(001)-CsI, whose energies are in ranges from −20.4 meV/Ų (I rich, Cs deficient) to 59.4 meV/Ų (I rich, Pb deficient) and −39.8 meV/Ų (I rich, Pb deficient) to 39.9 meV/Ų (I rich, Cs deficient), respectively. Therefore, the exact surface energy of one of the surfaces is even smaller than their averaged value of 9.6 meV/Ų, which leads to underestimated surface stability of the cubic phase. In terms of cubic CsPbBr₃, the CsPbBr₃ Cubic(001)-asym surface was used to consider the surface energy of cubic CsPbBr₃, which gives an surface energy of 10.9 meV/Ų, slightly larger than that of the cubic CsPbI₃. The orthorhombic phase of CsPbBr₃ appears differently from that of cubic CsPbI₃, for which, we used asymmetric (001) and (110) surfaces, see Figs. 3(g) and 3(h), since the energies of these surfaces are independent of the chemical potentials of the individual atoms. Interestingly, both surfaces have the surface energy of 13.9 meV/Ų.

Fig. 3.

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Fig. 3. Side views of surface configurations. (a) Surface (001)-asym of cubic CsPbI3 with the Pb, I atoms and Cs, I atoms at the different terminations. (b) Surface (010) of orthorhombic CsPbI3. (c) Surface (001)-CsI and (d) surface (100)-3I of orthorhombic CsPbI3. (e) Surface (001)-Cs and (f) surface (001)-4I of orthorhombic CsPbI3. (g) Surface (001)-asym and (h) surface (110)-asym of orthorhombic CsPbBr3 with Pb, Br atoms and Cs, Br atoms at the different terminations. (i) Surface (110) of cubic CsPbI3.

Given these surface energies, we are ready to discuss the stability of CsPbX₃ nanocrystals with different surface energies and surface/bulk ratios of the number atoms. In terms of one-dimensional nanostructures—i.e., nanowires—we considered five cross-section shapes for the cubic phase, as shown in Figs. 4(a)–4(d). Figures 4(a) and 4(b) show two considered triangular prisms. The one shown in Fig. 4(a) (0°-triangular prism) contains two (001) surfaces for right-angle sides and one (110) surface for the hypotenuse, which was experimentally found in a recent report,^[46] while the other one is a 90°-rotated one having two (110) surfaces and one (001) surface.

Fig. 4.

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Fig. 4. Top views of 1D NWs. (a)–(e) Schematic models of CsPbI3 NWs: (a) triangular prism containing two Cubic(001) surfaces for right-angle sides and one Cubic(110) surface for the hypotenuse (0°-triangular prism); (b) triangular prism having two Cubic(110) surfaces and one Cubic(001) surface (90°-triangular prism); (c) quadrangular prism with four Cubic(001) side surfaces (0°-quadrangular prism); (d) quadrangular prism with four Cubic(110) side surfaces (0°-quadrangular prism); (e) rectangular prism with Orth(010) and Orth(001)-CsI as the side surfaces. The ratio of longer and shorter edges of the rectangular cross-section is 1.22 using a criterion of maximizing the cross-section area under a given circumference. (f), (g) Schematic models of CsPbBr3 NWs: (f) quadrangular prism with Orth(001)-PbBr as all the side surfaces; (g) quadrangular prism with (110) and (001) as the side surfaces.

The (110) surface of the cubic phase is depicted in Fig. 3(i) and its surface energy of 16.2 meV (Pb rich, I deficient) to 53.8 meV (I rich, Pb deficient) is rather high. A straightforward strategy thus lies in avoiding the presence of this surface. There are also two quadrangular prisms under consideration. The one shown in Fig. 4(c) (0°-quadrangular prism) has four (001) surfaces while the 90°-rotated one shown in Fig. 4(d) is cleaved along (110) surfaces. Given the total energy of the cubic phase and all surface energies, we find that the 0°-quadrangular prism is the most stable shape for cubic CsPbI₃ NWs, which is highly consistent with the shapes found in experiment.^[47] We, therefore, use the 0°-quadrangular prism in comparison of stability with the orthorhombic phase. The shape of NWs in the orthorhombic phase is more complicated than that in the cubic phase. For simplicity, according to the experimental input,^[47] we only considered the rectangular ones shown in Fig. 4(e) with optimally selected surfaces to ensure the lowest surface energies. For 2D nano-plates, we used the most stable surfaces of both phases, respectively—i.e., Cubic(001)-asym and Orth(001)-CsI—for the surface area of nano-plates. Thus, the relative stability depends on the confined thickness of the nano-plates. In terms of 0D nanoparticles, we only considered cubic nanoparticles with the Cubic(001)-asym surface for the cubic phase and the Orth(010), Orth(001)-CsI and Orth(100)-3I surfaces for the orthorhombic phase. Since the case of CsPbBr₃ has experimental inputs, the cross-section of CsPbBr₃ nanowires is considered in a square pattern, as shown in Figs. 4(f) and 4(g).

As mentioned earlier, the exact surface energy of the Cubic(001) surface is over estimated. Therefore, we consider two extremes of the (001) surface of the cubic phase, namely 9.6 meV/Ų and 0 meV/Ų. The exact surface energy should be in between these two extremes while the former extreme corresponds to the minimum-size limit (red curves in Fig. 5) of a nanostructure and the zero surface energy stands for the maximum-size limit (blue curves). Figure 5 shows the phase diagrams for the cubic and orthorhombic phases as functions of temperature and size. The confined size is presented by a parameter R, which stands for the lengths of the right-angle side for triangles, the side for squares and the shorter sides of rectangles, as illustrated in Figs. 4(a) and 4(c). Here, the ratio for the longer and shorter edges of the rectangular cross-section (Fig. 4(e)) is estimated to be 1.22 using a criterion of maximizing the cross-section area under a given circumference. It explicitly shows in Fig. 5 that if R is smaller than 3.0 nm (minimum limit) to 10.0 nm (maximum limit), the cubic phase of CsPbI₃ becomes the most stable phase rather than the orthorhombic phase at room temperature. While the energetic difference between the two phases is smaller for CsPbBr₃, the critical size for phase transitions is thus larger than that of CsPbI₃, namely, 9.7 nm to 48.3 nm. For 2D nano-plates, the critical thicknesses to stabilize the cubic phases are 1.3 nm to 4.8 nm for CsPbI₃ and 5.4 nm to 24.6 nm for CsPbBr₃. While the critical thickness of CsPbI₃ is a bit too thin, that of CsPbBr₃ is highly feasible that could be practically adopted in potential applications. Nanoparticles are another form of nanostructures, which were mostly explored in experiments for both solar cell and light emission applications. It shows a relative high tolerance i.e., 7.8–18 nm for CsPbI₃, which is consistent with the quantum dot size observed with TEM (roughly 8–15 nm) and Rietveld refinement (about 9–17 nm).^[33] An even larger size of 14.0–71.9 nm is found in CsPbBr₃. These values are accessible in experiments, indicating the feasibility of this strategy of surface engineering for stabilizing the cubic phase of CsPbX₃.

Fig. 5.

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Fig. 5. Size and temperature dependent phase diagrams of CsPbI3 and CsPbBr3. Phase diagrams of (a) 0-dimensional, (b) 1-dimensional, and (c) 2-dimensional CsPbI3, and of (d) 0-dimensional, (e) 1-dimensional, and (f) 2-dimensional CsPbBr3. Two extremes of the (001) surface of the cubic phase were considered in the calculations. The former extreme corresponds to the minimum-size limit (surface energies of 9.6 meV/Å2 for CsPbI3 and 10.9 meV/Å2 for CsPbBr3, red curves) of a nanostructure while the maximum-size limit (blue curves) is derived by assuming a zero surface energy.

It is interesting that the critical sizes of structural phase transitions decrease with the increase of dimensions. This is straightforward to be understood that all these phase transitions are originated from energy competitions of bulk and surfaces of these two phases. The higher the dimension, the smaller the critical size. The numbers of exposed surfaces in 0D nanoparticles, 1D nanowires, and 2D nanoplates are 6, 4 and 2, respectively, leading to the decreased surface/bulk ratio with respect to the increasing dimensions. As a result, CsPbX₃ nanoparticles in the cubic phase shall show the excellent stability at room temperature. Indeed, there are several very recent experiments reporting potential applications of CsPbX₃ quantum dots or nanowires in solar cell and light emission.^[48] Another noticeable phenomenon lies in the critical sizes of phase transitions for CsPbBr₃ usually being larger than the corresponding CsPbI₃ counterpart, which could be, again, explained by the smaller energy difference between the two phases of their bulk forms; i.e., 0.11 eV (CsPbBr₃) versus 0.29 eV (CsPbI₃). The larger energy difference in CsPbI₃ means that it needs much more surface efforts to stabilize the cubic phase of CsPbI₃ than that of CsPbBr₃. The phase diagrams here give the reference parameters for the stabilization of cubic CsPbX₃ in different dimensions.

In summary, this work proposes a strategy that stabilizes the meta-stable cubic phase of CsPbX₃ at room temperature. Density functional theory calculations were employed to illustrate the feasibility of the strategy by comparing surface and bulk energies of the cubic and orthorhombic phases with varied surface/bulk ratios. The predicted critical thickness of 2D CsPbI₃ nano-plates that could stabilize the cubic phase—i.e., 1.3–4.8 nm—is slightly too thin for light absorption while that of CsPbBr₃—i.e., 5.4–24.6 nm—may be adopted in portable or flexible display. However, the moderate sizes of the nanoparticle (7.8–71.9 nm) and nanowire (3–48.3 nm) forms appear highly promising in applications of solar cells, flexible displays, among others. In light of this, our results go one-step towards solving the long-standing stability issue of CsPbX₃ and may promote practical applications of nanostructured CsPbX₃.