R. A.
Evarestov
^{a},
E. A.
Kotomin
^{bc},
A.
Senocrate
*^{b},
R. K.
Kremer
^{b} and
J.
Maier
^{b}
^{a}Institute of Chemistry, St. Petersburg State University, Petrodvorets, Russia
^{b}Max Planck Institute for Solid State Research, Stuttgart, Germany. E-mail: alessandro.senocrate@googlemail.com
^{c}Institute of Solid State Physics, University of Latvia, Riga, Latvia
First published on 27th January 2020
First principles Density Functional Theory (DFT) hybrid functional PBESOL0 calculations of the atomic and electronic structure of perfect CsPbI_{3}, CsPbBr_{3} and CsPbCl_{3} crystals, as well as defective CsPbI_{3} and CsPbBr_{3} crystals are performed and discussed. For the perfect structure, decomposition energy into binary compounds (CsX and PbX_{2}) is calculated, and a stability trend of the form CsPbBr_{3} > CsPbI_{3} > CsPbCl_{3} is found. In addition, calculations of the temperature-dependent heat capacity are performed and shown to be in good agreement with experimental data. As far as the defect structure is considered, it is shown that interstitial halide atoms in CsPbBr_{3} do not tend to form di-halide dumbbells Br_{2}^{−} while such dimers are energetically favoured in CsPbI_{3}, analogous to the well-known H-centers in alkali halides. In the case of CsPbBr_{3}, a loose trimer configuration (Br_{3}^{2−}) seems to be energetically preferred. The effects of crystalline symmetry and covalency are discussed, alongside the role of defects in recombination processes.
A few studies published so far include first principles calculations of native defects in MAPbI_{3},^{17} self-trapped holes therein,^{18} as well as native defects in CsPbI_{3}.^{12,13,19} Recently, we investigated point defect formation and basic properties in CsPbI_{3} crystals from first principles calculations.^{20} It was shown that interstitial iodine atoms tend to form I_{2}^{−} dimers (‘dumbbells’) within the crystal structure, in agreement with previous studies for MAPbI_{3},^{17,18,21} and with the well-known situation in alkali halides.^{22–26} In alkali halides, the X_{2}^{−} dumbbells (called H centers) can be created under UV light irradiation. A similar situation, but under visible light irradiation, has been hypothesized to be the cause for the light-enhanced ion conduction reported in MAPbI_{3}.^{16} Thus, studying interstitial halide ions is important to understand charge transport properties in halide perovskites, both in the dark and under illumination.
In this paper, we continue our first principles calculations^{20} by calculating the formations energies of CsPbI_{3}, CsPbBr_{3} and CsPbCl_{3} from their parent binary compounds, and include also vibrational effects by performing heat capacity calculations. We compare these with experimental heat capacities, showing an excellent agreement. We also expand our study of the defective structure of Cs-based halide perovskites by considering interstitial bromide ions in cubic CsPbBr_{3}. We show that the formation of Br_{2}^{−} dumbbells is energetically unfavorable, in contrast with the situation in CsPbI_{3}. Instead, the energetically more favorable configuration for an interstitial bromide is a loose trimer (Br_{3}^{2−}) (Fig. 1). Calculations of the one electron energies show that one electron levels of the interstitial defects in both CsPbI_{3} and CsPbBr_{3} lie in the middle of the bandgap, thus may act as traps for photo-generated carriers.
Fig. 1 Schematic view of the perfect CsPbI_{3} and CsPbBr_{3} crystals, and of the structures including interstitial halide atoms, before and after structural relaxation. |
Spin–orbital (SO) interactions were neglected in our calculations, since we expect these to have negligible influence on defect formation energies as well as phase stability. On the other hand, SO interactions significantly affect the band gap results. It has been shown for CsPbI_{3}^{17} that including SO interactions leads to the increase of the valence band top by 0.2 eV and to the decrease of the Pb-based conduction band bottom by 0.8 eV. For CsPbBr_{3} and CsPbCl_{3}, SO corrections were extracted from LMTO calculations.^{30} Taking these into account, our calculations are found in good agreement with experimental band gaps.
Gaussian atomic orbitals^{31} were used to expand the crystalline Bloch functions. The interaction between the core electrons and the valence and sub-valent electrons in iodine, lead and cesium atoms and the valence electrons was described by means of effective core pseudopotentials.
The Brillouin zone (BZ) sampling was performed with 8 × 8 × 8 k-point meshes generated according to the Monkhorst Pack scheme.^{32} Kohn–Sham equations were solved iteratively to self-consistency within 10^{−8} eV. Full geometry optimization was carried out both in perfect and defective crystals calculations. The Mulliken analysis was used for effective atomic charges.
In the point defects calculations, the supercell model^{33,34} was used and a supercell composed by 8 primitive cells (2 × 2 × 2) was chosen (the main conclusions were confirmed by also using a 16 primitive cells supercell). In the perfect cubic CsPbX_{3} crystal, three Wyckoff positions of the space group Pmm with the simple cubic lattice are occupied: Cs 1b (0.5,0.5,0.5), Pb 1a (0,0,0), X (= I, Br) 3d (0.5,0,0). The neutral iodine and bromine atoms X_{i} were initially placed into the interstitial c-position with the coordinates (0.25, 0.25, 0) in the supercell and allowed to move, in order to find the configuration with the minimum total energy. All symmetry restrictions were lifted in the calculations of the defective crystals, in order not to constrain the optimization of lattice parameters and atomic coordinates.
The X_{i} formation energies E_{f} were calculated using the following relation:
E_{f} = E_{def} − (E_{perf} + U_{i}) | (1) |
The CsPbX_{3} decomposition energies into binary compounds were taken from total energies of binary parent compounds (Table 1)
E_{dec} = E(CsPbX_{3}) − [E(CsX) + E(PbX_{2})], | (2) |
Space group | Z | a _{0}, b_{0}, c_{0} expt^{36} (Å) | a _{0}, b_{0}, c_{0} calc (Å) | |q| (e) | U _{Cs}, U_{Pb} (a.u.) | |
---|---|---|---|---|---|---|
CsI | Pmm | 1 | 4.57 | 4.66 | 0.99 | −20.011333 |
CsBr | Pmm | 1 | 4.29 | 4.39 | 0.99 | −20.023117 |
CsCl | Pmm | 1 | 4.12 | 4.19 | 0.99 | −20.052488 |
PbI_{2} | Pm1 | 1 | 4.56, 6.78 | 4.69, 6.57 | 0.84, −0.42 | −3.659705 |
PbBr_{2} | Pnam | 4 | 8.06, 4.73, 9.54 | 8.01, 4.75, 9.56 | 1.12, −0.54, −0.58 | −3.684755 |
PbCl_{2} | Pbnm | 4 | 7.61, 4.53, 9.03 | 7.54, 4.51, 9.03 | 1.32, −0.63, −0.69 | −3.745345 |
The heat capacity at constant volume (C_{v}) was calculated in a harmonic approximation according to ref. 34 and 35
(3) |
C_{p}(T) = C_{v}(T) + E(T)·T | (4) |
E(T) = α_{v}^{2}(T)·B·V_{mol} | (5) |
To fit the experimental heat capacities, a standard Debye behavior (with its characteristic Debye temperature Θ_{D}) has been initially considered. However, the experimental curves deviate quite strongly from this standard behavior, therefore this simple approximation has been complemented with the Einstein modes (corresponding to maxima in the phonon density of states due to acoustic phonon modes with low dispersion at zone boundaries, typically originating from vibrations of heavy atoms within compounds^{38,39}). These Einstein modes are generated with 2 parameters, a characteristic temperature (Θ_{E}) indicating the position of the maximum in the phonon density of states, and a weight (W_{E}) representing the fraction of atoms participating in this behavior.
Let us start by discussing the perfect structure. Our calculations, as already shown previously,^{20} are able to satisfactorily capture both the crystal phase and the electronic structure of CsPbI_{3}, CsPbBr_{3} and CsPbCl_{3}. From this basis, we move to calculate the decomposition energies of all these compounds (in the cubic phase) into the binary parent materials CsX and PbX_{2} (eqn (2)). Unsurprisingly, these energies are rather small (see Table 2): 0.23 eV, 0.26 eV and only 0.02 eV per formula unit for I, Br and Cl respectively. According to our calculations, the chloride perovskite is expected to be the least stable of all three compounds, while bromide and iodide show comparable stabilities. The strong deviation of the formation energy of CsPbCl_{3} from the trend may be connected with the more ionic nature of this compound (compared with the other two ones) and will require a more detailed study. The lack of experimental observations on the stability of these compounds makes any further conclusion at the moment unreliable.
Product (Pmm) | E _{pro} (a.u.) | Precursors | E _{pre} (a.u.) | Σ _{pre} (a.u.) | E _{dec} (a.u.) | E _{dec} (eV) |
---|---|---|---|---|---|---|
CsPbI_{3} | −57.847 | CsI + PbI_{2} | −31.402 −26.436 | −57.838 | 0.0086 | 0.23 |
CsPbBr_{3} | −63.725 | CsBr + PbBr_{2} | −33.359 −30.357 | −63.715 | 0.0094 | 0.26 |
CsPbCl_{3} | −68.417 | CsCl + PbCl_{2} | −34.925 −33.491 | −68.417 | 0.0007 | 0.02 |
We have also performed the calculations for the orthorhombic CsPbI_{3}, CsPbBr_{3} and CsPbCl_{3}, which now have four formula units per unit cell. Table 3 shows the obtained decomposition energies, which are larger than for the cubic crystal, consistently with the orthorhombic phase being more stable at the temperature represented by the calculations. The value for the decomposition energy of orthorhombic CsPbI_{3} also agrees well with recent experimental work,^{42} where the formation enthalpy of the phase from its solid halide precursors was estimated to be −0.18 eV.
Product (Pnma) | E _{pro} (a.u.) | Precursors | E _{pre} (a.u.) | Σ _{pre} (a.u.) | E _{dec} (a.u.) | E _{dec} (eV) |
---|---|---|---|---|---|---|
CsPbI_{3} | −57.849 | CsI + PbI_{2} | −31.402 −26.436 | −57.838 | 0.011 | 0.30 |
CsPbBr_{3} | −63.726 | CsBr + PbBr_{2} | −33.359 −30.356 | −63.715 | 0.011 | 0.30 |
CsPbCl_{3} | −68.419 | CsCl + PbCl_{2} | −34.925 −33.491 | −68.417 | 0.002 | 0.08 |
In further agreement is the fact that the experiments show the orthorhombic phase being more stable than the cubic one, albeit the experimental difference between the formation energy of cubic and orthorhombic CsPbI_{3} (0.15 eV)^{42} is found to be higher than our calculated one (0.08 eV).
In addition, we have calculated and measured the heat capacity of CsPbI_{3} and CsPbBr_{3} in the cubic crystal structure. Fig. 2a and b shows the experimental and theoretical dependence of the heat capacity on the temperature. The experimental curves tend to reach the classical limit given by the Dulong–Petit law (C_{p} = 3N_{A}·R, with R being the molar gas constant and N_{A} is the number of atom in the formula unit). This approximation yields limit values of ∼125 J (mol K)^{−1} for CsPbX_{3}. The calculation results depend on the number of k-points (N_{k}) used for the phonon calculations. Using only the gamma point (the simplest approximation) leads to a considerable underestimate (∼20%) of the heat capacity at high temperatures.
An increase of the number of k-points reduces the discrepancy, with 16 k-points providing sufficiently good agreement with experiments. Preliminary calculations also show that moving away from the cubic structure to consider an orthorhombic distortion yields no substantial difference in the calculated heat capacity values. This finding indicates that the crystal phase does not strongly affect the thermal behavior. More detailed calculations are on the way. The experimental heat capacities were also plotted in the coordinates useful to check for deviations from the Debye approximation (ω ∝ k), with C_{p}vs. T^{3} at low temperatures. As shown in Fig. 3, the peak at low temperatures in the quantity C_{P}/T^{3} indicates deviations from Debye behavior, as demonstrated by the comparison with the heat capacities (red solid lines) calculated assuming constant Debye temperatures as indicated. Such peaks typically result from maxima in the phonon density of states originating from acoustic phonon modes with low dispersion at the zone boundaries, originating from vibrations of heavy atoms in the compounds (see e.g.ref. 38 and 39).
We also noticed that the very low temperature behavior can be sample-dependent, likely due to structural defects and/or impurity which are difficult to control during the liquid-base synthesis. This aspect is discussed in more detail in the ESI.†
Let us now turn to important points concerning the modelling of the defective structure, namely the effect of the insertion of an interstitial halide in CsPbI_{3} and CsPbBr_{3}. Fig. 4 shows the initial and final crystalline structure for the interstitial halide atoms inserted into the cubic crystals. In the case of CsPbI_{3}, introduction of an interstitial I leads to the formation of an I_{2}^{−} dumbbell through interaction with one of the neighboring I ions. This process is accompanied by a considerable distortion of the lattice surrounding the defect (Fig. 4b), not only locally but also at long range (Table 4).
CsPbI_{3} | CsPbBr_{3} | |
---|---|---|
a _{0}, b_{0}, c_{0} (Å) | 12.630, 12.703, 12.595 (12.660) | 11.911, 11.799, 11.988 (11.880) |
Angles (°) | 90.09, 89.76, 89.01 (90) | 89.95, 89.85, 90.02 (90) |
q _{Cs}, q_{Pb}, q_{X} (e) | 0.99, 0.85, −0.62 | 0.99, 0.95, −0.65 |
E _{f} (eV) | −0.37 eV | +0.22 eV |
R, q (Å, e) | 3.35, −0.50 | 2.96, −0.50 |
ΔR (X_{i}, X_{1}, X_{2}, X_{3}) (Å) | −0.03, +0.48, +0.52 | +0.11, +0.18, +0.95 |
q (X_{i}, X_{1}, X_{2}, X_{3}) (e) | −0.35, −0.42, −0.61, −0.63 | −0.39, −0.54, −0.56, −0.63 |
We found in our previous study^{20} that the dumbbell (H center) formation is energetically favorable (E_{f} = −0.37 eV), notwithstanding the inclusion of the interstitial I atom at low temperatures. The I_{2}^{−} bond distance within the dumbbell (3.32 Å) is slightly shorter that in a free I_{2}^{−} ion (3.35 Å), as also observed for the H centers in alkali halides.^{22} In addition, the dumbbell formation causes a charge redistribution that leaves an almost identical charge on the two constituting I atoms (0.35e and 0.42e, with a small difference due to lattice distortions). Our previous calculations^{20} predicted also that the formation of larger aggregates e.g. trimers I_{3}^{2−}, is possible, but energetically less favorable (by 0.62 eV). A comparison with CsPbBr_{3} reveals a very different situation (Fig. 4d): the lowest in energy configuration after incorporating a neutral interstitial Br is a loose trimer, where distances between a central Br atom and two nearest halides are considerably larger (by 0.11 A and 0.18 A) than in a typical Br_{2}^{−} dimer as found in alkali bromides (the comparison with the dimer is carried out due to the lack of literature data on bromide trimers).^{22,26,43}
The formation energy of this complex defect is substantially larger (by 0.59 eV) than for the I_{2}^{−} dimer. In the trimer, a charge of 1.5e is spread over all three participating Br atoms. Considering that, due to some covalent character, the charge of the regular host Br ions (not participating in the trimer) is −0.65e, this indicates that the trimer is best denoted as Br_{3}^{2−}. To check the influence of the supercell size on the results obtained, we also performed time-consuming calculations doubling the supercell size (16 primitive unit cells instead of 8). The main conclusions are maintained, i.e. the H center formation energy is lower for CsPbI_{3} than CsPbBr_{3} (Br_{2}^{−} and I_{2}^{−} formation energy difference is 0.51 eV for a supercell of 16 unit cells). The X–X distance and Mulliken atomic charges are practically the same for supercells composed of 8 and 16 primitive cells. This substantial difference in behavior between the H center formation in CsPbI_{3} and CsPbBr_{3} could be related to the difference in the crystalline structure of the parent PbX_{2}, as PbI_{2} has a rhombohedral Pm1 symmetry (one formula unit per unit cell), while PbBr_{2} takes an orthorhombic Pnam space group (4 formula units per unit cell). As a consequence, the bromide ion sublattice has 4 atoms in the unit cell and their interatomic interactions become stronger in the perfect crystal, making the formation of a Br_{2}^{−} dumbbell more difficult when the interstitial bromide atom is inserted.
Considering the potential importance that this material has for photovoltaic and optoelectronic applications, we calculated positions of the energy levels of the interstitial defects in both crystals and found that in both cases these lie in the middle of the band gap (0.7 eV and 1 eV above the valence band maximum in CsPbI_{3} and CsPbBr_{3} respectively, with calculated bandgaps of 1.7 and 2.1 eV^{20}).
Due to their energy levels in the middle of the band gap, these defects could trap both electrons and holes. However, as such states are unstable (e.g. dimer or trimer dissociates after trapping an electron), these defects may not strongly affect the efficiency of the recombination processes in photovoltaic devices. The same is true for halide vacancies and V_{k} centers, but because they form shallow energy states.
The decomposition energies of CsPbX_{3} crystals in both cubic and orthorhombic phases were estimated using the results of PBESOL0 calculations of binary parent CsX and PbX_{2} crystals. The perovskite stability sequence was found as CsPbBr_{3} > CsPbI_{3} > CsPbCl_{3} for both crystalline phases.
The frozen phonon method was applied to obtain the temperature dependence of the heat capacity in the cubic phase. The good agreement obtained with the experimental data confirms the high accuracy of our calculations.
Concerning the defective structure, our first principles supercell calculations on CsPbBr_{3} show that the interstitial Br atoms (unlike the interstitial I atoms in CsPbI_{3}) do not form a Br_{2}^{−} dimer (commonly known as H center in alkali bromides),^{44} but rather a loosely bound trimer (Br_{3}^{2−}). Simple chemical arguments indicate that, due to the much higher polarizability, iodine atoms and ions are expected to form higher aggregates (e.g. polyiodides). The same situation is not expected (at least to a similar extent) in bromides.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9cp06322f |
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