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Effects of Defect on Work Function and Energy Alignment of PbI2: Implications for Solar Cell Applications

Two-dimensional (2D) layered lead iodide (PbI2) is an important precursor and common residual species during the synthesis of lead-halide perovskites. There currently exist some debates and uncertainties about the effect of excess PbI2 on the efficiency and stability of the solar cell with respect to its energy alignment and energetics of defects. Herein, by applying the first-principles calculations, we investigate the energetics, changes of work function and the defective levels associated with the iodine vacancy (VI) and interstitial iodine (II) defects of monolayer PbI2 (ML-PbI2). We find that the PbI2 has a very low formation energy of VI of 0.77 and 0.19 eV for dilute and high concentration, respectively, reflecting coalescence tendency of isolated VI, much lower than that of vacancies in other 2D materials like phosphorene. Similar to VI, a low formation energy of II of 0.65 eV is found, implying a high population of such defects. Both defects generate in-gap defective levels which are mainly due to the unsaturated chemical bonds of p-orbitals of exposed Pb or inserted I. Such rich defective levels allow the VI and II as the reservoir or sinks of electron/hole carriers in PbI2. Our results suggest that the remnant PbI2 in perovskite MAPbI3 (or FAPbI3) would play dual opposite roles in affecting the efficiency of the perovskite: (1) Forming Schottky-type interface with MAPbI3 (or FAPbI3) in which the built-in potential would facilitate the electron-hole separation and prolong the carrier lifetime; (2) Acting as the recombination centers due to the deep defective levels. To promote the efficiency by the Schottky effect, our work reveals that the II defect is favored, and to reduce the recombination centers the VI defect should be suppressed. Our results shall be beneficial in improving strategies for the related optoelectronics applications.


Introduction
Hybrid organic-inorganic perovskite materials have recently attracted widespread attention in photoelectric fields due to their superior features of high absorption coefficients 1 , desirable optical band gaps 2 , long carrier lifetime and diffusion length 3 . Specifically, the power conversion efficiency (PCE) of perovskite solar cells (PSCs) has reached ~25.5% 4 , while the key to realizing a good performance of PSCs is indispensable with the uniform and wellcrystallized perovskite absorbed films 5 . During the one-step reaction, changing the stoichiometry of the precursors PbI2 provides a simple means of varying the chemical potential and the outcome of the reaction, including the defect concentration, doping, and final composition 6 . In two-step synthesis methods, the lead iodide (PbI2) film is deposited and subsequently exposed to solutions of the organic salts thus obtaining the perovskite thin films after thermal annealing 7,8 . However, the PbI2 in the bottom layer cannot completely interpenetrate with the organic solution in the upper layer, resulting in uncontrollable PbI2 residue. The reaction process in solution proceeds by topotactic nucleation followed by grain growth by dissolution-reconstruction. There is a strong orientational and structural relationship between the final stage of the solution-reacted MAPbI3 product and the initial PbI2 crystallite [9][10][11] . As the precursor material used to fabricate perovskite thin films 12,13 , PbI2 is critical for efficient PCE due to the 6s 2 electronic configuration of Pb atoms 14 . More notably, a slight excess of PbI2 in the perovskite film has been observed to be beneficial [15][16][17][18] . The remnant PbI2, randomly distribute in the perovskite layer, on the surface or the grain boundary, can suppress charge recombination and reduce the intrinsic halide vacancy concentration thereby passivate trap states 8,15,[19][20][21][22] . It was also theoretically confirmed that the slight excess PbI2 is able to reduce surface cation concentration and the corresponding nonadiabatic coupling decreased by one order of magnitude, hence the charge carrier lifetime extended 23 . However, the excess/unreacted PbI2 will decompose into lead and iodine under illumination, which trigger and accelerate the degradation of PSCs 24 .
Actually, PbI2, with layered crystalline structure composed of covalently bonded I-Pb-I atomic planes, is a promising candidate as two-dimensional (2D) semiconductor material owing to its superior optoelectronic properties [25][26][27] . The 2H phase is the most thermal stable structure for bulk PbI2 at room temperature 28 , but the stacking forms of the adjacent (I−Pb−I)n layers are diverse due to the weak van der Waals interactions 29,30 . In contrast to other layered semiconductors like MoS2 and WS2, which prefer to n-type behavior, the PbI2 tends to have a p-type feature with tunable bandgap which strongly depends on the thickness [31][32][33] . As theoretical and experimental researches shown, the band structure of PbI2 would change from a direct bandgap of ~2.38 eV to an indirect bandgap of ~2.5 eV as the thickness decrease from bulk to monolayer 27,34,35 . These unique properties trigger many studies on the growth, exfoliation, and applications of the PbI2 nanostructure 34,[36][37][38][39][40] in optoelectronic devices, photodetectors, and photon detection [41][42][43] . In this work, we prefer to focus on monolayer-PbI2 (ML-PbI2), according to the following reasons: (1) The evidence for the ML-PbI2 has been discovered by electrocrystallisation 44 and the crystalline state of PbI2 has the hexagonal CdI2 type structure.
(2) The excess PbI2 species are distributed as nanosheets between the grain boundaries of the perovskite film 20,45 . (3) In 2D PbI2, low dimensionality provides a sensitive conductive channel for carriers and reduces phonon thermal transport at the same time 46 . (4) Moreover, it is possible to fabricate ML-PbI2 with other 2D materials, such as graphene, by epitaxial growth and layercontrol in photovoltaic and thermoelectric systems 39,47,48 .
Point defects, which are probably to be unintentionally introduced during fabrication and applications, are common in high specific surface area 2D materials. Besides, it is known that defective states, which are situated within the bandgap, affect dramatically carrier mobility and photocatalysis of 2D material. Such defective states can tune band gaps for enhancing light absorption, act as active sites for separating photogenerated carriers, and serve in molecular chemical adsorption with significantly decreased energy barrier of reactions 47,[49][50][51] . In previous studies, volatile species (for example I2) were found to rapidly decompose perovskites to causes severe degradation 52,53 . Besides, single-point iodine vacancy (VI) of sublattice sites in ML-PbI2 has also been observed in the previous experiments 48 . Related properties of ML-PbI2 with VI and nonmetallic atoms doping were theoretically investigated 54 . The results pointed that spinorbit coupling (SOC) does not alter the electron occupation of the bands near the Fermi level.
However, the effect of VI, in particularly its concentration, and interstitial iodine (II) on ML-PbI2 remains unexplored. Considering the prominent role of PbI2 in organic-inorganic perovskite, further understanding of the defect mechanisms of ML-PbI2 that are predominantly involved with VI and II is thus critically important for practical applications.
In this work, we explored the energetics, changes of work function, and the electronic properties of the pristine ML-PbI2, defective levels associated with the VI in different concentrations and II in two interacted sites by first-principles calculations. Our calculations reveal that the ML-PbI2 has a relatively low formation energy of VI and II, implying a high population of such defects under thermodynamic equilibrium conditions. Moreover, the in-gap defective levels in both VI and II allow the reservoir or sinks of electron/hole carriers in PbI2. It is suggested that the remnant PbI2 in perovskite MAPbI3 (or FAPbI3) would play dual opposite roles of the perovskite: (1) Forming Schottky-type interface with MAPbI3 (or FAPbI3) in which the built-in potential would facilitate the electron-hole separation and prolong the carrier lifetime; (2) Acting as the recombination centers due to the deep defective levels. The II is favored in order to promote the efficiency by the Schottky effect, and the VI would be suppressed for reducing the recombination centers.

Method
The theoretical calculations were performed using density functional theory (DFT) method as implemented in the Vienna ab initio simulation package (VASP) 55,56 . The generalized gradient approximation (GGA) of Perdew−Burke−Ernzerhof (PBE) exchange-correlation functional was adopted for geometry optimization and electronic properties 57,58 , and the Heyd-Scuseria-Ernzerhof (HSE06) functional 59 was further employed in the density of state (DOS) and band structure calculations for describing the electronic interactions more precisely. The plane wave basis was applied with a kinetic cutoff energy of 300 eV. The Brillouin zone was sampled with a 2×2×1 and a 1×1×1 Gamma centered Monkhorst-Pack grids for PBE and HSE calculations, respectively. The dispersion correction of DFT-D3 was also considered 60 . All structures were fully relaxed under the convergent criterion of 0.01 eVÅ -1 for atomic force. The relaxed lattice constant of PbI2 unit cell is a = b = 4.593 Å, which is in good agreement with the experiment result (4.557 Å) 34 . Meanwhile, a larger 6×6×1 supercell (36 Pb atoms and 72 I atoms contained) of ML-PbI2 was also considered. An 18 Å vacuum space was added on the z direction to eliminate the effect of image atoms. The work function is calculated via the formula: where is the energy in vacuum level, and is the Fermi energy. The band diagram of defect states is aligned through calibrating .
Formation energy of defects in ML-PbI2 per single VI or single II was calculated as: where ( 36 72+ ) is total energy of the ML-PbI2 supercell with vacancies or interstitials, ( 36 72 ) is the total energy of prinstic ML-PbI2, is the number of I atoms that have been added ( > 0) to or removed ( < 0) from the supercell when the VI or II doping is introduced, and is the chemical potentials. The value of the of I is defined within the range of values corresponding to Pb-rich or I-rich conditions. Under I-rich condition, the is equal to the energy of a I atom in a I2 molecule. Under Pb-rich condition, is determined as half of the energy difference between one formula unit of supercell ML-PbI2 and a Pb atom in bulk.

Results and discussion
The original 1T phase ML-PbI2 nanosheet, obtained from the 2H-PbI2 bulk 28 , has a hexagonal structure with space group P3 ̅ 1. Its atomic structure is similar to the 1T-TMDs        highly localized at the three exposed Pb atoms and the VI core, with slight contributions from the next nearest I atoms from the opposite side.
With increasing the content of iodine vacancies, corresponding to the medium-level VI-2 and high-level VI-3, the VBs and CBs tend to shift slightly to the Fermi level ( Fig. 3b and c). As   corresponding to the NN Pb atoms, with a smaller contribution of 5 states (blue lines) from the NN I atoms and the least for 6 states (black lines) of NN Pb atoms. As seen from Fig. 4a and b, these impurity levels near the Fermi level are localized and spin-dependent. In contrast, as shown in Fig. 4c, the impurity levels become delocalized and have no spin polarization when I-vacancy concentration increases further, developing a band-like extending defective states.
One striking feature of highly VI concentrated PbI2 is that the defective bands, evoluted from single VI, keeps a semiconducting behavior, which is different from the metallic behavior of other 2D materials like MoS2 66 .

I interstitial
Next, we examined the energetics and electronic properties associated with the II defect.
Here we only consider dilute adsorption of the iodine atom since higher uptake of iodine would lead to similar electronic and structural properties due to the large separation between each two adsorbates. As shown in Fig. 1a  The of II-I and II-Pb was listed in Table 1    As for II-Pb system, the interstitial iodine atom produces two defect levels, one is an occupied and spin-up state with a shallow nature (0.16 eV above VB by HSE) and another is empty and spin down state with a relatively deep location (1.70 eV above VB by HSE), as shown in Fig. 5b. The real space representation of the state shows that it is mainly contributed by the 5 orbital coupling between adsorbed I atom and the I atom from the broken Pb-I bond.  6p orbitals is red lines, I 5s orbitals is green lines, and I 5p orbitals is blue lines.

Band alignment and implications for solar cell application
Finally we would like to analyze the band alignment between perfect and defective PbI2 together with other common PSCs materials like the MAPbI3, FAPbI3 and electron hole transport materials. Figure Fig. 7, the interface would be Schottky type. Figure 8a shows the schematic diagram of Schottky junction between p-type PbI2 and perovskite MAPbI3. The highly dense low-lying defective states in PbI2, mainly from the II, would raise the Fermi level and reduce the Schottky barrier. The leakage of the electrons from the FAPbI3 or MAPbI3 into the PbI2 would generate the built-in potential across the interface.
Such built-in potential would help to facilitate the electron-hole separation and promote the carrier lifetime. (2) Acting as traps or recombination center: As shown in Fig. 7, the high-lying defective states associated with the VI locate below the CBM of FAPbI3 and MAPbI3 which are empty and allows carriers trapping or recombination. This would reduce the carrier lifetime in the host FAPbI3 or MAPbI3. Therefore, PbI2 would play both negative and positive roles in PSCs, which have a good agreement with previous works 20, 21,24 . To promote the efficiency by the Schottky effect, the II defect is favored, and to reduce the recombination centers the VI defect would be suppressed.

Conclusion
In summary, we have investigated the defective states in the PbI2 for the iodine vacancies and interstitial defects by using the first-principles study. Under Pb-rich and moderate growth condition, the iodine vacancy is the dominant defect, whereas under I-rich condition, the iodine interstitial becomes more popular. A noticeable trend was the steady decrease in bandgap with the growth of iodine vacancy, due to the consistent increase in the number of defective levels.