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Inorganic Lattice Fluctuation Induces Charge Separation in Lead Iodide Perovskites: Theoretical Insights Hiroki Uratani, and Koichi Yamashita J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b08855 • Publication Date (Web): 17 Nov 2017 Downloaded from http://pubs.acs.org on November 18, 2017

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The Journal of Physical Chemistry

Inorganic Lattice Fluctuation Induces Charge Separation in Lead Iodide Perovskites: Theoretical Insights Hiroki Uratani∗,†,‡ and Koichi Yamashita∗,†,‡ †Department of Chemical System Engineering, School of Engineering, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, 113-8656 Tokyo, Japan ‡CREST-JST, 7 Gobancho, Chiyoda-ku, 102-0076 Tokyo, Japan E-mail: [email protected]; [email protected]

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Abstract The high performance of recently emerged lead halide perovskite-based photovoltaic devices has been attributed to remarkable carrier properties in this kind of material: long carrier diffusion length, long carrier lifetime, and low electron–hole recombination rate. However, the charge separation mechanism underlying such carrier properties is still debated in this research field. In this work, using first-principles molecular dynamics simulations, we have demonstrated that the charge separation is induced by the structural fluctuation of the inorganic lattice, assuming that the charge carriers occupy the band-edge. It is shown that the charge separation is attributed to the electrostatic potential fluctuation coupled to the inorganic lattice dynamics, on the basis of both simple tight-binding model-based analyses and first-principles calculations. These results suggest that the organic cations, which are often used as components of lead halide perovskites, are unlikely to be essential for the above-mentioned carrier properties. Hence, it is expected that all-inorganic lead halide perovskite-based photovoltaics might be able to rival organic-inorganic lead halide perovskite-based ones in performance.

INTRODUCTION Lead halide perovskites, which have the general formula APbX3 (A is a monovalent cation called A-site cation, and X is a halogen), are attracting much interest for applications in photovoltaic devices known as perovskite solar cells. This kind of photovoltaic device, which has achieved a maximum photoconversion efficiency of 22.1%, 1 is considered a potential game changer in the field of solar energy utilization. In response to the achievement of such excellent photovoltaic performance, the underlying mechanism has also attracted a lot of attention among both experimental and theoretical researchers. Recent studies reported exceptionally long carrier lifetime and diffusion length in lead halide perovskites such as methylammonium lead iodide (MAPbI3 ; MA+ = 2 ACS Paragon Plus Environment

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CH3 NH3+ ). 2–6 In addition, the rate of electron–hole recombination is orders-of-magnitude smaller than that expected by the Langevin model, which simply assumes that recombination occurs when a hole and an electron encounter each other. 4,7 These properties, which may be keys to the high photovoltaic performance, suggest that charge carriers (holes and electrons) are separated so that recombination is prevented in those materials. However, the mechanism of the charge separation is still not fully understood. Some previous studies tackling this problem have proposed that the organic cations such as MA+ play the key role. 8–10 The organic cations contained in lead halide perovskites have orientational disorder at room temperature. 11 Because of the electric dipole moment of the organic cations, the disorder generates orientation-dependent fluctuation in the electrostatic potential inside the materials. This potential fluctuation has often been considered the crucial factor for the charge separation. Nevertheless, the need for the organic cations is still under debate in this research field. For example, Kulbak et al. 12 reported all-inorganic, CsPbBr3 -based cells which have a comparable photovoltaic performance to MAPbBr3 -based cells. Zhu et al. 13 investigated band edge carriers in organic cation-containing and all-inorganic lead halide perovskites by timeresolved spectroscopic techniques and found similar properties, for example, low carrier trapping and radiative recombination rate constants. Based on this result, they claimed that the organic cations might not be essential for these remarkable band edge carrier properties in lead halide perovskites. In this work, lead iodide perovskites, the most commonly studied type of lead halide perovskites, are the target of discussion. We show that the charge separation is mainly induced by the structural fluctuation of the inorganic PbI3– lattice itself, and that the contribution from the organic cations may not be the dominant factor. This paper is organized as follows. First, through first-principles molecular dynamics (FPMD) simulations, it is demonstrated that the charge separation originates from the structural fluctuation of the PbI3− lattice, and the A-site cations do not enhance the charge



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separation. Second, the mechanism of the lattice fluctuation-induced charge separation is discussed; it is shown that the charge separation is induced by the fluctuation in the internal electrostatic potential coupled to the PbI3– lattice motion, through both simple tight-binding model-based and first-principles-based analyses.

METHODS Structural Models We considered four types of lead iodide perovskites differing in A-site cations: MA+ , formamidinium cation (FA+ = CH(NH2 )2+ ), guanidinium cation (GA+ = C(NH2 )3+ ), and cesium cation (Cs+ ). In terms of the strength of the molecular electric dipole moment, MA+ > FA+ > GA+ ≈ 0. 8,14 Prior to FPMD simulations, model structures for each perovskite composition (APbI3 , for A = MA, FA, GA, and Cs) were constructed. All the model structures (Figure 1) were made of tetragonal simulation cells (a = c < b, α = β = γ = 90◦ ), so that their valence-band maxima (VBM) and conduction-band minima (CBM) are located at the Γ-point. Each model contains 32 PbI3– units and 32 A-site cations in its simulation cell. For MAPbI3 , the model structure was constructed as a 2 × 2 × 2 supercell of its tetragonal unitcell structure, because it is the most stable phase at room temperature. 15,16 For FAPbI3 , GAPbI3 , and CsPbI3 , the models were transformed from the cubic structures with rotation around the b-axis by 45◦ . Further details of the model construction including the cell parameters are described in the Supporting Information §1.1.

Computational Details All density-functional theory calculations including FPMD simulations were performed at the Γ-point, with Perdew–Burke–Ernzerhof formalism of the generalized gradient approximation, 17 a plane-wave cutoff of 400 eV, using the projector augmented-wave method 18 (for H, Pb, C, I, N and Cs, 1, 14, 4, 7, 5, and 9 electrons were explicitly treated, respectively), 4 ACS Paragon Plus Environment

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Figure 1: The initial structure of the simulation cells. Gray, purple, brown, blue, pink, and green atoms indicate Pb, I, C, N, H, and Cs, respectively. The cell boundaries are indicated by blue lines.



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as implemented in the VASP 5.4.1 code. 19–22 For visualization, VESTA software 23 was used.

First-Principles Molecular Dynamics Simulations FPMD simulations were performed with the time step of 0.25 fs, under the NVE ensemble. Before the simulations, the systems were thermalized so that the kinetic temperature was around room temperature during the simulations (for details, please see the Supporting Information §1.2). For each model, the simulations were conducted in two types of conditions, namely, Type I and Type II. In the Type I condition, the PbI3– frame is fixed to the initial geometry and only A-site cations (MA+ , FA+ , GA+ , and Cs+ ) have kinetic degrees of freedom. By contrast, in the Type II condition, no constraints are imposed on atomic motions, thus all atoms move. The simulations were conducted for 10 ps in the Type I condition or 15 ps in the Type II condition. To check the reliability of our FPMD simulations, we evaluated the kinetic temperature, rotational time scale of the organic cations, time-averaged structure of the PbI3– frame, and vibrational density of states (VDOS). Because the main focus of this paper is charge separation and not structural dynamics itself, details of those analyses are described in the Supporting Information §1.3–1.6 and only some essential points are noted below. During the simulations, the kinetic temperature fluctuated around room temperature. The rotational time scale of the organic cations was determined by the autocorrelation function of molecular orientation. The rotational motion of FA+ and GA+ was sufficiently fast so that the molecular orientation is completely randomized throughout the simulations; therefore, we can safely neglect the artifact originating in the starting orientation of these cations. The orientational autocorrelation functions of MA+ have been previously reported 24–26 and our result reasonably agrees with those. Because the rotation of MA+ was somewhat slower than that of FA+ and GA+ , we tested a different starting orientation of MA+ cations and confirmed that the conclusion is not likely to be affected. The time-averaged structures of MAPbI3 and FAPbI3 have tetragonal and quasi-cubic PbI3– frames, respectively, reproduc6 ACS Paragon Plus Environment

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ing the previous experimental 15,16 and theoretical 27 results. By contrast, the time-averaged structure of CsPbI3 has large octahedra-tilting, forming a quasi-orthorhombic polymorph. This is consistent with the fact that some imaginary phonon bands are found in perfect-cubic CsPbI3 , which indicates that cubic is not the most stable polymorph of CsPbI3 . 28,29 Similar to FAPbI3 , the averaged structure of GAPbI3 has a quasi-cubic PbI3– frame. While a few studies reported GAPbI3 in a nonperovskite phase (KCdCl3 type), 30,31 crystallographic characterization of perovskite-type GAPbI3 has not been performed as far as we know, whereas it is expected to be able to exist. 14 VDOS was obtained as the Fourier transform of the velocity autocorrelation function. For MAPbI3 , VDOS and infrared/Raman spectra have been reported by previous experiments and simulations, 32,33 and our results are considered to be consistent with them.

Evaluation of Charge Separation Following the approach of Ma et al., 9 to quantify the charge separation, the overlap integral q was defined as: ∫

∫ |ψVBM (r)||ψCBM (r)|dr =

q= Vcell

√ ρVBM (r)ρCBM (r)dr,

(1)

Vcell

where Vcell , ψVBM (r), ψCBM (r), ρVBM (r), and ρCBM (r) are the whole space inside the simulation cell, Kohn–Sham orbitals corresponding to the VBM and the CBM, and charge densities corresponding to the VBM and CBM, respectively. During the simulations, we calculated q every 0.1 ps, and averaged it over the whole simulation time (10 or 15 ps). Here 0 ≤ q ≤ 1, and smaller q indicates less overlap of VBM and CBM. Assuming that holes and electrons mainly occupy VBM and CBM respectively, q can be an indicator of the charge (electron–hole) separation. Because one may suspect that q can be significantly affected by changes in energetical order of the orbitals associated with the structural dynamics (e.g., VBM−1 at a snapshot could be VBM at the next snapshot),



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we also calculated the overlap integral regarding the bands below VBM and/or above CBM (at the Γ-point), and weighed them by the Boltzmann factor. The difference between the obtained value and q was negligible (for details, please see the Supporting Information §2). Also, to see the characteristics of the time evolution of VBM and CBM, autocorrelation functions of |ψVBM (r)| and |ψCBM (r)| were calculated and reported in the Supporting Information §3.

RESULTS AND DISCUSSION Charge Separation Induced by PbI3– Lattice Fluctuation Figure 2 shows the calculated q in Type I (yellow) and in Type II (blue). To validate the results, we tested a larger supercell in the case of CsPbI3 and obtained qualitatively the same results (please refer the Supporting Information §4). In Figure 2, q in the ”Type 0” condition, namely, PbI3– frame only (without A-site cations) fixed at the initial geometry, is also shown (black). It should be noted that Type 0 is not the optimized structure of the PbI3– frame; it is just extracted from the initial geometry of the FPMD simulations. Here, Type 0 and Type I have exactly the same, fixed PbI3– frame geometry, and their difference is the presence of the A-site cations. Because the electronic bands in the near-VBM and nearCBM regions have essentially no overlap with the A-site cations’ orbitals, 34 the difference in q between Type 0 and Type I should come from the electrostatic effect of A-site cations, as pointed out by previous researches. 8,9 However, Figure 2 shows that this is also the case in CsPbI3 , i.e., anyway this phenomenon is not unique to the organic cation-based perovskites. Changing the focus to a comparison between Type I and Type II, we find that Type II has smaller VBM–CBM overlap q (Figure 2). Considering that both Type I and Type II have A-site cation dynamics and they are different in PbI3– frame dynamics (fixed in Type I, and moving in Type II), this fact suggests that the structural fluctuation of the PbI3– frame induces the charge separation. 8 ACS Paragon Plus Environment

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We can see more directly that the structural fluctuation of the PbI3– frame is the dominant factor for the charge separation; even without A-site cations, the charge separation occurs. From the Type II trajectory, only the PbI3– frame was extracted, and we name the extracted trajectory as the Type III condition. In other words, Type II and Type III have exactly the same dynamics of the PbI3– frame, but Type III does not contain A-site cations. Note that in the Type III (and Type 0) condition, each simulation cell is negatively (-32) charged. The calculated q in Type III is shown in Figure 2 (magenta). Remarkably, q is smaller in Type III than in Type II. This result indicates that, with the structural fluctuation of the PbI3– frame, A-site cations are not necessary for the charge separation; the electrostatic effect of A-site cations does not enhance the charge separation.

1

Type 0 (fixed PbI3- frame only) Type I (PbI3- frame is fixed, A-site cations move) Type II (all atoms move)

0.9 0.8 0.7

Type III (PbI3- frame only, extracted from Type II trajectory)

0.6 q

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0.5 0.4 0.3 0.2 0.1 0

MAPbI3

FAPbI3

GAPbI3

CsPbI3

Figure 2: Calculated q for each perovskite composition in the Type I condition (PbI3– frame is fixed and A-site cations move) as yellow, Type II condition (all atoms move) as blue, and Type III condition (only PbI3– frame is extracted from the Type II trajectory) as magenta. For comparison, q of the ”Type 0” condition (fixed PbI3– frame without A-site cations) is shown as black. Note that the reported q values are averaged over the simulation time. As an example, the charge density distribution of VBM and CBM of MAPbI3 at 2.5 ps is shown graphically in Figure 3. In the Type I condition, the VBM and CBM are spread over the whole simulation cell. By contrast, in the Type II condition, the VBM is localized so



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that the overlap between VBM and CBM is reduced. In the Type III condition, localization of the VBM is further strengthened. Note that the spatial distribution of VBM and CBM depends on the time and the perovskite compositions. In the Supporting Information §5, VBM and CBM at other times and for other perovskite compositions are also drawn.

Figure 3: Spatial distribution of VBM and CBM of MAPbI3 at 2.5 ps in Type I, Type II, and Type III conditions.

One-Dimensional Tight-Binding Model-Based Analyses To understand the origin of the PbI3– lattice dynamics-induced charge separation, we introduce a one-dimensional periodic tight-binding model, in which 100 sites are placed on a straight line with interval a. The Hamiltonian matrix elements of this system are described


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as: ⟨ ⟩ ˆ n H n = En , ⟨ ⟩ ˆ n H n + 1 = tn eika ,

(2) (3)

ˆ |n⟩, En , tn , and k are the Hamiltonian, pseudo-atomic orbital of the n-th site, onwhere H, site energy of |n⟩, transfer integral between neighboring sites, and wave vector, respectively. En and tn have the dimension of energy. For simplification, we set k = 0, because VBM and CBM are located at k = 0 in this system. The i-th band (at k = 0) ψi is described as:

ψi =

∑

cin |n⟩ ,

(4)

n

where [ci0 , ..., ci100 ] can be obtained as the i-th eigenvector of the Hamiltonian matrix. In lead iodide perovskites, VBM and CBM are mostly composed of I 5p and Pb 6p orbitals, respectively, because the latter have higher orbital energy. To reproduce this set of evidences, we set En = −1.0 for odd n and En = −2.0 for even n, and tn = −1.0 for all n. Reflecting the different En set alternately, the calculated band energy has a gap between i = 50 and i = 51; i = 50 is VBM and i = 51 is CBM. Figure 4a shows that the population (|cin |2 ) of the VBM and the CBM is located on the sites with odd and even indexes, respectively, consistent with the fact that VBM and CBM of lead halide perovskites are mainly located on the halogen and Pb atoms, respectively. To include the effect of the lattice dynamics, we can consider two scenarios: the fluctuation in orbital–orbital interaction and the induced electrostatic potential fluctuation. The former and the latter can be modeled by the fluctuation in tn , and the fluctuation in En , respectively. Here, we calculated the population of VBM and CBM in each case. Figure 4b shows the population with the fluctuation in tn ; 0.5 sin(2πn/100) is added to tn . While the fluctuation in tn induces localization of the VBM and the CBM, they are localized in essentially the same region and not separated. Hence, the fluctuation in tn , namely, the fluctuation in the orbital–orbital interaction, is unlikely to be the origin 11 ACS Paragon Plus Environment


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of the charge separation. By contrast, the population calculated with the fluctuation in En , 0.1 sin(2πn/100) added to En , is described in Figure 4c showing clear separation of the VBM and the CBM. Hence, in this model, the charge separation is induced by the fluctuation in En , not tn . These qualitative analyses suggest that the PbI3– lattice dynamics-induced charge separation is caused by the electrostatic potential fluctuation.

Figure 4: Population of VBM and CBM in the one-dimensional tight-binding model, calculated (a) without any perturbation, (b) with 0.5 sin(2πn/100) added to tn , and (c) with 0.1 sin(2πn/100) added to En .


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Why PbI3– Lattice Dynamics Drives Charge Separation?–from the Viewpoint of Electrostatic Potential Fluctuation We confirmed the above discussion that the charge separation originates in the electrostatic potential fluctuation, by first-principles analyses. Using Hartree atomic units, the electrostatic potential VES (r) is described as: ∫ VES (r) =

ρ(r′ ) dr′ + Vei (r), ′ |r − r |

(5)

where ρ(r′ ) is the charge density at r′ . Vei (r) is the Coulomb potential from the ions. h Practically, Vei (r) is replaced by pseudopotentials. EES defined as follows represents the

electrostatic part of the energy of a hole: ∫ h EES

∗ ψVBM (r)VES (r)ψVBM (r)dr

=− Vcell

∫ =−

VES (r)ρVBM (r)dr

(6)

Vcell

Similarly, the electrostatic energy of an electron is described as ∫ e EES

=

VES (r)ρCBM (r)dr.

(7)

Vcell

h e The sum EES = EES + EES can be understood as the electrostatic contribution for the total

energy of the hole and the electron. In particular, small EES means that the charge carriers are stabilized by the electrostatic potential fluctuation; the hole and the electron are localized on a high VES (r) region and a low VES (r) region, respectively. The calculated q and EES every 0.1 ps for Type II simulations are shown in Figure 5. Evidently, the time evolution of q (red) and EES (blue) have quite similar patterns, which indicates that stabilization of the charge carriers by the electrostatic potential fluctuation is the determining factor of q. From the above, we can conclude that the fluctuation in the electrostatic potential inside the material separates holes and electrons. Finally, we show that the electrostatic potential fluctuation mainly comes from the PbI3– 13 ACS Paragon Plus Environment


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Figure 5: Time evolution of q (red) and EES (blue) for each perovskite composition.


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frame lattice dynamics. Here, VES (r) is defined as VES (r) calculated from atomic geometries

and self-consistent charge density in the Type III system, i.e., the system consist only of frame the PbI3– frame extracted from each Type II snapshot. EES calculated as below can be

understood as the contribution to EES from the PbI3– frame. h,frame e,frame frame EES = EES + EES

(8)

∫ h,frame EES

=−

frame VES (r)ρVBM (r)dr

∫ e,frame EES

(9)

Vcell frame VES (r)ρCBM (r)dr

=

(10)

Vcell

Note that ρVBM (r) and ρCBM (r) are calculated in the full systems containing both PbI3– frames and A-site cations; in other words, ρVBM (r) and ρCBM (r) are identical with those in frame Equation 6 and Equation 7. The calculated EES and EES for each snapshot are compared frame in Figure 6. We can find that EES and EES have almost the same values, indicating

that EES is dominantly governed by the structure of the PbI3– frame. Note that we also cation calculated VES (r), defined as VES (r) calculated in systems containing only A-site cations cation extracted from each snapshot; the resulting EES , which is the A-site cation contribution frame to EES , was far smaller than EES (please see the Supporting Information §6). Notably,

a recent combined experimental and theoretical study by Yaffe et al. 35 reported that local polar fluctuations are observed in both CsPbBr3 and MAPbBr3 , suggesting that the local polar fluctuations are general features of lead halide perovskites and not specific to organic cation-containing lead halide perovskites.

CONCLUSIONS We have demonstrated that the charge separation in lead iodide perovskites is induced by the structural fluctuation of the inorganic (PbI3– ) lattice, based on the FPMD simulations, though it should be noted that our discussion is assuming that the charge carriers occupy the 15 ACS Paragon Plus Environment


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frame (blue dots) for each perovskite composition. Figure 6: Comparison between EES and EES Diagonal lines are drawn as dashed lines. For MAPbI3 , FAPbI3 , and CsPbI3 , the region between 8 eV and 9.5 eV is shown in the insets.

band-edge. It is proposed that the charge separation is attributed to the electrostatic potential fluctuation based on the tight-binding model-based analyses, and that was confirmed by the first-principles calculations. Moreover, the electrostatic potential fluctuation was found to originate in the structural fluctuation of the inorganic lattice. Hence, we can conclude that the structural fluctuation of the inorganic lattice entails electrostatic fluctuation inside the material, and the electrostatic fluctuation induces the charge separation. Our result indicates that the A-site cations are unlikely to be the crucial factor for the exceptional carrier properties in lead halide perovskites, suggesting the possibility of all-inorganic lead halide perovskite-based solar cells whose performance rivals that of organic-inorganic lead halide perovskite-based solar cells.

Supporting Information Available The Supporting Information is available free of charge on the ACS Publications website. The file includes the following content. 16 ACS Paragon Plus Environment

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1. Detailed protocols and analyses of the FPMD simulations: initial structure construction, thermalization, kinetic temperature, rotational time scale of organic cations, timeaveraged structure, and VDOS. 2. Evaluation of the charge separation including the bands below VBM and above CBM.

3. Autocorrelation functions of |ψVBM (r)| and |ψCBM (r)|. 4. Validation of the results using a larger supercell. 5. Spatial distribution of VBM and CBM. 6. Estimation of the A-site cation contribution to the electrostatic energy of the charge carriers.

AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected] (H.U.). *E-mail: [email protected] (K.Y.).

Notes The authors declare no competing financial interests.

Acknowledgement This work was supported by JST CREST Grant Number JPMJCR12C4, Japan. Computational resources were provided by Reserch Center for Computational Science, National Institutes of Natural Sciences, Okazaki, Japan.



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