Nanoscale Charge Localization Induced by Random Orientations of

Dec 10, 2014 - Nanoscale Charge Localization Induced by Random Orientations of Organic Molecules in Hybrid Perovskite CH3NH3PbI3. Jie Ma and Lin-Wang ...
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Nanoscale Charge Localization Induced by Random Orientations of Organic Molecules in Hybrid Perovskite CH3NH3PbI3 Jie Ma and Lin-Wang Wang* Joint Center for Artificial Photosynthesis and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States S Supporting Information *

ABSTRACT: Perovskite-based solar cells have achieved high solar-energy conversion efficiencies and attracted wide attentions nowadays. Despite the rapid progress in solar-cell devices, many fundamental issues of the hybrid perovskites have not been fully understood. Experimentally, it is wellknown that in CH3NH3PbI3 the organic molecules CH3NH3 are randomly orientated at the room temperature, but the impact of the random molecular orientation has not been investigated. Because of the dipole moment of the organic molecule, the random orientation creates a novel system with long-range potential fluctuations unlike alloys or other conventional disordered systems. Using linear scaling ab initio methods, we find that the charge densities of the conduction band minimum and the valence band maximum are localized in nanoscales due to the potential fluctuations. The charge localization causes electron−hole separation and reduces carrier recombination rates, which may contribute to the long carrier lifetime observed in experiments. KEYWORDS: Perovskite solar cells, charge localization, electron−hole separation, linear scaling ab initio calculations CH3NH3PbI3 was synthesized in 1978 at high temperatures.1 In 2009, it was employed in dye-sensitized solar cells as visible light sensitizers to mesoporous TiO2, yielding a solar energy conversion efficiency of 3.8%.2 In recent few years, the hybrid perovskites have emerged as a new class of light absorbers, and the field of perovskite-based solar cells has been rapidly growing.3−8 It was shown that, by adding a hole conducting layer spiro-MeOTAD, the solid-state dye-sensitized perovskite solar cell efficiency can exceed 9.7%.9 Further studies found that the mesoporous TiO2 may not be essential. By replacing TiO2 with Al2O3, the energy conversion efficiency can reach 10.9%.10 In this “meso-superstructured solar cell”, the hybrid perovskite behaves as both a light absorber and an electron conductor. Besides these nanostructured dye-sensitized solar cells, a simple thin film perovskite layer sandwiched within a planar heterojunction also works extremely well, with an energy conversion efficiency of over 15%.11−14 In these thin film solar cells, the perovskite thin film performs the tasks of charge separation and charge transports of both electrons and holes. Careful interface and grain boundary engineering can further improve the efficiency.15,16 Recently, a hole-conductor-free mesoscopic perovskite solar cell has been demonstrated as stable for >1000 h in ambient air under full sunlight, with an energy conversion efficiency of 12.8%.17 Despite the rapid progress in device performance, the reason for the excellent performance is still an open question, and many fundamental questions of this material remain to be answered. © XXXX American Chemical Society

There are many theoretical studies on the structural and electronic properties of this material, mostly using a small periodic unit cell. It was found that CH3NH3PbI3 has a direct band gap of ∼1.6 eV.18 The conduction band minimum (CBM) consists of Pb p-state, and the valence band maximum (VBM) consists of Pb s-state and I p-state.18−20 The effective masses of free electrons and holes are 0.23 and 0.29, respectively,21 which are comparable to the values of silicon. Defect calculations show that both the shallow donors and shallow acceptors have low formation energies, which indicates flexibility in both n-type and p-type doping.22 The material also has a large absorption coefficient,23 due to the p−p transition and large density of states near the band edges.24 Experiments show that, although it is unstable in the presence of moisture, it may be stable in dry environment and that Al2O3 could successfully protect it from moisture.25 Both the electron and hole diffusion lengths are ∼1 μm in both thin film and mesostructured cells, and the minority carrier lifetime is ∼100 ns.26−28 Nevertheless, the origin of such long diffusion lengths and lifetimes has not been well understood. Strictly speaking, CH3NH3PbI3 is not a typical crystal. At the room temperature, it is in a tetragonal phase. Although the inorganic PbI3 framework is distorted from the perfect perovskite structure, it is nevertheless ordered and periodic; Received: September 11, 2014 Revised: December 8, 2014

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but the organic molecules CH3NH3 are randomly distributed along eight ⟨111⟩ orientations as observed by experiments.29−31 This random feature can cause some interesting phenomenon: on one hand, because the CBM and VBM are both composed of the atomic orbitals from the PbI3 framework with little contributions from the organic molecules, the random orientations of the molecules may not affect the CBM or VBM directly; on the other hand, because the organic molecule has an electric dipole moment, it induces an electric field that can affect the electrostatic potential and thus the related electronic structures. However, previous theoretical studies only considered ordered molecular orientations or ferroelectric multidomains,18−22,24,32 which are not the experimentally observed systems. The effect of the random molecular orientations on the electronic structures has not been addressed before and will be the topic of this Letter. Compared to other conventional random/disordered systems, e.g., alloy, amorphous, or defect systems, the current system is unique. The dipole induced electrostatic potential has a 1/R2 long-range decay, which can cause large and long-range-correlated fluctuations compared to conventional alloys. Using the ab initio linear scaling three-dimensional fragment (LS3DF) method, we have calculated the electronic structures of the tetragonal phase of CH3NH3PbI3 in large supercells (up to ∼20 000 atoms). All the organic molecules are randomly orientated along eight ⟨111⟩ equivalent directions. We find that (i) both the CBM and VBM charge densities are localized in nanoscale regions with very small overlaps between them. Because the localization size is several nanometers, the localization effect cannot be observed in small supercell calculations. (ii) The charge localization is caused by the potential fluctuation, which is induced by the random orientations of the molecular dipoles. A simple model is developed to represent the potential fluctuation. (iii) The screening effects from the molecular and ionic contributions only slightly reduce the potential fluctuation and increase the charge localization size. (iv) The excitons will be dissociated into free electrons and holes due to the charge separation. The charge separation can significantly reduce the electron−hole recombination rates, which may contribute to the long minority carrier lifetime and diffusion length observed in experiments. Charge Localization. The calculated charge densities of the CBM and VBM states in three different supercells are displayed in Figure 1. The isosurface contains ∼90% of the charge. In the top panel of Figure 1, the supercell is 25.6 × 25.6 × 25.6 Å3 in size and contains in total 768 atoms, which includes 64 randomly orientated organic molecules. Both the CBM and VBM charge densities are rather delocalized over the whole supercell. However, by increasing the isosurface value, we can still observe some charge localization (Supporting Information Figure S1). The LS3DF calculated results are almost identical to the direct density functional theory (DFT) calculations (Supporting Information). Apparently in such a small supercell, the charge localization is rather weak. In the middle panel of Figure 1, the supercell is 51.2 × 51.2 × 51.2 Å3 in size and contains in total 6144 atoms, which includes 512 randomly orientated organic molecules. In this supercell, the charge densities have some localization effects, especially for the CBM. In the bottom panel of Figure 1, the supercell is 76.8 × 76.8 × 76.8 Å3 in size and contains in total 20 736 atoms, which includes 1728 randomly orientated organic molecules. In this large supercell, the CBM and VBM charge densities are strongly localized in different regions. The size of the

Figure 1. Charge densities of the CBM and VBM states in the 768atom supercell (top panel), 6144-atom supercell (middle panel), and 20 736-atom supercell (bottom panel). The organic molecules are randomly orientated along the eight ⟨111⟩ directions. The isosurfaces contain ∼90% charge. Both the CBM and VBM charge densities are strongly localized in the large supercell.

localization can be judged by [1/∫ ρi2(r)d3r]1/3, with ρi the charge density of the ith state. They are 36 and 61 Å for the CBM and VBM, respectively. Note the stronger localization of the CBM than VBM in Figure 1 could be accidental for this particular configuration. For other configurations the CBM and VBM localization sizes could be similar. Because the localization size is larger than the sizes of the small supercells, the localization effect cannot be estimated adequately in the small supercell calculations. The charge separation between the CBM and VBM can be quantified by the overlap integral defined as

∫ |ψCBM(r)||ψVBM(r)|d3r

(1)

Here ψi is the wave function of the ith state, which satisfies ∫ |ψi|2d3r = 1. Apparently, if the CBM and VBM charge densities are completely localized in separated regions, the overlap integral will be zero. The calculated overlap integrals are displayed in Figure 2. In the 768-atom supercell where the CBM and VBM charge densities are delocalized, the overlap integral is ∼0.52, while in the 20 736-atom supercell where the charge densities are localized, the overlap integral is ∼0.06. As B

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region (blue), while the VBM is localized at the high potential region (red). Figure 4 shows the local density of states (LDOS)

Figure 2. Overlap integral between the CBM and VBM states. The xaxis indicates the three supercells. The overlap tends to zero, which indicates the charge separation.

the supercell size increases, the overlap integral decreases, which indicates stronger charge separations. We expect the overlap integral approaches zero when the supercell size continues to increase. The charge localization is caused by the potential fluctuation in the material. The local electrostatic potential on the Pb sites (as both the CBM and VBM are composed of the Pb orbitals) in the 20 736-atom supercell is displayed in Figure 3. The blue isosurface indicates the low electrostatic potential region, and the red isosurface indicates the high electrostatic potential region. The potential difference between the two regions is greater than 0.5 eV. Compared to Figure 1 (bottom panel), it can be seen that the CBM is localized at the low potential

Figure 4. Local density of states in the high potential region (red color in Figure 3) and in the low potential region (blue color in Figure 3). The LDOS shifts due to the electrostatic potential difference.

in the high potential region and low potential region, respectively, calculated by the generalized moments method33 based on the potential from the LS3DF calculation. Because of the potential energy difference, the LDOS in the low potential region shifts downward in energy compared to the LDOS in the high potential region. The overall shift is about 0.3 eV, slightly smaller than the potential difference due to the kinetic energy confinements. Next, we discuss the origin of the potential fluctuation. Because of the asymmetry, the CH3NH3 molecule has a small dipole moment. If all the molecules are randomly orientated, the dipole moments can induce a fluctuation of the electrostatic potential. We can calculate the electrostatic potential induced by the random orientated molecules using a simple dipole moment model (see the computational methods section). We calculate the electrostatic potentials on every Pb site using this dipole moment model and compare the results with the local potentials calculated by LS3DF. The comparison is shown in Figure 5. It is clear that there is an approximate linear relationship between the potentials calculated by the model and by LS3DF, proving that the potential fluctuation is mainly caused by the random orientations of the dipole moments of the organic molecules. Screening Effects. In reality, there are three screening mechanisms in this material: electronic screening, organic molecule configurational screening, and inorganic framework ionic relaxation screening. The self-consistent-field LS3DF calculation already captures the electronic screening as described by the ratio between the potentials calculated by the dipole model and LS3DF. This ratio εeff,∞ ≈ 4 (Figure 5) is an effective electronic screening, which is slightly smaller than the DFT calculated bulk ε∞ ≈ 6 of this material due to the finite size effect (finite reciprocal-vector q effect) of the potential fluctuations. In the following, we consider the other two screening effects. Because of the fluctuation of the electrostatic potential, there exists a local electric field at every organic molecule site. This can cause the molecule to align against the electric field to

Figure 3. Electrostatic potential at the Pb sites in the 20 736-atom cell. The red and blue colors represent the isosurfaces of 0.066 and 0.046 Hartree, respectively. Comparing with the bottom panel of Figure 1, it is clear that the CBM charge density is localized in the low potential region (blue), and the VBM charge density is localized in the high potential region (red). C

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disordered structure (Figure 4). The CBM−VBM band gap of this random structure is ∼0.17 eV lower than that of the ordered structure. The optical transition rate of the VBM state to the conduction band states is shown in the Supporting Information Figure S2. The potential fluctuation also induces a long-range electric field, which can cause ionic relaxations to screen this electric field. This is the ionic contribution to the dielectric constant ε0. However, this screening may sensitively depend on the length of the spatial variation of the electric field (reciprocal vector q dependence). Because the charge densities are localized in nanoscales, instead of using the macroscopic dielectric constant, we must consider the microscopic ionic screening effect at the wavevector q ≈ 2π/L, where L is the approximate length of the CBM/VBM charge density localization (see the computational methods section). We find that the atomic relaxation only slightly reduces the total potential by a factor of ∼1.2, which is smaller than the calculated macroscopic ionic screening20 due to the strong q dependence. This indicates that the ionic screening on the nanoscale potential fluctuation is very small, and it should only slightly increase the CBM and VBM localization sizes. Implications. Our results can have significant implications on the exciton dissociation and carrier dynamics in this material. In solar cells, the electron and hole generated by the photon can form exciton. The exciton binding energy can be estimated using the hydrogenic model

Figure 5. Comparison between the dipole moment model and the LS3DF calculation. The x-axis is the electrostatic potential on the Pb site calculated by LS3DF, and the y-axis is the electrostatic potential on the same site calculated by the dipole moment model. All the points approximately fall on a line, which indicates the linear relationship between the potentials. The slope of the line is ∼4.

lower the total energy, which induces an effective molecule configurational screening (dipole moment self-screening), and as a result, the molecule orientations may not be completely random. To address this issue, we employ the Monte Carlo simulation (see the computational methods section) in the 20 736-atom supercell, based on the dipole moment model. The mean-square-root potential fluctuation is 0.13 eV for the completely random configuration and 0.09 eV for the Monte Carlo simulated configurations at the room temperature. A snapshot of the atomic configuration from the Monte Carlo simulation is used to redo the LS3DF calculation, with the resulting CBM and VBM charge densities shown in Figure 6. It

Eb =

e2 ε and a* = a 2εa* m*

(2)

where m* = m*e m*h /(m*e + m*h ) is the electron−hole reduced effective mass, a (a*) is the Bohr radius for hydrogen (exciton), and ε is the dielectric constant. Using m* = 0.128 and ε = 6, we have a* = 25 Å and Eb = 48 meV.34,35 In order to form an exciton, the electron and hole must locate at the same region. As a result, they cannot take the advantage of the electron−hole separation at the low potential and high potential regions. The LDOS shift between the low potential and high potential regions in the equilibrium structure is about 0.22 eV, as discussed above. After considering the ionic relaxation screening effect, the LDOS shift can decrease to 0.22/1.2 = 0.18 eV, which is still much larger than the exciton binding energy. We expect that the exciton will be dissociated into separated electron and hole. Note the thermodynamic energy kBT and possible band bending in the bulk can also dissociate the exciton, but the electrostatic potential fluctuation revealed in the current study can accelerate such dissociation by providing a microscopic electron−hole separating electric field. If the exciton is intact, the electron and hole recombination lifetime (at the room temperature, thus ignoring the triplet dark exciton) can be calculated as36 1 4 αωn = |⟨e|p|h⟩|2 (3) τ 3 m2c 2

Figure 6. Charge densities of the CBM and VBM states of an equilibrium structure, which is taken from a snapshot of the Monte Carlo simulation at T = 300 K in the 20 736-atom supercell. Both the CBM and VBM are still strongly localized and separated, although the localization sizes increase.

where n = 2.4 is the refractive index,37 ω is the photon frequency, α is the fine-structure constant, and ⟨e|p|h⟩ is the momentum matrix element between the electron and hole states, which is calculated in the small 96-atom cell where the electron and hole stay in the same region. The calculated lifetime τ = 3 ns is much shorter than the experimentally observed lifetime of ∼100 ns,26−28 which confirms that the exciton must be dissociated. The CBM and VBM separation can reduce radiative recombinations between electrons and

is observed that the charge localization is slightly weaker than the one in Figure 1, but nevertheless, the CBM and VBM localization and separation still exist, with an overlap of ∼0.11, and the calculated localization sizes [1/∫ ρi2(r)d3r]1/3 are 83 and 56 Å for the CBM and VBM, respectively. The LDOS shift between the low potential and high potential regions in this structure decreases to 0.22 eV compared to 0.3 eV in the fully D

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localized in low and high electrostatic potential regions, respectively, which causes the charge separation. The charge localization and separation can dissociate excitons and reduce all possible carrier recombination rates, which thus contribute to the experimentally observed long carrier lifetime. A simple dipole moment model is shown to describe the potential fluctuations adequately. We find that the organic molecule configurational screening can reduce the potential fluctuation by a factor of ∼1.4 at the room temperature, while the ionic relaxation contributes to a possible screening by a factor of ∼1.2. This random orientation of the dipole moment creates an interesting system with a variational potential landscape. The itinerary state (should it exist) scattering rate becomes infinity at long length scale, which proves from another aspect that the wave functions must be localized. Our study sheds some light on the electronic structure and carrier dynamics of this currently intensely studied important material. Computational Methods. We employ the linear-scaling three-dimensional fragment (LS3DF) method.40 The LS3DF method divides the system into fragments, calculates each fragment separately, and patches them into the original system with a novel boundary cancellation technique. The Coulomb potential, based on the global charge density, is solved on the whole system, so it includes all the long-range self-consistent effects. It yields almost the same results as direct DFT calculations as shown in the Supporting Information. The CBM and VBM states are calculated by the folded spectrum method41 based on the potential obtained from the LS3DF calculations. Plane-wave norm-conserving pseudopotentials are used with an energy cutoff of 64 Ry. PBE exchange-correlation functional is used. To construct the fragments for LS3DF calculations, we rotate the tetragonal structure by 45° along the z-axis. Therefore, the smallest unit cell in our LS3DF calculation, which is 12.8 × 12.8 × 12.8 Å3 in size and contains 96 atoms, is a √2 × √2 × 1 primitive cell of the tetragonal structure. Through structural relaxations in small supercells, we find that the inorganic PbI3 framework only has a small dependence on the nearby CH3NH3 molecule orientations. Thus, in this work, we keep a rigid PbI3 framework in its tetragonal structure, which is calculated by ab initio atomic relaxation in the 96-atom unit cell, and place CH3NH3 randomly along the eight ⟨111⟩ orientations when constructing the atomic structures of the large supercells. From small supercell (384-atom) calculations, we find that further atomic relaxations do not change the charge densities significantly. To further consider the microscopic screening effect of ionic relaxations, we have built a 76.8 × 12.8 × 12.8 Å3 long supercell and applied a sine-like external potential along the x direction. The atomic relaxation is carried out using the ab initio code PEtot,42 which is the kernel of the LS3DF calculations. We calculate the response of the total potential to the external potential with/without atomic relaxations. The calculation shows further relaxations can only slightly reduce the potential fluctuations and thus cannot change our conclusions in large supercell calculations. In the dipole moment model, we ignore the periodic PbI3 inorganic framework and regard the molecule as a point dipole moment along the molecular orientation. The electrostatic potentials are calculated by an Ewald summation algorithm. In the Monte Carlo simulation, for a given molecular configuration, we use the above dipole model to calculate the total electrostatic energy (with the εeff,∞ screening included). We then randomly flip the orientations of ∼2% molecules to

holes (i.e., majority and minority carrier recombinations). Potentially, it can also reduce the nonradiative recombinations induced by trap states or deep levels. Imaging an electron falls into a midgap trap level in a low potential region; as there is no hole in that region, the filled midgap state cannot be released by recombining with a hole. As a result, the midgap state will no longer be active for further electron trapping. The charge localization and separation thus play significant roles in the carrier dynamics and lifetimes.38,39 The carrier transport in this system will be very interesting, as one knows the localized state transport and the extended Blö ch wave function transport can have very different properties. While a detailed transport analysis is outside the scope of the current study, here we will discuss this problem from one particular aspect. It is well established that a disordered system such as an alloy can exert a carrier scattering rate to an itinerary state and hence cause a finite carrier mobility. Since our system is a disordered system, one natural question is whether we can apply the same technique to calculate the itinerary state scattering rate. In the alloy treatment, the wave function scattering is caused by the coupling between the extended virtual-crystal Blöch wave ⃗ functions ψ(k)⃗ and ψ(k′): M(k ⃗ , k ′⃗ ) = ⟨ψ (k ⃗)|V |ψ (k ′⃗ )⟩

(4)

where V = Vtot − Vave is the perturbation potential. Here Vtot is the total potential of the system and Vave is the virtual-crystal potential, which is the averaged primitive cell potential of Vtot. ⃗ ′) ⃗ becomes In alloy systems, for small k⃗ and k′,⃗ the M(k,k independent of k⃗ and k′⃗ due to the lack of correlations of V at different sites. This will result in a finite scattering rate. For this perovskite system, V has a long-range correlation. As derived in the Supporting Information, for small k⃗ and k′,⃗ we have 16π 2 |p|2 1 |M(k ⃗ , k ′⃗ )|2 = 2 2 ⃗ 3Nε Ω |k − k ′|⃗ 2

(5)

Here |p| is the amplitude of the dipole moment, ε is the dielectric constant, N is the total number of the primitive cell, ⃗ ′)| ⃗ 2 and Ω is the volume of the primitive cell. When this |M(k,k is used in the Fermi Golden rule to calculate the scattering rate ⃗ 1/τ(k)⃗ for the wave function ψ(k): 1 τ (k ⃗ )

=

2π ℏ

∫ |M(k ⃗ , k ′⃗ )|2 δ[E(k ⃗) − E(k ′⃗ )] (2NπΩ)3 d3k ′⃗ (6)

we will have a divergent integral for k′⃗ around k,⃗ which means an extremely small lifetime. One can consider this as an indication that the whole extended itinerary wave function picture is not correct, or say, an extended wave function will be immediately localized by the scattering. The fact that the coupling between k⃗ and k′⃗ getting stronger when both the k⃗ and k′⃗ are small (eq 5) indicates a potential landscape with large potential variations at long length scales (|k|⃗ ≈ 2π/L with L the length scale). Such a potential landscape naturally leads to the wave function localization and electron−hole separation at large distances. In summary, we have studied the carrier localization effect due to the random orientation of the organic molecule CH3 NH 3 in CH3NH3 PbI3 . We find that this random orientation causes an electrostatic potential fluctuation, which is large enough to localize the CBM and VBM states within sizes of a few nanometers. The CBM and VBM states are E

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get a trial configuration and use Monte Carlo algorithm to accept or reject the trial configuration. In this way, the thermal equilibrium molecular configurations at a temperature can be obtained. The potential fluctuation is calculated as (⟨V2⟩ − ⟨V⟩2)1/2, where “⟨⟩” indicates the average over all Pb sites.

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ASSOCIATED CONTENT

S Supporting Information *

The comparison between the LS3DF calculation and the direct DFT calculation, the optical transition rate of the VBM state to the conduction band states, the charge densities of another atomic configuration from the Monte Carlo simulation, and the derivation of the coupling constant of scattering. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work performed by the Joint Center for Artificial Photosynthesis, a DOE Energy Innovation Hub, supported through the Office of Science of the U.S. Department of Energy under Award No. DE-SC0004993. Computations are performed using resources of the National Energy Research Scientific Computing Center (NERSC) at the LBNL and Oak Ridge Leadership Computing Facility (OLCF) at the ORNL that are supported by the Office of Science of the U.S. Department of Energy under Contracts No. DE-AC0205CH11231 and No. DE-AC05-00OR22725, respectively. The computational time at OLCF is allocated by Innovative and Novel Computational Impact on Theory and Experiment project.

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REFERENCES

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