The Nature of Electron Mobility in Hybrid Perovskite CH3NH3PbI3

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The nature of electron mobility in hybrid perovskite CHNHPbI Jie Ma, and Lin-Wang Wang Nano Lett., Just Accepted Manuscript • Publication Date (Web): 18 May 2017 Downloaded from http://pubs.acs.org on May 18, 2017

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The nature of electron mobility in hybrid perovskite CH3NH3PbI3 Jie Ma†,‡ and Lin-Wang Wang∗,‡ †Beijing Institute of Technology, Beijing 100081, China ‡Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA E-mail: [email protected]

Abstract CH3 NH3 PbI3 is one of the most promising candidates for cheap and high-efficiency solar cells. One of its unique features is the long carrier diffusion length (> 100 µm), but its carrier mobility is rather modest. The nature of the mobility is unclear. Here, using non-adiabatic wave function dynamics simulations, we show that the random rotations of the CH3 NH3 cations play an important role in the carrier mobility. Our previous work showed that the electron and hole wave functions were localized and spatially separated due to the random orientations of the CH3 NH3 cations in the tetragonal phase. We find that the localized carriers are able to conduct random walks, due to the electrostatic potential fluctuation caused by the CH3 NH3 random rotations. The calculated electron mobilities are in the experimentally measured range. We thus conclude that the carrier mobility of CH3 NH3 PbI3 is likely driven by the dynamic disorder that causes the fluctuation of the electrostatic potential. KEYWORDS: electron mobility, random walk, time-dependent tight-binding simulations, perovskite solar cells

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Organic-inorganic hybrid perovskite CH3 NH3 PbI3 has emerged as a very promising solar cell material. 1 It has a low cost to synthesize, a strong light absorption, and a nearly ideal band gap (∼1.6 eV) for the solar cell application. 2,3 Ever since CH3 NH3 PbI3 was first used as light sensitizers for dye-sensitized solar cells in 2009, 4 the field of perovskite-based solar cells has exploded. 5–15 After a few years of its development, the power conversion efficiency of the perovskite solar cell has reached 22.1%. 16 There remain many mysteries about the underlying mechanisms of CH3 NH3 PbI3 . What makes this material unique and suitable for the solar cell application is its unusually long carrier diffusion length, e.g., over 1 µm in thin film or nanostructured systems, 17–19 and > 100 µm in single crystals. 20 However, the long carrier diffusion length does not imply a high carrier mobility of CH3 NH3 PbI3 . 21 The room-temperature carrier mobility of this material has been measured by different groups using different techniques. 17,18,20,22–30 Field-effect transistor measurement for dc conductivity showed that the electron and hole mobilities were about 1 cm2 V−1 s−1 , 23 so were the photoluminescence quenching experiments. 17,18 The Hall effect measurements gave mobilities ranging from 1 to 100 cm2 V−1 s−1 for polycrystalline and single-crystal samples. 20,22,26,31 The THz conductivity measurements gave mobilities from 10 to 35 cm2 V−1 s−1 , 24,25,28,29 and the microwave conductivity measurements gave mobilities from 6 to 60 cm2 V−1 s−1 . 27,30 Note, some of the reported mobilities above are actually the sum of both electron and hole mobilities. Overall, the measured mobility of CH3 NH3 PbI3 is rather modest, at lease one order of magnitude lower than that of an inorganic semiconductor with a similar carrier effective mass. 21 Several THz and microwave spectroscopy studies have showed that the mobility had a T−1.3 to T−1.6 temperature dependence, which was interpreted mainly as the acoustic phonon scattering. 27–30 However, theoretical calculations based on the deformation potential predicted that the carrier mobility of an itinerary wave function under the acoustic phonon scattering should be several thousand cm2 V−1 s−1 , 32–34 ∼2 orders of magnitude higher than the experimentally measured values. It indicates that the itinerary wave function scattered by phonons might not be the correct mobility mechanism


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for CH3 NH3 PbI3 . The mechanism of the modest mobility is still a puzzle. The long diffusion length of CH3 NH3 PbI3 is mainly the result of the long carrier lifetime. The carrier lifetime of CH3 NH3 PbI3 can be as long as several hundred nanosecond (even microsecond), 17,20,22 which indicates the lack of nonradiative recombination centers and trap states. The electron-hole recombination rates in this material defy the Langevin recombination limit by orders of magnitude. 25 The below Langevin-limit electron-hole recombination rate 25,27 and the temperature activation of such recombination 27 might indicate the localization and spatial separation of electrons and holes. Besides the carrier localization, the I-V curve of the solar cell shows significant hysteresis, and a ramping up period of the photovoltaic efficiency after the light is turned on. Ferroelectric domains have been invoked to explain the carrier localization/separation and the hysteresis. However, the existence of ferroelectricity in this material is highly debated. In Ref. 35, using piezoresponse force microscopy (PFM), Hermes et al. showed that the hybrid perovskite material was piezoelastic, and some polarization pattern can form in response to the internal stress of the material, but the ferroelectric switching experiments did not show a switchable out-ofplane ferroelectric polarization. In Ref. 36, Fan et al. also failed to find ferroelectricity in the perovskite thin film using polarization-electric field hysteresis measurement and PFM measurement. In an oxide perovskite, the ferroelectricity is often caused by a distortion of the metal-oxygen octahedron. Such distortion does not exist in CH3 NH3 PbI3 probably due to the random rotations of CH3 NH3 cations. 36 On the other hand, there is no energetic benefit for the CH3 NH3 cations to align along the same direction. The lack of ferroelectricity domain calls for different explanations for the electron-hole localization/separation and the hysteresis phenomena. For example, the ramping up of the photovoltaic efficiency could be an indication of the defect-state filling or shallow-state charge trapping. The hysteresis might indicate shallow-state charge trapping and de-trapping. They can also be the results of ion (e.g., I− ion) diffusions. The carrier localization and separation can be a consequence of the CH3 NH3 random orientations as shown in our previous work. 37


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CH3 NH3 PbI3 has three phases: below 160 K it is in the orthorhombic phase, from 160 K to 320 K it is in the tetragonal phase, and above 320 K it is in the cubic phase. 38,39 In the tetragonal phase, the inorganic PbI3 framework is periodic and ordered; the CH3 NH3 cation is randomly orientated, and can rotate around its C-N axis, as well as changing the orientation of the C-N bond. 38,40–42 Molecular dynamics simulations in a 384-atom supercell 40 showed some dynamic localizations of the conduction and valence states, likely associated with the dynamics of disordered CH3 NH3 cations. In our previous work, 37 we have shown the nanoscale electron and hole localizations and separations in a 20736-atom supercell, because the randomly orientated CH3 NH3 dipoles induce the electrostatic potential fluctuation. The amplitude of the potential fluctuation is above 100 meV, greater than the exciton binding energy in this material. The optical absorption from the valence band maximum to the conduction bands has a strong peak ∼0.1 eV above the conduction band minimum. 37 The electrons are excited from local valence states to corresponding local conduction states, which have large oscillator strengths. Then the excited electrons and holes can relax to other local states with lower energies (e.g., the conduction band minimum and valence band maximum states), causing the charge separation. The localized and spatially separated electron and hole wave functions can explain the below Langevin-limit recombination rate, as well as the high Auger recombination rate, 25,43 and thus are responsible for the long carrier lifetime. Localized states, such as small polarons, generally have rather low roomtemperature mobilities (< 1 cm2 V−1 s−1 ), whereas large polarons can have relatively large mobilities. 31 Can the localized states of CH3 NH3 PbI3 have mobilities in the experimental range of 1-60 cm2 V−1 s−1 ? Moreover, what is the right picture to describe the carrier mobility in this material: is it an itinerary wave function scattering, or localized state diffusion? These are the questions we like to address in the current work. Our localized-state picture 37 is different from localizations caused by ferroelectric domains or conventional localized states (e.g., electron-phonon coupling induced polaron states, or impurity induced localized bound states) due to its nature of dynamic disorder. 44 The


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eigen states of CH3 NH3 PbI3 are all localized in real-space due to the random orientations of CH3 NH3 cations, 37 but the cation orientations are randomly varying with time. 38 The randomly rotating CH3 NH3 cations could induce mobilities of localized carriers in two possible ways: first, as their orientations are varying, the electrostatic potential will change accordingly and the occupied adiabatic eigen wave function can randomly drift in real-space; second, the rotating cations can induce electron transitions from the initially occupied adiabatic eigen state to other adiabatic eigen states located at nearby positions. In this Letter, we will focus on the effect of the rotating CH3 NH3 cations on the mobility of the localized carrier of CH3 NH3 PbI3 in the tetragonal phase. Since we are studying the intrinsic mobility, which by definition is in the low electric field limit, possible nonlinear effects of the alignments of CH3 NH3 cations under strong electric fields are ignored in the current work. The evolution of the carrier wave function is solved by the time-dependent Schrödinger’s equation. We will use a tight-binding (TB) Hamiltonian that is fitted to the ab initio band structure of CH3 NH3 PbI3 . The time dependence of the TB Hamiltonian is caused by the random fluctuation of CH3 NH3 orientations: The orientational fluctuation is modelled by a kinetic Monte Carlo procedure, which causes the randomly fluctuating potential and changes the Hamiltonian with time. The diffusion of the carrier wave function from its initial location is used to calculate the diffusion coefficient and the carrier mobility. In this work, we will ignore possible vibrational effects of the inorganic PbI3 framework. The exact effects of the PbI3 framework are still unclear and are deserved to be studied in the future. In the current study, we find that the electron diffusion is not originated from the self-drifting of the localized wave function; instead, it is mainly due to the electron transitions between eigen states when their eigen energies cross each other. The calculated room-temperature electron mobility and its temperature dependence are both in good agreement with the experiments; on the other hand, the itinerary-state-scattering picture gives too high mobilities or wrong temperature dependence. 32–34 Our dynamic-disordered carrier-localization picture provides one plausible explanation to the long carrier lifetime, the low electron-hole recombination


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rates, and the modest carrier mobility of CH3 NH3 PbI3 . We first discuss one example of the time-dependent diffusions of the electron. The electron starts as the lowest adiabatic eigen state at t = 0 ps, with its wave function localized in realspace. Figure 1 (a) shows the diffusion distance square R2 (t) as a function of time (see Methods in Supporting Information). We can see two distinct diffusion periods: before 0.6 ps, the electron diffusion is rather slow, whereas it becomes fast after 0.6 ps. The charge density of the electron (defined as |ψ(t)|2 with ψ(t) being the time-dependent wave function of the electron) at 0.25 ps is shown in Fig. 1 (b). The electron almost locates in the same region as that of its initial state. Within the first 0.6 ps, the charge density of the electron is always similar to Fig. 1 (b), which shows the localized region of the electron does not change significantly during the first 0.6 ps (or say the electron is still localized in the original region). During this time period, both the center of mass and the expansion of the localized charge density change little, so the R2 (t) of the electron increases slowly. Supporting Information Fig. S1 shows the center coordinates of the localized charge during the first 0.6 ps. The overall drifts are rather small (5∼10 ˚ A). Similar drifts were found in previous molecular dynamics simulations. 40 Those drifts of the localized charge contribute only a tiny part to the overall carrier mobility. After 0.6 ps, the electron starts to tunnel from the originally localized region to other regions. The charge density of the electron at 0.7 ps is shown in Fig. 1 (d). Besides the originally localized region (circled by red), there is also charge density located in a new region (circled by blue). The charge density appearing in the new region is responsible for the fast diffusion. We also find that the centers of mass of the charge densities in both the originally localized region (circled by red) and the new localized region (circled by blue) do not change significantly with time, and the main changes are the numbers of electrons inside those regions. The number of electrons outside the originally localized region as a function of time is shown in Fig. 1 (c). Its trend is exactly the same as the trend of the R2 (t) shown in Fig. 1 (a). It demonstrates that the diffusion is mainly due to the electron tunneling from the originally localized region to new localized regions, rather


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than the self-drifting of the localized charge densities in those regions. Figure 2 shows the projections of the time-dependent wave function ψ(t) on adiabatic eigen states. We have calculated the summation of the projection squares, which is defined as ∑ 4 i | < ϕi (t)|ψ(t) > | (ϕi (t) is the adiabatic eigen state described in Methods in Supporting Information), as shown in Fig. 2 (a). This quantity is always ≤ 1; smaller the value, broader the electron distribution among adiabatic eigen states. We find that before 0.6 ps (the slowdiffusion period), the values are always close to one, which shows the electron mainly occupies one eigen state; however, after 0.6 ps (the fast-diffusion period), the values decay quickly, which shows the electron must have been scattered to several eigen states. The adiabatic eigen energies as a function of time are shown in Fig. 2 (b), with the sizes of the red dots representing the occupation numbers on the eigen states. As the time evolves, all the eigen energies are varying rapidly on a sub-ps time-scale, due to the fluctuation of the electrostatic potential induced by the randomly rotating CH3 NH3 dipoles. Initially (t = 0), the electron occupies the lowest eigen state. During the first 0.6 ps (the slow-diffusion period), the lowest eigen energy is far away from other eigen energies, and there is no electron transition to other eigen state. One example of the density of state (DOS) within this time period (at t = 0.25 ps) is shown in Fig. 2 (c) with the red color indicating the occupied DOS. It shows the electron only occupies the original eigen state. The localized wave function of the occupied eigen state does not significantly drift away in real-space with time, despite its eigen energy fluctuation. Because there is no electron transition to other eigen state during the first 0.6 ps, the charge is still localized in the original region (as shown in Fig. 1 (b)), and the electron diffusion is slow. Around 0.6 ps, the eigen energy of the occupied state is shifted up, coming closer to other eigen energies (Fig. 2 (b)). When the occupied eigen state crosses with other eigen states in energy, there exist electron transitions to those eigen states, following the Laudau-Zener tunneling. 45 After 0.6 ps (the fast-diffusion period), the electron occupies several eigen states. The dominant occupation is not on the lowest eigen state, which shows the system is no longer in the ground state as would be described under


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the Born-Oppenheimer approximation. The DOS (black curve) and the occupied DOS (red) at 0.7 ps are shown in Fig. 2 (d). The electron mainly occupies the 3rd eigen state and also has small occupations on the first two eigen states. The charge density of the 3rd eigen state is shown in Fig. 2 (e), which mainly locates in the same region (circled by red) as the initially lowest eigen state. It indicates the 3rd eigen state at 0.7 ps has similar components to the lowest eigen state before 0.6 ps; i.e., they are essentially the same eigen state before and after the level crossing, which the electron always mainly occupies. There are also some charge densities locating outside the red circle in Fig. 2 (e), which shows the originally occupied eigen state mixes with other eigen states during the level crossing around 0.6 ps (as described in the Laudau-Zener model 45 ). We would like to stress that the center of mass of the charge density in the originally localized region (circled by red) does not change significantly compared with that shown in Fig. 1 (b) and (d), which shows no self-drifting of the localized charge density in that region. The charge densities of the first two eigen states at 0.7 ps are shown in Fig. 2 (f), demonstrating that the localized charge density inside the blue circle of Fig. 1 (d) is consisted with those two eigen states. Because there are electron transitions to other eigen states after 0.6 ps, the charge appears in new regions (as shown in Fig. 1 (d)), which induces the fast diffusion. We may conclude that the charge density in the originally localized region (circled by red) in Fig. 1 (b) and (d) comes from the originally occupied adiabatic eigen state (the 1st eigen state before 0.6 ps and the 3rd at 0.7 ps), and the charge density in the new localized region (circled by blue) in Fig. 1 (d) mainly comes from the other eigen states and the charge crossing term from the product of two adiabatic eigen states. Thus, the electron diffusion is not originated from the self-drifting of the localized adiabatic eigen state; instead, it is mainly due to the level crossing and the electron transitions between eigen states. It is worth noting that the above discussions are based on one case as an example. Other cases show similar behaviors, although the exact time for the electron tunneling varies in a random fashion. The 0.6 ps shown in Fig. 1 and 2 is just one case. In Supporting Information


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Fig. S2, we show another example, in which the first tunneling happens at 0.05 ps and there are several subsequent tunnelings within 1 ps. Nevertheless, the overall picture is the same as the one discussed above. Please also note that in the above example we have taken the lowest adiabatic eigen state at t = 0 as the initial state. One may take a thermal equilibrium state, e.g., choosing it according to the Boltzmann distribution. We find that the initial choice will not change the picture or the final mobilities in any significant way. After establishing the diffusion mechanism of the localized electron, we now calculate its mobility. The mobility is determined by the CH3 NH3 rotation. To yield a quantitative result for the mobility, it is necessary to first determine the rotation rate of the CH3 NH3 orientation. 46 The dynamic nature of the CH3 NH3 cation was first investigated in 1987 by Poglitsch and Weber 39 using frequency-dependent dielectric constant, and it was found that at room temperature (300 K) the CH3 NH3 relaxation time was about 5.4 ps. Using quasielastic neutron scattering measurements combined with Monte Carlo simulations, Leguy et al. determined that the rotation around the C-N axis was at sub-ps time-scale, and the rotation of the C-N orientation needed 14 ps. 47 A more recent ultrafast 2D vibrational spectroscopy and ab initio molecular dynamics simulations by Bakulin et al. showed the C-N reorientation lifetime was < 3 ps 48 at room temperature. Such a time-scale (3-14 ps) is also supported by several other experimental measurements and theoretical simulations. 49–51 Thus, to get the room-temperature electron mobility, we need to establish the dependence of the diffusion coefficient on the CH3 NH3 rotation rate. To obtain the diffusion coefficient, we need the statistically averaged diffusion distance square < R2 (t) >, to sample possible electron tunnelings (fast diffusions). For a given rotation rate, we have performed 20 independent random time-dependent simulations to get their average < R2 (t) >. The simulated results at the rotation rate of ∼0.25 ps−1 (a lifetime of ∼4 ps) are shown in Fig. 3 (a). The calculated diffusion distance squares for the 20 time-evolutions of the electron are shown as the black curves, and we observe fast diffusions happening from 0 ps all the way to 1 ps (i.e., they are well sampled). The red bold curve is the


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average of those black curves, which is almost a straight line, indicating a random diffusion as described by < R2 (t) >= 6Dt (see Methods in Supporting Information). The calculated diffusion coefficient is D ≈ 0.2 cm2 s−1 . We have also calculated the diffusion coefficients at several other cation rotation rates (the results at the rotation rate of 0.04 ps−1 are shown in Supporting Information Fig. S3), and the calculated diffusion coefficients are shown in Fig. 3 (b). Roughly speaking, the diffusion coefficients are proportional to the rotation rate γ (inverse of the lifetime). In Fig. 3 (b), the dashed line is a linear fitting D = kγ, and the calculated values mostly fall on the line. Because in the literature, the room-temperature CH3 NH3 rotation lifetime is from 3 to 14 ps as discussed above, 46 the corresponding diffusion coefficient is in the range of 0.06-0.27 cm2 s−1 . According to the Einstein relation, the roomtemperature electron mobility µ = D/kB T (kB is the Boltzmann constant) is in the range of 2.3-10 cm2 V−1 s−1 , which falls within the experimentally measured range discussed above. We would like to emphasize that in many experimental works, the reported mobility is the sum of both electron and hole mobilities, so we must divide those experimental values by 2 (assuming equal electron and hole mobilities) when comparing with our theoretical results. To study the temperature dependence of the mobility, we need to know the dependence of the CH3 NH3 rotation rate on the temperature, which is usually described as γ ∝ exp(−Ea /kB T )

(1)

where Ea is the activation energy of the rotation. From Fig. 3 (b), we notice that the diffusion coefficient linearly depends on the rotation rate (D ∝ γ). Thus, the temperature dependence of the mobility (µ = D/kB T ) is µ∝

exp(−Ea /kB T ) kB T

(2)

However, the activation energy Ea has not been well established. It ranges from 10 to 100 meV both experimentally and theoretically. 39,47,50–55 10 ACS Paragon Plus Environment

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To estimate the temperature dependence, we have tested two extreme cases, with activation energies of 10 and 100 meV, respectively. Because the CH3 NH3 rotation lifetime also has some uncertainty (e.g., 3-14 ps at 300 K as discussed above), we assume the rotation lifetime is ∼3.7 ps at 320 K (∼47 C◦ ). Thus, the electron mobility at 320 K is ∼7.8 cm2 V−1 s−1 . Because it is a rather conservative rotation rate, the mobility may be a little underestimated. The temperature dependence of the mobility is shown in Fig. 4 (a). The red and blue symbols are the mobilities calculated from the simulated diffusion coefficients (black squares in Fig. 3 (b)), and the dashed lines are based on Eq. (2). For comparison, we have replotted the experimental mobilities reported by Milot et al. 28 (black squares in Fig. 4). Their results were based on the THz measurement that does not depend on the connection of CH3 NH3 PbI3 to electrodes (i.e., without the surface contact resistance problem), and that measures directly the intrinsic bulk mobility. They have also reported the temperature dependence of the mobility, which provides a valuable data set to compare with our calculations. In Ref. 28, the reported carrier mobilities were the sum of both electron and hole mobilities. To compare with our calculated electron mobilities, we divide their reported mobility values by 2 (assuming equal electron and hole mobilities) in Fig. 4. Because in the low-temperature orthorhombic phase the CH3 NH3 cation orientations are fixed, our dynamic-disordered picture cannot be applied; for the high-temperature cubic phase, the PbI3 framework is also disordered, 56,57 which is not considered in our model. Thus, we only focus on the tetragonal phase (160 K to 320 K). If the activation energy is 100 meV (red), the mobility decreases as the temperature decreases, and the mobility at 160 K will be reduced to ∼1.3 cm2 V−1 s−1 . The decrease of the mobility is due to the slowdown of the CH3 NH3 rotation. The decreased mobility, however, does not agree with the experimental trend. On the other hand, if the activation energy is 10 meV (blue), the mobility increases as the temperature decreases, and the mobility at 160 K will be increased to ∼12 cm2 V−1 s−1 . The increase is due to the 1/kB T factor when the diffusion coefficient is converted to the electron mobility. Note that, with other small activation energies, we will get the same trend. The calculated temperature


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dependence of the mobility agrees qualitatively with the experimentally determined trend. 28 We thus take it as an evidence for the activation energy to be small. It will be extremely interesting for future experiments to more accurately determine the CH3 NH3 rotation rate as a function of temperature. A major question regarding to the carrier mobility is to determine whether the mobility is due to localized state diffusion (as we study here) or itinerary state (running wave) scattering. The carrier mobility based on the itinerary-state-scattering has been calculated for CH3 NH3 PbI3 by other groups. 32 Two types of scatterings have been considered: the acoustic phonon scattering and the charged impurity scattering. The acoustic phonon scattering can be obtained by calculating the elastic constant, the deformation potential, and the carrier effective mass. Standard formula also exists for the charged impurity scattering. 32 In Fig. 4 (b) we have replotted the data from Ref. 32, comparing with the experimental results 28 and our calculated results (with Ea = 10 meV). If we only consider the acoustic phonon scattering, the mobility (red curve) is above 2000 cm2 V−1 s−1 , which is almost 2 orders of magnitude higher compared with the experimental values. Charged impurity scattering can reduce the mobility. However, to yield similar mobilities as measured in experiments, one would need an impurity density at least above 1019 cm−3 . Even though such a high impurity density could exist due to the low impurity formation energy, 58 the mobility should decrease as the temperature decreases, which disagrees with the experimental trend. One may say that the scattering induced by the random orientations of the CH3 NH3 dipoles should be included. However, as analyzed in our previous work, 37 those scattering matrix elements diverge, indicating the itinerary state must become localized. Thus, we conclude that the itinerary-state-scattering picture does not agree with the experimental results. On the other hand, our calculated mobilities of the localized electron (induced by the randomly rotating CH3 NH3 cations) are in good agreement with the experiments. The mobility mechanism of CH3 NH3 PbI3 is similar to what we found in the monolayer pentathiophene butyric acid organic molecule system, 59 where the random thermofluctua-


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tions of the organic molecules (the dynamic-disorder driving force) cause the carrier localization and mobility. In that study, 59 the ab initio Hamiltonian was used during the whole time-evolution process, and the main movement of a carrier also comes from the electron transition between spatially nearby eigen states, when they have a crossing in their eigen energies (i.e., a Landau-Zener tunneling 45 ). This is also similar to the picture underlying the Marcus charge transfer theory. 60 However, the reasons for the eigen energy fluctuations are different: in the pentathiophene butyric acid organic molecule system, it is due to the wiggling of the organic molecule chain and the resulting change in the coupling strength between molecules; in a typical Marcus theory scenery, it is due to the electron coupling to the local harmonic phonon vibrations; in the current CH3 NH3 PbI3 case, it is due to the electrostatic potential fluctuation induced by the random rotations of dipoles (which is more long-ranged). In a Marcus theory scenery, the localization is caused by other physical conditions, e.g., an impurity or a state in a molecule. The monolayer pentathiophene butyric acid organic molecule system and the current CH3 NH3 PbI3 system belong to the dynamicdisordered system, 44 where the localization is also caused by the same physical condition that drives the eigen energy fluctuations. The dynamic-disordered picture can also be applied to other hybrid perovskites that (1) have randomly orientated dipoles to induce charge localizations and (2) those dipoles are rotating rapidly to induce localized-carrier diffusions. This is the case for CH3 NH3 PbBr3 . In a recent experimental work, 61 charge transport measurements in a steady state have been performed in CH3 NH3 PbBr3 single crystal. The measured temperature dependence of the mobility can be well explained by our model Eq. (2) with Ea = 10 meV (Supporting Information Fig. S4), which indicates our mobility mechanism works for the tetragonal CH3 NH3 PbBr3 . In our model, the effects of PbI3 are ignored. As the randomly orientated dipoles induce the electrostatic potential fluctuation, the PbI3 may move accordingly to screen the potential fluctuation. The screening sensitively depends on the length-scale of the potential fluctua-


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tion. Although the macroscopic screening (length-scale tends to infinity) is strong, we found the screening on nanoscale (∼7.7 nm, similar to the charge localization size) is rather weak in our previous work. 37 The screening only slightly increases the charge localization size, without changing the overall picture. If we include the nanoscale ionic screening 37 in our model, the calculated diffusion coefficient increases only 0.02 cm2 s−1 at the CH3 NH3 rotation rate of 0.25 ps−1 . Furthermore, the 0.02 cm2 s−1 change is resulted from the static screening. In reality, the on-site potentials fluctuate on a sub-ps time-scale (Supporting Information Fig. S5). The corresponding dynamic screening effect should be even weaker, due to the large atomic masses of Pb and I. There is however another possible effect of the PbI3 framework: the thermal fluctuation induced electron-phonon coupling, which has been ignored in the current work. The exact role of such effect is still not clear. One needs other computational method to study such effect. Our work shows the random rotations of CH3 NH3 cations are sufficient to provide a carrier mobility in the range of experimentally observed values. However, it might not exclude the possible roles of the PbI3 framework. The roles of the inorganic and organic parts may be glimpsed by comparing the mobilities of CH3 NH3 -based organic-inorganic hybrid perovskites and Cs-based pure inorganic perovskites. Experimentally, Chung et al. reported the hole mobility of CsSnI3 to be ∼585 cm2 V−1 s−1 , 62 and Yettapu et al. reported the carrier mobility of CsPbBr3 to be ∼4500 cm2 V−1 s−1 , 63 which are 1∼2 orders of magnitude higher that mobilities of CH3 NH3 -based perovskites; however, Zhu et al. reported similar mobilities of CH3 NH3 PbBr3 and CsPbBr3 in the near surface region. 64 It will be extremely interesting to study the effects of the inorganic framework in the future. In summary, we have used non-adiabatic wave function dynamics simulations to study the electron mobility of CH3 NH3 PbI3 in the tetragonal phase. The random orientations of the CH3 NH3 cause the electron and hole localized and spatially separated in this material; the rapid rotations of the CH3 NH3 cause the electrostatic potential fluctuating with time, which drives the carrier movements. This is a dynamic-disorder-driven carrier-localization


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and diffusion picture. Within this picture, the carrier diffusion is mainly due to the electron transitions between adiabatic eigen states when their eigen energies cross each other; the real-space locations of the localized adiabatic eigen states do not drift significantly. The calculated amplitude of the electron mobility is in the same range of the experimental values at room temperature, and its temperature dependence has the same trend as that in the experiments if we assume a low activation energy for the CH3 NH3 rotation.

Supporting Information Available The self-drifting of the localized charge, other time-dependent diffusion results, the diffusions at a low CH3 NH3 rotation rate, the mobility in CH3 NH3 PbBr3 , the on-site potential fluctuations, and the details of Methods. This material is available free of charge via the Internet at http://pubs.acs.org/.

Notes The authors declare no competing financial interest.

Acknowledgement This work is supported by the U.S. Department of Energy, Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract No. DE-AC02-05CH11231, through the Material Theory program [KC2301] in Lawrence Berkeley National Laboratory. This work uses the resource of National Energy Research Scientific Computing center (NERSC).


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(51) Chen, T.; Foley, B. J.; Ipek, B.; Tyagi, M.; Copley, J. R. D.; Brown, C. M.; Choi, J. J.; Lee, S.-H. Phys. Chem. Chem. Phys. 2015, 17, 31278–31286. (52) Frost, J. M.; Butler, K. T.; Brivio, F.; Hendon, C. H.; van Schilfgaarde, M.; Walsh, A. Nano Lett. 2014, 14, 2584–2590. (53) Arrighi, V.; Higgins, J. S.; Brugess, A. N.; Howells, W. S. Macromolecules 1995, 28, 2745–2753. (54) Motta, C.; El-Mellouhi, F.; Kais, S.; Tabet, N.; Alharbi, F.; Sanvito, S. Nat. Commun. 2015, 6, 7026. (55) Carignano, M. A.; Kachmar, A.; Hutter, J. J. Phys. Chem. C 2015, 119, 8991–8997. (56) Mashiyama, H.; Kurihara, Y.; Azetsu, T. J. Kor. Phys. Soc. 1998, 32, S156–S158. (57) Worhatch, R. J.; Kim, H.; Swainson, I. P.; Yonkeu, A. L.; Billinge, S. J. L. Chem. Mater. 2008, 20, 1272–1277. (58) Walsh, A.; Scanlon, D. O.; Chen, S.; Gong, X. G.; Wei, S.-H. Angew. Chem. Int. Ed. 2015, 54, 1791–1794. (59) Ren, J.; Vukmirovic, N.; Wang, L.-W. Phys. Rev. B 2013, 87, 205117. (60) Marcus, R. A. Rev. Mod. Phys. 1993, 65, 599–610. (61) Yi, H. T.; Wu, X.; Zhu, X.; Podzorov, V. Adv. Mater. 2016, 28, 6509–6514. (62) Chung, I.; Song, J.-H.; Im, J.; Androulakis, J.; Malliakas, C. D.; Li, H.; Freeman, A. J.; Kenney, J. T.; Kanatzidis, M. G. J. Am. Chem. Soc. 2012, 134, 8579–8587. (63) Yettapu, G. R.; Talukdar, D.; Sarkar, S.; Swarnkar, A.; Nag, A.; Ghosh, P.; Mandal, P. Nano Lett. 2016, 16, 4838–4848. (64) Zhu, H.; Trinh, M. T.; Wang, J.; Fu, Y.; Joshi, P. P.; Miyata, K.; Jin, S.; Zhu, X.-Y. Adv. Mater. 2017, 29, 1603072. 20 ACS Paragon Plus Environment

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Figure 1: An electron diffusion in real-space. (a) The calculated R2 (t) as a function of time for one time-dependent simulation. The slope of R2 (t) determines the diffusion velocity. It shows a slow diffusion before 0.6 ps and a fast diffusion after 0.6 ps. The charge density of the electron at 0.25 and 0.7 ps are shown in (b) and (d), respectively. The charge density is given by |ψ(t)|2 , where ψ(t) is the time-dependent wave function of the electron. The length of the supercell is ∼310 ˚ A. At 0.25 ps, the charge density is still localized in the same region as that at t = 0; however, at 0.7 ps there are charges in both the original region (circled by red) and a new region (circled by blue), which shows the electron diffusion. The number of electrons in the new region as a function of time is plotted in (c), and its trend is similar to that of R2 (t) in (a). It shows that the increase of R2 (t) is mainly due to the electron tunneling from the original region to the new region, rather than the self-driftings of the localized charge densities in those regions.


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Figure 2: The projections of the time-dependent wave function on the adiabatic eigen states. (a) The sum of the projection squares is almost 1 before 0.6 ps and decays after 0.6 ps, which shows the electron is on one eigen state before 0.6 ps and is distributed on several eigen states after 0.6 ps. (b) The lowest 10 eigen energies as a function of time. The sizes of the red dots represent the electron occupations on the eigen states. (c) and (d) show the DOS (black) and occupied DOS (red) at 0.25 and 0.7 ps, respectively. The charge density of the 3rd eigen state at 0.7 ps is shown in (e), and that of the first two eigen states is shown in (f). The red and blue circles, which are exactly the same as those in Fig. 1 (d), show the original and new localized regions. The 3rd eigen state mainly locates in the original region, and the first two eigen states mainly locate in the new region of Fig. 1 (d).


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Figure 3: The diffusion coefficient of the electron. (a) The calculated R2 (t) of 20 independent time-dependent simulations at the CH3 NH3 rotation rate of ∼0.25 ps−1 are shown as black curves. The red bold one is the average of the 20 curves. The red curve is almost linear and the slope determines the diffusion coefficient. (b) We have also calculated the diffusion coefficients at other rotation rates, and the calculated diffusion coefficients (black squares) are almost a linear function of the rotation rate. The dashed line is a linear fitting of the calculated diffusion coefficients.


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Figure 4: The estimated electron mobility and temperature dependence. (a) Because the activation energy of the CH3 NH3 rotation is not well established, we calculate the temperature dependence of the mobility with two activation energies, respectively. The red and blue symbols are obtained from the calculated diffusion coefficients (black squares in Fig. 3 (b)), and the dashed lines are obtained using Eq. (2). Because the experimental values from Ref. 28 are the sum of both electron and hole mobilities, they are divided by 2 (black squares) to compare with our calculated electron mobilities. For the high activation energy (100 meV), the mobility decreases as the temperature decreases; for the low activation energy (10 meV), the mobility increases as the temperature decreases, which agrees with the experiment. (b) The solid curves are the mobilities calculated based on itinerary wave functions, which are replotted from Ref. 32. The red curve only includes the acoustic phonon scattering, and the black curves include both phonon and ion scatterings at three different impurity densities. All those curves do not agree with the experiment. The experimental data and our calculated ones with Ea = 10 meV in (a) are replotted for comparison.


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Graphical TOC Entry

diffusion


The Nature of Electron Mobility in Hybrid Perovskite CH3NH3PbI3

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