Electron-Phonon Coupling and Polaron Mobility in Hybrid Perovskites

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Electron-Phonon Coupling and Polaron Mobility in Hybrid Perovskites From First-Principles Carlo Motta, and Stefano Sanvito J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b10163 • Publication Date (Web): 27 Dec 2017 Downloaded from http://pubs.acs.org on December 27, 2017

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The Journal of Physical Chemistry

Electron-Phonon Coupling and Polaron Mobility in Hybrid Perovskites from First-Principles Carlo Motta∗ and Stefano Sanvito School of Physics, AMBER and CRANN Institute, Trinity College, Dublin 2, Ireland E-mail: [email protected] Abstract Density functional theory electronic structures, maximally localized Wannier funcitons and linear response theory are used to compute the electron and hole mobility, µ, of both inorganic, Cs-containing, and hybrid, CH3 NH3 -containing, lead bromide perovskites. When only phonon scattering is considered we find hole mobilities at room temperature in the 40-180 cm2 V−1 s−1 range, in good agreement with experimental data for highly-ordered crystals. The electron mobility is about an order of magnitude larger, because low-energy phonons are ineffective over the Pb 6p shell. Most importantly, our parameter-free approach, finds a T −3/2 power-law temperature dependence of µ, which is a strong indication of polaronic transport in these compounds. Our work then offers an independent theoretical validation of the many hypotheses about the polaronic nature of the charge carriers in lead halide perovskites.

December 23, 2017

Introduction In the last few years there has been a growing effort aimed at understanding the working principles of hybrid perovskites, a novel set materials with exceptional photovoltaic prop1

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erties. 1,2 Leaving aside extrinsic features such as the low cost and ease of fabrication, the key factors determining the success of these materials are threefold: 1) a useful bandgap in the visible range, 2) an unprecedented light absorption efficiency, and 3) a long photocarrier recombination time. The first two properties are a consequence of an electronic structure, which is comparable with that of the best performing semiconductors, such as Si or GaAs, with the advantage of a quasi-direct bandgap. 3–5 In fact, the presence of an heavy element such as Pb, whose strong spin-orbit coupling reduces the bangap, in addition to a large cation providing an ideal scaffold for optimal bond lengths in the inorganic framework, 6 set the bandgap just to the right value. As long as the third effect is concerned, a long recombination time is an essential quality for a solar energy harvesting material, since it allows the photoexcited electrons (e) and holes (h) to migrate to their respective electrodes before recombining. While generally this property is associated to very high mobilities, µ, as in the case of Si (µ >1000 cm2 V−1 s−1 ) and GaAs (µ >8000 cm2 V−1 s−1 ), the same does not hold true for hybrid perovskite, where the mobilities assume much more modest values in the range 1-10 cm2 V−1 s−1 and 10-100 cm2 V−1 s−1 for poly- and mono-crystalline films, respectively. 7–14 Such mobility range agrees well with estimates based on solving the Boltzmann equation in the constant relaxation time approximation. 15 The coexistence of long recombination times with modest mobilities is indeed peculiar of this materials class and it is still subject of intense investigation. As it was recently argued, 6 the low mobilities, µ = (τ e/m∗ ), and the small effective masses, m∗ , characterizing hybrid perovskites can be explained with a short scattering lifetime, τ (e is the electron charge). However, the recombination lifetime is found to be exceptionally long, 7,8 which explains why the diffusion lenght is several orders of magnitude larger than the absorption depth, resulting in efficient carrier collection. In addition, the inverse power dependence of the mobility on temperature is a strong indication for negligible impurity scattering. 16 These evidences suggest that the mechanism governing the recombination is unlikely to be impurity-driven, an argument also supported by the absence of abundant deep trap defects

2


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in these materials. 6 In light of the mechanical softness of hybrid perovskites, it was recently proposed that the main source of scattering could be the interaction with phonons. 6,16 In this respect, hybrid perovskite is similar to typical III-V and IV semiconductors. Furthermore, a T

−3/2

temperature dependence of the mobility, which is characteristic of acoustic phonon

scattering, has been observed in hybrid perovskite. 9,17,18 However, recent experimental and theoretical results suggest that the electron-phonon interaction is dominated by longitudinal optical (LO) phonons via the Froelich mechanism. 19 Hence, the question arises on how to reconcile the T

−3/2

temperature dependence of the mobility typical of interaction with

acoustic phonons with the dominant scattering character which occurs by optical ones. The strong dynamic disorder typical of hybrid perovskite could generate the formation of polarons, which could explain its limited mobility as well as its long recombination lifetime. Indeed, polarons often display modest mobilities, and their lifetime is enhanced due to the phonon cloud screening the electronic charge. As pointed out recently, although the T

−3/2

temperature dependence is generally expected for acoustic phonon scattering, the converse may not necessary also hold, 19 and the exact temperature dependence is the result of complex scattering mechanisms that have to be analyzed case by case and are not derivable from general considerations. Hence, it would be desirable to show that polarons originated from optical modes would generate a mobility with a T

−3/2

behavior. This should be done by

employing a model that does not make any a priori assumption about the temperature dependence. Motivated by the need of developing theoretical models to investigate the microscopic nature of charge transport in organics perovskites and so to resolve some of the issues hereby discussed, 16 in this work we show a first-principle calculation of the polaron mobility of APbBr3 , with A being either Cs and CH3 NH3 . We employ a methodology, previously developed for organic crystals, 20 which allows us to directly compute the mode-dependent hole- and electron-phonon couplings. This offers an ab initio perspective on how individual phonons affect positive and negative charge carriers. Our method is based on constructing

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the maximally localized Wannier functions (MLWFs) from density functional theory band structure, 21 so to define an effective tight-binding Hamiltonian, which is then subsequently used to compute the mobility via the Kubo formula. Importantly, it does not make any assumption neither about the polaron character of the mobility (small or large polarons) nor about the dependence with the temperature.

Methods We now briefly recall how the charge mobility is evaluated. The starting point is the following Hamiltonian describing the relevant electronic bands and their interaction with the phonons

H=

X mn

εmn a†m an

+

X λ

X 1 † † λ ~ωλ gmn bλ + bλ a†m an . + ~ωλ bλ bλ + 2 mnλ

(1)

Here, am (a†m ) and bλ (b†λ ) represent respectively the annihilation (creation) operators for an electron located at the site m and for the λ-th phonon mode of frequency ωλ (~ is the Heisenberg constant). The sum is carried out over the indices m, n, which label the basis orbital. In general, they are collective indices of the basis orbital (the MLWFs, in our case) and of the cell vector. However, in this study we consider a single MLWF per cell, therefore the index represents the cell vector. The three terms in the Hamiltonian describe respectively the electronic system, the phonon energy, and the electron-phonon interaction. λ The dimensionless quantity gmn is the electron-phonon coupling constant for the mode λ.

The Hamiltonian is written on the basis of the MLWFs constructed for the valence band (VB) in the case of hole mobility and for the conduction band (CB) when electron transport is considered. The electron-phonon couplings are computed by numerical differentiation as √ λ gmn = (ωλ ~ωλ )−1 ∂εmn /∂Qλ after computing the MLWFs of the crystal displaced along each phonon mode by a phonon normal coordinate Qλ . In this work, only phonons at the Brillouin zone center Γ are considered. This is expected to provide meaningful results for the materials class explored here. In fact, as discussed previously, in hybrid lead halide 4


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perovskites scattering from longitudinal optical phonons provides the dominant contribution to the electron-phonon coupling near room temperature, while scattering off acoustic phonons is negligible. 19 A Γ point sampling only includes optical modes which are less dispersive than the acoustic ones. A polaronic Hamiltonian can be obtained by applying a canonical transformation to the many-body Hamiltonian, H, and by thermal averaging over the resulting operators. This effectively describes the temperature-dependent bandwidth narrowing due to the polaron dressing of the effective mass. By using the linear-response Kubo formula the charge mobility tensor can then be expressed as Z +∞ X P e0 − λ 2Gλ [1+2Nλ −Φλ (t)]−Γ2 t2 (R − R )(R − R ) dte µαβ (T ) = αm αn βm βn 2kB T ~2 mn −∞ 1X λ × [(εmn − ∆mn )2 + (~ωλ gmn )2 Φλ (t)] , 2 λ λ where Rαm is the α−th coordinate of the m-th site, Gλ = (gmm )2 +

λ 2 k6=m (gmk )

P

(2)

is the effective

electron-phonon coupling constant, Nλ = (e~ωλ /kB T − 1)−1 is the phonon occupation number for the λ-th mode (kB is the Boltzmann constant), Φλ (t) = (1+Nλ )e−iωλ t +Nλ eiωλ t is the incoP P λ λ λ λ λ gkn )]. 22–24 herent scattering factor and finally ∆mn = λ ~ωλ [gmn (gmm + gnn ) + 21 k6=m,n (gmk

The phenomenological polaron lifetime Γ accounts for static disorder and is set to ~Γ = 0.1 meV, a value that is suitable for ultra-pure crystals. 22 The sum over the electronic orbitals m, n is performed on first-neighbors. The exponential factor in Eq. (2) describes contributions from both coherent and incoherent scattering processes, and therefore it does not make any a priori assumption about the hopping or the band-like nature of the charge transport. 22–24

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Computational Details Our calculations have been performed by density functional theory (DFT). The all-electron code fhi-aims 25 was used for structural relaxations, while Quantum Espresso was employed as the electronic structure interface to be used with the post-processing scheme for computing the Wannier functions. In more detail, the electronic and vibrational properties of APbBr3 have been calculated with fhi-aims, with the exchange-correlation energy being described by the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA). 26 Long-range van der Waals interactions have been taken into account with the Tkatchenko and Scheffler (TS) scheme. 27 The reciprocal space integration was performed over a (8×8×8) Monkhorst-Pack grid. 28 A pre-constructed high-accuracy all-electron basis set of numerical atomic orbitals was employed, as provided by the fhi-aims “tight” default option. Structure optimization was performed with the Broyden-Fletcher-Goldfarb-Shanno algorithm, with a tolerance of 10−3 eV/˚ A. The Phonopy 29 code was employed to compute phonons at the Γ point with the frozen phonon approach, setting a finite displacement of 0.001 ˚ A. The relaxed crystal structure was then input in the calculation of the MLWFs of the system. To this end, the electronic structures of all the geometries obtained in the previous step were calculated self-consistently with the plane-wave pseudopotential code Quantum Espresso. Kinetic energy cutoffs of 40 Ryd and 300 Ryd were used for the wavefunctions and the charge density, respectively. A dense (10×10×10) k−mesh was adopted. The MLWFs were obtained with Wannier90, 30 minimizing their spread until the fractional change was less than 10−12 .

Results and Discussion In this work we analyze lead bromide perovskites, APbBr3 , which have a cubic unit cell at room temperature. We compare the entirely inorganic compound, A=Cs, with the hybrid 6


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inorganic one, where A=CH3 NH3 is the methylammonium cation (MA). While the first has a centrosymmetric unit cell with an optimal lattice parameter of 5.80 ˚ A, the latter looses the cell symmetry due to the presence of a polar molecule. In particular the methylammonium cation points towards one of the unit cell faces 31 and induces a slight distortion, such that the relaxed lattice parameters are 6.00 ˚ A, 6.00 ˚ A and 6.16 ˚ A. In Figure 1 we show the electronic bandstructure of the inorganic CsPbBr3 compound (for a more detailed analysis of the electronic properties see Figures S1 and S2). The Wannier bands are plotted on top of the DFT ones. The almost perfect superposition of the two warrants the fact that MLWFs can be used to describe the conducting h and e’s, which after an ultrafast thermalization process lie at the band edges. As expected, the VB MLWF orbital is mostly localized on the halide Br, and it retains the p-character. On the contrary, the CB MLWF orbital originates from the p-orbital of Pb. The phonon density of states (DOS) computed at the primitive cell Γ point is depicted in Figure 2. Here, the colors encode the projection of the phonon eigenvector onto the two sublattices of the system, namely the PbBr3 octahedral cage and the A lattice. For the hybrid compound, MAPbBr3 , we notice three regions. All the vibrations below 50 cm−1 are related to rotations of the inorganic octahedra, those in the range 50-100 cm−1 involve both subsystems, namely the vibration of the PbBr3 network and the libration of the molecules, while for frequencies ν > 100 cm−1 there are only intra-molecular vibrations. The behavior of the Cs-containing compound is similar, although two main differences are notable. Firstly, the presence of Cs makes the low-lying part of the spectrum less broadened than in the case of MA. Secondly, owing to the larger volume occupied by Cs, the vibrations of a sublattice perturb the other, therefore the projected spectrum is more evenly distributed between the two parts, namely most of the modes are mixed in nature (a full list of the vibrations eigenvectors and frequencies can be found in Figures S3 and S4). The peak at 130 cm−1 involves an antiphase vibration of Pb and Br, and in the hybrid system it splits into two peaks at 72 and 77 cm−1 , owing to the strain induced by MA along one of the three lattice

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directions. λ In order to obtain the charge-phonon coupling coefficients gmn , we recompute the MLWFs

after distorting the crystal along each phonon mode. The effective couplings Gλ are depicted in Figure 3 as vertical bars over the phonon DOS. This provides a clear picture of how each vibrational modes couples to the e’s and h’s. Both the hybrid and the inorganic perovskites offer a large number of low-energy modes with strong e/h-phonon coupling. In the hybrid case, the presence of a non-symmetric molecules splits the phonon degeneracies present in CsPbBr3 , resulting in a more broadened Gλ distribution. Overall, the Gλ intensities are comparable in the two cases. In the hybrid compound, a key aspect is that only those modes involving a considerable vibration of the inorganic cage have a strong coupling. Indeed, by comparing Figure 3 and Figure 2, it is clear that modes with a large projection on the organic cation have a negligible coupling, which virtually disappears in the range of the spectrum above 200 cm−1 , not shown here. As a curiosity related to the tight-binding nature of the model, in all cases the on-site energy of the MLWF Hamiltonian is the parameter that is mostly affected by the phonons. This means that the relevant phonon modes have a Holstein instead of a Peierls nature. Having derived from first principles the complete set of parameters needed by Eq. 1, we can now compute the charge carrier mobility through Eq. 2. The sum over the electronic orbitals m, n is performed over 10 elements, which are shown in Figure S6. All the other Hamiltonian parameters are smaller than these by at least one order of magnitude, and fall off the energy accuracy of the Wannier function construction procedure. Our results are shown in Figure 4. For each compound we calculate both the h and the e mobility µ along the three directions of the unit cell vectors. From the figure one can note a clear difference between the inorganic and hybrid perovskite, which is related to the presence of the molecule. Since CsPbBr3 is centrosymmetric, there is no variation in the mobility along the three directions in space. We point out anyway that the mobility curves display a slightly anisotropic character, resulting from numerical noise in the Hamiltonian parameters obtained

8


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after the Wannier function construction procedure. In contrast to the full-inorganic case, the strain induced by MA to the crystal introduces variations in the Pb-Br bond lengths, reflecting in direction-dependent mobilities. Note, however, that this result reflects the fact that our unit cell is the primitive one, so that configurations where different molecules point into different directions cannot be represented. In real samples the rotational timescale of the molecules is fast and at any given time one deals with crystals where the molecules have an average random orientation. This means that the actual mobility of MAPbBr3 is also isotropic and it will be an average of our calculated direction-dependent mobilities. Note also that, even though we compute the mobilities in a temperature range down to 10 K, our calculations are valid only up to about 250 K, below which the materials undergo a structural phase transition from the cubic phase studied here to a tetragonal one. 32 The calculated room temperature magnitudes of µ are summarized in Table 1 and match the experimental data for pure crystals, in particular in the case of h transport. We keep in mind that our calculations provide an upper estimate of µ, since we are solely considering the phonon contribution to the scattering. In general, our results suggest a trend where µe > µh . This could not be anticipated by simply looking at the e and h effective masses, since they are comparable. A possible reason is related to the spatial configuration of the MLWFs. While the e-MLWFs are localized around Pb, the h-MLWFs display a significant projection both on Pb and Br. By inspecting the phonon eigenvectors (included in the Figures S3 and S4), we notice that in all cases they introduce a substantial variation of the Pb-Br bond length. Hence, it is expected that the e’s are less perturbed by phonon disorder than h’s, resulting in less scattering and higher mobility. Nevertheless, we remind that the picture that we provide is that of a frozen atomic configuration. At room temperature, due to configurational disorder, the most realistic picture is given by a spatial average of the mobility over the three cartesian directions, which would return a less pronounced difference between h and e. We expect that this result remains valid also when considering spin-orbit coupling, which is neglected here due to technical reasons. In fact the spin-orbit interaction does not induce

9



any drastic change in the effective masses, 15 and it does not alter the orbital nature of either the conduction or the valence band. Moreover, spin-orbit coupling partially lifts the band degeneracies at the VB and CB edges, reducing the impact of interband scattering, which is neglected in the present single-band model. The most remarkable feature of our results is the T −3/2 power-law dependence of µ(T ) in the region at around 300 K. This is in full agreement with experiments, and provides a fully ab initio validation of the polaronic picture of charge transport in hybrid perovskites. We remind once again that our computational workflow does not make any assumption on the transport regime and on its T dependence, or take any parameters from experiments. Previously, the electron mobility has been studied by means of deformation potential theory (DPT). 33,34 Here we briefly point out the differences with the polaron method used in this work. The main assumption of DPT is that electron/hole scattering is only limited by scattering to long-wavelength acoustic phonons. As such, no information about the effect of all the crystal phonon modes is considered. Our work shows that a considerable electron-phonon interaction is present for many (low-energy) modes, which are included explicitly in the polaron model. Furthermore, deformation potential theory accounts only for a shift of the band edge, which results from the crystal deformation. On the contrary, by considering a Hamiltonian based on the crystal’s MLWFs, our polaron model is devoid of such a restriction and it includes the phonon perturbation of the entire valence/conduction bands. Our calculated mobility is within the same range of those obtained with DPT, 33 with the difference that the T −3/2 temperature dependence is an output of the calculation and not an assumption of the model. Once more, we stress the fact that our model is limited to the Γ point only.

10


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Conclusions In conclusion we have presented a fully ab initio analysis of the hole and electron mobility of inorganic and hybrid lead bromide perovskites. This is based on density functional theory electronic structures, maximally localized Wannier functions and a linear response evaluation of the temperature-dependent mobility. Our results return a picture where the room temperature hole mobilities are in the 40-180 cm2 V−1 s−1 range, which is well within the experimental window for high-purity crystals. The electron one is significantly higher mostly because the phonons are less active on the Pb 6p orbitals. Importantly, we find a clear T −3/2 temperature dependence, which points to the polaronic nature of the charge transport in these materials.

Figure 1: The electronic bandstructure of CsPbBr3 . The solid lines are the DFT calculated bands, while the circles are those obtained from the MLWFs. The insets show isosurfaces of the VB (lower picture) and CB (upper picture) Wannier functions.

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Phonon DOS

MA

0

50

100 ν (cm-1)

150

Cs

200 PbBr3

Phonon DOS

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A 0

50

100 ν (cm-1)

150

200

Figure 2: Phonon density of state at Γ for MAPbBr3 (top panel) and CsPbBr3 (bottom panel) with broadening of 0.1 eV. The color palette indicates the phonon projection over the PbBr3 cage (blue) and A-site cation MA or Cs (red). Note that we only show the range 0 < ν

Electron-Phonon Coupling and Polaron Mobility in Hybrid Perovskites

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