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1KAUST Solar Center, Division of Physical Science and Engineering, King AbdullahUniversity of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom ofSaudi Arabia. 2Center for Organic Photonics and Electronics, School of Chemistryand Biochemistry, Georgia Institute of Technology, Atlanta, GA 30332–0400, USA.*Corresponding author. Email: omar.abdelsaboor@kaust.edu.sa (O.F.M.); jean-luc.bredas@chemistry.gatech.edu (J.-L.B.)

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Molecular behavior of zero-dimensional perovskitesJun Yin,1 Partha Maity,1 Michele De Bastiani,1 Ibrahim Dursun,1 Osman M. Bakr,1

Jean-Luc Brédas,2* Omar F. Mohammed1*

Low-dimensional perovskites offer a rare opportunity to investigate lattice dynamics and charge carrier behavior inbulk quantum-confined solids, in addition to them being the leading materials in optoelectronic applications. In par-ticular, zero-dimensional (0D) inorganic perovskites of the Cs4PbX6 (X = Cl, Br, or I) kind have crystal structures withisolated lead halide octahedra [PbX6]

4− surrounded by Cs+ cations, allowing the 0D crystals to exhibit the intrinsicproperties of an individual octahedron.Usingboth experimental and theoretical approaches,we studied theelectronicand optical properties of the prototypical 0D perovskite Cs4PbBr6. Our results underline that this 0D perovskite be-haves akin to a molecule, demonstrating low electrical conductivity and mobility as well as large polaron bindingenergy. Density functional theory calculations and transient absorption measurements of Cs4PbBr6 perovskite filmsreveal the polaronband absorption and strongpolaron localization features of thematerial. A short polaron lifetimeof~2 ps is observed in femtosecond transient absorption experiments, which can be attributed to the fast lattice relaxa-tion of the octahedra and the weak interactions among them.

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INTRODUCTIONAll-inorganic metal-halide perovskites have attracted a great deal ofattention over the last few years owing to their tunable band gap (1, 2),high photoluminescence quantum yield (3), and narrow emission line-width, enabling promising optoelectronic applications, such as light-emitting diodes (4), low-threshold lasers (5, 6), and photodetectors(7). Among these perovskite materials, three-dimensional (3D) in-organic perovskites with the general formula APbX3 (A = Cs+ or Rb+

and X = Cl, Br, or I) have been extensively explored (8–13). Three-dimensional perovskites are composed of an extended network ofcorner-sharing [PbX6]

4− octahedra with A residing in the cavities ofthis network. However, under operating conditions, these 3D perov-skites suffer from phase transformation and instability, including sur-face hydration and ion migration; thus, their reduced-dimensionalitycounterparts are being increasingly investigated, especially for opto-electronic applications (14, 15). Under Cs-rich synthesis conditions,the 0D phase of Cs4PbX6 (X = Cl, Br, or I) can be obtained. This phaseexhibits a crystal structure with isolated octahedra [PbX6]

4− separatedby Cs+ ions (16). It should be noted that the 0D perovskites definedhere are different from the morphological 0D nanoparticles or quantumdots obtained from 3D perovskites. This specific structure of 0D pe-rovskites is expected to give rise to interesting electronic and opticalproperties that largely remain to be characterized (17).

Early work on 0D perovskites focused mainly on their fundamentaloptical absorption and photoluminescence properties and attemptedto distinguish their emission properties from those of 3D-like com-pounds (18–24). These studies have demonstrated that, unlike 3Dand other low-dimensional perovskites, the optical characteristics of 0DCs4PbX6 are determined by transitions between electronic states of thePb2+ ions, and their photoluminescence results from the radiative de-cay of Frenkel-type excitons at Pb2+ sites (19). Thus, the separated[PbX6]

4− octahedra in 0D perovskites prevent significant interactionsamong Pb2+ ions, leading to the generation of optical properties sim-ilar to those of individual octahedra. Recently, the intriguing crystal

phase and optical properties of 0D perovskites motivated many ex-perimental groups to fabricate pure single crystals and nanocrystals.For example, the nanocrystals of 0D Cs4PbBr6 perovskites have beenobtained with controlled sizes by using solution-processed synthesis(25–28).

The complete isolation of the octahedra is expected to lead to strongquantum confinement and electron-phonon interactions that can causeexciton localization and self-trapping (29), as well as small polaron for-mation upon charging. Thus, these 0D perovskites represent ideal plat-forms for fundamental photophysical studies of photogeneratedexcitons, polarons, and charge carriers. Here, we show that Cs4PbBr6exhibits molecular behavior in terms of charge carrier transport andlarge polaron binding energies. The calculated positive/negative polaronabsorption bands of charged 0D units are consistent with experimentalbroadband photoinduced absorption (PIA) data in the same spectralregion indicating a short lifetime of 2 ps. Such a polaron lifetime is re-produced via an analysis of the decay of charge populations for polaro-nic states, using nonadiabaticmolecular dynamics (NAMD) simulations.

RESULTS AND DISCUSSIONSTo better understand the zero-dimensionality effects on the structuraland electronic properties of Cs4PbBr6, we also include results on the3D CsPbBr3 perovskite, which contains the same elements. As shownin Fig. 1 (A and B), CsPbBr3 has the orthorhombic phase (Pnam) withcorner-sharing PbBr6 octahedra, whereas Cs4PbBr6 shows the rhom-bohedral phase (R�3c), which can be described as isolated [PbBr6]

4−

octahedra bridged by Cs+ cations with an average distance of ~10 Å.Starting from the experimental lattice parameters of CsPbBr3 (a =8.24 Å, b = 11.74 Å, and c = 8.20 Å) and Cs4PbBr6 (a = b = 13.72 Åand c = 17.30 Å), we further optimized their crystal structures atthe density functional theory (DFT) generalized gradient approximation(GGA)/Perdew-Burke-Ernzerhof (PBE) level; the resulting lattice con-stants for CsPbBr3 are, a = 8.54 Å, b = 11.91 Å, and c = 8.21 Å; andfor Cs4PbBr6, a = b = 14.08 Å and c = 17.56 Å. The calculated electronicbands and projected density of states are shown in Fig. 1 (C and D). Wefirst compare the band gaps calculated at the GGA/PBE level (thismethod has been shown to reproduce the band gaps of 3D perovskiteswell, although this is related to error cancellations) (30). CsPbBr3 iscalculated to exhibit a direct band gap (Eg = 2.13 eV) at the G point,

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which is slightly smaller than the experimental optical band gap of2.25 eV (31). Cs4PbBr6 also displays a direct band gap at the Gpoint but with a much larger value (Eg = 3.88 eV). This latter resultagrees well with both the recently reported experimental band gapof 3.95 eV (20, 25) and the value of 3.99 eV calculated at the GGA/PBE level (25). The valence and conduction bands are seen to flattenconsiderably when switching from the 3D to the 0D perovskite, whichis consistent with the band gap increase upon decreasing structuraldimensionality (29). We note that, when spin-orbit coupling (SOC)effects are included in the electronic band structure calculations, thevalence band curvatures are almost unchanged but with increasedconduction band degeneracies and largely reduced band gaps; forCs4PbBr6 in particular, several conduction bands appear at approx-imately 3.2 eV, which are ~0.8 eV below the other conductionbands. In analogy to CsPbBr3, the conduction band minimum(CBM) of Cs4PbBr6 is mainly composed of Pb-6p and Br-4p states,whereas the valence band maximum (VBM) consists of both Pb-6sand Br-4p states. However, these Pb-6s, 6p, and Br-4p atomic orbitalsare disconnected between octahedra because of the isolation of the[PbBr6]

4− octahedra in the electronic zero dimensionality of the struc-ture, which is also fundamentally responsible for the large band gapsof 0D perovskites.

The different dispersive features in the conduction and valencebands of the 3D CsPbBr3 and 0D Cs4PbBr6 have important implica-tions regarding carrier effective masses. On the basis of the VBM and

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CBM around the G point, for CsPbBr3, the calculated hole effectivemass is mh* = 0.131 m0 along the G→Y (−0.5, 0.0, 0.0) direction andmh* = 0.135 m0 along G→Z (0.0, 0.0, 0.5); the electron effective massis slightly larger, with me* = 0.149 m0 (G→Y) and me* = 0.142 m0

(G→Z), which agrees well with earlier theoretical results (32). In con-trast, Cs4PbBr6 shows nearly nondispersive transport along alldirections in the electronic bands, which results in heavy chargecarriers. In addition, Cs4PbBr6 shows 8.68 × 10−8 S/m electrical con-ductivity (s) in the absence of light irradiation, almost one order ofmagnitude smaller than CsPbBr3 (7.71 × 10−7 S/m) (fig. S1A), whichagrees well with calculated values of s/t for both holes and electrons[Cs4PbBr6, 2 × 1018 to 5 × 1018 (ohms·ms)−1 versus CsPbBr3, 3 × 1019

to 6 × 1019 (ohms·ms)−1] under the same charge carrier concentrationof 1021 cm−3 (see fig. S1B). This is also consistent with the ultralow pho-tocurrent observed in recent experimental work (28), and shows thation transport has little contribution to the electrical conductivity in 0Dperovskites; thus, the low electrical conductivity of Cs4PbBr6 is domi-nated by electrons and/or holes. Overall, the intrinsic large band gap,heavy charge carriers, and low electrical conductivity of 0D perovskiteslimit their application in photovoltaic devices.

The differences in crystal phases and band structures of CsPbBr3and Cs4PbBr6 also generate different absorption spectra and excitonwave function features. Figure 1 (E and F) compares the optical ab-sorption spectra of CsPbBr3 and Cs4PbBr6 calculated at the PBE/SOClevel with and without electron-hole interactions [details can be found

Fig. 1. Electronic and optical properties of CsPbBr3 and Cs4PbBr6. (A and B) Optimized crystal structures. (C andD) Electronic bands and projected density of states (PDOS)calculated at the PBE and PBE + SOC levels (the zero of energy is defined by the VBM in both instances). (E and F) Calculated absorption spectra with and without accounting forelectron-hole (e/h) pair interactions together with (see inset) the exciton wave functions (green area) corresponding to the excitonic peaks. a.u., arbitrary units.

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in the DFT calculations section and in the study of Yin et al. (33)]. Inboth cases, the main interband electronic transitions related to Pb2+(6s)Br−(4p)→Pb2+(6p) at the G point contribute to the higher-energyabsorption continuum. Once electron-hole pair interactions are in-cluded, in the case of CsPbBr3, the excitonic absorption is weak andthe exciton wave function is delocalized over several crystal lattices,which is consistent with its small exciton binding energy (19 to 62 meV)(34, 35). For Cs4PbBr6, the electronic zero dimensionality results in strongquantum confinement, which increases electron-hole interactions andleads to a strong excitonic peak at 3.3 eV and exciton localizationon a single [PbBr6]

4− octahedron (Fig. 1F). This excitonic peak hasbeen observed in previous experimental absorption measurementsat low temperatures (22).

The idea that the weak interactions between neighboring octahedrain 0D crystals cause their properties to be similar to those of individualunits is confirmed by time-dependent DFT (TDDFT) calculations ona single Cs4PbBr6 unit. These calculations can reproduce the exper-imental absorption and capture the absorption features of the Cs4PbBr6crystal (see Fig. 2A), with a main absorption peak at 320 nm and con-tinuous absorption below 300 nm. Thus, this confirms that a single[PbBr6]

4− octahedron reflects the electronic and optical properties ofthe 0D crystals. In addition, a single unit is an ideal model to study the“molecular” behavior of 0D perovskites, for instance, with regard tothe polaronic properties. Upon addition of a charge to ionic and highlypolar crystals, such as perovskites (36, 37), the strong Coulomb inter-actions between the excess charge and the lattice ions enhance theelectron-vibration couplings (38). When the excess charges are spatiallyconfined to a volume of approximately one unit cell or less, they arereferred to as small polarons, which have been observed in manyconjugated polymers upon photo-, electro-, or chemically induced dop-ing (39–41). As shown in Fig. 2B, once an electron is removed from asingle Cs4PbBr6 unit, the Pb–Br bond length greatly shortens (from

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3.10 to 2.69 Å), and the hole spin density localizes on the Pb atomand the surrounding six Br atoms, indicating the formation of a pos-itively charged polaron (Pb3+). The addition of electrons to the singleunit leads to a strongly localized spin density around Pb2+ and largerepulsive interactions between Pb and two Br atoms, which markedlyelongates these two Pb–Br bonds (from 3.10 to 5.34 Å). Adding an elec-tron to the continuous absorption band below 400 nm, these positive/negative polaronic states (see Fig. 2, B and C) present a new broad ab-sorption feature in the spectral range of 450 to 800 nm, which canthus be assigned to polaron absorption.

To further confirm the presence of polarons in 0D perovskites, weperformed femtosecond transient absorption (TA) measurements onCs4PbBr6 and CsPbBr3 thin films. The TA spectra of the CsPbBr3 andCs4PbBr6 films under identical experimental conditions at differenttime delays are shown in Fig. 3 (A and B). Here, negative featuresin the TA spectra correspond to ground-state bleaching (GSB), andbroad positive features correspond to PIA, mainly because of the for-mation of polarons. Similar to the TA spectrum of CsPbBr3, followingthe excitation wavelength at 310 nm (that is, above the band gap ofboth samples), the negative band at approximately 515 nm can beattributed to the GSB of the films owing to the depletion of theground-state carriers to excited states. A new positive broadbandappears above 530 nm in the spectrum of the Cs4PbBr6 film owingto polaron absorption, which is completely absent in the CsPbBr3film. In a 3D perovskite, the excitons can immediately dissociate intofree charge carriers because of the small exciton binding energy, whichleaves no time for polaron formation. On the other hand, in a waysimilar to crystalline inorganic materials, adding a charge ontothe 3D lattice does not significantly change the surroundings be-cause of the rigid inorganic framework. In this case, a large polaron(Fröhlich polaron) forms because of strong electron-optical phononcoupling, which has optical signatures in a terahertz frequencyrange (42, 43). However, because a 0D perovskite has a larger excitonbinding energy, excitons can hardly dissociate into free charge carriers.Therefore, in a way similar to a polymer system, a 0D perovskite is“soft” (that is, the individual octahedra are easily perturbed by thephotoexcitation), which leads to polarons as a new feature presentin low-dimensional perovskites. As shown in Fig. 3C, the correspondingkinetics of the polaron absorption band probed at 600 nm shows alifetime of ~2 ps. This is consistent with the lifetime of polarons ob-served in pristine polymers (44). Here, the polarons are highly loca-lized on [PbBr6]

4− octahedral, and they are likely not to escape but torecombine geminately over a short time.

To mimic the experimental pump-probe measurements and eluci-date the charge carrier (polaron) relaxation processes in the 0D perov-skite, we performed ab initio NAMD simulations on a single Cs4PbBr6unit by describing the spontaneous transitions among different electronicstates (for details, see the “MD calculations” section). As illustrated inFig. 3D, we assumed that the photogenerated electron could bepumped into unoccupied energy levels through polaronic transitionsPS1 and PS2, following an excitation energy of 4 eV. Starting fromthese two polaronic states in the 0D system, the charge population de-cays to 10% within a relaxation time of 1.5 ps, which agrees well withthe experimental polaron lifetime of ~2 ps. This fast recombination isdue to the similar charge distribution features of molecular orbitalsinvolved in the excited electron relaxations (see the insets of Fig. 3D).

Thus, in contrast to the rigid structure of CsPbBr3, the individual[PbBr6]

4− octahedra in the 0D crystal can be easily perturbed by photo-excitation processes, leading to polaronic states through structural

Fig. 2. Ground-state andpolaronbandabsorption. (A) Ground-state absorptionofa single Cs4PbBr6 unit and its charged-state absorption for (B) positive and (C) negativepolarons, as calculated at the TDDFT level, together with experimental absorptionspectra of Cs4PbBr6 thin film (gray line). The spin density distributions are given inthe insets.

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deformation potentials (local molecular arrangements). Moreover,the discontinuous character of the 0D crystal significantly increasesthe charge carrier effective masses and strengthens electron-phononcoupling, facilitating small polaron generation, andproviding an impor-tant optical fingerprint for the polaron band absorption in molecular0D perovskites.

The single-unit model also allows us to characterize the actual dis-tortions induced by the presence of small polarons. Here, we haveevaluated the polaron binding energy (Epolaron) and charge carrier mo-bility based on molecular Cs4PbBr6 dimers. The positive/negative po-laron binding energies (Epolaron) are calculated as 785/943 meV, whichare values that are much larger than those in typical p-conjugatedsystems (45) but smaller than those in some 3D and 2D hybrid pe-rovskites (33, 46). Because the average distance between neighboring[PbBr6]

4− octahedra is quite large (~10 Å), a band-like transport mech-anism is not suitable for describing the charge carrier mobility of 0Dperovskites. Thus, we applied Marcus theory to derive the charge carriermobilities by describing the charge hopping between neighboringunits in molecular dimers. As shown in Fig. 4A, three possible chargehopping paths in the 0D crystal were considered, while the resultinghole mobility of Cs4PbBr6 is calculated to be 1.4 × 10−9 cm2/V · s,whereas the electron mobility is 2.2 × 10−11 cm2/V · s. These low hole/electron mobilities are due to the huge reorganization energies for both

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holes (1.70 eV) and electrons (2.90 eV), as well as the weak electroniccouplings between neighboring units (see table S1). The significantlylarger negative polaron binding and reorganization energies of molec-ular Cs4PbBr6 are related to strong structural deformations (47). Asillustrated in Fig. 2, the positive polaronic state preserves the symmetricgeometry of the neutral state, while the negative polaronic state leads toextreme elongation of two Pb–Br bonds, breaking the symmetry of thenegatively charged [PbBr6]

4− octahedron. These calculated values areorders of magnitude smaller than those of the 3D CsPbBr3 perovskite,which has a hole mobility of 52 cm2/V · s and an electron mobility of11 cm2/V·s (48), and are even lower than those in typical conjugatedpolymers (for example, P3HT, ~10−4 cm2/V · s). Low charge carriermobilities become another electronic feature of molecular 0D pe-rovskites, again limiting their applications in optoelectronic devices.

To examine the small polaron features in bulk Cs4PbBr6 perov-skites, following the same computational strategy as in our previouswork on 2D hybrid perovskites (33), we exerted local perturbations on2 × 2 × 2 Cs4PbBr6 supercells, where the selected Pb–Br bond lengths ofthe central [PbBr6]

4− unit were shortened or enlarged to create localstructural deformations. Figure 4B illustrates the resulting electroniccharge densities at the VBM and CBM after local structural perturba-tions. In the unperturbed systems, the charge densities of the VBMand CBM are highly delocalized in all octahedra (see fig. S2). Once

Fig. 3. TA spectra and polaron kinetics. TA spectra of (A) Cs4PbBr6 and (B) Cs4PbBr3 thin films with different delay times at a photoexcitation wavelength of 310 nm. (C) Photo-excitation kinetics of the Cs4PbBr6 film probed at 600 nm about PIA (the solid blue line is the fit for the kinetics using a monoexponential function). (D) Simulated phonon-inducedrelaxationof thepolaronic state (PS1 andPS2) of a singleCs4PbBr6 unit. The frontiermolecular energy levels andchargedensity involved in theexcitedelectron relaxations aregiven inthe inset. HOMO, highest occupied molecular orbital; LUMO, lowest unoccupied molecular orbital; GS, ground-state.

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the Pb–Br bonds of the central [PbBr6]4− unit are shortened, the

hole charge density (blue area) localizes at the central site, leadingto the formation of Pb3+ centers coupled with the lattice deforma-tion. Similarly, in the case of enlarged Pb–Br bonds, the electroncharge density (red area) is also localized at the central [PbBr6]

4−.As we already demonstrated in the model for the polaronic state ofa single unit, the strong interaction between charge carriers and pho-nons yields the formation of both self-trapped electron and hole statesin the 0D crystal, which is in good agreement with previous experi-mental observations in low-dimensional lead halides (49–51).

On the basis of the perturbed supercell model for describing po-laron localization in the 0D crystal, we further performed ab initioMD calculations to study local structural changes and the evolutionof the charge density at a central [PbBr6]

4− octahedron. As shownin Fig. 4C and fig. S3, upon release from the initial state (for example,polaronic state with the longest Pb–Br bonds), the average Pb–Brbond length decreased, and the CBM energy level increased withinthe first 1 ps. During this time, the CBM charge density is still loca-lized at the central octahedron, and the Pb–Br bond lengths and CBMenergy undergo thermal fluctuations at room temperature. After 1.2 ps,the central octahedron recovers to the neutral state. Therefore, afterphotoexcitation, the structure deformation of single octahedra leadsto the formation of localized polarons with short lifetime and limitedtransport in the 0D crystal. In addition, although the positive and neg-ative polarons have similar absorption features (Fig. 2), the lifetime ofa positive polaron could be even shorter than the negative one becauseof the faster recovery (within 0.1 ps) of Pb–Br bonds and correspondingVBM energy (fig. S3).

In summary, we studied the molecular behavior of the 0DCs4PbBr6 perovskite by combining DFT calculations and femtosecondtransition absorption measurements. Unlike 3D CsPbBr3, 0D Cs4PbBr6displays a large band gap, as well as low electrical conductivity andcharge carrier mobility. A clear spectral feature for polarons with a shortlifetime of ~2 ps is observed in the femtosecond-TA experiments,

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confirming the generation of small polarons with large binding energiesand tight localization at individual [PbBr6]

4− octahedra. These findingsprovide a better understanding of the fundamental photophysical prop-erties of 0D perovskites and open up new avenues for the rational de-sign of low-dimensional perovskites in optoelectronic devices.

MATERIALS AND METHODSDFT calculationsDFT calculations were performed to optimize the crystal structuresof CsPbBr3 and Cs4PbBr6 using the PWSCF code, as implementedin the Quantum ESPRESSO (QE) package (52). The PBE exchange-correlation functional with ultrasoft pseudopotentials was used withand without consideration of SOC. Plane-wave basis set cutoffs forthe wave functions and charge density were set at 50 and 300 rydberg(Ry), respectively. The crystal structures were fully relaxed until thetotal force on each atom was less than 0.01 eV/Å. A uniform grid of6 × 6 × 6 k-mesh in the Brillouin zone was used to obtain the electronicband structures and projected density of states for both crystals. Adense k-mesh (24 × 24 × 24) was used to obtain the electrical transportcoefficients with the BoltzTraP package (53). On the basis of theelectronic wave functions of the ground state obtained by the QE code,the optical transitions of CsPbBr3 and Cs4PbBr6 were calculated usingboth the random-phase approximation and the Bethe-Salpeter equa-tion approach implemented in the YAMBO code (54). In this case,PBE calculations were performed by using norm-conserving pseudo-potentials including SOC effects and plane-wave basis set cutoffs of80 Ry (400 Ry) for the wave function (charge density) (33).

For the single Cs4PbBr6 unit, the neutral and charged geometrieswere optimized with the hybrid range-corrected functional CAM-B3LYP using the Gaussian 09 code. The LANL2DZ basis set was usedfor Cs and Pb atoms, and the 6-31G** basis set was used to describeBr. On the basis of the optimized geometries in both the ground andpolaronic states, the PBE0 hybrid functional was used to obtain the

Fig. 4. Polaron transport and localization. (A) Schematic of charge carrier hopping paths and dimermodel for calculating the charge carriermobility of 0D Cs4PbBr6. (B) Chargedensity distributions for a 2 × 2 × 2 Cs4PbBr6 supercell with shortened Pb–Br distances (positive polaron) and enlarged Pb–Br distances (negative polaron) within the central [PbBr6]

4−

octahedron. (C) Charge densitymapping of CBM of the central octahedron at selected times (0, 0.01, 0.05, 0.1, 0.5, and 1.0 ps). The initial state at 0 ps was set as the negative-polaronstate and had the longest Pb–Br bonds in the central octahedron. The selected Pb–Br bond lengths are indicated in each time delay.

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optical absorption spectra using the TDDFT method. A total of 200singlet states were considered in describing the possible opticaltransitions.

The charge transfer rates between neighboring octahedra weredescribed by Marcus theory (55, 56) following the equation kh=e ¼V2h=e

ℏ

ffiffiffiffiffiffiffiffiffiffiffip

lh=ekBT

qe�

lh=e4kT , where T is the temperature, kB and ℏ refer to the

Boltzmann and Planck constants, Vh/e is the electronic coupling be-tween the HOMO or LUMO levels of neighboring octahedra in theCs4PbBr6 dimers, and lh/e is the hole or electron reorganizationenergy calculated via the adiabatic potential energy surfaces. Vij is theelectronic coupling, defined as ⟨fijF̂jfj⟩, which can be calculated byprojecting the Fock matrix of the dimer onto the molecular orbitalsof the respective donor (fi) and acceptor (fj) units with subsequentsymmetric orthogonalization, where F̂ is the Kohn-Sham Fockoperator for the dimer system. The hole or electron mobility was thendescribed by the Einstein relation m ¼ eD

kBT(57), where e is the electron

charge and D is the charge diffusion coefficient of charge carriers,which can be approximately described as D ¼ 1

2d ∑ir2i kiPi , where ki

and ri are the charge-transfer rate and the neighboring distance forthe dimer and Pi represents the probability for hole/electron hoppingin the ith pathway.

MD calculationsAdiabatic ab initio MD calculations were also performed using the QEcode with the PBE functional and ultrasoft pseudopotential. The opti-mized crystal structure of the 2 × 2 × 2 Cs4PbBr6 supercell was used asthe starting point for the adiabatic MD calculations, where a 5-ps tra-jectory of the system was obtained with 1-fs time steps. The Andersenthermostat was used to control the temperature of the system.

NAMD calculations on a single Cs4PbBr6 unit were per-formed using the PYXAID code (58, 59) developed by Akimovand Prezhdo. In this code, fewest-switches surface hopping (60)implemented within TDDFT was used to study the hybrid perov-skite system (61–63), starting from the time-dependent Schrödingerequation iℏ ∂

∂tYnðr; tÞ ¼ Hðr;R; tÞYn ðr; tÞ and Kohn-Sham orbi-tals Yn ðr; tÞ ¼ ∑k Cn

k ðtÞFkðr;R; ðtÞÞ. From the solutions of theseequations, the probability of transition between adiabatic states iand j can be calculated using the wave function expansion coeffi-cients and coupling, defined as dij ¼ �iℏ⟨Fi

∂∂t

�� ��Fj⟩. The detaileddescriptions of this NAMD theory can be found in the study ofAkimov and Prezhdo (58, 59). Here, we used the real-time TDDFT/NAMD method with decoherence effects on a single Cs4PbBr6unit. A total of 1000 geometries were randomly selected from theadiabatic trajectories and used as initial conditions in the NAMDcalculations.

Preparation of 0D and 3D perovskite filmsThree-dimensional CsPbBr3 and 0D Cs4PbBr6 thin films weredeposited on glass substrates precleaned with acetone/isopropanol.CsPbBr3 was deposited by spin-coating a 0.25 M solution of CsBrand PbBr2 in dimethyl sulfoxide, dripped with toluene, and followedby thermal annealing at 100°C for 10 min. The Cs4PbBr6 thin filmwas deposited through the vacuum thermal evaporation (1 × 10−6

mbar) of Cs4PbBr6 powder (16), with a final thickness of 100 nm.The samples were fabricated and stored in a nitrogen-filled glove box.The materials and solvents were provided by Sigma-Aldrich and usedwithout further purification. The experimental x-ray diffractionpatterns of CsPbB3 and Cs4PbBr6 thin films are shown in fig. S4.

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Electrical conductivity measurementsThe electrical conductivity was determined by investigating the darkcurrent/voltage characteristics of CsPbBr3 and Cs4PbBr6 thin filmson commercial interdigitated substrates with a channel width-to-length ratio of 2000 mm/2.5 mm (Fraunhofer Institute for PhotonicMicrosystems).

Femtosecond TA spectroscopyThe time-resolved experiments were conducted on ExciPro pump-probe spectrometers (CDP), in which the fundamental output camefrom a Ti:sapphire femtosecond regenerative amplifier operating at800 nm with 35-fs pulses and a 1-kHz repetition rate. Pump pulses at310 nm were generated after passing through a fraction of the 800-nmbeam into a spectrally tunable (240 to 2600 nm) optical parametricamplifier (TOPAS Prime, Spectra-Physics) and a frequency mixer(NirUVis, Light Conversion). To generate the probe pulses (ultraviolet-visible and near-infrared wavelength continuum, white light), anotherfraction of 800 nm–amplified pulses was focused onto 2-mm-thick cal-cium fluoride (CaF2) crystal. To achieve better signal-to-noise ratios, theresulting white light was split into two channels, denoted the probe andreference, and focused on the two different fiber optics. The pumppulses were overlapped on the sample with the probe pulses after pass-ing through a synchronized chopper (500 Hz), which blockedalternative pump pulses. Finally, the change in absorption (DA) of theexcited state was calculated by subtracting the absorptions of the excitedand unexcited samples.

SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/12/e1701793/DC1fig. S1. Experimental and calculated electrical conductivity of CsPbB3 and Cs4PbBr6.fig. S2. Charge density descriptions of Cs4PbBr6 supercell.fig. S3. Average Pb–Br bond length and orbital energy evolution.fig. S4. X-ray diffraction patterns of CsPbB3 and Cs4PbBr6 thin films.table S1. Calculated charge transfer parameters and mobility of the Cs4PbBr6 crystal.

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Acknowledgments: J.Y., J.-L.B., and O.F.M. acknowledge the IT Research ComputingTeam and Supercomputing Laboratory at KAUST for their computational and storageresources, as well as their assistance. Funding: This work was supported by the KingAbdullah University of Science and Technology (KAUST). The work at Georgia tech wasfunded by the Office of Naval Research, Award NO. N00014-17-1-2208. Authorcontributions: J.Y. generated the idea and designed the simulation work. O.M.B. andO.F.M. crafted the experimental plan and directed the research. J.Y. performed all the

Yin et al., Sci. Adv. 2017;3 : e1701793 15 December 2017

ab initio calculations. J.Y., J.-L.B., and O.F.M. analyzed the theoretical results. M.D.B.and I.D. prepared the thin film samples and measured the electrical conductivity. P.M.and O.F.M. planned and performed the TA spectrum measurements and analyzed thedata. J.Y., J.-L.B., and O.F.M. wrote the manuscript. All authors discussed and commentedon the manuscript. Competing interests: The authors declare that they have nocompeting interests. Data and materials availability: All data needed to evaluatethe conclusions in the paper are present in the paper and/or the SupplementaryMaterials. Additional data related to this paper may be requested from the authors.

Submitted 29 May 2017Accepted 13 November 2017Published 15 December 201710.1126/sciadv.1701793

Citation: J. Yin, P. Maity, M. De Bastiani, I. Dursun, O. M. Bakr, J.-L. Brédas, O. F. Mohammed,Molecular behavior of zero-dimensional perovskites. Sci. Adv. 3, e1701793 (2017).

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Molecular behavior of zero-dimensional perovskitesJun Yin, Partha Maity, Michele De Bastiani, Ibrahim Dursun, Osman M. Bakr, Jean-Luc Brédas and Omar F. Mohammed

DOI: 10.1126/sciadv.1701793 (12), e1701793.3Sci Adv 

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