Photoinduced Single- and Multiple-Electron Dynamics Processes

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Photoinduced Single- and Multiple- Electron Dynamics Processes Enhanced by Quantum Confinement in Lead Halide Perovskite Quantum Dots Dayton Jon Vogel, Andrei Kryjevski, Talgat M Inerbaev, and Dmitri S. Kilin J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b03048 • Publication Date (Web): 21 Mar 2017 Downloaded from http://pubs.acs.org on April 22, 2017

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The Journal of Physical Chemistry Letters

Photoinduced Single- and Multiple- electron Dynamics Processes Enhanced by Quantum Confinement in Lead Halide Perovskite Quantum Dots Dayton J. Vogel1, Andrei Kryjevski2, Talgat Inerbaev3,5, Dmitri S. Kilin1,4* 1. Department of Chemistry, University of South Dakota, Vermillion, SD, 57069 2. Department of Physics, North Dakota State University, Fargo, ND, 58102 3. L.N. Gumilyov Eurasian National University, Astana 010008, Kazakhstan 4. Department of Chemistry and Biochemistry, North Dakota State University, Fargo, ND, 58102 5.National University of Science and Technology “MISIS”, Moscow, 119049 Russian Federation

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Abstract: Methyl-ammonium lead iodide perovskite (MAPbI3) is a promising material for photovoltaic devices. A modification of the MAPbI3 into confined nanostructures is expected to further increase efficiency of solar energy conversion. Photo-excited dynamic processes in a MAPbI3 quantum dot (QD) have been modeled by many-body perturbation theory and nonadiabatic dynamics. A photoexcitation is followed by either exciton cooling (EC), its radiative (RR) or non-radiative recombination (NRR), or multi-exciton generation (MEG) processes. Computed times of these processes fall in the order of MEG < EC < RR < NRR, where MEG is in the order of a few femtoseconds, EC at the picosecond range while RR and NRR are in the order of nanoseconds. Computed timescales indicate which electronic transition pathways can contribute to increase in charge collection efficiency. Simulated mechanism relaxation rates show that quantum confinement promotes MEG in MAPbI3 QDs.


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The discovery of methyl-ammonium lead halide perovskite (MAPbI3) materials for application in photovoltaic devices has generated interest and led to a vast expanse of work to understand the characteristics of the materials.1,2,3,4,5 There is great interest in the use of hybrid organic-inorganic perovskite (HOIP) materials for solar energy applications due to rapidly rising device efficiencies and low production costs.6,7,8 As the application of the materials is intended for photovoltaics and optoelectronics, the understanding of electronic processes following photoexcitation within the HOIP materials is of utmost importance. Experimental results have provided many of the necessary details to construct optimal photovoltaic systems to obtain high power conversion efficiencies as well as covered important electronic transition processes within semiconductor quantum dot (QD) structures to increase quantum efficiency (QE).9,10

Modification of material properties can be generated via

interfacing, modified device architecture, and physical constraints to the size of the photoactive material.11,12 One such process harnessed by physical confinement of the light absorbing material is MEG (also referred to as carrier multiplication).13,14,15 One of the attractive properties for choosing MAPbI3 as a starting material to study confinement effects is bulk 3D MAPbI3 has a relatively low band gap energy.16,17 The reduced energy needed to excite charges across the band gap allows a larger portion of the electromagnetic spectrum to be harnessed as well as provides an excellent starting point considering the band gap energy will increase due to spatial size constraints. This manipulation of the electronic structure due to confinement may lead to an enhancement of a specific electronic transition mechanism. To identify the probable relaxation mechanism within a material one must considering multiple relaxation processes, such as nonradiative exciton cooling (EC), non-radiative recombination (NRR), radiative recombination (RR), Auger, electron transfer, and multiple-electron generation (MEG). As competing


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mechanisms are often decided by the timescales of relaxation, transition rates will indicate the most probable process. In photovoltaics and optoelectronics the optimal result is a QE greater than 1, an achievement reached through MEG in small bandgap semiconductor QDs.18,19,20 Atomistic analysis of the competing photoinduced processes is modeled by combination of many-body perturbation theory, nonadiabatic dynamics, and density matrix approaches, all in the basis of Kohn-Sham orbitals | (). The many-body perturbation theory (MBPT) approach used for the MEG rate calculations is a well-established high-precision method for the electronic structure of nanoparticles.21 The MEG rate procedure used in this work has been previously introduced in the literature.22 First, one solves Bethe-Saltpeter equation, which re-sums perturbative electron-hole Coulomb interactions to all orders, for the electron-hole bound state (exciton) energies and wave functions. The exciton energies and wave functions are incorporated into the exciton-to-biexciton →

and biexciton-to-exciton

→

rate calculations, where the electron-to-trion decay, which is

the core elementary process, is included to the leading order in the RPA Coulomb interaction. The RPA-screened Coulomb potential involves polarization insertion re-summed to all orders. So, MBPT, and the MEG technique, in particular, goes beyond naive perturbative approach. Indeed, the KS orbitals parameterize ground state electron density and cannot not be directly associated with single-particle states. But they approximate electrons bound to the ions and include some electron interaction effects. Therefore, KS orbital basis is a reasonable starting point for the standard many-body techniques, such as MBPT, which is then used to describe excited state properties, such as exciton-to-bi-exciton decay, including electron Coulomb interactions.


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The explicit evolution of the electronic state is propagated in time using density matrix approach in Redfield formulation.23 The electronic transition probabilities between two specified electronic states facilitated by nuclear vibrations24 are modeled using the nonadiabtic coupling (NAC) terms, = ∑

,

providing wavefunction overlaps, which are calculated

by an “on-the-fly” procedure, = ()

(), between each time step along the ground

state MD trajectory.25 Performing an autocorrelation of the NACs, followed by a Fourier transform of the autocorrelation provides the average second order electron-phonon interaction perturbation term, referred to as components of the Redfield tensor. To calculate the rate of charge carrier relaxation, the normalized energy expectation value for each charge carrier as a function of time is fit to a single exponential.26,27 Use of transition probability between two specified states, as calculated through IOA, allows for the RR rates to be found as defined by Einstein coefficient of spontaneous transition. The Einstein coefficient for spontaneous emission, , can be used to relate the oscillator strength of a specified transition and the lifetime of the corresponding transition. is defined as =

&' ! "# $

%

( &

" $% &

and the relation to the lifetime of the emission is represented by ) = "

%$

# * +, = ! # = = & . ' '

'

/

' -'

. Here 01 = 2

' "%

, 3 = 4$, and c represent the $

set of fundamental constants. The rates of spontaneous emission, , are calculated in atomic units and converted to picosecond timescales. The atomistic model was built using three cubic unit cells of MAPbI3 in a planar arrangement. Additional iodine anions and methyl-ammonium cations were added in six locations near the edge of the QD to stabilize and maintain the material structure.28 In adding the iodine

anions

and

methyl-ammonium

cations

the

following

stoichiometry

of


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MAX PbY I Z + W ( MA + I − ) , where x = y = 19, z = 3*x= 57, and W = 6, providing a total system of chemical

composition MA19 Pb19 I 57 + 6MA+ + 6I − = C25 N 25 H150 Pb19 I 63 .

The

unit

cell

has

dimensions of a=32.6 A, b=18.5 A, and c=31.0 A. Periodic cells were spaced with vacuum to negate any interaction between adjacent QDs. The ground state electronic structure, heating, and molecular dynamics (MD) simulations were calculated using Vienna ab initio Software Package (VASP).29,30 For the ground state electronic structure, optimization was carried out at 0K using density functional theory (DFT) with the generalized gradient approximation (GGA) PerdewBurke-Ernzerhof (PBE) functional,31 projector augmented wavefunction (PAW) potentials,32,33 plane wave basis set with an energy cutoff of 300 eV,34,35 and periodic boundary conditions. The effect of spin-orbit coupling (SOC) in total is a minimal shift in the electronic structure of a system, and has been neglected for this calculation. In regards to the fine electronic structure, SOC is seen to have an effect on energies at the band gap edges and deep within the valence band. In the case of any type of dynamic calculation, the small change in electronic structure results in a negligible force felt by any atom. Heating calculations were performed to simulate ambient temperatures at 100K and 300K. MD simulations were performed along the ground state trajectory for 4 ps with 1 fs time steps. Analysis of structural modifications along the MD trajectory has been performed based on computed

radial

distribution

functions

(RDF)

as

defined

as

.

In this study, a comparison of electronic properties and resulting calculated rates for RR, EC, NRR, and MEG mechanisms are considered specifically in a MAPbI3 QD, shown in Figure 1A-


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B. In discussing the highlighted electronic transition mechanisms, one must not exclude the process of hot carrier extraction, as it also has the potential to compete with the presented electronic transition processes. However, as the system is a single MAPbI3 QD in vacuum there is no interface to assist in hot carrier extraction.

Initial analysis of the MAPbI3 QD optimized ground state electronic structure provides the density of states (DOS) and absorbance spectrum. The DOS plot, Figure 1C, near the band edges shows the DOSVB is near four time higher than the DOSCB, prompting quicker relaxation for holes than electrons. The molecular orbitals at the band edges are comprised of primarily Pb and I, with contributions of atomic orbitals provided in SI, Table S1. Also from the DOS, the bad gap energy of the QD, E QD g , is found to be 2.1eV. The value of the confined material is predictably higher than measured experimental and previously published computational band gap is close to 1.75 eV.16,17 The increase of energies for periodic bulk MAPbI3 materials, where E bulk g is expected as the material is physically confined and showing signs of electronic state E QD g separation due to quantum size effects. The results of quantum size effects also manifest themselves in the absorbance spectrum, Figure 1D, as the absorption edge is blue shifted in comparison to experimentally produced bulk spectra.36

To correctly study charge carrier dynamics of competing relaxation mechanisms one must move beyond static ground state electronic structure calculations. The dependence of electronic structure on thermal motion of ions is provided in the SI. To reach the goal of understanding charge carrier dynamics, as discussed in the methodology section. Moving forward three methods are applied to find the relative rates of RR, EC, NRR, and MEG


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mechanisms. Thorough investigation of competing relaxation processes required the use of multiple methodologies to correctly describe the mechanism. Each of the methodologies utilized have been previously defined and tested for systems of similar QD dimensions.37,38,26,27 As the large scale goal is to correctly identify trends in a range of QD sized, we utilized multiple methods to complete timely and efficient analysis.

Nonradiative Channel: Micro-canonical molecular dynamic (MD) simulations at ambient temperatures of 100K and 300K were computed to provide atomic positions at each time step of their respective trajectories. The MD trajectories for 100K39 and 300K40 can be viewed online, as well as thermalization effects on KSO energies due to nuclear motion at each temperature, SI, Figure S1. Following a MD trajectory provides a dynamic electronic structure, allowing for calculation of a change of linear absorption, as shown in Figure S2, and the impact of thermalization on KSO energies. Predicated that EC relaxations are facilitated by phonon modes, comparing the total wavefunction at each time step of the MD trajectory allows for the calculation on-the-fly nonadiabatic coupling between electronic states. The nonadiabatic coupling values are used to compute Redfield tensor matrix elements which provide transition probabilities between any pair of electronic states and their corresponding rate of NRR.

24

To

confirm memory effects are negligible in the system, the autocorrelation of the NAC values is calculated and found to decay abruptly, Figure S3. A plot of the EC transition rates vs. the absolute energy difference between two specified orbitals is shown for 100K and 300K in Figure 2A-B. The transition rate data shows two regions of higher transition rates, corresponding to near adjacent states, due to availability of low energy vibrations to accommodate the transition energies. The trend is expected as gap law predicts the fastest EC transitions to occur between


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states closest in relative energy.41,42 The groupings of the fastest rates are below 0.25 eV, representing intra-band EC transitions. The insets in Figure 2A-B are graphical bar plots of elementary transition probabilities near the band edges as a function of initial and final orbitals. Things to note from the inset: 1) away from the diagonal the probabilities decrease to zero very quickly, showing transitions are more likely to occur between adjacent states 2) The transition probability in the VB is much greater than that in the CB, leading to expectations that holes will have greater relaxation rates than electrons, in agreement with DOSVB > DOSCB.

To find the rate of EC relaxation, individual transitions identify initial non-equilibrium electronic distributions at which to being propagation of the density matrix in time. The transition with the highest oscillator strength, and therefore probability, is used as an example initial state. The plot in Figure 2C provides a visual representation of the spatial distribution of electron density as a function of time and energy in relation to the ground state equilibrium. A single exponential function is fit to electron expectation energies, the dashed line in Figure 2C, producing the rate for EC relaxation (hot carrier cooling). No change in electronic population, gain in population (electron), and loss in population (hole) are represented by green, red, and blue areas, respectively. Initial photoexcitation occurs on the left-hand side of the plot indicating time zero. For each specific initial excitation, the electron lifetime, τe, can vary as changing excitation energies will populate electrons and holes into a different sequence of states within the conduction and valence band. After a period of time both charge carriers reach their respective band edges and, under an approximation of no recombination, the energy expectation value remains constant. Results from calculation of EC relaxation rates provide a time scale of electron relaxation on the order of 1ps at room temperature, with individual rates for specific photo-


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excited transitions given in Table S2 and in SI, Figure S4. A comparison shows that EC relaxation rates for holes are faster than those of electrons, agreeing with our previous expectation based on a combination of low transition probability and sparse DOSCB band edge. Rates provided in Table S2 represent initial excited populations according to calculated most probable optical transitions. The initial states populated for each transition do not necessarily represent an excitation identical to the one seen in experiment. Specifically, calculated rates for excitations related to electrons excited from the VB edge to the CB edge have relaxation time close to 1ps. Previous work by Piatkowski et al. showed electron cooling on the order of 750 fs for a given excitation at 450nm.43 A comparison of EC relaxation rates at 100K and 300K shows relaxation rates for both electrons and holes being slower at 100K in respect to the same transitions at 300K. This is expected as a smaller number of normal modes being activated at low temperature, which facilitate the EC transitions.

The EC relaxation rates shown in Figure 2D exhibit several trends: (i) All carriers show quicker relaxation at higher temperature. (ii) Holes relax faster than electrons, as prompted by higher DOSVB > DOSCB. For EC relaxation, there is often a direct correlation between dissipated energy and rate of relaxation.42 This is known more commonly as gap law and is the reason for (i). When looking at rates of EC relaxation at two temperature regimes there is discord between the two temperature data sets. For the 300K system, rates of electrons and hole follow the expected trend. However, for the low temperature system the calculated rates can be view in two regimes. (iii) For high energy excitations, the rates of electrons and holes at 100K, follow the expected near linear gap law. (iv) At low excitation energies, the rates for the 100K system deviate from a linear trend. Rather than the expected fastest rates the rates begin to slow, providing longer lifetimes with smaller excitation energies. One explanation for this change in trend is the


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dependence of relaxation pathways on spatial distribution of electronic structure in the system. When excited to higher states in the conduction band, coupling between the high-energy states and states very near the band edge is large. This would promote rapid relaxation, while bypassing numerous states within the conduction band. This type of relaxation channel would be very different from that of a low excitation channel. The low excitation channel would pass through a different set of electronic states to reach the band edge.24 Resulting electronic population in each state involved in relaxation for parallel channels can be seen in Figure S5. With the number of channels increasing with excitation energy a key value of the transition rate between HO and LU ( ∆ε ~ 2.1eV ) , must be kept in consideration. Off-diagonal values of the Redfield tensor, visualized in Figure 2(A-B) insets, are often small but can play an important role when the values are non-zero.

As EC relaxation rate dictates the rate of non-radiative recombination (NRR), the availability of multiple channels increases the probability of NRR. Loss of generated electrons through NRR not only decreases the efficiency of the device but also generate heat, potentially thermally degrading the photoactive material. The NRR time for the LUMO-HOMO transition is found to be 3.6 ns, which is slower than the rates discussed for EC, RR, and MEG transitions, validating the discussion of the three highlighted transitions. The use of the ground state electronic structure to propagate the MD trajectory is an approximation used to save computational time and resources. The approximation can be negligible for specific systems including quantum confined systems with no band dispersion, rigid inorganic systems, and systems without considerable nuclear reorganization. Nuclear reorganization is likely a more important factor with systems containing organic, light elements,


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or unbound components. For MAPbI3 the MA component is most likely to experience effects of excited state forces. There has been investigation in the field to identify the extent of the excited state forces on MA and the cumulative effect on the overall electronic structure.44,45,46 However, as orbitals near the gap are localized on Pb and I rather than MA, reorganization may have lesser effect. An ideal analysis would include time dependent excited state molecular dynamics to account for nuclear reorganization resulting from changing localization of electronic charge. However, with a small Stoke’s shift value known for MAPbI3,47 emission is likely to occur from states close to the band edges. This provides reasoning to believe thermalization to the band edges occurs before emission, allowing ground state trajectories to be an appropriate approximation.

Radiative relaxation. Along with NRR relaxation, the RR relaxation mechanism is a competing process for electronic energy dissipation following photoexcitation. A graphical depiction of the value of the energy of photoexcitation during charge carrier relaxation following excitation from HO LU+27 is seen in Figure 3A-B. The calculated energy of excitation depends on the electronic populations for specified states and the relative energy difference between the states with respect to the equilibrium energy. The time dependent energy of excitation graphed in Figure 3A shows the initially created electron and hole remain in their initially excited states for a time period close to 150 fs. As the charge carriers begin to migrate towards their respective band edges the energy of excitation is reduced. Recombination is not considered during calculation of the excitation cooling, resulting in the final energy equivalent to E QD being g reached by the end of the cooling period near 5-10ps, see details in Table S2. Displayed in Figure 3B is the time-integrated emission during electronic relaxation with the energy range


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shown corresponding to the visible region of the spectrum. In comparing Figures 3A and 3B, the largest intensity of RR emission and final excitation energy is in the visible region, equal to the energy of E QD g . The integrated emission also shows multiple RR transition intensities that have an energy value greater than the band gap, E QD g . The intensities of the RR transitions in the visible region show a high probability for QD photoluminescence at E QD is in agreement with the g . This higher intensity at the E g

photoluminescence (PL) calculated with the MDPL method, shown in Figure 1D. Rates of R relaxation are calculated using the relationship of the Einstein coefficient for spontaneous emission. As defined in the methods section, the rate of spontaneous emission is inversely proportional to the transition energy and oscillator strength, τ 21 ∝

C , where C is a set of v f 2 21 12

fundamental constants, v21 , is the frequency of the transition, and f12 is the oscillator strength of the specified transition which is calculated during analysis of the ground state electronic structure. The calculated rates for RR relaxation in the MAPbI3 QD are found to be on the order of 10-100ps, making the RR relaxation mechanism one to two orders of magnitude slower than EC relaxation. This result is expected as EC relaxation is generally considered to be faster than RR relaxation.48 The strongest integrated emission peak corresponds to an electronic transition across the band gap, allowing one to explicitly consider the RR lifetime for the LUMO-HOMO transition. The calculated RR lifetime of the LUMO-HOMO transition is found to be 102 ps, shown as the highlighted row of Table S3. With NRR found to be on the scale of ns, the RR lifetime is 30-40 times faster than the NRR, indicating an emission event is much more probable than NRR.


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Given that the rate of RR relaxation, κ R , and the rate of NRR recombination, κ NR , have been calculated, the quantum yield (QY), 56 = 7 (7 + 79 ), can be determined where 7 = ) , and 79 = )9 ,. At room temperature where ) = 102=>, )9 = 3.6C>, the QY is calculated to be just over 97%, which agrees with the range of quantum yield seen in fully inorganic types of CsPbX3 quantum dots.49,50 The QY for the MAPbI3 QD at 100K was calculated to be near 4%, which is extremely low relative to the calculation at 300K. As the RR lifetimes are calculated based on the ground state electronic structure the LUMO-HOMO lifetime does not change from the 300K to 100K model. However, analysis of the EC recombination lifetime is an indication factor. The LUMO-HOMO Redfield tensor matrix element, which is dependent on the MD trajectory, is three orders of magnitude larger for the 100K than the 300K. This increased value of transition probability at low temperatures disagrees with the known gap law relationship between temperature and EC transition rates. According to intuitive expectations nonadiabatic coupling is rather expected to increase for higher temperatures since it is proportional to the change of the wave function overlap. Interestingly, here results show an opposite relationship between temperature and rates of EC relaxation. Thermal fluctuations of ions can lead to a loss of wavefunction overlap for strong radiative transitions. This phenomena has been seen to increase PL in QDs at low temperatures.51 A second hypothesis to explain the reduced QY at 100K may be that upon cooling the MAPbI3 QD structure experiences a phase transition to the orthorhombic phase that is known for fast NRR.52

Multiple Exciton Generation. The final electronic relaxation mechanism under consideration in this work is MEG. Quantum confinement is expected to facilitate MEG as interacting particles are confined within a small volume in close proximity to each other. Rates for MEG, calculated


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according to many-body perturbation theory (MBPT),37 converting an exciton to a bi-exciton, 12 (carrier multiplication), or a bi-exciton to an exciton, 21 (Auger recombination), are shown as a function of the excition energy, Eexciton , in units of E QD in Figure 3C. The plots in g Figure 3C represent MEG rates for the processes of 12 (green) and 21 (blue), with more details provided in Figure S6. Here, quantum efficiency of 12 process can be defined as ratio 5D→ = ( + 2→ )( + → ),, where corresponds to photon-to-exciton rate and → stands for photon-to-biexciton rate.22 Thus, when each photon results in a single exciton, 5D→ =1, while when each photon results in a biexciton 5D→ =2. Computed values a found in the range 1< 5D→ 1012 s-1 > 1011 s-1 > 108 s-1, respectively. From these results, one can expect MEG to be a likely pathway of relaxation within the size confined MAPbI3 material. Signatures of size confinement were seen in the increased spacing of discretized energy states as well as localized electron density on the QD in comparison to bulk MAPbI3. The increased intraband energy separation between states decreases the rate of NR relaxation by requiring multiple active phonon modes to dissipate the hot electron energy. The computed MEG rates demonstrate dependence on the excitation energy Eex/ E QD g reflecting resonant conditions between single and bi-exciton energies. As expected, initial MEG rates begin to peak near Eex/ E QD > 2, as it is the minimum energy for conservation of energy g threshold. Calculated QYs qualitatively indicates efficiencies close to 1 at room temperature, which is to be expected in QD. As these values of efficiency do not take into account MEG, one can expect the QE to increase leading to efficiencies over 1, making the materials highly desirable for energy conversion. Future work for these materials includes investigation of the relaxation mechanisms when including spin-orbit coupling (SOC). However, as the largest effect of SOC is seen in the electronic band structure of the conduction band, one would expect decreased rates of EC and RR relaxation rates, resulting in an increased disparity in rates while maintaining the same mechanism trend of MEG > EC > RR. A second task will be to model the QD with an interface available to account for potential electron transfer pathways.


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Acknowledgements. This research has been supported by NSF CHE-1413614 for methods development. Authors thank DOE BES NERSC facility for computational resources, allocation award # 91202, “Computational Modeling of Photo-catalysis and Photo-induced Charge Transfer Dynamics on Surfaces” supported by the Office of Science of the DOE under contract no. DE-AC02-05CH11231. The authors would like to thank Douglas Jennewein for support and maintaining the High-Performance Computing system at the University of South Dakota. Thanks are also extended to Stephanie Jensen, Wendi Sapp, Adam Erck, Yulun Han, Brendon Disrud, Bakhtyor Rasulev for collective discussion and editing. DSK and DJV thank Svetlana Kilina, Sergei Tretiak, Amanda Neukirch, and Nikolay Makarov (LANL) for stimulating discussions. DK acknowledges support of Center for Nonlinear Studies (CNLS) and Center for Integrated Nanotechnology (CINT), a U.S. Department of Energy and Office of Basic Energy Sciences user facility, at Los Alamos National Laboratory (LANL). This research used resources provided by the LANL Institutional Computing Program. T.M.I. thanks the Center for Computational Materials Science, Institute for Materials Research, Tohoku University (Sendai, Japan) for their continuous support of the SR16000 M1 supercomputing system. T.M.I. gratefully acknowledges financial support of the Ministry of Education and Science of the Russian Federation in the framework of the Increase Competitiveness Program of NUST MISIS (No. K3-2016-021). The


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calculations were partially performed at supercomputer cluster "Cherry" provided by the Materials Modeling and Development Laboratory at NUST "MISIS" (supported via the Grant from the Ministry of Education and Science of the Russian Federation No. 14.Y26.31.0005).

ASSOCIATED CONTENT Supporting Information. Supporting Information available: Methodology for computing radiative lifetimes, summary of numerical data for nonradiative lifetimes, data on influence of nuclear motion on electronic structure of perovskite quantum dots, additional numerical details on multiple exciton generation calculation are provided in the Supporting Information document. AUTHOR INFORMATION Corresponding Author *E-mail addresses: [email protected], [email protected].


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FIGURES:

Figure 1. (A) A graphical representation of the electronic relaxation, facilitated by electronelectron, electron-photon, and electron-phonon interactions, mechanisms of interest in this work, following the minimum photoexcitation (PE), needed for multi-exciton generation (MEG) are presented. Red non-linear arrows represent non-radiative (NR) thermalization facilitated by vibrational modes, with NR recombination represented by the vertical dashed red line. An electronic transition from the conduction band to the valence band corresponding to the radiative (R) relaxation mechanism, is represented by a green vertical arrow. MEG is represented by an


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intra-band electronic relaxation (black) accompanied by generation of a second electron (blue). (B) Atomistic model representing the methyl-ammonium lead iodide perovskite quantum dot. The atoms for carbon, hydrogen, iodine, nitrogen, and lead are colored gray, white, purple, blue, and silver, respectively, details on geometry are provided in SI. (C) Ground state electronic density of states near the band edges with the occupied valence band and unoccupied conduction band represented by the shaded and unshaded region of the spectra, respectively. (D) Computed ground state electronic optical absorption spectra (green) and PL spectra computed by MDPL(red) and integrated emission (blue) methods.

100K

300K

A

B

C

D

LU+27 LU

HO

Figure 2. Numerical results on the nonradiative relaxation (NR) at different temperatures. Top panels (a), (b) NR elementary events transition probability vs. the absolute energy between two specified Kohn-Sham orbitals; an inset shows a visual representation of state-to-state electronic


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transitions near the band edges. Here the x-axis covers the range of [0-0.25] eV with the value at ~2.1eV being the NRR rate. Lower panel (c) shows change in distribution of electron density as a function of energy and time in respect to electronic equilibrium following photoexcitation from HO to LU+27. Lower panel (d) summarizes integrated rates of hot carrier cooling for a range of initial excitations.

A

B

C

MEG

R

12 21

F

Figure 3. Observables representing energy of excitation (A), R, time integrated emission spectrum (B), and MEG (C) pathways in the photoexcited MAPbI3 QD. (A) Dynamics of exciton energy distribution dissipating as a function of time on a logarithmic axis shows population of excitation at the specific energy where the maximum is red and minimum is blue. Dynamics are shown following the strongest optical transition for the systems. (B) Time integrated emission spectra representing intensities contributed by all probable radiative transitions, within the visible region, as a function of time along the dynamic trajectory. (C) Multi-exciton generation and Auger recombination rates as a function of energy with respect to the bandgap energy, Eg. Rate intensity for two processes, a single exciton biexciton (green, MEG) and two excitons a


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single exciton (blue, Auger recombination) are plotted together, showing rates on a scale of 1014 s-1.

FIGURES: Figure 1


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Figure 2


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Figure 3


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A

B

C

MEG

R

12 21

F

Graphical abstract / Table of Contents figure


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Photoinduced Single- and Multiple-Electron Dynamics Processes

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