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Letter pubs.acs.org/JPCL
Photoinduced Single- and Multiple-Electron Dynamics Processes Enhanced by Quantum Confinement in Lead Halide Perovskite Quantum Dots Dayton J. Vogel,† Andrei Kryjevski,‡ Talgat Inerbaev,§,⊥ and Dmitri S. Kilin*,†,∥ †
Department of Chemistry, University of South Dakota, Vermillion, South Dakota 57069, United States Department of Physics, North Dakota State University, Fargo, North Dakota 58102, United States § L.N. Gumilyov Eurasian National University, Astana 010008, Kazakhstan ∥ Department of Chemistry and Biochemistry, North Dakota State University, Fargo, North Dakota 58102, United States ⊥ National University of Science and Technology “MISIS”, Moscow, 119049 Russian Federation ‡
S Supporting Information *
ABSTRACT: Methylammonium lead iodide perovskite (MAPbI3) is a promising material for photovoltaic devices. A modification of MAPbI3 into confined nanostructures is expected to further increase efficiency of solar energy conversion. Photoexcited 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 nonradiative recombination (NRR), or multiexciton generation (MEG) processes. Computed times of these processes fall in the order of MEG < EC < RR < NRR, where MEG is on the order of a few femtoseconds, EC is in the picosecond range, while RR and NRR are on the order of nanoseconds. Computed time scales indicate which electronic transition pathways can contribute to increase in charge collection efficiency. Simulated mechanisms of relaxation and their rates show that quantum confinement promotes MEG in MAPbI3 QDs.
T
an excellent starting point considering spatial size constraints are expected to increase the band gap energy. This manipulation of the electronic structure due to confinement may lead to enhancement or supression of specific electronic transition mechanisms. To identify the most probable relaxation mechanism within a material, one must consider multiple relaxation processes, such as nonradiative exciton cooling (EC), nonradiative recombination (NRR), radiative recombination (RR), Auger, electron transfer, and MEG. As competing mechanisms are often decided by the time scales of relaxation, shorter transition times 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-band-gap semiconductor QDs.18−20 Atomistic analysis of the competing photoinduced processes is modeled by a combination of many-body perturbation theory (MBPT), nonadiabatic dynamics, and density matrix approaches, all in the basis of Kohn−Sham (KS) orbitals |φj(t)⟩, with energies εj(t). The 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
he discovery of methylammonium lead halide perovskite (MAPbI3) materials for application in photovoltaic devices has generated interest due to rapidly rising device efficiencies and low production costs.1−5 This has driven great interest in the larger class of hybrid organic−inorganic perovksite (HOIP) materials for solar energy applications.6−8 As the application of the materials is intended for photovoltaics and optoelectronics, understanding of electronic processes following photoexcitation (PE) within the HOIP materials is of utmost importance. Recent experimental results have provided details on electronic transition processes within HOIP structures to increase their quantum efficiency (QE) as well as provided several ways to optimize power conversion efficiencies of HOIP-based photovoltaic systems.9,10 Modification of material properties can be generated via interfacing, modified device architecture, and physical constraints to the size of the photoactive material introducing spatially confined semiconductor quantum dots (QD).11,12 One such process harnessed by physical confinement of the light-absorbing material is multiexciton generation (MEG) (also referred to as carrier multiplication).13−15 One of the specific properties for choosing MAPbI3 as a material to study quantum confinement effects is that bulk 3D MAPbI3 has a relatively low band gap energy.16,17 The smaller band gap energy allows a larger portion of the electromagnetic spectrum to be utilized and also provides © 2017 American Chemical Society
Received: December 28, 2016 Accepted: March 21, 2017 Published: March 21, 2017 3032
DOI: 10.1021/acs.jpclett.6b03048 J. Phys. Chem. Lett. 2017, 8, 3032−3039
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The Journal of Physical Chemistry Letters in this work has been previously introduced in the literature.22 First, one performs a DFT caluclation with the HSE06 hybrid functional to substitute for the GW single-particle energy corrections in this approach. Second, one solves the Bethe− Salpeter equation, (εe − εh)ψehα + Σe ′ h ′Wehe ′ h ′ψeα′ h ′ = E αψehα
added in six locations near the edge of the QD to stabilize charge neutrality and maintain the material structure.28 By adding the iodine anions and methylammonium cations, one gets the following stoichiometry of MAXPbYIZ + W(MA+I−), where X = Y = 19, Z = 3X = 57, and W = 6, providing a total system of chemical composition of MA19Pb19I57 + 6MA+ + 6I− = C25N25H150Pb19I63. The simulation cell has dimensions of a = 32.6 Å, b = 18.5 Å, and c = 31.0 Å. Periodic cells were spaced with vacuum to negate any spurious interaction between adjacent QDs. The ground-state electronic structure, heating, and molecular dynamics (MD) simulations were calculated using the Vienna ab initio Software Package (VASP).29,30 For the ground-state electronic structure, optimization was carried out at 0 K using density functional theory (DFT) with the generalized gradient approximation (GGA) Perdew−Burke− Ernzerhof (PBE) functional,31 projector augmented wave function (PAW) potentials,32,33 plane wave basis set with an energy cutoff of 300 eV,34 and periodic boundary conditions. For MEG calculations hybrid functional HSE06 was used. The effect of spin−orbit coupling (SOC) in total is a small 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 (VB).35 In the case of any type of dynamic calculation, the small change in electronic structure results in a negligible force felt by any atom and small fluctuations in energies εj(t). Ambient temperatures of 100 K and 300 K were modeled by velocity rescaling technique. 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 (RDFs), defined as 1 RDF(t , r ) = 2 ∑IJ δ(r − |RI⃗ (t ) − RJ⃗ (t )|), with RI⃗ (t ) being 4π r positions of the I-th ion. 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,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 that the DOSVB is near four time higher than the DOSCB, prompting quicker relaxation for holes than that for electrons. The molecular orbitals at the band edges are comprised of primarily Pb and I, with contributions of atomic orbitals provided in the Supporting Information (SI), Table S1. Also from the DOS, the band gap energy of the QD, EQD g , is found to be 2.1 eV. The value of the confined material is predictably higher than that measured experimental and previously published computational band gap energies for is close to 1.60 ± periodic bulk MAPbI3 materials, where Ebulk g is expected as the material is .05 eV.16,17 The increase of EQD g spatially confined and shows signs of electronic state separation due to quantum size effects. The results of quantum size effects also manifest themselves in the absorbance spectrum, Figure
(1)
which resums perturbative electron−hole Coulomb interactions Wehe ′ h ′ to all orders, for the electron−hole bound state (exciton) energies Eα and wave functions ψehα . The exciton energies and wave functions are incorporated into the exciton-to-biexciton R1→2 and biexciton-to-exciton R2→1 rate calculations, where the core elementary process of electron-to-trion decay is included to the leading order in the random phase approximation (RPA) Coulomb interaction. The RPA-screened Coulomb potential involves polarization insertion resummed to all orders. Therefore, MBPT and the MEG technique, in particular, go beyond naive perturbative approach. Indeed, the KS orbitals parametrize the ground-state electron density and cannot be directly associated with single-particle states. However, they approximate electrons bound to the ions and include some electron interaction effects. Therefore, the 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-biexciton decay, including electron Coulomb interactions. The electronic state is propagated in time using the density matrix approach in a Redfield formulation.23 The electronic transition probabilities between two specified electronic states facilitated by nuclear vibrations24 are modeled using the nonadiabatic coupling (NAC) terms, Vij = ∑I ∫ ⟨φi|
dR ⃗ d |φ ⟩ I , dRI⃗ j dt
providing wave function overlaps,
which are calculated by an on-the-fly procedure, d Vij = ⟨φi(t )| dt |φj(t )⟩, 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 a transition probability between two specified states, as calculated through independent orbital approximation (IOA), ψijα ≈ δi , i ′ (α)δj , j ′ (α) + o..., allows for the RR rates to be found as defined by the Einstein coefficient of spontaneous transition. The Einstein coefficient for spontaneous emission, Aij, can be used to relate the oscillator strength of a specified transition and the lifetime of the corresponding transition. Aij is defined as Aij =
8π 2νij 2e 2 gi
f , and the relation to the lifetime of
ε0mec 3 gj ij
the emission is represented by τji = [Aji ]−1 =
const . νji 2f ji
Here ε0, α,
me, and c represent fundamental constants, vij stands for transition frequency. The rates of spontaneous emission, A21, are calculated in atomic units and converted to picosecond units. The atomistic model was built using three cubic unit cells of MAPbI3 in a planar arrangement exposing ⟨111⟩ surface. Additional iodine anions and methylammonium cations were 3033
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modes, comparing the total wave function at subsequent time steps of the MD trajectory allows for the calculation of on-thefly NAC between electronic states. The NAC 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 that 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 100 and 300 K in Figure 2A,B. The transition rate data show
Figure 1. (A) Graphical representation of the electronic relaxation, facilitated by electron−electron, electron−photon, and electron− phonon interactions; mechanisms of interest in this work, following the minimum PE, needed for MEG are presented. Red nonlinear arrows represent NR thermalization facilitated by vibrational modes, with NRR represented by the vertical dashed red line. An electronic transition from the conduction band (CB) to the VB corresponding to the radiative (R) relaxation mechanism is represented by a green vertical arrow. MEG is represented by an intraband electronic relaxation (black) accompanied by generation of a second electron (blue). (B) Atomistic model representing the methylammonium lead iodide perovskite QD. The atoms for carbon, hydrogen, iodine, nitrogen, and lead are colored gray, white, purple, blue, and silver, respectively; details on the geometry are provided in the SI. (C) Ground-state electronic DOS near the band edges with the occupied VB and unoccupied CB represented by the shaded and unshaded regions 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.
Figure 2. Numerical results on the NRR at different temperatures. (A,B) NR elementary event transition probability vs the absolute energy between two specified KS orbitals; an inset shows a visual representation of state-to-state electronic transitions near the band edges. Here the x-axis covers the range of [0−0.25] eV, with the value at ∼2.1 eV being the NRR rate. (C) Change in the distribution of electron density as a function of energy and time with respect to the electronic equilibrium following PE from HO to LU+27. (D) Summary of the integrated rates of hot carrier cooling for a range of initial excitations.
1D, as the absorption edge is blue-shifted in comparison to experimentally produced bulk spectra.36 To 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. The goal is to reach understanding of charge carrier dynamics, as discussed in the Methodology section therein. To provide a thorough and correct description of multiple competing relaxation processes, three methods are applied to calculate the relative rates of RR, EC, NRR, and MEG mechanisms. Each of the methodologies utilized have been previously defined and tested for systems of similar QD dimensions and sizes.26,27,37,38 As the large-scale goal is to correctly identify trends in a range of QD sizes, we utilized multiple methods to complete timely and efficient analysis. Nonradiative Channel. Microcanonical MD simulations at ambient temperatures of 100 and 300 K were computed to provide atomic positions {RI⃗ (t )} at each time step of their respective trajectories. The MD trajectories for 100 and 300 K are provided in SI, and thermalization effects on KSO energies due to nuclear motion at each temperature are provided in Figure S1 (SI). An MD trajectory provides a time-dependent electronic structure, allowing for calculation of a change of linear absorption, as shown in Figure S2, and the impact of thermalization on KSO energies. Upon predicating that EC relaxation pathways are facilitated by interaction with phonon
two regions of higher transition rates, corresponding to nearadjacent states, due to availability of low-energy vibrations to facilitate the transitions. The trend is expected as gap law predicts the fastest EC transitions to occur between states closest in relative energy.39,40 The groupings of the fastest rates are below 0.25 eV, representing intraband 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 insets are (1) away from the diagonal, the probabilities decrease to zero very quickly, showing that transitions are more likely to occur between adjacent states, and (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, one propagates the density matrix in time with initial conditions for the non-equilibrium electronic distributions specified by individual electronic transitions. 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 distribution of electron density as a function of time and energy in relation to the ground-state equilibrium. A single-exponential 3034
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Figure 3. Observables representing the energy of excitation (A), R, the time integrated emission spectrum (B), and MEG (C) pathways in the photoexcited MAPbI3 QD. (A) Dynamics of the exciton energy distribution dissipating as a function of time on a logarithmic axis shows the population of excitation at the specific energy where the maximum is red and the 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) MEG and Auger recombination rates as a function of energy with respect to the band gap energy, Eg. The rate intensity for two processes, a single exciton → biexciton (green, MEG) and two excitons → a single exciton (blue, Auger recombination) are plotted together, showing rates on a scale of 1014 s−1.
function is used to fit the expectation values of electron energies (dashed line in Figure 2C), producing the rate for EC relaxation (hot carrier cooling). An equilibrium value of electronic population, the gain in population (electron), and the loss in population (hole) are represented by green, red, and blue areas, respectively. Initial PE occurs on the left-hand side of the plot at t ≤ 0. For each specific initial excitation, the electron lifetime, τe, can vary as changing excitation energies will make electrons and holes to populate a different sequence of states within the CB and VB. After a certain period of time, both charge carriers reach their respective band edges and, under an approximation of no recombination, the expectation value of excitation energy remains constant. Results from calculation of EC relaxation rates provide a time scale of electron relaxation on the order of 1 ps at room temperature, with individual rates for specific photoexcited transitions given in Table S2 and in Figure S4 (SI). 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 a sparse DOSCB band edge. Rates provided in Table S2 represent initial excitations corresponding to the most probable of calculated optical transitions. The initial states for each transition do not necessarily represent an excitation identical to the one seen in experiment. Specifically, calculated rates for excitations from the VB edge to the CB edge have relaxation times close to 1 ps. Previous work by Piatkowski et al. showed electron cooling on the order of 750 fs for a given excitation at 450 nm.41 A comparison of EC relaxation rates at 100 and 300 K shows relaxation rates for both electrons and holes being slower at 100 K in respect to the same transitions at 300 K. This is expected as a smaller number of normal modes are activated at low temperature, which affects the rates of 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.40 This is known more commonly as gap law and is related to the trend (i). When looking at rates of EC relaxation at two temperature regimes, there is discord between the two temperature data sets. For the 300 K system, rates of electrons and holes follow the expected trend. However, for the low-temperature system, the calculated rates can be viewed in two regimes. (iii) For high-energy excitations, the rates of electrons and holes at 100 K follow the expected near-linear
gap law. (iv) At low excitation energies, the rates for the 100 K system deviate from a linear trend. The rates begin to slow, providing longer lifetimes with smaller excitation energies. One explanation for this change in trend is as follows. When excited to higher states in the CB, the relaxation dynamics is affected by the coupling between the high-energy states and states very near the band edge. This would promote rapid relaxation via multiple parallel channels while bypassing numerous states within the CB. This type of relaxation would be very different from that in a low excitation channel. The low excitation channel would pass through a different sequence 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. Typically, the number of channels increase with excitation energy. Off-diagonal values of the Redfield tensor, visualized in Figure 2A,B, insets, are often small but nonzero thus can play an important role. As the EC relaxation rate dictates the rate of 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 generates 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. This approximation can be valid and eligible for specific systems including quantumconfined systems with no band dispersion, rigid inorganic systems, and systems without considerable nuclear reorganization. Nuclear reorganization is likely to be a more important factor with systems containing organics, light elements, 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.42−44 However, as orbitals near the gap are localized on Pb and I rather than MA, reorganization in models with randomly distributed orientations of MA cations is expected to have a lesser effect. An ideal analysis would include time-dependent excited-state MD to account for nuclear reorganization resulting from changing the localization of electronic charge. However, with a small Stoke’s shift value known for MAPbI3,45 emission is likely to occur from states 3035
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The Journal of Physical Chemistry Letters close to the band edges. This provides reasoning to believe that 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 PE. A graphical depiction of the value of the energy of PE 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 that the initially created electron and hole remain in their initially excited states for a time period close to 150 fs. As the charge carriers migrate toward their respective band edges, the energy of excitation decreases with time. Recombination is not considered during calculation of the excitation cooling, resulting in the final energy equivalent to EQD g being reached by the end of the cooling period near 5−10 ps; see details in Table S2. Displayed in Figure 3B is the timeintegrated emission during electronic relaxation, with the energy range shown corresponding to the visible region of the spectrum. Comparing panels A and B in Figure 3, the largest intensity of RR emission and the final excitation energy is in the visible region, equal to the energy of EQD g . The integrated emission also shows multiple RR transition intensities that have an energy value greater than the band gap, EQD g . The intensities of the RR transitions in the visible region show a high probability for photoluminescence (PL) at EQD g . is in agreement with the PL This higher intensity at the EQD g calculated with the molecular dynamics−averaged photoluminescence (MDPL) method, shown in Figure 1D. Rates of RR 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, const τji = 2 , where const is a product of fundamental constants,
QD at 100 K was calculated to be near 4%, which is extremely low relative to the calculation at 300 K. As the RR lifetimes are calculated based on the ground-state electronic structure, the LUMO−HOMO lifetime does not change from the 300 to 100 K 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 3 orders of magnitude larger for the 100 K model than that for the 300 K model. 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, NAC is rather expected to increase for higher temperatures because 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. Typically, thermal fluctuations of ions are expected to lead to a loss of wave function overlap for strong radiative transitions. This phenomena has been seen to increase PL in QDs at low temperatures.49 A second hypothesis to explain the reduced QY at 100 K may be that upon cooling the MAPbI3 QD structure experiences a phase transition to the orthorhombic phase that is known for faster NRR.50 Chances for such phase transition are checked in Figure S6. Thermal motion and phase transition affect the geometry and may change the relative energy offsets of states near the gap, thus affecting the NRR rates. Multiple Exciton Generation. The final electronic relaxation mechanism under consideration in this work is MEG. Quantum confinement is expected to facilitate MEG since interacting particles are confined within a small volume in close proximity to each other. Rates for MEG, calculated according to MBPT,37 converting an exciton to a biexciton, 1 → 2 (carrier multiplication), or a biexciton to an exciton, 2 → 1 (Auger recombination), are shown as a function of the excition energy, Eexciton, in units of EQD in Figure 3C. The plots in Figure 3C g represent MEG rates for the processes of 1 → 2 (green) and 2 → 1 (blue), with more details provided in Figure S7. Here, the QE of the 1 → 2 process can be defined as the ratio QE1→2 = (R1 + 2R2)(R1 + R2)−1, where R1 corresponds to photon-toexciton rate and R2 stands for photon-to-biexciton rate.22 Thus, when each photon results in a single exciton, QE1→2 = 1, while when each photon results in a biexciton, QE1→2 = 2. Computed values are found in the range 1 < QE1→2 < 2. From the calculated rates, generate any type of MEG process in the MAPbI3 QD, the initial value of excitation is required to be equivalent to twice the EQD g . This energy threshold required to create a second exciton in QDs is less than the predicted 3− 14,20 5 times the EQD Peaks in g for bulk semiconducting materials. the 1 → 2 MEG rate in Figure 3C correspond to favorable resonance conditions and strong Coulomb coupling between the initial single-exciton and final-biexciton states. The probability of increased MEG events with increasing initial excitation energy is the expected trend as the initial exciton now can potentially relax from the initially excited state via EC mechanisms to an electronic state that has a high probability for MEG. This may also be evident when considering the energies at which peak probabilities for MEG are found. The electronic relaxation events that provide the necessary energy for the second exciton may be more favorable between certain two states found based on electron density and orbital hybridization or spatial localization in a specific area of the QD structure. The resulting calculated probabilities of MEG as a function of Eexciton/EQD are found to be on the time scale of 1014 s−1, g
νji f ji
νji is the frequency of the transition, and f ji 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 10−100 ps, making the RR relaxation mechanism 1−2 orders of magnitude slower than EC relaxation. This result is expected as EC relaxation is generally considered to be faster than RR relaxation.46 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 0.102 ns, shown as the highlighted row of Table S3. With NRR found to be on the scale of nanoseconds, the RR lifetime is 30−40 times faster than the NRR, indicating that an emission event is much more probable than NRR. Given that the rate of RR recombination, κRR, and the rate of NRR recombination, κNRR, have been calculated, the quantum yield of PL (QY), QY = κRR(κRR + κNRR)−1, can be determined, −1 where κRR = τ−1 RR and κNRR = τNRR. At room temperature where τRR = 0.102 ns and τNRR = 3.6 ns, the QY is calculated to be just over 97%, which agrees with the range of QYs seen in fully inorganic types of CsPbX3 QDs.47,48 The QY for the MAPbI3 3036
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second task will be to model the QD with an interface available to account for potential electron transfer pathways.
making it the fastest mechanism under consideration in this study. The values of all rates must be kept in perspective as a single model of MAPbI3 QDs was studied. One expects the electronic structure to change with varying spatial confinement, and the direct effect of larger sizes of QDs will lead to faster EC relaxation and a potentially less effective MEG process. It is also important to note that the results as a whole were used to identify qualitative trends of competing relaxation processes. Even with potential shifts of quantitative values due to adopted approximations, qualitative trends for processes are expected to hold. The overall goal of this work is to benefit the PV and optoelectronic community, and the practical goal is to increase device charge collection efficiency. The rates provided allow for qualitative results of competing mechanisms to predict probable QYs. As this is computational work, the reported correlation between qualitative QYs has the potential to guide material choice to increase charge collection efficiency. One factor in PV applications utilizing MEG in systems silimar to the one analyzed here is that the MEG mechanism requires a noticeable energy to initiate. Absorption spectra computed within a bound exciton approach cover a larger portion of the solar spectrum, compared to the spectrum computed in the independent orbital approximation. However, consideration of the excited-state nuclear reorganization could contribute additional correction to the range of the absorbed portion of the solar spectrum. Given the higher energy needed to initiate MEG and the relative intensity of solar irradiation on earth above the energy threshold, further application may be available for space exploration. In summary, atomistic modeling of electronic relaxation processes within a MAPbI3 QD allows identification of mechanisms that are responsible for high QY and can be harnessed for increased efficiency in photovoltaic and optoelectronic devices. A comparison of relaxation pathways via MEG, EC, RR, and NRR mechanisms resulted in computed time scales following the trend of, τMEG < τEC < τRR < τNRR, with values of the order of 10−14 s > 10−12 s > 10−11 s > 10−8 s, 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 nonradiative EC relaxation by requiring multiple active phonon modes to dissipate the hot electron energy. The computed MEG rates demonstrate dependence on the excitation energy Eex/EgQD, reflecting resonant conditions between single and biexciton energies. As expected, initial MEG rates begin to peak near Eex/EQD g > 2 as it is the minimum energy for the conservation of energy threshold. Calculated QYs qualitatively indicate efficiencies close to 1 at room temperature, which is to be expected in QDs. 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 SOC. However, as the largest effect of SOC is seen in the electronic band structure of the CB, 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
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.6b03048. Methodology for computing radiative lifetimes, summary of numerical data for nonradiative lifetimes, data on the influence of nuclear motion on the electronic structure of perovskite QDs, and additional numerical details on multiple exciton generation calculations (PDF) Movie of molecular dynamics trajectories from microcanonical molecular dynamics simulations at ambient temperatures of 100 K (MPG) Movie of molecular dynamics trajectories from microcanonical molecular dynamics simulations at ambient temperatures of 300 K (MPG)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected];
[email protected]. ORCID
Dmitri S. Kilin: 0000-0001-7847-5549 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research has been supported by NSF CHE-1413614 for methods development. The authors thank the DOE BES NERSC facility for computational resources and 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 HighPerformance Computing system at the University of South Dakota. Thanks are also extended to Stephanie Jensen, Wendi Sapp, Adam Erck, Yulun Han, Brendon Disrud, and Bakhtyor Rasulev for collective discussion and editing. D.S.K. and D.J.V. thank Svetlana Kilina (NDSU), Sergei Tretiak (LANL), Amanda Neukirch, and Nikolay Makarov (LANL) for stimulating discussions. D.S.K. acknowledges support of the Center for Nonlinear Studies (CNLS) and the 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.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.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. K3-2016021). The calculations were partially performed at supercomputer cluster “Cherry” provided by the Materials Modeling and Development Laboratory at NUST “MISIS” (supported via a Grant from the Ministry of Education and Science of the Russian Federation No. 14.Y26.31.0005). D.S.K. thanks NSF 3037
DOI: 10.1021/acs.jpclett.6b03048 J. Phys. Chem. Lett. 2017, 8, 3032−3039
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