Photoinduced Single- and Multiple- electron. Dynamics Processes Enhanced by Quantum. Confinement in Lead Halide Perovskite Quantum.

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1 Supporting Information: Photoinduced Single- and Multiple- electron Dynamics Processes Enhanced by Quantum Confinement in Lead Halide Perovskite Quantum Dots Dayton J. Vogel 1, Andrei Kryjevski 2, Talgat Inerbaev 3,5, Dmitri S. Kilin 1,4 * 1. Department of Chemistry, University of South Dakota, Vermillion, SD, Department of Physics, North Dakota State University, Fargo, ND, L.N. Gumilyov Eurasian National University, Astana , Kazakhstan 4. Department of Chemistry and Biochemistry, North Dakota State University, Fargo, ND, National University of Science and Technology MISIS, Moscow, Russian Federation S-1

2 Methodology: For each nuclear configuration, {R I }, the optimized electronic ground state, calculated in the basis of Kohn-Sham orbitals, can be characterized by plotting the DOS and absorption spectra. The ground state absorption spectrum is calculated under the independent orbital approximation, considering only single electronic transitions between two specified orbitals, with details provided in SI. Calculations of dynamic optical properties including MDPL and time-integrated photoluminescence have been previously presented in detail. 1 Lifetimes of R transitions are calculated using the Einstein coefficient of spontaneous emission, A 21. As defined, A 21 = 8π2 v 2 e 2 g 1 f ε 0 m e c 3 g 12, where v is the frequency of transition, e, ε 0, 2 m e, and c are fundamental constants, f 12 is the oscillator strength for a specified transition, and g i is the degeneracy of population of state i. The rate of spontaneous emission is given as the inverse of the Einstein coefficient, τ 21 = [A 21 ] 1, which can be simplified as τ 21 = C 1 where C is the set of fundamental constants. This allows for the rates of spontaneous emission to be calculated from the transition energy and oscillator strength for a specified transition, which is available from the ground state electronic structure calculation. E 2 f 12 Estimation of Radiative lifetime: From the ground state electronic spectra one can calculate the transition energies and oscillator strengths for allowed transitions, which can be used to calculate radiative relaxation lifetimes. From the electronic ground structure one can first calculate the transition dipole moment between each set of electronic states, D ij = e φ KS rφ KS dr. The transition dipole moment, D ij, is then used to calculated the oscillator strength for the corresponding transition, f ij = 4πm eω ij D 3ħe 2 ij 2 where ω ij is the energy of the transition. S-2

3 The Einstein coefficient for spontaneous emission, A 21, can be used to relate the oscillator strength of a specified transition and the lifetime of the corresponding transition. A 21 is defined as A 21 = 8π2 v 2 e 2 ε 0 m e c 3 g 1 g 2 f 12 and the relation to the lifetime of the emission is represented by τ 21 = [A 21 ] 1 = ε 0m e c 3 g 2 1 = m ec 3 1 = C 8π 2 v 2 e 2 g 1 f 12 16π 2 v 2 e 2 f 12 v 21 2 f 12. Here ε 0 = πe2 2α 12 m 3 c e2, α =, and C represents the set of fundamental constants. The rates of spontaneous emission, A 21, are calculated in atomic units and converted to picosecond timescales. hc Nonadiabatic dynamics methodology: To correctly model charge carrier dynamics of a real system, one must correctly simulate ambient conditions. Interacting the system with a thermostat set at an ambient temperature simulates a constant temperature through velocity rescaling according to 2 M dr I I( dt ) N t=0 I=1 2 = 3 Nk 2 BT, where N is the number of ions in the system. After stabilization of the system temperature is reached, the recorded velocities are used as initial conditions for the MD simulation. Nonadiabtic coupling (NAC) terms, V ij = φ i d φ dr j dr I I, provide wavefunction overlap I dt which are calculated by an on-the-fly procedure, V ij = φ i (t) d φ dt j(t), between each time step along the ground state MD trajectory. As nuclei are repositioned along the MD trajectory, one can consider the effect of nuclear motion on the electronic degrees of freedom. Shifted nuclei perturb the localization of electron density in the system, providing new sets of probable transitions resulting in transition rate changes. Performing an autocorrelation of the NACs, T 0 M ijkl (τ) = T 1 dtv ij (t + τ)v kl (t), followed by a Fourier transform of the autocorrelation, Γ + ijkl = dτm ijkl (τ)e iω klτ and Γ ijkl = dτm ijkl (τ)e iωijτ, provide components of the S-3

4 Redfield tensor, R ijkl = Γ + ljik + Γ + ljik δ lj m Γ immk δ ik m Γ lmmj. The Redfield tensor is the average second order electron-phonon interaction perturbation term, providing electronic transition probabilities between two specified electronic states facilitated by nuclear vibration, ( dρ ij dt ) diss = kl R ijklρ kl. The Redfield tensor along with the Liouville operator, Lρ = [F, ρ ], where elements of Fock-matrix F ij = F ij 0 + V ij (t) D ij ε ij 0 cos(ωt), are used to parameterize the equation of motion of electronic degrees of freedom as seen in equation 1. dρ ij dt = i (F ħ ikρ kj ρ jk F ki ) + ( dρ ij k ) = (L + R)ρ (S1) dt diss Diagonalization of the Liouville-Redfield super-operator provides eigenvalues and eigenvectors of the density matrix, (L + R )ρ υ = Ω υ ρ υ. The superposition of eigenvectors,, is used to evolve the density matrix in time and solve for expansion coefficients, ρ ij (t) = c υ ij ρ υ e Ω υt υ. Expansion coefficients, c v ij = ρ ij (0) ρ v, are found at time zero for a specified transition, defined by initial excitation involving a photo-excited electron and hole pair. 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. The above discussion describes the methodology used in calculation of NR and R relaxation rates. 1,2 r u Computation of Multiple Exciton Generation: Rates of MEG were calculated using many body perturbation theory technique. 32 In the impact ionization process an energetic exciton with energy greater than the energy threshold of 2E g can decay into a pair of low-energy excitons and electrons. Conversely, a bi-exciton can recombine into a single exciton. The relative efficiency of the two processes is determined by the corresponding final state density, i.e., by the densities of bi-exciton and S-4

5 exciton states, respectively. The rate of MEG for the exciton state γ, R 1 2 E γ, is given in equation 2(a). R 1 2 (E γ ) = R p exch + R h exch + R p dir + R h dir, (S2a) and includes direct channel contributions R p dir and R h dir and exchange channel contributions R p exch + R h exch for particle(electron) and hole, respectively. The term for particle exchange channel contribution R exch p (E γ ) = 2 2π W α ħ h 3 e 1 h 1 h 2 (Ψ e2 h 2 ) β (Ψ e1 h 1 ) 2 γ αβ Ψ e2 h 3 δ(e γ E α E β ), (S2b) is expressed in terms of the wave function of the exciton state α = eh α Ψ eh e, h, in the basis on non-interacting KS electron e, and hole h states. Here exciton wave functions and energies, Ψ α eh, E α have been determined from the Bethe-Salpeter equation in Tamm-Dankoff approximation. Coul Dir e h Ψ α e h ([ε e ε h ] E α )Ψ α eh + (W ehe h + W ehe h ) = 0, Coul W ehe h = 8πe2 q n 0, V ρ eh(q n )ρ e h (q n ) 1 q n 2 Dir W ehe h 1 q n 0. q 2 n ( q n,q n ) = 4πe2 ρ V ee (q n )ρ hh (q n ) Delta function δ(e γ E α E β ) ensures energy conservation and provides resonance condition for the final bi-exciton state, in spirit of Fermi golden rule. The term for the hole exchange channel contribution is formulated in the similar way. R exch h (E γ ) = 2 2π W α β γ ħ e 3 h 1 e 1 e 2 Ψ e2 h 2 Ψ e1 h 1 (Ψ e3 h 2 ) 2 αβ δ(e γ E α E β ) (S2c) The screened Coulomb matrix element, W jlnk W jlnk = 4πe2 ρ V q ln(q n )ρ n 0 kj (q n ) (S2d) q 2 n ( q n,q n ) 1 S-5

6 is computed in momentum space and includes V the volume of the simulation cell, ρ ln (q n ) the transition density as a function of momentum ρ ji (p) = k φ j (k p)φ i (k), and ( q n, q n ) the RPA polarization function in the uniform medium approximation 32. The expressions for the direct channel contributions R p dir and R h dir are the same with W h3 e 1 h 1 h 2 and W e3 h 1 e 1 e 2 replaced with W h3 e 1 h 2 h 1 and W e3 h 1 e 2 e 1, respectively and multiplied by 1/2. The 1 2 rate as a function of energy is given by averaging over the initial exciton states with the 25 mev resolution. The recombination rate, 2 1, is given by the expressions similar to equation 2 where the initial and final states have been reversed. The NREL website provides the AM1.5 standard spectra for spectral irradiance. Integration of three different functions over the whole data range and also from the minimum (280 nm) to the equivalent of 4.2 ev (295 nm) provide relative percentages of spectral irradiance reaching the earth s surface at or above 4.2eV. It is found that a very minimal percentage reaching the earth s surface is able to initiate MEG. Given the region of the spectrum needed for MEG is small, PV may not be the best application. However, the percentage of spectra irradiance greater then 4.2eV increases when moving outside of the earth s atmosphere. This can give application to PV applications for satellites, or dictate a more directed application in optoelectronics or lasing on earth. As MEG is likely a low probability processes on the earth s surface, making NR relaxation the dominant process. As this is the case, electron collection efficiency will compete only with NRR and R emission. S-6

7 Atomic orbitals I(s) I(p) Pb(s) Pb(p) Pb(d) I(tot) Pb(tot) HOMO LUMO Table S1. Partial contributions of each type of atomic orbital to frontier molecular orbitals of perovskite quantum dot Transition States Transition Energy (ev) Nonradiative Lifetimes (ps) 100K 300K Initial Final ih ie t h t e t h t e Table S2. The top nine most probable electronic transitions are listed by decreasing oscillator strength, with their respective transition energies and charge carrier relaxation times for low and high temperature. State number 446 corresponds to the HOMO. S-7

8 Transition States Initial Final Transition Energy (ev) Radiative Lifetime (ps) 446, HO 474, LU , HO , LU , HO-7 465, LU , HO , LU , HO 467, LU , HO-8 465, LU , HO , LU , HO-9 465, LU , HO-1 457, LU , HO 447, LU Table S3. Radiative lifetimes, t ij, for the nine most probable electronic transitions, calculated as described below, are provided. The last row, which is highlighted, corresponds to the LUMO- HOMO radiative transition. Oscillator strength of the states slightly above the gap is higher than the one of the gap transition. This observation may lead to the following effect: A high intensity of the incident exciting light would populate lowers excitations. In this case the PL will be contributed by the bright states that are above the gap and thus the overall PL intensity would increase. So, it agrees with effect of PL brightening induced by excessive illumination. S-8

9 Figure S1. Time dependence of Kohn-Sham orbital energies of the MAPBI3 perovskite quantum dots along the micro-canonical MD trajectories at T=100K (a) and T=300K (b). Molecular dynamics trajectories are available online for 100K, 3 and 300K. 4 S-9

10 Figure S2. Snapshots of optoelectronic properties along T=100K and T=300K microcanonical trajectories. Snapshots are taken at t=100fs and farther with step of dt=200 fs. Both density of states and linear absorption spectra are perturbed by nuclear motion. S-10

11 Figure S3. Autocorrelation of non-adiabatic couplings along molecular dynamic trajectory at 300K. S-11

12 Figure S4. Numerical results on the nonradiative relaxation (NR) at different temperatures. Top panels (a), (b) represent elementary processes rates of transitins between pairs of orbitals. Lower panel (c) summarize integrated rates of hot carrier cooling for a range of initial excitations. The nonradiative relaxation rates exhibit several trends (i) All carriers show quicker relaxation at higher temperature. (ii) Holes relax faster than electrons, as prompted by higher DOS(VB) > DOS(CB). The dependence of the NR rates on excitation energy obeys less trivial trends: (iii) Electrons at high excitation energies demonstrate an expected trend, namely the higher the excitation the longer the relaxation (electron cooling). This trend matches gap law predictions and matches the trend in the elementary relaxation rates. This trend agrees with the fact of low density of states higher in the conduction band and signifies sequential character of the electron cooling cascade. (iv) at low excitation energies, both electrons and holes demonstrate shortening of the relaxation time with increase of excitation energy. This trend is opposite to the energy gap law. Possible explanation of this trend is that at higher density of states, the cooling may occur vial multiple parallel channels. The number of such channels increase with the excitation energy. S-12

13 300K 100K LU+4 LU+9 Figure S5. Electron population as a function of time as the excited charge carrier relaxes via non-radiative relaxation. Initial electron population for two transitions, LU+4 and LU+9, for both 100K and 300K systems. Cascade thermalization is clearly seen in the top right panel, having the excited electron passion sequentially through all states before returning to the conduction band edge. Identified relaxation channels show variation depending on initially excited state. S-13

14 100K MD Trajectory Figure S6. The left plot provides averaged Pb-I RDF values are shown for sections of the MD trajectory in 350 fs increments. The right plot is a comparison of the initial geometry (cubic), orthorhombic, and final 100K MD position. S-14

15 Figure S7. Auxiliary numerical data supporting the analysis of multiple exciton generation rate. (a) Density of excited states for single excitations (blue) and double excitations (green). (b) rates of carrier multiplication processes computed according to Eqs. 2(a)-(c). (c) Density of bound exciton states as explicit function of energy shows decrease of the gap in comparison to independent orbital approximation. (d) Spectrum of linear light absorption by bound excitonic states as explicit function of transition energy. The simultaneous modeling of bound exciton formation and nuclear reorganization has potential to modify the optical gap due to dielectric screening of the electron-hole interaction. S-15

1. Vogel, D. J.; Kilin, D. S. First-Principles Treatment of Photoluminescence in Semiconductors. J. Phys. Chem. C 2015, 119, 27954-27964. 2. Chen, J., Schmitz, A., Inerbaev, T., Meng, Q., Kilina, S.

16 1. Vogel, D. J.; Kilin, D. S. First-Principles Treatment of Photoluminescence in Semiconductors. J. Phys. Chem. C 2015, 119, Chen, J., Schmitz, A., Inerbaev, T., Meng, Q., Kilina, S., Tretiak, S.; Kilin, D. S. First- Principles Study of p-n-doped Silicon Quantum Dots: Charge Transfer, Energy Dissipation, and Time-Resolved Emission. J. Phys. Chem. Lett. 2013, 4, Vogel, D. J.; D., K. Molecular dynamics of MAPbI quantum dot at T=100K. (accessed 6/18/2016). 4. Vogel, D. J.; D., K. Molecular dynamics of MAPbI quantum dot at T=300K. (accessed 6/18/2016). S-16

Optical Properties of Semiconductors. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

Optical Properties of Semiconductors 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India http://folk.uio.no/ravi/semi2013 Light Matter Interaction Response to external electric