Photoinduced Localized Hole Delays Nonradiative ElectronâHole

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Photoinduced Localized Hole Delays Nonradiative Electron-Hole Recombination in Cesium–Lead Halide Perovskites: A Time-Domain Ab Initio Analysis Jinlu He, Meng Guo, and Run Long J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b01266 • Publication Date (Web): 19 May 2018 Downloaded from http://pubs.acs.org on May 19, 2018

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The Journal of Physical Chemistry Letters

Photoinduced Localized Hole Delays Nonradiative Electron−Hole Recombination in Cesium–Lead Halide Perovskites: A Time-Domain Ab Initio Analysis Jinlu He,1 Meng Guo,2 Run Long1∗ 1

College of Chemistry, Key Laboratory of Theoretical & Computational

Photochemistry of Ministry of Education, Beijing Normal University, Beijing, 100875, P. R. China 2

Shandong Computer Science Center (National Supercomputer Center in Jinan), Jinan, Shandong Province P. R. China, 250101

∗

Corresponding author, E-mail: [email protected] 1

ACS Paragon Plus Environment

The Journal of Physical Chemistry Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ABSTRACT: All-inorganic perovskites have attracted intense interest as promising photovoltaic materials due to their excellent performance. Using time domain density functional theory combined with nonadiabatic (NA) molecular dynamics, we demonstrate that photoinduced localized polaron-like hole greatly delays the nonradiative electron-hole recombination relative to the structure with delocalized free charge of the CsPbBr3. This is because localized charge carriers diminish overlap between electron and hole wave functions and decrease the NA coupling by a fact of 6. In addition, the polaron formation increases the band gap of CsPbBr3, slowing down recombination further. The smaller NA coupling and larger bandgap compete successfully with the longer decoherence time, extending the recombination to tens of nanoseconds. The calculated recombination times show excellent with experiment. Our study reveals the atomistic mechanisms underlying the suppression of recombination upon formation of localized polaron-like hole, and advances our understanding of the excited state dynamics of all-inorganic perovskite solar cells.

TOC

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Hybrid organic−inorganic perovskite (HOIPs), such as CH3NH3PbI3 and CH3NH3PbBr3, have been extensively studied due to the rapid increase in power conversion efficiency from 3.81 to 22.1%,2 arising from their excellent properties that include long excited state lifetime and low electron-hole recombination rate.3,4 Rearrangements of organic cations can either form long-lived energetic charge carriers4 or suppress electron-hole-recombination.5-7 Utilization of these advantages can further enhance the photon-to-electron conversion efficiency of the HOIPs solar cells.8,9 Unfortunately, both migration of CH3NH3+ cation and moisture degradation cause the issue of long-term thermal stability of the HOIPs,10,11 that constitutes a major barrier to commercialization under the working conditions of solar cells.12 As alternatives, all-inorganic cesium lead halide (CsPbX3, X = Cl, Br, and I) perovskites have become promising photovoltaic materials, because they exhibit excellent thermodynamic stability and resistance to humidity.13,14 In addition, the CsPbX3 perovskites carry other prominent advantages, including high photoluminescence quantum yield,15 low threshold lasing,16 high carrier mobility and long diffusion length.17 The excellent properties stimulate their applications in solar cells,18-22 lasers,23-25 and light-emitting diodes.26-30 In order to achieve high performance of electronic, optoelectronic, and photovoltaic solar cell devices, one must to minimize nonradiative electron-hole recombination because it constitutes the main factor for charge and energy losses. Recently, Zhu and Podzorov have demonstrated that the large polarons might protect the charge carriers, because organic cations screen electron-hole Coulomb 3



interaction to retard recombination in HOIPs.31-33 Other researchers proposed that either rotation/rearrangements of organic cations produce ferroelectric and antiferroelectric domains,5,6,34 or tilting of the inorganic sublattice leads to the positive and negative charges confine in different localizations of the perovskites (polaron formation),35 can explain the prolonged excited state lifetime in HOIPS. Subsequently, Miyata et al. have provided the first time-domain view of the large polaron formation in both CH3NH3PbBr3 and CsPbBr3 perovskites regardless of the cation type, due to the deformation of the Pb-Br inorganic sublattice, that inhibits charge carriers scattering.36 Fafarman and coauthors suggested a similar viewpoint that the organic cations are not prerequisites for inhibiting recombination.37 However, the debate remains on polaron behavior between theoretical and experimental works.36,38 On one side, theoretical calculations have suggested that the large electron and hole polarons in CsPbBr3 are delocalized.36 On the other side, Santomauro et al. have shown experimentally that inorganic CsPbBr3 can form localized hole polaron and delocalized electron polaron upon photoexcitation using the time-resolved X-ray near-edge absorption spectroscopy (XANES), extending the charge carrier lifetime over one hundred nanoseconds.38 The XANES is able to reflect both the valence element and configuration of spectrum coordination structure via detecting adsorption edges for each shell of core electrons. Santomauro et al. probed the Br K-edge and the Pb L3-edge, in which the most intense features are due to core transitions from 1s to p-like final states and due to 2p to d-like final states respectively, and confirmed that hole is localized, forming small polarons, while electron appears as delocalized in the 4


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conduction band.38 In contrast, charge lifetime is about nanosecond in the free charge structure of CsPbBr3.39 If both electron and hole polarons are delocalized onto the entire material, the overlap of positive and negative charge wave functions remains largely unchanged compared with free charge structure. As a result, nonradiative electron-hole recombination time is highly impossible to experience substantial decrease. To unravel the nature of positive and negative charge polarons and the influence of polaron formation on nanoradiative electron relaxation in the experimentally studied CsPbBr3 perovskites,38 we perform time-domain ab initio simulations of the electron-hole recombination dynamics, highlight the nature of electron and hole polarons, directly mimic the time-resolved experiment.38 Motivated by the recent experimental and theoretical works,35, 37-40 we employ a combination of time-dependent density functional theory (TDDFT) with nonadiabatic molecular dynamics (NAMD) to investigate the nonradiative electron−hole recombination in CsPbBr3 systems with delocalized free charge and localized polaron-like hole. The obtained sub-nanosecond time scale for electron-hole recombination in the free charge CsPbBr3 system is in excellent agreement with experimental data.39 Upon formation of localized polaron-like hole, the electron-hole recombination takes over ten nanoseconds. This effect can be rationalized by decrease in NA coupling by a factor of 6, due to formation of localized polaron-like hole and delocalized electron that diminishes the overlap of ground and excited state wave functions. The increase in band gap further slows electron-hole recombination down. In the both free charge and localized polaron-like hole structures, the electron-hole 5



recombination is slow because the NA coupling is small and the pure-dephasing is short. The nonradiative electron relaxation is primarily driven by the low-frequency vibrations below 100 cm-1, with small contributions from the high-frequency vibrations above 300 cm-1. The atomistic time-domain simulations rationalize key factors affecting the nonradiative relaxation, provide important insights and suggest design principles to improve further the performance of all-inorganic perovskite solar cells. The simulations are performed with the decoherence-induced surface hopping NAMD technique,41 implemented within time-dependent density functional theory (TDDFT) in the Kohn−Sham representation.42-44 The lighter and faster electrons are treated quantum mechanically, whereas the heavier and slower nuclei are described semiclassically. Decoherence, known as the pure-dephasing in the optical-response theory,45 is needed because the decoherence time is dramatically shorter than the electron-hole recombination time.38 The approach has been used to study photoinduced carrier dynamics in a broad range of systems,7,46-50 including perovskites containing polaron,7 grain boundaries,46 and dopants,47 interfaced with TiO2,48 interacting with water49 and passivation with Lewis bases.50 A detailed description of theoretical method can be found in refs.51,52 The geometry optimization, adiabatic molecular dynamics (MD), and NA coupling calculations were carried out using the Vienna Ab initio Simulation Package (VASP).53 The electron exchange and correlation interactions were described with the Pedrew-Burke-Ernzerhof (PBE) functional.54 The projector-augmented wave (PAW) 6


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method55 was applied to treat the interaction between the ionic cores and the valence electrons. The energy cutoff of 400 eV in the plane-wave basis was used to converge energies and wave functions. The geometry optimization and MD are carried out at the direct band gap Γ-point. In order to obtain accurate density of states, a much denser 8 × 8 × 4 Γ-centered Monkhorst-Pack k-point mesh was adopted.56 After relaxing the geometry at 0 K, repeated velocity rescaling was applied to heat the two systems to 300 K for 2ps. Then, a 6 ps adiabatic MD simulation was performed in the microcanonical ensemble with a 1 fs atomic time-step. The evolution of the total energies of 6ps MD trajectory for the two systems is presented in Figure S1 in the supporting information (SI), whichs indicate that both the geometries are in equilibrium after heating the systems at 300 K for 6ps because the energies are convergence. The van der Waals interactions are described via the Grimme DFT-D3 method to stabilize both the structures during the calculations.57 The first 2000 geometries of the adiabatic MD trajectories have been selected as initial configurations for the NAMD simulations of electron-hole recombination using the PYXAID code.51,52 The optimized lattice constant of cubic phase (space group Pm3m) of CsPbBr3 is 5.899 Å, agreeing well with the experimental value of 5.870 Å.58 Since the tolerance factor of Cs-based perovskites is smaller than ideal perovskite due to large size Cs ions, for instance, which is 0.86 for CsPbBr3, leading to the structures that tend to form either nanocrystals or undistorted cubic perovskites at or near room temperature.59 In order to achieve a balance between computational cost of NAMD 7



simulation and electron and hole polarons formation requiring large simulation cell,60,61 and hence, we created a 100-atom (2 × 2 × 5) supercell, Figure 1a. We obtain localization of hole and delocalization of electron via significantly changing partial Pb-Br bond lengths and Pb-Br-Pb angles at both edges of the supercell,62 Figure 1b. While the electron and hole polarons in CsPbBr3 were created via adding or removing an extra electron from the neutral system, in which the Pb-Br-Pb angles and Pb-Br bond lengths after occurs slight change after geometry optimization. Therefore, Miyata et al. have obtained delocalized electron and hole polarons arising from small geometry deformation.36 Figure 1a presents the optimized CsPbBr3 geometry with delocalized free charge, in which the calculated average Pb-Br bond length is 2.950 Å, showing an excellent agreement with the experimental value of 2.960 Å.63 The calculated average Pb-Br-Pb angel is 180°. In the polaron-like structure, some of Pb-Br bond lengths increase form 2.950 Å to 3.017 Å after geometry optimization, highlighted by the ovals in Figure 1b. At the same time, the corresponding Pb-Br-Pb angles decrease from 180° to 153.0°. The distortions of inorganic Pb-Br sublattice may modulate electronic structure and influence electron-hole recombination.

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Figure 1. Optimized simulation cell of (a) free charge and (b) localized hole structures of CsPbBr3. The localized local structure (b) is created by distorting the inorganic Pb-Br sublattice. The ovals denote the distorted regions in (b).

Figure 2 shows that the projected density of states (PDOS) of the CsPbBr3 struectures with delocalized free charge and localized polaron-like hole, calculated using the optimized geometries. The PDOS is split into the contributions from Br, Pb and Cs orbitals. The PDOS illustrates that the highest occupied molecular orbital (HOMO) arises primarily from Br atoms. The Pb atoms contribute to it secondarily. While the lowest unoccupied molecular orbital (LUMO) primarily comes from the Pb atoms, agreeing with the previous works.64 Apparently, both the Br and Pb orbitals affect nonradiative electron relaxation across the band gap because they consititue intial and final states for the electron-hole recombination. Although Cs atoms do not contribute either to the LUMO or to the HOMO, the movement of Cs atoms should impact the iorganic Pb-Br sublattice geometry, affecting electron-hole recombination indirectly. The calculated direct band gap of CsPbBr3 at Γ-point for the free charge 9



structure is 1.85 eV using the PBE functional, agreeing with previous DFT calculation64 but showing a underestimation relative to the experimental value of 2.36 eV65 due to the well-known DFT problem. In the localized hole structure the band gap increases to 1.92 eV due to geometry change. Spin-orbit coupling (SOC) is important in CsPbBr3 because it contains heavr Pb and Br elements. The band gap of cubic CsPbBr3 calculated by the HSE functional without considering the spin-orbit coupling SOC gives a better agreement with the experimental value, due to the fortuitous cancellation of errors in the band gap.66,67 Although GW+SOC gives good band gap.68 GW calculations are much more expensive than hybrid DFT. More importantly, the nonradiative electron-hole recombination rate depends on the band gap, NA coupling, and conherence time. In contrast to the band gap, the NA couplings and the coherence times cannot be deternimed by calculation with a single geomtetry. Typically, the NA coupling is trajectory dependent, and a good statistical sampling is needed. The coherence time is computed from the autocorrelation function of band gap fluctuations, which also necessiates a sufficiently long trajectory. Because GW+SOC calculations are extremely computationally expensive, it is impossible to study quantum dynamics with GW+SOC. Thereofore, we use simple PBE funcitoal and it has provided good results in our previous quantum dynamics studies for all inorganic perovskites.47

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Figure 2. Projected density of states (PDOS) of (a) free charge and (b) localized hole structures of CsPbBr3. Zero energy is set to the Fermi level. LUMO is constituted primarily by Pb atoms, and HOMO arises mainly from Br atoms.

The nonradiative electron-hole recombination rate relies on the NA coupling between the intial and final states. The strength of NA coupling is deternimed by the both wave functions overlap iħϕ ϕ between the HOMO and LUMO and the nuclear velocity d /dt. Since both systems have a direct band gap at Γ point, the charge densities are calculated at this special point. Shown in Figure 3a for the CsPbBr3 with free charge, HOMO is delocalized on both Br and Pb atoms and LUMO is distributed on the Pb atoms within the entire simulation cell. The situation provides good wave funcitons mixing between HOMO and LUMO and is beneficial for enhencement of NA coupling. In the loclaized polaron-like hole structure, however, HOMO is just localized on the Br and Pb atoms of both edges (polaron formation), while the LUMO is delocalized on the whole cell (Figure 3b), achieving good 11



agreement with experiment.38 This behavior remains true at room temperature because the inorganic Pb-Br sublattice is rigid and occurs small distrotions during thermal fluctuations, shown in Figure S2. Smaller overlap of wave funcitons between HOMO and LUMO leads to weaker NA electron-phonon coupling. In order to check the localization of hole originates from the fact that edges have different symmetry realtive to the (2 × 2 × 5) supercell, we constructed the polaron-like geometry with a (2×2×6) supercell via via changing of Pb-Br-Pb angles and Pb-Br bond length. The calculated the charge densities of HOMO and LUMO show that hole is localized on both edges of the simulation cell while electron is delocalized onto the whole system (Figure S3), agreeing well with Figure 3. Therefore, the key point for formation of localized hole is due to the inorganic Pb-Br sublattice distortions rather than simucation cell symmetry. In addition, we compute the canonically averaged standard deviation of the position of each atom i,

σi =

ur ur 〈 ( ri − 〈 ri 〉 ) 2 〉

ur r , here i represents for

the location of atom i at time t along the 6 ps MD trajectories. Then we averaged the standard deviations over all Pb and Br atoms of the two systems by the formula shown in the angular bracket. The data summarized in Table 1 demonstrate that the standard deviations in the positions of Pb and Br atoms decrease in the localized polaron-like hole structure compared with the free charge structure, decreasing the NA electron-phonon coupling further. Therefore, polaron formation increases the band gap and decreases NA coupling, Table 2, which should suppress nonradiative electron-hole recombination.

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Figure 3. Charge densities of the HOMO and LUMO charge densities in CsPbBr3 of (a) free charge and (b) localized hole structures. Localized polaron-like hole in (b) weakens NA coupling and delays nonradiative electron-hole recombination.

Table 1. Standard Deviations in the Positions of Pb and Br Atoms in the CsPbBr3 with Free Charge and Localized Polaron-like Hole. Pb

Br

Free Charge

0.304

0.573

Localized Hole

0.230

0.379

Electron-vibrational

interactions

induce

both

elastic

and

inelastic

electron-phonon scattering. Elastic scattering destroys the quantum coherence formed between the HOMO and LUMO. Inelastic electron-phonon scattering leads to the electronic transition from LUMO to HOMO accompanying with energy dissipating into heat. Both of them affect the nonradiative electron decay. In order to identify the phonon modes participating into charge recombination, we computed the spectral 13



densities by performing Fourier transforms (FT) of the fluctuations for HOMO-LUMO energy gaps. Figure 4 shows that the low-frequency vibrations primarily induce the nonradiative electron relaxation in the both systems. The dominant peak at 72 cm-1 in both the free charge and localized hole structures can be ascribed to the vibrations of the Pb-Br inorganic octahedron,69 which contributes to create the largest NA electron-phonon coupling. The second major peak at 160 cm-1 in the free charge system can be seen as the Pb-Br bond stretching mode.63,70 This mode creates mild NA coupling. Due to the symmetry breaking with distorted lattice, the intensity of the dominant peak at 72 cm-1 is suppressed and the second major peak vanishes in the localized hole structure of CsPbBr3, leading to a weaker NA coupling and slower dephasing, Table 2. For the both systems, the other peaks at 33 cm-1 and 116 cm-1 can be attributed to the Br6 distortion71 and the motion of the Cs+ ions respectively.69 The weak peak at 320 cm-1 is related to the second-order phonon mode of the octahedron.72 These vibrations modulate the geometries and contribute to marginal electron-vibrational coupling.

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Figure 4. Spectral density obtained from Fourier transforms of the autocorrelation functions for the HOMO−LUMO gap fluctuations of CsPbBr3 with free charge and localized polaron-like hole.

Figure 5 shows the pure-dephasing functions,45 calculated using the second-order cumulant approximation.73 The functions characterize elastic electron−phonon scattering. The pure-dephasing times, τ, obtained by fitting with the functions to a Gaussian, exp0.5t⁄τ , were summarized in Table 2. The pure-dephasing times are short, 5.8 fs and 10 fs. The factor of around 2 decrease in the pure-dephasing time for the CsPbBr3 with free charge can be attributed to higher intensity of the major phonon modes and more types of vibrational modes capable of interacting with the electronic subsystem. The computed unnormalized autocorrelation functions (un-ACF) for the fluctuations of the HOMO−LUMO energy gaps (inset of Figure 5) further reveals the factor responsible for the shorter pure-dephasing proceeding in free charge system than in localized polaron-like hole system. Under the cumulant approximation, the dephasing function on is computed by integrating un-ACF. Typically, a bigger initial value of un-ACF and its slower decay are responsible for a shorter coherence time.73 A the same time, the faster nuclear motion favors faster deaphsing.74 Free charge system exhibits larger initial value of un-ACF and slower decay (Table 1), leading to faster pure-dephasing. The pure-dephasing times are sub-10 fs, which are significantly smaller than the nonradiative electron-hole recombination times of a few nanoseconds, Table 2. 15



Figure 5. Pure-dephasing functions for the HOMO−LUMO transition in the free charge and localized hole structures. The inset shows the unnormalized autocorrelation functions (un-ACF). The initial value equals to the bandgap fluctuation squared. The larger initial value results in faster pure-dephasing. Thus, it is essential to incorporation of decoherence into the NAMD simulation.75-77

Figure 6. Nonradiative electron-hole recombination dynamics in CsPbBr3 with free charge and localized hole. The calculated time scales are summarized in Table 2. The free charge structure exhibits faster relaxation, due to stonger NA coupling and 16


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smaller band gap.

Figure 6 presents the evolution of the population of the LUMO for both the free charge and localized polaron-like hole structures. In order to compare directly with experiment, we scaled the calculated band gap 1.85 eV of CsPbBr3 with free charge to experimental value of 2.36 eV65 by adding a constant. The same constant was applied to the CsPbBr3 with localized hole by assuming an identical error generated by PBE functional.78 The calculated electron-hole recombination in the localized hole structure proceeds over 15 times slower than that in the structure with delocalized charge carriers. The recombination time scales, τ, summarized in Table 2, are obtained via fitting the population decay to an exponent, = exp /. The electron-hole recombination time in the CsPbBr3 system with delocalized charge carriers is 0.74 ns, agreeing well with the experimentally measured data.39 Upon formation of localized hole, the recombination time increases to 11.35 ns. Because of the necessarily small size of the simulation cell for time-domain ab initio simulation, localized polaron-like holes are spatially closer each other between two edges. One may expect that the NA electron-phonon coupling and the charge recombination rates are overestimated. Indeed, the calculated electron-hole recombination time is shorter than the experimental value,38 highly because experimental data contain charge-diffusion processes inside perovskite samples. The retardation of electron-hole recombination in the polaron-like structure relative to free charge structure can be rationalized by factors of the NA coupling, band gap, and pure-dephasing time, Table 2. The NA 17



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coupling in polaron-like structure is 6-fold smaller than that in free charge structure, which constitutes the major factor to make the recombination time longer. The larger bandgap further slows electron-hole recombination. Both the factors compete successfully with the longer pure-depahsing time. The recombination is slow because the NA coupling is small, sub-3 meV, and the dephasing time is short, sub-10 fs. The results presented in this study rationalize experimental observations that photoinduced localized hole polaron extends excited-state lifetime, extend our research on how distortions of inorganic sublattice influence electron-hole recombination, and advances our understanding the key factors affecting the performance of all-inorganic perovskite photovoltaic solar cells. The qualitative conclusions obtained in this work rely on the CsPbBr3, such as the geometry deformation, charge localization, and quantum dynamics, which should be preserved for other all-inorganic cesium lead halide CsPbX3 perovskites because they have same space symmetry and similar electronic properties although quantitative difference are expected.

Table 2. Bandgap, Average NA Coupling, Pure-Dephasing Time, and Nonradiative Electron-Hole Recombination Time for Free Charge and Localized Polaron-like Hole Forms of CsPbBr3.

Free Charge

Bandgap

NA coupling

Dephasing

Recombination

(eV)

(meV)

(fs)

(ns)

2.36

2.55

5.8

0.74

18


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Localized

2.43

0.43

10

11.35

Hole

In summary, we have simulated nonradiative electron−hole recombination of CsPbBr3 system with free charge carriers and localized polaron-like hole structures using a combination of NAMD with ab initio real-time TDDFT. The simulations demonstrated that localized polaron-like hole and delocalized polaron-like electron are formed, arising from distortions of organic sublattice upon photoexcitation. Structure with delocalized charge carriers exhibit a smaller band gap and stronger coupling due to better overlap of the electron and hole wave functions. In contrast, structure with localized polaron-like hole increases band gap, diminishes overlap of ground and excited state wave functions, and decreases electron-phonon coupling by a factor of 6. As a result, both factors compete successfully with the longer pure-dephasing time, and extend the charge carriers lifetime to hundreds of nanosecond. The calculated electron-hole recombination times show excellent agreement with experiments. The recombination is slow because the NA coupling is week, smaller than 3 meV, and pure-dephasing time is short, smaller than 10 fs. The recombination in both systems is primarily promoted by low-frequency, less than 100 cm-1 modes, which also rationalize the small electron-phonon coupling. The study clarifies the influence of inorganic lattice distortions on the excited-state dynamics of perovskite-based materials, promotes our understanding of the key factors affecting 19



the performance of all-inorganic perovskite solar cells, and suggests an efficient way to control of charge recombination.

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work was supported by the National Science Foundation of China, grant Nos. 21573022 and 51861135101. R. L. acknowledges financial support by the Fundamental Research Funds for the Central Universities, the Recruitment Program of Global Youth Experts of China, and the Beijing Normal University Startup.

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