Nonadiabatic Molecular Dynamics Simulation of Charge Separation

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C: Energy Conversion and Storage; Energy and Charge Transport

Nonadiabatic Molecular Dynamics Simulation of Charge Separation and Recombination at a WS/QD Heterojunction 2

Yaqing Wei, Weihai Fang, Qiu Fang, and Run Long J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b10058 • Publication Date (Web): 13 Mar 2018 Downloaded from http://pubs.acs.org on March 18, 2018

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The Journal of Physical Chemistry

Nonadiabatic Molecular Dynamics Simulation of Charge Separation and Recombination at a WS2/QD Heterojunction Yaqing Wei,1 Wei-Hai Fang,1 Qiu Fang,1* Run Long1† 1

College of Chemistry, Key Laboratory of Theoretical & Computational

Photochemistry of Ministry of Education, Beijing Normal University, Beijing, 100875, P. R. China ABSTRACT: Two-dimensional transition metal dichalcogenides (TMDs), such as WS2, are appealing candidates for optoelectronics and photovoltaics. The strong Coulomb interaction in TMDs is however known to prevent electron-hole pairs from dissociating into free electron and hole. Experiment demonstrates that combination of WS2 and quantum dots (QD) can achieve efficient charge separation and enhance photon-to-electron conversion efficiency. Using real-time time-dependent density functional theory combined with nonadiabatic (NA) molecular dynamics, we model electron and hole transfer dynamics at a WS2/QD heterojunction. We demonstrate that both electron and hole transfer are ultrafast due to strong donor-acceptor coupling. Photoexcitation of the WS2 leads to a 75 fs electron transfer, followed by a 0.45 eV loss within 90 fs. Photoexcitation of QD results in 240 fs hole transfer, but loses only 0.15 eV of energy within one picosecond. The strong charge-phonon coupling and a

*

Corresponding Author Email: [email protected]

†

Corresponding Author Email: [email protected] 1

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broad range of phonon modes involved in electron dynamics are responsible for the faster electron transfer than the hole transfer. The electron-hole recombination across the WS2/QD interface occurs several hundred picoseconds, ensuing long-lived charge-separated state. Particularly, the hole transfer is threefold magnitude faster than electron-hole recombination inside QD, ensuing that QD can be excellent light-harvester. The detailed atomistic insights into the photoinduced charge and energy dynamics at the WS2/QD interface provide valuable guidelines for optimization of solar light harvesting and photovoltaic efficiency in modern nanoscale materials.

KEYWORDS: WS2/CdSe quantum dot heterojunction, charge separation and recombination, energy relaxation, nonadiabatic molecular dynamics, time-domain density functional theory

2


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1. INTRODUCTION Two-dimensional transition metal dichalcogenides (TMDs) are promising for optoelectronic and photovoltaic applications due to their excellent transport, electronic,1 and optical properties,2,3 arising from strong light-matter interaction. Interesting properties such as phase transition in TMDs bring novel physics and has been attracting significant research efforts.4,5 Most monolayer TMDs are direct band gap semiconductors and exhibit excellent light-harvesting. The band gap often occurs direct-to-indirect transition with the number of layers increasing. Due to quantum confinement effect, Coulomb interaction is poorly screened in TMDs, resulting in tightly-bound electron-hole pairs that are hard to separate into free charges participating photoelectric effect. Rationally stacking two different TMDs monolayer can form a type-II van der Waals heterojunction that maintains direct band gap behavior and increases the driving force,6,7 determined by the offset between the donor and acceptor conduction band minimum (CBM) for electron transfer and the offset between the valence band maximum (VBM) for hole transfer, and is beneficial for charge separation at the interface. Since TMDs alone are absent of driving for charge transfer, there are no experimental and theoretical works focusing on charge separation in homogeneously isolated materials. Indeed, experiment reported that the hole transfer from MoS2 to WS2 occurs within 50 fs.8 Zhao et al. revealed that ultrafast electron and hole transfer takes place within a subpicosecond in a MoS2/MoSe2 type II heterojunction by transient absorption measurement.9 They also reported the electron-hole recombination in MoS2/MoSe2 heterojunction10 takes place 3



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slower than isolated MoS210 and MoSe211 due to electron and hole wave functions localizing on two different materials in the heterojunciton.12 We demonstrated quantum coherence facilitates charge delocalization and separation by time-domain simulations.12 However, TMDs heterojunction themselves can only absorb a certain wavelength of solar spectrum because each monolayer has fixed bandgap. Compared to TMDs monolayer, colloidal quantum dots (QD) exhibit excellent advantages including high photochemical stability,13-15 high electron mobilities,16 and large absorption cross sections.17,18 In particular, QD’s tunable bandgap, achieved by modifying the size, shape and composition,19-22 allows them to absorb a broad range of solar light starting from UV region to NIR region. Hot-carrier generation and carrier multiplication provide opportunities to improve a solar cell’s voltage and current by reducing the loss of high-energy carriers.23 Phonon-bottleneck effect can further increase photoexcited electron lifetime and reduce energy losses.24 Stimulated by the complementary properties of TMDs and QDs, composites of TMDs with QDs have recently received significant attention for photovoltaic and photocatalytic applications.24-27 Many experimental efforts are focusing on the fabrication

of

hybrid

TMDs-QD

nanocomposites,

showing

enhanced

photon-to-electron conversion efficiency. 28,29 Experiments demonstrated that a hybrid MoS2-PbS QD phototransistor shows fast photoresponsivity due to using their integrated advantages of strong light absorption of QDs and high carrier mobility of MoS2,28 and a WS2/CdSe QD photocatalyst achieves efficient water splitting under visible-light irradiation.29 Recently, Boulesbaa and co-workers reported that an 4


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ultrafast electron transfer takes within 45 fs from WS2 to a core/shell CdSe/ZnS QD upon photoexcitation of WS2 monolayer, while the subsequent electron back transfer from the CdSe/ZnS QD to WS2 requires several hundred picoseconds.30 These experimental observations provide strong motivation for the development of QD-sensitized TMDs solar cells. The ZnS shell does not contribute to the band edge states in the core/shell CdSe/ZnS QD and has negligible influence on the charge separation dynamics in the WS2/QD hybrid system.31 CdSe QD substituting for the core/shell CdSe/ZnS QD in the hybrid system maintains the physical properties of the nanocomposites while reduces significantly the computational cost. To be utilization, rapid charge separation31 and slow electron-hole recombination32 are necessary to achieve high power conversion efficiency but the dynamics processes remain largely elusive. Therefore, an atomistic time-domain understanding of the underlying mechanism form charge transfer and recombination at the WS2/QD interface turns into an emergent task for the purpose of offering valuable guidance for design of high-performance photovoltaic devices. The present work combines nonadiabatic molecular dynamics (NAMD)33,34 with time-domain density functional theory (TDDFT)35 to investigate the photoinduced charge transfer and nonradiative electron-hole recombination at a type II WS2/CdSe QD heterojunction. The simulation shows that both the electron and hole transfer occur in subpicosecond time scale, due to strong donor-acceptor interactions and a broad range of phonon modes involved. The 75 fs electron transfer time scale, in excellent agreement with experimental data,36 proceeds several times faster than the 5



hole transfer, due to larger NA electron-phonon coupling. More importantly, electron losses its energy to heat on the similar time scale relative to the electron transfer, while the hole energy dissipates five times slower that the hole transfer, resulting in “hot” hole which is good for long-distance bandlike transport. The time scale for electron-hole recombination in the WS2/QD hybrid system is half of it inside QD itself due to strong NA coupling and small bandgap. The recombination takes several hundred picoseconds, showing excellent agreement with experimental data.30 The efficient electron and hole transfer guarantee that both WS2 and QD can be used as light harvesters. The long free electronlifetime of the QD indicates that efficient photovoltaic devices can be achieved with a higher QD concentration.

2. THEORETICAL METHODOLOGY The NAMD simulation is carried out by the mixed quantum-classical approach30,36 implementing the fewest switching surface hopping (FSSH) technique within the single-particle time-dependent Kohn-Sham (KS) density functional theory.37 This approach has been extensively applied to a variety of systems,33,38-41 including TiO2 sensitized by a semiconducting42 and metallic QD,33 a perovskite,33 and MoS2 monolayer,39 a QD/polymer hybrid,40 a superconductor,39 and isolated metallic nanoparticles.41 The paper is organized as follows. The next section describes briefly the theoretical background and computational details of the NAMD simulations. The results and discussion focus on the geometric and electronic structure, charge separation dynamics, and electron-hole recombination dynamics. 6


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The simulation results are directly compared with the available experimental data. A detailed description of the approach can be found elsewhere.42,43

2.1. Time-Dependent Kohn-Sham Theory for Electron-Nuclear Dynamics DFT express the ground state energy as a functional of electron density, ρ(, t), which is constructed from the occupied single-electron KS orbitals,

ρ(, t) = ∑ (, )

(1)

where is the number of electrons. Applying time-dependent variational principle to the KS energy leads to a set of coupled single-electron equations for the evolution of (, ). iℏ

(,)

= (, , ) (, ); = 1,2, … ,

(2)

By expanding the time-dependent single-electron KS orbitals, (, ) can be expressed in the basis of adiabatic KS orbitals, #$ %, ( )&, which are calculated via the time-independent DFT calculation for the current atomic positions R(t), obtained from the MD trajectory

(, ) = ∑$ '$ ( )#$ %, ( )&

(3)

Insertion of eq 3 into eq 2 gives equations for the expansion coefficients: iℏ

' ( ) (

= ∑$ ) ( )%*$ +($ + -($ &

(4)

7



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Here, *$ is the energy of the adiabatic state k, and -($ is the NA coupling between adiabatic states k and j. The NA coupling arises from the orbital dependence of atomic motions and represents the electron-vibrational interaction, and is calculated numerically as the overlap between orbitals j and k at sequential time steps.44 -($ = −/ℏ0 1( ∇3 1$ 4 ≈−

;ℏ

∇

-5 8 = −/ℏ 6 1( 7 7 1$ 9 - 8

%0 1( ( ) 1$ ( + ∆ )4 − 0 1( ( + ∆ ) 1$ ( )4&

(5)

The numerical calculation of the time-derivative term on the right-hand-side of eq 5 significantly reduces computational cost. The many-particle generalization of the above equations is presented in Ref 40.

2.2. Fewest Switches Surface Hopping Tully’s FSSH is the most popular SH algorithm for modelling photoexcitation dynamics.45 Under the mixed quantum–classical dynamics scheme, the lighter and faster electrons are treated quantum mechanically, while the heavier and slower nuclei are described classically. FSSH provides a description for the back-reaction of electron onto the nuclei and satisfies detailed balance between transitions upward and downward in energy.46 The probability of transition from state j state k within a time step interval - is given by47 d>($ =

∗ ? 3@ABC EBC FG

ABB

- ; H($ = '( '$∗

(6)

If the calculated d>($ is negative, the hopping probability is set to zero. A hop from 8


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state j to state k can happen only when the electronic occupation of state j decreases and the occupation of state k increases. Every time-step, a uniform random number between 0 and 1 is generated and compared to d>($ to determine whether hop happens or not.

To keep the total electron-nuclear energy conservation after a hop, the nuclear velocities along the direction of the NA coupling are rescaled. If a NA transition to a higher energy electronic state is predicted by eq 6, while the kinetic energy available in the nuclear coordinates along the direction of the NA coupling is insufficient to accommodate the increase in the electronic energy, the hop is rejected. This step leads to detailed balance between the upward and downward transitions in energy, resulting in the Boltzmann statistics and quantum-classical thermodynamics equilibrium.46 The classical path approximation (CPA) to FSSH48 assumes that the electronic dynamics depends on the nuclear evolution, while the nuclear dynamics is weakly dependent on the electronic evolution. Therefore, CPA-FSSH does not consider the back-reaction of electron on nuclei. The approximation is usually satisfied in condensed-phase systems, in which nuclei vibrates around equilibrium positions due to thermal fluctuations, and the electronic energy is rapidly dissipated among all vibrational degrees of freedom. This approximation to the original FSSH technique significantly reduces computational cost, since the majority of computational efforts of NAMD relying on the calculations of NA coupling is resolved using a single MD trajectory. Then, the hop rejection rule embedded in the original FSSH is replaced 9



with multiplication of FSSH probability upward in energy by the Boltzmann factor in this slightly simplified and more computationally efficient version of FSSH. The detailed description can be found elsewhere.43

2.3. Decoherence-Corrected Surface Hopping. FSSH is overcoherent due to classical vibrations and ignoring of nuclear wave functions, and therefore, it neglects decoherence induced in the electronic degrees of freedom by the quantum nuclei. In the limit of infinitely fast decoherence, this phenomenon in quantum mechanics is known as quantum Zeno effect.48 In the present case, decoherence correction49,50 necessitates to be implemented into FSSH because decoherence is extremely faster than the electron-hole recombination at the WS2/QD interface that occurs in several hundred picoseconds.30 In the current simulation, the TD-KS wave function (, ) is collapsed to an adiabatic eigenstate #$ %, ( )&, eq 3, on the decoherence time scale, as implemented in Ref.30 The collapse procedure realizes by resetting the off-diagonal matrix elements H($ to zero, entering the SH transition rate, eq 3. The collapse times are obtained by a sequence of random numbers sampled from the Poisson distribution with the characteristic time determined by the decoherence time. The probability of collapse onto the eigenstate k is given by the square of the coefficient '$ ( ) at the collapse time. The deocherence time is calculated as pure-dephasing time in the optical response theory.51,52 The atomic motions induced fluctuations in the electronic energy gap, ∆I, between the electron and hole are characterized by the unnormalized autocorrelation 10


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function (un-ACF), which is defined as C(t) = 〈∆I( )∆I(0)〉N

(7)

The brackets indicate canonical averaging. The second-order cumulant expansion to the optical response function gives rise to the pure-dephasing function.53 OPQRS ( ) = exp (−g( ))

(8)

where g(t)

Z

g(t) = X\ -Y X\ [ -Y )(Y )

(9)

Fitting eq 9 by a Gaussian gives the pure-dephasing time. Typically for condensed phase systems, the pure-dephasing times are within sub-100 fs.53,54

2.4. Simulation Details

The CdSe QD used in our study contains 33 Cd and 33 Se atoms. Cd33S33 is the smallest cluster that preserves the bulk crystalline structure, identified with experiment using mass spectroscopy by Ksuya and co-authors.55 At the same time, Cd33Se33 is a stable “magic” size cluster that has the correct size and structure to eliminate defect states and to “heal” the surface. This was established in the works by Galli and co-authors56 and confirmed further by Prezhdo group calculations.31,57-59 We placed the 66-atom QD above the periodically repeated WS2 monolayer to simulate the WS2/QD heterojunction whose ground state minimum geometry is obtained with geometry optimization. To screen off the artificial interactions, a 15 Å vacuum was 11



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added perpendicular to surface to separate the system from its periodic images. The coordinates of optimized QD and WS2/QD structures are listed in the Supporting Information (SI).

The geometry optimization, electronic structure, and adiabatic MD calculations are carried out by the Vienna ab initio simulation package (VASP),31 which utilizes the

nonlocal

electron

exchange-correlation

terms

generated

within

the

Perdew-Burke-Ernzerhof (PBE) functional60 based on the generalized gradient approximation. The projector augment wave method was used to represent the valence electron-ion interaction.61 The van der Waals interaction is described by the DFT-D3 method of Grimme to stabilize the hybrid system during geometry relaxation and adiabatic MD.62 The plane-wave energy cutoff was set to 400 eV. A 2×2×1 Monkhorst-Pack k-mesh was used for the geometry optimization,63 and a much denser 10×10×1 k-mesh was used to calculate the density of states,64 and 30 k-point paths between adjacent k-points along the high symmetry lines within the first Brillouin zone were used to obtain the band structure. The calculated the band structure shown in Figure S1 suggests the combined WS2/QD is a direct bandgap material at Γ-point. Furthermore, our simulation supercell contains many replicas of the units cells WS2, which is equivalent to including several more k-points of the unit cells. Most importantly, the key charge transfer dynamics under investigation take place at high energies involving manifold of states. The high density of the electronic states and a relatively large supercell enable us to obtain good sampling of these states.

12


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After relaxing the geometries at 0 K, repeated velocity rescaling was used to heat up the temperature of the isolated QD and combined WS2/QD heterojunction to 300 K , corresponding to the temperature in the experiment.30 Then, a 3 ps adiabatic MD simulation is performed at Γ-point in the microcanonical ensemble with a 1 fs atomic time-step. The adiabatic state energies and NA couplings are calculated at Γ-point for each step of the MD run. To simulate the photoexcitation charge separation and recombination dynamics, 500 initial system geometries are selected randomly as initial configurations of the system from the 3 ps adiabatic MD trajectory. In particular, the decoherence-corrected FSSH is used to simulate the electron-hole recombination.

3. RESULTS AND DISCUSSION

The time-domain ab initio simulations of the charge separation and recombination in the WS2/QD system provide a detailed atomistic picture of the electron/hole energy transfer, energy relaxation, and electron-hole recombination at the hetero-interface. The electron injection into the QD from the excited WS2 occurs on 75 fs while the hole injection to the WS2 from the excited QD takes places on 240 fs, followed by a 0.45 eV electron energy loss within 90 fs and 0.15 eV of hole energy loss within 1 ps respectively. The electron-hole recombination either across the WS2/QD or inside the QD alone requires 475 and 935 ps separately. Taken together, the four simulations provide a comprehensive description of the excited state dynamics at the WS2/QD hetero-interface and generate valuable suggestions for 13



photovoltaic device optimization.

The energy levels involved in the photo-induced charge transfer and recombination dynamics at the type-II WS2/QD interface are depicted in Figure 1. Photoexcitation of WS2 leads to electron transfer, while QD excitation results in hole transfer, A. Competing with the separation, the electron and hole can recombine in each material, B. Following the separation, the electron and hole recombination can take place at the interface, C. These processes together with electron-phonon energy relaxation occur in parallel and compete each other.

Figure 1. Diagram of the energy levels involved in the photo-induced charge separation and recombination dynamics. Absorption of a photon ]^ (]^ ) by WS2 (CdSe QD) results in charge separation A due to electron or hole transfer, respectively. Competing with charge separation, the weakly bound electron and hole can undergo electron-hole recombination B inside either material. Following the charge separation, the electron and hole can recombine at the interface C.

3.1. Geometric and Electronic Structure 14


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The interfacial geometry characterizes the interaction strength between WS2 and QD, and then influences the rates of the electron and hole transfer, as well as the competition processes including energy relaxation and electron-hole recombination. Figure 2a shows the side view of the optimized structure of WS2/QD heterojunction at 0 K, while Figure 2b presents a snapshot taken form adiabatic MD run at 300 K. At 0 K, both the WS2 monolayer and QD remain intact due to purely weak van der Waals interaction. At room temperature, CdSe QD changes a little since Cd and Se atoms are heavy. Thermal fluctuations induce significant distortion on the WS2 monolayer due to in-plane and out-plane of W-S motions, leading to decease in the WS2-QD distance. In particular, the average distance between the WS2 and QD decreases from 3.60 Å at 0 K to the 2.94 Å at 300 K, suggesting a strong coupling between the donor and acceptor materials and facilitating fast charge separation at the interface. To justify the validity of the CPA approximation used in this study, we have computed the optimized ground state and excited state geometries of the WS2/QD system, and compared the geometry differences to the thermal atomic fluctuations in the ground state. The excited state was obtained using constrained DFT, in which an electron was promoted across the bandgap. The averaged W-S and Cd-Se bond lengths for the optimized ground state geometries were 2.411 Å and 2.677 Å, while they for the excited state geometries were 2.412 Å in WS2 and 2.680 Å. In comparison, canonically averaged ground state bond lengths, 2.416 Å for the W-S and 2.708 Å for the Cd-Se, were much more significant than the difference between the ground and excited state geometries. 15



Figure 2. Side views of the simulation cell showing (a) optimized WS2/QD geometry at 0 K and (b) a representative geometry during molecular dynamics run at 300 K. Thermal motions reduce the WS2-QD distance, and hence, increase the donor-acceptor coupling and facilitate charge transfer.

Figure 3a shows the projected density of states (PDOS) of the WS2/QD composite calculated using the optimized geometry at 0 K. The PDOS demonstrates formation of a type Ⅱ photovoltaic heterojunction between WS2 and QD. The calculated bandgaps at the PBE level for WS2 and CdSe QD are 1.75 eV and 1.40 eV, in agreement with previous theoretical works using the same functional.30,65 The canonically averaged CBM and VBM offsets are 0.45 and 0.15 eV. Photoexcitation of WS2 and QD leads to electron and hole transfer respectively, accompanying with the excess energies of 0.45 and 0.15 eV lost to vibrational motions. The CBM and VBM are localized on two different components, QD and WS2, ensuring that the electron and hole wave 16


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functions decouple each other and achieve long-lived charge-separated state. Figure 3b shows the charge density of key orbitals involved in the charge transfer process. The vertical arrows pointing from panel 3b to panel 3a indicate the energies of these states. The left-hand-side two pictures represents the donor and acceptor states for the hole transfer, while the right-hand-side two pictures give the acceptor and donor states for the electron transfer. The charge separation dynamics is determined by the donor-acceptor interaction. Here, the donor state for electron transfer is significantly delocalized between the WS2 and QD, while the acceptor state is distributed on the whole QD. The strong mixing between electron and hole wave functions increase donor-acceptor coupling. On the contrary, the donor state for the hole dynamics is localized on the part of QD while the acceptor state is spread nearly uniformly across WS2 monolayer. This situation minimizes wave functions overlap and decreases donor-acceptor coupling. Strong donor-acceptor coupling facilitates rapid charge transfer.

17



Figure 3. (a) Projected density of states (PDOS) of the interacting WS2 and QD subsystems in the heterojunction. (b) Charge densities of the donor and acceptor orbital for the electron and hole transfer. The electron donor state for the electron dynamics is significantly delocalized between WS2 and QD while the acceptor state is distributed on the whole QD, increasing the donor-acceptor coupling. On the contrary, the donor state for the hole dynamics is localized on part of the QD while the acceptor is spread uniformly across the WS2 monolayer, decreasing the donor-acceptor coupling. The vertical arrows between panels (a) and (b) relate the donor and acceptor orbital densities to the energies. The iso-surface value is set to 0.00023 e/bohr3.

Electrostatic interaction between WS2 and QD leads to a significant charge redistribution at the interface, Figure S2 a,b. The Cd atoms of CdSe QD attract substantial electron density from WS2. The electron density in the WS2 sheet at the interfacial regions gets notably depleted, whereas the electron density in the WS2 18


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regions between the S atoms gets significantly enhanced. Such charge redistribution can both allow charge transfer during the photoexcitation and help subsequent charge transfer dynamics in the excited state.

3.2. Charge Separation Dynamics at the WS2/QD Interface The phonons participated into the electron transfer have higher frequencies than the hole transfer, Figure 4, due to the electron donor state localized on WS2, containing light S atoms, to be compared to the hole donor states localized inside CdSe QD, with heavier Se atoms, Figure 3b. The spectral density shown in Figure 4 is obtained by computing a Fourier transforms (FT) of the fluctuations of CBM and VBM offsets between the donor and acceptor states. The electron transfer dynamics couples primarily to the peak at 450 cm-1, while the hole transfer dynamics is dominated by the frequency of 135 cm-1. In addition, the electron transfer couples to several

higher-frequency

vibrations,

while

the

hole

transfer

couples

to

lower-frequency phonons. Higher frequencies create stronger NA coupling due to E5

faster atomic motions, −/ℏ0 1( ∇3 1$ 4 E , entering the NA coupling matrix element. Compared to hole transfer in Figure 4(a), the electron transfer in Figure 4(b) couple to higher frequencies. The dominant peak for electron transfer can be attributed to the out-of-plane S-W A1g mode at 450 cm-1. 66 The second main peak at 200 cm-1 is very close to the longitudinal-optical phonon of CdSe QDs at 205 cm-1.67 The very-high mode at 550 cm-1 may be the overtone of these low frequencies. For the hole transfer, the prominent peak at 135 cm-1 can be seen as Se-Cd vibrations in the CdSe QD. 19


68


The other peaks at 180 cm-1 and 250 cm-1 correspond to transverse optical phonon modes69 and out of phase Cd-Se-Cd modes in CdSe QD.70 It is should be noted that bond length fluctuations directly reflect geometry change and provide straight evidence for mutual atomic motion. We randomly chose three kinds of S-W and Cd-Se bond lengths along the MD trajectory, in which they are longer, nearly identical to, and shorter than the canonically average S-W and Cd-Se bond lengths. By performing FT of the chosen S-W and Cd-Se bond lengths, we obtained the frequencies associated with atomic motion, Figure S3. Since the WS2 sheet has same chemical environment, the spectral densities show almost identical peaks. While the atoms of the Cd33Se33 cluster experience different chemical environment, the peaks of the spectral densities display notable difference. A linear combination of the spectral densities obtained from the three kinds of Cd-Se bond lengths give a certain agreement with Figure 4. Increasing the combination number of spectral densities of bond lengths will increase the agreement of peaks with Figure 4. In particular, the peaks at 450 cm-1 are almost identical obtained from the two kinds of W-S bond lengths along the MD trajectory (cf. Figure S3a). Fluctuations in the Cd-Se bond lengths produce a broad range of low frequency modes below 300 cm-1 (Figure S3b), agreeing somewhat with frequencies shown in Figure 4. Both WS2 and CdSe QD vibrations take part in the electron and hole transfer because the initial and final states for the electron transfer and hole transfer arising from the WS2 and QD, and

vice versa. The out-of-plane displacement of S and W affects significantly the electron, hole, and energy relaxation dynamics, since they modulate the 20


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donor-acceptor separation and the electronic energies of the WS2. At the same time, the low-frequency Se-Cd vibrations alter the morphology of QD and wave functions distribution. Overall, both the vibrations involved in WS2 and QD are responsible for generating NA coupling.

Figure 4. Spectral density obtained from the Fourier transforms of the fluctuations of energy offsets between the donor and acceptor states for (a) the electron transfer and (b) hole transfer.

The photoexcitation dynamics of the charge separation and energy relaxation are presented in Figure 5, in which CBM+4 and VBM-5 were set to initial state for electron and hole transfer respectively. Parts a and b give charge transfer, while parts c and d display energy relaxation. The time constants reported in Figure 5 are obtained by exponential fitting, I ( ) = Α ab(− /Y). The fitting parameters listed in Table S1 indicate that the fill is in good convergence. The calculated 75 fs electron transfer 21



time shows excellent agreement with the experimentally measured data,30 which is 3 times faster than the hole transfer, occurring on 240 fs, due to the larger NA coupling, 14.6 vs 13.4 meV, Table1, and additional phonon modes involved, Figure 4. The averaged absolute NA couplings for electron and hole transfer are computed by averaging the coupling between CBM and CMB+4 and VBM and VBM-5 respectively. The same factors are responsible for the electron-phonon energy transfer faster than the hole-phonon energy relaxation, Figure 5c and d. Figure 5c shows intraband electron-phonon energy relaxation occurs in parallel with the electron transfer. While the hole-phonon energy relaxation is consistently slower than the hole transfer, requiring nearly one picosecond. The relatively long energy relaxation time leads to formation of “hot” carriers that can facilitate charge transport at long distance. The combined system has infinitely random configurations arising from change in rotation degrees between WS2 and QD. The current configuration has the smallest formation energy with testing several other geometries. Mutual geometry orientation of WS2 and QD may affect the donor-acceptor coupling, NA coupling as well as the rate of charge transfer. In general, charger transfer is accelerated if the mixing between donor and acceptor wave functions increases. On the contrary, the charge transfer is retarded when the wave functions overlap decreases.

22


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Figure 5. Charge separation dynamics. Top panels (a, b) show time-evolution decay of the population of the electron and hole donor states. Bottom panels (c, d) show evolution of the electron and hole energies. The energy relaxation is slower than population decay.

3.3. Electron-Hole Recombination In addition to charge separation, charge recombination plays a central role in affecting the performance of photovoltaic solar cells because this process constitutes the major pathway for charge and energy losses. The rate of recombination particularly affects the open-circuit voltage of a photovoltaic solar cell, and therefore, slow recombination is often needed. The QD CBM and WS2 VBM constitutes the initial and final states for the electron-hole recombination of the WS2/QD, whose orbital densities are localized on 23



the QD and WS2 respectively, the middle two pictures of Figure 3b. This situation diminishes electron and hole wave functions overlap and creates small NA coupling, as a result, leading to a slow electron-hole recombination across the WS2/QD interface. Figure 6 shows the spectral density of the fluctuations of the CBM-VBM energy gap for the WS2/QD and CdSe QD systems. Electron-vibrational interactions are of importance to the elastic and inelastic electron-phonon scattering. Both elastic and inelastic electron-phonon scattering affects the recombination. Inelastic scattering accommodates the electronic energy lost and induces a transition from the CBM to the VBM. Elastic scattering is known as pure-dephasing in optical response theory,30 which leads to loss of quantum of coherence formed between CBM and VBM and has subtle influence on the electron-hole recombination. Figure 6a illustrates that major frequency at 135 cm-1 arising from the Se-Cd acoustic modes,71 which contributes to create the NA coupling, inducing electron-hole recombination in the QD. Figure 6b shows that the WS2/QD combined system couple primarily to higher-frequency at 300 cm-1 originating from S-W in-plane E1g phonon modes at 288 cm-1.67 The second main peak at 135 cm-1 due to Se-Cd vibrations inside QD also provides additional channels for the recombination.72 The computed frequencies are related directly to the pure-dephasing function. Generally, more and higher frequencies induce faster dephasing.

24


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Figure 6. Fourier transforms of the CBM-VBM energy gap, identifying the frequencies responsible for the electron-phonon recombination in (a) isolated CdSe QD and (b) the WS2/QD interfaces.

Based

on the optical

response

theory73

and

second-order cumulant

approximation, the pure-dephasing functions are computed and shown in Figure 7. Shorter

coherence

time

favors

longer

electron-hole

recombination.

The

pure-dephasing functions are fitted by Gaussian, giving a 4.9 and 7.1 fs pure-dephasing time for the WS2/QD and CdSe QD systems, Table 1. In addition to participation of the broader and higher frequencies, the unnormalized autocorrelation functions (un-ACF) of the fluctuations of the CBM-VBM energy gaps, inset of Figure 7, also characterize the decoherence rate. The greater initial value of the un-ACF favors faster dephasing because the un-ACF initial values give the energy 25



Page 26 of 43

gap fluctuations squared. The magnitude of the gap fluctuations has a strong influence on coherence time.53 Here, the magnitude fluctuations in the WS2/QD heterojunction is larger than that of the QD arising from light sulfur atoms significant fluctuations of the electronic energy levels, leading to an accelerated loss of coherence.

Figure 7. The pure-dephasing functions for the CBM-VBM energy gap. The insets show the unnormalized autocorrelation functions.

Table 1. Average NA coupling, Pure-Dephasing Time, Nonradiative electron-hole recombination Time for QD and WS2/QD heterojunction NA coupling (meV)

Dephasing (fs)

Bandgap (eV)

Recombination (ps)

QD

2.0

7.1

2.1

935

WS2/QD

5.1

4.9

2.0

475

26


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The time-resolved electron-hole recombination across the HOMO-LUMO energy gap is shown in Figure 8. The population changes are small. Therefore, we employed a huge number surface hopping trajectories of 1000000 to ensure statistical reliability of the calculated population decay. Such large number of trajectories gives a very small population change to be on the order of 10-6 per time step. Hence, the 5ps NAMD run with the 1 fs nuclear integration time step can enable us to resolve the recombination time scale around 0.01×5×109 fs = 50 ns with a 1% uncertainly. In addition to the dephasing time, the bandgap, averaged NA coupling between CBM and VMB, recombination times are listed in Table 1. Since the PBE function often underestimates the bandgap, the bandgaps of the QD and WS2/QD used in the nonradiative electron-hole recombination are scaled to experimental values, 2.1 eV74 and 2.0 eV75,76 respectively. The calculated 935 ps time for electron-hole recombination in CdSe QD, are obtained using the exponential fitting: > ( ) = ab(− /Y). The smaller size of CdSe QD used here with respect to experiment that leads a larger bandgap to separate electron and hole wave functions. As a result, the electron-hole recombination is suppressed due to weakened NA electron-phonon coupling. This argument is in line with experiment in which the carrier lifetime of bare CdSe QD decreases with size growing.76 Thus, the calculated 935 ps electron-hole recombination time can compare with the experimental data of 750 ps.77,78 The recombination in the WS2/QD heterojunction is closer to half of the QD’s and longer than monolayer WS2,78,79 suggesting that combination of WS2 with QD can consistently reduce nonradiative electron-hole recombination and achieve long-lived 27



excited electron lifetime. The recombination time is longer for the QD due to larger bandgap and smaller NA coupling that competes successfully with the slower decoherence, Table 1. Particularly, the NA coupling constitutes the main factor that leads to the disparity in excited electron lifetime. The above simulations assumed that electron and hole are relaxed to the bottoms of bands. Electronic states within kBT of band edges can be accessed by electron and hole after that relaxation. Therefore, we included states that are in the energy range of (VBM- 3kBT) and (CBM+3kBT), and checked that higher energy states are not populated and not involved in the dynamics. For example, the occurrence of such pairs is expected with the Boltzmann probability on the order of e-6 ≈ 6.8× 10-3, because the nuclear dynamics is sampled by isothermal ground state molecular dynamics. The calculated time scales of electron-hole recombination involved many states are similar to the times across the HOMO-LUMO energy gap, Figure S4. The electron-hole recombination rate depends on the bandgap, NA coupling, and decoherence time. The small QD used in the present work localize the wave function on itself, weakening QD and WS2 coupling strength and NA electron-phonon coupling decreases too. The electron-hole recombination is suppressed across the QD/WS2 heterojunction. Increase the size of QD brings the electron and hole wave closer to each other, leading to increase in NA electro-phonon coupling. The coherent formed between the electron and hole states should increase at a given temperature because the vibrational modes distribute overall large amount of nuclei. As a result, the electron-hole recombination is accelerated. On the other hand, PBE functional 28


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generally underestimates the energy gap and make NA coupling stronger, the rate of electron-hole recombination becomes faster. The use of a hybrid functional would produce smaller NA coupling and better agreement with the experiment.80 It is impossible to perform quantum dynamics calculations on the current systems with hybrid functional due to large computational efforts. Therefore, we use simple PBE functional.

This

approach

provided

good

results

for

multiple

nanoscale

systems,12,33,39-41,54 including those containing Cd33Se33 QD31,59 and WS259 subsystems. It is indeed true that the charge/atom ratio is high in the simulation compared to experiment, because of computational limitations on the size of the simulation cell. At the same time, there is only one electron-hole pair in the simulation. Therefore, the current simulation does not include Auger type processes, in which two electron-hole pairs will interact and annihilate one of the pairs, creating a fast recombination channel. Such fast channels are absent in the simulation, and only the slow electron-hole recombination process is considered.

29



Figure 8. Electron-hole recombination dynamics at the WS2/QD heterojunction and CdSe QD respectively. The recombination occurs faster at the heterojunction than that of isolated QD.

An explicit electron-hole interaction, such as described by the Bethe-Salpeter theory, would be desirable. Unfortunately, the type calculations are extremely expensive. Literature reported that the time-dependent Bethe-Salpeter theory has been applied only to systems with fixed nuclei.79 The used approach involves electron correlation effects implicitly in the DFT functional. Presently, it is the most rigorous ab initio method available for modelling the charge-phonon coupled dynamics. Kilin and co-authors have shown that the bound excitons in perovskite QDs result in an offset to excitation energy with respect to the independent orbital approximation.81 In the present work, we are focusing on transfer of high energy electrons and holes, which involve quasi-continuum states at energies above the bound exciton energies of WS2 and QD respectively. For the electron-hole recombination, the electron and hole localize in different materials, indicating a rather weak electron-hole interaction. These features suggest that the single-particle approximation provides a good description of the charge dynamics occurring at the WS2/QD hetero-interface under investigation.

4. CONCLUDING REMARKS The qualitative conclusions obtained on this work depend on the WS2/CdSe QD interfacial properties, such as the donor-acceptor coupling, energy alignment of 30


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conduction and valence bands of the donor and acceptor materials, and donor and acceptor state densities. Replacement of either WS2 with another TMDs or CdSe with another QD might maintain these properties if the key energy alignment remains unchanged, although quantitative difference is expected. In summary, we reported a time-domain ab initio study of photo-induced charge transfer, energy relaxation and charge recombination at the WS2/QD heterojunction. The simulations directly mimic the time-resolved experiment. The obtained electron transfer time scale shows excellent agreement with experiment. Electron transfer from the WS2 monolayer to the CdSe QD is faster than the hole transfer occurring in the opposite direction, due to stronger NA electron-phonon coupling and participation of higher and a broader range of phonon modes. The electron-hole recombination at the WS2/QD interface is 3-4 orders of magnitude slower than charge separation, suggesting that WS2/QD hybrid composite is an excellent candidate for photovoltaic solar cells. The electron energy relaxation proceeds in parallel with the electron transfer and has 0.45 eV energy losses with 90 fs. Hole energy relaxation is slow and dissipates 0.15 eV energy to vibrations within one picosecond, and results in “hot” hole that favors charge transport at long distance. In particular, the extremely slow electron-hole recombination time close to one nanosecond inside the isolated QD further reduces energy losses to heat, indicating that efficient photovoltaic devices can be achieved with a higher QD concentration. The rapid charge separation and slow charge recombination at the WS2-CdSe QD interface provide the valuable insights for design of high-performance TMDs/QD 31



photovoltaic and optoelectronic devices with a rational of choice of ideal materials.

ASSOCATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website. Band structure and charge density difference of the WS2/QD combined system, Fourier transforms of the W-S and Cd-Se bond lengths in the WS2/QD combined system along the MD trajectory, expression for the fitting functions and fitting parameters, coordinates of the optimized WS2/QD and QD alone structures, and the input files for NAMD simulations.

ACKNOWLEDGEMENTS The authors acknowledge the National Science Foundation of China, grant Nos. 21573022, 5171101561, 21688102, 21590801, and 21421003. R. L. is grateful to the Recruitment Program of Global Youth Experts of China, the Beijing Normal University Startup Package, and the Fundamental Research Funds for the Central Universities.

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Nonadiabatic Molecular Dynamics Simulation of Charge Separation

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