Modeling Electron Injection at Semiconductor-Molecule Interfaces

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Modeling Electron Injection at Semiconductor-Molecule Interfaces using First-Principles Dynamics Simulation: Effects of Nonadiabatic Coupling, Self-Energy, and Surface Models Lesheng Li, and Yosuke Kanai J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b01820 • Publication Date (Web): 03 May 2019 Downloaded from http://pubs.acs.org on May 3, 2019

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The Journal of Physical Chemistry

Modeling Electron Injection at Semiconductor-Molecule Interfaces using First-Principles Dynamics Simulation: Effects of Nonadiabatic Coupling, Selfenergy, and Surface Models Lesheng Li† and Yosuke Kanai* Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599

ABSTRACT: Excited electron transfer across semiconductor-molecule heterogeneous interfaces is central to various future electronic and optoelectronic devices. At the same time, firstprinciples modeling of such dynamical processes remains as a great challenge in theoretical chemistry and condensed matter physics for developing better understanding at the molecular scale. Excited electron transfer from a molecule to semiconductor surface is a particularly difficult case to model accurately because the initial state of such an electron injection process often lies deep within the dense manifold of the conduction band states in the semiconductor. Nonadiabatic couplings and energy level alignments at such interfaces as well as the finite size error of the surface model all play important roles in numerical modeling of electron injection via first-principles theory. Using representative interfaces between a well-defined hydrogenterminated Si(111) surface and series of covalently adsorbed conjugated molecules, we investigate the extent to which these theoretical and numerical considerations influence the description of electron injection at semiconductor-molecule interface.

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1. INTRODUCTION Electron transfer processes across semiconductor-molecule heterogeneous interfaces are ubiquitous in various future optical and electronic device applications.1-7 In particular, investigations of electron transfer from a adsorbed molecule to material surface at such heterogeneous interfaces are largely motivated by the development of photo-electrochemical cells in recent years.1-3,

5-6

The electron injection rate is often estimated using a theoretic

framework of simple donor-accepter model based on Fermi’s golden rule8 and Marcus theory.9 In the case of describing electron transfer from a molecule to material surface, commonly called electron injection, analytical model expressions are extended by including a large number of final states (acceptor state) that comprise the conduction band (CB) through integrating its density of states (DOS). A single-particle state description is widely employed for describing semiconductor-molecule interfaces,10-11 and the electronic coupling between initial and final states (i.e. CB states) of the electron transfer process is often assumed not to vary significantly even when the final state in the conduction bands changes (i.e. independent of electronic energy), for convenience.12 Therefore, relative energetic alignment of the donor molecular state with respect to the conduction band DOS plays a decisive role in determining the transfer rate in many cases.12-14 This simplification makes the analytical expression tractable and often used for analyzing experimental measurements and developing conceptual understandings. Nowadays, advanced first-principles simulations can be employed for determining many of the essential parameters (i.e. DOS, reorganization energy, etc) of the analytical model expression as we have demonstrated in the context of a new 4th-order kinetic model in our earlier work.12 Combined use of the analytical models with first-principles determination of their parameters helps us develop unbiased interpretation and understanding. At the same time, such an approach is still


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fundamentally limited by some key assumptions of the underlying kinetic model we use, such as having a constant electronic coupling that is independent of the conduction band energy, etc. With increasingly more powerful computers, it has become possible to perform numerical simulations of electron dynamics based on first-principles theory.11,

15-16

However, these first-

principles numerical simulations are also not without their own limitations. The adiabatic representation is widely used in numerical simulations, especially in the context of the fewestswitches surface hopping method.17-19 Unlike for a donor-accepter molecular system, at semiconductor-molecule heterogeneous interfaces, most of the adiabatic states (i.e. electronic energy eigenstates) of interest are spatially well localized on either side of the interface in most cases, which alleviates the inconvenient problem of defining mathematically unique diabatic states.20-21 Coupling between the initial and final states for the electron transfer process derives from the lattice/atom movement, and it is characterized by the nonadiabatic coupling (NAC) in this representation. NAC is also called derivative or vibronic coupling, and it is often obtained from quantum-mechanical electronic structure calculation nowadays. When using an analytical expression like the extended Marcus theory for studying electron injection at semiconductormolecule interface,12 it is common to assume that the coupling between initial and final states does not vary as a function of the conduction band energy as discussed above. However, recent first-principles simulations show that such an assumption may not be warranted. For instance, a recent work by Duncan and Prezhdo22 on electron injection from a molecule into TiO2 have indicated a strong dependence of NACs on conduction band energy, showing that the rate cannot be simply taken as proportional to the conduction band DOS. In our previous first-principles simulation work on the excited electron dynamics at H-Si(111):cyanidin interface,16 we also observed that NACs between the silicon surface and molecular state show significant variation


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with the conduction band energy. For first-principles simulations, having accurate description of the electronic energy level alignment between the molecular state and the conduction band DOS is an important challenge. Standard single-particle description from the Kohn-Sham (KS) density functional theory (DFT) calculation usually does not yield an accurate energy alignment. Our earlier work found that approximating the energy level alignments by using KS single-particle energies could result in as much as an order of magnitude error for the electron transfer rate in an extreme case of an interface between a single lithium atom and an extended boron nitride sheet.23 For studying semiconductor-molecule heterogeneous interfaces, another key approximation comes from the use of atomistic structural models to represent the semi-infinite bulk material surface. In any atomistic simulations of a semiconductor-molecule heterogeneous interface, the simulation could potentially suffer from the finite size error in representing the semi-infinite surface. This aspect has been rather challenging to examine in computationally expensive firstprinciples simulations. In this present work, we examine these key aspects of the popular firstprinciples simulation approach based on the fewest-switches surface hopping method for modeling electron injection process at representative semiconductor-molecule heterogeneous interfaces. In particular, we examine the extent to which NACs, electronic energy levels, and surface models influence the description of the electron injection, using the Si(111) surface and a series of phenyl-based organic molecules as the adsorbate molecule.

2. COMPUTATIONAL DETAILS We closely follow the first-principles dynamics approach based on the fewest-switches surface hopping method combined with first-principles molecular dynamics (FPMD) and GW method within the many-body perturbation theory based on Green’s function formalism as we have done


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in our recent works.16, 23-25 FPMD simulations were performed for 2 ps with a time step of 0.48 fs at 295 K, using the modified version of the Qbox code.26 The Kohn-Sham (KS) wavefunctions were represented in plane-wave basis using norm-conserving pseudopotentials27 with a kinetic energy cutoff of 50 Ry. Generalized gradient approximation parameterized by Perdew, Burke, and Ernzerhof (PBE)28 was used for the exchange-correlation (XC) functional. The KS singleparticle energies and NACs were obtained from the FPMD simulation using the numerical prescription by Hammes-Schiffer and Tully.18 In this work, we use the Γ-point only sampling of the Brillouin Zone (BZ) integration because the wavefunctions (i.e. Bloch states) can be made real due to the time reversal symmetry in the BZ, enabling us to enforce the phase continuity rather straightforwardly.16, 29 The use of k-points in calculating NACs, however, would introduce subtle technical difficulties because the wavefunctions are generally complex. When a numerical eigenvalue solver is used in computation, the eigenfunctions are determined only up to a complex phase, which makes NAC calculation not straightforward in practice. Fewest-switches surface hopping (FSSH) simulation was then performed within the classicalpath approximation (CPA) as described in Ref. 16. This allows us to use a large number of atomic trajectories for converging the ensemble-averaged quantities because the trajectories do not depend on the hops within the CPA (for details see Table S1 of the Supporting Information). First, an ensemble of 2116 trajectories (1 ps) was generated from various different temporal points in the FPMD simulation. Thus, each of these 1 ps-long trajectories starts with different positions and momenta for atoms. Then, 500 FSSH simulations were performed for each trajectory, converging the sampling of the hopping probability distribution using Monte Carlo method. Further details can be found in Ref. 16, 23-24.


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In the following Section 3.2, effect of the electronic self-energy within the single-particle description is discussed for modeling the electron injection. The KS single-particle energies from DFT calculations inherently do not account for changes on the single-particle states when an extra electron/hole is present in the system. In the quasi-particle (QP) description, this manybody effect is taken into account through the self-energy operator, which is energy-dependent and non-local (unlike the exchange-correlation potential of the KS equation). The QP energies were obtained using many-body perturbation theory based on the single-particle Green’s function, starting from the KS single-particle wavefunctions and energies, and the many-body corrections (MBCs) were computed on top of the KS single-particle energies to obtain the QP energies. The MBCs were calculated within the so-called “one-shot” G0W0 approximation30-32 to the self-energy, starting from the KS wavefunctions and eigenvalues (PBE XC approximation). The random-phase approximation was used for calculating the dielectric function, and the screened Coulomb interaction was calculated using the Godby-Needs plasmon-pole model.33-34 The G0W0 calculations were performed using the Yambo code,35 with the starting KS singleparticle states calculated using the Quantum Espresso code.36 Even within the G0W0 approximation, it is computationally prohibitively expensive to take into account the time/trajectory dependence of the MBCs. Therefore, we obtain the MBCs at the equilibrium geometry, and we apply the computed MBCs to correct the time/trajectory-dependent KS energies to obtain the QP energies along the atomic trajectories, i.e. 𝜀𝐺0𝑊0@𝑃𝐵𝐸 (𝑡) = 𝜀𝑃𝐵𝐸 (𝑡) +∆ 𝑖 𝑖 (𝑒𝑞𝑢𝑖𝑙.), where 𝜀𝑃𝐵𝐸 (𝑡) is the KS single-particle energy of state i at time t, and ∆ 𝜖𝑀𝐵𝐶𝑠@𝑃𝐵𝐸 𝑖 𝑖 (𝑒𝑞𝑢𝑖𝑙.) is the MBC of state i obtained from the equilibrium geometry. This approach 𝜖𝑀𝐵𝐶𝑠@𝑃𝐵𝐸 𝑖 has been successfully used in our previous works for investigating excited electron dynamics at heterogeneous interfaces.16, 23-24 In this work, we validated this approximation by calculating the


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MBCs at two different geometries from the FPMD simulation, and no differences were found. Further details of the G0W0 calculations including the time/trajectory dependence of MBCs and the convergence tests of the G0W0 calculations are discussed in the Supporting Information.

Figure 1. Top and side views of the simulation cell at the (a) H-Si(111):phenyl, (b) H-Si(111):phenylacetylene, (c) H-Si(111):styrene, and (d) H-Si(111):phenyldiacetylene interfaces. The H-Si(111) surface was modeled using a 3×3 super cell surface slab with 8 layers. The bottom three layers of the surface slab were held fixed in their bulk positions during the simulations.

A representative semiconductor-molecule heterogeneous interface between a hydrogenterminated Si(111) surface37 and a phenyl group was considered in this work. Several linker groups between the semiconductor surface and the phenyl ring were studied (phenyl, phenylacetylene, styrene, and phenyldiacetylene, as shown in Figure 1) for investigating how the proximity of the molecule to the surface influences the NAC and thus also the electron injection kinetics. The H-Si(111) surface was modeled using a 3×3 super cell surface slab with 8 layers. The bottom three layers were held fixed in their bulk positions during the simulations. The surface slab was separated from its periodic images in the surface normal direction by a vacuum


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region of at least 15 Å such that the interaction between the repeating slabs is negligible. Spatially-resolved density of states (DOS) for the conduction band states based on the KS singleparticle energies is shown in Figure 2, where the silicon surface conduction band minimum (CBM) is set as the reference energy of 0 eV. The KS states with the largest contribution from the phenyl ring of the adsorbed molecules, a π* state, are indicated with the white arrows in Figure 2, and their spatial characters are shown at the top of each figure. These electronic states are well localized within the adsorbed phenyl ring, and they are referred to as the molecular state for convenience throughout this paper.

Figure 2. Spatially-resolved density of states (DOS) for the conduction band states based on Kohn-Sham (KS) single-particle energies at the interface of (a) H-Si(111):phenyl, (b) H-Si(111):phenylacetylene, (c) HSi(111):styrene, and (d) H-Si(111):phenyldiacetylene. The spatial characters of the molecular states are shown at the top of each figure, and their energetic positions are indicated by white arrows in the DOS figures. The spatiallyresolved DOS is calculated by averaging electronic density in the surface plane, and the silicon surface CBM is set to 0 eV as the reference energy.


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3. RESULTS AND DISCUSSION 3.1 Nonadiabatic couplings Nonadiabatic couplings (NACs) are obtained from the FPMD simulations as discussed above and also in Ref. 18 and in our previous studies.16, 23, 29 Top panel of Figure 3 shows the timeaveraged magnitude of NAC matrix among all the conduction band (CB) electronic states and the molecular states above the Fermi level, where the state indices are ordered according to their time-average KS single-particle energies. The state index covers the energy range of approximately 3.7 eV, and the arrows in Figure 3 indicate where the molecular state is located. Most of the significant NACs are found for pairs between the electronic states with similar energies, near the diagonal elements of the NAC matrix.16 For clarity, the NAC values between the molecular state and CB states are shown at the bottom of Figure 3, and they are referred to as NACSi-Mol for convenience. The NACSi-Mol varies significantly as a function of the energy, and a rather small number of the CB states couple strongly to the molecular state for all the four interfaces.


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Figure 3. Time-averaged magnitude of the nonadiabatic coupling (NAC) matrix (in atomic unit) for the conduction band electronic states above the Fermi energy at the (a) H-Si(111):phenyl, (b) H-Si(111):phenylacetylene, (c) HSi(111):styrene, and (d) H-Si(111):phenyldiacetylene interfaces. Top panel: NAC matrices; bottom panel: NAC values between the molecular state and the semiconductor states. The maximum value of the NAC between the semiconductor states and molecular state is also shown in the bottom figures. The state index 1 corresponds to the conduction band minimum state of the semiconductor surface and the reference energy of 0 eV.


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Figure 4. Time evolution of the excited electron probability at the H-Si(111):phenyl, H-Si(111):phenylacetylene, HSi(111):styrene, and H-Si(111):phenyldiacetylene interfaces. The simulations were performed using (a) the correct NAC matrices and KS energies (NACCORRECT with ƐKS), (b) the maximum NAC matrices and KS energies (NACMAXIMUM with ƐKS), as well as (c) the correct NAC matrices and QP energies (NACCORRECT with ƐQP). The silicon surface CBM is set as the reference energy of 0 eV. The excited electron initially occupies the molecular state as indicated by P(t=0). See the text for details.


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Figure 5. Probability change for the excited electron in the initially occupied molecular state at the (a) HSi(111):phenyl, (b) H-Si(111):phenylacetylene, (c) H-Si(111):styrene, and (d) H-Si(111):phenyldiacetylene interfaces. The results calculated from the NACCORRECT with ƐKS, NACMAXIMUM with ƐKS, and NACCORRECT with ƐQP are shown in black, red, and blue, respectively.

We here examine the extent to which assuming a constant NAC across the semiconductor surface and the adsorbed molecule (i.e. no variation with the conduction band state) influences the electron injection kinetics as often done in using kinetic models for convenience.12 Figure 3 reveals that NACs between the semiconductor surface states and the molecule state (i.e. NACSiMol)

vary significantly as a function of the semiconductor’s conduction band energy, and it is

importance to assess the widely-used approximation of having a constant NAC across the


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interface. Although the largest NACSi-Mol derives from a semiconductor state that is energetically quite close to the molecular state, it is not always the closest semiconductor state that gives the largest NACSi-Mol. The energy difference between the molecular state and the semiconductor state that yields the largest NACSi-Mol value is 0.08 eV, 0.18 eV, 0.10 eV, and 0.01 eV at the interface of

H-Si(111):phenyl,

H-Si(111):phenylacetylene,

H-Si(111):styrene,

and

H-

Si(111):phenyldiacetylene, respectively. For comparison, we performed FSSH simulations also by intentionally making all the NAC matrix elements between the semiconductor states and the molecular state (i.e. NACSi-Mol) to be its maximum value, and we denote this change as NACMAXIMUM. This particular choice is motivated by the fact that the electron is injected predominantly into the semiconductor states that are close in energy from the molecular state, and these semiconductor states all show comparably large NACSi-Mol as seen in Figure 3. In the simulations, the excited electron was initially placed in the molecular state, and the KS singleparticle energies (ƐKS) were used here for the energy level alignments. The simulation results based on the correct NAC matrix with KS single-particle energies (NACCORRECT/ƐKS) and the maximum NAC matrix with KS single-particle energies (NACMAXIMUM/ƐKS) are shown in Figure 4a and Figure 4b, respectively. Figure 5 shows the probability change for the initially occupied molecular state, and the results based on the NACCORRECT/ƐKS and NACMAXIMUM/ƐKS simulations are shown in black and red, respectively. For the NACMAXIMUM/ƐKS simulation, the probability decay for the molecular state is extremely fast as seen in Figures 4b and Figure 5. We determine the electron injection time constant (𝜏𝐸𝐼, in fs) and rate (𝜆𝐸𝐼, in fs-1) by fitting the probability change for the molecular state to a single exponential decay (Figure 5), and there is no probability increase in other states that are mostly localized on the molecules. As summarized in Table 1, there is at least one order of magnitude difference between the NACCORRECT/ƐKS and


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NACMAXIMUM/ƐKS simulations. The NACMAXIMUM/ƐKS simulation in which all the NACSi-Mol values are set to the its maximum value results in a significant overestimation of the electron injection rate by a factor of 10.09, 50.79, 7.93, and 12.32 at the H-Si(111):phenyl, HSi(111):phenylacetylene,

H-Si(111):styrene,

and

H-Si(111):phenyldiacetylene

interfaces,

respectively. The NACCORRECT/ƐKS simulation shows that greater than 70% of the excited electron probability reaches the semiconductor states that are within the thermal energy kBT of the conduction band minimum in ~0.7 ps. In contract, it takes considerably shorter time of ~0.3 ps in the NACMAXIMUM/ƐKS simulation. Assuming the NACs between the semiconductor and the adsorbed molecule to be constant could result in a significant error even when the constant NAC value is calculated using a semiconductor state with close energy.

Table 1. Electron injection time constant (𝜏𝐸𝐼, in fs) and rate (𝜆𝐸𝐼, in fs-1, in the parenthesis) calculated by fitting the probability change in the initially occupied molecular state to a single exponential decay, according to different NAC matrices and energies (NACCORRECT, NACMAXIMUM, ƐKS, and ƐQP) for the 3×3 super cell with 8 layers interface models.

Chemical group

phenyl

phenylacetylene styrene

phenyldiacetylene

NACCORRECT, ƐKS

21.43 (0.047)

70.44 (0.014)

12.46 (0.080)

17.82 (0.056)

NACMAXIMUM, ƐKS

1.92 (0.521)

1.38 (0.725)

1.40 (0.714)

1.34 (0.746)

NACCORRECT, ƐQP

1.58 (0.633)

2.08 (0.481)

2.15 (0.465)

2.01 (0.498)

As shown in Table 1, we also note that there is no obvious trend for the electron injection rates as a function of the distance between the surface and phenyl group. The maximum NACSiMol

shown in Figure 3 does not simply follow a trend correlated to the inverse distance between

the surface and the phenyl group. At the same time, such a trend exists for the same pair of


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electronic states across the interface. The NACSi-Mol between the state at the bottom of the semiconductor CB (i.e. conduction band minimum) and the molecular state is shown in Figure 6, as a function of the distance from the H-Si(111) surface to the center of mass of the phenyl ring. Among the linker groups of phenyl, phenylacetylene, and phenyldiacetylene, the coupling strength decreases as expected with the increasing distance between the phenyl ring and the surface although the trend is not likely exponential as one might expect when the interaction is still perturbative.38-39 Our calculations show that the molecular state spatially overlaps quite strongly with the semiconductor state because of the relatively low work function of the surface although these states are still distinguishable. We also note that the H-Si(111):styrene interface does not fall into the overall trend because its linker group is of a different type in terms of the carbon atom hybridization. The carbon atoms in the linker group of the styrene are of sp2 hybridization while the carbon atoms in the phenylacetylene and phenyldiacetylene cases are sp hybridized.

Figure 6. Nonadiabatic couplings (NACs, in atomic unit) between the silicon surface conduction band minimum state and the molecular state as a function of the distance from the surface to the center of mass of the phenyl ring at the four interfaces.

3.2 Self-energy in the single-particle description


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Determination of accurate single-particle energy levels from DFT-KS calculation remains a grand challenge.23, 40-44 A more rigorous approach is to obtain the energy levels using the quasiparticle (QP) energies (ƐQP) instead of KS eigenvalues via a many-body perturbation approach in the framework of Green’s function theory. In the last few decades, GW calculation has become a practical computational scheme for this purpose. Mathematically, it amounts to replacing the local exchange-correlation potential of the KS single-particle equation with the energydependent non-local self-energy operator. The self-energy operator is approximated from the single-particle Green’s function, G, and the screened Coulomb interaction, W in the GW calculation. Using the so-called G0W0 approximation as we do here, the KS single-particle wavefunctions are assumed to be sufficiently accurate representations of the QP amplitudes/wavefunctions (i.e. Dyson orbitals), and this is generally a valid assumption for relatively simple systems like silicon surface and organic molecules.45-48 We examine the extent to which having more accurate energy level alignments influences the excited electron injection kinetics by employing QP energies in the FSSH simulation. The spatially-resolved DOS at the four interfaces based on QP energies are shown in Figure 7, and the time evolution of the probability for the excited electron that initially occupying the molecular state is shown in Figure 4c. The probability change of the molecular state from the NACCORRECT/ƐQP simulation is shown by the blue curves in Figure 5. The initial decay in the NACCORRECT/ƐQP simulation is an order of magnitude faster than the NACCORRECT/ƐKS simulation, Table 1. Compared to the simulation based on KS energies (NACCORRECT/ƐKS), the electron injection rate in the simulation based on the QP energies (NACCORRECT/ƐQP) is significantly faster by a factor of 12.47, 33.36, 4.81, and 7.89 at the H-Si(111):phenyl, H-Si(111):phenylacetylene, H-Si(111):styrene, and HSi(111):phenyldiacetylene interfaces, respectively. These results show that having accurate


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energy level alignments at such semiconductor-molecule interfaces is of great significance for obtaining the electron injection rate, and continued development of accurate first-principles electronic structure methods for modeling extended systems is indeed central to electron dynamics simulation.

Figure 7. Spatially-resolved DOS for the conduction band states based on the QP energies calculated using GW method at the (a) H-Si(111):phenyl, (b) H-Si(111):phenylacetylene, (c) H-Si(111):styrene, and (d) HSi(111):phenyldiacetylene interfaces. The spatial characters of the molecular states are shown at the top of each figure and their energetic positions are indicated by white arrows in the DOS. The spatially-resolved DOS is calculated by averaging electronic density in the surface plane, and the silicon surface CBM is set to 0 eV as the reference energy.

3.3 Semiconductor surface models For atomistic simulations of excited electron dynamics at semiconductor-molecule interfaces, having a realistic representation of the semi-infinite semiconductor surface is a practical


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challenge. The accuracy of numerical simulations is limited not only by the level of theory we employ but also by the surface model, particularly by the finite size error even when the periodic boundary conditions are adapted. The semiconductor surface is often modeled using a surface slab such that it has a finite thickness in the surface normal direction and the lateral directions of the surface are modeled using the periodic boundary conditions. We examine here how the semiconductor surface model influences the simulation results by reducing the finite size error. In addition to the 3×3 super cell surface slab with 8 layers (622 electrons) we have employed above, we now consider three relatively large surface slabs of 3×3 super cell with 12 layers (918 electrons), 4×4 super cell with 8 layers (1092 electrons), and 6×6 super cell with 8 layers (2412 electrons) as shown in Figure 8. Because of the great computational cost associated with these large surface calculations, we only considered the interface case of H-Si(111):phenylacetylene as a representative case, and the FSSH simulations were performed only with the KS energies because preforming G0W0 calculations to obtain the QP energies on these large surface slabs are currently not feasible. The spatially-resolved DOS of these three large surface models are detailed in Figure S3 of the Supporting Information.


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Figure 8. Top and side views of the simulation cell at the H-Si(111):phenylacetylene interface model with different surface slabs of (a) 3×3 supercell with 12 layers, (b) 4×4 supercell with 8 layers, and (c) 6×6 supercell with 8 layers. The bottom three layers were held fixed in the bulk position during the simulations.

The time evolution of the excited electron probability is shown in Figure 9 for these three large surface models. In the simulations, the excited electron initially occupies the same molecular state (see Figure S3 of the Supporting Information). The probability decay for the initially occupied molecular states is shown in Figure 9d, and Figure 9e shows the ensemble average energy of the excited electron.16,

23

By fitting the probability decay and the average

energy of the excited electron to a single exponential function, the electron injection time constant (𝜏𝐸𝐼, in fs) and electron relaxation time constant (𝜏𝑅𝑒𝑙𝑎𝑥𝑎𝑡𝑖𝑜𝑛, in fs) are obtained as shown in Table 2.


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Figure 9. Time evolution of the excited electron probability at the H-Si(111):phenylacetylene interface using the surface models of (a) 3×3 super cell with 12 layers, (b) 4×4 super cell with 8 layers, and (c) 6×6 super cell with 8 layers. (d) Probability change in the initially populated molecular state and (e) the ensemble-averaged energy of the excited electron at the H-Si(111):phenylacetylene interface with different surface models. The simulations were performed using NACCORRECT and ƐKS. The silicon surface CBM is set as the reference energy of 0 eV. The excited electron initially occupies the molecular state (t=0).

Table 2. Interfacial electron injection time constant (𝜏𝐸𝐼, in fs) and electron relaxation time constant (𝜏𝑅𝑒𝑙𝑎𝑥, in fs) calculated by fitting the probability change in the initially occupied molecular state and the ensemble-averaged energy of the excited electron to a single exponential function at the H-Si(111):phenylacetylene interface with different surface models.

Super cell

3×3 super cell

4×4 super cell

6×6 super cell

Layers

8

12

8

8

𝜏𝐸𝐼 (fs)

70.44

65.78

11.89

6.89

𝜏𝑅𝑒𝑙𝑎𝑥 (fs)

199.98

190.93

160.73

120.53


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Between the 3×3 super cell slab models with 8 and 12 layers, we do not observe particularly notable differences, and the 12-layer surface model shows the electron injection and relaxation being slower than the 8-layer surface model only by 7.08% and 4.74%, respectively. At the same time, compared to the 3×3 super cell with 8 layers, the simulation exhibits much faster electron injection by as much as an order of magnitude when the 4×4 super cell with 8 layers and 6×6 super cell with 8 layers are used for modeling the semiconductor surface (Table 2). The electron relaxation time 𝜏𝑅𝑒𝑙𝑎𝑥 is affected to a much lesser extent, but it is still decreased by 19.63% and 39.73% for the 4×4 super cell and 6×6 super cell models, respectively. These results show that the electron injection is quite susceptible to the surface model (particularly to the finite size effect in the lateral direction) while the excited electron relaxation is not. Even with the very large 6×6 super cell surface model with 2412 electrons, the electron injection timescale is not completely converged.

4. CONCLUSIONS In the framework of first-principles simulation based on the fewest-switches surface hopping method, we investigated the extent to which nonadiabatic couplings (NAC), energy level alignments at the semiconductor-molecule interface, and the semiconductor surface model influence the electron injection dynamics from the adsorbed molecule, using a representative interface between a hydrogen terminated Si(111) surface and a phenyl group. In particular, we consider the electron injection from a well-localized π* state of the phenyl group into the semiconductor surface. We considered several different linker groups between the surface and the phenyl group. The NAC between the adsorbed molecule and semiconductor was found to vary significantly, and only a small number of electronic states in the semiconductor conduction


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band couple strongly to the molecular state from which the electron is injected. Our work also shows that assuming a constant value for all the NACs across semiconductor and molecule is therefore problematic although such an assumption is widely used in analytical kinetic models for convenience.12 By employing quasi-particle (QP) energies from GW calculation instead of Kohn-Sham (KS) DFT single-particle energies, our work also demonstrates the importance of having accurate energy level alignments at semiconductor-molecule interface, as perhaps expected, for studying electron injection process. Another practical but important consideration in first-principles simulations of the electron injection dynamics at semiconductor-molecule interface is the surface model, and we examine how the finite size error influences the simulation results. We found that electron injection is more sensitive to the finite size error in the surface lateral directions than expected while it was found not very sensitive to the thickness of the surface slab model. These studies point to the central importance of continued development of accurate and also efficient (i.e. scalable in terms of the number of electrons) first-principles electronic structure methods for accurately modeling excited electron dynamics processes at semiconductor-molecule interface.

ASSOCIATED CONTENT Supporting Information. The Supporting Information is available free of charge on the ACS Publications website. Error introduced by classical-path approximation, on-time dependence of the many-body corrections, convergence tests of the G0W0 calculations, spatially-resolved DOS for the H-Si(111):phenylacetylene interface with large surface slabs. AUTHOR INFORMATION


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Corresponding Author *Corresponding author to whom correspondence should be addressed. Email: [email protected] Present Address †Department

of Mechanical and Aerospace Engineering, Princeton University, Princeton, New

Jersey 08544, United States (L.L.). Notes The authors declare no competing financial interest. ACKNOWLEDGMENT This work was funded by the UNC Energy Frontier Research Center (EFRC) “Center for Solar Fuels”, an EFRC funded by the U.S. Department of Energy, Office of Science, Office of Basis Energy Sciences, under Award DE-SC0001011. We thank National Energy Research Scientific Computing Center, which is supported by the U.S. Department of Energy, Office of Science, under Contract No. DE AC02-05CH11231 for computational resources.

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TOC Graphic


33


Figure 1. Top and side views of the simulation cell at the (a) H-Si(111):phenyl, (b) HSi(111):phenylacetylene, (c) H-Si(111):styrene, and (d) H-Si(111):phenyldiacetylene interfaces. The HSi(111) surface was modeled using a 3×3 super cell surface slab with 8 layers. The bottom three layers of the surface slab were held fixed in their bulk positions during the simulations. 82x57mm (300 x 300 DPI)


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Figure 2. Spatially-resolved density of states (DOS) for the conduction band states based on Kohn-Sham (KS) single-particle energies at the interface of (a) H-Si(111):phenyl, (b) H-Si(111):phenylacetylene, (c) HSi(111):styrene, and (d) H-Si(111):phenyldiacetylene. The spatial characters of the molecular states are shown at the top of each figure, and their energetic positions are indicated by white arrows in the DOS figures. The spatially-resolved DOS is calculated by averaging electronic density in the surface plane, and the silicon surface CBM is set to 0 eV as the reference energy. 82x87mm (300 x 300 DPI)



Figure 3. Time-averaged magnitude of the nonadiabatic coupling (NAC) matrix (in atomic unit) for the conduction band electronic states above the Fermi energy at the (a) H-Si(111):phenyl, (b) HSi(111):phenylacetylene, (c) H-Si(111):styrene, and (d) H-Si(111):phenyldiacetylene interfaces. Top panel: NAC matrices; bottom panel: NAC values between the molecular state and the semiconductor states. The maximum value of the NAC between the semiconductor states and molecular state is also shown in the bottom figures. The state index 1 corresponds to the conduction band minimum state of the semiconductor surface and the reference energy of 0 eV. 165x82mm (300 x 300 DPI)


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Figure 4. Time evolution of the excited electron probability at the H-Si(111):phenyl, HSi(111):phenylacetylene, H-Si(111):styrene, and H-Si(111):phenyldiacetylene interfaces. The simulations were performed using (a) the correct NAC matrices and KS energies (NACCORRECT with ƐKS), (b) the maximum NAC matrices and KS energies (NACMAXIMUM with ƐKS), as well as (c) the correct NAC matrices and QP energies (NACCORRECT with ƐQP). The silicon surface CBM is set as the reference energy of 0 eV. The excited electron initially occupies the molecular state as indicated by P(t=0). See the text for details. 165x132mm (300 x 300 DPI)



Figure 5. Probability change for the excited electron in the initially occupied molecular state at the (a) HSi(111):phenyl, (b) H-Si(111):phenylacetylene, (c) H-Si(111):styrene, and (d) H-Si(111):phenyldiacetylene interfaces. The results calculated from the NACCORRECT with ƐKS, NACMAXIMUM with ƐKS, and NACCORRECT with ƐQP are shown in black, red, and blue, respectively. 165x114mm (300 x 300 DPI)


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Figure 6. Nonadiabatic couplings (NACs, in atomic unit) between the silicon surface conduction band minimum state and the molecular state as a function of the distance from the surface to the center of mass of the phenyl ring at the four interfaces. 82x38mm (300 x 300 DPI)



Figure 7. Spatially-resolved DOS for the conduction band states based on the QP energies calculated using GW method at the (a) H-Si(111):phenyl, (b) H-Si(111):phenylacetylene, (c) H-Si(111):styrene, and (d) HSi(111):phenyldiacetylene interfaces. The spatial characters of the molecular states are shown at the top of each figure and their energetic positions are indicated by white arrows in the DOS. The spatially-resolved DOS is calculated by averaging electronic density in the surface plane, and the silicon surface CBM is set to 0 eV as the reference energy. 82x87mm (300 x 300 DPI)


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Figure 8. Top and side views of the simulation cell at the H-Si(111):phenylacetylene interface model with different surface slabs of (a) 3×3 supercell with 12 layers, (b) 4×4 supercell with 8 layers, and (c) 6×6 supercell with 8 layers. The bottom three layers were held fixed in the bulk position during the simulations. 82x79mm (300 x 300 DPI)



Figure 9. Time evolution of the excited electron probability at the H-Si(111):phenylacetylene interface using the surface models of (a) 3×3 super cell with 12 layers, (b) 4×4 super cell with 8 layers, and (c) 6×6 super cell with 8 layers. (d) Probability change in the initially populated molecular state and (e) the ensembleaveraged energy of the excited electron at the H-Si(111):phenylacetylene interface with different surface models. The simulations were performed using NACCORRECT and ƐKS. The silicon surface CBM is set as the reference energy of 0 eV. The excited electron initially occupies the molecular state (t=0). 165x67mm (300 x 300 DPI)


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TOC 73x44mm (300 x 300 DPI)


Modeling Electron Injection at Semiconductor-Molecule Interfaces

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