Strong Influence of Ti Adhesion Layer on Electron-Phonon Relaxation

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Strong Influence of Ti Adhesion Layer on Electron-Phonon Relaxation in Thin Gold Films: Ab Initio Nonadiabatic Molecular Dynamics Xin Zhou, Joanna Jankowska, Linqiu Li, Ashutosh Giri, Patrick E. Hopkins, and Oleg V. Prezhdo ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.7b12535 • Publication Date (Web): 14 Nov 2017 Downloaded from http://pubs.acs.org on November 15, 2017

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ACS Applied Materials & Interfaces

Strong Influence of Ti Adhesion Layer on Electron-Phonon Relaxation in Thin Gold Films: Ab Initio Nonadiabatic Molecular Dynamics Xin Zhou,1,2 Joanna Jankowska,2,3 Linqiu Li,2 Ashutosh Giri,4 Patrick E. Hopkins,4,5,6 Oleg V. Prezhdo*2,7 1

College of Environment and Chemical Engineering, Dalian University, Dalian, 116622, P. R. China

2

Department of Chemistry, University of Southern California, Los Angeles, California 90089, United

States 3

Institute of Physics, Polish Academy of Sciences, 02-668 Warsaw, Poland

4

Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville,

Virginia 22904, USA 5

Department of Materials Science and Engineering, University of Virginia, Charlottesville, Virginia

22904, USA 6

Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA

7

Department of Physics and Astronomy, University of Southern California, Los Angeles, California

90089, USA

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

ABSTRACT: Electron-phonon relaxation in thin metal films is an important consideration for many ultra-small devices and ultra-fast applications. Recent time-resolved experiments demonstrate a significant, more than a factor of five, increase in the electron-phonon coupling and an acceleration in the electron-phonon relaxation rate when a narrow Ti adhesion layer is introduced between an Au film and a non-metal substrate. Using nonadiabatic molecular dynamics combined with real-time time-dependent density functional theory, we identify the reasons that give rise to this strong effect. First, the Ti layer greatly enhances the density of states in the energy region starting 1 eV below the Fermi level and extending several eV above it. Second, Ti atoms are 4 times lighter than Au atoms, and therefore, generate larger nonadiabatic electron-phonon coupling. Since transition rates depend on coupling and density of states, both factors accelerate the dynamics. Showing good agreement with

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the experiments, the time-domain atomistic simulations provide a detailed mechanistic understanding of the electron-phonon relaxation dynamics in thin gold films and related nano-materials.

Keywords: metallic films, adhesion layer, electron-phonon relaxation, nonadiabatic molecular dynamics, real-time time-dependent density functional theory, quantum dynamics.


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1. Introduction Thermal transport1-12 forms an increasingly important element in various applications including micro-electronics, thermo-electrics and electro-optics. At interfaces comprised of at least one metal, the transfer of energy between electrons and phonons in the metal contributes an additional channel for thermal transport. If the other material forming the interface is a non-metal, the predominant energy transfer mechanism across the metal/non-metal interface at time-scales when the electrons and phonons in the metal are in thermal equilibrium is phonon mediated. If both materials are metals, the predominant mechanism involves electrons on the two sides.13 In both cases, electron-phonon coupling can be an important factor in energy transfer, and thus electron-phonon relaxation cannot be ignored. To describe the heat flow and resulting temperature changes associated with the electron-to-phonon thermal conductance, one commonly considers a two-temperature model (TTM),14-15 in which electrons at temperature Te exchange energy with phonons at temperature Tp. Under conditions of strong non-equilibrium between the electron and phonon subsystems, the energy transfer rate becomes proportional to the difference of the fifth power of the two temperatures.14-15 Using the TTM, Majumdar and Reddy16 derived an analytical expression to obtain the role of electron-phonon coupling in metals on the thermal boundary resistance across metal/non-metal interfaces. Gundrum et al. derived a version of the diffuse mismatch model applied to electron-electron scattering at metal/metal interfaces that suggests the distribution of energies around the Fermi energies of the metals dictates electron interface conductance.

This model has been used to describe the measured thermal

conductance across a variety of metal/metal interfaces, but only considers that the electrons and phonons are in near equilibrium conditions (i.e., Te ~ Tp).

Hopkins et

al. extended this metal/metal electron conductance model to describe conditions of strong electron-phonon nonequilibrium (i.e., Te >> Tp) on electron-electron transmission across interfaces.

However, in these aforementioned metal/metal

interface studies, the role of electron-phonon coupling has been ignored.

However,

electronic surface states that have been studied by detailed spectroscopic studies using



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angle-resolved photoemission experiments and supported by calculations17-24 have suggested that electron-phonon coupling at metal interfaces can not be ignored, albeit the majority of these works have focused on metal/non-metal interfaces.

Thus, this

begs the question: what role does electronic nonequilibrium have on electron-phonon coupling at metal/metal interfaces? This question has not been addressed in any of these aforementioned studies, and we seek to answer this question in this current work. The fundamental understanding of how electronic nonequilibrium affects electron-phonon relaxation at metal/metal interfaces can begin to be discussed through the qualitatively framework of how electrons relax and energetically equilibrate in homogeneous metals. Due to the sub-picosecond timescale of electron energy exchange, short-pulsed lasers have been widely used to advance this understanding.14 The absorption of a laser pulse by a metal surface and the subsequent energy relaxation processes can be described by three characteristic time intervals corresponding to: (i) thermalization of the free electron gas, (ii) coupling between electrons and the lattice; and (iii) energy transport driven by the spatial gradient of the electron and phonon temperatures, during which Te ~ Tp.25 The first observation of the nonequilibrium between the electronic and vibrational states with short-pulsed, time-domain thermoreflectance was carried out by Eesley,26 who confirmed the earlier theories that described electrons and the lattice by two separate temperatures, thus validating the TTM. These ultrafast transient electron relaxation dynamics can be influenced by the presence of interfaces and the resulting heterogeneity of the electronic and phononic states around the interface.

Indeed these interfaces are ubiquitous in a range of

modern technologies; for example, construction of metal-metal and metal-nonmetal interfaces is a key process in fabrication of electronic and optoelectronic devices. The atomistic features of the interfaces govern interactions between the electronic and phononic subsystems. A thin transition metal film is often inserted between an Au layer and a substrate to improve adhesive strength. Audino et al. studied three different thin evaporated film combinations Au/Cr, Au/Ti and Au/Pd/Ti and found that


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Ti-based layers gave better resistance to inter-diffusion with gold than Cr-based layers.27 Other researchers also reported that Ti is often more favorable than Cr for optical applications because it provides a larger transmittance.28-29 Recently, Giri et al.30 utilized time domain thermoreflectance (TDTR) experiments31 to report that the electron-phonon coupling increased by more than a factor of five with the inclusion of a thin Ti adhesion layer between a Au film and a nonmetal substrate, and thus the electron-phonon energy relaxation notably accelerated. They showed that the deposited thin Ti adhesion layer not only enhanced the bonding between the metal film and the dielectric substrate, but also provided a strong channel for electron-phonon energy exchange near the interface, relative to the weak electron-phonon coupling characteristic of Au. Chang and co-workers used ultrafast laser pulses to launch acoustic phonons in single gold nanodisks with variable Ti layer thicknesses, and observed an increase in phonon frequencies as an adhesion layer facilitated stronger binding to the glass substrate.32 With a combination of TDTR measurements of thermal boundary conductance, picosecond acoustic measurements of acoustic reflectivity, and Green’s function calculations of phonon interfacial transmission,

Duda et al. demonstrated that a Ti adhesion layer between

Au and non-metal substrates will enhance vibrational energy transmission across metal/non-metal interfaces. The present work reports an atomistic ab initio time-domain simulation of the electron-phonon coupling and energy exchange in thin Au films with and without the Ti adhesion layer. We apply the nonadiabatic (NA) molecular dynamics (MD) approach formulated within the framework of real-time time-dependent density functional (TDDFT) theory in the Kohn-Sham (KS) representation to investigate the dynamics of phonon-induced relaxation of excited electrons in Au films. The paper is constructed as follows. The next section describes the essential theoretical background and computational details underlying the NAMD simulation. The results and discussion section comprises three parts, focusing on the geometric and electronic structure, electron-phonon NA coupling (NAC), and electron relaxation dynamics. The simulation results are compared with the available experimental data. The key



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findings are summarized in the conclusions.

2. Theory and Simulation Methods To simulate the electron-phonon interactions in the thin gold films, first, we identify the key electronic states involved in the photoinduced dynamics on the basis of experimental results and ab initio density functional theory (DFT). Then, we perform ground state ab initio MD and compute electron-phonon NAC matrix elements. Finally, we perform time-domain simulations of the electron-phonon relaxation dynamics using a mixed quantum-classical technique, combing real-time TDDFT in the KS representation and fewest switches surface hopping (FSSH).

2.1. Ab Initio Electronic Structure Calculations and Molecular Dynamics All quantum-mechanical calculations and ab initio MD are performed with the Vienna Ab initio Simulation Package (VASP),33-34 which utilizes plane waves and the ultrasoft pseudopotentials generated within the Perdew-Burke-Ernzerhof (PBE) generalized gradient DFT functional.35 The projector-augmented wave (PAW) approach was used to describe the interaction of the ionic cores with the valence electrons.36 The basis set energy cutoff was set to 400 eV. Aiming to mimic the experimentally studied Au films with and without a narrow Ti adhesion layer30 and limited computationally by the simulation cell size, we build an Au (111) surface with seven layers of Au atoms to simulate the gold film and add one layer of Ti on top of the Au surface to model Au film with a narrow Ti adhesion layer, as shown Figure 1. We further investigate the electronic structure of the Au film interacting with a layer of Si substrate. We construct the Au-Ti model as a 2x2 Au(111) surface supercell with the cell surface dimensions a=b=5.76759, and a 2x2 Ti(001) layer with dimensions a=b=5.9012. The lattice mismatch between Au and Ti is only 2.3%. In the Au-Si model, we used a 4x4 Au(111) supercell with a=b=11.53518 and a 3x3 Si(111) layer with a=b=11.5203. Here, the lattice mismatch is only 0.1%. A vacuum layer of 20 Å is introduced in the direction perpendicular to the surface to avoid spurious interaction between periodic slabs. The 5×5×1 Monkhorst-Pack mesh


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was used for the structural optimization and MD, and the 7×7×1 mesh was employed for density of states (DOS) calculations. After relaxing the geometry at 0 K, repeated velocity rescaling was used to bring the temperature of the systems to 300 K, corresponding to the temperatures in the experiment.30 Following the 3 ps thermalization dynamics, adiabatic MD simulations were performed in the microcanonical ensemble with a 1 fs atomic time-step. The lengths of the adiabatic MD trajectories were 7 ps for Au and 3 ps for Au-Ti. The adiabatic state energies and NAC were calculated for each step of the MD runs. 1000 geometries were selected randomly from the adiabatic MD trajectories, and were used as initial conditions for nonadiabatic dynamics simulations.

2.2. Nonadiabatic Molecular Dynamics with Real-Time Time-Dependent Density Functional Theory in the Kohn-Sham Representation To model the electron relaxation process, one needs to employ a methodology capable of describing transitions between electronic states. NAMD and, in particular, trajectory surface hopping37-38 provides an efficient method that describes electronic transitions in nanoscale systems. In this work, we utilize FSSH38-39 in the classical path approximation, as implemented in the PYXAID package.40-41 DFT expresses the ground state energy in terms of the properties of electron density. The latter is obtained from single-particle KS orbitals, Ψ , ,

, = |Ψ , | .

(1)

Here, Ne is the number of electrons. The equations-of-motion for the time-dependent KS orbitals is derived using the time-dependent variational principle,

ħ Ψ , = [, ]Ψ , .

(2)

These equations are nonlinear and are coupled, since the Hamiltonian itself, [, ], depends on occupied KS orbitals, eq 1. The time-dependent KS orbitals,

Ψ , , are expressed in the adiabatic KS orbital basis, Φ ; , computed for a

given nuclear configuration, , Ψ , = ∑ Φ ; ,


(3)


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using standard DFT codes.40-42 Substitution of eq 3 into eq 2 produces equations-of-motion for the expansion coefficients:

ħ c! = ∑ "# $! + &! ',

(4)

where # is the energy of the adiabatic state k, and &! is the NAC between adiabatic states k and j. The later arises from the orbital dependence on the atomic motion, , and is calculated numerically as the overlap between wavefunctions k and j at sequential time steps &! = −ħ〈Φ! |∇|Φ 〉 ∙

- -

= −ħ .Φ! /

/Φ 0.

(5)

Numerical evaluation of the derivative on the right-most side of eq 5 provides significant computational savings, if the analytic gradient matrix elements, appearing in the middle of eq 5, are not available. The NAC is electron-phonon coupling, since it arises from dependence of the wavefunctions on nuclear coordinates. FSSH38 is an approach for modeling dynamics of quantum-classical systems. FSSH was implemented in the real-time KS-TDDFT,42 tested,43 released in public software,40 and applied to various nanoscale systems.41,

44-54

FSSH provides a

probability for hopping between quantum states based on the evolution of the expansion coefficients, eq 4. Specifically, the probability of a transition from state j to another state k within the time interval dt is given in FSSH by38 d2! =

3 456∗78 :78 ; 677

d;
(6)

If the calculated d2! is negative, the hopping probability is reset to zero. A hop is possible only when the occupation of the initial state j diminishes and the occupation of the final state k grows. This feature minimizes the number of hops, leading to “fewest switches”. A uniform random number on the [0,1] interval is generated every step to compare with d2! and decide whether to hop. The quantum-classical energy is conserved by nuclear velocity rescaling. Based on semiclassical arguments55 only the velocity component along the direction of the matrix element of the gradient vector, 〈Φ! |∇ |Φ 〉, is rescaled. If the magnitude of this component of the kinetic energy is too small to account for the electronic energy increase after a hop, the hop is rejected. The velocity rescaling procedure gives


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detailed balance between up and down transitions. The detailed balance is needed to achieve the Boltzmann statistics and thermodynamic equilibrium.39 The classical path approximation to FSSH40, 44 assumes that the nuclear dynamics is weakly dependent on the electronic evolution, for instance, as compared to thermally induced nuclear fluctuations. This approximation generates great computational savings, because the time-dependent potential that drives multiple FSSH realizations is obtained from single MD trajectory. It is assumed further that the nuclear subsystem is close to thermodynamic equilibrium, and in particular that the energy deposited into the nuclear modes along the direction of the 〈Φ! |∇ |Φ 〉 vector dissipates rapidly among all nuclear degrees of freedom. Then, the hop rejection feature is replaced with multiplication of FSSH probability upward in energy by the Boltzmann factor. This maintains the detailed balance and achieves thermodynamics equilibrium in the long-time limit.40, 44

3. Results and Discussion 3.1. Geometric and Electronic Structure The structures of the thin Au films with and without the Ti adhesion layer were first optimized at 0 K. The optimized structures shown in Figure 1a indicate that adding a layer of Ti has a very minor effect on the Au film geometry. The geometry of the Ti layer also changes little due to the interaction. The largest atomic displacement along the vertical direction is less than 0.06 Å. The distance between the Ti layer and the nearest Au layer is 2.270 Å. The calculated Au-Ti bond length is 2.791 Å, which is close to observed and calculated Au-Ti bond lengths in clusters and alloys.56-57 The intra-layer Au-Au bond length is 2.884 Å, in good agreement with the experimental data.58 The influence of the Au-Ti interaction on the system geometry is much weaker than thermal effects. The geometries of both slabs change considerably due to thermal expansion, when the temperature is elevated to 300 K, Figure 1b. The distance between the Au layers increases by about 0.2 Å, while the distance between the Ti adhesion layer and Au slab grows by 0.05 Å. In addition to the thermal expansion, the finite temperature



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geometries exhibit small-scale sliding motions of layers with respect to each other, including both Au and Ti layers. Even though the bonding patterns do not change at the finite temperature relative to the optimized structure, in contrast to some other nanoscale systems,59-60 the additional symmetry breaking introduced by thermal fluctuations influences localization of the wavefunctions, generating NAC. Figure 2 characterizes the electronic structure of the Au-Ti slabs by presenting the projected density of states (PDOS) obtained using the optimized geometry at 0 K. All three panels in Figure 2 show the same DOS in different representations. The Fermi level is set to zero. The system has no bandgap, since it is a metal. The most obvious observation is that the Au DOS is significantly larger than the Ti DOS in the energy range between -7 eV and -1.5 eV. This is the range dominated by Au 5d states which contribute much more than Au 6s, 6p and Ti 3d states to the overall DOS. The second and most important observation is that the Ti DOS is about equal to the Au DOS in the -1.5 eV to 4 eV energy range. This is quite remarkable, since the system contains 7 times more Au atoms than Ti atoms. The Ti DOS in this energy range arises from Ti 3d states, while the Au DOS is determined by about equal contributions of Au 6s, 6p states and the tail of the 5d states PDOS. One can conclude that in general d states produce much higher DOS than s and p states. Similarly to the -7 eV to -1.5 eV range that exhibits high DOS due to 5d contributions of Au, the 3d states of Ti contribute significantly in the -1.5 eV to 4 eV energy range. We focus on the previously mentioned pump-probe experiments performed by Giri et al. using a pump excitation energy of 3.1 eV.30 In the energy range from 0 to 3.1 eV, Figure 2c, the contribution of Ti states is comparable to and even more important than that of Au states. Considering that Ti atoms constitute only 1/8 of all atoms, we conclude that the Ti adhesion layer significantly influences the electronic states in the experimentally relevant energy region. Thin gold films are usually evaporated onto nonmetal substrates, such as crystalline, fused silica.30, 61 In order to explore the effect of a nonmetal substrate on the electronic states of the gold film, we constructed a model with one layer of Si adsorbed onto the same Au (111) surface slab and calculated the PDOS of every


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component, Figure 3. Part c of Figure 3 shows clearly that the contribution of Si to the overall DOS is negligible over the whole considered energy range between -7 eV and 6 eV. The PDOS in Figure 3b indicates that the contributions of Si 3s and 3p states are much smaller than the contributions of Au 5d, 6s and 6p states. We conclude that a nonmetal substrate, such as Si, has little effect on the electronic DOS of Au films in the relevant energy range. The electron relaxation rate depends on the phonon-induced NAC between electronic states. The strength of the coupling is related to the corresponding wavefunction overlap, eq. 5. Therefore, we investigate the shape of the electron densities at different energies, Figure 4. The plotted densities are squares of the adiabatic KS orbitals that enter eq. 5. The pump-probe experiments employ 3.1 eV excitation energy.30, 61 Accordingly, we consider approximately the 0-3.1eV energy range and divide it into three intervals, 0-1 eV, 1-2 eV and 2-3 eV. Figure 4 displays the charge densities of the highest occupied molecular orbital (HOMO) at the Fermi energy, as well as representative states picked randomly 1 eV, 2 eV and 3 eV above the Fermi level for both Au and Au-Ti. The wavefunctions of the pristine Au film are symmetrical and localized on the bulk layers with little surface contribution. The symmetry is broken in the Au-Ti system by the Ti layer. Surface layers now contribute to the majority of states, which can be localized either on the Ti side or on the opposite side of the film. The orbital localization changes as atoms move along the trajectory, and an averaging over initial conditions is required to reproduce all situations. The states localized on the Ti side have considerable contributions from Ti, as expected based on PDOS, Figure 2. Typically, electron-phonon NAC matrix elements, eq. 5, are larger for more localized states, since changes in wavefunctions induced by nuclear motions are more significant if wavefunctions are localized.

3.2. Electron-Phonon Interactions Electron-phonon relaxation appears in trajectory surface hopping as a sequence of NA transitions between electronic energy levels. Each elementary transition reflects the strength of the NAC and the detailed balance factor for transitions upward and



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downward in energy. The NACs are obtained along the trajectory based on the adiabatic wavefunctions for the current moment in time, according to eq. 5. Table 1 compares the absolute values of NAC averaged over all electronic states involved in the relaxation to those averaged only over the neighboring states. Different energy ranges in both Au and Au-Ti systems are considered. The most striking observation is that the NAC magnitudes are much larger for Au-Ti compared to Au. Even a thin Ti adhesion layer changes the NAC by a factor of 3-5 and even more. The difference arises because Ti is a factor of 4 times lighter than Au, and the nuclear velocities directly affect the equation for NAC, eq. 5; at a given temperature, which determines nuclear kinetic energy, the velocity is inversely proportional to the square root of atomic mass. This dependence predicts a factor of 2 difference in NAC arising from Ti compared to Au. Further, a single layer of Ti doubles the DOS of the 7-layer Au slab within the relevant energy range, Figure 2. Generally, NAC between states that are close in energy is larger. Indeed, the absolute NAC averaged over nearest neighbor states is a factor of 2-4 larger than the absolute NAC averaged over all states, compare the first two lines in Table 1. Similarly, NAC between denser states in Au-Ti is larger than between sparser states of pristine Au. Thus, the Au and Ti mass difference, together with the difference in the DOS, can explain why electron-phonon coupling is much larger in Au-Ti than Au, in agreement with experiment.30 One might argue that an adhesion layer of Si on a Au film should also increase NAC, because Si is much lighter than Au. This is not the case, because Si contributes very little to the overall DOS in the relevant energy region, Figure 3. Even though Si atoms are light, they do not affect the electronic wavefunctions and the NAC matrix elements that depend on the wavefunction overlap, eq. 5. This result is consistent with the experimental observation that semiconducting substrates have little effect on electron-phonon relaxation in Au films.30 In order to characterize the frequencies of the phonon modes that couple to the electronic subsystem and the relative strength of electron-phonon coupling for different modes, we compute Fourier transforms of the energy gaps between HOMO at the Fermi energy and the three states sampled at 1 eV, 2 eV and 3 eV above the


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Fermi energy, Figure 5. The densities of these states are shown in Figure 4 for the optimized slab geometries. The heights of the peaks in Figure 5 characterize the strength of electron-phonon interaction at the corresponding frequencies. The spectra shown in Figure 5 are known as influence spectra, or spectral densities.62-63 The phonon modes that couple to the electronic subsystem have low frequencies, with the lower frequency acoustic modes showing stronger coupling that the higher frequency optical phonons, consistent with semi-classical theory. Our simulations shine important insight into the quantum mechanical mechanism driving this electron-low frequency phonon interaction, more specifically, that the electronic wavefunctions are quite delocalized, Figure 4. Local displacements of atoms with respect to each other, characteristic of higher frequency optical modes, have little effect on the delocalized wavefunctions. (The effect averages out on the scale of wavefunction delocalization). In comparison, larger scale acoustic motions induce a stronger wavefunction change. Recall that it is the phonon-induced change in the wavefunction that causes the electron-phonon NAC, eq 5. The dominant peaks below 100 cm-1 seen in the influence spectra of pristine Au can be assigned to acoustic phonon modes involving core Au atoms.64 Introduction of the Ti layer increases the frequency range of active vibrations. The higher frequency modes appear because Ti atoms are much lighter than Au atoms. Our simulation agrees well with the experimental shift to higher phonon frequencies observed upon inclusion of adhesion layers on gold nanodisks.32

3.3. Nonadiabatic Relaxation Dynamics of Photoexcited Electrons The electrons photoexcited to about 3.1 eV, as in the experiment,30 were allowed to relax from the nonequilibrium states to the Fermi energy by coupling to phonons. The electron-phonon relaxation dynamics was simulated using FSSH implemented with real-time TDDFT in the KS representation, as described in section 2.2 and in more details in refs.40-41 The evolution of the photoexcited electron energies in the Au and Au-Ti systems is shown in Figure 6. Note that the x-axis scales are different in parts a and b. Here, the zero of energy is again set at the Fermi level. The reported



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time constants were obtained by exponential fitting. Note that the early time dynamics in Figure 6 is Gaussian rather than exponential, which is typical of quantum dynamical systems,65 however, the exponential regime is achieved relatively fast. The electron-phonon relaxation dynamics is a factor of 4 faster in Au-Ti than in Au. The difference between pristine Au and Au with a Ti adhesion layer is more pronounced in the calculation than in the experiment, compare Figure 6 of this paper with Figure 2 of Ref.30 This discrepancy could be due to the small size of the simulation cell compared to the experimental system. The narrow slab used in the simulation emphasizes the coupling of Au states to Ti atoms. In thicker slabs, Au wavefunctions should be less delocalized onto the Ti adhesion layer. This discrepancy could also be due to the chemistry of the Ti layer in the prior experimental work, and the potential for oxygen defects or TiOx phases in Ti layer. The thickness of the Au film used in the experiment is about 16 nm, and the thickness of the Ti adhesion layer is 2.8 nm (see Table 1 of Ref. 30). The experimental Au/Ti thickness ratio is 5.7. The current simulation uses 7 atomic layers of Au and 1 atomic layer of Ti, giving a similar ratio. If the thickness of the Ti adhesion layer is increased, one can expect that the electron-phonon relaxation in the Au-Ti system will be completely dominated by Ti. On the one hand, the PDOS of a single Ti layer is already similar in magnitude to the PDOS of 7 Au layers, at energies around and above the Fermi energy, Figure 2. The Ti PDOS will be dominant for a larger number of Ti layers. On the other hand, Ti atoms are significantly lighter than Au atoms, and therefore, they create stronger electron-phonon coupling than Au atoms. Fermi’s golden rule, employed successfully to rationalize other NAMD simulations,66 cannot explain the 4-fold difference in the relaxation times observed in this work. In general, the initial stage of quantum dynamics is Gaussian rather than exponential, because the time-derivative of the quantum population is 0 at t=0, see early times in Figure 6. The non-exponential initial stage of quantum dynamics gives rise to the so-called quantum Zeno effect,65 in which interaction with environment dramatically slows down the dynamics. According to Fermi’s golden rule, the transition rate is proportional to coupling squared. In the current systems, the NAC in


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the Au and Au-Ti films is already different by a factor of 3-5, see Table 1. The DOS on Au-Ti is also larger than the DOS of Au, see Figure 2. Thus, Fermi’s golden rule should predict a factor of 10-20 times faster dynamics in Au-Ti, as opposed to a factor of 4. A similar situation is observed experimentally: the difference in the relaxation times, Figure 2 of Ref.30 is much less than the difference between the squares of the corresponding electron-phonon couplings (with and without Ti), Figure 3 of Ref.30 The fact that rate expressions such as Fermi’s golden rule cannot explain the experimental observations emphasizes the need for quantum dynamics simulations coupled closely with well controlled experiments.

4. Conclusions By performing time-domain ab initio simulations, we investigated the electron-phonon relaxation process in thin gold films with and without a Ti adhesion layer, and established the factors responsible for the difference in the observed relaxation rates in the two systems. Understanding of these factors is important for successful application of thin gold films and related systems in various electro-optical devices. The simulation supports the experimental observation that the relaxation between electrons and phonons accelerates significantly when a Ti adhesion layer is deposited on a Au film. The difference stems from two factors. First, Ti contributes very significantly, about 7 times more per atom than Au, to the density of electronic states in the relevant energy window. Second, Ti atoms are about 4 times lighter than Au atoms, generating faster nuclear velocities that determine the electron-phonon nonadiabatic coupling.

In contrast, semiconducting substrates, such as Si, SiO2 or

Al2O3, have no effect on electron-phonon relaxation, even though they are also composed of light atoms. This is because they have no states near the Fermi energy of Au. The reported results show good agreement with the experiment data and provide a detailed atomistic understanding of the electron relaxation processes needed for design of novel materials and devices.



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Acknowledgments X. Z. acknowledges support of the National Natural Science Foundation of China, grant no. 21473183 and China Scholarship Council. O. V. P. and P.E.H. acknowledge financial support from the US Department of Defense, Multidisciplinary University Research Initiative, grant No. W911NF-16-1-0406.


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Figure 1. Side views of the simulation cells showing the geometries of the Au (111) slab and the Au (111) slab with absorbed Ti layer at 0 K (a) and during the molecular dynamics run at 300 K (b). The distances between neighboring layers are shown on the left side of the structures.



(a)

50

Projected Density of States

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

0 -10 10

0

50

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Au 6s Au 5d

-8

-6

-4

-2

0

Au 6p Ti 3d

2

4

6

(b)

-2

-1

0

1

2

3

4

(c)

0 -10

5

Au Ti

-8

-6

-4

-2

0

2

4

6

Energy (eV)

Figure 2. Projected density of states of the Au (111) slab with adsorbed Ti layer (a) over broad energy range from -10 eV to 6 eV, (b) over narrower energy range from -2 eV to 5 eV, and (c) split into Au and Ti contributions. The Fermi level is set to zero and shown by the dashed line.


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(a)

200

Projected Density of States

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


0 -10

-8

-6

-4

-2

0

2

Au 6p Si 3p

4

6

(b)

20

0

Au 6s Au 5d Si 3s

-1

200

0 -10

0

1

2

3

(c)

4

Au Si

-8

-6

-4

-2

0

2

4

6

Energy (eV)

Figure 3. Projected density of states of the Au (111) slab with adsorbed Si layer (a) over broad energy range from -10 eV to 6 eV, (b) over narrower energy range from -1eV to 4 eV, and (c) split into Au and Si contributions. The Fermi level is set to zero and shown by the dashed line.



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Figure 4. Charge density of representative states: HOMO, 1 eV, 2 eV and 3 eV above the Fermi level for bare Au (111) and Au (111) with adsorbed Ti layer in their optimized structures. The isosurface value is set to 0.001 e/A3.


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(a) 0-1 eV 0-2 eV 0-3 eV Spectral Density

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(b)

0

100

200

300

400

500

-1

Frequency (cm )

Figure 5. Fourier transforms of the energy gaps from the Fermi level to electronic levels at 1, 2 and 3 eV above it, in (a) Au and (b) Au-Ti systems.


3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

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(a) Au

1.83 ps

0

1000

2000

3000

4000

5000

6000

Time (fs)

Energy (eV)

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

Energy (eV)


3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

(b) Au-Ti 0.45 ps

0

500

1000

1500

2000

Time (fs)

Figure 6. Average electron energy decay at 300 K in the (a) Au and (b) Au-Ti systems.


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Table 1. Absolute value of nonadiabatic coupling averaged over nearest states (Nabsi,i+1) and over all states (Nabstotal), and electron-phonon relaxation times in the Au and Au-Ti systems at 300 K. Au Nabsi,i+1(meV) Nabstotal(meV) decay time(ps)

Au-Ti

0-1 eV

0-2 eV

0-3 eV

1-2 eV

7.15 2.60

6.64 1.84

6.19 1.26 1.83

4.99 2.30

2-3 eV 5.90 1.95

0-1 eV

0-2 eV

0-3 eV

1-2 eV

2-3 eV

20.31 7.48

20.40 8.11

27.32 8.46 0.45

21.76 11.81

40.28 19.11



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TOC


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Strong Influence of Ti Adhesion Layer on Electron-Phonon Relaxation

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