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Thin Ti adhesion layer breaks bottleneck tohot hole relaxation in Au filmsCite as: J. Chem. Phys. 150, 184701 (2019); https://doi.org/10.1063/1.5096901Submitted: 20 March 2019 . Accepted: 17 April 2019 . Published Online: 09 May 2019

Xin Zhou, Marina V. Tokina, John A. Tomko, Jeffrey L. Braun, Patrick E. Hopkins, and Oleg V. Prezhdo

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Thin Ti adhesion layer breaks bottleneckto hot hole relaxation in Au films

Cite as: J. Chem. Phys. 150, 184701 (2019); doi: 10.1063/1.5096901Submitted: 20 March 2019 • Accepted: 17 April 2019 •Published Online: 9 May 2019

Xin Zhou,1,2 Marina V. Tokina,2 John A. Tomko,3 Jeffrey L. Braun,4 Patrick E. Hopkins,3,4,5and Oleg V. Prezhdo2,6,a)

AFFILIATIONS1College of Environment and Chemical Engineering, Dalian University, Dalian 116622, People’s Republic of China2Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA3Department of Materials Science and Engineering, University of Virginia, Charlottesville, Virginia 22903, USA4Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, Virginia 22903, USA5Department of Physics, University of Virginia, Charlottesville, Virginia 22903, USA6Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA

a)Author to whom correspondence should be addressed: prezhdo@usc.edu

ABSTRACTSlow relaxation of highly excited (hot) charge carriers can be used to increase efficiencies of solar cells and related devices as it allows hotcarriers to be extracted and utilized before they relax and lose energy. Using a combination of real-time density functional theory and nona-diabatic molecular dynamics, we demonstrate that nonradiative relaxation of excited holes in an Au film slows down 30-fold as holes relaxacross the energy range −2 to −1.5 eV below the Fermi level. This effect arises due to sharp decreases in density of states (DOS) and reducedhole-phonon coupling in this energy range. Furthermore, to improve adhesion, a thin film of transition metal, such as Ti, is often insertedbetween the noble metal layer and its underlying substrate; we demonstrate that this adhesion layer completely eliminates the hot-hole bottle-neck because it significantly, 7-fold per atom, increases the DOS in the critical energy region between −1.5 eV and the Fermi level, and becauseTi atoms are 4-times lighter than Au atoms, high frequency phonons are introduced and increase the charge-phonon coupling. The detailedab initio analysis of the charge-phonon scattering emphasizes the nonequilibrium nature of the relaxation processes and provides importantinsights into the energy flow in metal films. The study suggests that energy losses to heat can be greatly reduced by judicious selection ofadhesion layers that do not involve light atoms and have relatively low DOS in the relevant energy range. Inversely, narrow Ti adhesion layersassist heat dissipation needed in electronics applications.

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I. INTRODUCTION

Understanding electron-vibrational relaxation at interfaces isimportant for optimizing the efficiency of systems designed fora variety of applications, including thermoelectricity, microelec-tronics, and electro-optics.1–4 Interfacial energy transfer can occurthrough energy exchange from the respective charge carriers andphonons in each material comprising the interface;5–10 the trans-fer of energy between charge carriers and phonons across inter-faces involving metals can contribute an additional channel for ther-mal transport, and corresponding energy losses cannot be ignored.Commonly, the two-temperature model (TTM) is used to describe

the magnitude of electron-to-phonon thermal conductance,1,11–13 inwhich electrons at temperature Te exchange energy with phononsat temperature Tp. Since thermalization of excited electrons andphonons occurs very rapidly, short-pulsed lasers have been widelyused to understand the scattering mechanisms of the fundamentalenergy carriers.6,14–25

The nonequilibrium dynamics following excitation of electronsby a laser pulse exhibits three characteristic time scales, includingthermalization of the charge carrier gas due to electron-electronscattering (∼tens of femtoseconds), energy exchange between theelectrons and the lattice (∼hundreds of femtoseconds to a fewpicoseconds), and energy transport driven by the gradient in the

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lattice temperature (>picoseconds).26,27 Eesley was the first toobserve the nonequilibrium between the electronic and vibrationaldegrees of freedom using the short-pulsed, time-domain (TD) ther-moreflectance (TR) technique28,29 and confirmed the earlier theoriesthat described electrons and the lattice by two separate tempera-tures,23,24 validating the TTM.30,31

Interfaces involving metal films and multilayer metal stacksconstitute an essential component of electronic and optoelectronicand magnetic devices. Interactions among the various electron andphonon subsystems at metal interfaces are controlled by atom-istic features of the interfaces. For example, to improve adhesivestrength, a thin transition metal film is often inserted between anoble metal layer and its supporting substrate. Hohlfeld et al.32 stud-ied three different evaporated thin film systems, Au/Cr, Au/Ti, andAu/Pd/Ti, and found that Ti-based layers gave better resistance tointerdiffusion with gold than Cr-based layers. Other researchersalso reported that Ti is often more favorable than Cr for opti-cal applications because it provides a larger transmittance.33,34 Interms of thermal transport, Kaganov et al.35 demonstrated that thephonon-phonon thermal boundary conductance across Au/siliconinterfaces can increase at over a factor of three with the inclusionof a Ti adhesion layer between the Au and silicon. More recentstudies by Anisimov et al.36 and Olson et al.37 found that the oxy-gen stoichiometry of this Ti adhesion layer can also play a pro-nounced role in phonon coupling across Au/Ti/substrate interfacesfor a wide array of substrates, including sapphire, MgO, quartz, andgraphene.

While the dynamics of phonon thermal conductance at inter-faces have been widely studied, including those with nanoscale thinfilms such as the aforementioned Ti adhesion layers, the mecha-nisms driving electron-phonon dynamics at and across interfaceshave received much less attention. Several theories have been devel-oped that treat this electron-phonon interface problem.19,36,38–41Notably, Duda et al.42 found that the electron-phonon couplingincreased by more than a factor of five with the inclusion of a thinTi adhesion layer between the Au film and a nonmetal substrate andthat the electron-phonon energy relaxation time notably accelerated.A following density functional theory (DFT)-based computationalstudy identified two reasons for these experimental phenomena: theinclusion of a Ti layer greatly enhances the density of states (DOS) inthe energy region starting at 1 eV below the Fermi level and extend-ing several eV above it, and arguing that Ti atoms are four timeslighter than Au atoms and therefore generate enhanced electron-phonon coupling.37 Both factors accelerate the electron-phonondynamics.

The nature and dynamics of charge carriers, e.g., electronsvs holes, in an electronic device depend on many factors, includ-ing chemical doping, operation and gate voltage, Fermi level align-ment, and band bending at an interface. Our previous aforemen-tioned DFT study focused on electrons,43 rationalizing the observedacceleration of electron-phonon relaxation in Au due to a Ti adhe-sion layer reported by Majumdar and Reddy.39 The properties ofholes in Au are quite different from those of electrons. In partic-ular, hole DOS is much higher than electron DOS. Additionally,holes are supported primarily by Au 5d orbitals, while electronshave strong contributions from both Au 6s and 6p orbitals. Further-more, Ti electronic DOS is much higher in the energy range corre-sponding to Au electrons than holes. The dynamics of the hole is

interesting by itself as it is a fingerprint of differences between elec-tron and hole wavefunctions. Direct information on hole dynamicscan be obtained experimentally by angle-resolved photoemissionspectroscopy.23,44–47 However, the role of these hole dynamics oncarrier relaxation with a metal lattice has not been explicitly stud-ied and is important to consider to garner a more holistic under-standing of nonequilibrium energy transfer in metals and at metallicinterfaces.

The present work reports an atomistic ab initio time-domainsimulation of phonon-induced nonradiative relaxation of holes inthin Au films with and without a Ti adhesion layer. The simula-tions are performed using a nonadiabatic (NA) molecular dynam-ics (MD) approach formulated within the framework of real-timetime-dependent density functional theory (TDDFT) in the Kohn-Sham (KS) representation. The paper is constructed as follows:Sec. II describes the essential theoretical background and compu-tational details underlying the NAMD and real-time TDDFT sim-ulations; Sec. III details the results and discussion, which is com-prised of three parts—the geometric and electronic structure of themetal films, hole-phonon interactions, and nonradiative hole relax-ation dynamics; finally, the key findings are summarized in theconclusions.

II. THEORY AND SIMULATION METHODSThe simulations are carried out using a mixed quantum-

classical technique. The electronic structure is described quantummechanically, using ab initio DFT. The electronic evolution is mod-eled using real-time TDDFT. Nuclear motions are described byclassical MD, and the electron-nuclear interactions are modeledusing fewest switching surface hopping48 (FSSH), which is the mostcommon NAMD methodology.

A. Ab Initio electronic structure calculationsand molecular dynamics

All quantum-mechanical calculations and ab initioMD are per-formed with the Vienna Ab initio Simulation Package (VASP),23,49which utilizes plane waves and the ultrasoft pseudopotentials gen-erated within the Perdew-Burke-Ernzerhof (PBE) generalized gra-dient DFT functional.50 The interaction of the ionic cores withthe valence electrons is described by the projector-augmented wave(PAW) approach.51 The plane wave basis energy cutoff is set to be400 eV.

Considering computational limitations, the experimentallystudied Au films with and without a narrow Ti adhesion layer52are modeled with an Au (111) surface containing seven layers ofAu atoms and one layer of Ti atoms (Fig. 1). 20 Å of vacuumare included perpendicular to the slabs to eliminate interactionsbetween periodic images. The 5 × 5 × 1 Monkhorst-Pack mesh isused for geometry optimization and MD, and the 7 × 7 × 1 mesh isused to compute DOS.

After relaxing the geometry at 0 K, repeated velocity rescalingis employed to bring the temperature of the systems to 300 K, corre-sponding to common experimental conditions.53 Following 3 ps ofthermalization dynamics, adiabatic MD simulations are performedin the microcanonical ensemble with a 1 fs atomic time step. The

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FIG. 1. Side views of simulation cellsshowing the geometries of the Au(111)slab without and with the adsorbed Tilayer at 0 K, and at different times alongthe trajectory at 300 K. Vibrations involv-ing sliding of layers are observed at roomtemperature.

adiabatic MD trajectories are 7 ps and 3 ps long for Au and Au–Ti,respectively.

B. Time-dependent Kohn-Sham theoryBy mapping an interacting many-body system onto a system

of noninteracting particles moving in an effective potential, DFTexpresses the ground state energy as a functional of electron den-sity. In practical implementations, the density is constructed fromsingle-particle Kohn-Sham (KS) orbitals, Ψn(r, t),

ρ(r, t) =∑Nen=1 ∣Ψn(r, t)∣2, (1)

where Ne is the number of electrons. The evolution of Ψn(r, t) isdetermined by the time-dependent variational principle, leading tothe following equations of motion:

ih̵∂

∂tΨn(r, t) = H[ρ(r, t)]Ψn(r, t). (2)

The equations are coupled and nonlinear since the Hamiltonian,H[ρ(r, t)], is a functional of the electron density, which is obtainedby summing over many occupied KS orbitals [Eq. (1)]. By expandingthe time-dependent KS orbitals, Ψn(r, t), in the adiabatic KS orbitalbasis, Φk(r; R(t)), obtained for a given nuclear configuration,R(t),

Ψn(r, t) =∑k Cnk(t)Φk(r; R(t)), (3)

one takes advantage of the computational efficiency of time-independent DFT codes.54–56 Substituting this expansion into Eq. (2)gives equations of motion for the expansion coefficients

ih̵∂

∂tcnj (t) =∑k C

nk(t)(εkδjk + djk), (4)

where εk is the energy of the adiabatic state k and djk is the NAcoupling between adiabatic states k and j. The latter arises fromthe orbital dependence on the atomic motion, R(t), and is calcu-lated numerically as the overlap between wavefunctions k and j atsequential time steps via

djk = −ih̵⟨Φj∣∇RH∣Φk⟩

Ei − Ej⋅ dR

dt= −ih̵⟨Φj∣∇R∣Φk⟩ ⋅

dRdt

= −ih̵⟨Φj|∂

∂t|Φk⟩. (5)

The numerical evaluation of the time-derivative term on the right-hand-side of Eq. (5) provides significant computational savings ifthe off diagonal matrix elements of the gradient with respect tothe nuclear coordinates, encountered in the middle of Eq. (5), arenot available analytically. This NA coupling is a form of electron-phonon coupling as it reflects the dependence of the electronicwavefunctions on nuclear coordinates.

C. Fewest switches surface hoppingTo model the hole relaxation process, one needs to employ a

methodology capable of describing transitions between electronicstates. NAMD and, in particular, trajectory surface hopping54,56–59provide an efficient method to describe electronic transitions innanoscale systems. In this work, we utilize FSSH60 in the classicalpath approximation (CPA), as implemented in the PYXAID pack-age.54,61 Note that solving real-time TDDFT equations, Eq. (2), isnot sufficient for the present purpose, in particular, since they arereversible in time and do not obey detailed balance between tran-sitions upward and downward in energy and thus cannot describeelectron-phonon relaxation and approaches that assume thermody-namic equilibrium.

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FSSH62 is an algorithm for modeling dynamics of mixedquantum-classical systems. It assigns a probability for hoppingbetween quantum states based on the evolution of the expansioncoefficients [Eq. (4)]. The probability of a transition from state j toanother state k within the time interval dt is given in FSSH by63

dPjk =−2Re(A∗jkdjkṘ)

Ajjdt, Ajk = cjc∗k . (6)

In cases where the calculated dPjk is negative, the hopping proba-bility is set to zero because, by construction, a hop from state j tostate k can take place only when the occupation of state k increasesand the occupation of state j decreases. This feature of the algorithmachieves “fewest switches” by minimizing the number of hops. Auniform random number between 0 and 1 is generated every timestep and compared to dPjk in order to decide whether or not tohop.

It is assumed that electrons and phonons exchange energyinstantaneously during a hop. The energy balance is achieved bynuclear velocity rescaling; based on semiclassical arguments,64 onlythe velocity component along the direction of the matrix elementof the gradient vector, ⟨Φj∣∇R∣Φk⟩, is rescaled. If the magnitude ofthis component of the kinetic energy is too small to accommodatean increase in the electronic energy after a hop to a higher energystate, the hop is rejected. This velocity rescaling step gives rise todetailed balance between transitions up and down in energy, lead-ing to Boltzmann statistics and quantum-classical thermodynamicequilibrium.65

The classical path approximation (CPA) to FSSH54,61 assumesthat the nuclear dynamics are weakly dependent on the electronicevolution, for instance, as compared to thermally induced nuclearfluctuations. The CPA approximation allows great computationalsavings since the time-dependent potential that drives multipleFSSH realizations of the NA dynamics of the electronic subsystemcan be obtained from a single MD trajectory. It is assumed furtherthat the energy deposited into the nuclear modes along the directionof the NA coupling vector, ⟨Φj∣∇R∣Φk⟩, dissipates rapidly among allnuclear degrees of freedom such that the vibrational subsystem isalways close to thermodynamic equilibrium. Then, the hop rejectionfeature of the original FSSH is replaced with multiplication of theFSSH probability up in energy by the Boltzmann factor. This main-tains detailed balance and achieves thermodynamic equilibrium inthe long time limit.54,60

The FSSH-CPA calculations are performed by first computingthe adiabatic state energies and NA coupling along the MD trajec-tories, and then randomly selecting 1000 geometries as initial con-ditions for NAMD; 100 realizations of the stochastic FSSH processare sampled for each initial condition. FSSH has been implementedwithin real-time KS-TDDFT,54 tested,65 adapted to the CPA,62,66released in public software,60 and applied to various nanoscalesystems.59–61,66–72

III. RESULTS AND DISCUSSIONA. Geometric and electronic structure

The geometries of the Au films with and without the Ti adhe-sion layer have been optimized at 0 K first. As shown in Fig. 1,

introducing a layer of Ti results in a very minor distortion of theAu film geometry. The largest increase in the distance between adja-cent Au layers is less than 0.06 Å. The average length of the bondbetween the nearest Au and Ti atoms is 2.791 Å; this value is close tothose observed and calculated for the Au–Ti bond lengths in alloysand clusters.73,74 The intralayer Au–Au bond length is 2.884 Å, ingood agreement with the experimental value.75

As the temperature increases from 0 K to 300 K, both slabsexpand and the average distance between the Au slab and the Tiadhesion layer increases by 0.05 Å, while the distance between Aulayers increases by at most 0.26 Å. The top layer of the Au filmdistorts notably due to interaction with the Ti layer. The metal lay-ers vibrate horizontally with respect to each other (Fig. 1). Heatingdoes not perturb the bonding pattern, as it does in other nanoscalesystems.76,77 At the same time, adding a Ti layer breaks the Auslab symmetry, influences wavefunction localization, and enhanceselectron-phonon coupling.

Figure 2 presents the hole DOS of the Au and Au–Ti slabs.Shown are the total DOS (TDOS) and partial DOS (PDOS) for theoptimized geometry. The zero of energy is set at the Fermi level.Since Au and Ti are metals, the systems have no bandgaps. Parts(a) and (b) of Fig. 2 demonstrate that the TDOS of Au undergoesa sharp drop in the energy range between −2 and −1 eV and thatthe holes are supported primarily by Au 5d atomic orbitals. Such adramatic decrease in the DOS should have a strong influence on thecharge-phonon energy exchange as holes at energies below −2 eVrelax to the Fermi level.

Adding a narrow Ti adhesion layer has a minor effect on theTDOS, whose overall shape does not change, parts (c) and (d) ofFig. 2. However, the adhesion layer has a strong effect on the charge-phonon relaxation dynamics, as demonstrated and discussed below.Upon a closer inspection of Figs. 2(c) and 2(d), one observes thatthe contribution of Ti to the TDOS rises exactly where the Au PDOSdecreases; Ti PDOS is negligible below −2 eV. However, at −1 eVand above, Au and Ti have nearly equal PDOS, even though thenumber of Ti atoms is seven times smaller than the number of Auatoms (Fig. 1).

Hole relaxation dynamics depends on NA coupling betweenelectronic states induced by nuclear motions, and coupling strengthis related to localization of hole wavefunctions [Eq. (5)]. Accordingto the excitation energy employed in many two-color pump-probeexperiments,23,24,78,79 we consider the energy range between −3 and0 eV. Since the DOS undergoes a rapid change around −1.5 eV(Fig. 2), we separate the total energy range further into the −3 to−2 eV interval of high DOS, the −1 to 0 eV interval of low DOS, andthe high-to-low DOS transition interval from −2 to −1 eV. We com-pute hole charge densities for each interval by averaging over chargedensities of all KS orbitals within each energy range (Fig. 3).

The wavefunctions of the pristine Au film are symmetric; how-ever, the density distribution between inner and surface regions ofthe slabs changes with energy (Fig. 3). Far below the Fermi level atenergies with high DOS (Fig. 2), the charge density is distributednearly evenly across the slab. In the region from −2 to −1 eV, inwhich the Au DOS decreases sharply and the Ti DOS grows, the Aucharge density is localized toward the surface of the slab, and there-fore, the Au wavefunctions can couple strongly to the Ti adhesionlayer. The Au charge density is localized toward the middle of theslab at energies close to the Fermi level.

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FIG. 2. Total density of states (TDOS)and partial density of states (PDOS) ofthe Au(111) surface. (a) Over energyrange from −8 to 2 eV, and (b) overa narrower energy range from −3.5 to0 eV (b). TDOS of the Au(111) slabwith an adsorbed layer of Ti and PDOSsplit into Au and Ti contributions (c) overenergy range from −8 to 2 eV and (d)over a narrower energy range from −3.5to 0 eV. The Fermi level is set to zero andshown by the dashed line. The Au DOSdecreases rapidly above −2 eV, whilethe Ti DOS starts to increase at aboutthe same energy. Above −1 eV, the DOSof a single Ti layer matches the DOS ofseven Au layers.

Adding a Ti layer breaks the symmetry of the wavefunctions.Most importantly, the system’s charge density has strong contribu-tions from the Ti layer in the regions where the Au DOS is low, i.e.,between −2 and 0 eV. At energies below −2 eV, where Au DOS ishigh, the charge density is localized away from the Ti adhesion layer.

FIG. 3. Average charge densities for all states within energy ranges from −3 to −2eV, −2 to −1 eV, and −1 to 0 eV in the Au and Au–Ti systems. Considering pureAu, the charge density is distributed evenly between −3 and −2 eV, when DOS ishigh (Fig. 2). Surface atoms contribute significantly to charge density in the regionof decreasing DOS from −2 to −1 eV. Between −1 and 0 eV, the charge density islocalized inside the Au slab. Considering the Au–Ti system, Ti has no contributionto the charge density below −2 eV and contributes significantly above −2 eV,matching the properties of the DOS (Fig. 2).

This situation favors rapid relaxation of holes because the lighter andfaster Ti atoms can couple to the holes exactly in the energy regionswith low Au DOS (>−2 eV), where hole relaxation in pure Au alonewould be slow.

B. Hole-phonon interactionsNonradiative relaxation of holes by coupling to phonons

appears in the simulations as a sequence of NA transitions betweenenergy levels occupied by the holes. Each elementary transitionreflects the strength of the NA coupling between the states. The NAcoupling fluctuates along the trajectory and depends on the time-derivative overlap between the corresponding adiabatic wavefunc-tions [Eq. (5)].

Table I summarizes the averaged absolute values of the NA cou-pling in the Au and Au–Ti systems. The canonical averaging is per-formed over both the overall energy range from −3 to 0 eV and overthe three energy windows considered previously (Fig. 3). Consider-ing the pristine Au slab, we observe that the NA coupling decreasessignificantly, by a factor of 5, in the region from −1 to 0 eV withlow Au DOS, relative to the energies farther away from the Fermilevel. This effect can be understood by considering the fact that theNA coupling is inversely proportional to the gap between the cor-responding energy levels [Eq. (5)]. Note that the DOS also changesby a factor of 5–6 between −2 and −1 eV (Fig. 2). The decrease inboth DOS and NA coupling results in a dramatic slowing down ofthe nonradiative hole relaxation.

Addition of a thin Ti adhesion layer has a strong effect on theNA coupling values (Table I), changing the hole relaxation dynam-ics. On the one hand, the overall NA coupling magnitudes are 5–9times larger in the presence of the Ti layer. On the other hand,the difference in the NA coupling values between the high and low

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TABLE I. Average absolute value of nonadiabatic coupling (NAC) and hole relaxation time (τ) for Au and Au–Ti at 300 K. The Fermi level is set to 0 eV.

Au Au–Ti

−3 to −2 eV −2 to −1 eV −1 to 0 eV −3 to 0 eV −3 to −2 eV −2 to −1 eV −1 to 0 eV −3 to 0 eV

NAC (meV) 6.31 5.10 1.12 5.33 23.41 26.68 9.55 22.36τ (ps) 0.53/16.29 0.23

energy regions is much smaller, roughly a factor of 2, compared tothe factor of 5 for pure Au. Furthermore, the NA coupling is largestin the energy region of rapidly changing DOS, −2 to −1 eV, in whichnonradiative relaxation of holes in pure Au is slow. The NA cou-pling grows significantly in the presence of Ti because the Ti atomis about 4 times lighter than the Au atom and the coupling is pro-portional to the nuclear velocity [Eq. (5)]. In addition, the statesbecome more localized in the presence of Ti (Fig. 3), and localizedstates exhibit stronger electron-phonon coupling than delocalizedstates. This is consistent with previous simulations of pure metalsthat predict that transition metals with Fermi energies intersect-ing the d-band PDOS have larger electron-phonon coupling factorsthan free electron metals with s- and/or p-band PDOS at the Fermilevel.80

In order to characterize the phonon modes that couple to theelectronic subsystem and the relative strengths of the hole-phononcoupling for different modes, we compute Fourier transforms ofthe energy gaps between states sampled at −3 eV, −2 eV, −1 eV,and the Fermi level. The spectra shown in Fig. 4 are known asinfluence spectra or spectral densities.80–82 The heights of the peaks

characterize the strength of hole-phonon interaction at the corre-sponding frequencies. As can be seen from Fig. 4(a), the phononmodes that couple to the electronic subsystem in pristine Au havelow frequencies, indicating that the acoustic modes exhibit strongercoupling than the higher frequency optical phonons. The domi-nant peaks below 100 cm−1 seen in the influence spectra of pris-tine Au can be assigned to acoustic phonon modes involving coreAu atoms.83 The strength of the hole-phonon coupling decreasesas holes get closer to the Fermi level, in agreement with the NAcoupling values shown in Table I. Introduction of the narrowTi layer changes the influence spectrum completely (Fig. 4(b)).In this case, the signal extends to 500 cm−1, with the domi-nant peaks seen around 300 cm−1. The change occurs because Tiatoms are much lighter than Au atoms and because the hole statesdelocalize from the Au slab to the Ti layer (Fig. 3). The fastermotions seen in the influence spectra of the Au–Ti system generatestronger NA coupling (Table I), which depends on nuclear velocity[Eq. (5)]. The simulation agrees with the shift to higher frequen-cies seen in experiments with gold disks in contact with adhesionlayers.23

FIG. 4. Fourier transforms of the energygaps between states at −3 and −2 eV,−3 and −1 eV, and −3 and 0 eV in(a) Au and (b) Au–Ti systems. Ti intro-duces higher frequency modes becauseit is lighter than Au.

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C. Nonradiative relaxation of photo-excited holesThe evolution of the average energy of the holes excited to

about −3.1 eV, according to the excitation energy employed in thepump-probe experiments,23,24,81 is shown for the Au and Au–Ti sys-tems in Fig. 5; note the difference in the x-axis scales in parts (a) and(b). The nonradiative relaxation of holes in the pristine Au slab isclearly bi-phasic. In the region with high Au DOS (Fig. 2), at energiesbetween −3 and −1.5 eV, the hole relaxation is fast. The relaxationslows down by a factor of 30 in the region of low Au DOS closeto the Fermi energy. Attempting to fit the curve in Fig. 5(a) with asingle exponent equation fails completely, while a bi-exponential fitdoes a good job describing the simulation data. The transition fromsubpicosecond relaxation to dynamics in tens of picoseconds can bequalified as the phonon bottleneck to hot-hole relaxation.84–87 Suchslow relaxation of hot charge carriers can be exploited to increase theefficiency of solar cells.88–90

The relaxation is much faster in the presence of the Ti adhe-sion layer (Fig. 5(b)). Thus, one can state that the Ti adhesion layerbreaks the phonon bottleneck and provides new, efficient path-ways for relaxation of hot holes. In this case, a single exponen-tial fit does a good job describing the data. Rigorously, quantumdynamics is always Gaussian at early times,85 as can be seen in thediscrepancy between the exponential fit represented by the dashedline and the raw data shown by the solid line. Exponential decaydevelops when quantum dynamics evolve to include multiple statesin the Hilbert space. Nevertheless, one can assign a single timescale to the hole relaxation in the Au–Ti system; by 0.5 ps, thevast majority of the charges have decayed to within 0.2 eV of theFermi level (Fig. 5(b)). In comparison, even after 6 ps, the aver-age hole energy is 1 eV below the Fermi level in the pure Au slab(Fig. 5(a)).

The qualitative and quantitative differences in the nonradiativehole relaxation in the Au and Au–Ti systems can be rationalized byconsidering DOS (Fig. 2) and NA coupling (Table I). The Ti layer,with 7 times fewer atoms than the Au slab, doubles the system’s holeDOS between −1.5 eV and the Fermi level. Ti atoms also contributesignificantly to the charge densities of the hole states in this energyrange (Fig. 3), especially between −2 and −1 eV where the Au DOSdrops dramatically and the relaxation bottleneck develops in pristineAu. Ti atoms are lighter and thus move faster than Au atoms (Fig. 4).Consequently, the Ti layer increases the NA coupling (Table I), espe-cially in the energy range around −1.5 eV, where hole relaxationundergoes a transition from fast to slow decay.

One can attempt to explain the differences in the calculatedtime scales using Fermi’s golden rule, whereby the relaxation rate is aquadratic function of the coupling and a linear function of the DOS.On comparing the 0.53 ps and 16.29 ps relaxation times in pure Auto the observed 0.23 ps characteristic time scale in Au–Ti, the hole-phonon relaxation dynamics is found to be a factor of 2.3 and 70.8faster in Au–Ti than in Au (Fig. 5). The NA coupling is different inthe Au and Au–Ti films by a factor of 3.7–8.5 (Table I). Ignoring thechanges in DOS (Fig. 2), which are much smaller than this differencein coupling, and squaring the NA coupling, we obtain a predictedfactor of 13–72 faster dynamics in Au–Ti than Au. This shows rea-sonable agreement with 2.3 and 70.8 estimated from the fitted timescales (Fig. 5). The deviations can be ascribed to several factors. First,the DOS is a function of energy and changes rapidly around −1.5 eV(Fig. 2). One needs to exclude this region of rapidly changing DOSfrom consideration and apply Fermi’s golden rule to energy rangeswith relatively constant DOS (or use integral versions of Fermi’sgolden rule); the NA coupling fluctuates along the trajectory, andcomparison of average absolute values or root-mean-square values

FIG. 5. Hole relaxation dynamics at300 K in (a) the Au(111) slab and (b)the Au(111) slab with an adsorbed layerof Ti. The relaxation in pure Au showsbi-phasic behavior, matching the sharpdrop in DOS below −2 eV (Fig. 2). A nar-row Ti adhesion layer greatly acceleratesthe relaxation.

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can lead to different conclusions. Additionally, the localization ofthe charge density depends on the energy and presence of the Tilayer (Fig. 3). For instance, between −3 and −2 eV, where DOS isrelatively constant (Fig. 2), the charge is localized on roughly twicefewer atoms in Au–Ti compared to Au (Fig. 3). Thus, the effectiveDOS is twice smaller in Au–Ti and Au for this energy range. The factthat rate expressions such as Fermi’s golden rule cannot fully explainthe calculated results emphasizes the need for quantum dynamicssimulations.

Comparing the hole relaxation dynamics studied here to theelectronic relaxation investigated previously in the same Au andAu–Ti systems,85,91 we observe that electrons in the Au film relaxuniformly across the whole energy range, in contrast to holes. Theelectron relaxation time is 1.8 ps, which is between the time scalesfor the fast and slow hole relaxation components. The electron DOSis similar to the hole DOS near the Fermi level (Fig. 2). However,the electron charge densities are distributed more evenly87,91 thanthose of the holes (Fig. 3), generating better wavefunction overlapand larger NA coupling. The significantly longer time scale asso-ciated with nonradiative relaxation of holes near the Fermi energymakes them better candidates for extraction of hot charge carriers,compared to electrons. The Ti adhesion layer accelerates both elec-tron and hole relaxation, leading to subpicosecond energy losses toheat and greatly reducing the possibility of utilization of hot chargecarriers.

IV. CONCLUSIONSBy performing time-domain ab initio simulations, we demon-

strate that nonradiative relaxation of excited holes in an Au filmundergoes a sharp transition from the subpicosecond to tens-of-picoseconds regime at 1.5 eV below the Fermi level. The bottleneckto this nonradiative relaxation arises due to a rapid decrease in theDOS and hole-phonon coupling in this energy range. Hole localiza-tion changes with energy as well; deep inside the valence band, thehole is delocalized over the whole Au film. In the energy region ofrapidly changing DOS, it is localized near the film surface. Close tothe Fermi energy, the hole is localized toward the middle of the film.The nonradiative relaxation of the hole is mediated by Au acousticmodes with frequencies below 100 cm−1.

The situation completely changes in the presence of a narrowTi layer that is used in applications to improve adhesion to a sub-strate. The bi-phasic fast-slow hole relaxation switches to a singleexponential decay that takes 0.23 ps. The relaxation is acceleratedbecause Ti contributes very significantly to the DOS in the criticalregion in which the dynamics in pure Au is slow because the Austates localized near the film surface strongly mix with Ti states andsince Ti atoms are much lighter and move faster than Au atoms.Hole relaxation in the Au–Ti system is facilitated by modes thatextend to 500 cm−1, and the hole-phonon coupling grows by a fac-tor of 3.7 at energies far from the Fermi level and 8.5 near the Fermienergy.

The hole relaxation bottleneck present in the pristine Au filmcan be utilized to extract hot charge carriers, reducing energy lossesand increasing efficiencies of optoelectronic devices. In order toextend the lifetime of high energy carriers in the presence of adhe-sion layers, these layers should be made of relatively heavy chem-ical elements, compared to Au, and/or have relatively small DOS

in the relevant energy range. Inversely, narrow adhesion layers ofTi atoms assist heat dissipation, providing advantages in electron-ics applications. The detailed atomistic time-domain analysis ofthe nonradiative charge relaxation reported in this work is directlyrelated to time-resolved spectroscopy experiments and emphasizesthe highly nonequilibrium nature of the relaxation dynamics. Byestablishing the mechanism of the nonradiative charge relaxation,and characterizing the electronic states and phonon modes involved,the study facilitates fundamental understanding of energy flow atengineered nanoscale interfaces, guiding future experiments andproviding principles for design of more efficient materials anddevices.

ACKNOWLEDGMENTSThe research was funded by the US Department of Defence,

Multidisciplinary University Research Initiative, Grant No. W911NF-16-1-0406. X.Z. acknowledges support of the National Natural Sci-ence Foundation of China, Grant No. 21473183.

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