Vibration-phonon Coupling Research Articles

Non-Markovian vibrational relaxation dynamics at surfaces.

Vibrational dynamics of adsorbates near surfaces plays both an important role for applied surface science and as a model lab for studying fundamental problems of open quantum systems. We employ a previously developed model for the relaxation of a D-Si-Si bending mode at a D:Si(100)-(2 × 1) surface, induced by a "bath" of more than 2000 phonon modes [Lorenz and P. Saalfrank, Chem. Phys. 482, 69 (2017)], to extend previous work along various directions. First, we use a Hierarchical Effective Mode (HEM) model [Fischer et al., J. Chem. Phys. 153, 064704 (2020)] to study relaxation of higher excited vibrational states than hitherto done by solving a high-dimensional system-bath time-dependent Schrödinger equation (TDSE). In the HEM approach, (many) real bath modes are replaced by (much less) effective bath modes. Accordingly, we are able to examine scaling laws for vibrational relaxation lifetimes for a realistic surface science problem. Second, we compare the performance of the multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) approach with that of the recently developed coherent-state-based multi-Davydov-D2 Ansatz [Zhou et al., J. Chem. Phys. 143, 014113 (2015)]. Both approaches work well, with some computational advantages for the latter in the presented context. Third, we apply open-system density matrix theory in comparison with basically "exact" solutions of the multi-mode TDSEs. Specifically, we use an open-system Liouville-von Neumann (LvN) equation treating vibration-phonon coupling as Markovian dissipation in Lindblad form to quantify effects beyond the Born-Markov approximation.

Density functional study of relaxation of adsorbate vibration modes: Dominance of anharmonic interaction.

Formulation and density functional workflow for calculating the lifetime of vibrational modes of molecular adsorbates on solid surfaces due to vibration-phonon coupling are presented. The anharmonic coupling is invoked to give the correct description of the origin of temperature dependence. Using pyrrolidine (C4H9N) absorbed on the Cu(001) surface as a concrete example, we show that the anharmonic coupling can be one to two orders more significant than the harmonic interaction for the broadening of vibrational spectra, especially as temperature increases. These results challenge the common assumption that the anharmonic interaction is weak and call for attention of considering its effect in quantum relaxation and related problems.

A hierarchical effective mode approach to phonon-driven multilevel vibrational relaxation dynamics at surfaces

We discuss an efficient Hierarchical Effective Mode (HEM) representation of a high-dimensional harmonic oscillator bath, which describes phonon-driven vibrational relaxation of an adsorbate-surface system, namely, deuterium adsorbed on Si(100). Starting from the original Hamiltonian of the adsorbate-surface system, the HEM representation is constructed via iterative orthogonal transformations, which are efficiently implemented with Householder matrices. The detailed description of the HEM representation and its construction are given in the second quantization representation. The hierarchical nature of this representation allows access to the exact quantum dynamics of the adsorbate-surface system over finite time intervals, controllable via the truncation order of the hierarchy. To study the convergence properties of the effective mode representation, we solve the time-dependent Schrödinger equation of the truncated system-bath HEM Hamiltonian, with the help of the multilayer extension of the Multiconfigurational Time-Dependent Hartree (ML-MCTDH) method. The results of the HEM representation are compared with those obtained with a quantum-mechanical tier-model. The convergence of the HEM representation with respect to the truncation order of the hierarchy is discussed for different initial conditions of the adsorbate-surface system. The combination of the HEM representation with the ML-MCTDH method provides information on the time evolution of the system (adsorbate) and multiple effective modes of the bath (surface). This permits insight into mechanisms of vibration-phonon coupling of the adsorbate-surface system, as well as inter-mode couplings of the effective bath.

Isotopic effects in vibrational relaxation dynamics of H on a Si(100) surface

In a recent paper [U. Lorenz and P. Saalfrank, Chem. Phys. 482, 69 (2017)], we proposed a robust scheme to set up a system-bath model Hamiltonian, describing the coupling of adsorbate vibrations (system) to surface phonons (bath), from first principles. The method is based on an embedded cluster approach, using orthogonal coordinates for system and bath modes, and an anharmonic phononic expansion of the system-bath interaction up to second order. In this contribution, we use this model Hamiltonian to calculate vibrational relaxation rates of H-Si and D-Si bending modes, coupled to a fully H(D)-covered Si(100)-(2×1) surface, at zero temperature. The D-Si bending mode has an anharmonic frequency lying inside the bath frequency spectrum, whereas the H-Si bending mode frequency is outside the bath Debye band. Therefore, in the present calculations, we only take into account one-phonon system-bath couplings for the D-Si system and both one- and two-phonon interaction terms in the case of H-Si. The computation of vibrational lifetimes is performed with two different approaches, namely, Fermi's golden rule, and a generalized Bixon-Jortner model built in a restricted vibrational space of the adsorbate-surface zeroth-order Hamiltonian. For D-Si, the Bixon-Jortner Hamiltonian can be solved by exact diagonalization, serving as a benchmark, whereas for H-Si, an iterative scheme based on the recursive residue generation method is applied, with excellent convergence properties. We found that the lifetimes obtained with perturbation theory, albeit having almost the same order of magnitude-a few hundred fs for D-Si and a couple of ps for H-Si-, are strongly dependent on the discretized numerical representation of the bath spectral density. On the other hand, the Bixon-Jortner model is free of such numerical deficiencies, therefore providing better estimates of vibrational relaxation rates, at a very low computational cost. The results obtained with this model clearly show a net exponential decay of the time-dependent survival probability for the H-Si initial vibrational state, allowing an easy extraction of the bending mode "lifetime." This is in contrast with the D-Si system, whose survival probability exhibits a non-monotonic decay, making it difficult to define such a lifetime. This different behavior of the vibrational decay is rationalized in terms of the power spectrum of the adsorbate-surface system. In the case of D-Si, it consists of several, non-uniformly distributed peaks around the bending mode frequency, whereas the H-Si spectrum exhibits a single Lorentzian lineshape, whose width corresponds to the calculated lifetime. The present work gives some insight into mechanisms of vibration-phonon coupling at surfaces. It also serves as a benchmark for multidimensional system-bath quantum dynamics, for comparison with approximate schemes such as reduced, open-system density matrix theory (where the bath is traced out and a Liouville-von Neumann equation is solved) or approximate wavefunction methods to solve the combined system-bath Schrödinger equation.

A novel system-bath Hamiltonian for vibration-phonon coupling: Formulation, and application to the relaxation of Si–H and Si–D bending modes of H/D:Si(100)-(2 × 1)

We present a rigorous method to set up a system-bath Hamiltonian for the coupling of adsorbate vibrations (the system) to surface phonons (the bath). The Hamiltonian is straightforward to derive and exact up to second order in the environment coordinates, thus capable of treating one- and two-phonon contributions to vibration-phonon coupling. The construction of the Hamiltonian uses orthogonal coordinates for system and bath modes, is based on an embedded cluster approach, and generalizes previous Hamiltonians of a similar type, but avoids several (additional) approximations. While the parametrization of the full Hamiltonian is in principle feasible by a first principles quantum mechanical treatment, here we adopt in the spirit of a QM/MM model a combination of density functional theory (“QM”, for the system) and a semiempirical forcefield (“MM”, for the bath). We apply the Hamiltonian to a fully H-covered Si(100)-(2×1) surface, using Fermi’s Golden Rule to obtain vibrational relaxation rates of various H–Si bending modes of this system. As in earlier work it is found that the relaxation is dominated by two-phonon contributions because of an energy gap between the Si–H bending modes and the Si phonon bands. We obtain vibrational lifetimes (of the first excited state) on the order of 2ps at T=0K. The lifetimes depend only little on the type of bending mode (symmetric vs. antisymmetric, parallel vs. perpendicular to the Si2H2 dimers). They decrease by a factor of about two when heating the surface to 300K. We also study isotope effects by replacing adsorbed H atoms by deuterium, D. The Si–D bending modes are shifted into the Si phonon band of the solid, opening up one-phonon decay channels and reducing the lifetimes to few hundred fs.

Dynamics of radiationless transitions in large molecules: 4. An ensemble of molecules occurring in condensed medium at a finite temperature

Statistically averaged vibration-phonon (VP) states of large molecules occurring in a condensed medium are represented by superpositions of lower excited states of high-frequency intramolecular vibrations and phonons in the state of thermal equilibrium. The dynamics of amplitudes of these states is described by an ensemble of Hamiltonians, wherein the distribution of their parameters characterizes the configuration distribution of the medium and determines the broadening and shift of absorption bands and the time evolution of the ensemble population. The equations of motion for statistically averaged amplitudes, as found by second-order cumulant expansion, coincide with those found by solving the secular equation, which takes into account both intramolecular and VP interactions. The relative contribution of the decay and decoherence to the evolution of the amplitudes has been considered. Critical values at which the transition from regular recurrence cycles to stochastic dynamics takes place have been found for the VP coupling parameters. The ergodicity of VP systems and the conditions of separation of the intramolecular and phonon-stimulated dynamics have been considered.

Dynamics of radiationless transitions in large molecules: 3. Decay of vibration-phonon states

The statistical operator of the ensemble of high-frequency intramolecular vibrations associated with the phonon reservoir depends on the phonon occupation numbers under thermal equilibrium conditions. The eigenvalues of energy of statistically averaged vibration-phonon (VP) states are complex quantities. In the case of weak VP coupling, only one- and two-phonon transitions are taken into consideration for calculating the decay rate constant, in which the difference of phonon energies compensates for the difference in energy between the initial and final intramolecular states. Although the fast evolution of amplitudes of VP states is due to intramolecular redistribution of energy and is not reduced to exponential decay of the initial state, the imaginary components of the eigenvalues coincide with those predicted by Fermi’s golden rule. The relative contribution of two-phonon (combination) transitions compared with one-phonon transitions increases with an increase in the density of intramolecular states and temperature, becoming prevalent for large molecules at T ≥ D ≫ Δ0 (D = 100–200 K (70–140 cm−1) is the Debye temperature and Δ0∼10 cm−1 is the spacing between neighboring intramolecular vibration levels). When T ≥ D, the decay rate constant is K ∝ T2.

Isotopic effect on the vibrational lifetime of the carbon-deuterium stretch excitation on graphene

The relaxation of vibrational energy in the H and D stretch modes has been studied on the graphene surface using ab initio calculations. The dissipation of the vibrational energy stored in the stretching modes proceeds through vibration-phonon coupling, while the dissipation through electronic excitations makes only minor contributions. Recently, we reported the fast relaxation of the H stretch energy on graphene [S. Sakong and P. Kratzer, J. Chem. Phys. 133, 054505 (2010)]. Interestingly, we predict the lifetime of the D stretch to be markedly longer compared to the relaxation of the H stretch. This is unexpected since the vibrational amplitudes at carbon atoms in the joint C-D vibrational modes are larger than in the joint C-H modes, due to the mass ratio m(D)/m(C) > m(H)/m(C). However, the vibrational relaxation rate for the D stretch is smaller than for the H stretch, because the energy is dissipated to an acoustic phonon of graphene in the case of C-D rather than an optical phonon as is the case in C-H, and hence, the corresponding phonon density of states is lower in the C-D case. To rationalize our findings, we propose a general scheme for estimating vibrational lifetimes of adsorbates based on four factors: the density of states of the phonons that mediates the transitions, the vibration-phonon coupling strength, the anharmonic coupling between local modes, and the number of quanta involved in the transitions. Mainly the first two of these factors are responsible for the differences in the lifetimes of the C-H and C-D stretches. The possible role of the other factors is illustrated in the context of vibrational lifetimes in other recently studied systems.

Mode-selective excitation of hydrogen atoms on a Si surface: Non-Markovian and Markovian treatment of infrared laser driven dissipative quantum dynamics

Mode-selective excitation of adsorbates by shaped infrared laser pulses is investigated here theoretically, for the example of a H atom on a hydrogen-covered $\mathrm{Si}(100)\text{\ensuremath{-}}2\ifmmode\times\else\texttimes\fi{}1$ surface. The mode-selective excitation is perturbed by the intermode coupling within the system (bending and stretching modes) and by system-bath coupling to substrate phonons. Using a force-field based model, vibration-phonon coupling was found and predicted to lead to vibrational relaxation of the H-Si stretching mode on a ns timescale, and of the Si-Si-H bending mode on a ps timescale [I. Andrianov and P. Saalfrank, J. Chem. Phys. 124, 034710 (2006)]. To address the question as to whether in such a dissipative situation mode-selective control of adsorbate vibrational dynamics is still possible, a system-bath ansatz is used to derive an open-system density matrix theory in which the H vibrations are driven either by ${\mathrm{sin}}^{2}$, or by freely optimized infrared ps laser pulses. Both for the Si-H stretching and Si-Si-H bending vibrations mode-selective excitation is predicted to be possible. It is also found that the Markov approximation works well in most of the applications, and that simple ${\mathrm{sin}}^{2}$ are nearly as effective as pulses which were freely optimized by optimal control theory.

Theoretical study of vibration-phonon coupling of H adsorbed on a Si(100) surface

In this paper a perturbation-theory study of vibrational lifetimes for the bending and stretching modes of hydrogen adsorbed on a Si(100) surface is presented. The hydrogen-silicon interaction is treated with a semiempirical bond-order potential. Calculations are performed for H-Si clusters of different sizes. The finite lifetime is due to vibration-phonon coupling, which is assumed to be linear or bilinear in the phonon and nonlinear in the H-Si stretching and bending modes. Lifetimes and vibrational transition rates are evaluated with one- and two-phonon processes taken into account. Temperature effects are also discussed. In agreement with the experiment and previous theoretical treatment it is found that the H-Si (upsilon(s) = 1) stretching vibration decays on a nanosecond timescale, whereas for the H-Si (upsilon(b) = 1) bending mode a picosecond decay is predicted. For higher-excited vibrations, simple scaling laws are found if the excitation energies are not too large. The relaxation mechanisms for the excited H-Si stretching and the H-Si bending modes are analyzed in detail.

Spectral line shapes of impurity centers in solids at low temperatures

The paper briefly overviews a master equation theory for the case of a single pseudolocal vibration interacting with an electronic molecular transition, a phonon bath, and laser irradiation. The theory is valid for a broad range of laser powers. At low laser powers it was compared to sudden-jump model. Qualitatively, both theories predict that in addition to the conventional Lorentzian, the line consists of a dispersive component in the core region and a side band. Quantitatively, the sudden jump model is applicable only if the vibration–phonon coupling is independent of the electronic wave functions. An interesting property of the complex line was found at strong excitation. In this case the line first behaves in a fashion intermediate to the saturation behavior of homogeneously and inhomogeneously broadened spectra. But as the laser intensity is growing, the interaction with optical field “suppress” the dephasing caused by the interaction with the matrix and the line becomes a Lorentzian.

Spectral line shapes of single molecules beyond the sudden jump model

Detailed analysis is done for the case of a single pseudo local phonon interacting with an electronic molecular transition and a phonon bath. Both the sudden-jump model and the master equation formalism predict a complex optical line shape for a single-molecule embedded in a solid matrix. In addition to the conventional Lorentzian, the line consists of a dispersive component in the core region and a side band. The sudden jump model is applicable only if the vibration-phonon coupling is independent of the electronic wave functions. However if the line wings are neglected, the sudden jump model can fit experimental data even when the condition of its applicability is not fulfilled. An interesting property of the complex line was found at strong excitation. In this case the line behaves in a fashion intermediate to the homogeneously and inhomogeneously broadened spectra.

Vibrational dephasing in crystals: Theory and experiment

Vibrational lineshapes of molecules in crystals are determined to a large extent by the vibration-phonon coupling. We show how our previously developed nonperturbative theory of optical pure dephasing can be applied to the vibrational dephasing problem. We consider two cases: dephasing in a pure crystal, and dephasing of a substitutional impurity. We compare our theoretical results to temperature-dependent linewidth and lineshift measurements on the 766 cm −1 mode in pure naphthalene, and the V 3 mode of ReO 4 − in KI.