Dynamics of Phononic Dissipation at the Atomic Scale: Dependence on Internal Degrees of Freedom

PHYSICAL REVIEW B 76, 205430 ͑2007͒

Dynamics of phononic dissipation at the atomic scale: Dependence on internal degrees of freedom

H. Sevinçli,1 S. Mukhopadhyay,1,* R. T. Senger,1,2 and S. Ciraci1,2,† 1Department of Physics, Bilkent University, 06800 Ankara, Turkey 2UNAM-Material Science and Nanotechnology Institute, Bilkent University, 06800 Ankara, Turkey ͑Received 1 March 2007; revised manuscript received 3 October 2007; published 21 November 2007͒ Dynamics of dissipation local vibrations to the surrounding substrate is a key issue in friction between sliding surfaces as well as in boundary lubrication. We consider a model system consisting of an excited nano-particle which is weakly coupled with a substrate. Using three different methods, we solve the dynamics of energy dissipation for different types of coupling between the nanoparticle and the substrate, where different types of dimensionality and phonon densities of states were also considered for the substrate. In this paper, we present a microscopic analysis of transient properties of energy dissipation via phonon discharge toward the substrate. Finally, important conclusions of our theoretical analysis are veriﬁed by a realistic study, where the phonon modes and interaction parameters involved in the energy dissipation from an excited benzene molecule to the graphene are calculated by using ﬁrst-principles methods. The methods used are applicable also to dissipative processes in the contexts of infrared Raman spectroscopy and atomic force microscopy of molecules on surfaces.

DOI: 10.1103/PhysRevB.76.205430 PACS number͑s͒: 68.35.Af, 63.22.ϩm

I. INTRODUCTION propose an Hamiltonian treatment of the problem. Since the number of physical ingredients determining the dynamics is Friction between two surfaces in relative motion involves considerably large, our strategy will be to focus on them many interesting and complex phenomena induced by the separately to reveal their role in energy dissipation. long- and short-range forces, such as adhesion, wetting, atom In this work, we present our analysis concerning the de- exchange, bond breaking and bond formation, and elastic 1–14 pendence of phononic dissipation on internal degrees of free- and plastic deformation. In general, nonequilibrium dom of the nanoparticle and the substrate by using two types phonons are generated in the expense of damped mechanical of coupling between the finite and extended systems. We energy.15–20 Dissipation of this excess energy is one of the consider three different theoretical methods; namely, the important issues in dry-sliding friction and lubrication.10,21–23 equation of motion ͑EoM͒ technique which involves Laplace Normally, the dissipation of mechanical energy is resulted in heating of parts in relative motion. Sometimes, it gives rise transforms for the solution of the coupled differential equa- ͑ ͒ to wear and failure due to overheating. In general, significant tions for phonon operators, the Fano-Anderson FA method amounts of resources ͑energy and material͒ are lost in the which is useful for diagonalizing quadratic Hamiltonians, ͑ ͒ course of friction. One of the prime goals of tribology is to and Green’s function GF method by which we can incor- minimize energy dissipation through lubrication. Recently, porate the effect of multimodes into the study. The first two several works have attempted to develop surfaces with su- methods have limited applicability for specific cases only perlow friction coefficients.24 and will be presented for completeness. The GF method, In the past, the energy dissipation during sliding has been being the most general method, will be extensively dis- usually investigated in the macroscopic scale by using cussed. We also note that the GF method allows solutions simple Tomlinson’s model.3 Hence, the dissipated energy and beyond linear response where the EoM and FA methods friction force have been revealed indirectly from stick-slip yield solutions within the linear response regime. motion. The objective of the present work is to develop a The organization of the paper is as follows. In Sec. II, we microscopic ͑or atomic scale͒ understanding of phononic en- describe the physical model. The theoretical methods to be ergy dissipation during sliding friction, especially to shed used and their limitations are presented in Sec. III. The ap- some light on the dynamics of discharge of excited phonons plications of theoretical methods to different types of cou- ͑ on a nanoparticle representing a lubricant molecule or an pling and substrates having different densities of states will ͒ asperity into the substrate. This problem has many aspects be presented and discussed in Sec. IV. These are mainly the and the solution will depend on a variety of physical param- dependence of the decay rate on the coupling constant, the eters which can be grouped into major categories, such as interaction-specific dependence of decay rate on the nanopar- internal degrees of freedom of the nanoparticle, density of ticle mode frequencies, and the effect of neighboring modes substrate phonon modes, the type and strength of coupling on the decay rate of each other. Finally, we present a specific between the nanoparticle and the substrate, and finally the and realistic example, where the dissipation of vibrational initial temperatures. One way of studying this problem could ͑ H ͒ to a graphene substrate be carrying out state-of-the-art molecular dynamics simula- modes of benzene molecule C6 6 ͑ ͒ tions which yields sample specific results only. However, to is analyzed by using density functional theory DFT calcu- explore the general features of the phononic dissipation, we lations. We summarize our conclusions in Sec. V.

pending on the strength of the interaction, the sharpness of the peaks and the overlap between the neighboring peaks will differ. If the coupling is weak enough, we may neglect the overlaps, namely, * → ͉ ͉2␦ ͑ ͒ WklWkj Wkl l,j. 5 For Lorentzian coupling, we assume that the coupling terms of different modes of the nanoparticle do not overlap, and hence, we can treat each nanoparticle mode separately. For the second type of coupling, we consider the coupling FIG. 1. ͑Color online͒ A nanoparticle with discrete density of coefficients which scale inversely as the square root of the phonon modes is coupled to a substrate having continuous density of modes. product of the frequencies of the coupled modes, i.e., ␣͑␻ ␻ ͒−1/2 ͑ ͒ Wkj = k j . 6 II. MODEL The coefficient ␣ stands for the strength of the coupling and We first consider a nanoparticle representing a lubricant will depend on the interaction between the nanoparticle and ␻ molecule or an asperity, which is weakly coupled to a sub- the substrate. In both coupling types, Wkj is a function of k ͑ ͒ ␻ strate Fig. 1 . The vibrational modes of the nanoparticle are and j, explicit dependence on the wave vectors is not in- excited initially and the excess phonons discharge to the cluded for the sake of simplicity. The effect of initial tem- bulk. The total Hamiltonian of the system can be written as peratures of the parties, besides from the effect of tempera- ͑ ͒ ture difference, is another major topic in its own and we H = HM + HS + HMS 1 leave that discussion to another paper. In the present paper,

where HM and HS are the free phononic Hamiltonians of the we consider the initial temperatures to be zero and limit our nanoparticle ͑or molecule͒ and the substrate ͑or bath͒, re- attention to the near-equilibrium case in the weak coupling spectively; the interaction between them is represented by regime. Strong coupling regime and nonequilibrium cases HMS. We also define H=H0 +HMS. We assume that the har- will also be treated separately. monic approximation is good enough for HM and HS, and their phonon spectra are known, i.e., III. THEORETICAL METHODS ប␻ + ͑ ͒ A. Equation of motion technique HM = ͚ jaj aj, 2 j The time dependent occupancies of the nanoparticle modes can be obtained using Heisenberg’s equation of mo- ប␻ + ͑ ͒ HS = ͚ kbkbk. 3 tion, namely, A˙ ͑t͒=i͓H,A͑t͔͒/ប. The equations of motion for k the phonon annihilation operators are Here, ␻ are the frequencies of the nanoparticle modes with j ͑ ͒ ␻ ͑ ͒ * ͑ ͒ ͑ ͒ + a˙ l t =−i lal t − i͚ W bk t , 7 aj and aj being the corresponding annihilation and creation kl ␻ k operators; k are the frequencies of the substrate vibration + modes of wave vectors k and bk and bk are the correspond- ˙ ͑ ͒ ␻ ͑ ͒ ͑ ͒ ͑ ͒ ing phonon annihilation and creation operators. We have bk t =−i kbk t − i͚ Wkjaj t , 8 omitted the constant terms as they do not contribute to the j dynamics of the system. Here, we consider a single phonon that is, we have coupled differential equations for each op- branch without loosing generality, but the formalism can be erator. Performing Laplace transformation to both equations, extended to include multiple branches. The interaction a pair of coupled algebraic equations is obtained which can Hamiltonian HMS is also assumed to be quadratic in phonon be decoupled algebraically, and by inverse transformation, operators, ͑ ͒ the time dependent operator al t is obtained as + H = ͚ ប͑W b a + H.c.͒, ͑4͒ ͑ ͒ st st ͑ ͒ MS kj k j al 0 e ds 1 e Jl s ds k,j a ͑t͒ = Ͷ − Ͷ . ͑9͒ l ␲ ␻ ͑ ͒ ␲ ␻ ͑ ͒ 2 i B s + i l + Il s 2 B s + i l + Il s with Wkj being the coupling coefficient which is a function ␻ ␻ of k and j and has the dimesion of angular frequency. We where the integrals are to be evaluated along the Bromwich ͑ ͒ ͑ ͒ disregard the double annihilation and double creation of contour, with Il s and Jl s being the substrate and interac- phonons in the present work. tion specific functions ͑see Appendix A͒. Here, we consider two types of coupling. The first one is The first and second terms in Eq. ͑9͒ stand for the contri- the Lorentzian coupling in which the coupling coefficient butions from the initial excitation of the nanoparticle and the ␻ Wkj is a Lorentzian with its peak located at j and has a initial temperature of the substrate, respectively. The second ⌫ width j. As long as the coupling between the nanoparticle term does not contribute to the time dependent occupations and the substrate is weak, Wkj will be a peaked function of of the nanoparticle mode, since the initial temperature of the ␻ ␻ k and a separate peak will be present around each j. De- substrate is considered to be zero.

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B. Fano-Anderson method ͑ Ј͒ ͗ ˆ ͑ ͒ˆ +͑ Ј͒͘ ͑ ͒ D k,t − t =−i Ttbk t bk t , 12 Since the Hamiltonian is quadratic in operators, its solu- where Tt is the time-ordering operator, and the operators in tion is equivalent to diagonalizing a matrix. Exact diagonal- Heisenberg picture are distinguished by a hat. ization of such quadratic Hamiltonians was shown to be pos- 25 26 Since each term in the interaction Hamiltonian includes sible by Fano and Anderson independently, and the odd number of nanoparticle operators, only the even order procedure is widely used in atomic physics, solid state phys- terms contribute in the expansion. The first contribution due ics, quantum optics, etc. Here, we will apply their method to to the interaction is the second order term, the problem of phononic dissipation. Solution of the Hamiltonian is equivalent to finding the d͑0͒͑␻ ͒2͚ W2 D͑0͒͑␻ ͒ = d͑0͒͑␻ ͒2⌺͑2͒͑␻ ͒, ͑13͒ ␣͑␻ ͒ j jk k j j dressed operators q such that the Hamiltonian is diago- k nal in terms of the dressed operators, H ͑ ͒ with ⌺ 2 ͑␻ ͒ being the second order self-energy. =͚ ប␻ ␣+͑␻ ͒␣͑␻ ͒. j q q q q First, we wish to limit our attention to the case of a nano- As described in AppendixB, one can write the time de- particle having a single mode. In obtaining the solution for a pendent occupancy of jth mode as single mode, we will relate it to the FA result for the sake of illustration and then we will generalize our solution for the 2 ␻ case of a nanoparticle having multiple modes. In doing so, ͗ ͑ ͒͘ ͗ ͑ ͒ͯ͘͵ ␻ ͑␻ ͉͒␮͑␻ ␻ ͉͒2 −i qtͯ nj t = nj 0 d qg q q, j e we will be able to take the interplay between neighboring modes during dissipation into account. For a single nanoparticle mode, the higher order terms can + ͵ d␻ g͑␻ ͒͗n ͑0͒͘ k k k be expressed in terms of the second order self-energy and the 2 free GF as ␻ ϫͯ ␻ ͑␻ ͒␮*͑␻ ␻ ͒␯*͑␻ ␻ ͒ −i qtͯ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͵ 0 0 2 0 2 2 d qg q q, j q, k e , ͑␻ ␻ ͒ ͑␻ ␻ ͒ ⌺ ͑ ⌺ ͒ d j, q = d j, q „1+d + d + ¯…. ͑ ͒ ͑10͒ 14 For weak coupling, the above series can be written as where ␮͑␻ ,␻ ͒ are the expansion coefficients for the mo- ͑ ͒ q j d 0 ͑␻ ,␻ ͒ lecular operator a in terms of the dressed operators ␣͑␻ ͒ d͑␻ ,␻ ͒ = j q , ͑15͒ j q j q ͑0͒͑␻ ␻ ͒⌺͑2͒͑␻ ␻ ͒ ␯͑␻ ␻ ͒ ͑ 1−d j, q j, q and q , k are that of the substrate operator bk see Ap- pendixB͒. hence, the retarded GF becomes Due to the finite range of substrate density of states ͑ ͒ ͑␻ ͒ 1 DOS g k , the integrals involved in the FA method are dR͑␻ ,␻ ͒ = . ͑16͒ j q ␻ ␻ ⌺͑2͒͑␻ ␻ ͒ bounded. The method allows us to perform calculations for q − j − j, q any g͑␻ ͒ and for any type of coupling with a single nano- k The real and imaginary parts of the second order self- particle mode. The time dependent occupation is again sepa- energy can be separated, rable as contributions from the initial temperature of the ␻ ͑␻ ͒ 2 nanoparticle and that of the substrate. However, it should be ͑ ͒ d kg k Wjk ⌺ 2 ͑␻ ,␻ ͒ = P ͵ − i␲g͑␻ ͒W2 , ͑17͒ noted that the FA method is applicable for any coupling type j q ␻ − ␻ q jq and any density of states for the substrate as long as we q k consider a single nanoparticle mode. where P is for principal part of the integral, and the spectral function is obtained by ͑2͒ C. Green’s function method −2Im⌺ ͑j,␻ ͒ A͑j,␻ ͒ = q . q ␻ − ␻ −Re⌺͑2͒͑j,␻ ͒ 2 + Im ⌺͑2͒͑j,␻ ͒ 2 The effect of neighboring modes of nanoparticle having „ q j q … „ q … multimodes cannot be resolved within the above methods. ͑18͒ That is, EoM and FA methods are restricted to the linear The real part of the second order self-energy ⌺͑2͒ is equal response regime. We use a more generalized method by ␴ to the shift in the jth mode of the nanoparticle, j, obtained which one can consider effects of neighboring modes. For previously using the FA method, and the square of the imagi- this purpose, we employ Green’s functions. Initially, the sub- ⌺͑2͒ ͓␲ ͑␻ ͒ ͔2 nary part of is g q Wqj . That is, the FA expansion strate temperature is zero and the phonon modes of the nano- coefficient ␮ finds its expression in terms of the spectral particle are empty except for the excitations which do not function as necessarily obey Bose-Einstein distribution. That is, the initial occupation of a nanoparticle mode is not a function of A͑j,␻ ͒ ͉␮͑␻ ,␻ ͉͒2 = q . ͑19͒ temperature. Therefore, we make use of zero temperature q j ␲ ͑␻ ͒ 2 g q Green’s functions instead of Matsubara formalism, The time dependent GF can be written in terms of the spectral function and the time dependent occupancy of the jth ͑ Ј͒ ͗ ͑ ͒ +͑ Ј͒͘ ͑ ͒ d j,t − t =−i Ttaˆ j t aˆ j t , 11 mode is obtained as

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⌺͑2͒͑j,␻ ͒ (a) ⌺͑ ␻ ͒ q ͑ ͒ j, q = ͑ ͒ . 24 ′ ⌺ 2 ͑ ␻ ͒ t t1 t2 t3 t4 t2n−1 t2n t j, q 2 ͑ ͒ ... 0 ͑ Ј ␻ ͒ 1− 2 ͚ WjЈqd j , q j k1 j k2 kn j Wjq jЈj Once the self-energy is found, the spectral function, therefore the time dependent occupancy of the nanoparticle modes can (b) ′ be calculated. t t1 t2 t3 t2n−2 t2n−1 t2n t ... A few remarks about the above expression for the self- j j j j k1 1 n−1 kn energy follow. First, it is exact in the sense that it includes ⌺͑ ␻ ͒ contributions from diagrams to all orders. Second j, q is FIG. 2. Diagrams of order 2n. Solid lines are the phonon lines of not a quadratic function of coupling Wkj anymore, as it was the nanoparticle where the dashed lines are that of the substrate. ͑a͒ in the single mode approximation case. Rather, the decay rate Diagram for the case of single nanoparticle mode j. ki stand for the of jth mode collects contributions from all other modes also. substrate modes. ͑b͒ Diagram of order 2n when there exists multiple Third, the spectral function is not of Lorentzian shape any- modes ͑ji͒ for the nanoparticle. more; extra peaks and dips in the spectral function are in question which will be analyzed numerically in the following ␻ 2 section. d q ␻ ͗ ͑ ͒͘ ͗ ͑ ͒ͯ͘͵ ͑ ␻ ͒ −i qtͯ ͑ ͒ nj t = nj 0 A j, q e . 20 2␲ IV. RESULTS AND DISCUSSIONS In order to incorporate the effect of neighboring modes, In this section, we will apply the above methods using we follow the diagrammatic technique. As long as the Lorentzian and square-root coupling to analyze the depen- Hamiltonian is quadratic, the primitive vertex, from which dence of the decay rate on the properties of nanoparticles and all diagrams are to be constructed, will consist of two pho- substrates having one-dimensional ͑1D͒ and two- non lines. That is, each interaction point is the intersection of dimensional ͑2D͒ Debye DOSs. The analytical results ob- two phonon lines. Since the interaction Hamiltonian relates a tained using EoM technique and numerical results obtained nanoparticle mode to a substrate mode only, each vertex con- using FA and GF methods are discussed separately. An ap- tains one nanoparticle phonon line and a substrate phonon plication for a real physical system using DFT and the GF line. So the diagram of any order can be constructed ͓see Fig. method will also be presented. 2͑b͔͒. Having obtained the diagrammatic expansion for any ͑i͒ As an application of the EoM method, we consider the order 2n, under certain conditions about the coupling type, nanoparticle that has a single mode coupled to a 1D or 2D the ͑2n͒th order self-energy term can be expressed in terms Debye substrate. The substrate is initially at zero tempera- of the second order term and the free GF of the nanoparticle ture. Assuming Lorentzian coupling, namely, modes. If the fraction W /W is independent of k , the 2 k1j1 k1j2 1 ␣ ⌫ 1 self-energy for the multimode case can be found exactly W2 = , ͑25͒ kj ␲ ͑␻ ␻ ͒2 ⌫2/ where the fourth and sixth order contributions can be written 2 k − j + 4 as ͑ ͒͑ ͒ we have for Ij s see Appendix A ␻ ␣2⌫ max ␻ ␻d−1 cD ͵ d k k 1 Ij = , ⌺͑4͒͑ ␻ ͒ ͑0͒͑ ␻ ͒ 2 ⌺͑2͒͑ ␻ ͒ 2 ␲ ⌫ ⌫ j, q = ΀͚ d j1, q W ΁ j, q , 2 i 0 2 qj1 „ … ͑␻ ͒ͩ␻ ␻ ͪͩ␻ ␻ ͪ W k − is k − j − i k − j + i qj j1 2 2 ͑ ͒ 21 ͑26͒

where d is the dimension of the substrate, cD is the corre- ͑␻͒ ␻d−1 ␻ 1 sponding Debye constant for DOS, and gd =cD . j ⌺͑6͒͑ ␻ ͒ ͑0͒͑ ␻ ͒ 2 2 ⌺͑2͒͑ ␻ ͒ 3 j, q = ΀͚ d j1, q W ΁ j, q . and ⌫ are real and positive, and by deﬁnition of Laplace W4 qj1 „ … qj j1 transformation, Re͑s͒Ͼ0. In the weak coupling regime, the ␻ ͑22͒ width of the Lorentzian will be much smaller than j and ␻ 28 max, so we can approximate the above integral by extending the limits of integration to ͑−ϱ,ϱ͒, in which case the By mathematical induction, the ͑2n͒th term is found as ␻ integral can be evaluated analytically on the complex k ͑ ͒ ␣2 ͑␻ ⌫/ ͒d−1/͑ ␻ plane with the result Ij s = cD j −i 2 s+i j ⌫/ ͒ 1 + 2 . Performing the inverse transformation, one ﬁnds ⌺͑2n͒͑ ␻ ͒ ͑0͒͑ ␻ ͒ 2 n−1 ⌺͑2͒͑ ␻ ͒ n j, q = ͑ ͒ ΀͚ d j1, q W ΁ j, q , ⌫ / W2 n−1 qj1 „ … e− t 2 qj j1 ͗n ͑t͒͘ = ͗n ͑0͒͘ ͉͑⌫ + ⌬͒e⌬t/2 − ͑⌫ − ⌬͒e−⌬t/2͉2, j j ͉⌬͉2 ͑23͒ 4 ͑27͒ ⌬2 ⌫2 / ␣2 ͑␻ ⌫/ ͒d−1 and hence, where = 4−4 cD j −i 2 .

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1 consequence of dependence on substrate DOS is that if the DOS at the peak of the spectral function tends to zero, the ͑ ͉␮͑␻ ␻ ͉͒2͒ ␦ 0.8 spectral function and q , j has the form of a function. In the language of dressed modes, this corresponds to a j n (t) 0.6 localized mode, i.e., it does not decay at all. Such localized states are also known to occur in, e.g., solid state physics26 and atomic physics.25 For weak coupling, the real part of the 0.4 self-energy is small, so the peak of the spectral function is ␻ not altered significantly from its original position j. In other Occupation 0.2 words, lying outside the continuum of substrate modes, it is unlikely to be shifted into the range where it can decay or 0 0 2 4 6 8 10 vice versa. Decay of such localized modes is possible, on the Time (picoseconds) other hand, by including the anharmonic terms in the Hamil- tonian. Localized modes of the harmonic approximation now ͑ ͒ will gain finite width due to multiphonon interactions. In FIG. 3. Color online Decay of a single nanoparticle mode j general, three-phonon interactions are weak, four-phonon coupled to a 2D-Debye substrate. The coupling is Lorentzian. Oc- processes are even weaker, and the first nonzero contribution cupation ͗nj͑t͒͘ at time t is given relative to the initial occupation ͗ ͑ ͒͘ from three-phonon processes arises at the second-order term nj 0 . of the diagramatic expansion. Another mechanism for decay of localized harmonic modes can be due to double annihila- It is noted that the domain of applicability of the EoM tion and creation terms in the interaction Hamiltonian which technique is quite limited due to the fact that the inverse are neglected in this study ͓see Eq. ͑4͔͒. Namely, though Laplace transformation is not always possible neither ana- localized modes are not truly localized considering anhar- lytically nor numerically. Nevertheless, in certain cases, the monic terms or double annihilation/creation terms, their de- EoM method enables us to get analytical results within some cay rates will be small. Another important effect about DOS approximations. Figure 3 shows the decay of a single nano- dependence takes place when the spectral peak coincides particle mode to a 2D-Debye substrate obtained by the EoM with a van Hove singularity of the substrate DOS, by which method. The oscillatory behavior is due to the splitting of the the decay rate is enhanced abruptly. molecular spectrum into two. ͑iii͒ We also investigate the effect of a neighboring mode ͑ii͒ Next, using FA and GF methods, we discuss the criti- within square-root coupling in 1D and 2D Debye substrate cal features in the dissipation of single mode energies of densities of states using GF method. We consider four nano- ␻ ␻ ␻ ␻ ␻ ␻ nanoparticle to 1D and 2D substrates. In the weak coupling particle modes, 1 =0.7 max, 2 =0.65 max, 3 =0.55 max, ͑ ␻ ͒ ␻ ␻ regime, the width of the spectral function A j, q will be and 4 =0.45 max. The effect is analyzed pairwise, namely, ␻ ␻ ͑␻ ␻ ͒ ͑␻ ␻ ͒ ͑␻ ␻ ͒ small compared to q, provided that j is not close to zero, we consider 1 , 2 , 1 , 3 , and 1 , 4 as the nanopar- which is already satisfied for nanoparticles. In this limit, the ticle modes, keeping other parameters unchanged. That is, ␻ imaginary part of the self-energy can be interpreted as twice we keep 1 constant while changing the second mode and ␥ ⌺͑ ␻ ͒/ ␻ the decay rate, j =Im j, q 2. Therefore, the dependence investigate dependence of decay of 1 mode on the separa- of decay rate on the interaction type and strength as well as tion from the second nanoparticle mode. For both 1D-Debye on the frequency of the nanoparticle can be obtained from ͓Figs. 4͑a͒–4͑c͔͒ and 2D-Debye ͓Figs. 4͑d͒–4͑f͔͒, cases, we the spectral function. observe that the decay of excited modes gain a retardation as For the case of Lorentzian coupling, the interaction the mode frequencies get closer. A second behavior is the ␣ strength is linear with Wkj, which shows that the decay rate enhancement of fluctuations during decay as the mode fre- increases with ␣2. If the coupling is only a function of the quencies get closer. Both behaviors can be understood in distance between interacting atoms of the substrate and the terms of the spectral functions. In Fig. 5͑a͒, spectral func- ͑ ͒ ␣ ␻ ␻ ͑ nanoparticle, the coupling has the form of Eq. 6 with tions of 1 and 3 modes are plotted for single mode dashed being proportional to a spring constant kint connecting the curves͒ and multimode ͑solid curves͒ obtained using GF cal- interacting atoms. Since the spectral function scales with ␣2, culations. It is seen that the overlap is negligible and the 2 decay rate increases with kint for inverse-square-root cou- spectra are not changed considerably. When the modes are 2 ͓ ͑ ͔͒ ͑ ͒ pling case. The kint law was previously obtained using elastic closer Fig. 5 b , the single-mode spectra dashed curves continuum model for phononic dissipation in physisorption have finite overlap; correspondingly, the multimode spectral systems.27 functions affect each other. The Lorentzian shape is distorted ␻ The dependence on the nanoparticle mode frequency is a and the peak of 2 mode is enhanced. These result in retar- key issue we wish to emphasize in phononic energy dissipa- dation and fluctuations during decay. More precisely, the fi- tion. In Lorentzian coupling case, the decay rate is deter- nite overlap of spectra allows the nanoparticle to gain mined by the width of the Lorentzian rather than the fre- phonons back which are previously discharged to the sub- quency. On the other hand, for inverse-square-root coupling strate. This phonon exchange process continues during the ͓see Eq. ͑6͔͒, it is inversely proportional to the nanoparticle dissipation and gives rise to retardations and fluctuations ob- ␻ ͑ ͒ ͑ ͒ mode frequency j. It is evident from Eqs. B12 and 18 served in Fig. 4. ␻ ͑ ͒ that phonons in mode j decay faster as the substrate DOS at iv In order to provide a comparison of the results ob- ␻ ⌺͑ ␻ ͒ the center of the peak, j −Re j, q , increases. A crucial tained from the quantum treatment with those obtained by

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ω =0.7 ω 1 (a) ω1=0.7 ωmax 1 (b) 1 max 1 (c) ω1=0.7 ωmax ω =0.55ω ω4=0.45ωmax 3 max ω2=0.65ωmax 0.8 0.8 0.8

0.6 0.6 0.6 (t) (t) (t) j j j n n n

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 Time (picoseconds) Time (picoseconds) Time (picoseconds)

1 (d) ω1=0.7 ωmax 1 (e) ω1=0.7 ωmax 1 (f) ω1=0.7 ωmax ω4=0.45ωmax ω3=0.55ωmax ω2=0.65ωmax

0.8 0.8 0.8

0.6 0.6 0.6 (t) (t) j (t) j j n n n 0.4 0.4 0.4

0.2 0.2 0.2

0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 Time (picoseconds) Time (picoseconds) Time (picoseconds)

FIG. 4. ͑Color online͒ Effect of neighboring modes. ͑a͒–͑c͒ are for 1D-Debye DOS and ͑d͒–͑f͒ are for 2D-Debye DOS with nanoparticle ␻ ␻ ␻ ␻ ␻ ␻ ␻ ␻ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ vibration frequencies 1 =0.7 max, 2 =0.65 max, 3 =0.55 max, and 4 =0.45 max. a and d , b and e , and c and f show dissipation ͑␻ ␻ ͒ ͑␻ ␻ ͒ ͑␻ ␻ ͒ of phonon occupation for the pairs 1 , 4 , 1 , 3 , and 1 , 2 , respectively.

classical treatment, we carry out calculations using classical different dimensionalities. The interaction is described by a molecular dynamics method. We use a simple but effective harmonic spring between an atom of the nanoparticle and a approach, where we consider the nanoparticle ͑substrate͒ as a substrate atom. Using the dynamical matrix, the eigenmodes cluster ͑lattice͒ of masses and harmonic springs in the ﬁrst of the isolated nanoparticle are determined and the initial nearest neighbor approximation, and with the lattice having energy is loaded to the desired modes by giving the initial velocities to the atoms in correspondence with the modes. In the presence of the interaction between the nanoparticle and ω ω 1 (a) 1 (b) the substrate, the differential equations and hence the motion ω3 ω2 ω ω of atoms are determined in discrete time steps which are on 1 1

ω3 ω2 the order of femtoseconds. Since the classical version of the problem, which is stated above, is not an exact analog to the quantum one and due to quantum versus classical natures of the two, we compare and contrast the basic features of the results emerging from them.

pectral Function In agreement with the earlier prediction based on the elas- S tic continuum model27 and with the result previously obtained using the single mode, the dependence of decay rate 2 0.4 0.6 0.8 0.4 0.6 0.8 on the interaction strength obeys kint law for weak coupling.

Frequency (in units of ωmax) Likewise, the dependence on vibrational mode frequency veriﬁes the previously obtained result, namely, while keeping FIG. 5. ͑Color online͒ Effect of a neighboring mode on the the coupling strength constant, higher frequency modes de- spectral function. Dashed curves are the single mode spectral func- cay slower. We should note that the substrate DOS in the tions, whereas the solid curves are spectral functions in the exis- neighborhood of the nanoparticle mode frequency also af- ͑ ͒ ␻ ␻ ␻ tence of a neighboring mode. a Spectra of 1 =0.7 max and 3 fects the decay rates. Using a 1D substrate and choosing the ␻ ͑ ͒ ␻ =0.55 max for both cases are almost the same. b Spectra of 1 nanoparticle modes away from the maximum frequency of ␻ ␻ ␻ =0.7 max and 2 =0.65 max get narrowed and distorted when single the substrate, the effect of substrate DOS is minimized. Al- mode condition is relaxed. though the density of substrate phonon modes is higher for

205430-6 DYNAMICS OF PHONONIC DISSIPATION AT THE… PHYSICAL REVIEW B 76, 205430 ͑2007͒ higher frequencies, the decay rate decreases due to nanoparticle mode frequency dependence. Another property of the dissipation process becomes apparent when the dynamics is analyzed for a nanoparticle having one and two vibrational modes. We consider a diatomic molecule and a linear triatomic molecule, which have one and two vibrational modes along the molecular axis, where the interaction is also along the molecular axis. The effect of neighboring mode can be be analyzed by setting one of the modes of the triatomic molecule at the same frequency with the frequency of the diatomic one. Exciting only the common frequency of both diatomic and triatomic molecules, we compare the decay rates, keeping the interaction and substrate parameters fixed. In the weak coupling regime, it is observed that the decay rate of the common mode does not change appreciably. Moreover, exciting both vibrational modes of the triatomic molecule does not effect the decay FIG. 6. ͑Color online͒ Relaxed geometry of benzene on rate of the common mode to a great extend. This property graphene. Blue ͑gray͒ lines show the graphene structure. becomes more apparent when the coupling strength is weak- ened. Since the vibrational modes of a triatomic molecule The interaction between the C6H6 molecule and the un- along the molecular axis are well separated, this result is derlying graphene is calculated by relaxing the geometry in a expected in the light of GF solution of the quantum Hamil- supercell of the same size used for free C6H6; the final ge- tonian. The mode localization effect is also tested using clas- ometry is schematically shown in Fig. 6. The equilibrium sical MD simulations. Unlike the quantum solution, a mo- distance between the molecule and graphene is 3.75 Å; that lecular mode whose frequency lies above the maximum is, the interaction is weak. For the sake of simplicity, we frequency of the substrate has a small but yet finite decay assume that the interaction between the molecule and the rate. substrate is achieved by the C atoms lying on top of each ͑v͒ Finally, we present a specific and more realistic study other, and varying the benzene-graphene distance vertically, of energy dissipation from the excited modes of a benzene we obtain a total energy versus distance curve to which we ͑ ͒ C6H6 molecule coupled to graphene using GF method. perform a quadratic fit to calculate this effective interaction Here, the multimode frequencies of C6H6, the continuous constant. The interaction constant between C atoms lying on phonon spectrum of graphene, and the coupling between top of each other is found to be 13.55 eV/Å2. them are calculated by using first-principles ultrasoft Since the interaction is weak and the C6H6-graphene dis- pseudopotential29 plane-wave method30,31 within DFT.32 The tance is large, the dissipation will occur mostly through the exchange correlation potential has been approximated by out-of-plane motions of C6H6 atoms due to their coupling to generalized gradient approximation33 using PW91 func- the transverse modes of graphene. Using the normal coordi- tional. nates of these vibrational modes, the contribution of each These calculations allow an accurate quantum mechanical atom to the interaction can be determined. When a single treatment. All atomic positions are optimized by the conju- atom of the molecule is interacting with the substrate, the gate gradient method and the system is considered to be at ␻2 /ͱ␻ ␻ ␻ coupling coefficient goes like Wkj =c int k j, where j equilibrium when Hellman-Feynman forces are below stands for the frequency of jth mode of the molecule and c is / 10 meV Å. A large supercell is used for the free C6H6 mol- the normal coordinate of the interacting atom in jth mode. ecule so that the distance to the nearest atom of the neigh- When there are more than one atoms interacting with the boring C H molecule is above 10 Å. A plane-wave basis set ͑ ͒ 6 6 substrate as is the case for C6H6 graphene , we sum over 2 2 with kinetic energy cutoff ប ͉k+G͉ /2m=350 eV has been those interacting degrees of freedom to find the effective used. Each atom is shifted by 0.01 Å in each direction from coefficient c which scales the coupling strength. The out-of- their equilibrium positions, and the resulting forces on each plane vibrational modes of benzene and the scaling coeffi- atom are used to construct the dynamical matrix such that cients are given in Table I. It is worth mentioning that C6H6 ␮␯+ ␮␯− ␯␮+ ␯␮− has doubly degenerate modes which is due to hexagonal ␮␯ 1 F␣␤ − F␣␤ + F␤␣ − F␤␣ symmetry of the molecule, i.e., ͑1,2͒, ͑5,6͒, and ͑7,8͒ of K␣␤ = , ͑28͒ 2 2d Table I form degenerate pairs for free C6H6. The coefficients of the degenerate modes are identical due to symmetry ␮␯± where F␣␤ denotes the force on atom ␤ along ␯ when ␣th grounds. ␮ ͑ ␻ ͒ atom is moved along in positive or negative direction; d is The spectral function A j, q for each mode is calculated the displacement imposed on a specific atom. The dynamical using the GF method. Since only the lowest six of the out- matrix is defined in terms of these forces as Dij ␮␯ of-plane mode frequencies lie within the range of transverse =D3͑␣−1͒+␮,3␤−1+␯ =K␣␤. In solving the dynamical matrix, the substrate phonons, they gain a finite width while the remain- vibration frequencies and the corresponding normal coordi- ing three modes stay localized. The spectra of the lowest nates are determined. lying modes and the DOS of transverse substrate phonons

205430-7 SEVINÇLI et al. PHYSICAL REVIEW B 76, 205430 ͑2007͒

TABLE I. The out-of-plane vibrational modes of C6H6 and the that of the first two modes. Although the third and fourth effective coefficients which scale the coupling strength. modes are close to each other, the dissipation of the fourth one is much faster than that of the third mode. The broaden- ␻ Frequency j ing of the fifth and sixth modes would be much larger if their ͑ ͒ THz Effective coefficient cj shifted peaks were matched the singularity of the DOS near 26 THz. However, they gain a shift which pushes the peak 1 11.99 0.962 far from singularity. We observe that there might appear 2 11.99 0.962 more than one peaks in the spectrum of a single mode. This 3 20.58 0.301 is due to two reasons, the contributions from the singularities 4 21.39 1.105 in the substrate DOS and the interaction stimulated anharmo- 5 26.03 0.489 nicities within the molecule. Although the molecule is treated 6 26.03 0.489 in the harmonic approximation, interaction Hamiltonian 7 29.66 0.566 gives rise to an indirect coupling between different vibra- 8 29.66 0.566 tional modes of the free molecule. We note that the results for the specific case of benzene 9 30.12 0.528 molecule weakly interacting with graphene are in agreement with the results obtained in part ͑iii͒. More specifically, de- are shown in Fig. 7. The coupling strengths of degenerate pendence of spectral width on coupling strength, effect of modes are equal; therefore, their spectral functions are iden- van Hove singularities, and the interplay between neighbor- tical. Dependence of the shift and broadening of the spectral ing modes are illustrated in this specific example. function on the coupling coefficient manifests itself as the narrow spectral peak of the third mode when compared to V. CONCLUSION The phononic dissipation from a nanostructure weakly coupled to a substrate has been analyzed using three different methods. The EoM technique is able to yield analytical results, but has a limited range of applicability because of the fact that inverse Laplace transformation is not always possible. On the other hand, FA diagonalization is possible for any type of substrate density of states and any type of coupling, but is restricted to considerations of single nanoparticle mode only. Using GFs, the effect of neighboring nanoparticle modes can also be investigated. It is found that the stronger the coupling is, the faster is the rate of dissipation. Since the width of the spectrum of a single nanoparticle mode scales with the value of the substrate DOS at the shifted frequency of the nanoparticle mode, we observe that a single nanoparticle mode coupled to a 2D-Debye substrate decays faster than the one coupled to a 1D-Debye substrate. This situation can be reversed for those frequencies for which 1D-DOS is higher than the 2D-DOS, namely, for low frequencies ͑larger nanoparticles͒. That is, at frequencies at which 1D-DOS has higher values than 2D-DOS, decay rate of a mode coupled to the 1D substrate will be higher than that of the mode coupled to 2D substrate, provided that the remaining factors are kept identical. The presence of neighboring nanoparticle modes affect each other’s decay rate when their spectral functions have an appreciable overlap. Transitions between nanoparticle modes take place via the substrate modes; therefore, retardation as well as fluctuations become important when the modes are close enough. Fur- thermore, using the results obtained from a first-principles study, interplay between molecular modes depending on the substrate DOS is demonstrated. FIG. 7. ͑Color online͒ Spectra of the six lowest lying vibrational ͑ ͒ modes of benzene when interacting with a graphene sheet a.1 – ACKNOWLEDGMENTS ͑a.6͒ and DOS of transverse phonons of the graphene substrate ͑b͒. The red ͑dashed͒ lines indicate the vibrational frequencies of the This work was supported by The Scientific and Techno- free benzene molecule. logical Research Council of Turkey through Grant No.

205430-8 DYNAMICS OF PHONONIC DISSIPATION AT THE… PHYSICAL REVIEW B 76, 205430 ͑2007͒

TBAG-104T537. R.T.S. acknowledges ﬁnancial support APPENDIX B: DETAILS OF FANO-ANDERSON METHOD from TÜBA-GEBİP. This research was supported in part by a b TÜBİTAK through TR-Grid e-Infrastructure Project ͑http:// Since the bare phonon operators j and k form a com- ͒ plete set of operators for the combined system, the dressed www.grid.org.tr . ␣͑␻ ͒ operators q can be expanded in terms of the bare opera- APPENDIX A: DETAILS OF EQUATION OF MOTION tors as TECHNIQUE ␣͑␻ ͒ ␮͑␻ ␻ ͒ ␯͑␻ ␻ ͒ ͑ ͒ q = ͚ q, j aj + ͚ q, k bk, B1 The Laplace transformed form of equations of motion are j k ͓see Eqs. ͑7͒ and ͑8͔͒ ͓␣͑␻ ͒ ͔ and they satisfy the eigenoperator equation q ,H ͑ ͒͑ ␻ ͒ ͑ ͒ * ¯ ͑ ͒ ͑ ͒ ប␻ ␣͑␻ ͒ ¯al s s + i l = al 0 − i͚ Wklbk s , A1 = q q . Conversely, we ﬁnd the bare operators by the k following expressions in terms of the dressed operators:

¯ ͑ ͒͑ ␻ ͒ ͑ ͒ ͑ ͒ ͑ ͒ a = ͚ ␮*͑␻ ,␻ ͒␣͑␻ ͒, ͑B2͒ bk s s + i k = bk 0 − i͚ Wkj¯aj s , A2 j q j q j q where s is the Laplace frequency. Solving for ¯a ͑s͒, one ob- l ␯*͑␻ ␻ ͒␣͑␻ ͒ ͑ ͒ tains bk = ͚ q, k q . B3 q a ͑0͒ W* b ͑0͒ ¯a ͑s͒ = l − i͚ kl k ͑ ͒ l ␻ ͑ ␻ ͒͑ ␻ ͒ Substituting Eq. B1 into the eigenoperator relation, one s + i l k s + i k s + i l ends up with a pair of equations, W* W ¯a ͑s͒ − ͚ kl kj j . ͑A3͒ ␮͑␻ ␻ ͒͑␻ ␻ ͒ ␯͑␻ ␻ ͒ ͑ ͒ ͑ ␻ ͒͑ ␻ ͒ q, j q − j = ͚ q, k Wkj, B4 kj s + i k s + i l k Considering the couplings to be nonoverlapping ͓see Eq. ͑5͔͒, we are left with the relation ␯͑␻ ␻ ͒͑␻ ␻ ͒ ␮͑␻ ␻ ͒ * ͑ ͒ q, k q − k = ͚ q, j Wkj, B5 * ͑ ͒ j ͚ Wklbk 0 ͑ ͒ s + i␻ which can be solved self-consistently to obtain ␻ . Using Eq. al 0 k k q ¯a ͑s͒ = − i . ͑B5͒, ␯ can be expressed in terms of ␮ as l ͉W ͉2 ͉W ͉2 s + i␻ + ͚ kl s + i␻ + ͚ kl l ␻ l ␻ k s + i k k s + i k P ␯͑␻ ,␻ ͒ = ΀ + ␦͑␻ − ␻ ͒z͑␻ ͒΁͚ ␮͑␻ ,␻ ͒W* , q k ␻ ␻ q k q q j kj ͑A4͒ q − k j ͑ ͒ ͑ ͒ ͑ ͒ ͑B6͒ Having obtained ¯al s in terms of al 0 and bk 0 , the inverse Laplace transform will yield a ͑t͒; thus, we can obtain the l where stands for the principal part, and the ␦-function term time dependent occupancy of the lth mode. P accounts for the contribution from the singularity. Inserting We convert the summations into integrals over the sub- ͑ ͒ ͑ ͒ ␮͑␻ ␻ ͒ Eq. B6 into Eq. B4 , the following relation for q , j strate modes and denote them as ͑␻ ͒ and z q is obtained: ͉W ͉2 d␻ g͑␻ ͉͒W ͉2 I ͑s͒ = ͚ kl = ͵ k k kl , ͑A5͒ l ␻ ␻ P * k s + i k s + i k ␮͑␻ ,␻ ͒͑␻ − ␻ ͒ = ͚ ␮͑␻ ,␻ ͒W W + ͚ ␦͑␻ q j q j ␻ ␻ q l kj kl q kl q − k kl * ͑ ͒ ␻ ͑␻ ͒ * ͑ ͒ * W bk 0 d kg k W bk 0 − ␻ ͒z͑␻ ͒␮͑␻ ,␻ ͒W W . ͑B7͒ J ͑s͒ = ͚ kl = ͵ kl , ͑A6͒ k q q l kj kl l ␻ ␻ s + i k s + i k k If we consider the nanoparticle to have a single mode, the ͑␻ ͒ ␮ where g k is the phonon density of states for the substrate, relation between and z can be written in a much simpler Il and Jl depend on s, and Jl is an operator. We can write Eq. form and the dissipation of each mode can be treated sepa- ͑A4͒ as rately. From this point on, we will use the subscript j where necessary denoting that we are working on the dynamics of a ͑0͒ J ͑s͒ ¯a ͑s͒ = l − i l . ͑A7͒ the jth mode of the nanoparticle. l ␻ ͑ ͒ ␻ ͑ ͒ ͑␻ ͒ s + i l + Il s s + i l + Il s Relying on the above reasoning, zj q can be expressed ͑ ͒ as The inverse transform of ¯al s is ␻ ␻ ␴ ͑␻ ͒ ͑ ͒ st st ͑ ͒ q − j − j q al 0 e ds 1 e Jl s ds z ͑␻ ͒ = , ͑B8͒ a ͑t͒ = Ͷ − Ͷ . j q ͑␻ ͉͒ ͉2 l ␲ ␻ ͑ ͒ ␲ ␻ ͑ ͒ g q Wqj 2 i B s + i l + Il s 2 B s + i l + Il s ␴ ͑␻ ͒ ͑A8͒ j q being the shift in the jth nanoparticle mode,

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2 ͉ ͉2 d␻ g͑␻ ͉͒W ͉ Wqj ␴ ͑␻ ͒ = ͵ k k kj . ͑B9͒ ͉␮͑␻ ,␻ ͉͒2 = . j q P ␻ ␻ q j ͓␻ ␻ ␴ ͑␻ ͔͒2 ␲2 2͑␻ ͉͒ ͉4 q − k q − j − j q + g q Wqj ͑ ͒ In order to obtain the expansion coefﬁcients ␮ and ␯, B12 ␣͑␻ ͒ phononic commutation relation for q is employed. Since the Hamiltonian is diagonal with annihilation and ␣͑␻ ͒ ␣+͑␻ ͒ ␦͑␻ ␻ ͒ creation operators q and q and eigenfrequencies q − qЈ ␻ ͓␣͑␻ ͒,␣+͑␻ ͔͒ = ␦ = . ͑B10͒ q, the time dependence of the dressed annihilation operator q qЈ q,qЈ ͑␻ ͒ g q is

͓ ͑ ͔͒ ␻ ␻ Using the expansion in terms of bare operators Eq. B1 ␣͑␻ ͒ ␮͑␻ ␻ ͒ −i qt ␯͑␻ ␻ ͒ −i qt q,t = q, j aje + ͚ q, k bke . and Poincare’s theorem, i.e., k ͑ ͒ P P P P P B13 = ͩ − ͪ ␻ ␻ ␻ ␻ ␻ ␻ ␻ ␻ ␻ ␻ q − k qЈ − k qЈ − q q − k qЈ − k Correspondingly, the time dependence of the nanoparticle annihilation operator reads ͓see Eq. ͑B2͔͒ ␲2␦͑␻ ␻ ͒␦͑␻ ␻ ͒ ͑ ͒ + q − k qЈ − k , B11 ␻ ␮͑␻ ␻ ͒ ͑ ͒ ͵ ␻ ͑␻ ͒␮*͑␻ ␻ ͒␣͑␻ ͒ −i qt ͑ ͒ the modulus square of q , j is found as aj t = d qg q q, j q e . B14

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