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

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.

Figure 1

(Color online) A nanoparticle with discrete density of phonon modes is coupled to a substrate having continuous density of modes.

Figure 2

Diagrams of order $2n$. Solid lines are the phonon lines of the nanoparticle where the dashed lines are that of the substrate. (a) Diagram for the case of single nanoparticle mode $j$. ${\mathbf{k}}_i$ stand for the substrate modes. (b) Diagram of order $2n$ when there exists multiple modes $\left(j_i\right)$ for the nanoparticle.

Figure 3

(Color online) Decay of a single nanoparticle mode $j$ coupled to a 2D-Debye substrate. The coupling is Lorentzian. Occupation $⟨n_j\left(t\right)⟩$ at time $t$ is given relative to the initial occupation $⟨n_j\left(0\right)⟩$.

Figure 4

(Color online) Effect of neighboring modes. (a)–(c) are for 1D-Debye DOS and (d)–(f) are for 2D-Debye DOS with nanoparticle vibration frequencies ${\omega }_1=0.7{\omega }_{\mathit{max}}$, ${\omega }_2=0.65{\omega }_{\mathit{max}}$, ${\omega }_3=0.55{\omega }_{\mathit{max}}$, and ${\omega }_4=0.45{\omega }_{\mathit{max}}$. (a) and (d), (b) and (e), and (c) and (f) show dissipation of phonon occupation for the pairs $({\omega }_1,{\omega }_4)$, $({\omega }_1,{\omega }_3)$, and $({\omega }_1,{\omega }_2)$, respectively.

Figure 5

(Color online) Effect of a neighboring mode on the spectral function. Dashed curves are the single mode spectral functions, whereas the solid curves are spectral functions in the existence of a neighboring mode. (a) Spectra of ${\omega }_1=0.7{\omega }_{\mathit{max}}$ and ${\omega }_3=0.55{\omega }_{\mathit{max}}$ for both cases are almost the same. (b) Spectra of ${\omega }_1=0.7{\omega }_{\mathit{max}}$ and ${\omega }_2=0.65{\omega }_{\mathit{max}}$ get narrowed and distorted when single mode condition is relaxed.

Figure 6

(Color online) Relaxed geometry of benzene on graphene. Blue (gray) lines show the graphene structure.

Figure 7

(Color online) Spectra of the six lowest lying vibrational modes of benzene when interacting with a graphene sheet (a.1)–(a.6) and DOS of transverse phonons of the graphene substrate (b). The red (dashed) lines indicate the vibrational frequencies of the free benzene molecule.