Long Lived Acoustic Vibrational Modes of an Embedded Nanoparticle

Classical continuum elastic calculations show that the acoustic vibrational modes of an embedded nanoparticle can be lightly damped even when the longitudinal plane wave acoustic impedances $Z_o=\rho v_L$ of the nanoparticle and the matrix are the same. It is not necessary for the matrix to be less dense or softer than the nanoparticle in order to have long lived vibrational modes. A corrected formula for acoustic impedance is provided for the case of longitudinal spherical waves. Continuum boundary conditions do not always accurately reflect the microscopic nature of the interface between the nanoparticle and the matrix, and a multilayer model of the interface reveals the possibility of additional reduction of mode damping.

Figure 1

Mean square displacement within the nanoparticle interior of CSM $(\mathrm{S}\mathrm{P}\mathrm{H},\ell =0)$ vibrational modes for Au (radius $R_p=10\mathrm{n}\mathrm{m}$) in diamond (radius $R_m=2000R_p$). Arrows indicate positions and FWHM obtained with the CFM.

Figure 2

Normalized CFM frequency (solid line, left axis) and $1/Q=\Delta \omega /\omega$ (dotted line, right axis) of the fundamental $(\mathrm{S}\mathrm{P}\mathrm{H},\ell =0)$ mode along $Z_{om}/Z_{op}={\rho }_m{v}_{Lm}/{\rho }_p{v}_{Lp}=1$, where $v_{Tp}/v_{Lp}=0.375$ (gold) and $v_{Tm}/v_{Lm}=0.6$. $\Delta \omega$ is the full width at half maximum.