Acoustic vibrations in free-standing double layer membranes

We report on a Brillouin light scattering study of acoustic excitations in free-standing polymethyl methacrylate $\left(\mathrm{PMMA}\right)∕{\mathrm{Si}}_3{\mathrm{N}}_4$ double layer membranes. For vanishing-wave vector transfer along the membrane, the observed excitations are a series of longitudinal standing wave harmonics whose frequencies depend on PMMA and ${\mathrm{Si}}_3{\mathrm{N}}_4$ layer thickness. The associated displacement profiles below $30\mathrm{GHz}$ are largely confined to the softer PMMA layer with mode amplitudes significantly higher at the PMMA surface than at the free surface of the high elastic modulus ${\mathrm{Si}}_3{\mathrm{N}}_4$ layer. At higher frequencies, in the vicinity and beyond the resonance associated with the ${\mathrm{Si}}_3{\mathrm{N}}_4$ layer, the mode amplitudes become comparable at both free surfaces. Excitations with transverse polarization as well as waves whose amplitudes within the ${\mathrm{Si}}_3{\mathrm{N}}_4$ are consistent with flexural and dilatational modes for a single layer are also observed. For certain ranges of wave vector the latter interact and hybridize with the harmonics arising from the PMMA layer.

Figure 1

(a) Typical Brillouin spectra recorded from a double layer structure consisting of $100\mathrm{nm}$ $\left(d_S\right)$ ${\mathrm{Si}}_3{\mathrm{N}}_4$ and $75\mathrm{nm}$ $\left(d_S\right)$ PMMA membranes. Peaks observed for an angle of incidence $\theta =0^{\circ }$, are due to the lowest lying longitudinal standing waves $1–{3\mathrm{LSM}}^P$ and ${1\mathrm{LSM}}^S$. With increasing $\theta$, transverse, flexural, and dilatational type modes emerge. The low frequency data connected by a solid line have been reduced by a factor of three. (b) Spectra recorded from the $d_P=55\mathrm{nm}$ structure over a narrow range of $\theta \left(18°⩽\theta ⩽24°\right)$ illustrating the hybridized H-SLW and ${1\mathrm{LSM}}^P$ modes around $12\mathrm{GHz}$.

Figure 2

(a) Dispersion of modes in the unsupported double layer structure with $d_S=100\mathrm{nm}$ $\left({\mathrm{Si}}_3{\mathrm{N}}_4\right)$ and $d_P=75\mathrm{nm}$ (PMMA). The experimental data are indicated by squares and the circular dots represent calculated mode frequencies. Thin solid lines are the SLW (higher frequency) and ASLW (lower frequency) modes associated with a $100\mathrm{nm}$ thick freestanding ${\mathrm{Si}}_3{\mathrm{N}}_4$ membrane. (b) Dispersion from the $d_P=55\mathrm{nm}$ structure showing hybridization between H-SLW and $1{\mathrm{LSM}}^P$ for special angles $\theta$. (c) The mode amplitudes $U_1\left(\theta \right)$ and $U_3\left(\theta \right)$ calculated for H-SLW at $\theta =10°$ (thin line) and $\theta =20°$ (dark line) as a function of the distance (in nanometer) from the free surface of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ membrane.

Figure 3

Calculated mode displacements $U_1$ (dotted line) and $U_3$ (solid line) across the thickness of the freestanding double layer for the structure $d_S=100\mathrm{nm}$ and $d_S=75\mathrm{nm}$ corresponding to Figs. 1, 2. (a), (c), and (d) are calculated for $\theta =12°$ while (b) and (e) are determined at $\theta =40°$ and (f) is evaluated at $\theta =30°$.