Pseudosurface acoustic waves in hypersonic surface phononic crystals

We present a theoretical framework allowing to properly address the nature of surfacelike eigenmodes in a hypersonic surface phononic crystal, a composite structure made of periodic metal stripes of nanometer size and periodicity of $1\mu \text{m}$, deposited over a semi-infinite silicon substrate. In surface-based phononic crystals there is no distinction between the eigenmodes of the periodically nanostructured overlayer and the surface acoustic modes of the semi-infinite substrate, the solution of the elastic equation being a pseudosurface acoustic wave partially localized on the nanostructures and radiating energy into the bulk. This problem is particularly severe in the hypersonic frequency range, where semi-infinite substrate’s surface acoustic modes strongly couple to the periodic overlayer, thus preventing any perturbative approach. We solve the problem introducing a surface-likeness coefficient as a tool allowing to find pseudosurface acoustic waves and to calculate their line shapes. Having accessed the pseudosurface modes of the composite structure, the same theoretical frame allows reporting on the gap opening in the now well-defined pseudo-SAW frequency spectrum. We show how the filling fraction, mass loading, and geometric factors affect both the frequency gap, and how the mechanical energy is scattered out of the surface waveguiding modes.

Figure 1

Schematic diagram of the surface phononic crystal. Isotropic nickel stripes of width $d$ and height $h$ are deposited on a crystalline silicon substrate. The grating has period $\lambda$. Propagation direction is along the $x$ axis, surface normal to the bulk is along the $y$ axis.

Figure 2

(Color online) Nickel stripe on silicon substrate. (a) The top $2\mu \text{m}$ portion of the 2D rectangular unit cell is reported. The cell geometry is divided in three parts: A. Ni stripe, B. $1\mu \text{m}$ top portion of Si substrate, C. $99\text{−}\mu \text{m}$-thick Si bulk portion. Bloch boundary conditions are set on sides 1 and 2 of the substrate. The periodicity is $\lambda =1\mu \text{m}$. The height of the Si cell is $100\mu \text{m}$. The Ni stripe is $h=50\text{nm}$ high and, in this figure, $d=320\text{nm}$ wide. In figures (b) trough (f), the deformation and arrows correspond to the displacement field, while the color scale refers to the normalized total displacement. (b) Sin-like SAW solution for a pure silicon slab. The calculated twofold-degenerate SAW eigenfrequency is $\tilde{\nu }=4.92\text{GHz}$. The orthogonal cos-like solution is not reported. (c) Cos-like and (d) sin-like pseudo-SAW solutions for $d=320\text{nm}$. The two pseudo-SAW eigenfrequencies are ${\tilde{\nu }}_g=4.60\text{GHz}$ and ${\tilde{\nu }}_u=4.27\text{GHz}$. (e) Cos-like and (f) sin-like SAW solutions for substrate full coverage. The twofold-degenerate calculated SAW eigenfrequency is $\tilde{\nu }=4.12\text{GHz}$.

Figure 3

(Color online) SAW-likeness coefficient $\alpha \left(\nu \right)$ versus the calculated eigenfrequencies. Two distributions arise, ${\alpha }_u\left(\nu \right)$ and ${\alpha }_g\left(\nu \right)$, corresponding to sin-like (empty circles) and cos-like (crossed circles) displacement profiles. Calculation for configurations with different filling fraction are reported: (a) $p=0.1$, (b) $p=0.32$, and (c) $p=0.9$. Solid lines are intended as a guide to the eye.

Figure 4

(Color online) Dispersion relations of the elastic surface modes for small values of $k_x$ (in $\pi /\lambda$ units) and $n=1$, for a pure silicon slab $\left(p=0\right)$, for silicon full coverage $\left(p=1\right)$ and for the configuration with Ni stripes over Si substrate $\left(p=0.32\right)$. Solid lines are fits to data. Dotted lines are a guide to the eye.

Figure 5

(Color online) Left axis: pseudo-SAW eigenfrequencies ${\tilde{\nu }}_g$ (crossed circles) and ${\tilde{\nu }}_u$ (empty circles) versus surface filling fraction $p$ (or stripes width $d$ in nm, top axis). Right axis: pseudo-SAW frequency gap $\Delta \tilde{\nu }$ (triangles). The horizontal line is the $\Delta \tilde{\nu }=0$ line and refers to right axis only. Inset: magnification of the graph for high filling fractions $p$ is shown. Lines are a guide to the eye only.

Figure 6

(Color online) Geometric and mass loading effects on $\tilde{\nu }$. The reference system configuration is $p=0.2$, $h=50\text{nm}$, and $\rho ={\rho }_{\text{Ni}}$, hence $m$ is fixed to $m^{\left(ref\right)}$. The pseudo-SAW frequencies, calculated in the reference configuration, are ${\tilde{\nu }}_g^{\left(ref\right)}$ and ${\tilde{\nu }}_u^{\left(ref\right)}$. (a) Exploring the geometric effect: the mass loading factor $\left\{m,\rho \right\}$ is kept at the reference value, the variable being the geometric factor $\left\{p,h\right\}$. (b) Exploring the mass loading effect: the geometric factor $\left\{p,h\right\}$ is kept at the reference value, the variable being the mass loading factor $\left\{m,\rho \right\}$. Left axis: pseudo-SAW eigenfrequencies ${\tilde{\nu }}_g$ (crossed circles) and ${\tilde{\nu }}_u$ (empty circles). Right axis: pseudo-SAW frequency gap $\Delta \tilde{\nu }$ (triangles). The horizontal line is the $\Delta \tilde{\nu }=0$ line and refers to right axis only. Lines are a guide to the eye only.

Figure 7

(Color online) Analysis of the spatial distribution of the mechanical energy in the composite system over different filling fraction configurations normalized against the total energy inside the cell. In the inset, the circles represent the fraction of the mechanical energy confined in the nickel stripes region A, the triangles are for the top $1\mu \text{m}$ portion B of Si substrate and the squares are associated with the silicon bulk C. The dotted lines are a guide to the eye only.