In this work, we propose and explore a double-stage phononic crystal (PnC) consisting of a top suspended $\mathrm{ZnO}$ layer, an array of sandwiched $\mathrm{Si}$ pillars, and a bottom $\mathrm{ZnO}$ layer on a $\mathrm{Si}$ substrate. The proposed double-stage PnC exhibits interesting behavior of routing the incident surface acoustic waves (SAWs) based on polarization and frequency, so that it can be considered as an efficient building block for alternative multistage SAW components. Here, we design the double-stage PnC to behave as an efficient vertical SAW splitter in a Mach-Zehnder interferometer, using the excitation of appropriate surface-coupled modes through the periodic locally resonating pillars. Benefiting from the acoustoelectric effect in $\mathrm{ZnO}$, the exposed top $\mathrm{ZnO}$ layer serves as the sensing branch, and the bottom $\mathrm{ZnO}$ layer serves as the reference branch in the presented vertical Mach-Zehnder interferometer. The vertical configuration of the proposed design allows exposure of the top $\mathrm{ZnO}$ layer to external stimulations, such as exposure to ultraviolet illumination or a hydrogen environment, while protecting the underlying parts from being exposed. This structural aspect simplifies the experimental implementation of our design in comparison with the conventional planar Mach-Zehnder interferometers by avoiding the necessity of in-plane focusing of external stimulation on the sensing branch. Our optimized Mach-Zehnder interferometer shows an output transmission signal of \ensuremath{-}8 dB and a high extinction ratio of 23 dB at 8.6 GHz when the conductivity of the top $\mathrm{ZnO}$ layer is increased from about 1 S/m to 100 S/m via external stimulation. Here, we propose a highly sensitive miniature vertical surface acoustic Mach-Zehnder interferometer, which may open up alternative horizons for future multistage SAW components.
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