## 1 Introduction  
Optical semiconductor microcavities are widely used in nanophotonic applications. Based on the concept of a Fabry-Perot resonator, the metallic mirrors are replaced by distributed Bragg reflectors (DBRs), where two materials with contrasting indices of refraction are arranged in a periodic stack. A widely used material combination for optical DBRs is GaAs/AlAs and their alloys [1], which have been extensively studied in electronics, optoelectronics, optics, and more recently in phononics [2]. Other applications of GaAs/AlAs also include solar cells, lasers, and, in general, any application requiring optical cavities working in the near-infrared transparency band of GaAs. Optical semiconductor cavities have been widely applied to the study of nonlinear optical effects, polariton physics [3; 4; 5; 6] and in the development of single photon sources for quantum optical technologies [7; 8]. The nanophononic equivalent of a Fabry-Perot resonator can be constructed by enclosing a resonant spacer between two acoustic DBRs, which present a periodic modulation of the acoustic impedance [9; 10; 11]. Similar to optical DBRs, these resonant structures can control the propagation of acoustic phonons in the technologically relevant GHz-THz range by making use of minigaps in their band structure at the center and the edge of the Brillouin zone. The properties that determine the behavior of a nanophononic DBR-based device are the thickness (\(d\)) of the layers, the mass density (\(\rho\)) and the speed of sound (\(v\)). The mass density and the speed of sound are usually condensed into a single parameter, the acoustic impedance (\(Z=\rho v\)) [11; 12]. Acoustic cavities have been used to study fundamental phenomena such as the confinement of acoustic phonons [10; 13; 14], their lifetimes [15] to control phonon propagation dynamics [10], and as efficient coherent phonon generators [13; 16; 17]. By coupling several acoustic resonators, it is furthermore possible to generate effective phononic potentials and to study, e.g., Bloch oscillations of acoustic phonons [12; 18; 19]. From a more applied point of view, acoustic DBRs and cavities have been useful to demonstrate the enhancement of magnetic measurements [20], the control of emission of quantum-dot based lasers [21], and the amplification and stimulated emission of acoustic waves [16; 22; 23]. In a nutshell, the Fabry-Perot resonator demonstrated to be a versatile and useful structure in nanophononics. Alternative proposals to confine a nanophononic mode have been explored by adapting concepts previously applied in optics such as adiabatic cavities, Tamm resonators and topological cavities [24; 25]. It was recently realized that an optimized optical cavity in the near infrared is at the same time an optimized acoustic cavity in the 20 GHz range [11]. This concept was extended into GaAs/AlAs micropillars showing unprecedented quality factors at room temperature and high vacuum optomechanical couplings [2; 26]. The possibility of simultaneously confining light and sound make the GaAs/AlAs microstructures a very attractive platform for optomechanics and optoelectronics where interactions with phonons can be used as an additional engineering tool. In a previous work [27], we introduced the concept of topological invariants to nanophononics and experimentally implemented a nanophononic system supporting a robust topological interface state at 350 GHz. The confined nanophononic state was constructed through band inversion, i.e., by concatenating two semiconductor superlattices with inverted spatial mode symmetries at the Brillouin zone center. This led to the generation of robust interface states between two topologically different superlattices. To induce the exchange of spatial mode symmetries, i.e., the band inversion, we changed the thickness ratio between the two materials forming the individual DBRs.