## 1 Introduction
Indeed, the spatial-peak temporal-average intensity (or ISPTA) allowed by American Food and Drug Administration is 7.2 mW/mm\({}^{2}\) for diagnostic US applications, while the exposure limit for Electro-Magnetic (EM) systems, set by the Federal Communications Commission and IEEE, is only 10-100 \(\mu\)W/mm\({}^{2}\)[22]. As a result, about two orders of magnitude higher power can be transmitted using US, which is particularly useful for deep IMDs. Moreover, because acoustic waves do not directly interact with EM waves, they are not as susceptible to EM interference nor do they significantly affect the EM fields  
Figure 2: Block diagram of an implantable device with energy harvesting. of the surroundings. Finally, US waves have smaller wavelengths in tissue (e.g., \(\sim\)1.5 mm at 1 MHz) and low tissue attenuation (\(\sim\)0.5-1 dB/cm/MHz) allowing highly directive focusing down to millimeter spots at great depths and high acoustic-electrical efficiency with sub-millimeter-sized receivers [19; 23]. The adoption of US energy harvesting systems relies on piezoelectric devices. The piezoelectric phenomenon was discovered in 1880 by the French physicists [PERSON] and [PERSON] [24]. They discovered that certain materials generate an electrical polarization proportional to an applied mechanical stress via the piezoelectric effect (Figure 3). When an external force is applied, positive and negative charged surfaces are generated, called polar surfaces. The piezoelectric potential created by these polar surfaces can be used to drive electrons in an external circuit, realizing the conversion of mechanical energy into electricity. During the operations of the piezoelectric devices, polar surfaces have to be maintained in order to generate continuously the electric energy. At this purpose, US waves are the widely used means to induce a stress on the piezoelectric material. Thus, the output voltage of a piezoelectric transducer results to be an AC voltage. The frequency of the US wave is set equal to the resonant frequency of the transducer, which is inversely proportional to its size. As mentioned above, from an electrical point of view a piezoelectric device generates an AC voltage. Therefore, a system that provides AC-DC conversion is required to provide a DC supply to the electronic systems of the IMD. At the same time, due to possible misalignment between the external generator and internal transducer, the amplitude of the output voltage could unpredictably vary and, consequently, it is not suitable to directly feed the circuits. As shown in Figure 4, a typical power management integrated circuit (PMIC) consists of an AC-DC converter and a power converter, whose output voltage is controlled to provide a specific DC voltage value. Since the piezoelectric devices are often chosen from those commercially available, the architecture of the PMIC is the main diversification factor among the various proposed IMDs. In decades of evolution, the PMIC topology has been made more complex to meet the more and more stringent specifications imposed by device miniaturizations and ultra-low-power constraints. The output voltage of the rectifier is always less than the maximum input voltage, therefore, it may be too low to supply any electronics system. To solve this problem a boost converter is necessary. The DC-DC boost converter in Figure 4 can be implemented using switched inductor (SI) or switched capacitor (SC) topologies. SI converters are suitable for applications requiring high power but requires bulky off-chip components (inductors and/or transformers), resulting in a cost increase of the entire system. Moreover, magnetic components strongly reduces scalability and compactness of the IMD, making it unsuitable for applications requiring minimal invasive systems with mm-scale form factor [25]. On the other hand, in low-power low-area applications, SC converters represents a better  
Figure 3: Concept of piezoelectricity.