## 1 Introduction  
Metasurfaces are types of artificial surfaces with subwavelength meta-atoms, organized 2D spatially to achieve the light manipulation of amplitude, phase, and polarization, as well as light enhancement [1, 2]. Among them, the all-dielectric metasurface based on Mie resonances can efficiently localize and enhance the light in the nanostructures and find wide applications in nonlinear optics [3], optical switching [4], sensing [5, 6], and nanomaterial imaging [7]. The resonant excitation of magnetic dipolar (MD) resonance can realize low-power ultrafast (65 fs) all-optical switching, whose two-photon absorption is enhanced by a factor of 80 with respect to the unstructured film [8], while electric dipolar (ED) resonance [9] and anapoles [10, 11] can not only enhance the second-harmonic generation efficiency of nanostructures, but also increase the third-harmonic generation efficiency of nanostructures to as high as 1% [12, 13]. Beyond the capabilities of metasurfaces, optical bistability is another crucial aspect in the field of optics. Optical bistability is a process in which there are two possible optical output states for a single light input. It allows for rapid switching between these states through the flexible adjustment of the input light [14], and offers wide-ranging applications in all-optical switching, optical computers, transistors, memory devices, etc. [15, 16, 17]. A common method to realize optical bistability is to pump the dielectrics with intensity-dependent permittivities, which is known as the Kerr effect. It relies on the third-order nonlinear optical process, which is intrinsically weak. The nanophotonic structures can not only contribute to device miniaturization and the integration of multiple functionson a chip, but also strongly localize and enhance optical fields within the Kerr dielectric to improve the bistability performance. The grating nanostructures with guided mode resonances [18; 19] and quasi-bound states in the continuum [20], the photonic crystals [21], and the film-coupled plasmonic metasurfaces [22; 23] and metamaterials [24] were utilized to reduce the all-optical bistability switch threshold, as well as to enlarge the on/off ratio. The Kerr medium used in these works was mostly polymers that have high third-order nonlinear coefficients, e.g., polyDCHD-HS with \(\chi^{(3)}\) up to 4.4 \(\times\) 10\({}^{-17}\) m\({}^{2}\)/V\({}^{2}\)[25]. However, they had disadvantages including a low refractive index and low endurance, which make them not suitable for the switch devices that work in a high-intensity input laser and have a high repetition rate. On the other hand, silicon has a high laser damage threshold and is complementary metal oxide semiconductor (CMOS)-compatible for large-scale production at a low cost [8; 13]. However, its third-order nonlinear coefficient is at least two orders of magnitude lower than that of polyDCHD-HS, and switching thresholds of \(>\)75 MW/cm\({}^{2}\) can be found for Si-based optical bistabilities [26; 27; 28]. As the counterpart of electric anapole, the magnetic anapole state (MAS) in resonant dielectric nanostructures is also a nonradiative resonance originating from the destructive interference between Cartesian magnetic dipole [MD(C)] and magnetic toroidal dipole (MTD) moments [29]. However, it finds rare practical applications [30] due to its relatively weak field enhancement and localization. The traditional dielectric structure has a single structure and a slow response speed, which limits the data processing speed of optical bistable devices [31]. The Kerr effect has a fast response speed, but the nonlinearity coefficient of Kerr materials is small, and high light intensity is required to realize the optical bistable function [24]. Therefore, in terms of achieving optical bistability, the further research and development of novel materials and device structures is needed to overcome the above challenges and limitations. In this work, our objective is to optimize the metal-dielectric hybrid metasurface structure.