Optical fingerprints of two-dimensional interlayer-sliding multiferroic materials

Recently, two-dimensional (2D) interlayer-sliding multiferroic materials, consisting of two nonferroelectric monolayer ferromagnetic (FM) layers with specific interlayer stacking and sliding, have garnered significant interest for their potential in expanding the range of 2D multiferroics. However, the low-dimensional nature of 2D multiferroic materials results in relatively weak magnetic and ferroelectric (FE) properties, posing challenges for traditional detection methods in characterizing these two order parameters. In this work, we study the optical fingerprints of 2D interlayer-sliding multiferroic materials. By utilizing an abstract bilayer model, we demonstrate that the four multiferroic states $({P}_{\ensuremath{\uparrow}}{N}_{\ensuremath{\uparrow}},$ ${P}_{\ensuremath{\uparrow}}{N}_{\ensuremath{\downarrow}},$ ${P}_{\ensuremath{\downarrow}}{N}_{\ensuremath{\downarrow}},$ and ${P}_{\ensuremath{\downarrow}}{N}_{\ensuremath{\uparrow}})$ are interconnected via the horizontal mirror symmetry ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}}_{z}$, time-reversal symmetry $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}$, and their combination symmetries ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}}_{z}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}$. Subsequently, we analyze the tensor elements associated with the optical Kerr effect and second harmonic generation (SHG) effect in multiferroic states, revealing a strong correlation between both the Kerr effect and SHG effect and the multiferroic orders of 2D interlayer-sliding multiferroic materials, as well as the presence of distinct optical characteristics across different multiferroic states. Using ${\mathrm{VSe}}_{2}$ and ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ as two representative 2D interlayer-sliding multiferroic materials, we further validate the intricate correlations between the symmetries and optical properties. Our research provides theoretical guidance for characterizing 2D interlayer-sliding multiferroic materials by means of optical methods.