Large bulk photovoltaic effect and second-harmonic generation in few-layer pentagonal semiconductors PdS$_2$ and PdSe$_2$

Recently, atomically thin PdSe$_2$ semiconductors with rare pentagonal Se-Pd-Se monolayers were synthesized and were also found to possess superior properties such as ultrahigh air stability, and high carrier mobility, thus offering a new family of two-dimensional (2D) materials for exploration of 2D semiconductor physics and for applications in advanced opto-electronic and nonlinear photonic devices. In this work, we systematically study the nonlinear optical (NLO) responses [namely, bulk photovoltaic effect (BPVE), second-harmonic generation (SHG) and linear electric-optic (LEO) effect] of noncentrosymmetric bilayer (BL) and four-layer (FL) PdS$_2$ and PdSe$_2$ by applying the first-principles density functional theory with the generalized gradient approximation plus scissors-correction. First of all, the shift current conductivity is in the order of 130 $mu$A/V$^2$, being very high compared to known BPVE materials. Similarly, their injection current susceptibilities are in the order of 100$times$10$^8$A/V$^2$s, again being large. Secondly, the SHG coefficients ($chi^{(2)}$) of these materials are also large, being one order higher than that of the best-known few-layer group 6B transition metal dichalcogenides. For example, the maximum magnitude of $chi^{(2)}$ can reach 1.4$times$10$^3$ pm/V for BL PdSe$_2$ at 1.9 eV and 1.2$times$10$^3$ pm/V at 3.1 eV for BL PdS$_2$. Thirdly we find significant LEO coefficients for these structures in the low photon energy. All these indicate that 2D PdX$_2$ semiconductors will find promising NLO applications. Fourthly, we find that the large BPVE and SHG of the few-layer PdX$_2$ structures are due to strong intralayer directional covalent bonding and also 2D quantum confinement.