Theoretical investigation of noncentrosymmetric Ruddlesden-Popper germanium perovskites Cs 2 Ge X 4 ( X = Cl , Br, and I) for second harmonic generation

Inorganic halide perovskites possessing noncentrosymmetric (NCS) structures are highly desirable for anomalous photovoltaic, ferroelectric, and nonlinear optical (NLO) applications. In this work, we investigate the Ruddlesden-Popper (RP) germanium (Ge) halide perovskites ${\mathrm{Cs}}_{2}\mathrm{Ge}{X}_{4}$ ($X=\mathrm{Cl}$, Br, I) to identify low-energy NCS phases and evaluate their NLO properties via first-principles calculations. Starting from the centrosymmetric $I4$/mmm parent phase, we systematically introduce 40 symmetry-allowed structural distortion modes, including 37 octahedral rotation/tilt modes and 3 polar displacement modes, to generate a set of low-symmetry configurations. Our results reveal that NCS configurations are energetically favored in ${\mathrm{Cs}}_{2}\mathrm{Ge}{X}_{4}$ relative to their centrosymmetric counterparts, in contrast to analogous lead-based RP perovskites, which tend to stabilize centrosymmetric structures. The lowest energy NCS structure is identified as the polar $\mathit{Pc}$ phase. Symmetry-mode analysis indicates that the polar ${\mathrm{\ensuremath{\Gamma}}}_{5}^{\ensuremath{-}}$ distortion in these Ge-based systems originates from the stereochemically active lone-pair electrons of $\mathrm{G}{\mathrm{e}}^{2+}$, which drives inversion-symmetry breaking and stabilizes the NCS structure. Furthermore, we compute the second-order nonlinear susceptibility and second-harmonic-generation response of $\mathit{Pc}$-phase ${\mathrm{Cs}}_{2}\mathrm{Ge}{X}_{4}$, confirming its promising NLO performance. This study provides mechanistic insight into lone-pair-induced noncentrosymmetry in layered germanium halide perovskites and underscores their potential for advanced nonlinear optical applications.