Diversity of structural phases in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>A</mml:mi><mml:mi>Ge</mml:mi><mml:msub><mml:mi>X</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:math> halides

Utilizing first-principles and particle swarm optimization methods, we performed a systematic investigation of crystal structural stability and electrical properties of $A\mathrm{Ge}{X}_{3}$ compounds $(A=\mathrm{Li},\mathrm{Na},\mathrm{K},\mathrm{Rb},\mathrm{Cs};X=\mathrm{F},\mathrm{Cl},\mathrm{Br},\mathrm{I})$. Our work not only found the perovskite phases, but also discovered the post-perovskite, post-post-perovskite, and octahedra face-sharing structure phases, which are usually stable under high pressure, large strain, or exist in the inorganic--organic compound. These diverse structural phases provide abundant and promising properties. One striking phase is the $P{2}_{1}{2}_{1}{2}_{1}$ perovskite in ${\mathrm{KGeCl}}_{3}$ and ${\mathrm{KGeBr}}_{3}$, which possesses intrinsic small polar vortices and antivortices that may lead to a gyrotropic phase transition. The ferroelectric phase of $R3c$ $(\mathrm{LiGe}{X}_{3}, \mathrm{NaGe}{X}_{3})$, $R3m$ $(\mathrm{CsGe}{X}_{3})$ and $Pna{2}_{1}$ $(\mathrm{KGe}{X}_{3}, \mathrm{RbGe}{X}_{3})$ were also found. The new $Pna{2}_{1}$ phase possesses both proper and hybrid improper polarization, and the latter is evidenced by two original energetic couplings: one coupling polarization with the antipolar distortion from Ge atoms and antipolar distortion from $A$-site atoms, and another coupling polarization with out-of-phase halide octahedral tilting, in-phase halide octahedral tilting, and antipolar distortion from Ge atoms. The diverse structural phases and electrical properties make Ge-based halide a promising ${ABX}_{3}$ structure family in ferroics and piezoelectrics.

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