Title: Theory of ultrathin ferroelectrics: the case of CsGeBr$_3$

Abstract: Ferroelectricity has recently been demonstrated in germanium-based inorganic halide perovskites. We use atomistic first-principles-based simulations to study ultra-thin CsGeBr$_3$ films with thicknesses of 4-18 nm and develop a theory for ferroelectric ultrathin films. The theory introduces (i) a local order parameter, the local polarization, which allows the identification of phase transitions into both monodomain and polydomain phases, and (ii) a dipole pattern classifier, which allows efficient and reliable identification of unique dipole patterns. Application of the theory to both halides CsGeBr$_3$ and CsGeI$_3$, as well as oxide BiFeO$_3$ ultrathin ferroelectrics, which undergo paraelectric cubic to ferroelectric rhombohedral phase transition in bulk, reveal two distinct scenarios for ultrathin films. In the first one, the films transition into a monodomain phase, which is allowed below a critical value of the residual depolarizing field. Above this critical value, the second scenario occurs, and the film undergoes a phase transition into a nanodomain phase. The two scenarios are associated with the opposite response of Curie temperature to thickness reduction. As the film's thickness decreases, the transition temperature into the monodomain phase increases while the transition temperature into the nanodomain phase decreases. The surface effects are responsible for the Curie temperature enhancement, while the stripe domain pattern is the origin of the transition temperature suppression. Application of dipole pattern classifier reveals a rich variety of nanodomain phases in halide films: nano-stripes, labyrinths, zig-zags, pillars, and lego-domains. Our work could lead to both a deeper understanding of nanoscale ferroelectrics and discoveries of unusual nanoscale dipole patterns.