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Improper ferroelectrics, in which electric polarization emerges as a secondary effect of primary order parameters such as nonpolar structural distortions, are theoretically predicted to exhibit no intrinsic thickness limit. Yet, this prediction is inconsistent with experiment where ferroelectricity is observed to disappear in improper ferroelectric films under six formula units thick. We transcend this limitation by moving beyond the conventional epitaxial design paradigm that focuses largely on in-plane lattice mismatch and chemical bonding. By considering structural compatibility in the out-of-plane direction, we realize undiminished improper ferroelectricity at the monolayer limit. Our findings confirm the theoretically predicted absence of a critical thickness for improper ferroelectrics and establish a broadly applicable strategy for designing additional ultrathin ferroelectric materials. 

Abstract Optical second harmonic generation (SHG) is a nonlinear optical effect widely used for nonlinear optical microscopy and laser frequency conversion. Closed-form analytical solution of the nonlinear optical responses is essential for evaluating materials whose optical properties are unknown a priori. A recent open-source code, ♯SHAARP.si, can provide such closed form solutions for crystals with arbitrary symmetries, orientations, and anisotropic properties at asingleinterface. However, optical components are often in the form of slabs, thin films on substrates, and multilayer heterostructures with multiple reflections of both the fundamental and up to ten different SHG waves at each interface, adding significant complexity. Many approximations have therefore been employed in the existing analytical approaches, such as slowly varying approximation, weak reflection of the nonlinear polarization, transparent medium, high crystallographic symmetry, Kleinman symmetry, easy crystal orientation along a high-symmetry direction, phase matching conditions and negligible interference among nonlinear waves, which may lead to large errors in the reported material properties. To avoid these approximations, we have developed an open-source package named Second Harmonic Analysis of Anisotropic Rotational Polarimetry in Multilayers (♯SHAARP.ml). The reliability and accuracy are established by experimentally benchmarking with both the SHG polarimetry and Maker fringes using standard and commonly used nonlinear optical materials as well as twisted 2-dimensional heterostructures.