The Classical-to-Quantum Crossover in strain-induced ferroelectric transition in SrTiO$_3$ membranes

Mechanical strain presents an effective control over symmetry-breaking phase transitions. In quantum paralelectric SrTiO3, strain can induce the ferroelectric transition via modification of local Ti potential landscape. However, brittle bulk materials can only withstand limited strain range (~0.1%). Taking advantage of nanoscopically-thin freestanding membranes, we demonstrated in-situ strain-induced reversible ferroelectric transition in a single freestanding SrTiO3 membranes. We measure the ferroelectric order by detecting the local anisotropy of the Ti 3d orbital using X-ray linear dichroism at the Ti-K pre-edge, while the strain is determined by X-ray diffraction. With reduced thickness, the SrTiO3 membranes remain elastic with >1% tensile strain cycles. A robust displacive ferroelectricity appears beyond a temperature-dependent critical strain. Interestingly, we discover a crossover from a classical ferroelectric transition to a quantum regime at low temperatures, which enhances strain-induced ferroelectricity. Our results offer a new opportunities to strain engineer functional properties in low dimensional quantum materials and provide new insights into the role of the ferroelectric fluctuations in quantum paraelectric SrTiO3.