Theory of Ferroelectric ZrO$_{2}$ Monolayers on Si

We use density functional theory and Monte Carlo lattice simulations to investigate the structure of ZrO$_{2}$ monolayers on Si(001). Recently, we have reported on the experimental growth of amorphous ZrO$_{2}$ monolayers on silicon and their ferroelectric properties, marking the achievement of the thinnest possible ferroelectric oxide [M. Dogan et al. Nano Lett., 18 (1) (2018)]. Here, we first describe the rich landscape of atomic configurations of monocrystalline ZrO$_{2}$ monolayers on Si and determine the local energy minima. Because of the multitude of low-energy configurations we find, we consider the coexistence of finite-sized regions of different configurations. We create a simple nearest-neighbor lattice model with parameters extracted from DFT calculations, and solve it numerically using a cluster Monte Carlo algorithm. Our results suggest that up to room temperature, the ZrO$_{2}$ monolayer consists of small domains of two low-energy configurations with opposite ferroelectric polarization. This explains the observed ferroelectric behavior in the experimental films as a collection of crystalline regions, which are a few nanometers in size, being switched with the application of an external electric field.