  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.
