Causes of ferroelectricity in HfO$_{2}$-based thin films: An $\textit{ab initio}$ perspective

We present a comprehensive first principles study of doped hafnia in order to understand the formation of the ferroelectric orthorhombic [001] grains. Assuming that tetragonal grains are present during the early stages of growth, matching plane analysis shows that tetragonal [100] grains can transform into orthorhombic [001] during thermal annealing, when they are laterally confined by other grains. We show that among 0%, 2% and %4 Si doping, 4% doping provides the best conditions for the tetragonal [100] to orthorhombic [001] transformation. This also holds for Al doping. We also show that for Hf$_{x}$Zr$_{1-x}$O$_{2}$, where we have studied ${x}=1.00,0.75,0.50,0.25,0.00$, the value ${x}=0.50$ provides the most favorable conditions for the desired transformation. In order for this transformation to be preferred over the tetragonal [100] to monoclinic [100] transformation, out-of-plane confinement also needs to be present, as supplied by a top electrode. Our findings illuminate the mechanism that causes ferroelectricity in hafnia-based films and provide an explanation for common experimental observations for the optimal ranges of doping in Si:HfO$_{2}$, Al:HfO$_{2}$ and Hf$_{x}$Zr$_{1-x}$O$_{2}$. We also present model thin film heterostructure computations of Ir/HfO$_{2}$/Ir stacks in order to isolate the interface effects, which we show to be significant.