  The ferroelectric properties of nanoscale silicon doped HfO$_2$ promise a
multitude of applications ranging from ferroelectric memory to energy-related
applications. The reason for the unexpected behavior has not been clearly
proven and presumably include contributions from size effects and doping
effects. Silicon incorporation in HfO$_2$ is investigated computationally by
first-principles using different density functional theory (DFT) methods.
Formation energies of interstitial and substitutional silicon in HfO$_2$ paired
with and without an oxygen vacancy prove the substitutional defect as the most
likely. Within the investigated concentration window up to 12.5 formula unit %,
silicon doping alone is not sufficient to stabilize the polar and orthorhombic
crystal phase (p-o-phase), which has been identified as the source of the
ferroelectricity in HfO$_2$. On the other hand, silicon incorporation is one of
the strongest promoters of the p-o-phase and the tetragonal phase (t-phase)
within the group of investigated dopants, confirming the experimental
ferroelectric window. Besides silicon, the favoring effects on the energy of
other four-valent dopants, C, Ge, Ti, Sn, Zr and Ce, are examined, revealing Ce
as a very promising candidate. The evolution of the volume changes with
increasing doping concentration of these four-valent dopants shows an inverse
trend for Ce in comparison to silicon. To complement this study, the
geometrical incorporation of the dopants in the host HfO$_2$ lattice was
analyzed.
