Large and tunable spin-orbit effect of 6p orbitals through structural cavities in crystals

We explore from first-principles calculations the ferroelectric material ${\mathrm{Pb}}_{5}{\mathrm{Ge}}_{3}{\mathrm{O}}_{11}$ as a model for controlling the spin-orbit interaction (SOC) in crystalline solids. The SOC has a surprisingly strong effect on the structural energy landscape by deepening the ferroelectric double well. We observe that this effect comes from a specific Pb Wyckoff site that lies on the verge of a natural cavity channel of the crystal. We also find that a unique cavity state is formed by the empty $6p$ states of another Pb site at the edge of the cavity channel. This cavity state exhibits a sizable spin splitting with a mixed Rashba-Weyl character and a topologically protected crossing of the related bands. We also show that the ferroelectric properties and the significant SOC effects are exceptionally robust in the presence of n-type doping at levels of up to several electrons per unit cell. We trace the provenance of these original effects to the unique combination of the structural cavity channel and the chemistry of the Pb atoms with $6p$ orbitals localizing inside the channel.