  Interactions between electrons can strongly affect the shape and
functionality of multi-electron quantum dots. The resulting charge
distributions can be localized, as in the case of Wigner molecules, with
consequences for the energy spectrum and tunneling to states outside the dot.
The situation is even more complicated for silicon dots, due to the interplay
between valley, orbital, and interaction energy scales. Here, we study
two-electron wavefunctions in electrostatically confined quantum dots formed in
a SiGe/Si/SiGe quantum well at zero magnetic field, using a combination of
tight-binding and full-configuration-interaction (FCI) methods, and taking into
account atomic-scale disorder at the quantum well interface. We model dots
based on recent qubit experiments, which straddle the boundary between strongly
interacting and weakly interacting systems, and display a rich and diverse
range of behaviors. Our calculations show that strong electron-electron
interactions, induced by weak confinement, can significantly suppress the
low-lying, singlet-triplet (ST) excitation energy. However, when the
valley-orbit interactions caused by interfacial disorder are weak, the ST
splitting can approach its noninteracting value, even when the
electron-electron interactions are strong and Wigner-molecule behavior is
observed. These results have important implications for the rational design and
fabrication of quantum dot qubits with predictable properties.
