Energy spectra of three electrons in Si/SiGe single and vertically coupled double quantum dots

We study three-electron energy spectra in Si/SiGe single and vertically coupled double quantum dots where all the relevant effects, such as, the Zeeman splitting, spin-orbit coupling, valley coupling and electron-electron Coulomb interaction are explicitly included. In the absence of magnetic field, our results in single quantum dots agree well with the experiment by Borselli {\em et al.} [Appl. Phys. Lett. {\bf 98}, 123118 (2011)]. We identify the spin and valley configurations of the ground state in the experimental cases and give a complete phase-diagram-like picture of the ground state configuration with respect to the dot size and valley splitting. We also explicitly investigate the three-electron energy spectra of the pure and mixed valley configurations with magnetic fields in both Faraday and Voigt configurations. We find that the ground state can be switched between doublet and quartet by tuning the magnetic field and/or dot size. The three-electron energy spectra present many anticrossing points between different spin states due to the spin-orbit coupling, which are expected to benefit the spin manipulation. We show that the negligibly small intervalley Coulomb interaction can result in magnetic-field independent quartet-doublet degeneracy in the three-electron energy spectrum of the mixed valley configuration. Furthermore, we study the barrier-width and barrier-height dependences in vertically coupled double quantum dots with both pure and mixed valley configurations. Similar to the single quantum dot case, anticrossing behavior and quartet-doublet degeneracy are observed.