  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.
