  We investigate the energy spectra of three electrons in SiGe/Si/SiGe
equilateral triangular and symmetric linear triple quantum dots in the presence
of magnetic (in either Faraday or Voigt configuration) and electric fields with
single valley approximation by using the real-space configuration interaction
method. The strong electron-electron Coulomb interaction, which is crucial to
the energy spectra, is explicitly calculated whereas the weak spin-orbit
coupling is treated perturbatively. In both equilateral triangular and
symmetric linear triple quantum dots, we find doublet-quartet transition of
ground-state spin configuration by varying dot size or interdot distance in the
absence of external fields. This transition has not been reported in the
literature on triple quantum dots. In the magnetic-field (Faraday
configuration) dependence of energy spectra, we find anticrossings with large
energy splittings between the energy levels with the same spin state in the
absence of the spin-orbit coupling. This anticrossing behavior originates from
the triple quantum dot confinement potential. In addition, with the inclusion
of the spin-orbit coupling, we find that all the intersections shown in the
equilateral triangular case become anticrossing whereas only part of the
intersections in symmetric linear case show anticrossing behavior in the
presence of magnetic field in either the Faraday or Voigt configuration. All
the anticrossing behaviors are analyzed based on symmetry consideration.
Moreover, we show that the electric field can effectively influence the energy
levels and the charge configurations.
