The band structure of crystalline silicon (having a covalently-bonded, cubic, diamond lattice structure) is known to exhibit in the conduction band, electron states that show six equivalent (degenerate) minimum energies, associated with crystal momenta (k) that are 0.85 of the way to the Brillouin-zone boundary; these six states are termed valleys . In nanoscale devices the degeneracy of the valleys is broken by various eects, including strain, connement eects (such as lattice mismatch and/or abruptness of the interface between the nano-feature and the conning material) and electriceld eects. Due to strain in Si/SiGe quantum wells and (interfacial) quantum dots (in particular, in heterostructure semiconductors) , and higher subband quantization energy in MOS devices , the energies of the (four) in-plane valleys are raised, resulting in a remaining double (two-fold) degeneracy (in the direction normal to the dot's plane), which itself has been shown to be lifted (2,0) O, and ⊖ indicate states with both electrons in the low-energy valley, with the electrons in dierent valleys, and with both electrons in the high-energy valley, respectively. The 5 near-horizontal lines correspond to the (2,0) charge conguration (as indicated). The three quasi-parallel lines with a larger slope correspond to the (1,1) charge conguration (as indicated). The valley-orbit and spin-isospin couplings were neglected, and as a result all states are associated with good total spin [S (S + 1),Sz] and valley isospin [V (V + 1),Vz] quantum numbers (as indicated). The symbols made from a combination of the capital letters S and T have the following meaning: A capital S denotes a singlet state, whereas a capital T denotes a triplet state. The subscripts ± denote an Sz = ±1 spin projection or a Vz = ±1 valley projection, respectively, whereas a subscript 0 denotes an Sz = 0 or Vz = 0 projection. A superscript s denotes the regular spin, whereas a superscipt v denotes the valley isospin. xn (with n = 1, 2, or 4) denotes the degeneracy associated with a given energy curve; red color is used for the ⊕ states (with Vz = +1), green color for the 0 O states (with Vz = 0), and blue color for the ⊖ ones (with Vz = -1). The dots in this gure and in all subsequent gures are equidistant from the origin. SU(2) algebra generated by the i-multiplied Pauli matrices] with that of the regular spin of the electrons (replacing S with V when referring to the VDOF). Obviously, in the absence of two-body interactions, the occupation of the single-particle energy states in the dots would depend on an interplay between the connement (includ-ing an applied magnetic eld) and the valley eects. The symbols ⊕, 0 O, and ⊖ indicate states with both electrons in the low-energy valley, with the electrons in dierent valleys, and with both electrons in the highenergy valley, respectively. The spin-isospin coupling was neglected, and as a result all states are associated with good total spin (S , S z ) and valley isospin (V , V z ) quantum numbers, as indicated via the symbols made from a combination of the capital letters S and T. These symbols have the following meaning: A capital S denotes a singlet state, whereas a capital T denotes a triplet state . The subscripts ± denotes an S z = ±1 or V z = ±1 projection, respectively, whereas a subscript 0 denotes an S z = 0 or V z = 0 projection. A superscript s denotes the regular spin, whereas a superscipt v denotes the valley isospin. The symbol xn (with n = 1, 2, or 4)