  We investigate the spin of a multielectron GaAs quantum dot in a sequence of
nine charge occupancies, by exchange coupling the multielectron dot to a
neighboring two-electron double quantum dot. For all nine occupancies, we make
use of a leakage spectroscopy technique to reconstruct the spectrum of spin
states in the vicinity of the interdot charge transition between a single- and
a multielectron quantum dot. In the same regime we also perform time-resolved
measurements of coherent exchange oscillations between the single- and
multielectron quantum dot. With these measurements, we identify distinct
characteristics of the multielectron spin state, depending on whether the dot's
occupancy is even or odd. For three out of four even occupancies we do not
observe any exchange interaction with the single quantum dot, indicating a
spin-0 ground state. For the one remaining even occupancy, we observe an
exchange interaction that we associate with a spin-1 multielectron quantum dot
ground state. For all five of the odd occupancies, we observe an exchange
interaction associated with a spin-1/2 ground state. For three of these odd
occupancies, we clearly demonstrate that the exchange interaction changes sign
in the vicinity of the charge transition. For one of these, the exchange
interaction is negative (i.e. triplet-preferring) beyond the interdot charge
transition, consistent with the observed spin-1 for the next (even) occupancy.
Our experimental results are interpreted through the use of a Hubbard model
involving two orbitals of the multielectron quantum dot. Allowing for the spin
correlation energy (i.e. including a term favoring Hund's rules) and different
tunnel coupling to different orbitals, we qualitatively reproduce the measured
exchange profiles for all occupancies.
