We use Pauli-spin blockade in two-electron vertical double quantum dots to quantitatively estimate the exchange energy
J in a wide range of interdot level detuning $\ensuremath{\Delta}$ and fully compare it with calculations. Pauli-spin blockade is lifted via a singlet- (
S-)triplet (
T) transition mediated by hyperfine coupling, which abruptly occurs in our devices when the $S\mathrm{\text{\ensuremath{-}}}T$ transition energy or
J is compensated by the Zeeman energy. We use this feature to derive
J depending on $\ensuremath{\Delta}$ between the $S\mathrm{\text{\ensuremath{-}}}S$ and $T\mathrm{\text{\ensuremath{-}}}T$ resonances. The obtained
J versus $\ensuremath{\Delta}$ including the resonance effect is perfectly reproduced by Hubbard model calculations.
We perform a Hartree-Fock calculation in order to describe the ground state of a vertical double quantum dot in the absence of magnetic fields parallel to the growth direction. Intradot and interdot exchange interactions determine the singlet or triplet character of the system as the tunneling is tuned. At finite Zeeman splittings due to in-plane magnetic fields, we observe a quantum phase transition from a ferromagnetic to a symmetric phase through a canted antiferromagnetic state. The latter is obtained even at zero Zeeman energy for an odd number of electrons.