  We study theoretically spin relaxation during phonon-assisted tunneling of a
single electron in self-assembled InAs/GaAs quantum-dot molecules formed by
vertically stacked dots. We find that the spin-flip tunneling rate may be as
high as 1% of the spin-conserving one. By studying the dependence of spin
relaxation rate on external fields, we show that the process is active at a
considerable rate even without the magnetic field, and scales with the latter
differently than the relaxation in a Zeeman doublet. Utilizing a multiband
$\boldsymbol{k}\cdot\boldsymbol{p}$ theory, we selectively investigate the
impact of various spin-mixing terms in the electron energy and carrier-phonon
interaction Hamiltonians. As a result, we identify the main contribution to
come from the Dresselhaus spin-orbit interaction, which is responsible for the
zero-field effect. At magnetic fields above $\sim$ 15 T, this is surpassed by
other contributions due to the structural shear strain. We also study the
impact of the sample morphology and determine that the misalignment of the dots
may enhance relaxation rate by over an order of magnitude. Finally, via virtual
tunneling at nonzero temperature, the process in question also affects
stationary electrons in tunnel-coupled structures and provides a Zeeman-doublet
spin relaxation channel even without the magnetic field.

Mesoscale and Nanoscale Physics