  Nuclear spin polarization can be pumped into spin-blocked quantum dots by
multiple Landau- Zener passages through singlet-triplet anticrossings. By
numerical simulations of realistic systems including approximately $10^7$
nuclear spins during $10^5$ sweeps, we uncover a mechanism of dynamical
self-quenching which results in a fast saturation of the nuclear polarization
under stationary pumping. This is caused by screening of the field of the
nuclear spins. In systems with moderate spin-orbit coupling, self-quenching
also screens the spin-orbit interaction. The mechanism is generic and remains
robust under a moderate noise. Our finding explains low polarization levels
achieved experimentally and calls for developing new protocols that break the
self-quenching limitations.
