  The cross section of the $^{62}$Ni($n,\gamma$) reaction was measured with the
time-of-flight technique at the neutron time-of-flight facility n_TOF at CERN.
Capture kernels of 42 resonances were analyzed up to 200~keV neutron energy and
Maxwellian averaged cross sections (MACS) from $kT=5-100$ keV were calculated.
With a total uncertainty of 4.5%, the stellar cross section is in excellent
agreement with the the KADoNiS compilation at $kT=30$ keV, while being
systematically lower up to a factor of 1.6 at higher stellar temperatures. The
cross section of the $^{63}$Ni($n,\gamma$) reaction was measured for the first
time at n_TOF. We determined unresolved cross sections from 10 to 270 keV with
a systematic uncertainty of 17%. These results provide fundamental constraints
on $s$-process production of heavier species, especially the production of Cu
in massive stars, which serve as the dominant source of Cu in the solar system.
