  The low-energy structure and $\beta$ decay properties of the neutron-rich
even-mass nuclei near the neutron number $N=60$ that are experimentally of much
interest are investigated within the framework of the nuclear density
functional theory and the interacting boson-fermion-fermion model. By using the
results of the constrained self-consistent mean-field calculations based on the
relativistic energy density functional, the interacting-boson Hamiltonian
describing the even-even core nuclei, the boson-fermion, and the
fermion-fermion interactions are determined. The Gamow-Teller transition
strengths, with the corresponding operator being constructed without
introducing further phenomenological adjustment, are computed by using the wave
functions of the initial and final nuclei of the $\beta$ decay. The triaxial
quadrupole potential energy surfaces computed for the $N=60$ even-even isotones
suggest a pronounced $\gamma$ softness. The calculated energy spectra for the
even-even and odd-odd nuclei in the Rb to Cd isotopic chains exhibit an abrupt
change in nuclear structure around $N=60$, as suggested experimentally. The
predicted $\beta$-decay $\log{}_{10}ft$ values underestimate the measured
values for the nuclei with low $Z$ and with $N \leqslant 60$, exhibit a rapid
increase for $N>60$, reflecting the nuclear structure evolution, and agree
rather well with the measured values for those nuclei with $Z$ being not far
from the proton major shell closure $Z=50$. Sensitivity of the predicted
$\beta$-decay properties to the model assumptions and parameters employed in
the nuclear structure calculations is discussed, specifically, by comparing
results obtained based on the different choices of the underlying energy
density functional.
