The delayed decay of $^{77(\mathrm{m})}$Ge, produced by neutron capture on $^{76}$Ge, is a potential background for Germanium based neutrinoless double beta decay search experiments such as GERDA or the future LEGEND-1000 experiment.
In this work we present a search for $^{77}$Ge in the full GERDA Phase$\,$II data.
We employ a delayed coincidence method to identify the decay of $^{77}$Ge via the isomeric state of $^{77}$As (9/2+, $475.5\,$keV, $t_{1/2} = 114\,\mu$s).
New digital signal processing methods were employed to select and analyze pile-up signal.
We found no signal and are able to set an upper limit on the production rate of $^{77}$Ge at $<0.235\,$nuc/(kg$\cdot$ yr) (90$\%$ CL), which translates into $<0.40\,$nuc/(kg$\cdot$yr) (90$\%$ CL) for the total $^{77(\mathrm{m})}$Ge production assuming equal production of $^{77}$Ge and $^{77\mathrm{m}}$Ge.
Given the very similar configuration - bare germanium detectors in liquid argon - this limit benchmarks our LEGEND-1000 predictions.
Scaled with the production rate obtained from simulations and using a new delayed coincidence cut condition on $^{77}$Ge, we estimate a $^{77(\mathrm{m})}$Ge background index contribution in LEGEND-1000 at LNGS (Laboratori Nazionali del Gran Sasso) of $4.0^{+3.0}_{-2.9}\times 10^{-7}\,$cts/(keV$\cdot$kg$\cdot$yr), which is about $4\%$ of the target background budget.