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GeCo$_2$O$_4$ is a unique system in the family of cobalt spinels ACo$_2$O$_4$(A= Sn, Ti, Ru, Mn, Al, Zn, Fe, etc.) in which magnetic Co ions stabilize onthe pyrochlore lattice exhibiting a large degree of orbital frustration. Due tothe complexity of the low-temperature antiferromagnetic (AFM) ordering andlong-range magnetic exchange interactions, the lattice dynamics and magneticstructure of GeCo$_2$O$_4$ spinel has remained puzzling. To address this issue,here we present theoretical and experimental investigations of the highlyfrustrated magnetic structure, and the infrared (IR) and Raman-active phononmodes in the spinel GeCo$_2$O$_4$, which exhibits an AFM ordering below theN\'eel temperature $T_N$ ~21 K, followed by a cubic ($Fd{\bar 3}m$) totetragonal ($I4_{1}/amd$) structural phase transition at $T_S$ ~16 K. Ourdensity-functional theory (DFT+U) calculations reveal that one needs toconsider magnetic-exchange interactions up to the third nearest neighbors toget an accurate description of the low-temperature AFM order in GeCo$_2$O$_4$.At room temperature three distinct IR-active modes ($T_{1u}$) are observed atfrequencies 680, 413, and 325 cm$^{-1}$ along with four Raman-active modes$A_{1g}$, $T_{2g}(1)$, $T_{2g}(2)$, and $E_{g}$ at frequencies 760, 647, 550,and 308 cm$^{-1}$, respectively, which match reasonably well with our DFT+Ucalculated values. All the IR-active and Raman-active phonon modes exhibitsignatures of moderate spin-phonon coupling. The temperature dependence ofvarious parameters, such as the shift, width, and intensity, of theRaman-active modes, is also discussed. Noticeable changes around $T_N$ and$T_S$ are observed in the Raman line parameters of the $E_{g}$ and $T_{2g}$modes, which are associated with the modulation of the Co-O bonds in CoO$_6$octahedra during the excitations of these modes.

Lattice dynamics and magnetic exchange interactions in GeCo2O4, a spinel with S = 1/2 pyrochlore lattice