Lattice dynamics and magnetic exchange interactions in GeCo2O4 : A spinel with S=12 pyrochlore lattice

${\mathrm{GeCo}}₂{\mathrm{O}}₄$ is a unique system in the family of cobalt spinels $A{\mathrm{Co}}₂{\mathrm{O}}₄$ ($A$= Sn, Ti, Ru, Mn, Al, Zn, Fe, etc.) in which magnetic Co ions stabilize on the pyrochlore lattice exhibiting a large degree of orbital frustration. Due to the complexity of the low-temperature antiferromagnetic (AFM) ordering and long-range magnetic exchange interactions, the lattice dynamics and magnetic structure of a ${\mathrm{GeCo}}₂{\mathrm{O}}₄$ spinel have remained puzzling. To address this issue, here we present theoretical and experimental investigations of the highly frustrated magnetic structure, and the infrared (IR) and Raman-active phonon modes in the spinel ${\mathrm{GeCo}}₂{\mathrm{O}}₄$, which exhibits an AFM ordering below the N\'eel temperature ${T}_N\ensuremath{\sim}21$ K and an associated cubic ($Fd\overline{3}m$) to tetragonal ($I{4}₁/amd$) structural phase transition whose location at ${T}_N$ vs at a lower ${T}_S\ensuremath{\sim}16$ K is controversial. Our density functional theory ($\mathrm{DFT}+U$) calculations reveal that one needs to consider magnetic-exchange interactions up to the third-nearest neighbors to get an accurate description of the low-temperature AFM order in ${\mathrm{GeCo}}₂{\mathrm{O}}₄$. At room temperature, three distinct IR-active modes (${T}_{1\mathrm{u}}$) are observed at frequencies 680, 413, and 325 ${\mathrm{cm}}^{\ensuremath{-}1}$ along with four Raman-active modes ${A}_{1\mathrm{g}}, {T}_{2\mathrm{g}}$(1), ${T}_{2\mathrm{g}}$(2), and ${E}_{\mathrm{g}}$ at frequencies 760, 647, 550, and 308 ${\mathrm{cm}}^{\ensuremath{-}1}$, respectively, which match reasonably well with our $\mathrm{DFT}+U$ calculated values. All the IR-active and Raman-active phonon modes exhibit signatures of moderate spin-phonon coupling. The temperature dependence of various parameters, such as the shift, width, and intensity, of the Raman-active modes is also discussed. Noticeable changes around ${T}_N\ensuremath{\sim}21$ K and ${T}_S\phantom{\rule{4pt}{0ex}}\ensuremath{\sim}16$ K are observed in the Raman line parameters of the ${E}_{\mathrm{g}}$ and ${T}_{2\mathrm{g}}$(1) modes, which are associated with the modulation of the Co-O bonds in ${\mathrm{CoO}}₆$ octahedra during the excitations of these modes.