Lattice dynamics and magnetic exchange interactions in ${\mathrm{GeCo}}_2{\mathrm{O}}_4$: A spinel with $S=\frac{1}{2}$ 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 Néel temperature $T_N\sim 21$ 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 $T_S\sim 16$ 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 ${\mathrm{cm}}^{-1}$ 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 ${\mathrm{cm}}^{-1}$, 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 $T_N\sim 21$ K and $T_S\sim 16$ 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.

Figure 1

(a) Crystal structure of GCO. Dashed black lines mark the boundaries of the structural primitive cell. The primitive cell consists of two ${\mathrm{GeO}}_4$ tetrahedra and one ${\mathrm{Co}}_4$ tetrahedral unit. Here Co, Ge, and O ions are presented by the blue, gray, and red color, respectively. (b) Schematic representation of $\mathbf{q}$ = ($\frac{1}{2},\frac{1}{2},\frac{1}{2}$) AFM ordering in GCO. Magnetic moments at the Co sublattice are shown using arrows in a collinear setting, i.e., majority or up ($+$) and minority or down ($-$) spin states are denoted using up and down arrows, respectively. Alternating kagome (KGM) and triangular (TRI) planes are highlighted using dashed horizontal lines. A $T_{+}{K}_{+}{T}_{-}{K}_{-}{T}_{+}{K}_{+}{T}_{-}{K}_{-}\cdots$-type AFM spin configuration of TRI ($T_{\pm }$) and KGM ($K_{\pm }$) layers can be noticed along the [111] direction.

Figure 2

(a) Definition of all four magnetic-exchange interactions, i.e., first- ($J_1$), second- ($J_2$), and third- ($J_3$ and $J_3^{\prime }$) NN, considered in this work. Co atoms are shown in blue color. Ge and O atoms are omitted for clarity. Note that $J_3^{\prime }$ passes through an intermediate Co atom (see text). (b) Fitting of the DFT ($\mathrm{PBEsol}+U$) energy values computed for various different spin configurations in a doubled primitive cell, as shown in (a), with our model spin Hamiltonian. Here, we decide to choose the $\mathrm{PBEsol}+U$ method since it predicts better lattice parameters compared to the $\mathrm{PBE}+U$ predictions.

Figure 3

Atomic vibration patterns for all four IR-active phonon modes: (i) $T_{1\mathrm{u}}\left(1\right)$, (ii) $T_{1\mathrm{u}}\left(2\right)$, (iii) $T_{1\mathrm{u}}\left(3\right)$, and (iv) $T_{1\mathrm{u}}\left(4\right)$. The color coding of atoms is the same as in Fig. 1. These modes are listed here in the order of decreasing frequency (see Table 1).

Figure 4

Atomic vibration patterns for all five Raman-active phonon modes: $A_{1\mathrm{g}}, T_{2\mathrm{g}}\left(1\right), T_{2\mathrm{g}}\left(2\right), E_{\mathrm{g}}$, and $T_{2\mathrm{g}}\left(3\right)$. These modes are listed here in the order of decreasing frequency (see Table 2).

Figure 5

Fourier-transform infrared spectrum of GCO polycrystalline sample recorded at room temperature. The $\mathrm{DFT}+U$ simulated IR spectrum is given in the SM [54].

Figure 6

Raman spectra of GCO recorded at temperatures $T$ = 5, 10, 12, 14, 18, 20, 21, 22, 24, 26, 30, 40, 60, 80, 100, and 300 K.

Figure 7

Comparison of the $\mathrm{DFT}+U$ predicted phonon frequencies (${\omega }_{\mathrm{theory}}$) for the simulated paramagnetic phase with the experimentally measured frequencies at 300 K (${\omega }_{\mathrm{expt}}$) for the IR- and Raman-active modes. The data plotted in this figure were obtained from Tables 1 and 2.

Figure 8

The temperature dependence of Raman intensity ($\mathrm{I}\times {10}^4$), full width at half maximum ($\mathrm{\Delta }F$), and Raman-peak position (RPP) for the $A_{1\mathrm{g}}, T_{2\mathrm{g}}\left(1\right), T_{2\mathrm{g}}\left(2\right)$, and $E_{\mathrm{g}}$ Raman-active modes. The lines connecting the data points are visual guides. The $T_N$ and $T_S$ mark the transition temperatures corresponding to the antiferromagnetic and the cubic-to-tetragonal structural phase transitions, respectively.

Figure 9

The temperature dependence of the line separation $\mathrm{\Delta }\omega$ of the low-frequency shoulder from the position of the $T_{2\mathrm{g}}\left(1\right)$ peak. Inset: the shoulder appearing on the low-frequency side of the $T_{2\mathrm{g}}\left(1\right)$ peak.