Systematic studies of magnetic ordering, magnetic-field-induced transitions, electronic structure, and optical properties of the frustrated spinel ${\mathrm{GeCo}}_2{\mathrm{O}}_4$ (GCO) are reported. Our results reveal that GCO orders antiferromagnetically (AFM) at $T_{\mathrm{N}}=20.4$ K but with significant short-range ferromagnetic (FM) order up to $T$ $\sim 5T_{\mathrm{N}}$. The paramagnetic susceptibility ($\chi$) fits the modified Curie–Weiss law, $\chi ={\chi }_{\mathrm{o}}$ + $C$/($T-$ $\theta$), with $\theta =+51$ K for 100 K < $T<800$ K. The fit to high-temperature-series expansion of $\chi (T$) yields J${}_1$/${\mathrm{k}}_{\mathrm{B}}=14.7$ K as the dominant FM exchange coupling for the pyrochlore lattice of ${\mathrm{Co}}^{2+}$ spins consisting of alternate planes of Kagomé (KGM) and Triangular (TRI) spins lying perpendicular to [111] direction. From the analysis of the M-H plots at 2 K and published results, three critical fields are identified: $H_d\sim 11$ kOe due to AFM domains, $H_{C1}\approx$ 44 kOe related to spin-flips and FM ordering of the TRI spins, and $H_{C2}\approx 97$ kOe related to FM ordering of the KGM spins. For $H$ > $H_{C2}$, GCO is a forced ferromagnet with some canting of the spins. Magnetic field dependence of $T_{\mathrm{N}}$ follows the relation $T_{\mathrm{N}}\left(H\right)=T_{\mathrm{N}}$(0)–$D_1{H}^2$ valid for antiferromagnets with $D_1=6.63\times {10}^{-10}$ K/${\mathrm{Oe}}^2$. This magnitude of T${}_{\mathrm{N}}$(H) along with the temperature dependence of H${}_d$, H${}_{C1}$, and H${}_{C2}$ are used to construct the H-T phase diagram. From the magnitudes of the Curie constant ($C$) and the saturation magnetization at 2 K it is shown that ${\mathrm{Co}}^{2+}$ ions in GCO have the ground state with effective spin $S$ = 1/2. High resolution x-ray photoelectron spectra of 2p and 3d orbitals of Co and Ge confirm the divalent and tetravalent electronic states of Co and Ge, respectively, in GCO. The energy band gap ($E_g=3.28$ eV) evaluated using DFT+U calculations is in good agreement with the experimental results ($E_g=3.16$ eV) obtained from the diffuse reflectance spectroscopy.

We report the low-temperature anomalous magnetic behavior of ferrimagnetic spinels cobalt orthotitanate (Co₂TiO₄), which exhibits magnetic compensation behavior across 31.74 K, and cobalt orthostannate (Co₂SnO₄) exhibiting two sequential magnetic transitions, namely (i) ferrimagnetic to paramagnetic transition with Néel temperature T_N ∼ 41 K and reentrant spin-glass behavior with glass transition temperature T_SG ∼ 39 K. The Arrott plot (H/M versus M²) criterion has been used to extricate the order of sequential magnetic transitions occurring below T_N. Negative slopes of the Arrott plots below 32 K, metamagnetic-like character of the M-H isotherms, anomalies in the specific-heat (C_P T⁻¹ versus T) below 15 K, and a zero-crossover of isothermal magnetic-entropy-change (ΔS) signify the presence of pseudo first-order discontinuous magnetic phase transition in the low-temperature regime 5 K ≤ T ≤ 32 K. The dc- and ac-susceptibilities of both Co₂TiO₄ and Co₂SnO₄ are interpreted in terms of frozen-spin-clusters, which are responsible for very large magnitudes of the coercivity H_C ∼ 20 kOe and bipolar-exchange bias H_EB ∼ −20 kOe observed below 10 K.

Static and dynamic magnetic properties of normal spinel Co ₂ RuO ₄ = (Co ²⁺ ) A [ C o 3 + Ru 3 + ] B O 4 are reported based on our investigations of the temperature ( T ), magnetic field ( H ) and frequency ( f ) dependence of the ac-magnetic susceptibilities and dc-magnetization ( M ) covering the temperature range T = 2 K–400 K and H up to 90 kOe. These investigations show that Co ₂ RuO ₄ exhibits an antiferromagnetic (AFM) transition at T _N ∼ 15.2 K, along with a spin-glass state at slightly lower temperature ( T _SG ) near 14.2 K. It is argued that T _N is mainly governed by the ordering of the spins of Co ²⁺ ions occupying the A -site, whereas the exchange interaction between the Co ²⁺ ions on the A -site and randomly distributed Ru ³⁺ on the B -site triggers the spin-glass phase, Co ³⁺ ions on the B -site being in the low-spin non-magnetic state. Analysis of measurements of M ( H , T ) for T < T _N are used to construct the H – T phase diagram showing that T _SG shifts to lower T varying as H ^2/3.2 expected for spin-glass state whereas T _N is nearly H -independent. For T > T _N , analysis of the paramagnetic susceptibility ( χ ) vs. T data are fit to the modified Curie–Weiss law, χ = χ ₀ + C /( T + θ ), with χ ₀ = 0.0015 emu mol ⁻¹ Oe ⁻¹ yielding θ = 53 K and C = 2.16 emu-K mol ⁻¹ Oe ⁻¹ , the later yielding an effective magnetic moment μ _eff = 4.16 μ _B comparable to the expected value of μ _eff = 4.24 μ _B per Co ₂ RuO ₄ . Using T _N , θ and high temperature series for χ , dominant exchange constant J ₁ / k _B ∼ 6 K between the Co ²⁺ on the A -sites is estimated. Analysis of the ac magnetic susceptibilities near T _SG yields the dynamical critical exponent zν = 5.2 and microscopic spin relaxation time τ ₀ ∼ 1.16 × 10 ⁻¹⁰ sec characteristic of cluster spin-glasses and the observed time-dependence of M ( t ) is supportive of the spin-glass state. Large M – H loop asymmetry at low temperatures with giant exchange bias effect ( H _EB ∼ 1.8 kOe) and coercivity ( H _C ∼ 7 kOe) for a field cooled sample further support the mixed magnetic phase nature of this interesting spinel. The negative magnetocaloric effect observed below T _N is interpreted to be due to the AFM and SG ordering. It is argued that the observed change from positive MCE (magnetocaloric effect) for T > T _N to inverse MCE for T < T _N observed in Co ₂ RuO ₄ (and reported previously in other systems also) is related to the change in sign of (∂ M /∂ T ) vs. T data.