Antiferromagnetic short-range order and cluster spin-glass state in diluted spinel ZnTiCoO₄

The nature of magnetism in the doubly-diluted spinel ZnTiCoO₄= (Zn²⁺)[Ti⁴⁺Co²⁺]O₄is reported here employing the temperature and magnetic field (H) dependence of dc susceptibility (χ), ac susceptibilities (χ' andχ″), and heat capacity (C_p) measurements. Whereas antiferromagnetic (AFM) Néel temperatureT_N= 13.9 K is determined from the peak in the ∂(χT)/∂TvsTplot, the fit of the relaxation timeτ(determined from the peak in theχ″ vsTdata at different frequencies) to the Power law:τ=τ₀[(T-T_SG)/T_SG]^-yields the spin glass freezing temperatureT_SG= 12.9 K,zν∼ 11.75, andτ₀∼ 10^-12s. Since the magnitudes ofτ₀andzνdepend on the magnitude ofT_SG, a procedure is developed to find the optimum value ofT_SG= 12.9 K. A similar procedure is used to determine the optimumT₀= 10.9 K in the Vogel-Fulcher law:τ=τ₀ exp[E_a/k_B(T-T₀)] yieldingE_a/k_B= 95 K, andτ₀= 1.6 ×  10^-13s. It is argued that the comparatively large magnitude of the Mydosh parameter Ω = 0.026 andk_BT₀/E_a= 0.115 (≪1) suggests cluster spin-glass state in ZnTiCoO₄below T_SG. In theC_pvsTdata from 1.9 K to 50 K, only a broad peak near 20 K is observed. This and absence ofλ-type anomaly nearT_NorT_SGcombined with the reduced value of change in magnetic entropy from 50 K to 1.9 K suggests only short-range AFM ordering in the system, consistent with spin-glass state. The field dependence ofT_SGshows slight departure (ϕ∼ 4.0) from the non-mean-field Almeida-Thouless lineT_SG(H) =T_SG(0) (1 -AH^2/). Strong temperature dependence of magnetic viscositySand coercivityH_Cwithout exchange bias, both tending to zero on approach toT_SGfrom below, further support the spin-glass state which results from magnetic dilution driven by diamagnetic Zn²⁺and Ti⁴⁺ions leading to magnetic frustration. Magnetic phase diagram in theH-Tplane is established using the high-field magnetization dataM(H,T) forT<T_Nwhich reveals rapid decrease ofT_SGwith increase inHwhereas decrease inT_Nwith increase inHis weaker, typical of AFM systems. ForT>T_N, the data ofχvsTare fit to the modified Curie-Weiss law,χ=χ₀+C/(T+θ), withχ₀= 3.2 × 10^-4emu mol^-1Oe^-1yieldingθ= 4 K andC= 2.70 emu K mol^-1Oe^-1. This magnitude ofCyields effective magnetic moment = 4.65μ_Bfor Co²⁺, characteristic of Co²⁺ions with some contribution from spin-orbit coupling. Molecular field theory with effective spinS= 3/2 of Co²⁺is used to determine the nearest-neighbor exchange constantJ₁/k_B= 2.39 K AFM and next-nearest-neighbor exchange constantJ₂/k_B= -0.66 K (ferromagnetic).