${\mathrm{CaCo}}_{2\ensuremath{-}y}{\mathrm{As}}_{2}$ is a unique itinerant system having strong magnetic frustration. Here, we report the effect of electron doping on the physical properties resulting from Ni substitutions for Co. The single crystals of $\mathrm{Ca}{({\mathrm{Co}}_{1\ensuremath{-}x}{\mathrm{Ni}}_{x})}_{2\ensuremath{-}y}{\mathrm{As}}_{2}$ were characterized by single-crystal x-ray diffraction, energy-dispersive x-ray spectroscopy, magnetization $M$ versus temperature $T$, magnetic field $H$, time $t$, and heat capacity ${C}_{\mathrm{p}}(H,T)$ measurements. The A-type antiferromagnetic (AFM) transition temperature ${T}_{\mathrm{N}}=52$ K for $x=0$ decreases to 22 K with only 3% Ni substitution and is completely suppressed for $x>0.16$. For $0.11\ensuremath{\le}x\ensuremath{\le}0.52$ strong ferromagnetic (FM) fluctuations develop as revealed by magnetic susceptibility $\ensuremath{\chi}(T)=M(T)/H$ measurements. For $x=0.11$ and 0.16 competing AFM and FM interactions result in a reentrant spin-glass behavior below ${T}_{\mathrm{N}}$, as evidenced by the observations of thermomagnetic hysteresis and magnetic relaxation. Enhanced FM fluctuations are also found for the $x=0.21$ and 0.31 crystals, where ${\ensuremath{\chi}}_{c}$ increases significantly at low $T$. A large $\ensuremath{\chi}$ anisotropy in these compositions where ${\ensuremath{\chi}}_{c}$ is up to a factor of two larger than ${\ensuremath{\chi}}_{ab}$ suggests that the FM spin fluctuations are quasi-1D in nature. Weak ferromagnetic contributions to the magnetization are found at $T=2$ K for $x=0.11$--0.31. Heat-capacity ${C}_{\mathrm{p}}(T)$ measurements reveal the presence of FM quantum spin fluctuations for $0.11\ensuremath{\le}x\ensuremath{\le}0.52$, where a logarithmic $T$ dependence of ${C}_{\mathrm{p}}(T)/T$ is observed at low $T$. The suppression of AFM order by the development of strong FM fluctuations in $\mathrm{Ca}{({\mathrm{Co}}_{1\ensuremath{-}x}{\mathrm{Ni}}_{x})}_{2\ensuremath{-}y}{\mathrm{As}}_{2}$ crystals suggests the presence of a FM quantum-critical point at $x\ensuremath{\approx}0.20$. Our density-functional theory (DFT) calculations confirm that FM fluctuations are enhanced by Ni substitutions for Co in ${\mathrm{CaCo}}_{2\ensuremath{-}y}{\mathrm{As}}_{2}$. The Sommerfeld electronic heat-capacity coefficient is enhanced for $x=0$, 0.21, and 0.42 by about a factor of two compared to DFT calculations of the density of states (DOS) at the Fermi energy, suggesting an enhancement of the DOS from electron-phonon and/or electron-electron interactions. The crystals with $x>0.52$ do not exhibit FM spin fluctuations or magnetic order at $T\ensuremath{\ge}1.8$ K, which was found from the DFT calculations to arise from a Stoner transition. Superconductivity is not observed above 1.8 K for any of the compositions. Neutron-diffraction studies of crystals with $x=0.11$ and 0.16 in the crossover regime $(0.1\ensuremath{\lesssim}x\ensuremath{\lesssim}0.2)$ show no evidence of A-type ordering as observed in the parent compound with $x=0$. Furthermore, no other common magnetic structures, such as ferromagnetic (FM), helical stacking of in-plane FM layers, or in-plane AFM structure, are found with an ordered moment greater than the uncertainty of $0.05\phantom{\rule{0.28em}{0ex}}{\ensuremath{\mu}}_{\mathrm{B}}$ per transition-metal atom.
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