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The trilayer nickelate Nd$_4$Ni$_3$O$_{10-\delta}$ ($\delta \approx$ 0.15)was investigated by the measurements of x-ray diffraction, electricalresistivity, magnetic susceptibility, and heat capacity. The crystal structuredata suggest a higher Ni valence in the inner perovskite-like layer. At ambientpressure the resistivity shows a jump at 162 K, indicating a metal-to-metaltransition (MMT). The MMT is also characterized by a magnetic susceptibilitydrop, a sharp specific-heat peak, and an isotropic lattice contraction. Below$\sim$ 50 K, a resistivity upturn with a log$T$ dependence shows up,accompanying with a negative thermal expansion. External hydrostatic pressuresuppresses the resistivity jump progressively, coincident with the diminutionof the log$T$ behavior. The low-temperature electronic specific-heatcoefficient is extracted to be $\sim$ 150 mJ K$^{-2}$ mol-fu$^{-1}$, equivalentto $\sim$ 50 mJ K$^{-2}$ mol-Ni$^{-1}$, indicating an unusual heavy-electroncorrelated state. The novel heavy-electron state as well as the logarithmictemperature dependence of resistivity is explained in terms of the Ni$^{3+}$centered Kondo effect in the inner layer of the (NdNiO$_3$)$_3$ trilayers.

Metal-to-metal transition and heavy-electron state in Nd$_4$Ni$_3$O$_{10-\delta}$