Investigation of the Paramagnetic State of the Kagome Kondo Lattice Compound YbV$_6$Sn$_6$: a $^{51}$V Nuclear Magnetic Resonance Study

YbV$_6$Sn$_6$ is a recently discovered kagome-lattice metal that orders at $T_{\rm N}\approx0.4$~K. Its layered structure combines a triangular Kondo lattice of Yb$^{3+}$ ions with vanadium-based kagome planes, which may host an interplay between strong correlations and band topology. We report a $^{51}$V nuclear magnetic resonance (NMR) study of the paramagnetic state of YbV$_6$Sn$_6$. Detailed field-angular dependence of single-crystal NMR spectra determined the principal-axis directions of the electric field gradient tensor at the $^{51}$V sites, as well as their nuclear quadrupole frequency, $\nu_{\rm Q}$, and asymmetry parameter, $\eta$. The Knight shift, $K$, was measured for different field orientations, and the analysis of $K$ against magnetic susceptibility to extract anisotropic hyperfine couplings. Accurate spectral assignments further enabled measurements of the nuclear spin-lattice relaxation rate, $1/T_1$, for both in-plane and out-of-plane field directions. The temperature dependence of $1/T_1$ shows that out-of-plane spin fluctuations are suppressed below $\sim$20~K, whereas in-plane fluctuations are markedly enhanced, which might be understood by thermal depopulation of the low-lying crystalline electric field excited state. The notable anisotropy in $1/T_1$ indicates that the paramagnetic state of YbV$_6$Sn$_6$ is strongly affected by in-plane spin dynamics.