Crystal electric field excitations and spin dynamics in a spin-orbit coupled distorted honeycomb magnet BiErGeO$_5$

The magnetic properties and crystal electric field (CEF) scheme of BiErGeO$_5$ are investigated via magnetization, heat capacity, muon spin relaxation (muSR), and inelastic neutron scattering (INS) experiments on a polycrystalline sample. The Er$^{3+}$ ions form a quasi-two-dimensional distorted honeycomb network with a Kramers doublet ground state. Magnetic susceptibility and heat capacity reveal short-range antiferromagnetic correlations, manifested as a broad maximum around 1.4 K. Heat-capacity data further confirm the onset of a magnetic long-range order at $T_ N = 0.4$ K. The INS spectra exhibit eight CEF excitations and the CEF analysis yields the $g$-factor anisotropy with $g_{xy}/g_{z} = 1.38$ and exchange anisotropy with $J_{xy} = 2.96$ K and $J_{z} = 1.56$ K. The experimental temperature and field dependent magnetization and heat capacity are also reproduced by the simulation using CEF energy scheme. Zero-field muSR measurements down to 30 mK, do not exhibit coherent oscillations or a static 1/3 tail. The spectra are well described by two exponential relaxation components, indicating two magnetically inequivalent muon environments. The relaxation rates display a nearly temperature-independent plateau below $T_{\rm N}$ and follow an Orbach-type activated behavior at higher temperatures involving excited CEF levels, consistent with the INS results. Longitudinal-field $\mu$SR measurements reveal only weak decoupling up to 1.5 T, indicating persistent slow spin fluctuations below $T_{\rm N}$.