  Strongly charged head-to-head (H2H) domain walls (DWs) that are purposely
engineered along the [110] crystallographic orientation into ferroelectric
BaTiO$_3$ single crystals have been proposed as novel 2-dimensional electron
gases (2DEGs) due to their significant domain wall conductivity (DWC). Here, we
quantify these 2DEG properties through dedicated Hall-transport measurements in
van-der-Pauw 4-point geometry at room temperature, finding the electron
mobility to reach around 400~cm$^2$(Vs)$^{-1}$, while the 2-dimensional charge
density amounts to ~7$\times$10$^3$cm$^{-2}$. We underline the necessity to
take account of thermal and geometrical-misalignment offset voltages by
evaluating the Hall resistance under magnetic-field sweeps, since otherwise
dramatic errors of several hundred percent in the derived mobility and charge
density values can occur. Apart from the specific characterization of the
conducting BaTiO$_3$ DW, we propose the method as an easy and fast way to
quantitatively characterize ferroic conducting DWs, complementary to previously
proposed scanning-probe-based Hall-potential analyses.
