The electrical conductivity tensor of β-Ga2O3 analyzed by van der Pauw measurements: Inherent anisotropy, off-diagonal element, and the impact of grain boundaries

The semiconducting oxide $\beta$-Gallium Oxide ($\beta$-Ga$_{2}$O$_{3}$) possesses a monoclinic unit cell whose low symmetry generally leads to anisotropic physical properties. For example, its electrical conductivity is generally described by a polar symmetrical tensor of second rank consisting of four independent components. Using van der Pauw measurements in a well-defined square geometry on differently-oriented high-quality bulk samples and the comparison to finite element simulations we precisely determine the ratio of all elements of the $\beta$-Ga$_{2}$O$_{3}$ 3-dimensional electrical conductivity tensor. Despite the structural anisotropy a nearly isotropic conductivity at and above room temperature was found with the principal conductivities deviating from each other by less than 6% and the off-diagonal element being $\approx3$% of the diagonal ones. Analysis of the temperature dependence of the anisotropy and mobility of differently doped samples allows us to compare the anisotropy for dominant phonon-scattering to that for dominant ionized-impurity scattering. For both scattering mechanisms, the conductivites along the $a$ and $b$-direction agree within 2%. In contrast, the conductivity along $c$-direction amounts to $0.96\times$ and up to $1.12\times$ that along the $b$-direction for phonon and ionized impurity scattering, respectively. The determined transport anisotropies are larger than the theoretically predicted effective mass anisotropy, suggesting slightly anisotropic scattering mechanisms. We demonstrate that significantly higher anisotropies can be caused by oriented extended structural defects in the form of low-angle grain boundaries for which we determined energy barriers of multiple 10 meV.