  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.
