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 $\ensuremath{\beta}$-gallium oxide ($\ensuremath{\beta}\ensuremath{-}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$) 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 $\ensuremath{\beta}\ensuremath{-}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ three-dimensional electrical conductivity tensor. Despite the strong structural anisotropy, a weakly anisotropic conductivity at and above room temperature was found. In the ${a}^{*}bc$ coordinate system, the diagonal elements deviate from each other by no more than 6%. Based on these results and the off-diagonal element being $\ensuremath{\approx}5%$ of the diagonal ones, the direction of highest conductivity is rotated ${(59\ifmmode\pm\else\textpm\fi{}15)}^{\ensuremath{\circ}}$ from the $c$ direction towards the ${a}^{*}$ direction with a conductivity of $(1.12\ifmmode\pm\else\textpm\fi{}0.09)\ifmmode\times\else\texttimes\fi{}$ that in the perpendicular direction of lowest conductivity. 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 conductivities along the $a$ and $b$ direction agree within 2%. In contrast, the conductivity along the $c$ direction amounts to $0.96\ifmmode\times\else\texttimes\fi{}$ that along the $b$ direction for dominant phonon scattering, and up to $1.12\ifmmode\times\else\texttimes\fi{}$ for ionized-impurity scattering. The transport anisotropies are determined to be 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 up to 93 meV.