Vertical $pn$ heterojunction diodes were prepared by plasma-assisted molecular beam epitaxy of unintentionally-doped $p$-type SnO layers with hole concentrations ranging from $p=10^{18}$ to $10^{19}$cm$^{-3}$ on unintentionally-doped $n$-type $beta$-Ga$_{2}$O$_{3}$(-201) substrates with an electron concentration of $n=2.0times10^{17}$cm$^{-3}$. The SnO layers consist of (001)-oriented grains without in-plane expitaxial relation to the substrate. After subsequent contact processing and mesa etching (which drastically reduced the reverse current spreading in the SnO layer and associated high leakage) electrical characterization by current-voltage and capacitance-voltage measurement was performed. The results reveal a type-I band alignment and junction transport by thermionic emission in forward bias. A rectification of $2times10^{8}$ at $pm1$V, an ideality factor of 1.16, differential specific on-resistance of 3.9m$Omegathinspace$cm$^{2}$, and built-in voltage of 0.96V were determined. The $pn$-junction isolation prevented parallel conduction in the highly-conductive Ga$_{2}$O$_{3}$ substrate (sheet resistance $R_{S}approx3thinspaceOmega$) during van-der-Pauw Hall measurements of the SnO layer on top ($R_{S}approx150$k$Omega$, $papprox2.5times10^{18}$cm$^{-3}$, Hall mobility $approx1$cm$^{2}$/Vs). The measured maximum reverse breakdown voltage of the diodes was 66V, corresponding to a peak breakdown field 2.2MV/cm in the Ga$_{2}$O$_{3}$-depletion region. Higher breakdown voltages that are required in high-voltage devices could be achieved by reducing the donor concentration in the $beta$-Ga$_{2}$O$_{3}$ to increase the depletion width as well as improving the contact geometry to reduce field crowding.
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