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Vertical diamond Schottky diodes with blocking voltages $V_{\text{BD}} > 2.4 \text{ kV}$ and on-resistances $R_{\text{On}} < 400 \text{ m}Ω\text{cm}^{2}$ were fabricated on homoepitaxially grown diamond layers with different surface morphologies. The morphology (smooth as-grown, hillock-rich, polished) influences the Schottky barrier, the carrier transport properties, and consequently the device performance. The smooth as-grown sample exhibited a low reverse current density $J_{\text{Rev}} < 10^{-4} \text{ A}/\text{cm}^{2}$ for reverse voltages up to $2.2 \text{ kV}$. The hillock-rich sample blocked similar voltages with a slight increase in the reverse current density ($J_{\text{Rev}} < 10^{-3} \text{ A}/\text{cm}^{2}$). The calculated 1D-breakdown field, however, was reduced by $30 \text{ } \%$, indicating a field enhancement induced by the inhomogeneous surface. The polished sample demonstrated a similar breakdown voltage and reverse current density as the smooth as-grown sample, suggesting that a polished surface can be suitable for device fabrication. However, a statistical analysis of several diodes of each sample showed the importance of the substrate quality: A high density of defects both reduces the feasible device area and increases the reverse current density. In forward direction, the hillock-rich sample exhibited a secondary Schottky barrier, which could be fitted with a modified thermionic emission model employing the Lambert W-function. Both polished and smooth sample showed nearly ideal thermionic emission with ideality factors $1.08$ and $1.03$, respectively. Compared with literature, all three diodes exhibit an improved Baliga Figure of Merit for diamond Schottky diodes with $V_{\text{BD}} > 2 \text{ kV}$.