This letter reports the demonstration of lateral AlN Schottky barrier diodes (SBDs) on single-crystal AlN substrates by metalorganic chemical vapor deposition (MOCVD) with an ultra-low ideality factor (η) of 1.65, a high Schottky barrier height of 1.94 eV, a breakdown voltage (BV) of 640 V, and a record high normalized BV by the anode-to-cathode distance (LAC). The device current was dominated by thermionic emission, while most previously reported AlN SBDs suffered from defect-induced current with higher η (>4). This work represents a significant step towards high-performance ultra-wide bandgap AlN-based high-voltage and high-power devices.