We demonstrate ultralow leakage current AlGaN/GaN Schottky-barrier diodes (SBDs) based on a 3-D anode contact. In contrast to conventional AlGaN/GaN SBDs, this new device forms a Schottky contact directly to the 2-D electron gas (2-DEG) at the sidewalls of the 3-D anode structure to improve its turn-on characteristics. In addition, this device integrates an insulated trigate MOS structure to reduce its reverse-bias leakage current. By optimizing this new technology, we demonstrate SBDs with 3-D anode structures with turn-on voltage of 0.85 V, ON-resistance of 5.96 Omega mm and ideality factor of 1.27. The reverse-bias leakage current was significantly reduced by nearly four orders of magnitude, down to 260 pA/mm, with a breakdown voltage of up to 127 V for a distance of 1.5 mu m between the cathode and anode electrodes. To the best of our knowledge, this is among the lowest leakage currents reported in lateral AlGaN/GaN SBDs fabricated on silicon substrate.