With a non-linear element (e.g., a diode) with small reverse saturation current, a power diode or rectifier is provided. A non-linear element includes a first electrode provided over a substrate, an oxide semiconductor film provided on and in contact with the first electrode and having a concentration of hydrogen of 5×1019 atoms/cm3 or less, a second electrode provided on and in contact with the oxide semiconductor film, a gate insulating film covering the first electrode, the oxide semiconductor film, and the second electrode, and third electrodes provided in contact with the gate insulating film and facing each other with the first electrode, the oxide semiconductor film, and the second electrode interposed therebetween or a third electrode provided in contact with the gate insulating film and surrounding the second electrode. The third electrodes are connected to the first electrode or the second electrode. With the non-linear element, a power diode or a rectifier is formed.