Electroluminescence (EL) measurements are carried out on a two-terminal carbon nanotube (CNT) based light-emitting diode (LED). This two-terminal device is composed of an asymmetrically contacted semiconducting single-walled carbon nanotube (SWCNT). On the one end the SWCNT is contacted with Sc and on the other end with Pd. At large forward bias, with the Sc contact being grounded, electrons can be injected barrier-free into the conduction band of the SWCNT from the Sc contact and holes be injected into the valence band from the Pd electrode. The injected electrons and holes recombine radiatively in the SWCNT channel yielding a narrowly peaked emission peak with a full width at half-maximum of about 30 meV. Detailed EL spectroscopy measurements show that the emission is excitons dominated process, showing little overlap with that associated with the continuum states. The performance of the LED is compared with that based on a three-terminal field-effect transistor (FET) that is fabricated on the same SWCNT. The conversion efficiency of the two-terminal diode is shown to be more than three times higher than that of the FET based device, and the emission peak of the LED is much narrower and operation voltage is lower.