We propose a design of single-mode orbital angular momentum (OAM) beam laser with high direct-modulation bandwidth. It is a microcylinder/microring cavity interacted with two types of second-order gratings: the complex top grating containing the real part and the imaginary part modulations and the side grating. The side grating etched on the periphery of the microcylinder/microring cavity can select a whispering gallery mode with a specific azimuthal mode number, while the complex top grating can scatter the lasing mode with travelling-wave pattern vertically. With the cooperation of the gratings, the laser works with a single mode and emits radially polarized OAM beams. With an asymmetrical pad metal on the top of the cavity, the OAM on-chip laser can firstly be directly modulated with electrical pumping. Due to the small active volume, the laser with low threshold current is predicted to have a high direct modulation bandwidth about 29 GHz with the bias current of ten times the threshold from the simulation. The semiconductor OAM laser can be rather easily realized at different wavelengths such as the O band, C band, and L band.