Recent experiments are showing possibilities to generate strong magnetic fields on the excess of 500 T with high-energy nanosecond laser pulses in a compact setup of a capacitor connected to a single turn coil. Hot electrons ejected from the capacitor plate (cathode) are collected at the other plate (anode), thus providing the source of a current in the coil. However, the physical processes leading to generation of currents exceeding hundreds of kiloamperes in such a laser-driven diode are not sufficiently understood. Here we present a critical analysis of previous results and propose a self-consistent model for the high current generation in a laser-driven capacitor-coil assembly. It accounts for three major effects controlling the diode current: the space charge neutralization, the plasma magnetization between the capacitor plates, and the Ohmic heating of the external circuit-the coil-shaped connecting wire. The model provides the conditions necessary for transporting strongly super-Alfvenic currents through the diode on the time scale of a few nanoseconds. The model validity is confirmed by a comparison with the available experimental data.