 Current continuity between a metal and an intrinsic semiconductor

In a steady series circuit, both electron and hole conventional currents contribute in the intrinsic semiconductor, while one electron population carries the metal current. Taking magnitudes along the current direction gives the displayed relation, with $n=p=n_i$ in the intrinsic region. The large metal carrier density can yield a much smaller drift speed despite its smaller cross-section. Interface conversion between electron and hole transport does not violate total current continuity.