Abstract: An undoped GaAs layer is epitaxially grown on a substrate in a crystal growth device. An undoped Al.sub.x Ga.sub.1-x As layer is then epitaxially grown to form an undoped hetero-junction structure. After this, a sample is transferred to a focused ion beam (FIB) apparatus. A dopant ion beam is focused and implanted into the Al.sub.x Ga.sub.1-x As layer in a dot-like or wire-like pattern so that it does not extend to the undoped GaAs layer or channel layer, and a zero- or one-dimensional carrier gas 8 is generated in the channel layer. The invention allows maskless ion implantation, and makes the fabrication process much easier because quantum wires and dots are drawn, patterned or formed directly by ion implantation. In addition, no etching process is required, so quantum wires and quantum dots can be fabricated precisely.