Embodiments of the invention are directed to a method of fabricating a semiconductor device. A non-limiting example of the method including performing first fabrication operations to form nanosheet field effect transistor (FET) devices in a first region of a substrate. The first fabrication operations include forming a first channel nanosheet, forming a second channel nanosheet over the first channel nanosheet, forming a first gate structure around the first channel nanosheet, and forming a second gate structure around the second channel nanosheet, wherein an air gap is between the first gate structure and the second gate structure. A dopant is applied to the first gate structure and the second gate structure, wherein the dopant is configured to enter the air gap and penetrate into the first gate structure and the second gate structure from within the air gap.