A technique is described that allows the routine preparation of preselected areas, such as leaky p-n junctions on silicon wafers, for transmission electron microscopy. The method yields thick, mechanically stable specimens with an electron-transparent centre, the thinnest part being at the preselected position. The crystal is first thinned by local ultrasonic dimpling from the reverse side, and then is chemically etched to the required final thickness. The technique may possibly be used for other non-ductile materials such as germanium, gallium arsenide, glasses, garnet crystals, ceramics. etc.