The hardness of a TiN film 2.0 µm thick is measured by indentation with a Vickers indenter. To avoid influence from the substrate, the empirical guideline of keeping the indentation depth to one tenth of the film thickness or less is used. The depth of a Vickers indent is about one seventh of its diagonal length. Which is closest to the largest allowable diagonal length?

Answer: A. About 1.4 µm

The upper limit on indentation depth is 2.0/10 = 0.20 µm, and since the depth is about 1/7 of the diagonal, the upper limit on the diagonal is 0.20×7 = 1.4 µm. Indenting deeper also deforms the softer substrate beneath the film, giving a value lower than the hardness of the film alone. About 0.20 µm is the value obtained by mistaking the depth limit for the diagonal length, and about 14 µm is the value obtained by indenting all the way to the film thickness without using the one-tenth guideline. A 1.4 µm diagonal is hard to read accurately with an optical microscope, so the hardness of thin films is usually found by nanoindentation, which continuously measures load and indentation depth. The one-tenth value is a guideline and changes with the ratio of film to substrate hardness.