| A highly versatile simulation program is developed and used to examine how the resistivity of thin metal films and lines increases as their dimensions approach and become smaller than the mean fee path of electrons in metals such as copper (size effect). The simulation program: (1) provides a more accurate calculation of surface scattering effects than that obtained from the usual formulation of Fuchs' theory, (2) calculates grain-boundary effects that are consistent with the theory of Mayadas and Shatzkes, (3) includes the effects of surface and grain-boundary scattering either separately or together, and (4) simulates the effect on resistivity if a surface of a film or line has a different value for the scattering parameter. The increase in resistivity with decreasing thickness of thin, evaporated copper films (approximately 10 nm to 150 nm thick) was determined from sheet resistance and film thickness measurements. Good agreement between the experimental results with those of the simulation program was obtained when the measured mean grain sizes were used by the simulation program. The mean of the grain sizes tend to decrease with decreasing film thickness and thereby increase the impact of grain-boundary scattering on the effective resistivity of the film. Estimates of the mean grain size for each film were determined from using, in combination, the electron backscatter diffraction (EBSD) and the X-ray diffraction (XRD) methods. With values for the measured change in sheet resistance with temperature of these films, it is shown that measurements of the electrical film thickness, using Matthiessen's rule, agreed to within 3 nm of the physical measurements (profilometer) of these films. Hence, Matthiessen's rule can continue to be used to measure the thickness of a copper film and, by inference, the cross-sectional area of a copper line for dimensions well below the mean free path of electrons in copper at room temperature (39 nm). |
