Mark G. Vangel
Harry A. Schafft and Constance E. Schuster
The thickness of a metal film can, in principle, be measured by the ratio of the rate of change of sheet resistance with temperature to the rate of change of resistivity with temperature.
The relationship between resistivity (an intrinsic material property) and temperature has been carefully investigated, and fairly well modeled, for several materials. We have used three data sets for copper from the literature, and a slight modification of a nonlinear model used by Matula (J. of Phy. and Chem. Reference Data, 1979). The uncertainty in the derivative of resistivity with respect to temperature was approximated using a parametric bootstrap.
Corresponding data on sheet resistance were obtained from a 1994 NIST investigation, in which four sets of measurements were made over three days. A straight line provides an excellent fit to each of these datasets, leading to four measurements of the derivative of resistance. However, the standard errors from the regression fits substantially underestimate the `true' uncertainty, since all of these measurements were made at NIST, but the proposed measurement procedure will be used in different laboratories. The results of an interlaboratory study on sheet resistance (conducted by NIST) were used to appropriately `inflate' the resistance slope uncertainties. From this result, the uncertainty in the average slope of the resistance lines was obtained. A propagation-of-errors argument then leads to the final answer: the uncertainty in the ratio of the derivative estimates. To summarize these results, the relative uncertainty in the derivative of resistivity for copper is 2.85%, the relative uncertainty of sheet resistance is .584%, and hence the uncertainty in the ratio is 2.9%.
Figure 23: The top plot shows the fit of Matula's equation (fit to log resistivity, with a constant term added to account for residual resistivity), along with 100 bootstrap replicates of this fit. The bootstrap was performed by sampling the six parameters of the nonlinear model from a normal distribution having the estimated mean and covariance matrix. Each of these bootstrap curves was differentiated numerically, and the pointwise standard deviation for the the derivative of resistivity was calculated. The estimated derivative of the resistivity curve is displayed in the bottom plot, along with approximate 95% confidence intervals given by plus or minus twice the bootstrap standard deviation.
Date created: 7/20/2001