Estimating effective dielectric thickness for capacitors with extrinsic defects by a statistical method

Abstract

The maximum likelihood approach and the delta method were used to form confidence intervals on the reliability parameters for capacitors subjected to a step stress test. In particular, a fraction of devices had a dielectric thickness less than the nominal. The algorithm was tested on -300 artificial data sets and applied to a real data set. After trimming the artificial data of some obviously spurious observations, the 90% confidence intervals usually bracketed the known true values of the artificial data sets -90% of the time. The exception was h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> which had a success rate of only 66%. While its success rate was low, the confidence intervals were usually within 1 nm of the known value. Most importantly, the success rate for the confidence bounds on the FIT rate exceeded the intended confidence level. The results of using the ML approach on the real data set met with similar success. The method outlined here is perfectly general and can be applied to any real data set, allowing the reliability engineer to estimate the effective dielectric thickness of low V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> capacitors. In Summary The maximum likelihood approach and the delta method were used to form confidence intervals on the reliability parameters for capacitors subjected to a step stress test. In particular, a fraction of devices had a dielectric thickness less than the nominal. The algorithm was tested on -300 artificial data sets and applied to a real data set. After trimming the artificial data of some obviously spurious observations, the 90% confidence intervals usually bracketed the known true values of the artificial data sets -90% of the time. The exception was h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> which had a success rate of only 66%. While its success rate was low, the confidence intervals were usually within 1 nm of the known value. Most importantly, the success rate for the confidence bounds on the FIT rate exceeded the intended confidence level. The results of using the ML approach on the real data set met with similar success. The method outlined here is perfectly general and can be applied to any real data set, allowing the reliability engineer to estimate the effective dielectric thickness of low V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> capacitors. In particular, uncertainties in h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> and alpha can be quantified with confidence intervals and the effect of these uncertainties (along with uncertainties in the other reliability parameters) are taken into account when estimating a function of the MLEs like the FIT rate.particular, uncertainties in h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> and alpha can be quantified with confidence intervals and the effect of these uncertainties (along with uncertainties in the other reliability parameters) are taken into account when estimating a function of the MLEs like the FIT rate.