Reliability functions estimated from commonly used yield models

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Abstract

Reliability can be estimated from a semiconductor yield model once in-process measurements of manufacturing defects are obtained which cause both yield and reliability losses. Such reliability is more useful, than reliability estimated from field failure data, for determining during the early production stage whether a reliability requirement will be met or not. The purpose of this paper is to investigate and compare reliability functions estimated from previous yield models that are commonly used in literature. The results characterize the impact of defect clustering and environmental conditions on reliability estimated from yield.

Introduction

Generally, yield is defined as the ratio of the number of usable devices after the completion of a production process to the number of devices at the beginning of a process [1], [2], [3]. Yield is directly related to the profitability of any production line and predicting yield for new technologies is important to determine the cost of a new chip before production, to estimate the number of wafer starts required and to modify layout for the yield improvement. Three types of problems are often considered in semiconductor yield predictions: parametric yield, systematic yield and random yield. Parametric yield measures the quantification of chip performance due to process parameter variation and corner analysis, Monte Carlo analysis and response surface methodology are often used. Systematic yield is associated with non-random defects that affect every die in a region of the wafer, and thus is typically assumed to be constant. Finally, random yield is caused by particle defects that affect dies randomly but may be clustered in local areas of the wafer. Because random yield tends to improve much more slowly than systematic yield [4], modeling of random yield has been a focus in semiconductor yield modeling [1], [2], [3]. Practically, random yield is directly observed from the fraction of devices passing a yield test and then the mean number of random defects is estimated from the observed yield value and an assumed yield model. Various models of random yield such as Poisson and negative binomial models have been developed for at least forty years so as to relate the mean number of random defects in a device to the random yield. As the wafer’s dimension grows, random defects will continue to dominate yield loss because the probability that a defect will deposit on the device increases even with a manufacturer’s precautions. For excellent overviews of yield models, see Cunningham [1] and Ferris-Prabhu [2].

Reliability is formally defined as the probability that a device will perform its intended function for a specified mission time under the stated environmental condition. Two types of reliability problems are often considered in semiconductor devices: intrinsic failures and extrinsic failures. Extrinsic failures are due to manufacturing defects while intrinsic failures refer to the breakdown of devices free of manufacturing defects. Burn-in is useful for screening out extrinsic failures before they are released to the customer [3]. Traditionally, extrinsic reliability is estimated from a time-to-failure distribution such as mixed exponential or Weibull distributions and an optimal burn-in time is determined from the mixed distributions.

Random particle defects created during semiconductor manufacturing can be of the two types: killer defects (fatal defects or yield defects) and latent defects (non-fatal defects or reliability defects) [2], [5], [6], [7], [8], [9], [10]. A killer defect results in an immediate device failure as it lands in a location where it can produce an electrical short between two wires or it can break a wire. Such a killer defect can be detected at manufacturing yield test and thus random yield can be alternatively defined as the probability that a device has zero killer defect [1]. On the other hand, a latent defect is naturally same as the killer defect but small in size or lands in a location that does not cause an immediate failure. Therefore, a latent defect resides in a device when the device is released to the customer if a proper screening test such as burn-in is not implemented. Whether it causes the device failure in field depends on the field operating condition and a mission time specified. Therefore, extrinsic reliability can be defined as the probability of a device having zero latent defect realized before a specified mission time in the stated operating condition [11]. Well known is that extrinsic reliability can be estimated from a semiconductor yield model because random manufacturing defects result in both yield and reliability losses [11]. Extrinsic reliability estimated from yield is more useful, than reliability estimated from time-to-failure data, for determining during the early production stage whether a reliability requirement will be met or not [8], [9].

The purpose of this paper is to investigate the relationship between random yield and extrinsic reliability. We will use terms yield and reliability for abbreviation rather than random yield and extrinsic reliability, respectively. Using a statistical hierarchical model, we derive general expressions for yield and reliability. Then reliability functions are derived corresponding to yield models commonly used in literature. Properties of reliability functions are investigated which characterize the impact of defect clustering and operating conditions on reliability estimated from yield. Finally, these properties are illustrated with gate oxide failures, which are important reliability concerns for VLSI CMOS circuits.

Section snippets

Related previous research

Jensen [16] is probably the first considering the relation of yield and reliability. Empirical data are presented to show that high manufacturing yield implies low burn-in failures and high field reliability. Subsequently, time-independent and time-dependent models between yield and reliability are developed assuming that the same distribution can be used for the number of killer defects and for the number of latent defects in a device [5], [6], [7], [8], [11], [17], [18], [19], [20], [21], [22]

Expressions for yield and reliability

Let N be the total number of random defects introduced during fabrication on a device. Let Ny be the number of killer defects in a device. Assume that each defect in a device is independent having the same probability θ of being a killer defect [5], [9], [12], [13], [14], [15]. Then we haveNy|Nbionomial(N,θ).A killer defect results in an immediate device failure and thus is removed by a yield test. However, any latent defect is released to the field and may or may not cause the device failure

Properties of reliability estimated from yield

Based on the model presented in Section 3, we investigate reliability functions estimated from yield. Lower bounds for yield and reliability can be easily obtained in terms of the mean number of defects in a device. In practice, such lower bounds can be estimated during the early production stage if in-process measurements of manufacturing defects are obtained to estimate θ, pt and D0.

Proposition 1

It holds thatY(1-θ)E(N),R(t)(1-pt)E(N),where Y and R(t) are given in Eqs. (1), (2), respectively.

Proposition 2

Examples with gate oxide failures

The reliable functioning of gate oxide is crucial to the performance of a field effect transistor. Suppose that the lifetime of a device is completely determined by the lifetime of the gate oxide in it. Consider oxide with a top gate area A and a thickness of w in one device.

Conclusions

In this paper, we investigate a general relationship of random yield and extrinsic reliability using a statistical hierarchical model. Extrinsic reliability can be estimated from random yield if in-process measurements of manufacturing defects are obtained during the early stage of production.

We show analytical results on reliability functions estimated from yield and the results are illustrated with gate oxide failures. First, both yield and reliability are bounded in terms of mean number of

Acknowledgements

This work was supported by the faculty research fund of Konkuk University in 2005. The authors wish to thank the referees for their constructive comments which improved the presentation of this paper.

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