Paper Info
Title
Reliability results of HBTs with an InGaP emitter
Abstract
Accelerated lifetest results are presented on HBTs with InGaP emitters. An Arrhenius plot indicates the existence of a temperature dependent activation energy, E a . A low E a mechanism dominates above T j ∼380 °C and a high E a mechanism dominates at lower temperature. The critical transition temperature between regimes is determined using the method of maximum likelihood. The difference in E a ’s between low and high temperature regimes is statistically significant. A comparison is made between lifetimes determined from at temperature vs. 40 °C data. No significant difference is observed indicating that beta degradation can be monitored at temperature only and cooling to low temperature is not necessary. Other comparisons indicate that junction temperatures up to 367 °C can still provide good estimates of lower temperature behavior. By the method of maximum likelihood, the predicted MTTF at T j = 125 °C is 7.6 × 10 9 h with 95% CBs of [6.4 × 10 8 , 8.9 × 10 10 ]. Given the typical industry standard of 1 × 10 6 h, the reliability requirements are easily met. It is suggested that the standard of 1 × 10 6 h does not adequately capture failure time variation and that a better specification is in terms of fails in time (FITs). The 10 year average FIT rate at 125 °C is found to be negligible. Assuming a much higher junction temperature of 210 °C, the average failure rate climbs to ∼5 FITs with an upper 95% confidence bound of ∼40 FITs. 1 Introduction The reliability of HBTs with InGaP emitters has been studied over the past several years by many authors [1–8] . The reported median time to failure (MTTF) has varied over a wide range, from <10 7 to 3 × 10 9 h at 125 °C [1,5] . Junction temperatures in previous studies have typically varied from 275 to 325 °C, but have been reported as low as ∼225 °C [2] . The degradation of β = I c / I b (where I c is the collector current and I b is the base current) is the response of interest. In this work, the failure criterion was degradation in β by 20%. The Arrhenius rule is assumed to hold and is given by (1) ln t f = ln A + E a kT j where ln t f is the log of the failure time, ln A is the intercept term, E a is the activation energy, T j is the junction temperature and k is Boltzmann’s constant. Assuming failure times are lognormally distributed, the estimates of the reliability parameters ln A and E a can be used to predict the log mean, μ , or the MTTF ( e μ ) at the junction temperature of interest. From the Arrhenius equation, higher T j reduces the MTTF. The lifetests in this study were conducted at very high junction temperatures in an effort to decrease test time. One product of this analysis is the determination of a maximum T j that can be used that will still allow valid extrapolations to use conditions. The method of maximum likelihood (ML) was used for all reliability calculations unless otherwise noted. Details of the method are described elsewhere [9,10] . One benefit of using the ML approach is that the estimates are normally distributed with a mean of the estimate itself and a variance found from an advanced likelihood approach discussed in Appendix A and described in [11] . The standard error (SE) of the estimate is the square root of the variance. Given the mean and SE, a simple z -test can be performed to compare two estimates. The form of the test is given by (2) Z = ∣ T ^ 1 - T ^ 2 ∣ SE 1 2 + SE 2 2 where Z is a test statistic, T ^ 1 and T ^ 2 are MLEs (maximum likelihood estimates, e.g., ln A 1 and ln A 2 ), and the SEs are their associated standard errors. 1 Eq. (2) assumes the two SEs are not significantly different and so can be combined or “pooled”. 1 If Z exceeds 1.96, then the two estimates are significantly different at the 95% confidence level. If not, then the two estimates are not considered significantly different. 2 Experimental Group A was composed of a total of 115 individual HBT test structures that were stressed at ambient temperatures of 209, 245, 255, 265, 275 and 285 °C from wafer lots 1–11. Although every lot was not represented at each temperature (see Table 1 ), all 11 lots were produced in the same time period. Testing across all temperatures could not be done simultaneously due to space limitations. Devices were stressed at ambients of 275 and 285 °C in one round, at 245, 255 and 265 °C in a different round and at 209 °C at yet a third time. The test structure was an individual four finger HBT with a total emitter area of 80 μm 2 . The HBTs were biased at 5 V V ce and 16 mA I c (20 kA/cm 2 ) in a common emitter configuration. The median value of β at the highest temperature was ∼45, corresponding to an I b of ∼0.35 mA. Since this is well within the measurement resolution of our lifetester, the uncertainty in the failure time is not unduly affected by the high T j . The self-heating temperature ( T sh ) was determined using a technique described by Yeats [12] . The junction temperatures that correspond to the above ambient temperatures were found to be 312, 355, 367, 379, 391, 413 °C respectively. This is also given in Table 1 . The base current and base-emitter voltage ( V be ) were measured every hour at temperature and every 5 h at 40 °C. The value of β at t = 0 was determined both at 40 °C and at the stress temperature. Thus, β from either measurement type at t > 0 could be compared to the appropriate initial value. Group A HBTs had structures with two orientations, one horizontal (H), the other vertical (V). Note that only the “wiring” was rotated, not the HBT itself. The use of two different structures was done to conserve space on the wafer mask. No significant difference was expected. Group B was composed of 63 devices tested at ambients of 220, 240 and 260 °C ( T j ’s of 285, 305 and 325 °C) from lots 12–18 which were produced in 2003. The HBTs were tested similarly to group A except V ce = 3 V. 3 Results 3.1 Separating groups A low and A high Figs. 1 and 2 show probability plots for the A low and A high data. The failure times were interval censored. The individual failure times plotted in Figs. 1 and 2 are the low end of the interval. The straight lines are the fit of the Arrhenius equation which assumes a constant shape factor across temperatures. Since these lines are from a global fit over all three temperatures, the lines do not always intersect the data exactly. For the A low plot, the data lie on parallel lines, suggesting the failure mechanism does not change in this range of temperature. The failure times of the lowest ambient of 209 °C are slightly longer than predicted by the Arrhenius model. The A high data in Fig. 2 reveals a tolerable fit to the Arrhenius equation. So, the Arrhenius model should be permissible for groups A low and A high . Further, a chi-square test of homogeneity [13] was performed on the log means and shape factors within each group. In each case, the log means were significantly different but the shape factors were not. Fig. 3 is an Arrhenius plot of A low and A high data where failure times were determined from the degradation of β at temperature. 2 Results reported on groups A low and A high combined the failure times for both vertical and horizontal structures. It will be shown later that there is no statistical significance between these two groups, justifying this approach. 2 Clearly, there is a change in slope at ∼380 °C. Although a rough guess can be made by eye, a better estimate can be determined with ML. If the Arrhenius model holds within each group, then the following equations will be valid for low and high temperature regimes: (3) ln t f , low = ln A low + E a , low kT j (4) ln t f , high = ln A high + E a , high kT j where all terms have been previously defined. The parameters were estimated subject to the constraint, (5) ln A low = ln A high + E a , high - E a , low kT crit where T crit is the critical junction temperature at which the transition occurs. Eq. (5) falls out of the requirement that Eqs. (3) and (4) agree at T crit . The MLE for T crit = 378 °C. Fig. 3 shows fitted lines for the two regions. Clearly, the projected reliability at 125 °C is much higher for group A low . The relevant Arrhenius parameters and their associated standard errors for groups A and B are given in Table 1 . 3 The SEs for A low and A high at temperature were calculated using a non-parametric bootstrap method described in [16] . 3 The values of E a and log(MTTF) are compared using a z -test and the differences between A high and A low are both found to be statistically significant with a p -value 4 In this context, a p -value is the probability of no difference. If the p -value is less than 0.05, the difference is said to be significant. 4 of ∼0.03. The existence of a low E a at high temperature coupled with a high E a at low temperature may appear counter-intuitive. If the failure mechanism is diffusion related, then typically more diffusion paths become open with higher temperature so the diffusivity should increase along with E a . Further, if both mechanisms are in operation over the entire temperature regime, then the lower E a mechanism should dominate at lower temperature. This is not observed. Although the actual failure mechanism is not known, we can speculate as to the cause of this behavior. First, it has been noted that E a can be a function of the degree of order in an order–disorder alloy and can in fact decrease with increasing temperature [14] . Such may be the case here. Further, if two failure mechanisms were operating at temperatures below ∼380 °C, we would expect the shape factor to increase [15] . From Table 1 , we see the shape factor actually decreases and the decrease is statistically significant to >90% confidence. Lastly, initial attempts to fit the life data used a bimodal mixture model that assumed two failure modes are operating across all temperatures. However, estimates of this model were not stable. Since the significant curvature observed in Fig. 3 is not adequately fit by a single activation energy, a fit assuming a transition from one mechanism to another was necessary. Figs. 4–6 show the “at temperature” % change in β with time for ambient temperatures of 209, 245 and 285 °C. respectively. Across all temperatures, β is observed to degrade slowly at first and then drop dramatically. This behavior is consistent with the observations of several other investigators [1,4–6,8] . Although it has been suggested that a failure criterion of 50% degradation should be used to capture the catastrophic drop in β [4,5,8] , Figs. 4–6 reveal that the sudden decline is easily captured with the more conservative criterion of 20%. 3.2 Group A low at temperature vs. 40 °C data As noted earlier, I b was measured every hour at temperature and every 5 h at 40 °C. The amount of β degradation was calculated using both data sets. Fig. 7 is an Arrhenius plot of 40 °C and at temperature data for group A low . Little difference in the MTTF can be detected. A z -test was performed comparing ln A , E a and MTTF at 125 °C for the two groups. No significant difference was observed. Approximately 2 h are required to cool from temperature to 40 °C. So, cooling added approximately 30% onto the time of testing. Since the estimates from each group are comparable, either data set can be used to predict reliability under use conditions. Future tests can be conducted without cooling, greatly reducing test time. 3.3 Group A low vs. group B The A low data was gathered at junction temperatures of 312, 355 and 367 °C. Although 312 °C is not terribly high, 355 and 367 are more extreme. The use of such high T j ’s may appear questionable. Fig. 8 shows an Arrhenius plot of the at temperature data for groups A low and B. Group B had T j ’s of 285, 305 and 325 °C. Both the intercept terms and E a ’s are quite close (see Table 1 ). In fact, a z -test confirmed that there is no significant difference between the value of ln A , E a or log(MTTF) for the two groups. Group B HBTs were from an earlier qualification. These results indicate that our reliability is stable over time. We have found that using predictions from very high junction temperatures are practically the same as those from much lower temperatures. Thus, assuming no low E a mechanism is operating at temperatures below 285 °C, lifetesting can be conducted at very high temperatures, greatly decreasing test time. Notice that the lowest T j used here is in the same range as earlier investigations. Another check on the equivalence between groups A low and B is to compare the values of “s” (shape factor) from the individual fits for each temperature. Using a chi-square test of homogeneity [13] , no significant difference was observed between the shape factors for the two groups across all six temperatures. This result is consistent with a single failure mechanism within this temperature range. 3.4 Comparison between ML and least squares As has been previously noted [10] , least squares is a popular yet inefficient method for extrapolating lifetest data to use conditions. Fig. 9 is an Arrhenius plot for the group A low at temperature data showing both a least squares and ML fit. Note that the 95% confidence interval on the predicted MTTF is much wider for least squares. This is the result of only having one degree of freedom for the t -value that is used to form the confidence interval. Since the ML approach is more efficient, the resulting confidence intervals are much narrower for the same data. 3.5 Comparing A low V and H structures As stated earlier, both vertical (V) and horizontal (H) structures were stressed in both A low and A high groups. For the group A low data, the predicted MTTF at 125 °C for the H structure was not significantly different than the V structure when using either the at temperature or 40 °C data. Fig. 10 is an Arrhenius plot showing both at temperature groups. Little difference is observed. The results of a z -test confirmed the graphical result, allowing the failure times of both structures to be pooled. A similar result was found for the A high data. 3.6 FIT rate calculation The average FIT rate 5 The average FIT rate should be distinguished from the instantaneous FIT rate or bathtub curve. For the lognormal distribution, the instantaneous FIT rate is not constant with time. Eq. (6) finds the overall average over the time period chosen. 5 of a component is defined as (6) FIT = - ln ( 1 - F ) × 10 9 time where F is the probability of failure and the time is in hours. Since the probability of failure uses the log mean and the shape factor (assuming the lognormal distribution), the FIT rate makes use of both parameters. In contrast, a standard of 1 × 10 6 h at 125 °C does not take the shape factor into account. Assuming a shape factor of 0.5 and a MTTF of 1 × 10 6 h, the 10 year FIT rate is a negligible 5 × 10 −3 . However, if the shape factor was just slightly higher, say 0.8, the 10 year FIT rate would jump to a non-negligible 12. Clearly, a better specification is required than the MTTF alone. For this data set, the average 10 year FIT rate at 125 °C was estimated to be ∼0. If the junction temperature were raised to 210 °C, the FIT rate would climb to 4.8 FITs. Assuming the projected probability of failure at a given time and temperature is small (<10%), the average FIT rate can be interpreted as the average number of failures per hour over the time of interest out of a population of 1 billion components. Note this is only an approximation since the instantaneous FIT rate for the lognormal is not constant. So, a 10 year average FIT rate of 4.8 FITs means that we would expect an average of 4.8 failures/hour over 10 years out of one billion components in the field. The probability of failure by 10 years can be estimated by taking the average FIT rate by the total time divided by 1 billion, or (7) P ( failure ) = 4.8 × 24 × 365 × 10 1 × 10 9 = 0.042 % The estimated probability of failure by direct calculation from the reliability parameters from the Arrhenius equation gives essentially the same result. A likelihood approach illustrated in Appendix A (see also [11] ) is used to estimate the asymptotic standard error of the FIT rate. The upper confidence limit is approximated by (8) upper 95 % CL ≈ F I ^ T + 1.96 × S . E . where F I ^ T is the estimated average FIT rate. This approximation should be satisfactory here, since the sample size is 53. The upper limit is estimated to be ∼40 which would correspond to ∼0.35% failures in 10 years. The choice for a FIT rate specification for an HBT is difficult and depends on the reliability of other components. Making a crude guess of 1 FIT in 10 years at 125 °C, our results indicate that an individual HBT with an InGaP emitter would meet both the MTTF and FIT specifications. Even so, it is easy to imagine circumstances where the MTTF is lower than 1 × 10 6 h but the shape factor is small enough such that the FIT rate is still negligibly small. In this case, even though the MTTF was a little low, the customer would likely not experience a problem. Clearly, the FIT rate provides a more rational approach to specifying component reliability. 4 Summary This study has several key findings. First, a critical junction temperature of ∼380 °C has been determined, above which a low E a mechanism dominates β degradation. Below this temperature, the failure mechanism appears to make a transition to a high E a regime and lifetests performed below T crit can be used to predict reliability behavior at use conditions of 125 °C. The value of T crit is high, so that stress time can be greatly reduced. The predicted MTTF at 125 °C is of 7.6 × 10 9 h with 95% CBs of [6.4 × 10 8 , 8.9 × 10 10 ] which is very close to the MTTF predicted for an earlier low temperature test. Thus, the industry standard of 1 × 10 6 h is clearly satisfied. Second, degradation rates determined at temperature and at 40 °C provide indistinguishable reliability estimates. Therefore, cooling to low temperature during testing is not necessary and elimination of this step will help reduce test time. Third, a comparison is made between least squares and ML approaches. Since the ML method is more efficient, the confidence intervals associated with any reliability parameter will be smaller than least squares, for the same data. Lastly, a FIT rate calculation is performed and the predicted FIT rate at 125 °C is negligible. Since both the log mean and shape factor are used in the FIT calculation, it is a superior measure of reliability performance than just the MTTF. Appendix A Calculating confidence intervals One convenient aspect of likelihood methods is their ability to provide confidence intervals. The likelihood function can be considered an un-normalized probability and is a function of the (unknown) distribution parameters and the observed failure times. For the case of interval censored Arrhenius lognormal data, the likelihood function takes the form: (A.1) L = ∏ i = 1 n Φ ln t i , high - ( ln A + E a / kT j ) σ - Φ ln t i , low - ( ln A + E a / kT j ) σ where Φ is the cumulative distribution function of the normal distribution and t i ,low and t i ,high are the low and high ends of the failure interval for the i th out of n observations. Eq. (A.1) assumes the observations are independent. Using a log transformation, Eq. (A.1) becomes (A.2) log L ≡ Λ = ∑ i = 1 n log Φ ln t i , high - ln A + E a / kT j σ - Φ ln t i , low - ln A + E a / kT j σ The Fisher information is the symmetric matrix formed by the negative partial derivatives of Λ shown below: (A.3) F = - ∂ 2 Λ ∂ ln A 2 ∂ 2 Λ ∂ ln A ∂ E a ∂ 2 Λ ∂ ln A ∂ σ ∂ 2 Λ ∂ ln A ∂ E a ∂ 2 Λ ∂ ln E a 2 ∂ 2 Λ ∂ ln E a ∂ σ ∂ 2 Λ ∂ ln A ∂ σ ∂ 2 Λ ∂ ln E a ∂ σ ∂ 2 Λ ∂ ln σ 2 The inverse of the Fisher information is the variance–covariance matrix given by (A.4) F - 1 = Σ = var ( ln A ) cov ( ln A , E a ) cov ( ln A , σ ) cov ( ln A , E a ) var ( E a ) cov ( E a , σ ) cov ( ln A , σ ) cov ( E a , σ ) var ( σ ) Thus, the diagonal elements of Σ provide the variances of the estimates. The square root of the variance is the estimated standard error (SE). A 95% confidence interval for, say, E a , is given by [ E a − 1.96 × SE( E a ), E a + 1.96 × SE( E a )]. This interval assumes the sample size is large enough that the approximate distribution of the MLEs is normal. Often, we desire a confidence interval for a function of the estimates. In this case, we find the SE by multiplying the vector of the partial derivatives of the function with respect to the parameters by Σ . For an arbitrary function, G , the variance of G is given by (A.5) var ( G ) = - ∂ G ∂ ln A ∂ G ∂ E a ∂ G ∂ σ var ( ln A ) cov ( ln A , E a ) cov ( ln A , σ ) cov ( ln A , E a ) var ( E a ) cov ( E a , σ ) cov ( ln A , σ ) cov ( E a , σ ) var ( σ ) ∂ G ∂ ln A ∂ G ∂ E a ∂ G ∂ σ So, in the case of the FIT rate, G is given in (6) . The expressions for the partial derivatives in the above can be lengthy. Therefore, the software package MATLAB was used since it can handle symbolic differentiation. References [1] Cheskis D et al. Production InGaP HBT reliability. In: GaAs Rel Workshop, 2000. p. 167–9. [2] Cheon S et al. Reliability of manufacturing 6 inch InGaP HBTs. GaAs Rel Workshop, 2002. p. 155–6. [3] N. Pan High reliability InGaP/GaAs HBT IEEE Electron Dev Lett 19 4 1998 115 117 [4] Feng K et al. Reliability of InGaP/GaAs HBTs under high current acceleration. In: GaAs IC Symposium, 2001. p. 273–6. [5] Feng K et al. Determination of reliability on MOCVD grown InGaP/GaAs HBTs under both thermal and current acceleration stresses. In: GaAs Rel Workshop, 2001. p. 159–62. [6] Gupta A et al. InGaP makes HBT reliability a non-issue. In: GaAs Mantech Conference, 2001. p. 203–6. [7] O. Ueda Current status of reliability of InGaP/GaAs HBTs Solid State Electron 41 10 1997 1605 1610 [8] Yeats B et al. Reliability of InGaP emitter HBTs. In: GaAs Mantech, 2000. p. 131–5. [9] W. Nelson Accelerated testing 1990 John Wiley & Sons New York [chapter 5] [10] C. Whitman Accelerated life test calculations using the method of maximum likelihood: an improvement over least squares Microelectron Reliab 43 2003 859 864 [11] W. Nelson Accelerated testing 1990 John Wiley & Sons New York p. 294–6 [12] Yeats B. Inclusion of topside metal heat spreading in the determination of HBT temperatures by electrical and geometrical methods. In: GaAs IC Symposium, 1999. p. 59–62. [13] W. Nelson Applied life data analysis 1982 John Wiley & Sons New York p. 525–33 [14] R. Borg G. Dienes An introduction to solid state diffusion 1988 Academic Press p. 103–10 [15] F. Nash Estimating device reliability: assessment of credibility 1993 Kluwer p. 179 [16] B. Efron R. Tibshirani An introduction to the bootstrap 1993 Chapman & Hall
Year
DOI
Venue
2006
10.1016/j.microrel.2006.02.004
Microelectronics Reliability
Keywords
DocType
Volume
maximum likelihood,statistical significance,activation energy,failure rate
Journal
46
Issue
ISSN
Citations 
8
Microelectronics Reliability
0
PageRank 
References 
Authors
0.34
1
1
Name
Order
Citations
PageRank
Charles S. Whitman185.83