- Email: [email protected]
Percolative approach for failure time prediction of thin film interconnects under high current stress
Microelectronics Reliability 45 (2005) 391–395 www.elsevier.com/locate/microrel
Research Note
Percolative approach for failure time prediction of thin film interconnects under high current stress E. Misra a, Md M. Islam b, Mahbub Hasan c, H.C. Kim a, T.L. Alford a
a,b,*
Department of Chemical and Materials Engineering, Arizona State University, Tempe, AZ 85287-6006, USA b Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287-6006, USA c Standard Analog Business Line, Philips Semiconductors Inc., Tempe, AZ 85284, USA Received 7 September 2003; received in revised form 1 July 2004 Available online 8 December 2004
Abstract The present study deals with the use of a rapid and non-destructive technique based on percolation theory to predict failure times during the reliability analysis of thin film interconnects under high current stress. Al–Cu test structures were used for this purpose. Small populations of these structures of similar geometry were subjected to extremely high current density conditions (30.6 and 46.6 MA/cm2) and their corresponding failure times were noted. The critical exponent (lB) for the Al–Cu structures stressed at both the current densities was calculated to be 0.16. The value of the lB showed that the structures undergo biased percolation and have similar failure mechanisms (due to Joule heating) at both current densities. The calculated value of lB was used to predict the failure times of the fuses under each of the current stresses. The discrepancy between the experimental failure time and the predicted failure time was significantly low (<12%) in both cases thus expressing the strength of this prediction technique. Ó 2004 Published by Elsevier Ltd.
1. Introduction The constant miniaturization of device size in integrated circuits (ICs) has resulted in their being subjected to extremely high current density conditions leading to their early failure as well as other reliability related issues. Reliability of the thin film metallization in interconnects is of importance in IC technology thus leading to the search for a rapid, accurate, and cost efficient failure time prediction technique. However, most of the methodologies being used currently in the reliabil*
Corresponding author. Address: Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287-6006, USA. Tel.: +1 480 965 7471; fax: +1 480 965 8976. E-mail address: [email protected] (T.L. Alford). 0026-2714/$ - see front matter Ó 2004 Published by Elsevier Ltd. doi:10.1016/j.microrel.2004.09.009
ity study of thin film interconnects are destructive [1]. These techniques involve subjecting a representative group of randomly selected samples to an overstressed condition of either constant current or voltage until failure occurs [2]. Since failure under normal device operation conditions takes a long time to occur, the samples are therefore subjected to accelerated testing conditions of higher temperature and current density [2]. The failure times detected for the samples are then analyzed using a suitable statistical distribution to determine the various factors and distribution parameters (viz. activation energy and current density exponent) leading to failure which are then extrapolated to predict reliability of the interconnects under normal operating conditions [1,2]. However, the main disadvantage involved with these techniques is that as the number of samples to be
392
E. Misra et al. / Microelectronics Reliability 45 (2005) 391–395
destroyed is only a small fraction of entire population; it can produce significant uncertainty in the definition of distribution parameters [3,4]. Moreover extrapolation of accelerated test data leads to inflation of the statistical errors involved which need to be minimized [2]. Also, sample to sample variations in testing conditions would lead to introduction of errors which would then cause improper interpretation of test data [2]. These methodologies also involve more cost in destroying a portion of the samples and also take a long time for test purposes. So the best methodology for prediction of failure time is by adapting non-destructive techniques. However, the main requirements of such a prediction technique are that it should be rapid, accurate, and reliable. In the past, percolation theory has been successfully used to study the failure in disordered materials [5]. Studies have also been conducted for the use of biased percolation model for understanding the failure of electrical devices [1,5–7]. If failure occurs by standard percolation then the damage would be more uniform and dependent on a single test parameter as is observed in the case of agglomeration of Ag metallization [8]. However, in case of biased percolation, failure occurs by the formation of defect channels perpendicular to the direction of current flow as is observed in the case of electromigration induced failure mechanisms [1,4]. Biased percolation model has been used in the above mentioned studies because it has been assumed that the failure of the devices occurs due to generation of Joule heating induced defects [5]. These studies are based on the assumption that the thin film under study is in the form of two-dimensional square lattice network of resistors having equal initial resistances, deposited on an insulating substrate at a constant temperature, and contacted on either sides for application of a constant external current or voltage [1,7]. A failure of such a network of resistors occurs by the formation of a continuous path of defects between the two contacts; this leads to an increase in overall resistance of the network [7]. At this point the network can be treated as a mixture of conductors and insulators. Hence, based on percolation theory the resulting overall resistance of a large two-dimensional random resistor network (RRN) can be represented as [1,9]: R j q qc jl
toring the fraction of defects on a large network is not feasible under normal and routine tests, thus Eq. (1) cannot be used for practical purposes of reliability analysis [1]. In order to overcome this difficulty, Pennetta et al. [1,7] have suggested the use of the scaling approach in which resistance is related directly with time [1]: RðtÞ j t tf jl
ð2Þ
Here tf is the failure time. Interestingly, the critical exponent l in Eq. (2) is the same as it is in Eq. (1). The value of l for biased percolation was determined to be 0.26 under constant current conditions and 0.57 under constant voltage conditions [7]. In the present article we use Al–Cu single line fuses to experimentally show that the biased percolation theory and scaling relations can be used as a rapid, nondestructive, and cost efficient technique for prediction of lifetime of thin film interconnects.
2. Experimental details Single line test structures of pure Al–Cu metallizations were used in the present study. The test structures were fabricated by a three metal standard 0.5 lm CMOS process. These structures consisted of two contact pads connected by a thin top metal line that acts as a fuse. The length and the width of the fuse were 11.6 lm and 1.4 lm, respectively and the total thickness of the Al– Cu top metal was 980 nm. Fig. 1 depicts an optical micrograph of the as fabricated test structure. The fabricated fuses were subjected to high current stress and their corresponding failure times were determined using a Credence ASL2000 tester. At the onset, the tester was used to charge a capacitor, which was then subsequently discharged through the fuse to blow it. The current through the probe needles connecting the capacitor and the fuse pads is limited by connecting a resistor in series with the fuse. This also protects the needles from being damaged from excessive current flow. Two
ð1Þ
where q is the fraction of broken resistors, qc is the percolation threshold, and l is a dimensionless statistical parameter known as conductivity critical exponent that describes the status of the biasing in a system [9]. Pennetta et al. [1] have proposed the theoretical aspects of such a technique based on percolative models and scaling relations for prediction of failure time of thin film interconnects [1,9,10]. For unbiased or standard percolation l has a constant value of 1.299, but for biased percolation the value is not constant and depends on the degradation environment [9]. Since moni-
Fig. 1. Optical micrograph of the Al–Cu fuse test structure.
Capacitor
5V
Al-Cu Fuse
E. Misra et al. / Microelectronics Reliability 45 (2005) 391–395
393
CurrentVoltage Source
Limiting Resistor (5Ω / 10Ω)
Fig. 2. Schematic of the tester set-up.
different resistors (5 and 10 X) were used in this study. A voltage/current (VI) source with a maximum DC current capability of 30 mA was used to determine the status of the lines after stressing with the current pulse. The failure time of the fuses was determined from the current versus time curves obtained. Fig. 2 is the schematic of the tester set-up used in this case.
3. Results and discussion In this study a small population of Al–Cu fuse structures are subjected to high current conditions until failure occurs by open circuit. Figs. 3 and 4 are the typical plots of current as a function of time obtained for the Al–Cu fuse structures in series with the 5 X and 10 X resistors, respectively. As has been mentioned earlier, the resistors control the magnitude of current flow through the test structures. The plots indicate that the current density flowing through the structures in case of the lower resistor is 46.6 MA/cm2 (0.64 A) and for the larger resistor is 30.6 MA/cm2 (0.42 A). The time to failure under each current condition is measured and reported in Table 1 (30.6 MA/cm2) and Table 2 (46.6 MA/ cm2). It has been reported in past literature that Joule heating is observed in Al structures at current densities greater than 1 MA/cm2 [11]. Since the current densities involved in the present case are quite high (>30 MA/ cm2), we therefore assume Joule heating induced failure under both current densities and use the biased percola-
Fig. 4. Plot of a typical current versus time curve for Al–Cu structures in series with a 5 X resistor (J = 46.6 MA/cm2).
Table 1 Table for comparison of measured and predicted failure times of Al–Cu fuses subjected to current density of 30.6 MA/cm2 Fuse
tf,measured (ls)
tf,predicted (ls)
% error (D%)
1 2 3 4 5 6 7
1.76 2.19 2.19 1.90 2.00 1.48 1.33
1.92 2.12 2.37 1.75 2.04 1.40 1.18
8.74 3.16 8.30 8.34 2.08 5.05 11.58
Table 2 Table for comparison of measured and predicted failure times of Al–Cu fuses subjected to current density of 44.6 MA/cm2 Fuse
tf,measured (ls)
tf,predicted (ls)
% error (D%)
1 2 3 4 5 6 7
0.62 0.52 0.76 0.62 0.90 0.67 0.62
0.60 0.50 0.71 0.65 0.87 0.69 0.65
3.12 4.05 7.38 5.36 3.88 3.84 4.76
tion model [1] for prediction of failure time of the Al–Cu structures. In case of biased percolation, the probability of breaking a resistor depends on both its position and time. Pennetta et al. [1] have used MonteCarlo simulations to show that the resistance evolution for biased percolation follows a scaling relation as R(t) jt tfjlB; where the value of lB depends on the Joule heating effects (current density and temperature) on the sample [1] and tf is the failure time of the sample due to Joule heating. If the initial value of resistance is R (0), then Eq. (2) can be represented as [1]: RðtÞ=Rð0Þ ¼ elB ¼ ð1 t=tf ÞlB :
Fig. 3. Plot of a typical current versus time curve for Al–Cu structures in series with a 10 X resistor (J = 30.6 MA/cm2).
ð3Þ
Thus, Eq. (3) can be used to predict the failure time of the thin film interconnects from early time measurements of resistance evolution. By measuring the resistance values R(t1) and R(t2) at two successive time intervals, t1 and t2, the failure time tf can be calculated as
394
rt1 t2 r1 where 1=lB Rðt1 Þ r¼ Rðt2 Þ tf ¼
E. Misra et al. / Microelectronics Reliability 45 (2005) 391–395
ð4Þ
ð5Þ
Thus, in the present study a number of test structures having similar geometry and material characteristics and located at the edge of the die (e.g., in the scribe line or test bar) were subjected to high current density conditions (30.6 and 46.6 MA/cm2) and their corresponding failure times are noted. For the group of samples subjected to current density of 30.6 MA/cm2 the parameter r is calculated using Eq. (4), the measured failure times and the corresponding current versus time plots. Then Eq. (5) is used to calculate the value of the critical exponent, lB. The average value of lB for the samples subjected to 30.6 MA/cm2 was determined to be 0.16. Finally, this calculated average lB can be used to predict the failure time of the rest of the structures in a nondestructive manner using Eq. (4). Table 1 shows the measured failure time (tf,measured) and the calculated failure time (tf,predicted) for the group of Al–Cu fuses (having similar geometry and material characteristics) and subjected to 30.6 MA/cm2 current density. The percentage errors are also shown in Table 1 (percentage error, D% = (tf,predicted tf,measured)/tf,measured). In the table, negative D% represents under prediction and positive D% represents over prediction. Similarly for the Al–Cu structures subjected to a current density of 46.6 MA/cm2, the parameters r and lB are calculated using the corresponding current versus time plots (Fig. 4) and Eqs. (3) and (4). The average value of lB in this case also is found to be 0.16. The value of lB obtained in both cases of current densities is thus indicative of biased percolation and supports the initial assumption of Joule heating being the dominant mechanism for the failure of the structures. Moreover, the similar lB values indicates that the failure mechanism in both cases is similar i.e., by vaporization of the metallization due to the large amount of heat produced (Joule heating) by flow of high current density through the metal lines. Table 2 lists the tf,measured values of the Al–Cu fuses subjected to the greater current density (46.6 MA/ cm2). The percentage errors are also calculated for the structures subjected to the higher current density (Table 2). The difference between the predicted failure time and the measured failure time in this case also is found to be significantly low. Tables 1 and 2 show that the discrepancy between the predicted failure time and the measured failure time is significantly low for structures subjected to both current densities and further justifies the strength of this technique. Thus, the present study is used to experimentally prove that for similar structures (i.e., same materials and geometry) undergoing similar degradation mecha-
nism the value of lB is a constant and can be extracted from a small population of samples. It can also be stated that the biased percolation theory can be used for rapid and non-destructive prediction of failure time and reliability analysis of large number of such samples under high current conditions.
4. Conclusions In this paper, we have used biased percolation theory and scaling models to predict the failure time of Al–Cu structures of similar geometry and materials characteristics when subjected to extremely high current density (30.6 and 46.6 MA/cm2). The salient features of this technique are that it is a rapid, non-destructive, and a less expensive methodology that can be adapted for assessing the reliability of thin film interconnects used for specific high current fuse applications. Most of the lifetime prediction techniques being currently used in the IC industry are destructive and involve significant amounts of money and time. However, in the reliability technique being used in this study only a few test samples need be initially destroyed to determine the average value of the exponent, l and after which the entire method of predictions is non-destructive. In the present study, the calculated value of l under both the current densities were found to be similar (0.16) thus indicating that the failure mechanism in both cases were similar and was largely governed by Joule heating generated due to the flow of the large current densities through the metal lines. Moreover, the predicted results were found to be satisfactory in comparison to the experimental test results for both the current densities. Thus, based on the experimental results obtained in the present study it can be said that the testing technique based on biased percolation theory can be used in industrial applications for development of interconnects for specific high current fuse applications where cost and time are of great concern.
Acknowledgments This work is supported by a grant from Philips Semiconductors Inc., Standard Analog, Tempe, AZ 85284, USA. The authors are grateful to E. Joseph of Philips Semiconductors Inc. for their support towards this project.
References [1] Pennetta C, Reggiani L, Trefa´n Gy. Scaling and universality in electrical failure of thin films. Phys Rev Lett 2000;84:5006–9.
E. Misra et al. / Microelectronics Reliability 45 (2005) 391–395 [2] Kim CU, Michael NL, Jiang Q-T, Augur R. Efficient electromigration testing with a single current source. Rev Sci Inst 2001;72:3962–7. [3] Ohring M. Reliability and failure of electronic materials and devices. San Diego: Academic Press; 1998. [4] Fantini F, Lloyd JR, De Munari I, Scorzoni A. Electromigration testing of integrated circuit interconnections. Microelectron Eng 1998;40:207–21. [5] Gingl Z, Penetta C, Kiss LB, Reggiani L. Biased percolation and abrupt failure of electronic devices. Semicond Sci Technol 1996;11:1770–5. [6] Pennetta C, Gingl Z, Kiss LB, Reggiani L. Biased percolation and electrical breakdown. Semicond Sci Technol 1997;12:1057–63.
395
[7] Pennetta C, Reggiani L, Kiss L. Thermal effects on the electrical degradation of thin film resistors. Physica A 1999;266:214–7. [8] Sieradzki K, Bailey K, Alford TL. Agglomeration and percolation conductivity. Appl Phys Lett 2001;79: 3401–3. [9] Stauffer D, Aharony A. Introduction to percolation theory. London: Taylor & Francis; 1992. [10] Stanley HE. Scaling, universality, and renormalization: three pillars of modern critical phenomena. Rev Mod Phys 1999;71:s358–66. [11] Sim SP. Procurement of specification requirements for protection against electromigration failures in aluminum metallizations. Microelectron Reliab 1979;19:207–18.
Research Note
Percolative approach for failure time prediction of thin film interconnects under high current stress E. Misra a, Md M. Islam b, Mahbub Hasan c, H.C. Kim a, T.L. Alford a
a,b,*
Department of Chemical and Materials Engineering, Arizona State University, Tempe, AZ 85287-6006, USA b Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287-6006, USA c Standard Analog Business Line, Philips Semiconductors Inc., Tempe, AZ 85284, USA Received 7 September 2003; received in revised form 1 July 2004 Available online 8 December 2004
Abstract The present study deals with the use of a rapid and non-destructive technique based on percolation theory to predict failure times during the reliability analysis of thin film interconnects under high current stress. Al–Cu test structures were used for this purpose. Small populations of these structures of similar geometry were subjected to extremely high current density conditions (30.6 and 46.6 MA/cm2) and their corresponding failure times were noted. The critical exponent (lB) for the Al–Cu structures stressed at both the current densities was calculated to be 0.16. The value of the lB showed that the structures undergo biased percolation and have similar failure mechanisms (due to Joule heating) at both current densities. The calculated value of lB was used to predict the failure times of the fuses under each of the current stresses. The discrepancy between the experimental failure time and the predicted failure time was significantly low (<12%) in both cases thus expressing the strength of this prediction technique. Ó 2004 Published by Elsevier Ltd.
1. Introduction The constant miniaturization of device size in integrated circuits (ICs) has resulted in their being subjected to extremely high current density conditions leading to their early failure as well as other reliability related issues. Reliability of the thin film metallization in interconnects is of importance in IC technology thus leading to the search for a rapid, accurate, and cost efficient failure time prediction technique. However, most of the methodologies being used currently in the reliabil*
Corresponding author. Address: Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287-6006, USA. Tel.: +1 480 965 7471; fax: +1 480 965 8976. E-mail address: [email protected] (T.L. Alford). 0026-2714/$ - see front matter Ó 2004 Published by Elsevier Ltd. doi:10.1016/j.microrel.2004.09.009
ity study of thin film interconnects are destructive [1]. These techniques involve subjecting a representative group of randomly selected samples to an overstressed condition of either constant current or voltage until failure occurs [2]. Since failure under normal device operation conditions takes a long time to occur, the samples are therefore subjected to accelerated testing conditions of higher temperature and current density [2]. The failure times detected for the samples are then analyzed using a suitable statistical distribution to determine the various factors and distribution parameters (viz. activation energy and current density exponent) leading to failure which are then extrapolated to predict reliability of the interconnects under normal operating conditions [1,2]. However, the main disadvantage involved with these techniques is that as the number of samples to be
392
E. Misra et al. / Microelectronics Reliability 45 (2005) 391–395
destroyed is only a small fraction of entire population; it can produce significant uncertainty in the definition of distribution parameters [3,4]. Moreover extrapolation of accelerated test data leads to inflation of the statistical errors involved which need to be minimized [2]. Also, sample to sample variations in testing conditions would lead to introduction of errors which would then cause improper interpretation of test data [2]. These methodologies also involve more cost in destroying a portion of the samples and also take a long time for test purposes. So the best methodology for prediction of failure time is by adapting non-destructive techniques. However, the main requirements of such a prediction technique are that it should be rapid, accurate, and reliable. In the past, percolation theory has been successfully used to study the failure in disordered materials [5]. Studies have also been conducted for the use of biased percolation model for understanding the failure of electrical devices [1,5–7]. If failure occurs by standard percolation then the damage would be more uniform and dependent on a single test parameter as is observed in the case of agglomeration of Ag metallization [8]. However, in case of biased percolation, failure occurs by the formation of defect channels perpendicular to the direction of current flow as is observed in the case of electromigration induced failure mechanisms [1,4]. Biased percolation model has been used in the above mentioned studies because it has been assumed that the failure of the devices occurs due to generation of Joule heating induced defects [5]. These studies are based on the assumption that the thin film under study is in the form of two-dimensional square lattice network of resistors having equal initial resistances, deposited on an insulating substrate at a constant temperature, and contacted on either sides for application of a constant external current or voltage [1,7]. A failure of such a network of resistors occurs by the formation of a continuous path of defects between the two contacts; this leads to an increase in overall resistance of the network [7]. At this point the network can be treated as a mixture of conductors and insulators. Hence, based on percolation theory the resulting overall resistance of a large two-dimensional random resistor network (RRN) can be represented as [1,9]: R j q qc jl
toring the fraction of defects on a large network is not feasible under normal and routine tests, thus Eq. (1) cannot be used for practical purposes of reliability analysis [1]. In order to overcome this difficulty, Pennetta et al. [1,7] have suggested the use of the scaling approach in which resistance is related directly with time [1]: RðtÞ j t tf jl
ð2Þ
Here tf is the failure time. Interestingly, the critical exponent l in Eq. (2) is the same as it is in Eq. (1). The value of l for biased percolation was determined to be 0.26 under constant current conditions and 0.57 under constant voltage conditions [7]. In the present article we use Al–Cu single line fuses to experimentally show that the biased percolation theory and scaling relations can be used as a rapid, nondestructive, and cost efficient technique for prediction of lifetime of thin film interconnects.
2. Experimental details Single line test structures of pure Al–Cu metallizations were used in the present study. The test structures were fabricated by a three metal standard 0.5 lm CMOS process. These structures consisted of two contact pads connected by a thin top metal line that acts as a fuse. The length and the width of the fuse were 11.6 lm and 1.4 lm, respectively and the total thickness of the Al– Cu top metal was 980 nm. Fig. 1 depicts an optical micrograph of the as fabricated test structure. The fabricated fuses were subjected to high current stress and their corresponding failure times were determined using a Credence ASL2000 tester. At the onset, the tester was used to charge a capacitor, which was then subsequently discharged through the fuse to blow it. The current through the probe needles connecting the capacitor and the fuse pads is limited by connecting a resistor in series with the fuse. This also protects the needles from being damaged from excessive current flow. Two
ð1Þ
where q is the fraction of broken resistors, qc is the percolation threshold, and l is a dimensionless statistical parameter known as conductivity critical exponent that describes the status of the biasing in a system [9]. Pennetta et al. [1] have proposed the theoretical aspects of such a technique based on percolative models and scaling relations for prediction of failure time of thin film interconnects [1,9,10]. For unbiased or standard percolation l has a constant value of 1.299, but for biased percolation the value is not constant and depends on the degradation environment [9]. Since moni-
Fig. 1. Optical micrograph of the Al–Cu fuse test structure.
Capacitor
5V
Al-Cu Fuse
E. Misra et al. / Microelectronics Reliability 45 (2005) 391–395
393
CurrentVoltage Source
Limiting Resistor (5Ω / 10Ω)
Fig. 2. Schematic of the tester set-up.
different resistors (5 and 10 X) were used in this study. A voltage/current (VI) source with a maximum DC current capability of 30 mA was used to determine the status of the lines after stressing with the current pulse. The failure time of the fuses was determined from the current versus time curves obtained. Fig. 2 is the schematic of the tester set-up used in this case.
3. Results and discussion In this study a small population of Al–Cu fuse structures are subjected to high current conditions until failure occurs by open circuit. Figs. 3 and 4 are the typical plots of current as a function of time obtained for the Al–Cu fuse structures in series with the 5 X and 10 X resistors, respectively. As has been mentioned earlier, the resistors control the magnitude of current flow through the test structures. The plots indicate that the current density flowing through the structures in case of the lower resistor is 46.6 MA/cm2 (0.64 A) and for the larger resistor is 30.6 MA/cm2 (0.42 A). The time to failure under each current condition is measured and reported in Table 1 (30.6 MA/cm2) and Table 2 (46.6 MA/ cm2). It has been reported in past literature that Joule heating is observed in Al structures at current densities greater than 1 MA/cm2 [11]. Since the current densities involved in the present case are quite high (>30 MA/ cm2), we therefore assume Joule heating induced failure under both current densities and use the biased percola-
Fig. 4. Plot of a typical current versus time curve for Al–Cu structures in series with a 5 X resistor (J = 46.6 MA/cm2).
Table 1 Table for comparison of measured and predicted failure times of Al–Cu fuses subjected to current density of 30.6 MA/cm2 Fuse
tf,measured (ls)
tf,predicted (ls)
% error (D%)
1 2 3 4 5 6 7
1.76 2.19 2.19 1.90 2.00 1.48 1.33
1.92 2.12 2.37 1.75 2.04 1.40 1.18
8.74 3.16 8.30 8.34 2.08 5.05 11.58
Table 2 Table for comparison of measured and predicted failure times of Al–Cu fuses subjected to current density of 44.6 MA/cm2 Fuse
tf,measured (ls)
tf,predicted (ls)
% error (D%)
1 2 3 4 5 6 7
0.62 0.52 0.76 0.62 0.90 0.67 0.62
0.60 0.50 0.71 0.65 0.87 0.69 0.65
3.12 4.05 7.38 5.36 3.88 3.84 4.76
tion model [1] for prediction of failure time of the Al–Cu structures. In case of biased percolation, the probability of breaking a resistor depends on both its position and time. Pennetta et al. [1] have used MonteCarlo simulations to show that the resistance evolution for biased percolation follows a scaling relation as R(t) jt tfjlB; where the value of lB depends on the Joule heating effects (current density and temperature) on the sample [1] and tf is the failure time of the sample due to Joule heating. If the initial value of resistance is R (0), then Eq. (2) can be represented as [1]: RðtÞ=Rð0Þ ¼ elB ¼ ð1 t=tf ÞlB :
Fig. 3. Plot of a typical current versus time curve for Al–Cu structures in series with a 10 X resistor (J = 30.6 MA/cm2).
ð3Þ
Thus, Eq. (3) can be used to predict the failure time of the thin film interconnects from early time measurements of resistance evolution. By measuring the resistance values R(t1) and R(t2) at two successive time intervals, t1 and t2, the failure time tf can be calculated as
394
rt1 t2 r1 where 1=lB Rðt1 Þ r¼ Rðt2 Þ tf ¼
E. Misra et al. / Microelectronics Reliability 45 (2005) 391–395
ð4Þ
ð5Þ
Thus, in the present study a number of test structures having similar geometry and material characteristics and located at the edge of the die (e.g., in the scribe line or test bar) were subjected to high current density conditions (30.6 and 46.6 MA/cm2) and their corresponding failure times are noted. For the group of samples subjected to current density of 30.6 MA/cm2 the parameter r is calculated using Eq. (4), the measured failure times and the corresponding current versus time plots. Then Eq. (5) is used to calculate the value of the critical exponent, lB. The average value of lB for the samples subjected to 30.6 MA/cm2 was determined to be 0.16. Finally, this calculated average lB can be used to predict the failure time of the rest of the structures in a nondestructive manner using Eq. (4). Table 1 shows the measured failure time (tf,measured) and the calculated failure time (tf,predicted) for the group of Al–Cu fuses (having similar geometry and material characteristics) and subjected to 30.6 MA/cm2 current density. The percentage errors are also shown in Table 1 (percentage error, D% = (tf,predicted tf,measured)/tf,measured). In the table, negative D% represents under prediction and positive D% represents over prediction. Similarly for the Al–Cu structures subjected to a current density of 46.6 MA/cm2, the parameters r and lB are calculated using the corresponding current versus time plots (Fig. 4) and Eqs. (3) and (4). The average value of lB in this case also is found to be 0.16. The value of lB obtained in both cases of current densities is thus indicative of biased percolation and supports the initial assumption of Joule heating being the dominant mechanism for the failure of the structures. Moreover, the similar lB values indicates that the failure mechanism in both cases is similar i.e., by vaporization of the metallization due to the large amount of heat produced (Joule heating) by flow of high current density through the metal lines. Table 2 lists the tf,measured values of the Al–Cu fuses subjected to the greater current density (46.6 MA/ cm2). The percentage errors are also calculated for the structures subjected to the higher current density (Table 2). The difference between the predicted failure time and the measured failure time in this case also is found to be significantly low. Tables 1 and 2 show that the discrepancy between the predicted failure time and the measured failure time is significantly low for structures subjected to both current densities and further justifies the strength of this technique. Thus, the present study is used to experimentally prove that for similar structures (i.e., same materials and geometry) undergoing similar degradation mecha-
nism the value of lB is a constant and can be extracted from a small population of samples. It can also be stated that the biased percolation theory can be used for rapid and non-destructive prediction of failure time and reliability analysis of large number of such samples under high current conditions.
4. Conclusions In this paper, we have used biased percolation theory and scaling models to predict the failure time of Al–Cu structures of similar geometry and materials characteristics when subjected to extremely high current density (30.6 and 46.6 MA/cm2). The salient features of this technique are that it is a rapid, non-destructive, and a less expensive methodology that can be adapted for assessing the reliability of thin film interconnects used for specific high current fuse applications. Most of the lifetime prediction techniques being currently used in the IC industry are destructive and involve significant amounts of money and time. However, in the reliability technique being used in this study only a few test samples need be initially destroyed to determine the average value of the exponent, l and after which the entire method of predictions is non-destructive. In the present study, the calculated value of l under both the current densities were found to be similar (0.16) thus indicating that the failure mechanism in both cases were similar and was largely governed by Joule heating generated due to the flow of the large current densities through the metal lines. Moreover, the predicted results were found to be satisfactory in comparison to the experimental test results for both the current densities. Thus, based on the experimental results obtained in the present study it can be said that the testing technique based on biased percolation theory can be used in industrial applications for development of interconnects for specific high current fuse applications where cost and time are of great concern.
Acknowledgments This work is supported by a grant from Philips Semiconductors Inc., Standard Analog, Tempe, AZ 85284, USA. The authors are grateful to E. Joseph of Philips Semiconductors Inc. for their support towards this project.
References [1] Pennetta C, Reggiani L, Trefa´n Gy. Scaling and universality in electrical failure of thin films. Phys Rev Lett 2000;84:5006–9.
E. Misra et al. / Microelectronics Reliability 45 (2005) 391–395 [2] Kim CU, Michael NL, Jiang Q-T, Augur R. Efficient electromigration testing with a single current source. Rev Sci Inst 2001;72:3962–7. [3] Ohring M. Reliability and failure of electronic materials and devices. San Diego: Academic Press; 1998. [4] Fantini F, Lloyd JR, De Munari I, Scorzoni A. Electromigration testing of integrated circuit interconnections. Microelectron Eng 1998;40:207–21. [5] Gingl Z, Penetta C, Kiss LB, Reggiani L. Biased percolation and abrupt failure of electronic devices. Semicond Sci Technol 1996;11:1770–5. [6] Pennetta C, Gingl Z, Kiss LB, Reggiani L. Biased percolation and electrical breakdown. Semicond Sci Technol 1997;12:1057–63.
395
[7] Pennetta C, Reggiani L, Kiss L. Thermal effects on the electrical degradation of thin film resistors. Physica A 1999;266:214–7. [8] Sieradzki K, Bailey K, Alford TL. Agglomeration and percolation conductivity. Appl Phys Lett 2001;79: 3401–3. [9] Stauffer D, Aharony A. Introduction to percolation theory. London: Taylor & Francis; 1992. [10] Stanley HE. Scaling, universality, and renormalization: three pillars of modern critical phenomena. Rev Mod Phys 1999;71:s358–66. [11] Sim SP. Procurement of specification requirements for protection against electromigration failures in aluminum metallizations. Microelectron Reliab 1979;19:207–18.