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A new method for the lifetime determination of submicron metal interconnects by means of a parallel test structure
Microelectronics Reliability 45 (2005) 753–759 www.elsevier.com/locate/microrel
Research Note
A new method for the lifetime determination of submicron metal interconnects by means of a parallel test structure q K. Vanstreels
b
a,*
, M. DÕOlieslaeger a,b, W. De Ceuninck J. DÕHaen b, K. Maex c
a,b
,
a IMO/LUC, Wetenschapspark 1, B-3590 Diepenbeek, Belgium IMOMEC/IMEC, Wetenschapspark 1, B-3590 Diepenbeek, Belgium c IMEC, Kapeldreef 75, B-3100 Leuven, Belgium
Received 28 May 2004; received in revised form 26 October 2004
Abstract Simulation experiments on both series and parallel electromigration (EM) test structures were carried out under current (or voltage) stress and further analysed by means of the total resistance (TR) analysis and a software package ‘‘failure’’ in order to calculate and to compare the behaviour of both EM test structures. These simulation experiments show that the parallel EM test structure is a correct approach for the determination of the failure time of submicron interconnects, the activation energy and the current density exponent n of the thermally driven process, therefore leading to a very substantial reduction of the number of samples that are needed to perform the EM tests. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Scaling is the magic word in the microelectronics industry. It refers to the miniaturization of active components and connections on a chip and has followed MooreÕs law for many decades. Despite many advantaq An earlier version of this paper was published in Proceedings of 24th International Conference on Microelectronics (MIEL 2004), 16–19 May 2004, Nisˇ, Serbia and Montenegro, vol. 2, pp. 633–636. * Corresponding author. E-mail addresses: [email protected] (K. Vanstreels), [email protected] (M. DÕOlieslaeger), [email protected] (W. De Ceuninck), [email protected] (J. DÕHaen), [email protected] (K. Maex).
ges, scaling also strongly influences the reliability and lifetime of interconnects. Electromigration is one of the most severe failure mechanisms of on-chip interconnects [1]. It is the mass transport of a metal due to the momentum transfer between conducting electrons and diffusing metal ions. A major problem when testing the reliability of new components is that their lifetime under real life conditions (Tmax = 125 C; j = 1–3 · 105 A/cm2 for a standard type IC) is always extremely long (in order of years). For that reason, the physical failure mechanisms are studied and methods are established for accelerating these mechanisms. The failure times of the devices operating are measured and models are developed for extrapolating these results to life time conditions [2–4]. The accelerated conditions for these tests are the temperature T and the current density j. In order
0026-2714/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2004.12.010
754
K. Vanstreels et al. / Microelectronics Reliability 45 (2005) 753–759
to gain enough statistical information out of these tests, a multi-link approach is desirable. The importance and utility of such an approach in electromigration reliability analysis was already mentioned by Ogawa et al., using dual-damascene interconnect ensembles in conjunction with statistical analysis based on ‘‘weakest link’’. Through the use of multi-linked interconnect ensembles, statistical evidence of two distinct failure modes in dualdamascene copper/oxide interconnects was first reported here [5]. In the present work, we will focus on the theoretical validation of a multi-linked interconnect approach in electromigration reliability analysis using mathematical simulations. In contrast to multi-level structures like for instance a dual-damascene structure, we will only focus on one-level interconnect structures. From a statistical point of view, a series connection of metal interconnects would be the optimum configuration for these tests at accelerated conditions, because the same current passes through all the series connected metal lines. However, this test structure is not desirable in practice due to technical limitations of the measurement equipment. Another possibility is a parallel configuration. Mathematical simulation is an ideal tool to validate the parallel test structure approach. In order to calculate and to compare the behaviour of parallel and/or series test structures, simulation experiments were carried out and further analysed by means of both the total resistance (TR)-analysis and a software package ‘‘failure’’. In practice, only the TR-analysis can be used for the series or parallel test structures. This analysis depends on the cross-cut method of the total resistance development of the structure. The software package ‘‘failure’’ [6] on the other hand, estimates parameters by means of the Maximum Likelihood Estimation (MLE).
2. Experimental details The studied parameter during the accelerated conditions is the relative resistance change, defined as DRðtÞ RðtÞ Rðt ¼ 0Þ R0 Rðt ¼ 0Þ
ð1Þ
The resistance R(t) is the resistance of the interconnect at time t and at accelerated conditions j (current density) and temperature T. Experiments show that DR(t)/R0 changes linearly as a function of time [7]. Define FC as the failure criterion of an interconnect. This means that interconnects where the drift DR(t)/R0 exceeds FC, are considered as failed. Subsequently, DRðtÞ FC ¼ t R0 tf
ð2Þ
where tf is the failure time of the interconnect. The studied quantity is the median of the failure times, i.e. the time where 50% of the interconnects failed, according to the failure criterion. For extrapolation of the simulation results to more real time conditions, we used the Black-model, which is the most intensively used extrapolation model. This model relates the median lifetime l of a line with the temperature T (in K) and the current density j (in MA/cm2) [8]. Ea ð3Þ l ¼ Ajn exp kB T where A is a material constant, kB is the Boltzmann constant, Ea is the activation energy (in eV) of the thermally driven process and n, the current density exponent, which usually has values between 1 and 3. A typical test goes as followed. Take a set of N interconnects with accelerated conditions j and T. It is assumed that both j and T are constant and that the failure times of the interconnects obey to a monomodal distribution. Moreover, a lognormal distribution is taken, because it is the far most common used distribution [9], i.e. tf / log n(l, r), where l (=median life time) and r are respectively the scale and shape parameter of the distributed failure times. The mean is noted as m. Due to transformations of statistical distributions, the resistance change DR / log n(l 0 , r), l0 ¼
R0 FC l
ð4Þ
where we note l 0 and m 0 as respectively the median and mean of the lognormally distributed DR. The shape parameter r is the same as for the failure times. At higher accelerated conditions (different j or T), the resistance change per unit of time is also lognormally distributed. Using the Black equation (3), the scale parameter l 0 (j, T) at higher accelerated conditions can be written as n j Ea 1 1 ð5Þ l0 ðj; T Þ ¼ l0 ðj1 ; T 1 Þ 1 exp j kB T T 1 where l 0 (j1, T1) is the scale parameter at accelerated conditions j1 and T1 and the shape parameter is the same for all accelerated conditions. The simulation of a series EM test structure of N interconnects is quite simple. For this structure, we assume that the current density j in Eq. (3) is constant as a function of time and as a consequence for each interconnection, the resistance change per unit of time DRi(t) = DRi(1) is also constant as a function of time. Moreover, DRi(t) / log n(l 0 , r). Taking for simplicity Ri(t = 0) at a constant level R0 "i, the resistance for each interconnection per unit of time is given by Ri ðtÞ ¼ R0 þ t DRi ð1Þ
ð6Þ
Using Eqs. (2) and (6), the relative resistance change for the total structure is
K. Vanstreels et al. / Microelectronics Reliability 45 (2005) 753–759
PN
i¼1 DRi ð1Þ
Rtot ð0Þ
PN t ¼
i¼1 DRi ð1Þ
N R0
t ffi
m0 t R0
ð7Þ
0
where m is the mean of the lognormally distributed DRi(1) values. Using Eq. (4), subsequently the failure time of the total series structure can be approximated by tf ffi
FC R0 l0 l ffi m0 m0
ð8Þ
For parallel structures, where the current density ji is not constant, the situation is far more complicated. The resistance Ri and the current density ji of the ith interconnection after 1 time unit t1 are respectively given by Ri ðt1 Þ ¼ R0 ji ðt1 Þ ¼
FC t1 þ R0 tf
jtot Rtot ðt1 Þ Ri ðt1 Þ
jtot ¼ N j0
ð9Þ ð10Þ ð11Þ
where i = 1 . . . N (number of interconnects), jtot is the constant current density of the structure and Ri(t1) is the resistance of the parallel structure at time t1. At time t2 = 2t1, it is assumed that the current density does not change during this step, so n j ðt1 Þ ð12Þ Ri ðt2 Þ ¼ Ri ðt1 Þ þ ðRi ðt1 Þ R0 Þ i j0 In contrast to a series structure, it is not easy to derive an equation for the failure time of the total structure, so this has to be determined iteratively.
3. Results and discussion For the simulation experiments on both series and parallel structures, a set of N values for the resistance Ri of each interconnect and the corresponding current density ji is generated using DR / log n(l 0 , r) and formula (4). Further evolution in time of both resistance and current density for each interconnect in the structure can than be calculated, using formulas (6),(10) and (12). Depending on the value for the failure criterion FC, the failure times of the individual interconnects can be extracted from the simulated data. The results of the simulation are further analysed by means of the TRanalysis and a software package ‘‘failure’’ [6], which estimates parameters by means of the MLE. These estimations are based on the median-determination, while for the TR-analysis, the determination of the failure times is based on the mean. The 4 estimated parameters are respectively l and r for tf / log n(l, r), the activation energy Ea and the current density exponent n. For the simulation experiments, N interconnects in a series and/or parallel structure are taken with length L =
2000 lm, thickness d = 0.5 lm, width b = 0.5 lm, resistivity qAl = 2.68 lX cm, current density j1 = 2 MA/cm2, temperature T1 = 200 C, current density exponent n = 2, activation energy Ea = 0.8 eV, failure times tf / log n(200, 0.5), failure criterion FC = 1, number of 500 time steps of 1 h as standard input values for the simulation experiments. A good agreement between the input parameters l, r and Ea for the simulation experiments (on series and parallel structures) and the estimated values from the simulated data using both the ‘‘Failure’’analysis and TR-analysis would suggest that the determination of lifetime and/or activation energies with suchlike structures is correct. Moreover, an agreement between the results for both applied methods on series/ parallel structures gives an additional confirmation of the observed trends. 3.1. Relative resistance change for series and parallel interconnects Fig. 1 shows the relative drift (DR/R) for 3 series and parallel interconnects as a function of time. For the series interconnects, the drift increases linearly, while for the parallel interconnects the drift of the individual interconnects bend towards each other. The drift velocity of interconnects that start with a relatively high rise of resistance decreases, while the drift velocity of the interconnects that start with a relatively low rise of resistance increases. This is because a lower current will run through the interconnects with the highest rise of resistance, decreasing the accelerated conditions for these interconnects and therefore making these interconnects less sensitive for electromigration at this point in time, the current will then distribute over the other interconnects, making these interconnects more sensitive to electromigration. In Fig. 2 the current is shown for the interconnects from Fig. 1. Moreover, it appears that as the current density exponent n increases, the drift of the individual interconnects bow more towards each other.
6 0.50 5
Parallel interconnects Series interconnects
0.25
4 DR/R
DRtot ðtÞ ¼ Rtot0
755
0.00
3
0 10 20 30 40 50
2 1 0
0
100
200
300
400
500
Time (hours)
Fig. 1. The relative drift for 3 series and parallel interconnects as a function of time for n = 2.
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K. Vanstreels et al. / Microelectronics Reliability 45 (2005) 753–759 0.0065
Parallel Series
Current I (A)
0.0060 0.0055 0.0050 0.0045 0.0040 0.0035 0
100
200
300
400
500
Time (hours)
Fig. 2. The current through 3 series and 3 parallel interconnects as a function of time.
3.2. Influence of the current density exponent on the median life time l and shape parameter r Table 1 shows the scale parameters l and shape parameters r estimated by ‘‘failure’’ for 80 parallel and series interconnects as a function of the current density exponent n, using FC = 1. All failure times are determined by the cross-cut method of the TRanalysis.
Notice that l for both series and parallel interconnects agree with the proposed failure time of 200 h used as an input parameter for the simulated data. For series interconnects, the shape parameter is in agreement with the expected value of 0.5, while for parallel interconnects the shape parameter is smaller. This is due to the fact that for the parallel interconnects, the drifts of the individual interconnects bow towards each other, as mentioned before. For increasing n, the drifts of the parallel interconnects bend even more towards each other, thereby decreasing the shape parameter. For FC < 1, the shape parameters of the parallel interconnects will agree more the values for series interconnects, while for FC > 1, the opposite is true, because the interconnections bow more towards each other. Moreover, Table 1 shows that for every current density exponent n, the failure time of the total parallel structure (with FC = 1) agrees with the initially proposed value of 200 h for the scale parameter l of the individual interconnects. This means that a parallel EM test structure is a correct approach for the determination of the failure time of submicron interconnects. For the total series structure, the failure time of the total structure deviates from the proposed value for l of the individual interconnects. This can easily be explained by the asymmetry of
Table 1 Scale parameters l and shape parameters for 80 parallel and series interconnects as a function of the current density exponent, estimated by ‘‘failure’’ using FC = 1 Failure analysis
TR-analysis
Series
Parallel
n
1,2,3
1
Sim
l
r
l
r
l
r
l
r
Tf
tf
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
193 187 204 202 209 196 203 220 200 202 214 209 196 193 186 194 182 203 204 188
0.522 0.464 0.402 0.453 0.476 0.504 0.437 0.457 0.505 0.44 0.472 0.481 0.491 0.484 0.469 0.454 0.511 0.399 0.466 0.453
191 184 204 200 207 194 202 218 198 202 213 209 195 190 185 191 181 201 202 185
0.403 0.355 0.323 0.358 0.373 0.393 0.339 0.358 0.394 0.35 0.366 0.38 0.384 0.368 0.367 0.347 0.409 0.302 0.364 0.344
191 184 204 200 207 192 203 218 198 202 213 209 195 190 185 191 181 201 202 185
0.317 0.279 0.254 0.283 0.293 0.262 0.274 0.282 0.31 0.275 0.286 0.3 0.302 0.289 0.295 0.273 0.322 0.237 0.286 0.27
191 185 204 201 208 195 203 220 199 202 213 209 195 191 186 192 182 202 203 186
0.256 0.225 0.204 0.229 0.236 0.252 0.222 0.227 0.25 0.222 0.23 0.242 0.244 0.233 0.237 0.22 0.26 0.191 0.231 0.217
168 167 189 180 187 172 185 197 176 184 193 187 174 171 167 174 159 188 183 169
196 188 207 205 212 199 207 223 203 206 217 214 199 194 189 195 186 204 207 189
Mean Error
199 9
0.47 0.032
198 7
0.364 0.026
198 6
0.284 0.021
198 5
0.231 0.016
179 10
202 10
2
3
Series
Parallel
1,2,3
1,2,3
K. Vanstreels et al. / Microelectronics Reliability 45 (2005) 753–759 1.2
Series
1.0
t f /200
0.8 0.6 0.4 0.2 0.0 0.0
Parallel 0.5
1.0
1.5
2.0
2.5
3.0
Shape parmeter σ
Fig. 3. The failure time divided by the proposed failure time of 200 h for both series and parallel structures, using n = 2 and N = 1000 as a function of r.
the distributed failure times and using Eqs. (7) and (8). Fig. 3 shows the failure time for the total series/parallel structure divided by 200 (proposed value for l). The failure time fore the parallel structure is independent of r, while for the series structure, clearly a dependence can be found. For higher values of r, the failure time for the total series structure deviates even more from the proposed value, confirming our statement. For the moment it is not clear why for the parallel structure no dependence can be found. 3.3. Influence of the current density exponent n on rser/r and rpar/r As shown in Table 1, the shape parameter rpar of the simulated data for the parallel structures was lower than the expected input value r for the simulation experiments. Moreover, for higher values of n, its value decreased even more. For a further understanding of this phenomenon, the influence of several input parameters on rpar was investigated in more detail. Fig. 4 shows respectively rpar and rser divided by the input shape
757
parameter r for the simulation experiments as a function of n for several values of r and with a fixed failure criterion FC = 1. All experiments were done on both series and parallel structures containing 1000 interconnects. For the series structure, rser/r is clearly independent of n for several values of r, as mentioned before. For the parallel structure on the other hand, the same relationship of rpar/r as a function of n can be found, independent of the values for r. This means that for a fixed FC value, a certain relationship of rpar/r as a function of n can be found independent of the input value r. 3.4. Influence of FC on rser/r and rpar/r To determine the influence of the failure criterion FC on rpar/r as a function of n, simulation experiments were carried out on both series and parallel structures containing 1000 interconnects with a fixed r = 0.5 and with FC varying from 0.5 to 3. The results are shown in Fig. 5. The results show that the shape of the curves for different FC values does not change. However, for FC < 1, the curves shift move more towards 1, while the opposite is true for FC > 1. Fig. 6 shows the relationship between rpar/r and FC for different values of n. It is shown that rpar/r decreases in the same way, but that for different n, the relationship shifts more towards (lower n values) or more away from 1 (higher n values). Both observed phenomena in Figs. 5 and 6 are in agreement with our earlier observations, where higher FC values and n lead to a bigger difference between the shape parameters of both series and parallel structures. As a consequence of these results, using both Figs. 5 and 6, and knowing both n and FC out of the experiments, it is possible to give an estimation of the correct shape parameter for parallel test structures. However, the accuracy of this estimation method in practice is still under investigation. Moreover, also other correction methods are oncoming.
Series
Series
1.0
0.8
0.8
σser /σ ; σpar /σ
σser /σ ;σpar /σ
1.0
0.6 0.4 0.2
Parallel
0.0 0
5
10
15
20
n Fig. 4. rpar/r and rser/r as a function of n for several input values of r (0.1–1.0) and with a fixed FC = 1.
FC = 0.5 FC = 1.0 FC = 1.5 FC = 2.0 FC = 3.0
0.6 0.4
Parallel
0.2 0.0 0
5
10
15
20
n Fig. 5. rpar/r and rser/r as a function of n for r = 0.5 and with FC varying from 0.5 to 3.
758
K. Vanstreels et al. / Microelectronics Reliability 45 (2005) 753–759 1.0
σser /σ ; σpar /σ
0.8 n=1
0.6
n=2
0.4
n=3 n=4 n=5 n=6 n=7 n=8
0.2 0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
FC Fig. 6. rpar/r as a function of FC for r = 0.5 and with n varying from 1 to 20. Fig. 7. Cumulative failure plot for the parallel interconnects of simulation 9 in Table 2.
3.5. Determination of activation energies Consider that the failure time tf of the interconnects at accelerated conditions j1 = 2 MA/cm2 and T1 = 200 C is lognormally distributed with l1 = 200 and r = 0.5 using FC = 1. At higher temperatures 220 and 240 C, the scale parameters can be calculated using Eq. (7), where we assume that the system is driven by only 1 activation energy (Ea = 0.8 eV). Table 2 Table 2 The activation energy determined with ‘‘failure’’ and the TRanalysis for 80 interconnects ‘‘Failure’’ analysis
TR-analysis
n
2
2
2
2
Sim
Series
Parallel
Series
Parallel
Ea
Ea
Ea
Ea
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.772 0.746 0.876 0.757 0.817 0.752 0.805 0.815 0.802 0.805 0.809 0.858 0.854 0.792 0.759 0.778 0.718 0.801 0.793 0.778
0.774 0.749 0.884 0.759 0.822 0.756 0.814 0.821 0.806 0.814 0.812 0.866 0.86 0.792 0.765 0.783 0.725 0.804 0.796 0.781
0.772 0.761 0.894 0.743 0.825 0.747 0.843 0.831 0.806 0.831 0.804 0.875 0.879 0.791 0.767 0.809 0.724 0.827 0.790 0.803
0.774 0.745 0.881 0.761 0.82 0.756 0.808 0.818 0.806 0.810 0.813 0.863 0.855 0.791 0.764 0.776 0.724 0.799 0.796 0.776
Mean Error
0.79 0.02
0.80 0.02
0.81 0.1
0.80 0.1
shows the calculated activation energies for both series and parallel test structures, using the TR-analysis and ‘‘failure’’ for 80 series and parallel interconnects (n = 2). For both used methods, the mean values of the activation energies agree very well the proposed value of 0.8 eV, used as an input parameter for the simulated data. This means that for monomodal lognormally distributed failure times, both series and parallel test structures can be used for the determination of both the activation energy and the current density exponent n. Fig. 7 shows the cumulative failure plot for the 80 parallel interconnects of simulation experiment 9 in Tables 1 and 2.
4. Conclusion Using simulation experiments, we showed that for monomodal lognormally distributed failure times, the parallel EM test structure is a correct approach for the determination of the failure time of submicron interconnects. Moreover, the simulation experiments showed that the calculated activation energies for both series/ parallel structures are in good agreement with the proposed value of Ea (0.8 eV), used as an input parameter for the simulation experiments. This means that for monomodal lognormal distributed failure times, the parallel EM structure can be used for the determination of both Ea and n, leading to a very substantial reduction of the number of samples that are needed. However, simulation experiments have shown that the shape parameter for the parallel structure is smaller than expected and therefore a correction of this value is necessary.
K. Vanstreels et al. / Microelectronics Reliability 45 (2005) 753–759
References [1] Scorzoni A, Neri B, Caprile C, Fantini F. Mater Sci Rep 1991;7:143. [2] Lloyd JR. MRS Symp Proc 1994;265:177. [3] Thomson CV, Kahn H. J Electron Mater 1993;22(6):581. [4] De Schepper L, De Ceuninck W, Lekens G, Stals L, Vanhecke B, Roggen J, et al. Qual Reliab Eng Int 1994;10:15. [5] Ogawa ET, Lee K-D, Matsuhashi H, Ko K-S, Justison PR, Ramamurthi AN, et al. 2001 IEEE Int Reliability Physics (IRPS; 39th annual), p. 341–9.
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[6] Failure. Software package for reliability data analysis, distributed by Destin nv, Wetenschapspark 1, B-3590 Diepenbeek. [7] Bobbio A, Saracco O. Microelectron Reliab 1975;14:431. 338. [8] Black JR. IEEE Trans Electron Dev 1969;10(4). [9] Birolini A. Quality and reliability of technical systems. Berlin Heidelberg: Springer Verlag; 1994.
Research Note
A new method for the lifetime determination of submicron metal interconnects by means of a parallel test structure q K. Vanstreels
b
a,*
, M. DÕOlieslaeger a,b, W. De Ceuninck J. DÕHaen b, K. Maex c
a,b
,
a IMO/LUC, Wetenschapspark 1, B-3590 Diepenbeek, Belgium IMOMEC/IMEC, Wetenschapspark 1, B-3590 Diepenbeek, Belgium c IMEC, Kapeldreef 75, B-3100 Leuven, Belgium
Received 28 May 2004; received in revised form 26 October 2004
Abstract Simulation experiments on both series and parallel electromigration (EM) test structures were carried out under current (or voltage) stress and further analysed by means of the total resistance (TR) analysis and a software package ‘‘failure’’ in order to calculate and to compare the behaviour of both EM test structures. These simulation experiments show that the parallel EM test structure is a correct approach for the determination of the failure time of submicron interconnects, the activation energy and the current density exponent n of the thermally driven process, therefore leading to a very substantial reduction of the number of samples that are needed to perform the EM tests. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Scaling is the magic word in the microelectronics industry. It refers to the miniaturization of active components and connections on a chip and has followed MooreÕs law for many decades. Despite many advantaq An earlier version of this paper was published in Proceedings of 24th International Conference on Microelectronics (MIEL 2004), 16–19 May 2004, Nisˇ, Serbia and Montenegro, vol. 2, pp. 633–636. * Corresponding author. E-mail addresses: [email protected] (K. Vanstreels), [email protected] (M. DÕOlieslaeger), [email protected] (W. De Ceuninck), [email protected] (J. DÕHaen), [email protected] (K. Maex).
ges, scaling also strongly influences the reliability and lifetime of interconnects. Electromigration is one of the most severe failure mechanisms of on-chip interconnects [1]. It is the mass transport of a metal due to the momentum transfer between conducting electrons and diffusing metal ions. A major problem when testing the reliability of new components is that their lifetime under real life conditions (Tmax = 125 C; j = 1–3 · 105 A/cm2 for a standard type IC) is always extremely long (in order of years). For that reason, the physical failure mechanisms are studied and methods are established for accelerating these mechanisms. The failure times of the devices operating are measured and models are developed for extrapolating these results to life time conditions [2–4]. The accelerated conditions for these tests are the temperature T and the current density j. In order
0026-2714/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2004.12.010
754
K. Vanstreels et al. / Microelectronics Reliability 45 (2005) 753–759
to gain enough statistical information out of these tests, a multi-link approach is desirable. The importance and utility of such an approach in electromigration reliability analysis was already mentioned by Ogawa et al., using dual-damascene interconnect ensembles in conjunction with statistical analysis based on ‘‘weakest link’’. Through the use of multi-linked interconnect ensembles, statistical evidence of two distinct failure modes in dualdamascene copper/oxide interconnects was first reported here [5]. In the present work, we will focus on the theoretical validation of a multi-linked interconnect approach in electromigration reliability analysis using mathematical simulations. In contrast to multi-level structures like for instance a dual-damascene structure, we will only focus on one-level interconnect structures. From a statistical point of view, a series connection of metal interconnects would be the optimum configuration for these tests at accelerated conditions, because the same current passes through all the series connected metal lines. However, this test structure is not desirable in practice due to technical limitations of the measurement equipment. Another possibility is a parallel configuration. Mathematical simulation is an ideal tool to validate the parallel test structure approach. In order to calculate and to compare the behaviour of parallel and/or series test structures, simulation experiments were carried out and further analysed by means of both the total resistance (TR)-analysis and a software package ‘‘failure’’. In practice, only the TR-analysis can be used for the series or parallel test structures. This analysis depends on the cross-cut method of the total resistance development of the structure. The software package ‘‘failure’’ [6] on the other hand, estimates parameters by means of the Maximum Likelihood Estimation (MLE).
2. Experimental details The studied parameter during the accelerated conditions is the relative resistance change, defined as DRðtÞ RðtÞ Rðt ¼ 0Þ R0 Rðt ¼ 0Þ
ð1Þ
The resistance R(t) is the resistance of the interconnect at time t and at accelerated conditions j (current density) and temperature T. Experiments show that DR(t)/R0 changes linearly as a function of time [7]. Define FC as the failure criterion of an interconnect. This means that interconnects where the drift DR(t)/R0 exceeds FC, are considered as failed. Subsequently, DRðtÞ FC ¼ t R0 tf
ð2Þ
where tf is the failure time of the interconnect. The studied quantity is the median of the failure times, i.e. the time where 50% of the interconnects failed, according to the failure criterion. For extrapolation of the simulation results to more real time conditions, we used the Black-model, which is the most intensively used extrapolation model. This model relates the median lifetime l of a line with the temperature T (in K) and the current density j (in MA/cm2) [8]. Ea ð3Þ l ¼ Ajn exp kB T where A is a material constant, kB is the Boltzmann constant, Ea is the activation energy (in eV) of the thermally driven process and n, the current density exponent, which usually has values between 1 and 3. A typical test goes as followed. Take a set of N interconnects with accelerated conditions j and T. It is assumed that both j and T are constant and that the failure times of the interconnects obey to a monomodal distribution. Moreover, a lognormal distribution is taken, because it is the far most common used distribution [9], i.e. tf / log n(l, r), where l (=median life time) and r are respectively the scale and shape parameter of the distributed failure times. The mean is noted as m. Due to transformations of statistical distributions, the resistance change DR / log n(l 0 , r), l0 ¼
R0 FC l
ð4Þ
where we note l 0 and m 0 as respectively the median and mean of the lognormally distributed DR. The shape parameter r is the same as for the failure times. At higher accelerated conditions (different j or T), the resistance change per unit of time is also lognormally distributed. Using the Black equation (3), the scale parameter l 0 (j, T) at higher accelerated conditions can be written as n j Ea 1 1 ð5Þ l0 ðj; T Þ ¼ l0 ðj1 ; T 1 Þ 1 exp j kB T T 1 where l 0 (j1, T1) is the scale parameter at accelerated conditions j1 and T1 and the shape parameter is the same for all accelerated conditions. The simulation of a series EM test structure of N interconnects is quite simple. For this structure, we assume that the current density j in Eq. (3) is constant as a function of time and as a consequence for each interconnection, the resistance change per unit of time DRi(t) = DRi(1) is also constant as a function of time. Moreover, DRi(t) / log n(l 0 , r). Taking for simplicity Ri(t = 0) at a constant level R0 "i, the resistance for each interconnection per unit of time is given by Ri ðtÞ ¼ R0 þ t DRi ð1Þ
ð6Þ
Using Eqs. (2) and (6), the relative resistance change for the total structure is
K. Vanstreels et al. / Microelectronics Reliability 45 (2005) 753–759
PN
i¼1 DRi ð1Þ
Rtot ð0Þ
PN t ¼
i¼1 DRi ð1Þ
N R0
t ffi
m0 t R0
ð7Þ
0
where m is the mean of the lognormally distributed DRi(1) values. Using Eq. (4), subsequently the failure time of the total series structure can be approximated by tf ffi
FC R0 l0 l ffi m0 m0
ð8Þ
For parallel structures, where the current density ji is not constant, the situation is far more complicated. The resistance Ri and the current density ji of the ith interconnection after 1 time unit t1 are respectively given by Ri ðt1 Þ ¼ R0 ji ðt1 Þ ¼
FC t1 þ R0 tf
jtot Rtot ðt1 Þ Ri ðt1 Þ
jtot ¼ N j0
ð9Þ ð10Þ ð11Þ
where i = 1 . . . N (number of interconnects), jtot is the constant current density of the structure and Ri(t1) is the resistance of the parallel structure at time t1. At time t2 = 2t1, it is assumed that the current density does not change during this step, so n j ðt1 Þ ð12Þ Ri ðt2 Þ ¼ Ri ðt1 Þ þ ðRi ðt1 Þ R0 Þ i j0 In contrast to a series structure, it is not easy to derive an equation for the failure time of the total structure, so this has to be determined iteratively.
3. Results and discussion For the simulation experiments on both series and parallel structures, a set of N values for the resistance Ri of each interconnect and the corresponding current density ji is generated using DR / log n(l 0 , r) and formula (4). Further evolution in time of both resistance and current density for each interconnect in the structure can than be calculated, using formulas (6),(10) and (12). Depending on the value for the failure criterion FC, the failure times of the individual interconnects can be extracted from the simulated data. The results of the simulation are further analysed by means of the TRanalysis and a software package ‘‘failure’’ [6], which estimates parameters by means of the MLE. These estimations are based on the median-determination, while for the TR-analysis, the determination of the failure times is based on the mean. The 4 estimated parameters are respectively l and r for tf / log n(l, r), the activation energy Ea and the current density exponent n. For the simulation experiments, N interconnects in a series and/or parallel structure are taken with length L =
2000 lm, thickness d = 0.5 lm, width b = 0.5 lm, resistivity qAl = 2.68 lX cm, current density j1 = 2 MA/cm2, temperature T1 = 200 C, current density exponent n = 2, activation energy Ea = 0.8 eV, failure times tf / log n(200, 0.5), failure criterion FC = 1, number of 500 time steps of 1 h as standard input values for the simulation experiments. A good agreement between the input parameters l, r and Ea for the simulation experiments (on series and parallel structures) and the estimated values from the simulated data using both the ‘‘Failure’’analysis and TR-analysis would suggest that the determination of lifetime and/or activation energies with suchlike structures is correct. Moreover, an agreement between the results for both applied methods on series/ parallel structures gives an additional confirmation of the observed trends. 3.1. Relative resistance change for series and parallel interconnects Fig. 1 shows the relative drift (DR/R) for 3 series and parallel interconnects as a function of time. For the series interconnects, the drift increases linearly, while for the parallel interconnects the drift of the individual interconnects bend towards each other. The drift velocity of interconnects that start with a relatively high rise of resistance decreases, while the drift velocity of the interconnects that start with a relatively low rise of resistance increases. This is because a lower current will run through the interconnects with the highest rise of resistance, decreasing the accelerated conditions for these interconnects and therefore making these interconnects less sensitive for electromigration at this point in time, the current will then distribute over the other interconnects, making these interconnects more sensitive to electromigration. In Fig. 2 the current is shown for the interconnects from Fig. 1. Moreover, it appears that as the current density exponent n increases, the drift of the individual interconnects bow more towards each other.
6 0.50 5
Parallel interconnects Series interconnects
0.25
4 DR/R
DRtot ðtÞ ¼ Rtot0
755
0.00
3
0 10 20 30 40 50
2 1 0
0
100
200
300
400
500
Time (hours)
Fig. 1. The relative drift for 3 series and parallel interconnects as a function of time for n = 2.
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K. Vanstreels et al. / Microelectronics Reliability 45 (2005) 753–759 0.0065
Parallel Series
Current I (A)
0.0060 0.0055 0.0050 0.0045 0.0040 0.0035 0
100
200
300
400
500
Time (hours)
Fig. 2. The current through 3 series and 3 parallel interconnects as a function of time.
3.2. Influence of the current density exponent on the median life time l and shape parameter r Table 1 shows the scale parameters l and shape parameters r estimated by ‘‘failure’’ for 80 parallel and series interconnects as a function of the current density exponent n, using FC = 1. All failure times are determined by the cross-cut method of the TRanalysis.
Notice that l for both series and parallel interconnects agree with the proposed failure time of 200 h used as an input parameter for the simulated data. For series interconnects, the shape parameter is in agreement with the expected value of 0.5, while for parallel interconnects the shape parameter is smaller. This is due to the fact that for the parallel interconnects, the drifts of the individual interconnects bow towards each other, as mentioned before. For increasing n, the drifts of the parallel interconnects bend even more towards each other, thereby decreasing the shape parameter. For FC < 1, the shape parameters of the parallel interconnects will agree more the values for series interconnects, while for FC > 1, the opposite is true, because the interconnections bow more towards each other. Moreover, Table 1 shows that for every current density exponent n, the failure time of the total parallel structure (with FC = 1) agrees with the initially proposed value of 200 h for the scale parameter l of the individual interconnects. This means that a parallel EM test structure is a correct approach for the determination of the failure time of submicron interconnects. For the total series structure, the failure time of the total structure deviates from the proposed value for l of the individual interconnects. This can easily be explained by the asymmetry of
Table 1 Scale parameters l and shape parameters for 80 parallel and series interconnects as a function of the current density exponent, estimated by ‘‘failure’’ using FC = 1 Failure analysis
TR-analysis
Series
Parallel
n
1,2,3
1
Sim
l
r
l
r
l
r
l
r
Tf
tf
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
193 187 204 202 209 196 203 220 200 202 214 209 196 193 186 194 182 203 204 188
0.522 0.464 0.402 0.453 0.476 0.504 0.437 0.457 0.505 0.44 0.472 0.481 0.491 0.484 0.469 0.454 0.511 0.399 0.466 0.453
191 184 204 200 207 194 202 218 198 202 213 209 195 190 185 191 181 201 202 185
0.403 0.355 0.323 0.358 0.373 0.393 0.339 0.358 0.394 0.35 0.366 0.38 0.384 0.368 0.367 0.347 0.409 0.302 0.364 0.344
191 184 204 200 207 192 203 218 198 202 213 209 195 190 185 191 181 201 202 185
0.317 0.279 0.254 0.283 0.293 0.262 0.274 0.282 0.31 0.275 0.286 0.3 0.302 0.289 0.295 0.273 0.322 0.237 0.286 0.27
191 185 204 201 208 195 203 220 199 202 213 209 195 191 186 192 182 202 203 186
0.256 0.225 0.204 0.229 0.236 0.252 0.222 0.227 0.25 0.222 0.23 0.242 0.244 0.233 0.237 0.22 0.26 0.191 0.231 0.217
168 167 189 180 187 172 185 197 176 184 193 187 174 171 167 174 159 188 183 169
196 188 207 205 212 199 207 223 203 206 217 214 199 194 189 195 186 204 207 189
Mean Error
199 9
0.47 0.032
198 7
0.364 0.026
198 6
0.284 0.021
198 5
0.231 0.016
179 10
202 10
2
3
Series
Parallel
1,2,3
1,2,3
K. Vanstreels et al. / Microelectronics Reliability 45 (2005) 753–759 1.2
Series
1.0
t f /200
0.8 0.6 0.4 0.2 0.0 0.0
Parallel 0.5
1.0
1.5
2.0
2.5
3.0
Shape parmeter σ
Fig. 3. The failure time divided by the proposed failure time of 200 h for both series and parallel structures, using n = 2 and N = 1000 as a function of r.
the distributed failure times and using Eqs. (7) and (8). Fig. 3 shows the failure time for the total series/parallel structure divided by 200 (proposed value for l). The failure time fore the parallel structure is independent of r, while for the series structure, clearly a dependence can be found. For higher values of r, the failure time for the total series structure deviates even more from the proposed value, confirming our statement. For the moment it is not clear why for the parallel structure no dependence can be found. 3.3. Influence of the current density exponent n on rser/r and rpar/r As shown in Table 1, the shape parameter rpar of the simulated data for the parallel structures was lower than the expected input value r for the simulation experiments. Moreover, for higher values of n, its value decreased even more. For a further understanding of this phenomenon, the influence of several input parameters on rpar was investigated in more detail. Fig. 4 shows respectively rpar and rser divided by the input shape
757
parameter r for the simulation experiments as a function of n for several values of r and with a fixed failure criterion FC = 1. All experiments were done on both series and parallel structures containing 1000 interconnects. For the series structure, rser/r is clearly independent of n for several values of r, as mentioned before. For the parallel structure on the other hand, the same relationship of rpar/r as a function of n can be found, independent of the values for r. This means that for a fixed FC value, a certain relationship of rpar/r as a function of n can be found independent of the input value r. 3.4. Influence of FC on rser/r and rpar/r To determine the influence of the failure criterion FC on rpar/r as a function of n, simulation experiments were carried out on both series and parallel structures containing 1000 interconnects with a fixed r = 0.5 and with FC varying from 0.5 to 3. The results are shown in Fig. 5. The results show that the shape of the curves for different FC values does not change. However, for FC < 1, the curves shift move more towards 1, while the opposite is true for FC > 1. Fig. 6 shows the relationship between rpar/r and FC for different values of n. It is shown that rpar/r decreases in the same way, but that for different n, the relationship shifts more towards (lower n values) or more away from 1 (higher n values). Both observed phenomena in Figs. 5 and 6 are in agreement with our earlier observations, where higher FC values and n lead to a bigger difference between the shape parameters of both series and parallel structures. As a consequence of these results, using both Figs. 5 and 6, and knowing both n and FC out of the experiments, it is possible to give an estimation of the correct shape parameter for parallel test structures. However, the accuracy of this estimation method in practice is still under investigation. Moreover, also other correction methods are oncoming.
Series
Series
1.0
0.8
0.8
σser /σ ; σpar /σ
σser /σ ;σpar /σ
1.0
0.6 0.4 0.2
Parallel
0.0 0
5
10
15
20
n Fig. 4. rpar/r and rser/r as a function of n for several input values of r (0.1–1.0) and with a fixed FC = 1.
FC = 0.5 FC = 1.0 FC = 1.5 FC = 2.0 FC = 3.0
0.6 0.4
Parallel
0.2 0.0 0
5
10
15
20
n Fig. 5. rpar/r and rser/r as a function of n for r = 0.5 and with FC varying from 0.5 to 3.
758
K. Vanstreels et al. / Microelectronics Reliability 45 (2005) 753–759 1.0
σser /σ ; σpar /σ
0.8 n=1
0.6
n=2
0.4
n=3 n=4 n=5 n=6 n=7 n=8
0.2 0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
FC Fig. 6. rpar/r as a function of FC for r = 0.5 and with n varying from 1 to 20. Fig. 7. Cumulative failure plot for the parallel interconnects of simulation 9 in Table 2.
3.5. Determination of activation energies Consider that the failure time tf of the interconnects at accelerated conditions j1 = 2 MA/cm2 and T1 = 200 C is lognormally distributed with l1 = 200 and r = 0.5 using FC = 1. At higher temperatures 220 and 240 C, the scale parameters can be calculated using Eq. (7), where we assume that the system is driven by only 1 activation energy (Ea = 0.8 eV). Table 2 Table 2 The activation energy determined with ‘‘failure’’ and the TRanalysis for 80 interconnects ‘‘Failure’’ analysis
TR-analysis
n
2
2
2
2
Sim
Series
Parallel
Series
Parallel
Ea
Ea
Ea
Ea
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.772 0.746 0.876 0.757 0.817 0.752 0.805 0.815 0.802 0.805 0.809 0.858 0.854 0.792 0.759 0.778 0.718 0.801 0.793 0.778
0.774 0.749 0.884 0.759 0.822 0.756 0.814 0.821 0.806 0.814 0.812 0.866 0.86 0.792 0.765 0.783 0.725 0.804 0.796 0.781
0.772 0.761 0.894 0.743 0.825 0.747 0.843 0.831 0.806 0.831 0.804 0.875 0.879 0.791 0.767 0.809 0.724 0.827 0.790 0.803
0.774 0.745 0.881 0.761 0.82 0.756 0.808 0.818 0.806 0.810 0.813 0.863 0.855 0.791 0.764 0.776 0.724 0.799 0.796 0.776
Mean Error
0.79 0.02
0.80 0.02
0.81 0.1
0.80 0.1
shows the calculated activation energies for both series and parallel test structures, using the TR-analysis and ‘‘failure’’ for 80 series and parallel interconnects (n = 2). For both used methods, the mean values of the activation energies agree very well the proposed value of 0.8 eV, used as an input parameter for the simulated data. This means that for monomodal lognormally distributed failure times, both series and parallel test structures can be used for the determination of both the activation energy and the current density exponent n. Fig. 7 shows the cumulative failure plot for the 80 parallel interconnects of simulation experiment 9 in Tables 1 and 2.
4. Conclusion Using simulation experiments, we showed that for monomodal lognormally distributed failure times, the parallel EM test structure is a correct approach for the determination of the failure time of submicron interconnects. Moreover, the simulation experiments showed that the calculated activation energies for both series/ parallel structures are in good agreement with the proposed value of Ea (0.8 eV), used as an input parameter for the simulation experiments. This means that for monomodal lognormal distributed failure times, the parallel EM structure can be used for the determination of both Ea and n, leading to a very substantial reduction of the number of samples that are needed. However, simulation experiments have shown that the shape parameter for the parallel structure is smaller than expected and therefore a correction of this value is necessary.
K. Vanstreels et al. / Microelectronics Reliability 45 (2005) 753–759
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