Solid-State Electronics 210 (2023) 108812 Contents lists available at ScienceDirect Solid State Electronics journal homepage: www.elsevier.com/locate/sse Modeling and simulation of successive breakdown events in thin gate dielectrics using standard reliability growth models E. Miranda a, *, F.L. Aguirre a, E. Salvador a, M.B. González b, F. Campabadal b, J. Suñé a a Universitat Autònoma de Barcelona, 08193 Cerdanyola del Valles, Spain b Institut de Microelectrònica de Barcelona, IMB-CNM, 08193 Cerdanyola del Valles, Spain A R T I C L E I N F O A B S T R A C T Handling Editor: Sorin Cristoloveanu The application of constant electrical stress to a metal–insulator-semiconductor (MOS) or metal–insulator-metal (MIM) structure can generate multiple breakdown events in the dielectric film. Very often, these events are Keywords: detected as small jumps in the current–time characteristic of the device under test and can be treated from the Dielectric breakdown stochastic viewpoint as a counting process. In this letter, a wide variety of standard reliability growth models for Oxide breakdown this process are assessed in order to determine which option provides the best simulation results compatible with MOS the experimental observations. For the generation of the breakdown event arrivals, two alternative stochastic MIM Oxide reliability methods for the power-law Poisson process are investigated: first, the inversion algorithm for the cumulative distribution function and second, an on-the-fly method based on the so-called rejection algorithm. Though both methods are equivalent, the first one is more appropriate for data analysis using spreadsheet calculations while the second one is highly suitable for circuit simulation environments like LTSpice. The connection of the selected nonhomogeneous Poisson process with the Weibull model for dielectric breakdown is also discussed. 1. Introduction distributions but for processes instead of data. Moreover, they can be identified with stochastic diffusive processes with deterministic average Application of constant or ramped voltage/current stress to a MIM or trends. In this work, we show the distinctive behavior exhibited by MIS structure can generate successive breakdown (BD) events in the different reliability growth models (polynomial, exponential, logarith­ dielectric film that are often detected as jumps in the conduction char­ mic, power-law), discuss their pros and cons and analyze in detail the acteristic of the device. This kind of experiment is the basis of thin oxide best option compatible with the experimental observations. We report reliability analysis and is often carried out for assessing the integrity of two methods for the generation of succesive BD events according to a the dielectric layer for a given technology. In this letter, we will exclu­ power-law Poisson process: first, the inversion algorithm for the cu­ sively focus the attention on constant voltage stress (CVS). In principle, mulative distribution function and second, an on-the-fly method based when the number of investigated devices is large enough, the most on the so-called rejection algorithm. Though both methods provide complete information about the arrival time of successive BD events is similar results, the first one is more appropriate for data analysis using obtained from the order statistics analysis. It has been demonstrated that spreadsheet calculations while the second is highly suitable for circuit the time-to-BD for the first, second, third and so on events occurring in a simulators. In this work, LTspice from Analog Devices will be considered device subject to CVS can be succesfully modeled by the clustering to this purpose. We will also show how the proposed approach can be distribution or alternatively, when variability is under control, by one of extended for simulating different generation laws other than the one its limits, i.e. the Weibull distribution [1]. However, when the number of specifically addressed here. This opens the path to the investigation of measurements is small and simulated curves with average features are more complex generation process involving acceleration factors or self- required, a first order approach consists in considering standard reli­ regulatory mechanisms. ability growth models [2]. In general, these models describe how the system reliability changes over time during the testing process and in a broad sense they are equivalent to the parametric probability * Corresponding author. E-mail address: enrique.miranda@uab.cat (E. Miranda). https://doi.org/10.1016/j.sse.2023.108812 Available online 28 October 2023 0038-1101/© 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). E. Miranda et al. Solid State Electronics 210 (2023) 108812 Fig. 1. a) i-v and b) i-t curves after and before the first bd event. notice the effect of electrical stress. Fig. 2. Evolution of the current as a function of time for a constant applied bias Fig. 3. Two-phase degradation showing the non-volatility character of the (7.5 V). The inset shows a detail of the initial steps and the arrival time of the generated failures. The degradation was stopped for several seconds and BD events. then resumed. 2. Devices and measurements until the occurrence of the failure event. Fig. 2 shows the staircase evolution of the current during degrada­ Measurements shown in this work were obtained with Al/Al2O3/p- tion at constant voltage (7.5 V). As illustrated in the inset of Fig. 2, ti type Si capacitors. The Al2O3 layer was grown (10 nm-thick) by the corresponds to the BD time for event i. Additionally, Fig. 3 shows that atomic layer deposition (ALD) technique. Details about the sample the failure events are non-volatile and irreversible. In this case, the fabrication, characterization and reliability characteristics can be found degradation was stopped (V = 0 V) for several seconds and then in [3,4]. Fig. 1.a and 1.b illustrate typical current–voltage (I-V) and resumed. The current magnitude at the onset of the second phase cor­ current–time (I-t) curves for the device under test before and after the responds to the final current magnitude of the first phase. first BD event, respectively. As shown in Fig. 1.a, the application of successive voltage ramps with increasing maximum voltage generates 3. Exploratory analysis of reliability growth models electron trapping in the structure that ends with the formation of a BD path and therefore with a huge increase of the leakage current that flows In order to deal with the stepwise behavior of the I-t characteristics through the device. This corresponds to a hard BD failure mode. In the exhibited by our devices, we consider the random arrival of BD events as case of Fig. 1.b, where a constant voltage is applied, the leakage current a point process with Λ(t) the number of events at time t. In particular, initially decreases in agreement with the results shown in Fig. 1.a, but non-homogeneous Poisson processes (NHPP) with time-varying rate then starts increasing, presumably because of electron traps generation, 2 E. Miranda et al. Solid State Electronics 210 (2023) 108812 Fig. 5. inversion algorithm for the generation of bd times. b) number of events as a function of time. the grey lines corresponds to 50 runs. function (intensity function) λ(t) = dΛ(t)/dt are assumed here. Since NHPP is a Markovian process, we are implicitly assuming that the sys­ tem has no memory (past events are not correlated with future events) and disjoint time intervals are independent. Six standard models are analyzed in this work as possible candidates for Λ(t) with the aim of determining the best option for the available data set: polynomials of degree 1 to 3, exponential: a⋅exp(b⋅t), logarithmic: a⋅ln(t) + b, and power function: a⋅tb, the latter so-called Duane/Crow-AMSAA model or power- law process (PLP). Parameters were first extracted using the least- squares method (LSM). Notice that these models refer to the number of BD events and not directly to the current magnitude. Fig. 4.a shows fitting results with the polynomials. Clearly, the best result is obtained with the cubic expression. Linear and quadratic expressions not only depart from the experimental data but also yield negative values for the first few events. The problem with the cubic interpolation is that if not used with care it can generate unexpected results both for low and high number of events (see Fig. 4.b). This strongly depends on the number of data points considered for the fitting exercise. As shown in Fig. 4.c, exponential and logarithmic models can be ruled out for obvious rea­ sons. The obtained curvatures are incompatible with the experimental dataset. Moreover, for the exponential case, the number of events at the outset of the experiment is different from zero. For the logarithmic case, the number of events is negative at the outset. According to the results shown in Fig. 4.b and 4.c, PLP model offers the most acceptable repre­ sentation of the available data. 4. Analysis of the power-law model As shown in the previous Section, the PLP model (Λ(t) = a⋅tb) de­ serves special attention. It does not only provide a high correlation co­ efficient (R2) but also always positive values for the number of events. Extracted parameters from Fig. 4.c are: a = 5.4⋅10-2 and b = 1.6 for the time expressed in seconds. In particular, b > 1 indicates reliability degeneration and an increasingly arrival rate of BD events with time. Fig. 4. Model fitting and correlation coefficients: a) polynomial, b) cubic and Later, we will show the effects of the opposite behavior. Being PLP a power-law (less datapoints corresponding to a different measurement), c) log­ NHPP, the probability for a number k of events at time T can be calcu­ arithmic, exponential, and power-law. lated from: ak T bk [ ] P(N(T) = k ) = exp − aT b (1) k! so that the waiting time to the next event, given an event at time T, has 3 E. Miranda et al. Solid State Electronics 210 (2023) 108812 problem. In addition, the Cramér-von Mises goodness statistics can be used to identify whether the system strictly follows PLP or not. Confi­ dence intervals for the parameter estimates are also provided in [6]. 5. SPICE simulations of multiple BD events In this Section, we pay particular attention to the generation of BD events in the framework of the LTspice simulator. Since a NHPP is assumed here for the generation process, the probability for the occur­ rence of a new BD event is simply given by the Poisson rule: P(Λ(t + Δt) − Λ(t) = 1 ) = λ(t) × Δt (4) where Δt is the time interval considered and λ the intensity of the process. This generation rule can be expressed in LTspice as a rejection al­ gorithm (see Fig. 6): N(t) = int{SR × idt[SR × rand(SR × t + i) < ddt(Λ(t) ) ) ] } (5) Fig. 6. LTSpice script for a PLP process. 50 curves are generated with identical model parameters. Model parameters are those reported in Fig. 4.c. Output M where int is the integer part of the number, idt the time integral, ddt the corresponds to the average number of events while output PLP to the corre­ time derivative, rand the random number generator, and t the simulation sponding stochastic result. For the sake of completeness, two measurement time. SR is the sampling rate and i the random number generator seed. directives were included for the analysis. rand(SR⋅t + i) in (5) generates SR uniformly distributed random numbers in the range [0,1] per second. The index i shifts the argument of the cumulative distribution function: the function for every simulation run. ddt(Λ)/SR comes from the PLP { [ ]} process, Eqn. (4). When the logic expression < is true, a unity is added to FT (t) = 1 − exp − a (T + t)b − T b (2) the integral counter represented by idt. Now, this counter value times SR scales the number of events to the expected value. Fig. 7.a-c show the Expression (2) can be straightforwardly used for calculating the suc­ results obtained with the LTspice simulator. Fig. 7.a shows the stochastic cessive BD arrival times using the inversion algorithm described in process as well as its average trend, Fig. 7.b the time required to reach N Table 1. This approach is remarkably easy to implement in a spread­ BD events (ST) as a function of the simulation run and the number of sheet. Simulation results obtained with this method are shown in Fig. 5. events generated at time T (NE), and Fig. 7.c the shape of the stochastic paths around the average curve. This latter plot provides information TABLE 1. Inversion algorithm for the power-law model about the evolution of the dispersion of the generated characteristics when the average trend is eliminated (detrended time series). Just for Inversion algorithm for the power-law model comparison, we have included a second generation model (not related to 1 Input exp. data and number of events N our devices) in order to demonstrate the flexibility of the proposed 2 Fit Eq. (2) to the exp. data to define a and b approach. The model considered is: 3 define ti-1 = 0, i = 0 4 for i < N then Λ(t) = a × [1 − exp( − b × t) ] (6) 5 generate ui form Unif(0,1) 6 )1 which expresses a self-saturation process with amplitude a and gener­ 1 ln(ui ) b ation rate b. The corresponding results obtained with a = 100 and b = ( calculate ti = ti−b 1 − a 7 define ti-1 = ti 0.05 are illustrated in Fig. 7.d-f. This process is typical of a first-order 8 i=i+1 rate equation with limited number of events. 9 end 10 Output t1, t2, t3, t4,…,tN 6. Conclusions The generation of breakdown events in thin insulating layers (Al2O3) Interestingly, for PLP, the time-to-first BD (T = 0 in expression (2)) is was investigated. Different standard reliability growth models were Weibull-distributed which agrees with the weakest link character of evaluated and discussed. The best option compatible with the experi­ oxide breakdown. Moreover, PLP is also consistent with the generation mental data seems to be the Power-law Poisson process. We have also rate of defects in thin oxide films [4]. However, it is worth mentioning shown how to calculate the successive breakdown times using two that though LSM provides reasonable parameter values, the method alternative approaches: inversion of the cumulative distribution func­ requires further confirmation. In this regard, parameter extraction can tion (more appropriate for spreedsheet calculators) and the rejection also be carried out using the maximum likelihood estimates (MLE). For method (suitable for circuit simulators). We also discussed how the time-truncated data, the PLP parameters read: rejection method can be easily applied to other reliability growth models N N representing different failure mechanisms. a= b= ( ) (3) tNb ∑N tN i=1 ln ti Declaration of Competing Interest In our case, a = 3.4⋅10-2 and b = 1.7 are obtained for N = 83 and t83 = The authors declare the following financial interests/personal re­ 88.1 s, which closely agree with the extracted parameters using LSM. lationships which may be considered as potential competing interests: Notice that both the scale parameter a and shape parameter b depend on Enrique Miranda reports financial support was provided by European the last observed BD time so that they should be strictly treated as un­ Metrology Programme for Innovation and Research (EMPIR). known random variables. However, as reported in [5], Bayesian esti­ mates using the Higgins-Tsokos loss function can help to overcome this 4 E. Miranda et al. Solid State Electronics 210 (2023) 108812 Fig. 7. a) and d): number of generated events using the plp and self-saturation processes, respectively. b) and e): results from the measurement directives, st: time-to- 50 events, ne: number of events at 50 s. c) and f): Stochastic paths around the average trend calculated as V(PLP)-V(M), see Fig. 6. Data availability Ciencia e Innovación, Spain. M.B.G. acknowledges the Ramón y Cajal grant No. RYC2020-030150-I. E.S acknowledges the Departament de The authors do not have permission to share data. Recerca i Universitats de la Generalitat de Catalunya for the 2020 FISDU 00261 scholarship. Acknowledgements References E.M. and J.S. acknowledge the support provided by the European project MEMQuD, code 20FUN06, which has received funding from the [1] Muñoz J, et al. IEEE EDL 2020;41:1770–3. [2] Jokiel-Rokita A, et al. in Selected Stochastic Models in Reliability. Kapital Ludzki 2021. EMPIR programme co-financed by the Participating States and from the [3] Campabadal F, et al. J Vac Sci Technol B 2011;29. European Union’s Horizon 2020 research and innovation programme. [4] Conde A, et al. Solid St Electron 2012;71:48–52. The authors also thank projects 20225AT012 from CSIC, PID2022- [5] F. Alenezi et al., DOI:10.48550/arXiv.2002.00351. [6] J. Chumnaul. Theses and Dissertations 2700 (2019). Mississippi State University. 139586NB-C41 and PID2022-139586NB-C42 from the Ministerio de 5