An Improved Description of the Dielectric Breakdown in Oxides Based on a Generalized Weibull distribution

E. A. de Vasconcelos; E. F. da Silva; jr; Jr.; L. C. Malacarne; R. S. Mendes; S. Picoli; U. M. S. Costa; V. N. Freire

REVIEW

An Improved Description of the Dielectric Breakdown in Oxides Based on a Generalized Weibull distribution

U. M. S. Costa, V. N. Freire, L. C. Malacarne, R. S. Mendes, S. Picoli, Jr., E. A. de Vasconcelos, E. F. da Silva

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Materials Science · Submitted 22 Feb 2006 · 22 references

Pith's verdict: A mathematically clean but empirically under-supported application of q-Weibull to oxide breakdown; the extrapolation claim needs direct validation.

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CitationE. A. de Vasconcelos, E. F. da Silva, jr, Jr., L. C. Malacarne, R. S. Mendes, S. Picoli, U. M. S. Costa, V. N. Freire. "An Improved Description of the Dielectric Breakdown in Oxides Based on a Generalized Weibull distribution." arXiv:cond-mat/0602515 (2006). https://pith.science/paper/BUY2QZOR

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Claim paper

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Editor's note

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Colleague,

This is a short application paper: the authors re-derive the q-Weibull distribution as a gamma scale-mixture of a Weibull, introduce a q-Weibull plot for linearized fitting, and apply it to three published dielectric-breakdown datasets (experimental and simulated). The distribution itself and the gamma-mixture derivation are not new—Ref. [14] by three of the same authors already has them. The new piece is the application to tBD data and the q-Weibull plot. That is a modest but real contribution: the plot gives reliability engineers an easy way to test whether a dataset is consistent with a gamma mixture of Weibulls, and the fits in Figs. 1–3 are visibly better than plain Weibull fits.

The math in Sec. II checks out. Eq. (4) follows from the gamma mixture, and the CDF in Eq. (6) with the q-exponential is consistent. The connection to Tsallis statistics is interpretive rather than derivational; the paper itself hedges it as "implicit," which is honest.

The soft spots are in the empirical claims. The headline benefit—correct tBD extrapolation from the conventional experimental window without large samples—is asserted, not demonstrated. Fig. 1(b) fits a q-Weibull with q=1.32 to the full 4000-device dataset, then infers that extrapolation to 0.001% is correct. That is an extrapolation from a model fit, not a validation. The paper never fits q and β on high-percentile data and checks predicted low-percentile failure times against held-out data. Because q-Weibull has an extra parameter, a better in-sample fit is expected even if the tail shape is wrong. The straightness of the q-Weibull plot over the observed range does not establish the unobserved tail.

The area-scaling section has a separate weakness: the two areas are fitted with different q values (1.71 and 1.87 in Fig. 2). That undercuts the claim that a single fluctuation mechanism explains both. The gamma assumption for the fluctuation distribution is plausible but untested; the paper does not compare it with the Gaussian thickness fluctuation model that Wu et al. used to generate the same curvature.

There are no error bars on q or β, no model-selection statistic, and no discussion of how sensitive the extrapolation is to q. These are fixable, but they are central to the paper's practical claim. The citation pattern looks fine; Ref. [14] is the natural prior source and is cited.

Who is this for? Reliability physicists who work on oxide breakdown and want a simple parametric alternative to Weibull when thickness fluctuations are present. It deserves serious peer review, but the referee should ask for direct validation of the extrapolation claim and quantitative model comparison. As is, I would not accept it without those additions.

Recommendation: send to referees, with the expectation of revision.

Referee report

Significance

If the central claim were fully supported, the paper would be significant for reliability projection of ultra-thin oxides because it offers a simple parametric family with a plausible mixture interpretation and a straight-line plotting method. The algebraic derivation of the q-Weibull distribution from gamma fluctuations of the Weibull scale parameter is a useful and correct contribution, and the connection to Tsallis statistics is clearly laid out. However, the empirical validation is limited: q and beta are fitted to each dataset, the q-Weibull contains the Weibull as a special case, no error bars or model-selection statistics are provided, and the headline claim of correct extrapolation from a conventional window is not demonstrated because the model is fitted to the full observed percentile range. These deficiencies are addressable, so the manuscript's promise is clear but its current evidence is incomplete.

Major objections (3)

[Section III, Fig. 1(b) and text after Fig. 1] The central claim that the q-Weibull distribution permits improved tBD extrapolation from a conventional experimental window is not supported by the evidence presented. In Fig. 1, q=1.32 and beta are fitted to the full dataset containing 4000 devices and extending down to roughly 0.025% failure; the figure therefore demonstrates interpolation over the observed range, not extrapolation to 0.001% or below. A supporting test would fit the q-Weibull only to the high-percentile window (for example, 1% to 63%) and compare its predicted quantiles at 0.1%, 0.01%, and 0.001% against the observed tail, with the same test performed for the ordinary Weibull. Without such a restricted-window validation, the paper's advertised benefit is asserted rather than demonstrated.
[Section II, Eq. (2)] The gamma distribution for theta = x0^{-beta} is assumed without an empirical or mechanistic justification from oxide thickness fluctuations. The text references Wu et al. (Ref. [5]) as showing that Gaussian thickness fluctuations produce the observed slope variation, but the paper does not show that a gamma distribution for theta follows from Gaussian thickness statistics. The shape of the q-Weibull tail depends on this premise, so the straight-line q-Weibull plot and the extrapolation claims would not follow if the true fluctuation statistics are not gamma. The authors should test the gamma assumption directly against thickness-fluctuation data or derive it from a physical model.
[Section III, Figs. 2 and 3] The improved area scaling and improved fits are judged only by visual straightness, with q and beta fitted separately to each area or injection-polarity dataset. Since the q-Weibull reduces to the Weibull when q=1, an improved fit with an additional fitted parameter is expected even if the q-Weibull is not the correct model. Moreover, the fitted q values differ across the compared conditions (q=1.87 and q=1.71 for the two areas in Fig. 2; q=1.44 and q=1.93 for the two injection polarities in Fig. 3), which suggests the improvement is dataset-specific rather than a universal oxide-breakdown law. The paper should report parameter uncertainties, goodness-of-fit statistics, or a model-selection criterion, and should clarify whether q is expected to be a universal constant for area scaling.

Minor objections (6)

[Title] The title contains a typo: "Oxid es" should be "Oxides."
[Eqs. (5)-(6)] The notation shifts from q' and tilde{x0} in Eq. (5) to q and alpha in Eq. (6); the relations q = 1/(2-q') and alpha = tilde{x0}/(2-q')^{1/beta} should be stated explicitly before Eq. (6) to avoid confusion.
[Fig. 2 caption and axis label] The caption of Fig. 2 indicates ln[-ln_q[1-F_d(t)]] on the vertical axis, while the axis label in panel (a) appears as ln[-ln[1-F(t)]]; the notation should be made consistent.
[Section II, after Eq. (1)] The text gives the modal lifetime as "63,2%" but the standard decimal notation is "63.2%."
[Section III, q-Weibull plot procedure] The statement that the q parameter is obtained through a linear adjustment is underspecified, because computing ln_q requires knowing q in advance; the authors should describe the actual estimation procedure, such as a nonlinear fit or a search over q that maximizes linearity.
[References and data acquisition] The footnote in Ref. [19] indicates that experimental data were obtained by scanning published figures; this digitization step should be mentioned as a potential source of noise in the analysis. Reference [16] also contains typographical errors in its title.

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The paper

The paper claims

q-Weibull statistics, produced by gamma-fluctuating Weibull scale parameters, straighten oxide breakdown plots and enable low-percentile lifetime projections.

The reading

The paper sets out to show that the standard Weibull distribution misreads time-to-breakdown ($t_{\mathrm{BD}}$) data of ultra-thin oxides because it leaves out sample-to-sample fluctuations in oxide thickness. It derives a q-Weibull distribution by averaging the Weibull density over a gamma-distributed scale parameter, and shows that its cumulative plot, $\ln[-\ln_q(1-F_q(t))]$ versus $\ln t$, is a straight line. Fitted to experimental $t_{\mathrm{BD}}$ data from 4000 MOS capacitors, to two-area data from 6.9 nm oxides, and to percolation-model simulations, the q-Weibull fits better than the Weibull distribution and gives a straight-line projection down to failure fractions near 0.001%. If the claim holds, lifetime projection and area scaling for advanced gate stacks can be done from conventional experimental windows without thousands of devices.

What carries the argument

The machinery is a scale-mixture identity. The paper starts from the Weibull density $p(x;\theta)=\beta\theta x^{\beta-1}e^{-\theta x^\beta}$, treats $\theta=x_0^{-\beta}$ as random with gamma density $p(\theta)=\delta^r\theta^{r-1}e^{-\delta\theta}/\Gamma(r)$, and averages. The mixture is $P_{q'}(x)=\beta(2-q')\tilde x_0^{-\beta}x^{\beta-1}\exp_{q'}[- (x/\tilde x_0)^\beta]$, where $\exp_{q'}(a)\equiv[1-(1-q')a]^{1/(1-q')}$ for $1-(1-q')a\geq 0$ and zero otherwise, with $q'=(2+r)/(1+r)$ and $\tilde x_0^\beta=\delta/(r+1)$. Using the q-logarithm $\ln_q x\equiv(x^{1-q}-1)/(1-q)$ and the cumulative $F_q(t)=1-\exp_q[-(t/\alpha)^\beta]$, the identity $\ln[-\ln_q(1-F_q(t))]=\beta\ln t-\beta\ln\alpha$ is the entire source of the q-Weibull plot: it converts a curved Weibull projection into a straight line whose slope is the Weibull slope $\beta$ and whose intercept sets the lifetime scale $\alpha$. Since $q=1/(2-q')$, the fitted $q$ is a direct measure of the gamma shape parameter $r$, so the same parameter controls both the fluctuation width and the curvature of the breakdown plot.

Extended reading notes

Core claim

The central claim is that the curvature seen in standard Weibull plots of time-to-breakdown for ultra-thin oxides is a signature of modal-parameter fluctuations, and that averaging the Weibull distribution over a gamma distribution of the scale parameter produces a q-Weibull distribution that removes that curvature. With cumulative form $F_q(t)=1-\exp_q[-(t/\alpha)^\beta]$, the plot of $\ln[-\ln_q(1-F_q(t))]$ against $\ln t$ is a straight line of slope $\beta$. The paper shows that 4000-device $t_{\mathrm{BD}}$ data, which bend in an ordinary Weibull plot, fall on a q-Weibull line with $q=1.32$; that breakdown data from devices of area $10^{-4}$ and $10^{-5}\,\mathrm{mm}^2$ collapse better under the q-Weibull plot; and that percolation-model simulations for gate and substrate injection are better fitted by q-Weibull than by Weibull. The authors conclude that fluctuation of the modal value explains the deviation from Weibull behavior, connects the fitting parameter $q$ to Tsallis statistics, and permits correct extrapolation to very low failure percentiles from the conventional experimental window.

Load-bearing premise

The load-bearing premise is that the sample-to-sample fluctuations of the Weibull scale parameter are governed by a gamma distribution; if the real oxide-thickness fluctuations follow a different distribution, the q-Weibull straight-line plot and the extrapolation to 0.001% failures would not hold.

Editorial extensions

If this is right

Correct time-to-breakdown extrapolation to failure fractions as low as 0.001% can be made from the conventional experimental window, removing the need to test thousands of devices.
Area scaling becomes consistent: failure data from devices of different areas lie on the same q-Weibull straight line, as shown with two-area data, making reliability comparisons across device sizes reliable.
The fitted q parameter provides a physical measure of the spread of modal-parameter fluctuations, linking device-to-device variability to the entropic index q.
Ordinary Weibull analysis is recovered as the q=1 limit, so the framework generalizes rather than replaces existing reliability practice.
The same q-Weibull treatment should apply to high-k gate dielectrics, where equivalent-thickness fluctuations are expected to be significant.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not stated in the paper: fitting q independently for different oxide thicknesses or electric fields and using $r=1/(q-1)$ would predict the gamma shape of the thickness-fluctuation distribution, a testable link between the model and direct roughness measurements.
The same gamma-mixture construction should also describe charge-to-breakdown ($Q_{\mathrm{BD}}$) and breakdown-voltage statistics, since the argument is about modal-parameter fluctuations generally rather than about $t_{\mathrm{BD}}$ specifically.
The derivation implies a pure power-law tail $x^{-(1+r\beta)}$ for large times; therefore the low-failure-percentile behavior is fixed by the same parameter that sets the curvature of the q-Weibull plot, a relation that large breakdown experiments could check directly.
If the scale parameter drifts during stress rather than being fixed per device, the q-Weibull plot would bend in long-time data; looking for such bending would distinguish static sample-to-sample variability from progressive degradation.

Original abstract

In this work, we address modal parameter fluctuations in statistical distributions describing charge-to-breakdown $(Q_{BD})$ and/or time-to-breakdown $(t_{BD})$ during the dielectric breakdown regime of ultra-thin oxides, which are of high interest for the advancement of electronic technology. We reobtain a generalized Weibull distribution ($q$-Weibull), which properly describes $(t_{BD})$ data when oxide thickness fluctuations are present, in order to improve reliability assessment of ultra-thin oxides by time-to-breakdown $(t_{BD})$ extrapolation and area scaling. The incorporation of fluctuations allows a physical interpretation of the $q$-Weibull distribution in connection with the Tsallis statistics. In support to our results, we analyze $t_{BD}$ data of SiO$_2$-based MOS devices obtained experimentally and theoretically through a percolation model, demonstrating an advantageous description of the dielectric breakdown by the $q$-Weibull distribution.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.07 · human participants 0.02 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.33 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In this work, we address modal parameter fluctuations in statistical distributions describing charge-to-breakdown $(Q_{BD})$ and/or time-to-breakdown $(t_{BD})$ during the dielectric breakdown regime of ultra-thin oxides, which are of high interest for the advancement of electronic technology. We reobtain a generalized Weibull distribution ($q$-Weibull), which properly describes $(t_{BD})$ data when oxide thickness fluctuations are present, in order to improve reliability assessment of ultra-thin oxides by time-to-breakdown $(t_{BD})$ extrapolation and area scaling. The incorporation of fluctuations allows a physical interpretation of the $q$-Weibull distribution in connection with the Tsallis statistics. In support to our results, we analyze $t_{BD}$ data of SiO$_2$-based MOS devices obtained experimentally and theoretically through a percolation model, demonstrating an advantageous description of the dielectric breakdown by the $q$-Weibull distribution.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this work, we address modal parameter fluctuations in statistical distributions describing charge-to-breakdown $(Q_{BD})$ and/or time-to-breakdown $(t_{BD})$ during the dielectric breakdown regime of ultra-thin oxides, which are of high interest for the advancement of electronic technology.	background 0.66	0.31	0.06	asserted 0.66
s1	We reobtain a generalized Weibull distribution ($q$-Weibull), which properly describes $(t_{BD})$ data when oxide thickness fluctuations are present, in order to improve reliability assessment of ultra-thin oxides by time-to-breakdown $(t_{BD})$ extrapolation and area scaling.	contribution 0.94	0.94	0.59	asserted 1.00
s2	The incorporation of fluctuations allows a physical interpretation of the $q$-Weibull distribution in connection with the Tsallis statistics.	interpretation 1.00	0.72	0.12	asserted 0.98
s3	In support to our results, we analyze $t_{BD}$ data of SiO$_2$-based MOS devices obtained experimentally and theoretically through a percolation model, demonstrating an advantageous description of the dielectric breakdown by the $q$-Weibull distribution.	result 0.46	0.92	0.32	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.62 · evidence from analysis of existing data 0.70.

What was examined, and what was not

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arXiv listing abstract v1

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full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

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References

Works cited (22)

[14]

S. Picoli Jr., R. S. Mendes, and L. C. Malacarne, Physica A 324, 678 (2003)

View on Pith 2003
[5]

E. Y. Wu, J. H. Stathis and L-K. Han, Semicond. Sci. Technol. 15, 425 (2000)

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[1]

The International Technology Roadmap for Semiconduc- tors (Semiconductor Industry Association, San Jose, CA, 2002), see also http://public.itrs.net for the most recent updates

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[2]

G. D. Wilk, R. M. Wallace, and J. M. Anthony, J. Appl. Phys. 89, 5243 (2001); R. A. MacKee, F. J. Walker, M. F. Chisholm, Phys. Rev. Lett. 81, 3014 (1998); R. A. MacKee, F. J. Walker, M. F. Chisholm, Science 293, 468 (2001); V. Fiorentini and G. Gulleri, Phys. Rev. Lett. 89, 6101 (2002); D. Muller, T. Sorsch, S. Moccio, F. Baumann, K. Evans-Lutterdodt, an...

View on Pith 2001
[3]

L. Niemeyer, L. Pietronero, and H. J. Wiesmann, Phys. Rev. Lett. 52, 1033 (1984); M. A. F. Gomes, E. F. da Silva Jr., and J. A. Aguiar, Semicond. Sci. Techn., 10, 1037 (1995)

View on Pith 1984
[4]

H. J. Wiesmann, and H. R. Zeller, J. Appl. Phys. 60, 1770 (1986)

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[6]

R. Degreave, B. Kaczer and G. Groeseneken, Microelec- tron. Reliab. 39, 1445 (1999)

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[7]

M. L. Green, E. P. Gusev, R. Degreave and E. L. Gar- funkel, J. Appl. Phys. 90, 2057 (2001)

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[8]

D. R. Wolters, and J. F. Verwey, in Instabilities in Silicon Devices, edited by G. Barbottin and A. Vapaile (Elsevier, North Holland,1986), p. 315

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[9]

B. V. Gnedenko and V. Yu. Korolev, Random Summa- tion: Limit Theorems and Applications (CRC, Boca Ra- ton, 1996)

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[10]

G. Wilk and Z. Wlodarczyk, Phys. Rev. Lett 84, 2770 (2000)

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[11]

G. Wilk and Z. Wlodarczyk, Physica A 290, 55 (2001)

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[12]

C. Beck, Phys. Rev. Lett. 87, 180601 (2001)

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[13]

G. M. Batanov et al , Plasma Phys. Rep. 28, 111 (2002)

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[15]

R. Degraeve, B. Kaczer, and G. Groeseneken, Semicond. 5 Sci. Technol. 15, 436 (2000)

View on Pith 2000
[16]

N. L. Johnson and S. Kotz, Distribution in Statistics: Continous Univariante Distribution-1 , New York, John Wiley and Sons, 1970

View on Pith 1970
[17]

C. Tsallis, J. Stat. Phys. 52, 479 (1988); C. Tsallis, R. S. Mendes and A. R. Plastino, Physica A 261, 534 (1998); C. Tsallis, in: S. Abe, Y. Okamoto (Eds.), Nonex- tensive Statistical Mechanics and Its Application , Lec- ture Notes in Physics, Springer, Heidelberg, 2001; See http://tsallis.cat.cbpf.br for periodically updated bib li- ography on the subject

View on Pith 1988
[18]

Strictly speaking, this average process only leads to q > 1,which is just the relevant case to this work

View on Pith
[19]

The experimental data were obtained through direct scanning of the ﬁgures and then recovering the data using the Opti-Graph program

View on Pith
[20]

A. Teramoto,H. Umeda, K. Azamawari, K. Kobayashi, K. Shiga, J. Komori, Y. Ohno, and A. Shigetomi, Micro- electron. Reliab., 41, 47 (2001)

View on Pith 2001
[21]

for gate (squares) and substrate (dots) injection adju sted with Weibull distributions; (b) the same data as in (a) but depicted in a q-Weibull plot showing improved ﬁtting. and exp q′ (−a x) ≡ 0 if 1 − (1 − q′) a x < 0, with q′ = (2 + r)/ (1 + r) and ˜xβ 0 = [δ/ (r + 1)]. By comparing Eq.(1) with Eq.(5) we verify that Pq′ (x) is a generalization of the W...

View on Pith
[22]

S. S. Sombra, U. M. S. Costa, V. N. Freire, E. A. de Vas- concelos, E. F. da Silva Jr., Physica A 305, 351 (2002)

View on Pith 2002

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [14]

S. Picoli Jr., R. S. Mendes, and L. C. Malacarne, Physica A 324, 678 (2003)

View on Pith 2003

[2] [5]

E. Y. Wu, J. H. Stathis and L-K. Han, Semicond. Sci. Technol. 15, 425 (2000)

View on Pith 2000

[3] [1]

The International Technology Roadmap for Semiconduc- tors (Semiconductor Industry Association, San Jose, CA, 2002), see also http://public.itrs.net for the most recent updates

View on Pith 2002

[4] [2]

G. D. Wilk, R. M. Wallace, and J. M. Anthony, J. Appl. Phys. 89, 5243 (2001); R. A. MacKee, F. J. Walker, M. F. Chisholm, Phys. Rev. Lett. 81, 3014 (1998); R. A. MacKee, F. J. Walker, M. F. Chisholm, Science 293, 468 (2001); V. Fiorentini and G. Gulleri, Phys. Rev. Lett. 89, 6101 (2002); D. Muller, T. Sorsch, S. Moccio, F. Baumann, K. Evans-Lutterdodt, an...

View on Pith 2001

[5] [3]

L. Niemeyer, L. Pietronero, and H. J. Wiesmann, Phys. Rev. Lett. 52, 1033 (1984); M. A. F. Gomes, E. F. da Silva Jr., and J. A. Aguiar, Semicond. Sci. Techn., 10, 1037 (1995)

View on Pith 1984

[6] [4]

H. J. Wiesmann, and H. R. Zeller, J. Appl. Phys. 60, 1770 (1986)

View on Pith 1986

[7] [6]

R. Degreave, B. Kaczer and G. Groeseneken, Microelec- tron. Reliab. 39, 1445 (1999)

View on Pith 1999

[8] [7]

M. L. Green, E. P. Gusev, R. Degreave and E. L. Gar- funkel, J. Appl. Phys. 90, 2057 (2001)

View on Pith 2001

[9] [8]

D. R. Wolters, and J. F. Verwey, in Instabilities in Silicon Devices, edited by G. Barbottin and A. Vapaile (Elsevier, North Holland,1986), p. 315

View on Pith 1986

[10] [9]

B. V. Gnedenko and V. Yu. Korolev, Random Summa- tion: Limit Theorems and Applications (CRC, Boca Ra- ton, 1996)

View on Pith 1996

[11] [10]

G. Wilk and Z. Wlodarczyk, Phys. Rev. Lett 84, 2770 (2000)

View on Pith 2000

[12] [11]

G. Wilk and Z. Wlodarczyk, Physica A 290, 55 (2001)

View on Pith 2001

[13] [12]

C. Beck, Phys. Rev. Lett. 87, 180601 (2001)

View on Pith 2001

[14] [13]

G. M. Batanov et al , Plasma Phys. Rep. 28, 111 (2002)

View on Pith 2002

[15] [15]

R. Degraeve, B. Kaczer, and G. Groeseneken, Semicond. 5 Sci. Technol. 15, 436 (2000)

View on Pith 2000

[16] [16]

N. L. Johnson and S. Kotz, Distribution in Statistics: Continous Univariante Distribution-1 , New York, John Wiley and Sons, 1970

View on Pith 1970

[17] [17]

C. Tsallis, J. Stat. Phys. 52, 479 (1988); C. Tsallis, R. S. Mendes and A. R. Plastino, Physica A 261, 534 (1998); C. Tsallis, in: S. Abe, Y. Okamoto (Eds.), Nonex- tensive Statistical Mechanics and Its Application , Lec- ture Notes in Physics, Springer, Heidelberg, 2001; See http://tsallis.cat.cbpf.br for periodically updated bib li- ography on the subject

View on Pith 1988

[18] [18]

Strictly speaking, this average process only leads to q > 1,which is just the relevant case to this work

View on Pith

[19] [19]

The experimental data were obtained through direct scanning of the ﬁgures and then recovering the data using the Opti-Graph program

View on Pith

[20] [20]

A. Teramoto,H. Umeda, K. Azamawari, K. Kobayashi, K. Shiga, J. Komori, Y. Ohno, and A. Shigetomi, Micro- electron. Reliab., 41, 47 (2001)

View on Pith 2001

[21] [21]

for gate (squares) and substrate (dots) injection adju sted with Weibull distributions; (b) the same data as in (a) but depicted in a q-Weibull plot showing improved ﬁtting. and exp q′ (−a x) ≡ 0 if 1 − (1 − q′) a x < 0, with q′ = (2 + r)/ (1 + r) and ˜xβ 0 = [δ/ (r + 1)]. By comparing Eq.(1) with Eq.(5) we verify that Pq′ (x) is a generalization of the W...

View on Pith

[22] [22]

S. S. Sombra, U. M. S. Costa, V. N. Freire, E. A. de Vas- concelos, E. F. da Silva Jr., Physica A 305, 351 (2002)

View on Pith 2002