weibull breakdown characteristics and oxide thickness uniformity

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 47, NO. 12, DECEMBER 2000 2301

Weibull Breakdown Characteristics and OxideThickness Uniformity

Ernest Y. Wu, Member, IEEE, Edward J. Nowak, Member, IEEE, Rolf-Peter Vollertsen, and L.-K. Han, Member, IEEE

Abstract—In this work, we investigated both experimentally andnumerically the impact of macroscopic oxide thickness uniformityon Weibull breakdown characteristics for both Weibull parame-ters, namely, the characteristic times and Weibull slopes over awide range of oxide thicknesses. We report the abnormal charac-teristics of the Weibull time-to-breakdown distributions and non-Poisson area scaling behavior observed on ultrathin oxides. Twonumerical methods using the parameters obtained from two inde-pendent sets of experimental results are developed to quantitativelyexplain these effects in the context of current modulation due tooxide thickness variation. The relationship between time-to-break-down and charge-to-breakdown distributions has been clarifiedand established. It is found that without proper treatment of theseeffects, the use of Weibull slopes at higher failure percentiles canlead to erroneous and pessimistic reliability projection. Further-more, we perform a detailed full-scale Monte Carlo analysis toevaluate the impact of thickness variation on low-percentile break-down distributions and their sensitivity to the thickness depen-dence of the times-to-breakdown and Weibull slopes.

Index Terms—Gate dielectric, oxide, reliability.

I. INTRODUCTION

RECENT work has indicated the Weibull slopes be-come shallower with decreasing oxide thickness

[1], [2]. The shallower Weibull slopes impose a further con-straint on oxide scaling [2], [3]. Severe impact of Weibull slopeson reliability projection has recently been discussed [3], [4]. Forreliability projection, depending on the specified requirement, asmaller can result in a decrease of two to three orders of mag-nitude in projected lifetime relative to thick oxides with a muchlarger . Based on the percolation theory, the smallerfor ultra-thin oxides is attributed to a drastic decrease as well as a largerspread in critical defect density at which breakdown occurs [1],[2]. Therefore, accurate determination of Weibull slopes is alsoessential to the fundamental understanding of the oxide break-down process [1], [2]. Furthermore, time-to-breakdowndistributions rather than charge-to-breakdown have beentraditionally used for oxide reliability characterization. The re-lationship between and distributions has never beeninvestigated in particular for ultrathin oxides in terms of theWeibull slopes.

Manuscript received March 20, 2000; revised June 16, 2000. The review ofthis paper was arranged by Editor G. Groeseneken.

E. Y. Wu and E. J. Nowak are with the IBM Microelectronics Division, EssexJunction, VT 05452 USA (e-mail: [email protected]).

R.-P. Vollertsen is with the Infineon Technologies, Essex Junction, VT 05452USA.

L.-K. Han is with the IBM Semiconductor Research and Development Center,Hopewell Junction, NY 12533 USA.

Publisher Item Identifier S 0018-9383(00)10392-2.

Currently, there is considerable confusion regarding whetheroxide thickness uniformity can fundamentally improve oxidereliability [5], [6]. Several important aspects regarding theimpact of uniformity on oxide reliability should firstbe clarified. First, the macroscopic uniformity, such assample-to-sample thickness variation, is different from themicroscopic uniformity such as surface roughness and inhomo-geneity within a sample. Both our work [4] and the work of [5]deal with the macroscopic variation,not the microscopicuniformity. Second, the overall effect of macroscopic unifor-mity on oxide reliability depends on the failure percentiles ofpractical interest [4]. Third, to correctly account for the effectof macroscopic thickness uniformity, a detailed knowledge ofhow fast and degrade with decreasing should bedetermined. Fourth, in a manufacturing environment, a certain

tolerance must be considered. This particular issue hasrecently been treated by Hunter for 4.0-nm oxide, where it wasconcluded that a tight thickness control is unwarranted [7].Many of the above issues have not been adequately addressed,both theoretically and experimentally in [5] and [6].

In this paper, after a brief description of experimental condi-tions, we present in Section III the experimental observationsof and distributions with and without variationsfor both thin and thick oxides. A discussion of numerical sim-ulation and the results will be given in Section IV. Section Vcompares the experimental data with simulation results. Then,in Section VI, we explore the impact of variation on oxidereliability for a wide range of thicknesses and sensitivity pa-rameters. Finally, we conclude that although very uniformdistribution eases the accurate determination of Weibull slopesat high percentiles, at low percentiles realistic variation indo not significantly degrade the reliability. Hence, we demon-strate that further reduction of variation beyond reasonabletolerances cannot necessarily improve gate oxide reliability. Itshould be emphasized that this work is not intended to addressthe issues such as microscopic uniformity within a sample andits impact on oxide reliability.

II. EXPERIMENTAL

A variety of capacitors with both ppoly/n-substrate capac-itors as well as nFET and pFET devices were fabricated usingthe standard CMOS process for ranging from 2.4 nm to4.2 nm. The nFET and pFET structures are designed with dif-fusions surrounding the gate area. During the stress, a bias isapplied to the gate while both diffusions and the substrate orn-well are grounded to form inversion layers. For pn capaci-tors, substrate injection is used for stress. Both NO oxides and

0018–9383/00$10.00 © 2000 IEEE

2302 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 47, NO. 12, DECEMBER 2000

Fig. 1. T andQ distributions stressed atV = 3.9 V and 140 C inNFET inversion mode.

oxides grown on nitrogen implanted Si substrates (NI/I ox-ides) were used in this work. A constant voltage stress techniquewas adopted with initial breakdown defined as oxide breakdownwhether soft or hard. Both breakdown modes have been shownto have the same failure mechanism [8], [9]. Each breakdowndistribution contains at least 40 to 100 samples, and sometimesas many as 4000 samples.

III. EXPERIMENTAL OBSERVATIONS

Fig. 1 shows the Weibull and distributions (NFETinversion) for N O oxides with a typical oxide thickness of 2.48nm. Four different areas were used with a sample size from 60to 100 capacitors for each case. Fig. 1 indicates that Weibullslopes of distributions are larger than those of dis-tributions by about 23%. Using the maximum likelihood esti-mation for 95% confidence level [10], we obtained the inter-vals of Weibull slopes to be about 30% which is comparable tothe difference between and distributions. Therefore,one cannot definitively conclude that the distributions aresteeper than the distributions although there seems to besuch a trend. Using Poisson random statistics [3], [11]–[13], wecan translate the and distributions for different areasto a reference area of cm as given in Fig. 2. It canbe seen that the distributions do not merge into a singledistribution as expected. However, the translated distribu-tions for different areas overlay relatively well indicating thatthe property of random statistics is well preserved under areascaling. These results appear to suggest the normalizeddis-tributions fail to follow random statistics while the datado. Notice that at lower percentiles , data appear to con-verge. Furthermore, a close inspection of the distributionsin Fig. 2 reveals that there is a change of slope in the dis-tribution from higher to lower percentile in contrast to thedistributions for which such effects are less evident. A similarnon-Poisson area scaling effect of distributions for PFETdevices in inversion is shown in Fig. 3. Note that reasonablyconverged distributions are also observed in this case. Itshould be pointed out that the breakdown distributions exhibita variety of features in terms of shapes and apparent distortions

Fig. 2. TranslatedT andQ distributions of Fig. 1 using Poisson randomstatistics.

Fig. 3. TranslatedT andQ distributions stressed atV = �3.9 V and140 C for PFET (inversion) for the structures with the same areas as those inFig. 1.

which do not appear in the distributions with much larger samplesize as discussed later.

To further investigate these phenomena, we plot the normal-ized and distributions in Fig. 4 for N I/I oxides of2.45 nm. With the small sample size of about 50 chips for eachdistribution, it is probably safe to conclude that these distri-butions show similar slopes and follow well-behaved Poissonrandom statistics. It is clear the normalized distributiondoes not exhibit any of the abnormal characteristics seen inFigs. 2 and 3. It is known that the nitrogen implant (NI/I)process can reduce oxide growth rate, which in turn leads tomuch better uniformity and less spread in the tunnelingcurrent [14]. The normalized distributions of four differentareas are given in Fig. 5 for NO oxides with a of 4.2nm. Once again all four distributions overlay well underPoisson area scaling. Fig. 6 attempts to compare the breakdowndistributions for NO and N I/I oxides using a sample size of upto 100 capacitors. One could conclude that distribution ofN O oxides is shallower than that of NI/I oxides. It is known

WU et al.: WEIBULL BREAKDOWN CHARACTERISTICS 2303

Fig. 4. TranslatedT andQ distributions stressed atV = 3.9 V and140 C for N I/I oxide in NFET inversion mode.

Fig. 5. TranslatedT distributions for N O oxide withT = 4.2 nmstressed atV =5.4 V and 140 C.

but often not appreciated that Weibull slopes are notoriously dif-ficult to measure to a satisfactory accuracy with a very limitednumber of samples [3], [10]. We will show that the above con-clusion based on even this large sample size is unwarranted.

The current density distributions are compared in Fig. 7 forthe N O oxide and the N I/I oxide presented in Figs. 2 and 4with 2.4 to 2.5 nm. We also include in the figure the caseof thick N O oxide (4.2 nm) for comparison. The currents weretaken to be the initial on-stress currents, and normalized to theircorresponding mean values. The histograms are approximatedwith Gaussian distributions. It can be seen that the spread of thecurrent density distribution for the thin NO oxide is about tentimes as large as those of thin NI/I oxide and the thicker NOoxide. As explained in the next section, it is this current modu-lation effect due to variation which plays an important rolein causing the abnormal characteristics in the distributionsfor ultrathin oxides.

Fig. 6. T distributions for N O and N I/I oxides stressed atV = 4 V and140 C with limited 100 samples of p=n capacitors.

Fig. 7. Normalized current density distributions for NO and N I/I oxides aswell as thick oxide of 4.2 nm.

IV. NUMERICAL SIMULATION

For ultrathin oxides, it is well known that the trapezoidalbarrier for the direct tunneling regime gives rise to a strongexponential dependence of direct tunneling current on oxidethickness [15], [16]. Therefore, even a small thickness varia-tion of nm can result in significant changes in tunnelingcurrents. On the other hand, for thick oxides often stressed inFowler-Nordheim regime, the current sensitivity to thicknessvariation is much reduced because tunneling currents stronglydepend on oxide field rather than oxide thickness due to the tri-angle barrier. This explains why for thicker oxide as in Fig. 5, thedistortions in the distributions are absent. To obtaindistribution, we use the measured tunneling current to infer the

for each sample, using a well-calibrated current-thicknesscurve [15], [16]. In this calibration, values varying from1.65–2.67 nm are first determined using capacitance–voltage(C–V) measurements in accumulation with proper correction forsurface quantization effect [17]. distribution is obtained insuch a manner as given in Fig. 8, indicating that a Gaussian func-tion is a reasonable choice [7].

In our first Monte Carlo simulation [4], ’s and currentdensities are generated using a Weibull function and an approx-imate Gaussian function, respectively. and Weibull


Fig. 8. Oxide thickness distribution over 4000 samples obtained from acalibrated current-thickness curve.

Fig. 9. SimulatedT distributions with various standard deviations forcurrent density distribution.

slopes can be determined experimentally or selected as parame-ters in the simulation. The average current density and its stan-dard deviation are also determined experimentally as given inFig. 7. Then, values are simply calculated by dividingby the current density. A typical simulated case is given in Fig. 9for three values of the standard deviation but a fixed averagecurrent density. Typical values of and the Weibull slope areselected. It is seen that at large standard deviation of currentdensity, not only can the distribution be distorted at higher per-centiles with a much shallower slope, but also the characteristicparameter can be affected. It is also interesting to note thatat lower percentiles, which are of great practical importance, thedistribution converges to an ideal distribution. This meansthat the thinner oxides with higher currents have almost no im-pact on the overall distribution. The thicker oxides withlower tunneling currents, however, tend to fail later in time, thusthey affect the higher failure percentiles only. Therefore, it is im-portant to note that it is the thicker population in the entire dis-tribution which skews the distribution at higher percentilesrather than thinner oxides.

To understand this effect in detail, let us consider twodistributions, and , with two different values as illus-trated in Fig. 10 by the dotted lines. In this figure, avalue of1.7 is fixed for both distributions, and a value of the distri-bution, is also fixed at 10 s. The total distribution can be

Fig. 10. Illustration of the effect of combining two individual distributionswith two differentT values.

obtained by combining the two distributions. The relationis used [18] where is the frac-

tion of sample population of the distribution . The solid linein Fig. 10 represents the total distribution with the seconddistribution at a different value of and a fixed value of0.5. The dashed line shows the total distribution forwith a value of 100 s for the second distribution. Let us focuson the two distributions with values apart. It is seen athigh percentiles above 40%, the total distribution bends over andgradually converges to the second distribution. It is interestingto see that at low percentiles, the total distribution is shifted fromthe first distribution. This is because all cumulative fail percent-ages of the first distribution become smaller when the two distri-butions are combined, i.e., a shift along vertical axis. This ver-tical shift in the cumulative fail percentiles is insensitive to the

of the second distribution beyond a certain difference.of 20 s case still shows a sensitivity. This can be easily

seen by considering the difference between the first and totaldistributions, . Onegets if at the same time when

. If two distributions are apart by , i.e.,, a shift of units can occur in Weibull scale. This

expression explains why the shift, , is parallel with respectto by the amount of shift mainly controlled by , therelative contribution of samples of distribution. For the rela-tive contribution of 0.97, the shift is reduced to 3%, hence, thetotal distribution at low percentiles overlays well with the firstdistribution. Thus, to a first order, the amount of shift, , incumulative fail percentage is only determined by the additionalsamples of distribution , not the fail times themselves. How-ever, the distortion at high percentiles is determined by both thenumber of samples and fail times of distribution.

With this understanding, we can now examine the effect of theGaussian distribution on the total distribution. To simplify theanalysis, we consider only three distributions ( , and

) in Fig. 11(a) with the sample counts of , and, respectively obtained by using a simple Monte Carlo

generation. The distribution can be thought of as corre-sponding to an average while distributions corre-spond to the minimum and maximum values. As discussedabove, with respect to the first distribution , the total distri-


Fig. 11. TotalT distributions obtained by combining three individualdistributions withT values7:5� apart for (a)� = 2:0 and (b)� = 1:2. Thearrows mark the shift of the total distribution from the distribution(D ).

bution is shifted downward to lower percentiles, thus, closer tothe second distribution . For a Gaussian distribution, thisis why the contribution of minimum to the total distribu-tion drastically diminishes at low percentiles as shown in Fig. 9.In Fig. 11(b), we also plot the results of the simulation for ashallower of 1.2 with all other conditions and parameters re-maining fixed. It is interesting to note that the deviation of thetotal distribution from the distribution is larger for steeper

’s than for shallower ones. As explained above, the shift ofthe total distribution from the distribution along verticalaxis is mainly affected by the total number of samples addedregardless of the values ofas shown in Fig. 11(b). Therefore,the relative increase in fail percentage for the total distributionfrom the distribution actually becomes smaller for smaller

values. In other words, shallower’s reduce the sensitivity ofthe total distribution to the presence of various individual dis-tributions as first reported in [4]. This property has importantconsequences on oxide thickness scaling in the presence ofvariation, as we will discuss later. The same conclusions can beobtained using a similar relation:with . These results are in disagreement withthe recent claim that thinner oxides with smaller’s are moresusceptible to variation than thicker oxides with larger’s[6].

The above analysis should provide insight and understandingof experiments and simulations discussed later. We shall discussnow a complementary simulation scheme to the first method asdiscussed previously [4]. The assumed area scaling law foris valid for small variation of about 0.1 nm as supportedby the excellent agreement between the simulation and experi-mental results [4]. However, for a larger variation, this as-sumption is not valid. It was found that for a fixed gate voltageof around 4 V, varies roughly about 1.3 decade per nm[4], [19]. Therefore, a different simulation is developed but inthe same spirit as the method described above. In this approach,

and are treated as two independent random variables.Similar to the previous method, in this Monte Carlo simulation,a value is first randomly generated with a specified average

and the standard deviation. Its correspondingand arecalculated based on the their relation with . Then, a break-down time is randomly generated using a Weibull function.

Fig. 12. NormalizedT andQ distributions stressed atV = 4.0 V and140 C using p =n capacitors for NO oxide. TheT andQ data yieldthe same slope except for skewedT distribution at original high percentilesdue toT variation.

All breakdown times are randomly generated in such a manner.In order to apply this method, a detailed knowledge of the thick-ness dependence of the and slope is required. In contrast tothis method, the first method does not require a prior knowledgeof the dependence of and and proves to be usefulin revealing the mechanism behind the effect of macroscopicthickness uniformity on the oxide breakdown distribution [4].However, the second method in principle does not have the con-straint of applicable range.

V. COMPARISON OFSIMULATION WITH EXPERIMENTS

To verify the simulation results at lower percentiles of a hun-dred ppm (parts per million), unlike a conventional TDDB mea-surement with limited samples, an extremely large number ofthousand samples is required. In this case, a different set of hard-ware with p n-substrate capacitors was used for stress undersubstrate injection for a sample size up to 4000 identical capac-itors (250 ppm) at an area of cm . For other areas, we usethe sample size from 500 to 2000 samples. Both NO and NI/I oxides were used here. We plotted the normalized and

distributions using Poisson area scaling for NO and N I/Ioxides in Figs. 12 and 13, respectively. It is seen for NO oxidesat both high and low percentiles data follows Poisson areascaling with well-behaved distributions. On the other hand,distributions for NO oxides show nonlinear behavior at higherpercentiles because of the strong effect of tunneling current. Atlow percentiles, they become parallel to the distributionsas the effect of variation diminishes as explained previ-ously. Therefore, the use of distributions is more accurateto measure Weibull slopes as done previously [1], [4], [16] be-cause they are much less sensitive to variation. We willreturn to this point later.

None of these abnormal characteristics are found in thedistributions for N I/I oxide because the variation is min-imal. It is important to point out that for NI/I oxide, bothand distributions exhibit the same slopes. To our knowl-edge, this clear relationship in terms of Weibull slopes between


Fig. 13. NormalizedT andQ distributions stressed atV = 4.0 V and140 C using p =n capacitors for N I/I oxide. TheT andQ data yieldthe same slope with no distortion inT distribution because of minimizedT variation.

and distributions has never been established previ-ously. This is perhaps because electron and hole trapping oc-curring in thicker oxides alter the stress current, thus leadingto a rather complicated relationship. In addition, the and

distributions display the same’s for either N O or NI/I oxides except for the distortion at high percentiles for NO

data. This precisely illustrates that the dependence ofWeibull slopes is an intrinsic statistical property [1], [2]. There-fore, the appearance that the distributions of the N I/Ioxide are steeper than those of the NO oxide (Fig. 6) is merelyan artifact because of limited sample size. The results pre-sented in Figs 12 and 13 also illustrate two important points.First, one can clearly appreciate that well-behaved distributionsas a result of using extremely large sample size as compared toFigs. 2–5. Second, the power of Poisson normalization schemeis clearly demonstrated in revealing the detailed differences inbreakdown distributions [3], [10]–[12]. Unfortunately, the im-portance of sample size is sometimes overlooked, and conclu-sions based on very few samples is still a common practice intoday’s oxide reliability work.

Since the simulation results suggest the distortions indistributions at high percentiles are due to variation, weshould be able to experimentally verify this hypothesis with alarge sample size. We divide the total population of samplesinto subgroups by selecting a specific region of the entire cur-rent density distribution. By fixing the standard deviation butvarying the value of mean current density, we select thevalues for each subgroup as illustrated in the inset of Fig. 14.Fig. 14 shows the Weibull distributions for these four groupsof stressed samples. Their corresponding and valuesas well as their values are given in Fig. 15 as a function of

. is determined from each mean value of initial cur-rent using the same relation between tunneling current andas discussed earlier. The error bars were obtained from the in-terval at the 95% confidence level using the maximum likeli-hood estimation [10]. It is evident from these two figures thatonce different oxide thickness values are properly sorted out,the thickness dependence of and is revealed. Fig. 15clearly shows an increase of by from 2.62 nm (region

Fig. 14. SubgroupedT distributions using different mean values of currentdensities. The original distribution is the same as the one in Fig. 12 with an areaof 10� cm (V = 4.0 V and 140 C).

Fig. 15. T ; Q , and� versusT after dividing the total distribution intofour subgroups (Fig. 14). Note theT sensitivity,�, is�5 dec/nm. The filledand open squares represent the Weibull slopes derived fromT andQdistributions, respectively, whereas the filled and open circles are forT andQ at 63%, respectively.

1) to 2.74 nm (region 4) while increases only by 40%. Forthis small range of variation of nm, this change in

is much smaller as compared to . The large change inis a result of the strong exponential dependence of direct

tunneling current on oxide thickness [15].Furthermore, Fig. 15 provides important empirical relations

of the dependence of and although it only covers asmall range of values. These relations are required for thesecond simulation. The dependence of in Fig. 15 is con-sistent with more extensive experimental results obtained overa much larger range of values from 1.6 to 5.0 nm [19]. For

dependence on , a simple relation, ,can be used as shown in Fig. 15 because only a small range( nm) of is of interest here. is referred to as thesensitivity in the following section. Recent published data in-dicated that this quantity can change to a much large value of

dec/nm [20] for thinner oxides 1.4 to 2.0 nm. However,it should be pointed out that different stress voltages and tem-peratures were used in [20] due to constrains of experimentaltime-window. Fig. 16 shows an excellent agreement betweensimulation and experiments for both higher and lower failure


Fig. 16. Measured and simulatedT distributions where the lowerpercentiles are magnified. The simulation was done using a standard deviationof 0.03 nm andhT i = 2:67 nm with aT sensitivity of 5.2 dec/nm in goodagreement with experimentally derived value of 5.0 dec/nm (Fig. 15).

percentiles. It is seen in Fig. 16 that at high percentiles the cumu-lative fails bend over as simple calculation shows (Figs. 10 and11). On the other hand, at lower percentiles, the experimentaldata clearly shows a change in slope indicating thinner oxidesearlier on the total breakdown distribution diminishes as shownFig. 12. Because oxide intrinsic reliability must be projected toa much lower failure percentile, the use of measured slopes athigher percentiles can lead to erroneous and pessimistic relia-bility predictions.

VI. OXIDE THICKNESSVARIATION AND POSSIBLERELIABILITY

IMPROVEMENT

In this section, we will critically examine the role ofmacroscopic thickness uniformity in gate oxide reliabilityimprovement using the second Monte Carlo simulation methoddescribed earlier. We will investigate this issue under twoextreme conditions, i.e., very tight thickness uniformity anda large degree of thickness variation, in conjunction with thesensitivity of thickness dependence. To study the possiblereliability improvement by reducing variation, we useexperimental data by selecting 750 samples from an initialof 4000 samples with a resultant variation of only 0.01nm and with a standard deviation as small as 0.0027 nm. Thisscreened sample distribution is shown in Fig. 17. Toachieve such tight uniformity of 0.01 nm means one hasto reject 80% of the total population of the samples. However,as clearly shown in Fig. 17, such stringent uniformity does notlead to any improvement in reliability at low percentiles ascompared to the total breakdown distribution.

As oxide reliability margin drastically shrinks, it becomes in-creasingly important to accurately evaluate oxide reliability in-cluding all the key aspects, such as variation. First, formanufacturing, a certain tolerance of thickness variation mustbe considered [7]. Second, as already pointed out above, thesensitivity of thickness dependence may change overa different thickness regime [3], [19], and it is not clear nowwhether can also depend on gate voltages. The depen-dence of has been reported indicating a change inup to 6dec/nm for thinner oxides down to 2.3 nm [19]. An increase inthe sensitivity means a fast decrease in with decreasing

. Now, we can perform our thought experiments using the

Fig. 17. Screened and overallT distributions for N O oxide. 750 samplesselected from a total of�4000 samples to achieve a standard deviation of 0.0027nm show no improvement in oxide reliability at low percentiles!

Fig. 18. SimulatedT distributions for various values ofT sensitivityparameters for 2.7 nm oxide. Experimentally measuredT sensitivity for 2.7nm is�5 dec/nm (Fig. 15).

second Monte Carlo simulation to investigate the overall effectsof variation on distribution by varying andvalues (also Weibull slopes).

Fig. 18 shows several simulated Weibull distributions of 2.7nm oxide using sensitivity as a theoreticalrunning parameter as well as the distribution withoutvariation. We will use the termideal distributionto refer the

distribution without variation to avoid confusion fromnow on. Note as varies as a result of changingfor a fixed

, each distribution with variation and its correspondingideal distribution simply shifts by the same amount. Thus, theycan simply be normalized to their ideal distributions. The stan-dard deviation of oxide thickness is selected to be 0.2 nmas done in [7]. As expected, it is seen that the distribu-tions become broader whenincreases because decreasessharply with decreasing . However, at low percentiles ofpractical interest, all distributions become nearly parallel witha reduced or an increase in cumulative fail as a result offast decrease in with . We define a quantity calledrel-ative-fail-increase, which is the difference between the cumu-lative fail and that of the ideal case, then normalized to that ofthe ideal case, ideal ideal . As expected, we havefound this quantity is very sensitive to, and sensitivity,and variation as we discussed below. The experimentallymeasured value for 2.7 nm oxide is dec/nm (Fig. 15). Ac-cording to Fig. 18, a lifetime reduction for a given low percentileis only about . Therefore, the effect of nonuniformity


Fig. 19. Theoretical 3-D plot of relative-fail-increase as a function ofT sensitivities and Weibull slopes. The filled circles are the results usingexperimentally measuredT sensitivities forT �2.3 to 4.2 nm. The filledsquare is obtained using� 7.2 dec/nm derived from recent published data forT � 1.4 to 2.0 nm [20].

at the low-percentile distribution is greatly reduced evenwith a tolerance as large as 0.4 nm.

Notice in Fig. 18, for large values of sensitivities,distributions tend to become concave. Because thinner oxidesexhibit a fast reduction of with decreasing as com-pared to thicker oxides [3], [20], this concave characteristic of

distribution is expected to appear more likely for thinneroxides than for thicker oxides for the same amount of thick-ness variations. As pointed out by Hunter [21] and confirmedexperimentally [19] by using large number of samples about athousand, a Weibull distribution would show a convex curvaturein the lognormal plot whereas a lognormal distribution appearswith a concave curvature in the Weibull plot [21]. However, asexperimentally proven [3], [12] for both thick and ultrathin ox-ides, oxide breakdown shows the weakest-link characteristics.Therefore, the Weibull distribution is the appropriate choice forthe description of oxide breakdown. The appearance of the con-cave-shape distribution in the presence of thickness variationseen in Fig. 18 can easily give rise to a straight line fit if thesame data are plotted in the lognormal plot, thereby leading tothe conclusion that such a breakdown distribution follows thelognormal distribution. As explained above, such a conclusionis unwarranted and can lead to erroneous reliability projectionas discussed in [3] and [19]. The concave curvature in break-down distribution is merely an artifact as a result of thicknessvariations manifested more pronouncedly for ultrathin oxides.

Fig. 19 displays a theoretical simulation (3D plot) of rela-tive-fail-increase as a function of Weibull slopeand sen-sitivity . values are implicit because of its relationshipwith and . It is surprising to see that the relative-fail-in-crease changes more drastically for largevalues of thick ox-ides than for small values of thinner oxides for the same valueof sensitivity. This result appears to be counter-intuitive.As discussed in the previous section, because the lack of sensi-tivity for the overall distribution with shallower Weibull slopesto variations, the detrimental effect of degradation withdecreasing is compensated. This understanding can pro-vide significant relief for tolerance control from reliabilityviewpoint. While the simulations results of the study cases inthe three-dimensional (3-D) plot are interesting, in reality theexperimentally determined sensitivity varies up to about 6dec./nm down to 2.3 nm [19]. Therefore, relative-fail-increase

Fig. 20. Relative-fail-increase vs. thickness variation for� = 1.15 of 1.6 nmoxide with three differentT sensitivities, 5, 7, and 9 dec/nm.

only remains in valley and flat regions of the 3-D plot as indi-cated by the filled circles in Fig. 19. The filled square is obtainedusing a sensitivity of dec./nm derived from recentlypublished data for 1.4 to 2.0 nm [20]. For thesethinner oxides, the relative-fail-increase is about 2.5 for 1.1to 1.2 for a tolerance of 0.4 nm. We estimate the effect of

tolerance on overall oxide reliability as shown in Fig. 20for 1.15 of 1.6 nm oxide. The thickness variation on thehorizontal axis corresponds to the difference between the min-imum and maximum thickness. It is seen that a very sharp rise inrelative fail increase as the variation becomes very large,particularly for a large sensitivity of 9 dec/nm. However,as soon as thickness variation is reduced to 0.3 nm, its effectis greatly reduced to a negligible level. Therefore, we can con-clude the sample-to-sample uniformity cannot fundamen-tally improve oxide reliability. In fact, the understanding fromthis work allows us to decouple oxide reliability from yield foran allowed tolerance.

VII. CONCLUSIONS

It is found that small oxide thickness variations can cause thenonlinearity in Weibull distributions with a change of slopefrom high to low percentiles. The use of Weibull slope at higherpercentiles can lead to an erroneous and pessimistic projection.It is evident that reduction of variation can simplify relia-bility evaluation. However, based on our extensive experimentalresults and a detailed full-scale Monte Carlo analysis, a reason-able tolerance can be allowed, and improvement invariation beyond realistic tolerances cannot fundamentallyenhance oxide reliability. We show it is imperative to take allthe effects such as Weibull slopes, sensitivity, toler-ance, and low percentile breakdown behaviors into account toanswer the questions for the impact of uniformity on re-liability. We found that shallower Weibull slopes significantlycompensate the increase in sensitivity for ultrathin ox-ides. Our analysis and understanding of the interplay of theseparameters mentioned above provide not only important relieffor ultrathin oxide reliability but also a proper methodology fortreating the issue of macroscopic uniformity in the contextof oxide reliability.


ACKNOWLEDGMENT

We are indebted to J. Stathis, A. Strong, and J. Aitken for crit-ical reading of the manuscript and interesting discussions. Weare also thankful to E. Crabbe, S. Crowder, L. Su, P. Varekemp,P. Agnello, and S.-H. Lo for encouragement and interesting dis-cussion, and to R. Dufrense, D. Brochu, and B. Morse for theirconstant technical support. E.Wu is grateful to Prof. J. Sune forinteresting discussions regarding the impact of macroscopic andmicroscopic thickness uniformity on oxide reliability.

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Ernest Y. Wu received the M.S. and Ph.D. degrees in physics from the Univer-sity of Kansas, Lawrence, in 1986 and 1989, respectively.

He joined the IBM Storage Product Division in 1989. In 1994, he transferredto the IBM Microelectronics Division, Essex Junction, VT. He is currently a Se-nior Engineer in the Technology Reliability Department, IBM MicroelectronicsDivision.

Dr. Wu has served on the device dielectric committee as a co-chair for2000 International Reliability Physics Symposium (IRPS). He is a memberof CMOS and Interconnect Reliability committee of International ElectronDevice Meeting (IEDM) for 1999 and 2000.

Edward J. Nowak joined IBM in 1981. He is currently with the IBM Microelec-tronics Division, Essex Junction, VT, where he works on CMOS device design.

Rolf-Peter Vollertsen received the Dipl.-Ing. and Dr.-Ing. degrees in materialscience engineering from the University of Erlangen, Germany, in 1982 and1987, respectively.

After joining the Microelectronics Division, Siemens AG, Germany, his re-search focused on thin single and multilayer dielectric quality and reliability.In 1992, he transferred to the DRAM Development Alliance, a cooperationbetween IBM and Infineon Technologies Corp. (formerly Siemens Microelec-tronics, Inc.) at IBM, Essex Junction, where he is responsible for front end of linetechnology reliability. His research interests include reliability of thin dielectricsand device reliability for the Gigabit generation of DRAMs. He is currently in-volved in process and product qualification. He is the author or co-author ofseveral refereed and conference papers.

Dr. Vollertsen served on the technical program committees of the IRPS 2000and the IRW.

L.-K. Han , photograph and biography not available at the time of publication.

weibull breakdown characteristics and oxide thickness uniformity

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