Determining appropriateness of sampling integrated circuit test data in the presence of manufacturing variations

Methods and systems determine an original statistical variance of an original failure distribution of a component (that is common to all chips tested) that occurs during manufacturing of wafers containing such chips. These methods and systems determine a first statistical variance of a reconstructed failure distribution, relative to sample size; and determine a second statistical variance of a mean time to failure of the component, relative to sample size. The first and second statistical variances are combined into a total reconstruction variance. Methods and systems determine whether the original statistical variance is less than the total reconstruction variance to identify whether the process of creating the reconstructed failure distribution can be used. Therefore, these methods and systems prohibit testing of the additional wafers manufactured using the specific wafer design and manufacturing process when on the original statistical variance is less than the total reconstruction variance.

FIELD OF INVENTION

The present invention relates to performing sample testing during manufacturing and/or qualification a wafer using a wafer design and a manufacturing process that uses wafer manufacturing machines.

BACKGROUND

Sampling methodologies allow less than all components on an integrated circuit device to be tested, thereby substantially reducing testing time as well as saving computer resources that are associated with calculating results based upon the actual physical testing of the integrated circuit devices. One sampling-based technique is a methodology to reconstruct a Weibull distribution to solve variability issues related to breakdown (BD) statistics. While this method has been shown to be successful in SiO2with small thickness (TOX) variation, its applicability remains questionable, particularly for middle of line/back end of line (MOL/BEOL) dielectrics with substantial spacing variation and intrinsic line-edge roughness (LER).

SUMMARY

According to various embodiments of the present invention herein, methods and systems herein manufacture a wafer using a wafer design and a manufacturing process that uses wafer manufacturing machines. The wafer design can contain identically designed integrated circuit chips. Various methods, systems, and computer program products herein test the manufactured wafer to produce test data. Using the test data, such methods, systems, and computer program products herein determine an original failure distribution of a component of the wafer (that is common to all of the integrated circuit chips, such as insulator thickness) based on the test data; and determine an original statistical variance of the original failure distribution of the component that occurs during the manufacturing of the wafer. This original statistical variance can be a chip-to-chip variance, a device-to-device variance, etc., and is based on empirical (actual physical) testing of the manufactured integrated circuit chips.

Additionally, such methods perform a process of creating a reconstructed failure distribution of the component using a sample size of the test data that are less than all of the test data. More specifically, in the sampling sizes these methods select the number of integrated circuit chips per wafer to sample (that are less than all of the integrated circuit chips in the wafer design) and select the number of devices per integrated circuit chip to sample (that are less than all of the devices in the integrated circuit chips in the wafer design). Thus, while both the test data and the sample sizes comprise the test data from a limited number of integrated circuit chips per wafer and a limited number of devices per integrated circuit chip, all of the test data is much larger (a large multiple of) the sampling sizes; and use of the much smaller sampling sizes can save computer resources and time when performing sample-based testing the wafer design and manufacturing process.

In order to evaluate whether it is appropriate to perform testing of the specific wafer design and manufacturing process using the reconstructed failure distribution of the component generated with the smaller sampling sizes, these methods determine a first statistical variance of the reconstructed failure distribution relative to sample size; and determine a second statistical variance of a mean time to failure of the component relative to sample size, again using all of the test data. These methods then combine the first statistical variance and the second statistical variance into a total reconstruction variance.

Further, such methods determine whether the original statistical variance is less than the total reconstruction variance to identify whether additional wafers manufactured according to the specific wafer design and manufacturing process can be subjected to sample-based testing using the process of creating the reconstructed failure distribution with the smaller sampling sizes. Therefore, these methods prohibit such sample-based testing of the additional wafers when the original statistical variance is less than the total reconstruction variance, but permit such sample-based testing of the additional wafers when the original statistical variance is not less than the total reconstruction variance.

In additional embodiments, these methods can alter the sample sizes to include more samples based on identifying that the process of creating the reconstructed failure distribution of the component cannot be used.

DETAILED DESCRIPTION

As mentioned above, a sampling-based technique can be used as a methodology to reconstruct a Weibull distribution to solve variability issues related to breakdown (BD) statistics. While this method has been shown to be successful in SiO2with small thickness (TOX) variation, its applicability remains questionable, particularly for MOL/BEOL dielectrics with substantial spacing variation and intrinsic line-edge roughness (LER) because this method requires four assumptions to be validated (in Table 1) see below.

TABLE 1Assumptions of Reconstruction Methodology:1).Local (in-die) BD distributions must be Weibull-distributed andNon-Weibull or bimodal distributions within chips are not valid2).Poisson are-scaling is assumed for all local (in-die) distributions.3).TBD data are collected with sufficient accuracy.4).Constant Weibull slopes for all in-die TBD distributions.

In other words, only T63varies from dies to dies but β is fixed. This means this is an approximation procedure and only valid for very small TOX variation since percolation theory inherently links T63with β through thickness/spacing dependence of T63and β.

As noted above, a sampling-number dependence of Weibull slope (β) has been recently reported; however, its applicability remains questionable. As a result, to obtain accurate β value, a large number of samples are required. Moreover, it is shown that the reconstructed distribution with a small sampling number can be misleading as it masks the non-Weibull/non-Poisson area-scaling nature of underlying in-die TBD distributions, revealed by extraordinary statistics of ˜10,000 samples. Methods described herein, reveal statistical scaling property for the β-sampling curve, along with a new result of sampling-number dependence of T63variation. These methods can report a fundamental mathematical formulation for these two sampling-number dependencies of Weibull slope and T63variations. As a result, a quantitative criterion for the applicability of reconstruction methodology is developed.

FIGS. 1A-Bshow some typical TBD distributions of front-end-of-line (FEOL) 27 Å SiO2at chip level, the combined TBD data across the wafer, and the reconstructed distribution using sampling-based methodology. In this case, a large sampling number of 44 devices per chip are used with a total of ˜4200 samples. The sampling-number (n) dependence of Weibull slope (β) is shown inFIG. 2Afor two cases with and without TOXvariation. It can be seen inFIG. 2Athat β sampling-number dependence is comparable for these two cases, supporting the original concept of this methodology.

To understand its root-cause, methods plot the standard deviation (σ) of local T63(in-die) inFIG. 2Bshowing remarkable differences in these two cases along with the corresponding log normal plots of local T63values given inFIGS. 3A-B. As seen inFIG. 3B, for no TOXvariation case, the local T63values varies with the sampling number as ˜n0.5consistent with the central-limited theory. In contrast, with TOXvariation shown inFIG. 3A, the σ of local T63decreases slightly with increasing sampling-number (n) but remains almost saturated towards large n values.

To deepen the understanding, methods perform Monte Carlo (MC) simulations with different TOXdistributions (uniform vs. normal distributions with different a Tox) with a mean TOXvalue of 2.7 nm and also a case without any TOXvariation. Both sampling-number dependencies of β and σ T63are consistent with the MC simulation as seen inFIGS. 4A-B. Table 2 (see below) summarizes the equations methods and system herein use for statistical models of sampling number (n) dependence of β and T63variation (σT63).

Total variance of two independent sources of variations can be written as where VarWBis the variance of Weibull distribution and the variance of central-limited theory are given respectively: since no Tox variation after reconstruction, the variance is that of Weibull, and also that it is fixed quantity and independent of sampling number. The sampling-number dependence of Weibull-slope can be shown as a graph of VarWBof Equation 2 is given inFIG. 13A. Equation 4 is numerically solved by inverting Equation 2 to obtain Equations 4A given inFIG. 13Bwith various σ0values. The sampling-number dependence of T63variation assuming a log normal distribution can be shown: where σT63∞is the σ of log normal distribution. The total reconstructed variance for general applications where η is the reconstructed T63and for log normal distribution for T63variation, we have σ0=σ1=σ2. For general cases with σ0≠σ1≠σ2Equation (9) still holds as n→∞. Equation 10 provides a quantitative criterion for applicability of reconstruction methods. Varoriginalis the variance of an original distribution can be determined.

Since methods deal with Weibull, and/or log normal, and other distributions here, the variance analysis provides a general framework to characterize the spread of any arbitrary distributions. The agreement of statistical model to SiO2data with and without TOXvariation is excellent as shown inFIG. 2A. The σ0 value of 0.60 is also comparable to 0.71 extracted from the central limit theory inFIG. 2Bdue to a small 3σ=0.01 nm even for no TOXvariation case. These results explain the root-cause of β-sampling curve β(n), arising from subgroup sampling as dictated by central limit theory. This is a systematic effect rather than stochastic fluctuation effect from conventional sample-size. The β value at n→∞ corresponds to the Weibull slope of reconstructed distribution by properly taking this systematic effect of sampling-number dependence into account.

Further, Equation 4 provides an accurate method to determine the β∞using only a few small sampling numbers. This effect should be properly included in reliability extrapolation since β values can strongly affect the projection results.FIG. 5shows its comparison of statistical model with BEOL/MOL data.

In the case of larger TOXvariation, the σ for T63variation inFIG. 2Bshows a much weaker dependence on sampling-number (n). To understand its root-cause, methods perform MC simulation assuming T63variation follows a log normal distribution but changing σLog nwith a fixed β=1.5. The MC results inFIG. 6Areveal the variance of T63from chips to chips gradually approach the case of no T63variation as σLog nis reduced. In contrast, for larger σLog nvalues, σT63only weakly depends on n, similar to the case with TOXvariation inFIG. 2B. The lines inFIG. 6Aagree well with the analytical solution of Equation 5.FIG. 6Bshows the effect of changing β coincides with the σ0value in Equation 5 while the T50magnitude is insensitive to the T63variance.

FIG. 2Bshows the comparison of Equation 5 with SiO2data including TOXvariation. These results suggest as T63variation is included, a non-vanishing σT63∞exists regardless of sampling-number and this should be taken into account in this reconstruction methodology.

FIG. 7compares the relative contributions of two terms of Equation 6 for SiO2with and without TOXvariation, showing β-sampling effect (1st term) dominates over T63variation. Notice that variance of β change increases with n due to the inverse relation between VarWBand β (FIG. 14A).FIG. 8shows comparison of two variance components (Var1, Var2) for extrinsic BD case and intrinsic BD case using SiO2oxides (BEOL Low-k 14 ground rule, BEOL low-k 32 ground rule, and FEOL SiO2).

FIG. 8shows the both variance terms for extrinsic BD dominate in comparison with intrinsic BD case. The variance methodology outlined in Table 2 (above) also allows one to compare FEOL data with BEOL data.FIG. 9reveals as n→∞, both variance terms approach comparable for BEOL 32 ground-rule data but remain much higher than those of FEOL SiO2, suggesting that thickness variation for BEOL is more much worse than those of FEOL dielectrics as expected.FIG. 10illustrates the variance squared over sampling number, where the top diamond line represents total reconstructive variance, the triangle line represents one variance (Var2), the dotted line represents a different variance (Var1), and the dashed line represents the variance of the original distribution. As can be seen, when the two variances (Var1 and Var2) are added together, to achieve the total reconstructed variance, it positions the total reconstructed variance above the variance of the original distribution. As scaling continues to 14 ground-rule technology, both variance contributions terms remain higher than 32 ground-rule, pointing out more challenging tasks for variability reduction for 10 nm and 7 nm technologies ahead. With these fundamental understandings, the systems and methods herein establish a quantitative criterion for its applicability to BEOL/MOL/FEOL dielectrics, as discussed above, the reconstruction (FIG. 1B) is an approximation procedure to reconstruct a Weibull distribution as if no TOXvariation from generally complicated and unknown sources of variations although thickness variation is often considered a primary source. Since the variance of the original distribution is a fixed quantity and does not depend on sampling number.

A minimum quantitative criterion is that reconstructed total variance cannot exceed that of this original distribution as given by Equation 10. The original distributions and reconstructed distributions for three examples are given inFIGS. 1A,11B, and12B. A full comparison for the different contributions of variance terms including the variances of the original distributions are given inFIGS. 11A,11B,12A for three cases. For FEOL SiO2of intrinsic BD with small TOXvariations, this method works extremely well as shown inFIG. 11A. The cases of FEOL SiO2with extrinsic BD and BEOL intrinsic BD fail to meet this criterion suggesting method cannot be applied even though it gives an attractive β values or there constructed distribution appears to Weibull (FIG. 12A).

Methods develop fundamental understanding for statistical properties of reconstruction methodology in full agreement with experimental data. Thus, a quantitative criterion for the applicability of this methodology is provided to guide reliability assessment and technology, particularly relative for future technology generations.

InFIG. 14demonstrates the main features and effects of the reconstruction method and how raw data and reconstructed data are related over time. InFIG. 15the small sampling number gives a good impression but masks the underlying skewed TBDdata. For additional information on masking underlying skewed data, see Wu, et al. IEEE International Reliability Physics Symposium, p. 3A2.2, 2015.

FIG. 16is a flowchart illustrating some aspects of exemplary methods, systems, and computer program products herein. In Item100methods manufacture a wafer using a specific a wafer design and a manufacturing process that uses wafer manufacturing machines. The wafer design can contain identically designed integrated circuit chips. In item102, these various methods, systems, and computer program products herein test the manufactured wafer to produce test data.

In item104, using a first, relatively very large sampling size of the test data, such methods, systems, and computer program products determine an original failure distribution of a component of the wafer (that is common to all of the integrated circuit chips, such as insulator thickness). In item106, these methods, systems, and computer program products determine an original statistical variance of the original failure distribution of the component that occurs during the manufacturing of the wafer. This original statistical variance calculated in item106can be a chip-to-chip variance, a device-to-device variance, etc., and is based on empirical (actual physical) testing of the manufactured integrated circuit chips that occurs in item102.

Additionally, in item108such methods perform a process of creating a reconstructed failure distribution of the component using a second, relatively much smaller sampling sizes of the test data that is hundreds to tens of thousand times less than all of the test data. More specifically, when selecting the sampling sizes in item108, these methods select the number of integrated circuit chips per wafer to sample (that are less than all of the integrated circuit chips in the wafer design) and select the number of devices per integrated circuit chip to sample (that are less than all of the devices in the integrated circuit chips in the wafer design). Thus, while both the test data and the sample sizes comprise the test data from less than all the integrated circuit chips per wafer and less than all the devices per integrated circuit chip, all of the test data is much larger (a large multiple of (e.g., 100×, 1000×, 10,000×, 100,000×, etc.) the sampling sizes; and use of the much smaller sampling sizes can save computer resources and time when performing sample-based testing the wafer design and manufacturing process.

In order to evaluate whether it is appropriate to perform testing of the specific wafer design and manufacturing process using the reconstructed failure distribution of the component generated with the smaller sampling sizes, in item110these methods determine a first statistical variance of the reconstructed failure distribution relative to sample size; and, in item112determine a second statistical variance of a mean time to failure of the component relative to sample size, again using all of the test data. These methods then combine the first statistical variance and the second statistical variance into a total reconstruction variance in item114.

In item116, such methods determine whether the original statistical variance is less than the total reconstruction variance to identify whether additional wafers manufactured according to the specific wafer design and manufacturing process can be subjected to sample-based testing using the process of creating the reconstructed failure distribution with the smaller sampling sizes. Therefore, these methods prohibit such sample-based testing of the additional wafers when the original statistical variance is less than the total reconstruction variance in item120, but permit such sample-based testing of the additional wafers when the original statistical variance is not less than the total reconstruction variance in item118.

In additional embodiments, in item122, these methods can alter the sample sizes to include more samples when item116identifies that the process of creating the reconstructed failure distribution of the component cannot be used120, and processing loops back to item108to repeat the reconstruction process with a larger sample size.

FIG. 17is a flowchart demonstrating systems and methods herein. In item150the number of chips across a wafer is selected and the number of devices per chip. In item152, the systems and methods execute stress with the TDDB data collection for the devices. As shown in item154, the systems and methods then perform a reconstruction analysis and obtain β and σT63sampling number dependence. Further in item156, the systems and methods determine Weibull Slope β (∞) using Eq. 4 and 4a with Eq. 2 and determines VarT63∞using Eq. 5 or 8. Then in item158, the systems and methods determine Vartot—rec=T63—Rec2(var1+var2) using Eq. 11 and compare with Var (original) using Eq. 12. In item160, the systems and methods process, Var(Original)≧Vartot—rec. If the process determines that this is correct then, β(∞) is correct as seen in item162. However if no then the reconstruction cannot be used, as shown in item164.

Therefore, as shown in the comparison ofFIGS. 2A and 2B, while the reconstructed Weibull distribution with (dots inFIG. 2A) and without (triangles inFIG. 2A) oxide thickness variation does not show significant differences as sampling number increases (FIG. 2A), there are differences with (dots inFIG. 2B) and without (triangles inFIG. 2B) oxide thickness in variance (standard deviation (sigma)) with sampling number (FIG. 2B). This demonstrates that a reconstructed distribution with a small sampling number can be misleading (FIG. 2A) as it masks the non-Weibull/non-Poisson area-scaling nature of underlying in-die thickness breakdown distributions (FIG. 2B).

In view of the fact that a reconstructed distribution with a small sampling number can be misleading, methods herein determine an original failure distribution of a component of the wafer (that is common to all of the integrated circuit chips, such as insulator thickness) using a first, relatively very large sampling size of the test data. These methods also determine a first statistical variance of the reconstructed failure distribution relative to sample size, which is shown as item180(Original Var) inFIGS. 18 and 19. The propriety of use of the reconstructed distribution with a small sampling number is judged against this original variance level180.

More specifically, these methods create a reconstructed failure distribution of the component using a second, relatively much smaller sampling sizes of the test data (that is hundreds to tens of thousand times less than all of the test data); and determine a first statistical variance of such a reconstructed failure distribution relative to sample size, which is shown as item182(Rec_Var_Beta) inFIGS. 18 and 19. Additionally, these methods determine a second statistical variance of the mean time to failure of the component relative to sample size, again using all of the test data, which is shown as item184(Rec_Var_T63) inFIGS. 18 and 19. Then, such methods combine the first statistical variance182and the second statistical variance184into a total reconstruction variance186(Rec_Var_Tot, inFIGS. 18 and 19).

As can be seen by further comparison ofFIGS. 18 and 19; inFIG. 18, the original statistical variance180is not less than the total reconstruction variance186; while, to the contrary, inFIG. 19, the original statistical variance180are less than the total reconstruction variance186. Therefore, with the results inFIG. 18, because the original statistical variance180is not less than the total reconstruction variance186, additional wafers manufactured according to the specific wafer design and manufacturing process can be subjected to sample-based testing using the process of creating the reconstructed failure distribution with the smaller sampling sizes. To the contrary, with the results inFIG. 19, because the original statistical variance180are less than the total reconstruction variance186, additional wafers manufactured according to the specific wafer design and manufacturing process cannot be subjected to sample-based testing using the process of creating the reconstructed failure distribution with the smaller sampling sizes.

Referring now toFIG. 20, a schematic of an example of a system10is shown. System10is only one example of a suitable system and is not intended to suggest any limitation as to the scope of use or functionality of embodiments of the invention described herein. Regardless, system10is capable of being implemented and/or performing any of the functionality set forth hereinabove.

In this application, the definition of mean variance are defined for any arbitrary distributions as follows in these equations:

μ=∫x⁢⁢f⁡(x)⁢ⅆx=∫x⁢ⅆFⅆx⁢ⅆx⁢⁢Var=σ2=∫(x-μ)2⁢f⁡(x)⁢ⅆx=∫x2⁢f⁡(x)⁢ⅆx-μ2⁢⁢and⁢(13)=∫x2⁢ⅆFⅆ⁢x⁢ⅆx-μ2(14)
Where f(x) is the probability density function, F(x) is the cumulative probability function so that

f⁡(x)=ⅆFⅆx.(15)
The mean is shown in this equation

μ=∫x⁢⁢f⁡(x)⁢ⅆx=∫x⁢ⅆF=∫ⅆ(F⁢⁢x)-∫F⁢ⅆx(16)
Furthermore, variance is shown in this equation

Equations 16 and 17 can be solved numerically for breakdown measurements for arbitrary failure distributions with the cumulative failure fraction using Equation 18 or 19 below. In these equation I is the ith sample and N is the total number of samples

β⁡(n)≡β⁡[Var-1⁡(β)]=Var⁡(β∞)⁡[1-1n](20)
The one parameter (β∞) model is shown in Equation 21 below andFIG. 21.

Single-parameter model for β-sampling number dependence, β(n) is shown inFIG. 22. The lines are a one-parameter model and the symbols use MC simulation. The Weibull distribution is generated randomly and divided into 100 dies and the reconstruction analysis applied to obtain β(n). The self-consistency of a single-parameter β(n) is shown inFIG. 23A-23B. The σ T63—WBfollows the central limit theory and the Var(β) should equal σ02are shown inFIG. 23B.

Piece-wise approximation of Var(β) for fast determination of Weibull slope is shown inFIG. 24. In many applications Var(β) (Eq. 3) can be approximated by Equation 23 and 24 so that β(n) can be solved analytically.

β⁡(n)≡β-1⁡[Var⁡(β)]=Var⁡(β∞)-σ02n⁢⁢⁢β=α1/ρ⁡(α2β∞2⁢⁢ρ-σ02n)-1/2⁢⁢ρ⁢⁢β⁡(n)≡β-1⁡[Var⁡(β)]=Var⁡(β∞)⁢(1-1n)⁢⁢⁢β=β∞⁡(1-1n)-1/2⁢⁢ρ(25)
Extraction of 1 or 2-parameters in β(n) models is shown inFIG. 26where the interception and slope can yield two parameters in (β∞& σ0) Equation 21 and a force-fit using Equation 22 can yield1300.

It should be understood that the terminology used herein is for the purpose of describing the disclosed [systems, methods and computer program products] and is not intended to be limiting. For example, as used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. Additionally, as used herein, the terms “comprises” “comprising”, “includes” and/or “including” specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. Furthermore, as used herein, terms such as “right”, “left”, “vertical”, “horizontal”, “top”, “bottom”, “upper”, “lower”, “under”, “below”, “underlying”, “over”, “overlying”, “parallel”, “perpendicular”, etc., are intended to describe relative locations as they are oriented and illustrated in the drawings (unless otherwise indicated) and terms such as “touching”, “on”, “in direct contact”, “abutting”, “directly adjacent to”, etc., are intended to indicate that at least one element physically contacts another element (without other elements separating the described elements). The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or act for performing the function in combination with other claimed elements as specifically claimed.