of Qoss Section for Single Event wet: Model (HICUP)/67531/metadc792133/m2/1/high... · Further Development of the Heavy &n Qoss Section for Single Event wet: Model (HICUP ... dependent

Further Development of the Heavy &n Qoss Section for Single Event w e t : Model (HICUP)

,W., F. W. Sexton, A. K. Prinja+ Sandia National Laboratories

Albuquerque, New Mexico 87185

Abstract

The HICUP models are shown to be useful tools for both analyzing cross section data and performing upset rate calculations, thereby allowing the cross section concept to be used in both areas. The angular dependent HICUP model is developed from the RPP geometry and the Weibull density function. It is compared with angular cross section data, showing good agreement. The HICUP model is used to derive the correct scaling laws for transforming cross section data taken off-normal to normal incidence. The HICUP scaling reconciles two previously proposed inverse cosine scaling corrections which are shown to be asymptotic forms of the.HICUP scaling. The angle-integrated form, I-HICUP, is developed and used in Galactic Cosmic Ray (GCR) upset rate calculations on several devices. Results are nearly identical to SPACE RADIATION, calculations. I-HICUP is used to perform an uncertainty analysis of GCR upset rate, examining the sensitivity to uncertainties in the input parameters. The GCR upset rate shows the greatest sensitivity to upset threshold, device depth (and funnel depth if applicable), and saturation cross section, the least sensitivity to RPP length-to-width aspect ratio. The other Weibull parameters, width, W, and shape, b, are of intermediate Sensitivity.

1. Introduction

In a previous paper [l] we developed an expected value model for the angular dependent heavy ion upset cross section which was shown to agree well with measured data on SRAMs. This model, since named HICUP, is based on a Rectangular Parallelepiped (RPP) sensitive volume and accounts for the stochastic nature of the cross section by assuming that the upset threshold, E (energy or charge), is governed by a Weibull distribution whose parameters are obtained from the measured cross section data. In reference [l], HICUP was also used to derive an expression for scaling angular cross section data to normal incidence. This generalized expression, developed in the form of a bridging function, was able to reconcile two previous formulas [SI for correcting the inverse cosine scaling laws, collapsing to Petersen's correction, [cos(6)-(hlZ)sin(0)]-' near the LET threshold, and Sexton's, [ cos (6 )+(h l Z)sin(B)]-', near saturation. In addition to presenting more data comparisons, this paper will, (1) explain the relationship between HICUP and the RPP chord length distribution, (2) develop a HICUP formalism for the upset rate integral which employs the total (i.e. solid angle integrated) upset cross section, (3) Compare the HICUP formalism with SPACE RADIATION, one of the standard, chord length based SEU codes, and, (4) use the HICUP formalism to study the sensitivity of the upset rate to the input parameters.

II. The Angular Dependent HICUP Model. Relationship to the Chord Model.

The HICUP model was developed in two stages. In stage I , the point-value cross section model was developed. It is based on the standard RPP geometry for the sensitive volume, Figure 1, and a fixed upset threshold, E. This directional and LET dependent point-value cross section is defined as the sensitive area of the RPP projected onto the plane whose normal coincides with the direction, a, of the incoming beam of heavy ions.

+ University of New Mexico

This work was supnnrted by the United States Department of Energy under Contract DE- A C N - 94AlfZfjO08.

DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

Figure 1. RPP Geometry and Coordinate System Used to Derive the HICUP Model

A differential element of upset cross section is given as,

= Z w , s, WY’, a= (e ,+ do ={(bx’(y~,$ )-sC sine )case + ( h - ~ , mse )sine}dy’

where S, is the criiical, minimum chord length needed for upset,

H ( x ) = l for x 2 0 and 0 for xI;O

H (n) is the Heaviside step function, and S, is the funnel length. The point-value cross section model is obtained by Integration over y’,

c

Ap is the total area of the RPP projected normal to the heavy ion beam, S, is the maximum chord length through the RPP for the given direction, a, and L,,,,” is the local (Le. directional dependent) minimum LET. The removal cross section, g(a,Sc ) is defined as,

where the geometric coefficients, g f and g2 contain trigonometric terms only. It is termed the removal cross section because it represents that portion of the total projected area not satisfying the Sc requirement, thus necessitating its removal. A complete derivation of (3) can be found in reference [i]. An alternative derivation begins with the directional, differential RPP chord length distribution, c(Q,s), given by Petroff in reference [lo]. The quantity, c(Q,s) ds, gives the exact probability of encountering a chord of length s in the R direction. Appendix A shows that the point-value model, (3) can be expressed in terms of the directional, integral chord distribution C(n,s),

2

D

Thus, although initially derived independently and from a different approach, the point-value cross section model is firmly rooted to the RPP chord length distribution, the main difference being one of terminology and perspective. However, as is often true, a change of perspective can lead to new insights, and although (3) is simply the chord distribution repackaged, a new and useful result is obtained in stage II of the development.

Several investigators have examined the shape of the experimental, normally incident, heavy ion upset cross section curve [2,3,4,5,6]. It is now generally accepted that the departure from ideal, step function-like response can be attributed to stochastic variations in E, although the exact sources of E- randomness remain unclear. The cross section data curve, appropriately normalized, thus represents the integral probability distribution of the upset threshold. Therefore, In Stage II, we modify the point-value model by treating E as a random variable and performing an expected value calculation,

( Q S , ( L , E ) ) f ( E ) d E E, = & . ( h + S f ) L, =LETthreshokl (6)

The Weibull density function is used for j ( E ) because the Weibull integral has been successful in curve fitting the cross section data. Integration of (6) yields,

Z(Q L )= [Ap (at) -an, L)]F[L’(S,(W] I (7)

where F ( x ) is the Weibull integral function,

F( x ) = 1 - exp( - x b ) b = shape parameter, (8)

L’(t) is a dimensionless variable given by,

and g( Q, L ) is the average removal cross section,

These equations were initially developed in Appendix B of [l]. This development is repeated in Appendix B of this paper with additional elaboration and, in the interest of simplicity, some modification of terms. Equations (7) - (10) define the HICUP model. It provides a closed-form analytic expression for the angular dependent cross section which will now be compared to experimental data.

111. Comparing HICUP With Experimental Data

Figure 2 compares measured data with the HICUP model. The data are plotted uncorrected (i.e. no inverse cosine scaling). The RPP lateral dimensions were obtained from the saturation cross section while the depth, h, and the funnel length, S,, can be estimated from charge collection models [8].

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9 (c+3gr-s) . Figure 2. Comparison of the expected value model with the data (from Sexton [7]) for the Rkl-05 SRAM.

The HICUP model of the data in Figure 2 is plotted in Figure 3 versus azimuth angle, 4~ , for various polar angles, 0 , and for a fixed LET of 70 MeV-cm2 /mg. In computing the RPP lateral surface dimensions, the aspect ratio, a = I / w , was assumed to be unity.

50,

270

Oi

Figure 3. Angular Dependence of the Upset Cross Section Based On the HICUP Model of the Rkl-05 SRAM. For This Case: 81=0,82-lC/6,83-l4,84=n/3, mi, L=70.

As expected of a square surface, the angular variations are not large, amounting to only a few percent. Figure 4 presents the angular dependence assuming an aspect ratio of 3. While the variations appear quite dramatic, the net effect is only ten percent, small relative to the other uncertainties in cross section measurement and upset rate prediction. Based on this quick look, it would appear that aspect ratio is not a major factor governing the upset rate and that, in the absence of direct process knowledge, assuming a square RPP is reasonable. We will examine this issue further in the following section on upset rate sensitivity analysis.

4

90 A 2.1-10-~

0

270

+i

Figure 4. Angular Dependence of the Upset Cross Section Based On the HICUP Model of the Rkl-05 SRAM. For This Case: 01=0,02=IC/6,03=rc/4, M d 3 , -3, L=70.

IV HICUP Scaling Laws

The inverse cosine scaling rules, given below,

L Om Leff =- a,$ F-, COS0 a s e

are often used to map cross section data, om, taken at tilt angle, 8, onto a curve of data taken at normal incidence. Discontinuities have been observed when data are presented using (1 1). This has become a recent topic of concern, bringing into question the validity of this transform and that of the simple RPP construct. The discontinuities occur at transition points when a lower LET ion sweep {usually 0 < 8 < 60' ) is completed and replaced with a higher LET ion. Figure 5 demonstrates the problem, showing Sexton's data [7] transformed by (11). Many factors have been identified as potential contributors, (a) failure to account for track structure (b) charge collection by diffusion from ion strikes occurring within a diffusion corridor surrounding the depletion region (c) geometric effects associated with the breakdown of the thin RPP assumption, and (d) charge collection by funneling.

5

Xe * 40 60 80 100 120 140

I / , * , , , , * , , I , 1E-10

20 Effective LET (Mev - cm2/ rng)

Figure 5. Discontinuities in the heavy ion upset cross section.

The following corrections have been proposed to account for finite depth effects neglected by the thin RPP assumption,

“etr =m cos0 -- sine

Petersen derived (12) on the basis that only ion strikes piercing both top and bottom of the RPP cause upsets. As shown in Figure 6, this forces the ion to travel the maximum chord for the given orientation, increasing ocff relative to inverse cosine scaling. In contrast, the basis for Sexton’s correction, (13), also depicted in Figure 6, is that any ion striking the RPP causes upset, regardless of path length, which leads to a reduction in oCff relative to inverse cosine scaling. Petersen considered (12) and (13) to be limiting forms, which bound a more general, L-dependent correction. Following this line of thought, we used the HICUP model to derive the exact form for the bridging function connecting (12) and (13).

I I

Figure 6. Correcting for finite Depth Effects.

The complete derivation of the HICUP scaling laws were given in [i] and resulted in the following expressions for effective LET and cross section,

which falls out naturally from the definition of & and is consistent with Golke’s criterion [13], and ,

With,

Equations (14) through (16) represent the generalized cross section scaling rules, which correct the inverse cosine scaling for finite thickness, funneling, and E-randomness. Plots of the bridging function for Sexton’s data are shown in Figure 7 at two different tilt angles.

‘a 40 60 80 loo 120 140 160

L ( MW - cm2/ ny

Figure 7. The bridging function.

7

Model

Xe

4E-7

h -= 3E-7 - a \

01 E o 2E-7 - Y

1E-7 -

OE+O 20 40 60 80 100 120 140 160

Leff (MeV - cm2/ rng)

Figure 8. Corrected upset cross section using the HICUP scaling equations.

Figure 8 shows Sexton's data corrected using the HICUP scaling. The solid curve in Figure 8 represents the HICUP model, O(O,O,L), drawn for comparison. Normally, the Weibull parameters would not be known a prioriand an iterative approach would be necessary to obtain the best fit.

V. Upset Rate Prediction

The classic method for rate prediction employs the RPP differential chord length distribution.

where, xp is the average projected area of the RPP and @(L) is the integral flux. Since E is stochastic, the standard approach [6] is to perform an integral weighting of R ( E ) with f ( E ) ,

where L,- is the maximum LET for the flux spectrum and S,, is the maximum diagonal of the RPP. In practice, (18) is computed numerically and the weighting is actually performed over the integral Weibull distribution,

j= l j=1

With the HICUP model the error rate expression is given by the comparatively simple expression,

JLIl

where, cp(L) is the differential flux and E( L ) is the angle-integrated HICUP cross section defined by,

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which for brevity will be called the I-HICUP model as distinct from the angle dependent HICUP model, equation (7). Approximate closed form solutions to (21) are obtainable and make for efficient numerical computation of the upset rate. Figure 9 presents a comparison of the measured cross section curve, representing normal incidence, with the I-HICUP model.

4E-7

n cy

0

c

3E-7 v

E 2E-7 1 u) u) 2 1E-7 0

z I .r: I . l , I . I , I .

0 20 40 60 80 100 120 140 160 LET (Mev-cmWmg)

Figure 9. Comparison of the normal-cross section curve with the I-HICUP for the Rkl-05 SRAM.

This integral form is more drawn out and exhibits a significantly lower value for the LET threshold. This should come as no surprise since, as shown by equation (3), the threshold, L,,,,,, , is governed by the directionally dependent maximum chord length. Figure 10 presents a SPACERAD-HICUP upset rate comparison for several devices. The SPACERAD results were obtained from a previous comparison of upset rate methodologies on these same devices [6] and the HICUP integral cross section for each device was developed from the Weibull parameters given in Table I of 161, repeated below.

Table I . Device Parameters.

Device RPP Parameters Weibull Parameters h Cr,, Lo Sf WL b

1 (m2) (Mev-em'/ mg) (rvn> (Mev-cm'/ mg) HM 6508 4 4.9e-6 48 0 50 1.15 6504 RH 2 1.7e-6 30.75 0 40 1.4

R4-25 1 1.2e-5 23.7 0 70 1 .I TCS-130 .4 5e-8 40 0 40 1.4 RK1-05 2 3.4e-7 29.5 1.6 47 2.7 4042 .5 1.2e-7 16.2 0 25 1.4

AS 200 1.8 3.12e-6 16.1 6 3.5 30 3.2 651 6 4 2.2e-6 4 0 12.4 2.7

6508 RH 2 4.2e-6 38.8 0 57 .75

For all cases, the Galactic Cosmic Radiation (GCR) environment was obtained from SPACERAD using a geostationary orbit with 100 mils of aluminum shielding and solar minimum conditions. The excellent agreement of the two methods is indicative of the shared set of initial assumptions, e.g. an RPP

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geometry and an upset threshold whose stochastic behavior is captured by the cross section data curve at normal incidence. The main difference is the order with which the upset parameters are processed. HICUP preprocesses all the relevant inputs prior to computing the upset rate, collapsing them into a single property, the integral upset cross section, a. This greatly simplifies the upset rate integral and yields a more intuitive expression, since E is then expressed as the product of flux-timescross-section, a standard concept. Finally since, a' is a function of the inputs parameters, p = ( A,a,h,Sf ,Eo,b,W), it becomes a useful tool for examining upset rate sensitivities to uncertainties in p .

-

1 E-5

- 1E-6 P W

¶i a 1E-7 t n

2. 3 1E-8 U 3 W m

1E-9

IL- I"

6516 4042 Rk 1-05 6508 RH HM6508 R160-25 AS 200 R4-25 6504 RH TCS130 R50-25

Device

Figure 10. Comparison of HICUP and SPACERAD methods of computing the heavy ion upset rate for several devices.

VI. Uncertainty Analysis

The upset rate sensitivity to uncertainties in the cross section input parameters is determined from as follows,

where,

---I a Z 1 - d Q xi- and p=(A,a ,h ,S f ,Eo ,b ,W) . a p - 4 ~ n a p

Figures 11 and 12 show the upset rate sensitivrty, for the devices in Table I. Each curve represents the percent variation in the upset rate given a 10 % uncertainty in a single input parameter, all other parameters being held constant.

10

E E 1 r

0.1 ii

0.01 ' I I I I t I I I 1 651 6 4042 Rkl-05 6508RH HM6508

AS 200 R4-25 6504RH TCS130 Device

Figure 11. Upset rate sensitivity to variations in the RPP parameters: depth, h, lateral surface area, A, and aspect ratio, a.

Figure 11 presents the sensitivities to the RPP parameters, indicating that the upset rate is most sensitive to uncertainties in depth, with a 10 % error in h leading to a 15 - 20 percent error in E. The next most sensitive RPP parameter is the lateral area, A, which is given by the saturation cross section, o,, , and generally produces a 10 - 15 percent variation in E. The least sensitive RPP parameter is the aspect ratio, which induces a variation in of, at most, a few tenths of a percent. While it has generally been recognized that a is not a critical parameter [2], Figure 11 demonstrates this in a clear and quantitative manner. This also further confirms the tentative conclusions drawn in Section Ill regarding the insensitivity of the angular dependent HICUP model to aspect ratio. This implies that, for the purpose of upset rate analysis, little error is induced if distinct sensitive areas within a cell are added together to form a single, square RPP provided that the distinct areas are governed by the same threshold stochastics and provided that the aspect ratios of the sensitive areas are not too large (i.e. a bit line).

60

df3b dRW dREo 50 -

h g 40

.1 30

Y - L.

2 20

c .- 0 C

10

U

651 6 4042 Rkl-05 6508 RH HM6508 AS 200 R4-25 6504 RH TCS130

Device Figure 12. Upset rate sensitivity to variations in the Weibull parameters: shape, b, width, W, and threshold, Eo.

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Figure 12 presents the sensitivities in E to the Weibull parameters and shows greater fluctuation among the devices than occurred with the RPP parameters. For the most part, it appears that the upset threshold, Eo , engenders the most sensitivity while the shape and width parameters are about equally impoftant. Further analysis, demonstrates that the relative importance of the parameters are inter-related with the shape parameter playing a dominant role. This is demonstrated in Figure 13, which presents sensitivity results for a device whose parameters are given by the Rk 1-05 with the exception that the shape parameter is allowed to vary from 1 .O to 3.0 (the true shape parameter for the Rk 1-05 is 2.7).

t 50

n $40 - .L 30

v s dRE .e .e I g 20

10

0 1 1.5 2 2.5 3 3.5

Weibull Shape Parameter, b Figure 13. Influence of Weibull shape parameter on upset rate sensitivity to other Weibull parameters. All other parameters identical to the Rk 1-05 SRAM.

This clearly shows that R-sensitivity increases to all the Weibull parameters as b increases, with the greatest effect being on W. Figure 13 also shows sensitivity results for the funnel parameter, S’ , which was not given in Figure 11 because only two of the devices (Rk 1-05 and AS 200) exhibited a funnel effect. For both of these devices, a 10 % uncertainty in S’ produced roughly a 15 % variation in R .

VII. Summary

In addition to their utility in analyzing and scaling cross section data, the HICUP models can be used to perform upset rate analysis. The I-HICUP methodology was shown to yield results equivalent to the standard chord length model. However, the I-HlCUP approach maintains continuity with the concept of upset cross section and permits more detailed analysis of upset rate, such as the uncertainty analysis presented here. We plan to provide a subroutine of the HICUP models for inclusion into Sevem Communications’ SPACE RADIATION, code for those interested in testing them out.

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DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recorn- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

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