An Independent Journal Dedicated to the Advancement of Chip - Scale Electronics July - August 2000 Email the editor

Uniform Dispersal

To calculate the flux density of a particles emitted from a thin sheet of material per unit area and unit time, the following geometry is considered (Figure 3).

Figure 3. This illustration depicts the geometry used to calculate the flux density of alpha particles for the products analyzed.

The radiating impurities, uranium and thorium, are uniformly dispersed in the matrix material at low concentration; the sheet is very thin in comparison to its lateral dimensions; and half of the total flux of particles that are not stopped inside the sheet exits one surface of the sheet, that is, edge effects are neglected.

Let the impurity concentration c be measured as a weight fraction of the matrix material which has density p_m. Then the total a particle emission from 1 g of this material is

R_w = c R₀      (1)

In a small volume dV_m of this material there is a mass dm = p_m dV_m from which particles are emitted at a rate of

dR_v = dm R_w = c p_m R₀ dV_m      (2)

into the full sphere of 4¶ steradian.

Inside the material, the a particles have a mean range L over which they lose their energy, indicated by the circle in Figure 3. Only that conical fraction q of the sphere of radius L that penetrates the surface of the material contributes to the alpha particle flux through the surface.

If we consider the volume in the form of a thin sheet of thickness ds and area A so that dV_m = A.ds, inside the bulk of a material at a distance s from its surface, then the flux dr of a particles emitted per unit area and unit time through one of the surfaces of the sheet can be expressed as (dR_v/A) q(s) or

dr = c p_m R₀ q(s) ds      (3)

where

q(s) = 1/2L²(L - s)/L³ = 1/2(1 - s/L)      (4)

is the ratio of the volume of the cone reaching above the surface, to the volume of the sphere given by the mean range L, see Figure 4.

Figure 4. Derivation of a particle escape fraction.

The total flux from all depths of a sheet of material with thickness d is obtained by integrating equ. (3) over the mean range L assuming the thickness d of the material is larger than the alpha particle range in the material.

The alpha particle flux is essentially constant over this range, i.e., independent of position,⁷ with only the particle energy decreasing until it has travelled a distance L. With these conditions one obtains the following equation expressing the particle flux density at one of the surfaces as

r = 1/2 c p_mR₀ (1-s/L) ds = 1/4 c p_mR₀ x L      (5)

for d > L or d = L.

This expression depends only on materials parameters, including the a particle energy, which determine the mean range L in a material, a value which can be calculated for a specific material composition using tabulated data (6,7).

For sheets thinner than L, equ.(5) represents an upper limit since the alpha particle flux density will be smaller for less material contributing to it. The exact expression is found by integration between 0 and d which gives

r = 1/2 c p_m d(1- 1/2• d/L) R₀•      (6)

For d = L, the two expressions are equal.

To summarize, the variables used in the derivation are:

For a concentration of uranium of 15 ppb (i.e., 1.5x10^-8) in a matrix material of density p = 1.2 g cm^-3 having a mean alpha particle range of L = 20 µm, and forming a sheet of thickness d = 100 µm, the alpha particle flux density through each of its two surfaces is

r_example = 1/4 x 1.5x10^-8 x 2x10^-3 x 1.2 x 2.5x10⁴ = 2.3x10^-7 cm^-2 s^-1 .

In practical applications, this is more commonly expressed as an hourly flux density, either by using the elemental rates R'₀ in the appropriate unit, [g^-1 h^-1], or by multiplying equ. (5) or (6) by 3600 to give

r' = 900 c L p_m R₀      (7)

making the numerical example have an emission flux density of r'_example = 0.0008 cm^-2 h^-1.

The a particle emission rates from different source elements simultaneously present in a material are additive, that is, the total rate r_tot is calculated as the sum of the individual rates as

r_tot = _i r_i = 1/4 p_{m i} c_i L_i R_0i      (8)

where the subscript i refers to each atomic species. Usually, this applies to uranium and thorium which are commonly both present in the ppb to ppm range in natural and man-made materials.

Surface Densities

From the U and Th concentrations determined for the four Dow Corning products discussed earlier, we calculated the a particle flux densities at the surface of an elastomer sheet 5 mils thick (typical thickness in CSP packages), prepared from each material using a density value of p = 1.2 g cm^-3 and a mean range of L = 23 µm as shown in Table 2.

The latter value is calculated by comparison with air as reference⁶ using the atomic composition of the elastomer products according to

L = 3.2 x 10^-4A/pL₀      (9)

where A is a weighted average of the atomic numbers of all elements in the material.

Thus, the 6811 product, which has been formulated for DRAM applications, would emit only 2x10^-5 a particles from a 5-mil thick sheet every hour from an area of 1 cm²; to put this into perspective, that is one a particle emitted about every six years on average.

Also note that the value calculated above for the 6810 material compares fairly well to the measured value of r' = 12x10^-3 cm^-2 h^-1 reported earlier.

One possible explanation for the existing discrepancy between measured and predicted values of r0 stems from using monoenergetic a particles at 4.1 MeV in the theoretical flux density calculation, when, in actuality, U and Th in secular equilibrium are accompanied by other elements that emit a particles at higher energies.

Summary

In materials employed to package DRAM chips, low a particle emission rates are critical, in addition to the other accepted performance requirements for microelectronics.

Silicones formulated for DRAM applications offer very low U and Th content. This article⁸ has presented the equations derived for calculating the a particle flux densities from the particle emission rates of the elements and their concentrations, for predicting soft errors in DRAMs.

References

1. A. Norris and M. Gladstone, "Silicone Materials for Chip-Scale Packaging," Chip Scale Review, March-April 1998, pp 84-86.

2. T. May and M. Woods, "A New Physical Mechanism for Soft Errors in Dynamic Memories," Proc. 16^th Inernational Reliablity Physics Symposium," April 1978, p.33.

3. D. Douldein and A. Kumar, "Materials Analysis Aspects of theAlpha Particle Induced Error Phenomenon," Proc. 29^th Electronic Components and Technology Conference, May 1979, p. 265.

4. ILCO, alpha particle testing conducted at Spectrum Sciences, P.O. Box 3567, Saratoga, CA 95070.

5. M. White, J. Serpiello, et al., "The Use of Silicone TRV Rubber for Alpha Particle Protection on Silicon Integrated Circuits," Proc. 19^th Reliability Physics Symposium, April 1981, p. 43.

6. F. Kohlrausch, Praktische Physik, Vol. 3, B.G. Teubner, Stuttgart, Germany, 1968, p. 128.

7. I. Kaplan, Nuclear Physics, Addison-Wesley Publishing Co., Reading, Mass., 1955, p. 251.

8. Presented in part at the Pan Pacific Microelectronics Symposium, Maui, Hawaii, January 25, 2000.

Dr. Norris is an associate scientist in the Microelectronics Product Develop-ment Group at Dow Corning Corp. She joined Dow Corning in 1980 with a bachelor's degree in chemistry from the University of Wisconsin at LaCrosse. She later left the company to earn her Ph.D. in materials engineering science from Virginia Poly-technic Institute and State University in 1987. She rejoined Dow Corning upon completion of her doctorate. [a.norris@dowcorning.com]

Dr. Norris

	Dr. Pernisz is a research scientist in Midland, where he has worked in Central R & D since 1983. He received the degrees Dipl. Phys. (M.Sc.) and Dr. rer. nat. (D. Sc.) in exper-imental solid-state physics from UniversitÉt Struttgart, Germany. He currently focuses on the fundamental understanding of the physics of Si-containing materials. [udo.pernisz@dowcorning.com]
Dr. Pernisz

back