Table of Contents
CHAPTER: The PEARSON IV DISTRIBUTION ALGORITHM and its APPLICATION to ION IMPLANTED DEPTH PROFILES
Fig1. (-15) Gaussian profiles (g1 = 0)
Fig 2. (-16) Effect of kurtosis for zero skew
Fig 3. (-14) Flat-topped modified Gaussian profiles
Fig 4. (-13) Effect of skew for more pointed modified Gaussian profiles
Fig 5. (-12) Effect of skew for more flat-topped modified Gaussian profiles
Fig 6. (-1) Regions of validity and profiles shapes for kurtosos (b2) versus skew (g1)
CHAPTER: The PEARSON IV DISTRIBUTION ALGORITHM and its APPLICATION to ION IMPLANTED DEPTH PROFILES
The Pearson IV distribution algorithm is used to describe ion implanted depth distributions in the literature.1-5 The Pearson VI distribution is less frequently used but is generally also applicable, and even preferred by a few authors. The Pearson algorithms describe a statistical distribution in terms of four central moments, m, s, g1, and b2, based on a modified or skewed Gaussian distribution. Ion implanted depth profiles are basically modified Gaussian distributions, or can be described in such terms -- therefore the correspondence with, and use of, Pearson distributions to describe implanted depth profiles. Depth profiles in amorphous or amorphized materials can be fit very well by Pearson IV distributions. One difficulty that is sometimes encountered is that channeling tails on depth distributions in crystalline materials cannot be fit well, no matter how the crystal is oriented in an attempt to minimize channeling. Channeling in good crystalline materials cannot be eliminated because ions incident in any direction can be scattered into certain crystallographic directions that may have relatively large diameters, such as the <110> of Si and other diamond lattice crystals. The degree of channeling varies with the electronic stopping Se of the combination of incident element Z1 and the elements of the target matrix Z2, realizing that significant oscillations occur in the values of Se with Z1 and Z2. The magnitude of Se and the corresponding degree of channeling varies as the incident ion energy to the 1/2 power, so the degree of channeling increases with decreasing energy and decreasing mass, in general.
In the Pearson IV system, for a purely Gaussian distribution, the third moment is zero and the value of the fourth moment must be 3±e, where epsilon is less than ~ 10-2, and the first moment, m, is known as the mean, and the second moment, s, is known as the standard deviation, of the distribution, and the distribution is symmetric. For a modified Gaussian distribution with non-zero value of third moment and appropriate value of fourth moment, another descriptor is used, the mode of the distribution, or the most probable value of the distribution, Rm.
In ion implantation terminology, the first central moment is known as the projected range or Rp, and is the depth for which the integrals of the distribution for depths less than and greater than Rp are equal, or, the depth for which the number of implanted ions deeper than Rp is equal to the number of implanted ions at depths less than Rp. The word 'projected' means that the actual three-dimensional depth trajectories are 'projected' onto the axis (depth) normal to the 'surface' (depth equal to zero), and are not the total or actual ranges (distances) traveled by the ions during the statistical processes of losing energy through electronic and nuclear energy loss collisions. The total ranges or trajectories or paths are longer than the projected ranges. The second central moment is known as the range straggle (longitudinal) or DRp, and is also "projected" onto the axis normal to the 'surface.' The corresponding parameter that describes the 'width' of the distribution in the direction normal to the 'depth' axis or parallel to the 'surface.' is called the lateral DRp or DRp . The third central moment is called the skew of the distribution g1, and has positive and negative values. Positive skew or positive values of g1 means that the distribution is 'slumped' or 'cocked' or 'tilted' or 'pushed at the top' toward the surface (zero value of m or depth), and negative values of skew or negative values of g1 means that the distribution is slumped or tilted or pushed at the top or 'skewed' toward greater depth. The fourth central moment is called the kurtosis, and describes how much the distribution is 'pinched from the sides', to form a more sharply peaked shape (values of fourth moment greater than the number 3 for the Pearson IV distribution), or 'pushed down from the top' to form a more flatly topped distribution (value of fourth moment less than 3 for the Pearson IV distribution). For the Pearson IV algorithm, the numerical value 3 is a mathematical analytical point in the equations for the third moment, which affects the fourth moment as described above (See discussion below for a description of the regions of characteristic shape and validity specifically for the Pearson IV representation.)
For ion implantation depth distributions, the mode of the distribution is known as Rm, and is the depth of the maximum or peak of the depth distribution, and is different from, and not to be confused with, the value of Rp or projected range, for a distribution with non-zero value of third moment g1, or for a "skewed" or "modified" Gaussian distribution. Caveat scientor: Rp is NOT the depth of the peak of the depth distribution. This error has sometimes been made in the literature and causes confusion and errors in comparisons between experimental data and calculated depth distribution parameters such as those calculated using the TRIM (Monte Carlo) and MARLOWE programs. The difference between the values of Rp and the peak depth of a distribution can be significant if the absolute value of third moment or skewness of the distribution is significantly greater than about 0.1.
A discussion of the Pearson IV distribution and its application to ion implanted depth distributions is given in reference 3, and is repeated here in part for completeness of this treatment and to save the reader the need to search out the reference in the library. An important purpose of this treatment is to define the regions of combined values of the third and fourth central moments, g1 and b2, that yield 'valid' or practical or meaningful shapes for implanted depth distributions and that describe various shapes of depth distributions such as convex, concave, more sharply peaked or more flatly topped than Gaussian, the limits (magnitudes) beyond which the values of third and fourth moment do not cause any significant (measurable) change in shape, and regions where no relevant distribution occurs.
The mathematical equations for the four moments of the Pearson IV distribution can be found in appropriate books on statistical mathematics, and also in treatments specifically related to ion implanted depth distributions, for example, references 1 and 3. [We apologize for a few errors in reference 3, related to the distinction between mode and average depths, and Rp in reference 3.]
Referring to the basic differential equation, there exist three conditions on the coefficients b0, b1, and b2. We are concerned here with only the first of these conditions, b12 - 4b0b2 < 0. The distributions of interest have a single mode at x = a. The first moment m is about x = 0. The first central moment does not enter into the expressions for the shape of the distribution; it only defines the location x in depth of the distribution. The second, third, and fourth central moments are about <x>, where the x axis is parallel to the direction of incident ions (all assumed to be co-parallel) and x = 0 represents the surface.
An analytical point occurs at b2 = 3. Gaussian profiles are shown in Fig. 1 for a range of values of second moment from 0.03 to 0.40 (g1= 0 and b2= 3±e). For no skew (g1 = 0), the effect of kurtosis (fourth moment) is shown in Fig. 2 for distributions more pointed than or more sharply peaked than Gaussian (b2 > 3) for s = 0.10 and b 2 = 3+e, 4, 7, and >30. The effect of kurtosis that produces distribution more flatly topped than Gaussian is shown in Fig. 3 for s = 0.10, g1 = 0, and b2 = 2.99 (3-e), 2.50, 2.25, and 2.00. For g1 = 0, b2 < 2 gives no meaningful result, and b2 > 30 is indistinguishable from b2 = 30.
The effect of skew for a fixed second moment of s = 0.10 and a range of values of kurtosis where kurtosis produces distributions more pointed than Gaussian is shown in Fig. 4 for the following selected values:
g1 +0.3 +1.0 +2.0 +3.0
b2 3.4 5.0 15 50
Mirror images of these distributions would result from making the values of g1 negative.
The effect of skew for a fixed second moment of s = 0.10 and a value of kurtosis that produces distributions that are more flatly topped than Gaussian is shown in Fig. 5 for b2 = 2.50 and g1 = +0.1, +0.2, and + 3.0. Again, mirror images of these distributions would result from making the values of g1 negative.
As the magnitude of skew increases, the magnitude of kurtosis that corresponds to no actual kurtosis increases from 3±e at g1 = 0, being 4.96 at g1 = b2±1, ~ 10 at g1 = 10, and ~ 20 at g1 = ±3, etc. The curve labeled A in Fig. 6 gives this boundary value as a function of g1, up to g1 = ±3. The shaded region of Fig. 6 marked by the boundary line D shows the region where the modified Gaussians change from convex to concave as b2 increases. The boundary limit labeled B shows the appropriate values of b2 above which larger values of b2 do not produce significant changes in the distributions.
The region roughly bounded by the curve labeled E in Fig. 6 corresponds to the region within which flatly topped modified Gaussians can be obtained. The remainder of the region between curves A and C is an "excluded" region within which distributions do not correspond to implanted depth distributions. In the region below curve C, sharply pointed concave curves are obtained. They are more sharply pointed than experimental implanted depth distributions are, but they have the correct general nature and might be considered.
Reference to Fig. 6 can assist in judicious selection of initial values for g1and b2 when an experimental curve is to be fitted or matched to values of moments, and thereby can save time in reaching the final moments, if an iterative manual procedure is to be used.
A better approach is to program a computer with the Pearson equations, and develop or use a curve fitting program, and input digital data that describes the profile, either directly from a tabulation of digital data, or via the use of a digitizing tablet and tracing of the profile, or an appropriate computer program.
Relative kurtosis: The value of b2 or kurtosis in the Pearson IV system does not reflect the actual magnitude or amount of fourth moment present in a modified Gaussian because of the non-zero value of b2 (b2 = 3) for g1 =0 that corresponds to the absence of all fourth moment or kurtosis and the increasing value of b2 that corresponds to no fourth moment as the magnitude of the third moment g1 increases. The concept of a "relative" kurtosis has proved useful to see more clearly how much fourth moment actually exists in a modified Gaussian distribution. A relative kurtosis b2' is defined here as the calculated Pearson IV kurtosis b2 minus the value of b2 that corresponds to zero fourth moment for the skewness g1 in the same distribution. This no-kurtosis value increases with increasing third moment and is determined from curve A in Fig. 6.
References
1. W.K. Hofker, Philips Research Reports Supplements, No.8, 41-57 (1975)
2. W.K. Hofker, Radiat. Effects 25, 205-06 (1975)
3. R.G. Wilson, Radiat. Effects 46, 141-48 (1980)
4. F. Jahnel, H. Ryssel, G. Prinke, K. Hoffmann, K. Mueller, J. Biersack, and R. Henkelmann, Nucl. Intrum. Meth. 181/182, 223-29 (1981)
5. A.F. Burenkov, F.F. Komarov, and M.M. Temkin, Radiat. Effects 66, 115-18 (1982)
Figure captions
1. (-15) Gaussian profiles (g1 = 0)
2. (-16) Effect of kurtosis for zero skew
3. (-14) Flat-topped modified Gaussian profiles
4. (-13) Effect of skew for more pointed modified Gaussian profiles
5. (-12) Effect of skew for more flat-topped modified Gaussian profiles
6. (-1) Regions of validity and profiles shapes for kurtosos (b2) versus skew (g1)
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