# ASTM E1636-10

Designation: E1636 − 10Standard Practice forAnalytically Describing Depth-Profile and Linescan-ProfileData by an Extended Logistic Function1This standard is issued under the fixed designation E1636; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice describes a systematic method for analyz-ing depth-profile and linescan data and for accurately charac-terizing the shape of an interface region or topographic feature.The profile data are described with an appropriate analyticfunction, and the parameters of this function define theposition, width, and any asymmetry of the interface or feature.The use of this practice is recommended in order that theshapes of composition profiles of interfaces or of linescans oftopographic features acquired with different instruments ortechniques can be unambiguously compared and interpreted.1.2 This practice is intended to be used for two purposes.First, it can be used to describe the shape of depth-profilesobtained at an interface between two dissimilar materials thatmight be measured by common surface-analysis techniquessuch as Auger electron spectroscopy, secondary-ion massspectrometry, and X-ray photoelectron spectroscopy. Second, itcan be used to describe the shape of linescans across adetectable topographic feature such as a step or a feature on asurface that might be measured by a surface-analysistechnique, scanning electron microscopy, or scanning probemicroscopy. The practice is particularly valuable for determin-ing the position and width of an interface in a depth profile orof a feature on a surface and in assessments of the width as anindication of the sharpness of the interface or feature (acharacteristic of the material system being measured) or of theachieved depth resolution of the profile or the lateral resolutionof the linescan (a characteristic of the particular analyticaltechnique and instrumentation).1.3 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.4 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2E673 Terminology Relating to SurfaceAnalysis (Withdrawn2012)3E1127 Guide for Depth Profiling in Auger Electron Spec-troscopyE1162 Practice for Reporting Sputter Depth Profile Data inSecondary Ion Mass Spectrometry (SIMS)E1438 Guide for Measuring Widths of Interfaces in SputterDepth Profiling Using SIMS2.2 ISO Standards:4ISO 18115 Surface Chemical Snalysis – Vocabulary, 2001;Amd. 1:2006, Amd. 2:2007ISO 18516 Surface Chemical Analysis – Auger ElectronSpectroscopy and X-Ray Photoelectron Spectroscopy –Determination of Lateral Resolution, 20063. Terminology3.1 Definitions—For definitions of terms used in thispractice, see Terminology E673 and ISO 18115.3.2 Definitions of Terms Specific to This Standard:3.2.1 Throughout this practice, three regions of a sigmoidalprofile will be referred to as the pre-interface, interface, andpost-interface regions. These terms are not dependent onwhether a particular interface or feature profile is a growth ora decay curve. The terms pre- and post- are taken in the senseof increasing values of the independent variable X, the depth(for a depth profile) or the lateral position on the surface (for alinescan).1This practice is under the jurisdiction of ASTM Committee E42 on SurfaceAnalysis and is the direct responsibility of Subcommittee E42.08 on Ion BeamSputtering.Current edition approved Jan. 1, 2010. Published March 2010. Originallyapproved in 1999. Last previous version approved in 2004 as E1636 – 04. DOI:10.1520/E1636-10.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at service@astm.org. For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.3The last approved version of this historical standard is referenced onwww.astm.org.4Available from International Organization for Standardization (ISO), 1, ch. dela Voie-Creuse, Case postale 56, CH-1211, Geneva 20, Switzerland, http://www.iso.ch.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States14. Summary of Practice4.1 Depth-profile data for an interface (that is, signal inten-sity or composition versus depth) or linescan data (that is,signal intensity or composition versus position on a surface)are fitted to an analytic function, an extended form of thelogistic function, in order to describe the shape of suchprofiles.5,6Least-squares fitting techniques are employed todetermine the values of the parameters of this extended logisticfunction that characterize the shape of the interface. Theinterface width, depth or position, and asymmetry are deter-mined from these parameters.5. Significance and Use5.1 Information on interface composition is frequently ob-tained by measuring surface composition while the specimenmaterial is gradually removed by ion bombardment (see GuideE1127 and Practice E1162). In this way, interfaces are revealedand characterized by the measurement of composition versusdepth to obtain a sputter-depth profile. The shape of suchinterface profiles contains information about the physical andchemical properties of the interface region. In order to accu-rately and unambiguously describe this interface region and todetermine its width (see Guide E1438), it is helpful to definethe shape of the entire interface profile with a single analyticfunction.5.2 Interfaces in depth profiles from one semi-infinite me-dium to another generally have a sigmoidal shape characteristicof the cumulative logistic distribution. Use of such a logisticfunction is physically appropriate and is superior to otherfunctions (for example, polynomials) that have heretofore beenused for interface-profile analysis in that it contains theminimum number of parameters for describing interfaceshapes.5.3 Measurements of variations in signal intensity or surfacecomposition as a function of position on a surface giveinformation on the shape of a step or topographic feature on asurface or on the sharpness of an interface at a phase boundary.The shapes of steps or other features on a surface can giveinformation on the lateral resolution of a surface-analysistechnique if the sample being measured has sufficiently sharpedges (see ISO 18516). Similarly, the shapes of compositionalvariations across a surface can give information on the physicaland chemical properties of the interface region (for example,the extent of mixing or diffusion across the interface). It isconvenient in these applications to describe the measuredlinescan profile with an appropriate analytic function.5.4 Although the logistic distribution is not the only func-tion that could be used to describe measured linescans, it isphysically plausible and it has the minimum number ofparameters for describing such linescans.5.5 Many attempts have been made to characterize interfaceprofiles with general functions (such as polynomials or errorfunctions) but these have suffered from instabilities and aninability to handle poorly structured data. Choice of the logisticfunction along with a specifically written least-squares proce-dure (described in Appendix X1) can provide statisticallyevaluated parameters that describe the width, asymmetry, anddepth of interface profiles or linescans in a reproducible andunambiguous way.6. Description of the Analysis6.1 Logistic Function Data Analysis—The logistic functionwas first named and applied to population growth in the 20thcentury by Verhulst.7In its simplest form, this function may bewritten as:Y 5111e2x(1)in which Y progresses from 0 to 1 as X varies from −∞to +∞.The differential equation generating this function is:dY/dX 5 Y~1 2 Y! (2)and in this form describes a situation where a measurablequantity Y grows in proportion to Y and in proportion to finiteresources required by Y. Appropriate to an interface, thepropensity for change in the fractional composition of a speciesat a particular boundary is proportional to the concentration ofthat species at the boundary and the concentration of the otherspecies at the adjacent boundary. The logistic function as adistribution function and growth curve has been extensivelyreviewed by Johnson and Kotz.8Interface or linescan profiledata are usefully fitted to an extended form of the logisticfunction:Y 5 @A1As~X 2 X0!#/~11ez! (3)1@B1Bs~X 2 X0!#/~11e2z!where:z 5 ~X 2 X0!/D (4)and:D 5 2D0/@11eQ~X2X0!# (5)6.1.1 Y is a measured signal (for example, from a surface-analysis instrument, a scanning electron microscope, or ascanning probe microscope) or a measure of the elementalsurface concentration of one of the components and X, theindependent variable, is a measure of the sputtered depth,usually expressed as a sputtering time, or lateral position on thesurface. Pre-interface and post-interface signals or surfaceconcentrations are described by the parameters A and B,respectively, and the parameters Asand Bsare introduced toaccount for any time-dependent instrumental effects or other-wise to better describe the shape of the measured profile. X0isthe midpoint of the interface region (depth or time for a profileor of position for a linescan). The scaling factor D0is a5Kirchhoff, W. H., Chambers, G. P., and Fine, J., “An Analytical Expression forDescribing Auger Sputter Depth Profile Shapes of Interfaces,” Journal of VacuumScience and Technology A, Vol 4, 1986, p. 1666.6Wight, S. A. and Powell, C. J., “Evaluation of the Shapes of Auger- andSecondary-Electron Line Scans across Interfaces with the Logistic Function,”Journal of Vacuum Science and Technology A, Vol 24, 2006, p. 1024.7Verhulst, P. F., Acad. Brux, Vol 18 , 1845, p. 1.8Johnson, N. L., and Kotz, S., Distributions in Statistics: Continuous UnivariateDistributions, Chapter 22, Houghton Mifflin Co., Boston, 1970.E1636 − 102characteristic depth for sputtering through the interface regionof a depth profile or a characteristic width for a linescan; Q,anasymmetry parameter, is a measure of the difference incurvature in the pre- and post-interface ends of the interfaceregion. Conventional measures of the interface width can bedetermined from D0and Q. Fig. 1 shows examples of profileshapes from Eq 3-5 for illustrative values of D0and Q.56.2 Fitting of interface-profile data to the above function, Eq3, can be accomplished by using least-squares techniques.Because these equations are non-linear functions of the threetransition-region parameters, X0, D0, and Q, the least-squaresfit requires an iterative solution. Consequently, Y, as expressedby Eq 3, can be expanded in a Taylor series about the currentvalues of the parameters and the Taylor series terminated afterthe first (that is, linear) term for each parameter.Y(obs) − Y(calc) is fitted to this linear expression and theleast-squares routine returns the corrections to the parameters.The parameters are updated and the procedure is repeated untilthe corrections to the parameters are deemed to be insignificantcompared to their standard deviations. Values for interfacewidth, depth, and asymmetry can be calculated from theparameters of the fitted logistic function. The iterative solutionalso requires a robust means for making initial estimates of theparameter values.6.3 Implementation of this procedure can be readily accom-plished by making use of a specialized computer algorithm andsupporting software (logistic function profile fit (LFPF)) de-veloped specifically for this application and described inAppendix X1.6.3.1 The fitting can also be done in Excel, using the solveroption to determine the variables A, B, As, Bs, X0, D0, and Q.Write the definition of the logistic function (Eq 3-5) in Exceland calculate its values as a function of X. If the exponentialfunction ezproduces overflow when z 709, this problem caneasily be circumvented by writing EXP (min (z, 709)) insteadof EXP(z).6.3.2 The fitting can also be done with any suitable nonlin-ear least-squares software that is available.7. Interpretation of Results7.1 The seven parameters necessary to characterize theinterface-profile shape are determined by a least-squares fit ofthe interface data to the extended logistic function. Theseparameters are related to the three distinct regions of theinterface profile. Two parameters, an intercept A and a slope Asare necessary to define the pre-interface asymptote while twomore, B and Bs, define the post-interface asymptote. For theanalysis of many interface profiles, it may be satisfactory toassume that both of the slope parameters, Asand Bs, are zero.Two more parameters, D0and X0, define the slope and positionof the transition region. In addition, an asymmetry parameter Qthat causes the width parameter to vary logistically from 0 to2D0, is introduced as a measure of the difference in curvaturein the pre- and post-transition ends of the transition region. IfQ 0, the post-transition region has the greatestcurvature. If Q =0, D = D0and the transition profile issymmetric.The parameter Q has the dimensions of1⁄ X whereasFIG. 1 Plot of Eq 3-5 Showing Relative Intensity as a Function of Relative Position X with A = As= Bs= X0=0,B = 100, D0= 10 nm, andthe Indicated Values of Q (from the paper referenced in Footnote 5)E1636 − 103D0has the dimensions of X. The product QD0is dimensionlessand is a measure of the asymmetry of the profile independentof its width. If the absolute magnitude of QD0is less than 0.1,the asymmetry in the transition profile should be barelydiscernible. Fig. 1 shows illustrative plots of the logisticfunction (Eq 3-5) for values of QD00, 0.05, 0.1, 0.2, and 0.5.7.2 The final results should include the calculated values ofY and associated statistics, the values of the determinedparameters and their uncertainties, and statistics related to theoverall quality of the least-squares fit.7.3 The width of the interface region, If, is the depth (time)or distance required for the decay or growth curve to progressfrom a fraction f of completion to (1 − f) of completion. For thecase where Q =0,Ifis proportional to D0and is given by thesimple formula:If5 2D01n @~1 2 f!/f# (6)so that, for example, the traditional 16 % to 84 % interfacewidth is 3.32 D0. Similarly, the interface widths determinedfrom the 10 % to 90 %, 12 % to 88 %, 20 % to 80 %, and 25 %to 75 % intensity changes are 4.39D0, 3.99D0, 2.77D0, and2.20D0, respectively.7.4 Introduction of the asymmetry parameter Q into theextended logistic function makes the ca