1、Designation: E 1636 04Standard Practice forAnalytically Describing Sputter-Depth-Profile Interface Databy an Extended Logistic Function1This standard is issued under the fixed designation E 1636; the number immediately following the designation indicates the year oforiginal adoption or, in the case
2、of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (e) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice covers a systematic method for analyzingsputter-depth-profile interface data a
3、nd for accurately charac-terizing the shape of the interface region. Interface profile dataare described with an appropriate analytic function; the param-eters of this function define the interface width, its asymmetry,and its depth from the original surface. The use of this practiceis recommended i
4、n order that the shapes of compositionprofiles of interfaces acquired with different instruments andtechniques on different materials can be unambiguously com-pared and interpreted.1.2 This practice is intended to be used to describe the shapeof depth profile data obtained at an interface between tw
5、odissimilar materials for that case in which the measuredconcentration of the outer material goes from 100 to 0 % andthe inner material goes from 0 to 100 %.1.3 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user o
6、f 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:2E 673 Terminology Relating to Surface AnalysisE 1127 Guide for Depth Profiling in Auger Electron Spec-troscopyE 116
7、2 Practice for Reporting Sputter Depth Profile Data inSecondary Ion Mass Spectrometry (SIMS)E 1438 Guide for Measuring Widths of Interfaces in SputterDepth Profiling Using SIMS3. Terminology3.1 DefinitionsFor definitions of terms used in this prac-tice, see Terminology E 673.3.2 Definitions of Terms
8、 Specific to This Standard:3.2.1 Throughout this practice, the regions of the sigmoidalprofile will be referred to as the pre-interface, interface, andpost-interface regions. These terms are not dependent onwhether a particular interface profile is a growth or a decaycurve. The terms pre- and post-
9、are taken in the sense ofincreasing values of the independent variable X, the sputtereddepth.4. Summary of Practice4.1 Sputter depth profile interface data (composition versusdepth) is fitted to an analytic function, an extended form of thelogistic function, in order to describe the shape of suchint
10、erface profiles.3Least-squares fitting techniques are em-ployed to determine the values of the parameters of thisextended logistic function which characterize the shape of theinterface. Interface width, depth, and asymmetry are deter-mined by these parameters.5. Significance and Use5.1 Information o
11、n interface composition is frequently ob-tained by measuring surface composition while the specimenmaterial is gradually removed by ion bombardment (see GuideE 1127 and Practice E 1162). In this way, interfaces arerevealed and characterized by the measurement of compositionversus depth to obtain a s
12、putter-depth profile. The shape ofsuch interface profiles contains information about the physicaland chemical properties of the interface region. In order toaccurately and unambiguously describe this interface regionand to determine its width (see Guide E 1438), it is necessaryto define the shape of
13、 the entire interface profile with a singleanalytic function.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 Nov. 1, 2004. Published December 2004. Originallyappro
14、ved in 1999. Last previous version approved in 1999 as E 163694 (1999).2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe AST
15、M website.3Kirchhoff, W. H., Chambers, G. P., and Fine, J., “An Analytical Expression forDescribing Auger Sputter Depth Profile Shapes of Interfaces,” Journal of VacuumScience and Technology, , p. 1666, 1986.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 194
16、28-2959, United States.5.2 Although no general physical model currently exists fordescribing the shape of interface sputter-depth profiles, inter-face profiles do have a sigmoidal shape characteristic of thecumulative logistic distribution. Use of such a logistic functionis physically plausible and
17、is superior to other functions (forexample, polynomials) that have heretofore been used forinterface profile analysis in that it contains the minimumnumber of parameters for describing interface shapes.5.3 Many attempts have been made to characterize interfaceprofiles with general functions (such as
18、 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 describ
19、e the width, asymmetry, anddepth of interface profiles in a reproducible and unambiguousway.6. Description of the Analysis6.1 Logistic Function Data AnalysisIn its simplest form,the logistic function may be written as:Y 511 1 e2x(1)in which Y progresses from 0 to 1 as X varies from to +.The differen
20、tial equation generating this function is:dY/dX 5 Y1 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. The logistic function was first namedand applied to population growth in the 20th century byVe
21、rhulst.4The logistic function as a distribution function andgrowth curve has been extensively reviewed by Johnson andKotz.5Interface profile data is fitted to an extended form of thelogistic function:Y 5 A 1 AsX 2 Xo!#/1 1 ez!1 B 1 BsX 2 Xo!#/1 1 e2z! (3)where:z 5 X 2 Xo!/D (4)and:D 5 2 Do/1 1 eQX2X
22、o!# (5)6.1.1 Y is a measure of the elemental surface concentrationof one of the components and X, the independent variable, is ameasure of the sputtered depth, usually expressed as a sputter-ing time. Pre-interface and post-interface elemental surfaceconcentrations are described by the parameters A
23、and B,respectively, the parameters Asand Bsare introduced to accountfor time dependent instrumental effects. Xois the midpoint ofthe interface region (interface depth or time).The scaling factorDois the characteristic depth for sputtering through theinterface region; Q, an asymmetry parameter, is a
24、measure ofthe difference in curvature in the pre- and post-interface ends ofthe interface region.All measures of the interface width can bedetermined from Doand Q.6.2 Fitting of interface profile data to the above functions,Eq 3 , can be accomplished by using least-squares techniques.Because these e
25、quations are non-linear functions of the threetransition-region parameters, Xo, Do, 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 afterth
26、e first (that is, linear) term for each parameter. Y (obs) Y(calc) is fit to this linear expression and the least-squaresroutine returns the corrections to the parameters. The param-eters are updated and the procedure is repeated until thecorrections to the parameters are deemed to be insignificantc
27、ompared to their standard deviations. Values for interfacewidth, depth, and asymmetry can be calculated from theparameters of the fitted logistic function.6.3 Implementation of this procedure can be readily accom-plished by making use of a specialized computer algorithm andsupporting software (LOGIT
28、) developed specifically for thisapplication and described in Appendix X1.6.3.1 The fitting can also be done in Excel, using the solveroption to determine the variables A, B, As,Bs,Xo,Doand Q.Write the definition of the logistical function (equations 3through 5) in Excel and to calculate its values
29、as a function ofX. If the exponential function ezproduces overflow when z 709; this can easily be circumvented by writing EXP (min(z,709) instead of EXP(Z).7. Interpretation of Results7.1 The seven parameters necessary to characterize theinterface profile shape are determined by a least-squares fit
30、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 slopeAsare necessary to define the pre-interface asymptote while twomore, B and Bs, define the post-interface asymptote. For
31、 theanalysis of typical interface profiles, it is usual to assume thatboth of these slopes are zero. Two more parameters, Doand Xo,define the slope and position of the transition region. Inaddition, an asymmetry parameter Q that causes the widthparameter to vary logistically from O to 2Do, is introd
32、uced asa measure of the difference in curvature in the pre- andpost-transition ends of the transition region. If Q O, the post-transition region has the greatest curvature. IfQ =O, D = Doand the transition profile is symmetric. Theparameter Q has the dimensions of 1/X whereas Dohas thedimensions of
33、X. The product QDois dimensionless and is ameasure of the asymmetry of the profile independent of itswidth. If the absolute magnitude of QDois less than 0.1, theasymmetry in the transition profile should be barely discern-ible.7.2 The final results should include the calculated values ofY and associ
34、ated statistics, the values of the determinedparameters and their uncertainties, and statistics related to theoverall quality of the least-squares fit.4Verhulst, P. F., Acad. Brux. Vol 18, p. 1, 1845.5Johnson, N. L. and Kotz, S., “Distributions in Statistics: Continuous UnivariateDistributions,” Hou
35、ghton Mifflin Co., Boston, 2, Chapter 22, 1970.E16360427.3 The width of the interface region, If, is the depth (time)required for the decay or growth curve to progress from afraction f of completion to (1 f) of completion. For the casewhere Q =O,Ifis proportional to Doand is given by the simpleformu
36、la:If5 2 Do1n 1 2 f!/f (6)so that, for example, the traditional 16 to 84 % interfacewidth is 3.32 Do.7.4 Introduction of the asymmetry parameter Q into theextended logistic function makes the calculation of the 16 to84 % points of the interface more complicated. In particular,for fractions f and (1
37、f) of completion of the interfacetransition:Xf5 Xo1 2 Do1n f/1 2 f!#/1 1 eQXf2Xo!# (7)and:X12f!5 Xo1 2 Do1n 1 2 f!/f/1 1 eQX12f2Xo!# (8)Xfand X(1f)can be evaluated most readily by Newtonsmethod of successive approximations.8. Reporting of Results8.1 Interface profile shapes can be accurately charact
38、erizedby the extended logistic function and its parameters. Results ofsuch interface analysis should report these parameters (Xo, Do,Q) together with their uncertainties, the standard deviation ofthe fit, and an interface width obtained from Doand Q that isbased on some accepted definition (for exam
39、ple, 16 to 84 %concentration change).8.2 Sputtered depth, X, is often difficult to determine experi-mentally so that depth profile data are normally acquired withtime as the independent variable. This sputtered time can bereferenced with respect to a removal time obtained with acalibrated sputtering
40、 standard under the same sputtering con-ditions of ion energy, beam angle, current density, etc. as theinterface measurement itself. In this way, time can be trans-formed into an equivalent depth derived from a standardmaterial and this equivalent depth should be used in reportingthe interface param
41、eters and analysis results. Sputtering stan-dards are available from the National Institute of Standards andTechnology (SRM 2135c) and from the UK National PhysicalLaboratory (CRM 261) and from the Surface Analysis Societyof Japan (a multilayer GaAs/AIAs superlattice reference ma-terial).9. Example
42、of Interface Profile Data Analysis Using theMethod Suggested9.1 Sputter-depth-profile data obtained at an interface be-tween Cr and Ni has been analyzed by fitting the extendedlogistic function to this data using least-squares techniques.The results of this analysis are presented in Fig. 1; the soli
43、dlines are calculated values from Eq 3 . A separate analysis wasdone for each constituent to determine the parameters of the fit;these are listed in Table 1. Comparison of the chromium andnickel parameters indicates the high precision attainable indescribing the profile shape and in determining sput
44、tered depth(and, therefore, interface width) with this analysis method.10. Keywords10.1 logistic function; sputter-depth-profile interface dataNOTE 1The solid lines are the calculated values from Eq 3 .Parameters of the fit are given in Table 1.FIG. 1 Typical Depth Profile of Chromium Through a Chro
45、mium(x) and Nickel (o) InterfaceE1636043APPENDIX(Nonmandatory Information)X1. FITTING OF DEPTH PROFILE INTERFACE DATA TO THE LOGISTIC FUNCTION BY MEANS OF ASPECIALIZED COMPUTER ALGORITHM, LOGIT6X1.1 ScopeX1.1.1 This appendix describes a specialized computeralgorithm and supporting software (LOGIT) d
46、eveloped for thefitting of depth profile interface data to the extended logisticfunction in order to determine the parameters of this fittedfunction. These parameters characterize the shape of theinterface region and so define the interface width, its asymme-try, and its depth from the original surf
47、ace.X1.2 Significance and UseX1.2.1 LOGIT has been developed to fit interface profiledata to the extended logistic function. The specifically writtenleast-squares procedure used in LOGIT results in a rapid andreliable analysis.An important feature of LOGIT is that it doesnot require initial estimate
48、s to be made of the parameters; it is,therefore, simple and easy to use and can run without operatorintervention. LOGIT is robust in handling a wide variety ofdata of sigmoidal character and can deal effectively withextremely sharp profiles, noisy data, and pronounced outliers.X1.2.2 LOGIT has been
49、extensively tested on a variety ofinterface profile data; it has been found able to fit such data tothe extended logistic function to within the measurementuncertainty.X1.2.3 LOGIT is a suitable implementation procedure foruse with this practice.X1.3 Description of the Procedure, LOGITX1.3.1 LOGIT consists of a main program and 14 subrou-tines written in Fortran 77. The version available will run onIBM XT, AT, and compatible personal computers possessing amathematics coprocessor (Intel 8087 and 80287). Typically,250K of memory should be available
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