Monte Carlo Modeling for Electron Microscopy and Microanalysis (Oxford Series in Optical and Imaging Sciences)

Monte Carl o Modelin g fo r Electro n Microscopy an d Microanalysis

OXFORD SERIE S IN OPTICA L AND IMAGING SCIENCE S EDITORS

MARSHALL LAP P JUN-ICHI NISHIZAWA BENJAMIN B . SNAVELY HENRY STAR K ANDREW C . TA M TONY WILSO N

1. D . M . Lubma n (ed.) . Lasers and Mass Spectrometry 2. D . Sarid . Scanning Force Microscopy With Applications to Electric, Magnetic, and Atomic Forces 3. A . B . Schvartsburg . Non-linear Pulses in Integrated and Waveguide Optics 4. C . J. Chen . Introduction to Scanning Tunneling Microscopy 5. D . Sarid . Scanning Force Microscopy With Applications to Electric, Magnetic, and Atomic Forces, revised edition 6. S . Mukamel. Principles of Nonlinear Optical Spectroscopy 1. A . Hasegaw a an d Y . Kodama. Solitons in Optical Communications 8. T . Tani. Photographic Sensitivity: Theory and Mechanisms 9. D . Joy . Monte Carlo Modeling for Electron Microscopy and Microanalysis

Monte Carlo Modelin g for Electro n Microscop y and Microanalysis DAVID C . JO Y

New York

Oxford

OXFORD UNIVERSIT Y PRES S 1995

Oxford Universit y Pres s Oxford Ne w Yor k Athens Aucklan d Bangko k Bomba y Calcutta Cap e Tow n Da r e s Salaa m Delh i Florence Hon g Kon g Istanbu l Karach i Kuala Lumpu r Madra s Madri d Melbourn e Mexico City Nairob i Pari s Singapor e Taipei Toky o Toront o and associate d companie s i n Berlin Ibada n

Copyright © 199 5 b y Oxfor d Universit y Press , Inc . Published b y Oxfor d University Press, Inc. , 200 Madiso n Avenue , New York , New Yor k 1001 6 Oxford i s a registered trademar k o f Oxfor d Universit y Press All right s reserved. N o par t o f thi s publication ma y b e reproduced , stored i n a retrieval system , or transmitted , in an y for m o r b y any means , electronic, mechanical , photocopying, recording , o r otherwise , without th e prio r permissio n o f Oxfor d University Press. Library o f Congres s Cataloging-in-Publicatio n Dat a Joy, Davi d C, 1943 — Monte Carl o modelin g for electro n microscop y an d microanalysi s / David C . Joy. p. cm. — (Oxfor d series in optica l and imagin g sciences : 9) Includes bibliographical references . ISBN 0-19-508874- 3 1. Electro n microscopy—Compute r simulation . 2 . Electro n prob e microanalysis—Computer simulation . 3. Mont e Carlo method . I. Title . II . Series . QH212.E4J67 199 5 502'.8'25—dc20 94-3564 2 A disk , 3!/2-inc h MS-DO S format , containing al l th e sourc e cod e i n this boo k i s availabl e b y mail fro m Davi d C . Joy, P.O . Bo x 23616 , Knoxville, T N 37933-1616, U.S.A . The cost , includin g postage an d handling, i s $10.00 (U.S. and Canada ) o r $15.00 (elsewhere) . Payments accepte d i n the for m o f a check o r a mone y order .

135798642 Printed i n th e Unite d State s o f Americ a on acid-fre e pape r

PREFACE Electron microscop y an d electron bea m microanalysi s are techniques tha t are now in dail y use i n many scientifi c disciplines an d technologies. Thei r importanc e de rives fro m th e fac t tha t the information they generat e comes fro m highl y localize d regions of the specimen, the data produced are unique in scope, an d the images and spectra produced can be quantified to give detailed numerical data about the sample. For quantification to be possible, however , it is necessary to be able to describe how a bea m o f electron s interact s wit h a soli d specimen—an d suc h a descriptio n i s difficult t o provide because of the very varied and complex nature of the interactions between energeti c electron s an d solids . The purpose of this book i s to demonstrat e ho w thi s interactio n ca n be accu rately simulate d and studie d on a personal computer , b y applyin g simpl e physica l principles an d th e mathematica l techniqu e o f Mont e Carl o (o r random-number ) sampling. Th e ai m is t o provide a practical rathe r tha n a theoretical guid e t o this technique, and the emphasis is therefore on how to program and subsequently use a Monte Carl o model . Th e bibliography list s other books that cover the mathematics of Monte Carlo sampling an d the physical theory of electron scatterin g i n detail. To make the programs develope d her e a s accessible a s possible, a disk—for us e with MS-DOS-compatible computers—ha s bee n mad e available ; i t contain s al l o f th e source code described in this book together with executable (i.e., runnable) versions. To order se e facin g copyrigh t page. This book woul d not have been possibl e withou t the generou s cooperatio n o f many othe r people . I a m especiall y gratefu l t o Dr . Dale Newbur y of N.I.S.T. , fo r first introducin g me to Monte Carlo models, an d to him and Dr. Robert Myklebust, also o f N.I.S.T. , fo r sharin g thei r cod e wit h me . Dr . Hug h Bisho p o f A.E.R.E . Harwell, whos e Ph.D . wor k produce d th e firs t electro n bea m Mont e Carl o pro grams, kindly lent me a copy of his thesis and provided some invaluable background information. Dr . Peter Duncumb , of the University of Cambridge an d Tube Investments Ltd., whose pioneering wor k on Monte Carlo modeling usin g minicomputers ultimately made thi s book possible , len t me copies o f his earl y reports an d paper s and has bee n unfailingl y helpful i n answerin g man y questions abou t the develop ment o f the technique . Th e program s give n i n thi s volum e hav e bee n refine d an d improved through the efforts o f many colleagues wh o have used them over the past few years . Vita l improvements i n science , substance , an d style have been mad e by

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Drs. John Armstrong (California Institute of Technology), Ed Cole (Sandia), Zibigniew Czyzewsk i (Universit y of Tennessee) , Raynal d Gauvi n (Universit e d e Sher brooke), Davi d Howit t (Universit y o f California) , Davi d Hol t (Imperia l College , London), Pegg y Moche l (Universit y of Illinois) , Joh n Rus s (Nort h Carolin a Stat e University), an d Oliver Well s (IBM) . T o them, to my student s Suichu Luo, Xinle i Wang, an d Xia o Zhang , an d t o man y other s wh o hav e give n o f thei r tim e an d expertise, I am deeply grateful. An y errors and problems tha t remain ar e strictly my own responsibility. Finally, I dedicat e thi s boo k t o m y wif e Carolyn , withou t whos e lov e an d encouragement thi s manuscrip t woul d hav e remaine d jus t anothe r pil e o f flopp y discs. Knoxville D August 1994

. J.

CONTENTS 1. An Introduction to Monte Carlo Methods

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1.1. Electro n Bea m Interaction—The Problem 3 1.2. Th e Monte Carl o Method 4 1.3. Brie f Histor y o f Monte Carl o Modelin g 5 1.4. Abou t This Boo k 7

2. Constructing a Simulation 2.1. Introductio n 9 2.2. Describin g th e Problem 9 2.3. Programmin g th e Simulatio n 1 2.4. Readin g a PASCAL Program 1 2.5. Runnin g the Simulatio n 2 3

3. The Single Scattering Model

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3.1. Introductio n 2 5 3.2. Assumption s o f the Singl e Scatterin g Mode l 2 5 3.3. Th e Single Scattering Mode l 2 6 3.4. Th e Singl e Scatterin g Mont e Carl o Cod e 3 7 3.5. Note s on the Procedures an d Functions Use d in the Program 4 3.6. Runnin g the Program 5 0

4. The Plural Scattering Model

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4.1. Introductio n 5 6 4.2. Assumption s o f the Plural Scatterin g Mode l 5 6 4.3. Th e Plural Scatterin g Mont e Carl o Cod e 6 2 4.4. Note s on the Procedures an d Functions Use d in the Progra m 7 4.5. Runnin g the Program 7 5

5. The Practical Application of Monte Carlo Models 5.1. Genera l Consideration s 7 7 5.2. Whic h Type of Monte Carl o Model Shoul d Be Used? 7

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5.3. Customizin g the Generi c Programs 7 8 5.4. Th e "All Purpose" Progra m 7 9 5.5. Th e Applicabilit y of Mont e Carl o Technique s 7 6. Backscattered Electrons

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6.1. Backscattere d Electron s 8 1 6.2. Testin g th e Mont e Carl o Models o f Backscattering 8 6.3. Prediction s o f the Mont e Carl o Model s 9 0 6.4. Modelin g Inhomogeneou s Material s 9 7 6.5. Note s o n the Progra m 10 5 6.6. Incorporatin g Detector Geometr y an d Efficienc y 11

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7. Charge Collection Microscopy and Cathodoluminescence 114 7.1. Introductio n 11 4 7.2. Th e Principles o f EBIC and C/ L Imag e Formation 11 4 7.3. Mont e Carl o Modelin g o f Charge Collectio n Microscop y 11 8. Secondary Electrons and Imaging 8.1. Introductio n 13 4 8.2. Firs t Principles—S E Model s 13 8.3. Th e Fast Secondar y Mode l 14 8.4. Th e Parametri c Mode l 15 6

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9. X-ray Production and Microanalysis

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9.1. Introductio n 17 4 9.2. Th e Generatio n o f Characteristi c X-ray s 17 4 9.3. Th e Generatio n o f Continuum X-rays 17 5 9.4. X-ra y Production i n Thin Film s 17 7 9.5. X-ra y Production i n Bul k Sample s 19 1 10. What Next in Monte Carlo Simulations?

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10.1. Improvin g the Mont e Carl o Mode l 19 9 10.2. Faste r Mont e Carl o Modelin g 20 2 10.3. Alternative s t o Sequentia l Mont e Carl o Modelin g 20 10.4. Conclusion s 20 5 References 20 Index 21

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Monte Carl o Modelin g fo r Electro n Microscopy an d Microanalysi s

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1 AN INTRODUCTION TO MONTE CARLO METHODS 1.1 Electron beam interaction—the problem The interaction o f an electron bea m with a solid i s complex. Within a distance o f a few ten s t o a fe w hundred s o f angstrom s o f enterin g th e target , th e electro n wil l interact wit h th e sampl e i n som e way . The interactio n coul d b e th e resul t o f th e attraction between the negatively charged electron and the positively charged atomic nucleus (and equally the repulsion between the negatively charged atomic electrons and negativ e charg e o n th e inciden t electron) , i n whic h cas e th e electro n wil l b e deflected throug h som e angl e relativ e t o it s previou s directio n o f travel , bu t it s energy wil l remain essentially unaltered. This i s called a n elastic scattering event . Alternatively, the incident electron could cause the ionization of the atom by removing a n inner-shell electro n fro m it s orbit, s o producing a characteristic x-ra y or an ejected Auger electron; it could have a collision wit h a valence electron to produce a secondary electron ; it could interac t wit h the crystal lattice of the solid to generat e phonons; or , i n on e o f severa l othe r possibl e ways , i t coul d giv e u p som e o f it s energy t o the solid. These type s of interactions i n which the electron changes both its directio n o f trave l an d it s energ y ar e example s o f inelastic scatterin g events . After travelin g a further distance , the electro n wil l then agai n b e scattered , eithe r elastically o r inelastically a s before, and this process wil l continu e until either the electron gives up all of its energy to the solid and comes to thermal equilibrium with it o r until it manages to escap e fro m th e soli d i n som e way. While at a sufficiently atomisti c leve l this train of scattering events is presumably quit e deterministic—give n sufficien t informatio n abou t th e electro n an d th e parameters describin g it—t o a n observe r abl e t o watc h th e electro n a s i t travel s through th e solid , th e sequenc e o f events makin g up th e trajector y fo r an y give n electron woul d appear t o b e entirel y random . Ever y electro n woul d experience a different se t of scattering event s and every trajectory woul d be unique. Since, i n a typical electron microscope, ther e are actually about 1010 electrons impinging on the sample each second, it is clear that there is not likely to be any simple or compact way t o describ e th e spatia l distributio n o f th e innumerabl e interaction s tha t ca n occur or the various radiations resulting from these events. At best it will be possible to assig n probabilitie s t o specifi c events , suc h a s the chanc e o f a n electro n bein g 3

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backscattered (i.e. , bein g scattere d throug h more tha n 90°) o r of being transmitte d through the target; but any more detailed analysis of the interaction will be impossible. The Monte Carlo method described here uses such probabilities, together with the idea of sampling by using random numbers, to compute one possible set of scattering events for an electron as it travels into the solid. By repeating this process many times, a statistically valid and detailed picture of the interaction process can be constructed.

1.2 The Monte Carlo method One of the very earliest publishe d paper s o n "Monte Carlo" methods (Kahn , 1950 ) provides a n excellent statement of the basis o f the method—"By applyin g random sampling techniques t o the problem [o f interest] deductions about the behavior o f a large numbe r o f [electrons ] ar e mad e fro m th e stud y o f comparativel y few . Th e technique is quit e analogous t o public opinion polling of a small sample t o obtain information concernin g th e populatio n o f th e entir e country. " Th e us e o f rando m sampling to solve a mathematical problem can be characterized a s follows. A game of chance is played in which the probability of success P is a number whose value is desired. If the game i s played N time s with r wins then r/N i s an estimate of P. The "game o f chance" will be a direct analogy , or a simplified version, of the physical problem to be solved. To play the game of chance on a computer, the roulette wheel or dice are replaced b y random numbers. The implication of a "random" number is that any number within a specified range (usually 0 to 1 ) has an equal probability of being selected , and all the digits that make up the number have an equal probability (i.e., 1 in 10 ) of occurring. Thus to take a simple and relevant example, conside r a n electron tha t ca n b e scattere d elasticall y o r inelastically , th e probabilit y o f eithe r occurrence being determined by its total cross section. If the probability that a given scattering even t i s elasti c i s ps an d pi i s th e probabilit y o f a n inelasti c scatterin g eventoandityaoiueabet a two alternatives by picking a random number RND (0 < RN D < 1 ) and specifying that i f RN D ^ pe, the n a n elasti c even t occurs , otherwis e a n inelasti c even t i s assumed to have occurred. I f this selectio n procedur e i s applie d a large number of times, then the predicted rati o of elastic t o inelastic event s will match the expecte d probability derive d fro m th e give n probabilitie s p e an d p ;, sinc e a fractio n pe of th e rando m number s will be ^pe. Rando m number s can als o b e use d t o mak e other decisions . For example , i f the probability p(%) o f the electron bein g scattere d through som e angl e 6 i s known , eithe r experimentally o r fro m som e theoretica l model, then a specific scattering angl e a ca n be obtained o r picking another random number an d solving fo r a th e equation:

INTRODUCTION T O MONT E CARL O METHOD S 5

which equate s RN D t o th e probabilit y o f reachin g th e angl e a give n th e know n distribution o f p(Q). B y repeatin g thes e random-numbe r samplin g processe s eac h time a decision must be made, a Monte Carlo simulatio n of one particular electro n trajectory throug h the soli d ca n be produced. Th e result o f suc h a procedure i s not necessarily o r eve n probabl y a trajectory representin g on e tha t could b e observe d experimentally unde r equivalent conditions . However , b y simulatin g a sufficientl y large numbe r of suc h trajectories , a statisticall y significan t mixture of al l possible scattering event s will have been sample d an d the composite result will be a sensibl e approximation t o experimenta l reality .

1.3 A brief history of Monte Carlo modeling The firs t publishe d exampl e o f th e us e o f rando m number s t o solv e a problem i s probably that of Buffon, who—i n his 177 7 volume Essai d'Arithmetique Morale— described a n experiment i n which needles of equa l length wer e throw n a t random over a shee t marke d wit h paralle l lines . B y countin g th e numbe r o f intersection s between lines an d needles, Buffo n wa s abl e t o deriv e a value for TT . Subsequently, other mathematician s an d statistician s followe d Buffon' s lea d an d mad e us e o f random number s a s a wa y o f testin g theorie s an d results . Becaus e man y o f th e phenomena o f interest t o physicists in the early twentieth century , such a s radioactive decay or the transmission o f cosmic rays through barriers, displaye d a n apparently random behavior, it was also an obvious step to try to use random number s to investigate suc h problems. Th e procedure wa s to model, fo r example, a cosmic ray interaction b y permittin g th e "particle " to play a game o f chance, th e rule s o f the game being suc h tha t the actua l deterministi c an d random feature s of the physica l process wer e exactl y imitated , ste p b y step , b y th e gam e an d i n whic h rando m numbers determine d th e "moves." During the Manhattan Project, which led to the development o f the first atomic bomb, John von Neumann, Stanislav Ulam, and others made innovative use of both random-number samplin g an d game-playing situation s involvin g rando m number s as a wa y o f studyin g physica l processe s a s divers e a s particl e diffusio n an d th e probability of a missile strikin g a flying aircraft . It was during this period tha t these techniques wer e first dubbe d "Monte Carlo methods " (Metropoli s an d Ulam, 1949 ; McCracken, 1955) . Because th e Mont e Carlo metho d need s a large suppl y o f random number s a s wel l a s muc h repetitiou s mathematica l computation , th e late r development of the technique was closely geare d t o the development of "automatic computing machines. " A s firs t mechanica l an d the n electroni c machine s becam e available durin g the 1950s , the technique foun d increasin g applicatio n t o problem s ranging i n scop e fro m diffusio n studie s i n nuclea r physic s t o th e modelin g o f population growt h by th e Burea u of the Census . A valuable bibliograph y o f thes e early paper s an d techniques ca n be found i n Meye r (1956) . Although Monte Carl o methods ha d been applied to many other phenomena, it was no t unti l th e wor k o f Hebbar d an d Wilso n (1955 ) tha t th e metho d wa s sue -

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cessfully employe d for charged particles. Later work by Sidei et al. (1957) and Leiss et al . (1957 ) le d t o a majo r paper b y Berge r (1963 ) tha t lai d th e groundwor k for future developments . Simultaneously , in England , M . Green , a physic s graduat e student in Cambridge workin g for V . E. Cosslett , wa s persuaded t o investigate th e application of von Neumann's Monte Carlo method, and the university's EDS AC II computer, t o th e scatterin g o f electro n beams . Takin g experimenta l dat a o n th e scattering of electrons in a 1000- A film as a starting point, Green (1963) and later Bishop (1965 ) wer e abl e t o deriv e th e electro n backscatterin g coefficient , an d th e depth dependenc e o f characteristi c x-ra y production , fro m a bul k sampl e an d t o demonstrate good agreemen t wit h measured values . However, whil e this approac h showed th e validit y o f th e technique , i t wa s limite d i n it s applicatio n t o thos e situations where the suitably detailed initial experimental data were available. At the 4th Internationa l Conferenc e on X-ray Optics an d Microanalysi s i n Paris i n 1965 , however, two independent papers (Bishop, 1966; Shimizu et al., 1966) demonstrated how theoretically based electron scattering distributions could replace and so generalize experimental distributions; within a short time, groups in Europe, Japan , and the United States had produced working programs based on this concept. One of the most important of these was the one produced by Curgenven and Duncumb (1971) working a t the Tube Investments Laboratory i n England. Thi s progra m introduce d several ne w concepts , includin g th e so-calle d multipl e scatterin g approximatio n discussed i n Chapter 4 of this book, and was optimized t o run on a relatively smal l scientific computer . Copies of the FORTRAN code were generously made available to intereste d laboratorie s throughou t the worl d for thei r own use ; a s a result , thi s program came into widespread use and made a significant contribution to popularizing th e ide a tha t electron-soli d interaction s coul d b e modele d convenientl y an d accurately b y computer . By 1976 , th e us e o f this technique wa s sufficientl y commo n fo r a conferenc e entitled "Us e o f Mont e Carl o Calculation s i n Electro n Prob e Microanalysi s an d Scanning Electro n Microscopy " t o b e hel d a t th e Nationa l Burea u o f Standard s (NBS) in Washington, D.C. The proceedings o f that meeting (Heinrich et al., 1976 ) still for m on e o f th e basi c resource s fo r informatio n i n thi s field , an d programs , algorithms, an d procedures develope d b y th e NBS grou p have formed the startin g point fo r man y o f th e program s i n curren t use , includin g thos e describe d i n thi s volume. Apart from th e very earliest examples , which were run by hand, using random numbers generated b y spinning a "wheel o f chance" (Wilson, 1952) o r on mechanical desk calculators (e.g., Hay ward and Hubbell, 1954), Monte Carlo programs were designed t o be run o n the main-frame computers the n becoming availabl e in most government an d industria l laboratories an d universities . Althoug h suc h machine s were both large and expensive, their capabilities were very limited, and considerable ingenuity was necessary to produce workable programs within the limits set by the available memory (often as little as 2000 words) and operating time between crashes

INTRODUCTION T O MONT E CARL O METHOD S 7

of th e system . Nevertheles s Mont e Carl o program s wer e ofte n cite d a s a prim e example of the new analytical power made available throug h electronic computing . With the advent of personal computers (PCs), this power is now available to anyone who needs it. Monte Carlo programs are, in most cases, relatively short in length and can readily be run on any modern P C without encountering any problems with the lack o f memory. The programs are also computationally intensive, i n the sens e that once th e progra m ha s obtaine d al l th e necessar y data , i t perform s calculation s continuously until its task i s finished. This is not the ideal situatio n for a program that is run on a time-shared main-fram e compute r because it means that the actual computing time will depend directly on the number of users working on the machine at any given time (unlike programs such as word processing, where the majority of the computer's tim e is spent in waiting for the operator t o enter the next character and multipl e users produce littl e apparen t drop i n response speed) . Consequently , even o n relativel y powerfu l time-shar e systems , the computationa l spee d experi enced by a user when running a Monte Carlo program ca n seem very slow ; thus to calculate a sufficient numbe r of trajectories to produce an accurate result can cost a lot o f both time an d money. While PCs d o not, in general , perfor m the individual computations a s rapidl y a s th e mai n frame , the y are dedicate d t o on e task ; a s a result, their effective throughpu t can easily rival that of much larger machines. Also, since access to PCs is often fre e or at least very cheap, over-lunch or even overnight runs are no financial burden to the user. Finally, the interactive nature of PCs and the ready acces s t o graphical presentation tha t they provide offer s th e chance to make programs that ar e both mor e accessibl e an d more immediatel y useful .

1.4 About this book This volume is not intended t o replace standard textbooks on the general theory of Monte Carlo samplin g (such as those by Hammersley, 1964 , and Schreider, 1966) , nor i s it a substitut e for a comprehensive guid e to electro n beam microscopy an d microanalysis (suc h a s Goldstei n e t al. , 1992) . Rather , i t i s intende d t o provid e electron microscopists , microanalysts, and anyon e concerned wit h the behavior of electrons i n solid s wit h ready acces s t o the power o f th e Mont e Carl o method . I t therefore provide s workin g Mont e Carl o simulatio n routines fo r th e modelin g o f electron trajectorie s i n a soli d an d discusse s procedure s t o dea l wit h associate d phenomena suc h as secondary electron and x-ray production. These procedures can then be added to the basic simulatio n as required to produce a program customized to tackle particular problems in image interpretation o r microanalysis. The goal is to make availabl e simulation s whos e accuracy is at least a s good a s that likely t o be achieved in a comparable measuremen t or experiment o n a n electron microscope . The programs have been develope d an d have been designe d to be run on persona l computers rather than on scientific minicomputers or full siz e main-frame machines. Even given th e advanced PC designs no w available , thi s has occasionally mad e it

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necessary t o compromise betwee n th e completenes s o f the model an d the spee d of execution. In most cases the choice has been for the version tha t is fast in operation, since a goo d approximatio n availabl e rapidl y i s muc h mor e usefu l tha n a n exac t result that takes a day to compute. N o claim is made that these programs represent the bes t o r eve n th e onl y wa y t o d o th e job . Indeed , a larg e numbe r o f othe r approaches ar e cited in the text. This book wil l have achieved it s purpose if you— the user—feel ready , willing , and abl e t o use th e printed program s give n here , o r those availabl e on the accompanyin g disk, as the basis fo r your own experimenta tion an d development .

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CONSTRUCTING A SIMULATION 2.1 Introduction In thi s chapter , w e wil l develo p a Mont e Carl o simulatio n o f a rando m wal k (sometimes calle d a "drunken walk, " afte r it s most popular mode o f experimental investigation), Although this particular problem is only loosely related to the studies of electron beam interactions that follow, the model that we will develop provides a convenient wa y o f establishin g a framewor k fo r thos e subsequen t simulations . I t also illustrate s th e genera l principle s o f programming to b e followe d in thi s boo k and introduces some of the important practical detail s associated wit h constructing and running suc h models on a personal computer.

2.2 Describing the problem The random walk problem can be stated as follows: "How far from the starting point would a walke r b e afte r takin g N step s o f equa l lengt h bu t i n randoml y chose n directions?" In order to simulate this problem, we must break it down to a sequence of instructions, or algorithms, that allow us to describe it mathematically. Figure 2.1 shows the situation for one of the steps making up the walk. It commences fro m th e coordinate (x, v) reached a t the conclusion o f the previous step and is made at some random angl e A wit h the X-axis, so that:

where RND is a random number between 0 and 1 . The coordinates xn, yn of the end of th e ste p are then :

Equations (2.2 ) an d (2.3 ) ca n be cast in a more symmetrica l for m b y writing

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Figure 2.1. Coordinat e syste m fo r th e random wal k simulation .

so that

cos (A) and cos (B), the cosine s o f the angle s between the vecto r representin g th e direction of the step and the axes, are called the direction cosine s an d will in futur e be abbreviate d t o CA an d CB . Although Eqs. (2.1) , (2.5), and (2.6) give an accurate mathematical descriptio n of on e ste p o f th e random walk , the axe s o f th e coordinat e syste m ar e constantl y changing as the walker moves from one step to the next. It would intuitively be more satisfactory t o describe th e progress o f the walker with respect t o a fixed referenc e frame o f axe s (suc h a s th e wall s o f th e room) , because thi s make s i t possibl e t o predict when , fo r example , a collision migh t occur . Wit h thi s descriptio n then , a s shown in Fig. 2.2 :

where X, Y are the angles described b y the direction cosines CX, CY for the previous step, an d A, B ar e the angles fo r the new direction cosine s CA, CB. As befor e

CONSTRUCTING A SIMULATIO N 1

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Figure 2.2. Modifie d coordinat e description usin g fixe d axe s for th e random walk .

and fro m th e usua l trigonometri c expansion s we get

using the result that sin (X) = co s (Y) and sin (Y) = co s (X). Equations (2.7), (2.10), (2.11), (2.12) , an d (2.13 ) no w describ e ho w t o calculat e th e en d poin t o f a step , given its starting point. The next step can similarly be computed using the identical equations bu t resetting th e coordinate s s o that the ol d end point becomes the new starting point and the exit direction cosine s becom e the entry direction cosines :

The recipe fo r simulatin g the random walk is therefore a s follows: Given a starting point (x0, y0) an d a starting direction (CX, CY), the n Repeat the sequence Find th e deviation angl e 6 [Eq . (2.7)] Calculate the new directio n cosine s CA,CB [Eqs . (2.12) an d (2.13)] Calculate the new coordinates xn,yn [Eqs . (2.10 ) an d (2.11)] Then reset the coordinate s fo r the next step x = xn, y = yn, CX = CA, CY = CB until the required number of steps has been taken Then distanc e s from startin g point is s = V(* — x0)2 + ( y - y0)2

12 MONT

E CARL O MODELIN G

2.3 Programming the simulation To carry out the simulation on a computer, the recipe given above must be expresse d in a for m tha t th e compute r ca n understand . Thi s require s tha t w e choos e on e computer language , fro m amon g th e man y no w available , i n whic h t o cod e ou r program. Unfortunately , any discussion of programming language s is liable t o lead to acrimony , becaus e anyon e wh o regularl y use s a particula r languag e an d ha s become use d to its syntax an d particular strengths an d weaknesses ca n always fin d sufficient reason s t o prove that any other languag e is deficient i n power o r conve nience. However , stripped o f the theological overtone s s o often accompanyin g this sort of debate, the truth is that any of the languages now in common use on persona l computers coul d b e used t o cod e thi s an d the following , Monte Carl o simulation s without a noticeabl e effec t o n th e qualit y of th e fina l product . But sinc e i t i s no t practical t o provid e equivalen t cod e fo r al l possibl e languages , i t i s necessar y t o choose jus t one arbitrarily, fo r whateve r reasons see m appropriate , an d wor k with that. Even though it might not have been your firs t o r even your second choice, th e decision her e ha s been t o use PASCAL. The reasons fo r this decision wer e princi pally a s follows: 1. PASCA L is a good exampl e o f a modern language . I t allow s for structure d an d modular programming ; i t has a powerful ye t simpl e syntax ; and—since i t i s a compiled language—i t i s fas t i n execution . 2. Th e styl e o f a PASCA L program , i n particula r th e us e o f indentin g an d th e availability of long descriptiv e variabl e names , leads to code tha t is easy to read and understand . 3. Variant s of PASCA L ar e commerciall y availabl e fo r al l computer s likel y t o b e encountered i n current use. Although there ar e slight difference s betwee n them , the origina l definitio n o f the languag e wa s sufficientl y precis e that these varia tions rarel y pos e a proble m i f a progra m mus t b e move d fro m on e versio n o f PASCAL to another . 4. Finally , i f yo u canno t tak e PASCA L a t an y price , then—sinc e othe r moder n languages such as QUICKBASIC, ADA, MODULA-2, o r C now share s o many of the features of PASCAL—conversion from PASCAL to any other language of your choice i s straightforward. I n fact, softwar e is available tha t can effec t suc h transformations automaticall y in many cases . The program s i n thi s boo k ar e writte n i n TURB O PASCAL ™ (versio n 5.0) , which i s perhaps th e most widel y use d form o f PASCAL for MS-DOS computers . These programs wil l also compile an d run, without any changes being necessary, in

CONSTRUCTING A SIMULATION

13

Microsoft QUICK PASCAL™ version 1.0 and higher. Other variants of PASCAL and other types of computers may require some modifications to the code, especially for the graphics commands. A disk (IBM/MS-DOS format) containing all of the code discussed in this text, as both source code and as compiled and executable code, is available from the author. Ordering details are given on the copyright page of this book.

2.4 Reading a PASCAL program The PASCAL program that implements the mathematical description derived above for the random walk simulation is as follows: Program Random—walk ; {this stimulates a simple random walk with equal-length steps} uses CRT,DOS,GRAPH ; {resource s required }

var CA,CB,CX,CY:extended; {directio n cosines } x , x n , y , y n , t h e t a , d i s t a n c e : e x t e n d e d ; {ste p variables } step:extended; {displa y variables } hstart,vstart:integer; {scree n center } i , t r i e s : i n t e g e r ; {counte r variables } GraphDriver:Integer; {whic h graphic s card? } GRAPHMODE:Integer; {whic h displa y mode? } ErrorCode:Integer; {i s ther e a problem? } Xasp,Yaspiword; {aspec t rati o o f screen } cons twopi = 6 .28318;

{2TT constant}

Procedure set—up—screen; {gets the required input data to run the simulation} begin {ensures screen is clean} ClrScr; GoToXY(10,5) ; writeln('Random Walk Simulation'); GoToXY(33, 5) ; {get number of steps} write ('.............hhow many steps?'); readln(tries); end;

Procedure initialize; {identify which graphics card is in use and initialize it}

14

MONTE CARLO MODELING

var

InGraphicsMode: Boolean ; PathToDriver:String; begin DirectVideo: =False; PathToDriver: = ' ' ; GraphDriver: =detect; InitGraph(GraphDriver,GRAPHMODE,PathToDriver); SetViewPort (0, O.GetMaxX,GetMaxY,True) ; [clip viewport} hstart: =trunc (GetMaxX/2) ; vstart: =trunc(GetMaxY/2); step: = (GetMaxX/50) ;

{horizontal midpoint of screen} {vertical midpoint of screen} {a suitable increment}

GetAspectRatio(Xasp,Yasp);

{find aspect ratio of this display}

end;

Procedure initialize—coordinates; {set up the starting values of all the parameters} begin

x:=0; y:=0; CX:=1.0; CY : = 0 . 0 ; randomize; {and reset the random-number generator}

end;

Procedure new_coord ; {computes the new coordinates xn,yn given x,y, var

VI,V2:extended ; begin VI : =c o s ( t h e t a ); V2: =s i n ( t h e t a ); CA:=CX*V1 - CY*V2 ; CB:=CY*V1 + CX*V2 ; xn:= x + step*CA ; yn:= y + step*CB ; {ne

end;

{new direction cosines}

w coordinates }

CX,CY and theta}

CONSTRUCTING A SIMULATION Procedure plot_xy(a,b,c,d:extended); {plots the step on the screen. Since all screen coordinates are integers, this conversion is made first. The real X,Y coordinates are separated from the plotting coordinates, which put X=Y=0 at the point hstart,vstart} var

ih, iv, ihn, ivn: integer;

{plotting variables}

begin ih:=hstart + trun c (x*Yasp/Xasp) ; {correc iv:=vstart + t r u n c ( y ) ;

t fo r aspect ratio}

ihn:=hstart + trun c (xn*Yasp/Xasp) ; {ditto ivn:=vstart + t r u n c ( y n ) ;

}

line (ih, iv, ihn, ivn) ;

{and draw the line}

end;

Procedure reset—coordinates ; {shifts coordinat e reference X, Y to new coordinates XN,YN and resets the direction cosines} begin x:=xn; y:=yn; CX:=CA; CY:=CB; end;

Procedure how—far ; {computes the distance traveled in the walk} label hang ;

var a,b:integer; s:string; begin distance:=sqrt((x-hstart)*(x-hstart) + distance:=distance/step;

(y-vstart)*(y-vstart) ) ;

{in step-lengt h units }

15

16

MONTE CARLO MODELING

a: =t r u n c ( G e t M a x X * 0 . 2 ); {adjus b:=trunc (GetMaxY*0 . 9) ; {ditto Str (distance : 3 : l , s) ; {conver

t for your screen} }

t nutnber to text string}

s : = c o n c a t ( s , 'step s fro m o r i g i n ' ) ; OutTextXY(a,b,concat ( ' T h e walke r traveled ' s ) ) ; hang:

{label for goto Call}

if (no t keypressed) the n got o hang ; closegraph; {tur

{keep display on screen} n off graphics}

end;

****mainatahe*****

begin set—up—screen; initialize; {fin

d graphics card and set it

up}

initialize—coord loop starts here

for i : = l t o trie s d o begin theta: =twopi*RANDOM;

{get deviation angle}

new_coord; {comput

e new position}

plot^xy(x,y,xn,yn);

{plot

reset—coordinates;

{move origin to XN, YN position}

d e l a y ( l O O ) ; {t

this step on screen}

o slo w down

display}

CONSTRUCTING A SIMULATION

17

end;

*

loop ends here

how—far;

*

{find out distance traveled}

end.

All PASCAL programs have exactly the same layout, which aids in following the code even though the format itself seems , at first, rather unusual. To further hel p the reade r t o follo w th e structur e an d logi c o f th e program , thi s boo k wil l als o employ a consistent se t o f typeface conventions whe n listing th e program s in th e text. If you decide t o type any of these programs int o your computer t o run them , then these bold and italic effects should , of course, be ignored. The program will be set i n Courie r typeface ; quote s fro m th e progra m code , procedure , an d function names , etc., will also b e se t i n the sam e typeface to identif y the m i n th e text. The firs t lin e in the listing alway s identifies the program name, here RANDO M WALK. Th e keyword PROGRAM will be printed in bold type to identify it . Note that this statement , lik e al l PASCA L statements , end s i n a semicolo n (;) . This i s th e "delimiter" tha t separate s on e progra m statemen t fro m th e next . Th e PASCA L compiler ignore s spaces and line breaks, following the program logic only from th e delimiters. I f desired , furthe r detail s abou t th e progra m ca n the n b e adde d a s a comment. In PASCAL, this is done by enclosing the text within {. . .} brackets. Fo r clarity i n the printed listings, comments wil l be shown in italics. Next, the resources require d b y the program ar e listed. Th e keyword uses, here printe d bold , i s followe d b y a lis t o f th e librarie s require d b y th e cod e tha t follows. I n thi s cas e th e CRT , DOS, an d GRAP H librarie s ar e needed . O n othe r computers or in other variants of PASCAL, the list of resources may be different o r missing altogether . The keywor d var , whic h wil l b e i n bold , introduce s a listin g o f al l th e variables use d in th e program, groupe d b y their type . Every variabl e i n PASCA L must b e define d befor e i t i s use d an d b e o f a typ e (e.g. , integer , longinteger , extended, real , Boolean , etc. ) recognized b y th e program . Onc e a variabl e i s "typed," its properties are fixed; s o if for some reason a particular numbe r is needed in mor e tha n one for m (fo r example , a s both a real numbe r and a n integer) , tw o variables must be defined an d the value converted fro m on e type to the other, using one of the special functions provided fo r this by the language. Variables listed at the start o f a program ar e GLOBA L variables ; that is , ever y part o f the progra m ca n read and write these quantities. Variables can also be defined within the procedures

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E CARL O MODELIN G

and functions use d by the program (see below). In this case, the variables are local, or private, to the particular procedure or function i n which they appear an d are not available to other parts of the program. Note that PASCAL does not distinguish the case o f letters , s o dummy , DUMMY , an d Dumm y woul d al l refe r t o th e sam e quantity. CONST identifie s a lis t o f parameter s tha t wil l b e o f constan t valu e i n th e program, suc h as the length o f the ste p taken by the walke r and the coordinates o f the center o f the display screen. Onc e defined here, thes e quantities are also globa l in their effect. Constant s appearing in the CONST list do not appear in the VAR list, and their type is determined by the format assigned to them when they first appear. Thus CONST dumm

y=

5 ;

will defin e a constant o f integer type , while the statemen t CONST dumm

y=

5.0 ;

would defin e one wit h properties o f a real number . Onc e assigned , th e valu e of a constant canno t be changed; any attempt to do so will lead the compiler t o produce an error statement. Temporary constants—that is, ones whose values may be altered if required—are obtaine d b y a normal progra m lin e equatin g tw o variables : new-dummy : =5 .15 ; Note that assigning the value on the left-hand side of a statement t o value given on the right-hand sid e requires a ":=." The simple "=" implies a logical operation , a s in "i f A= B, " not th e ac t o f equatin g tw o values . Sinc e i t has bee n estimate d tha t 95% of all errors in PASCAL come from omissio n of the ";" and confusion betwee n " =" and ": =," care in copying these jots and tittle will be repaid by a reduction in the tim e it takes t o ge t a program running . Following thes e definitions , th e listin g contain s cod e fo r FUNCTIO N an d PROCEDURE routines. These pieces o f code play the role assigned to subroutines in some other languages . A function ha s the specifi c job o f performing som e defined mathematical operation . I t therefor e return s t o th e progra m lin e tha t calle d i t a n output o f som e type , an d this typ e mus t appear i n the definition . Thu s Function SUM: real; tells us that this is a function whos e output is a real number—fo r example , the sum of tw o o r mor e othe r numbers . I f th e functio n i s supplie d wit h dat a o n whic h t o

CONSTRUCTING A SIMULATIO N 1

9

work, the n bot h th e typ e of th e dat a supplie d an d o f th e outpu t returned mus t be given, so FUNCTION

black—box(a,b:integer):real;

takes i n data defined as integer but produce s as output a real number. Many func tions, such as sin or cosin or log, are already included in the language itself; but by writing the necessary code, as many new and specialized function s a s required can be added. A procedure, by comparison, can perform any legal set of operations and so has n o "type" of it s own. In either case , th e procedures an d function s mus t b e defined befor e they ar e calle d b y th e program , s o all o f th e cod e associate d with these items appears before the main body of the program. As far as possible, al l of the detai l i n a program i s carried ou t by procedure s an d functio n calls . Th e main program then simpl y lists the order i n which these ar e carried out . Thi s makes the logic of the program easy to follow, particularly if descriptive names are used for the functions an d procedures. Sinc e PASCA L imposes n o limitations o n the lengt h of names, it is worth choosing ones whose meaning will still be evident 6 months after a program is written (i.e., set_to_zero is descriptive and helpful, ST Z is not). Laying out program s i n thi s wa y als o make s i t eas y t o subsequentl y construc t ne w programs, sinc e th e components can be reused. The main program runs between the key words BEGIN and END, set in bold type. Fo r clarity , th e star t wil l als o usuall y b e identifie d b y a suitabl e lin e o f comment. In reading a program for the first time , therefore, the main program code should be identified, remembering that it is always at the end of the listing; then the procedures and functions should be studied as they are called b y the program. The program starts by calling th e procedure set_up_screen . which , as can be see n from the code for it, first clears the display screen, presents the program name ready for th e simulatio n to begin, an d asks for th e number of step s to be modeled . Since the purpose of this program is to plot the random walk on the screen, the next step is to set up the graphics display of the computer. This is complicated by the fact tha t MS-DO S machine s com e with a wid e variety o f graphica l displa y hardware. Th e procedure initializ e (take n directl y fro m th e TURB O PASCA L reference book s supplie d by Borland Inc.) determines automaticall y whic h type of graphics display (i.e., CGA, EGA, VGA, Hercules, etc.) is present in the computer at the time the program is run. I t then selects a "graphics driver program" (file s o n the dis k wit h th e extensio n .BGI ) tha t wil l interfac e th e compute r cod e t o th e hardware i n use. Provide d tha t our softwar e is written carefully enough, the sam e code will produce an acceptable output on any of the possible displays, even though these wil l differ i n parameters suc h as their resolution an d the shape of the screen . The starting point of the random walk is to be placed at the center of the screen, but th e coordinates o f this point will var y from on e type of display to another . We

20 MONT

E CARL O MODELING

therefore ask the machine to tell us how many picture points ("pixels") it can plot in the X (or horizontal) direction, using the call GetMaxX, an d how many in the Y (or vertical) direction, using GetMaxY. Th e center o f the scree n i s then, quite generally, a t th e coordinate s (GetMaxX/2 , GetMaxY/2) . I n plottin g o n th e screen , th e coordinates mus t b e give n a s intege r number , s o th e procedur e convert s th e rea l numbers (GetMaxX/2, GetMaxY/2 ) to integers (hstart, vstart) using the ' trunc ' function define d in PASCAL. Similarly, the step length is set to be one-fiftieth o f the width o f the screen i n the horizontal direction (i.e., GetMaxX/50). A final touc h is to fin d th e "aspec t ratio " o f th e display—tha t is , th e rati o o f a uni t ste p i n th e horizontal direction to a unit step in the vertical direction. This is obtained from th e call GetAspectRatio ( ) , whic h gives us the ratio Xasp/Yasp for the display in use. B y correcting for the aspect ratio of the screen, w e can ensure that the displays on the screen will look simila r for all the graphics card s supported by the software. The next procedur e used , initialize_coordinates , set s th e position variables x,y and the direction cosines CX,CY t o the desired startin g values. In some computer languages (e.g. , BASIC), all variables are automatically reset to zero each time the program i s run; bu t in PASCAL, the value assigned to a variable when the program start s is quite arbitrary, so it is necessary to explicitly set each to the value required. The procedure also resets the random number generator with the statement "randomize." The quality of a Monte Carlo simulation depends to a great extent on the quality of the random numbers with which it is supplied, since if these exhibit any patterns in their behavior or if a given number repeats within a limited number of calls, the output data will be flawed. The random-number generators i n TURBO PASCAL (and QUICK PASCAL) , summone d b y th e cal l RANDO M ar e quit e wel l behaved provide d tha t th e generato r i s periodicall y rerandomize d usin g th e cal l RANDOMIZE. Althoug h it is not a documented feature, it can also be noted that if the initial RANDOMIZE statemen t i s omitted (o r placed withi n {. .} comment brackets), the n th e sam e strin g o f rando m number s wil l b e generate d ever y tim e th e program i s run. This can, in some cases, b e usefu l fo r debuggin g a simulation and also i n examining th e effec t o f changing a singl e paramete r i n a simulation . The program now calls the procedure new_coord, which calculates the new coordinates xn, yn give n the startin g coordinate s x, y; the initial directio n cosine s CX, CY; th e deflectio n angl e 0 ; an d th e ste p length , using th e formula s derive d above. Because all of these variables have been declared to be global by being listed at the start of the program, no variables need be passed t o the procedure. Th e effec t of this step is then displaye d on the screen, usin g the procedure plot_xy, whic h draws the step on the screen. Note that since the graphics routines that place pixel s on th e scree n nee d intege r number s a s input , the procedure mus t first conver t th e actual rea l numbe r coordinates x, y, xn, yn t o loca l intege r variables ix, iy, ixn, iyn using the trunc function supplie d in TURBO PASCAL . Th e origin o f the simulation (x = 0 , y = 0) is plotted at the center of the screen (hstart,vstart) and the coordinat e system is suc h tha t a positive x valu e moves the point t o the right an d a positive y

CONSTRUCTING A SIMULATIO N 2

1

value moves the point downward. As discussed in the procedure Initialize , th e distance plotted in the x direction (horizontal ) is corrected fo r the aspect ratio of the display screen i n use, so that all screens will give roughly equivalent displays. Note that if the line being drawn has coordinates tha t would place it off the screen (tha t is if ih or ihn does not lie between 0 and GetMaxX, or if iv or ivn does not lie between 0 and GetMaxY), then the command SetViewPort, whic h appears in the procedure initialize, wil l automatically "clip" the plot at the edges of the CRT. The program will continue to run, but plotting wil l not resume until the line again lies in the visibl e regio n o f th e screen . Th e plot-xy procedur e i s on e o f the buildin g blocks tha t w e wil l b e usin g agai n i n late r chapters . Next , th e progra m call s th e procedure reset^coordinates, whic h shifts th e origin of the calculation fro m x t o xn, and y t o yn, an d reset s th e directio n cosines . A standar d TURB O PASCA L command Delay () i s then called. The purpose of this is simply to slow down the action of the loop so that it can be watched more easily on the screen. The delay time is approximately equal to the parameter passed to the function i n milliseconds, thus Delay (100) wil l hold u p the loop for about one-tenth o f a second . Th e loop i s then repeated unti l th e desired numbe r o f step s has bee n calculated . The program the n exits fro m th e loop, an d the distance between th e start and finish coordinate s i s computed using the procedure how_ f ar. Thi s use s Pythagoras's theore m t o find th e distanc e o f the walke r from th e startin g point in units a step i n length. Sinc e th e compute r i s now i n graphics mode , thes e dat a canno t b e simply printed to the screen bu t must be drawn on the CRT as if it were a piece of a graph. First, the numerical valu e is converted to a string (i.e., a list of letters, digits , or symbols ) usin g th e standar d TURB O PASCA L command St r ( ) . Th e ":3:1" which follows the distance variable formats the result as three digits, one of which is after the decimal point. The string is assigned to a variable s, which is declared at the start o f th e procedure ; i t i s therefor e a loca l variabl e (existin g onl y withi n thi s procedure). Th e string s is then concatenated (i.e. , added ) to other string s of text to form th e messag e draw n o n th e scree n b y th e comman d OutTextX Y ( ). Th e message i s draw n startin g a t th e coordinate s a,b, whic h are , again, loca l t o thi s procedure. Sinc e w e wis h on e piec e o f cod e t o serv e fo r al l types o f display , th e values of a and b are chosen a s fractions of GetMaxX and GetMaxY. Since personal preferences abou t th e appearanc e o f th e displa y ma y vary , however , th e actua l values can readily b e changed to place the text at any other desired position o n the screen. In fact, changing the values of a and b and then recompiling an d rerunning the program t o se e the result is a n excellent wa y for users withou t much practical experience wit h computer s t o gai n proficienc y i n modifyin g th e cod e an d s o t o overcome an y fear of the dire results of "interfering" with the program. Note that if a and/o r b ar e chose n t o b e large r tha n GetMax X o r GetMaxY , th e tex t ma y disappear becaus e th e comman d SetViewPort , i n th e procedur e initialize , "clips" the displa y a t th e edge s o f the screen . I n orde r t o hol d th e displa y o n th e screen so that it can be examined or printed out, this procedure finishes with a loop.

22 MONT

E CARL O MODELING

The command keypressed call s a standard function tha t checks whethe r o r not any key has been touched. The output o f the function i s a Boolean (i.e., it has the value tru e or false). In PASCAL, the code statemen t if (function) then (operation) will lea d t o th e operatio n bein g performe d i f th e functio n evaluate s to a positiv e number or produces a Boolean variabl e wit h the value "true" (i.e., +1) . If, on the other hand , th e functio n evaluates t o zer o o r to a negative numbe r o r produces a Boolean variable with the value "false" (i.e. , 0) then the operation is not performed. Here, i f keypressed i s "false" (i.e. , n o key wa s touched) the n th e expression no t keypressed has the value "true," so the program goes back to the label hang and continues t o cycle around until eventually some key on the keyboard is struc k and the procedure i s exited . It may seem to be overelaborate t o present a simple program in this structured way, bu t adherenc e t o thi s forma t produce s cod e tha t i s eas y t o follow . A s th e programs in this book become longe r and more complex, the benefits of its use will be mor e evident . Debuggin g th e progra m i s als o mad e easie r becaus e a whol e procedure o r function ca n b e dropped out by simply placing {. . .} brackets around the call t o it, allowing th e errant region o f the program to be quickly identified. In addition, by doing all of the real work in a program in the procedures an d functions, a librar y o f progra m module s i s buil t u p tha t make s i t possibl e t o pu t togethe r a special-purpose program wit h the minimum of effort. Onc e a procedure or function has been debugge d and tested i n one program, i t can safel y b e moved into anothe r program wit h the certai n knowledg e tha t it will wor k as expected . Finally i t must be pointed ou t that the code, a s given here, doe s violate on e of the accepte d principle s o f "good programming" i n that global variable s (i.e. , variables declared at the beginning of the program and so visible to all of the procedures and function ) ar e used . "Good " practice trie s t o avoi d th e possibl e sid e effect s o f global variables, suc h as the inadvertent corruption of a variable by a faulty proce dure, by only passing copies o f the variables to the procedures a s they are required. However, in the electron bea m simulations that follow, as many as 15 or 20 differen t variables ma y be required t o describ e th e current statu s of the electron , an d al l of these may be needed b y an y given procedure o r function. Writin g the code t o pass this number of variables t o a procedure is cumbersome and the resulting operation i s also slow, so the code give n here respects th e spirit of the injunction against globa l variables bu t use s them anywa y in th e interest s o f convenienc e an d speed . Fo r a more detailed discussion o f this topic, for other questions related to programming in PASCAL, o r fo r informatio n o n compiling , editing , o r runnin g TURB O PASCA L programs, see the documentation supplied with the program or read one of the many books o n this language.

CONSTRUCTING A SIMULATIO N 2

3

2.5 Running the simulation If the programs ar e available on disk, then this simulation ca n be run by booting the computer in the usual way from either a hard disk or a floppy disk and obtaining the ">" prompt. I f th e computer wa s booted fro m a floppy, then remove thi s syste m disk and insert the disk with the Monte Carlo programs on and type Randomwalk. at the ">" prompt. Be sure that the disk (if it is a copy) has the .BGI graphics driver files o n it or else th e progra m wil l crash. I f the compute r wa s booted fro m a hard disk, then put the program floppy in the first available drive (usually labeled A:) and type A:Randomwalk. I f th e progra m ha s bee n type d int o th e compute r fro m th e listing, the n sav e i t t o dis k before compilin g an d running it , usin g the instructio n provided b y th e relevan t manua l for th e languag e chosen . I f yo u ar e plannin g t o make an y changes t o th e program, always wor k fro m a copy rathe r tha n fro m a n original. When th e welcom e scree n appears , ente r th e numbe r o f step s required , hi t return, an d the simulatio n wil l run. To generate anothe r run , hit retur n (t o get th e ">" prompt onc e more) . I f your keyboard ha s function keys , then hittin g F l wil l redisplay th e origina l instructio n (i.e. , "A:Randomwalk") , an d hitting Retur n will cause th e progra m t o restart . Otherwis e repea t th e proces s a s describe d above . Figure 2.3 shows some typical results for 200-step simulations. Every run will give a different track , and each will give a different distanc e between the start and finish of the walk. As is evident fro m Fig . 2.4, whic h plots distances recorded for severa l

Figure 2.3. Si x random walks , each o f 200 steps .

24 MONT

E CARL O MODELIN G

Figure 2.4. Summar y o f random wal k data .

successive run s a t a give n numbe r o f steps , the radia l distanc e travele d fro m th e starting poin t ca n var y ove r a wid e range . Thi s i s t o b e expected , becaus e th e simulation is, in effect, takin g samples of a random walk and there will therefore be a statistica l scatte r i n th e dat a point s obtained . I f th e sample s ar e assume d t o b e independent, whic h implies among other things that the random numbers used in the simulation are really random and not in any way related, the n the relative precisio n of the result will be proportional t o the inverse of the squar e root of the number of samples taken. Thus a relative precision of 10 % would be achieved o n 100 samples, while t o reac h a 1 % precision woul d requir e 10,00 0 samples . S o whil e a singl e simulation of a random walk of a given number of steps would produce little usefu l information, a serie s o f suc h simulations , whe n averaged , woul d giv e u s a mea n distance traveled an d an estimate of the relative precision o f this mean value. Thus plotting (see Fig. 2.4) the average value of the distance traveled, determined fro m all the simulations using the same number of steps, as a function o f the number of steps taken give s a series o f values that falls clos e to the predicted theoretica l resul t that the distanc e travele d i s equa l t o th e squar e roo t o f the numbe r o f step s taken. A n essential requiremen t o f any Mont e Carl o simulation , therefore , i s the necessit y t o repeat the simulatio n a sufficientl y larg e numbe r o f times to obtain a statisticall y meaningful result .

3

THE SINGLE SCATTERING MODEL

3.1 Introduction In thi s chapte r w e wil l develo p a Mont e Carl o mode l fo r th e interactio n o f a n electron beam with a solid using the "single scattering " approximation . Withi n the limits discussed below, this model is the most accurate representation of the electro n interaction that we can construct, an d it is capable of giving excellent result s over a wide range of conditions and materials. Thi s program , together wit h the faster but less detailed "plura l scattering" mode l developed in the following chapter, will form the framewor k aroun d whic h specifi c application s o f Mont e Carl o modelin g t o problems in electron microscop y an d microanalysis will be generated i n subsequent chapters.

3.2 Assumptions of the single scattering model The single scatterin g Monte Carl o simulation calculates the passage of the electron through the solid by tracking it from on e interaction to the next. In principle, these interactions coul d b e eithe r elasti c o r inelastic , s o th e resultan t change s i n th e direction of motion and the energy o f the electron would be computed o n the basis of the specific sequenc e o f scattering event s that each electron suffered . Whil e this is a feasible way of tackling the problem, it has the disadvantage that because not all scattering events are equally important—in terms of their effect o n the trajectory or energy—or probable , th e amoun t o f computatio n require d become s excessivel y large. Th e procedure described her e makes two significant assumptions to provid e an accurat e simulatio n while minimizing the amoun t of calculation required . 1. Only elastic scatterin g events ar e considered i n determining th e path taken by an y given electron. Elasti c scatterin g i s the net deviation that the incident elec tron undergoe s a s a resul t o f th e coulombi c attractio n betwee n th e negativel y charged electro n an d the positively charge d atomi c nucleus and the correspondin g repulsion betwee n th e orbitin g electron s an d th e inciden t electron . Thi s process , which is described mathematicall y by the screened Rutherford cross section, results in angular deflections of from typicall y 5° to 180° . (For an excellent concis e discussion o n electron scatterin g phenomen a se e Egerton, 1987. ) Inelasti c scattering , on the othe r hand , produces deflection s o f the orde r A£/ £ [i.e. , the rati o o f AE , th e 25

26 MONT

E CARL O MODELIN G

energy los t i n the event , to E, th e energy o f the electro n (Egerton , 1987)] . Thus a 10-keV electron whic h deposits 1 5 eV in an inelastic collisio n i s deflected throug h an angl e o f abou t (15/10,000 ) radian s (i.e. , 1. 5 milliradian s o r abou t 0.1°) . A n electron that loses more energy in an inelastic event is, of course, deflected through a larger angle, but the probability o f these events is proportional t o about 1/AE , so only a smal l fractio n o f inelasti c event s wil l produce significan t deviations of th e trajectory. Consequentl y elasti c scatterin g event s ar e th e one s tha t dominat e i n determining th e spatia l distributio n o f th e interaction ; ignorin g th e effect s o f th e inelastic scatterin g produce s littl e error . 2. The electron i s assumed to lose energy continuously along its path at a rate determined b y th e Beth e relationshi p rathe r tha n a s th e result o f discret e inelasti c events. While some energy loss mechanisms are continuous, such as the production of Bremsstrahlung ("brakin g radiation" ) b y the slowin g down of the incident elec tron a s a consequenc e o f electron-electro n interactions , mos t energ y losse s ar e associated wit h a specific event such as the production of an inner shell ionization . However, th e averagin g o f al l suc h effect s alon g th e pat h take n b y th e electron s leads to a convenient simplificatio n in which all possible types of inelastic event are taken accoun t of b y on e simpl e expressio n involvin g onl y a singl e variable . Thi s model of continuous energy loss i s therefore in almost universal use—although, as discussed later , significan t modifications to its functional form ma y become neces sary a t lo w bea m energies .

3.3 The single scattering model The basic geometry for the simulation (Fig. 3.1) assumes that the electron undergoes an elastic scatterin g even t at some point Pn, having traveled t o Pn fro m a previous scattering even t at Pn~l. The fundamental tas k of the simulation is to compute th e coordinates o f th e poin t Pn+l t o whic h th e electro n travel s a s a resul t o f th e scattering event at Pn. The parameters that describ e the instantaneous situatio n o f the electron ar e its energy E and the direction cosine s o f the trajectory segmen t that brought it from Pn-1 to Pn. The direction cosine s ex, cy, cz are defined in the same way as the direction cosine s employed in the random walk simulation except that in this three-dimensiona l situatio n ther e ar e no w thre e rathe r tha n tw o axe s t o b e considered. Th e X, Y, and Z axes in all the simulations in this book are defined with the conventio n tha t th e positiv e Z axi s i s norma l t o th e specime n surfac e an d directed int o the specimen, the X axis is parallel t o the tilt axis, and the X-Y plane is the surfac e plane o f th e untilte d sample . Whe n th e sampl e i s tilted , th e positiv e direction o f the Y axis i s down the surfac e o f the specimen . To calculate th e position o f the new scatterin g poin t Fn+1, we first requir e t o know th e distance , o r "step, " betwee n Pn+l an d the precedin g poin t Pn. A s dis cussed above , a n assumption of this particular model is that only elastic scatterin g events are explicitly considered. Th e distance between successiv e scattering events

THE SINGL E SCATTERIN G MODE L 2

7

Figure 3.1. Definitio n o f coordinate system use d i n the Monte Carl o simulation progra m i n this chapter .

is therefore related to the elastic mea n free path—which, in turn, is a function o f the elastic cros s sectio n a £. The total screene d Rutherfor d elasti c cros s sectio n cr £ is given by the relation (Newbur y an d Myklebust, 1981) :

where E is the electron energy i n kilo-electron volts , Z is the atomic number of the target, an d a i s a "screenin g factor " tha t account s fo r th e fac t tha t th e inciden t electron does not see all of the charge on the nucleus because of the cloud of orbiting electrons, a i s usuall y evaluate d usin g a n analytica l approximation , suc h a s that given by Bisho p (1976) :

28 MONT

E CARL O MODELIN G

The cros s sectio n crB define s a mea n fre e pat h X , which i s give n b y th e formula :

where Na i s Avogadro's number , p is the density of the target i n g/cm3, and A is the atomic weigh t o f th e targe t i n g/mole . X represents th e averag e distanc e tha t a n electron wil l travel between successiv e elastic scatterin g events . Calculating X from the formulas above fo r a selection of elements show s (Tabl e 3.1 ) that X is typically of the orde r o f a few hundre d angstrom s a t 10 0 keV an d abou t one-tenth o f that at 10 keV. Material s wit h highe r atomi c number s caus e mor e elasti c scatterin g tha n elements o f lower atomi c number , therefor e the y hav e a shorte r mea n fre e path . The actual distance that a n electron travels betwee n successive elastic scatterings will , of course , var y in a random fashion . Th e probabilit y p(s) o f a n electro n traveling a distance s whe n th e mea n fre e pat h i s X is

An estimat e fo r th e distanc e actuall y travele d ca n the n b e foun d b y choosin g a random numbe r RN D an d solving Eq . (1.1 ) i n the for m

which give s

Table 3.1 Elastic mean free paths Element

Z

Carbon 6 Silicon 1 Copper 2 Gold 7

131 4 111 9 29 98

X at 100 keV 0 A 17 2 A 12 7A 3 9A 1

X at 10 keV 0A 7A 5A 0A

THE SINGL E SCATTERIN G MODE L 2

9

Figure 3.2. Variatio n o f step length, in units of the mean fre e path, wit h the random numbe r chosen.

and hence

(since RN D i s a rando m numbe r betwee n 0 an d 1, 1 — RND i s als o a rando m number i n the sam e rang e an d s o can be replaced b y ye t anothe r random number RND). Figure 3.2 plots the variation of step length, in units of X , as a function o f the random number chosen, using the result of Eq. (3.7). Since the random numbers are uniformly distribute d betwee n 0 an d 1 , there is , fo r example , a 10 % chanc e o f drawing a number suc h tha t RND s 0. 1 and so (from Fig . 3.2 ) o f getting a step length such that s ^ 2. 3 X. Equally there is a 10% chance of drawing a number such that RND 3: 0.9, in which case the step length s < 0.1 X. The step lengths therefor e vary ove r a wide range o f values, depending o n the random number picked by th e computer. However, as is readily shown from Eq. (3.7), the average value of the step length s resulting from a large number of tries will be X, as would be expected fro m the definitio n o f X as th e average distance betwee n successiv e scatterings . In th e scatterin g even t a t Pn, which marks th e star t o f th e step , th e electro n is deflecte d throug h som e angl e (j > relativ e t o it s previou s direction o f trave l (se e Fig. 3.1) . Th e wa y i n whic h thi s scatterin g occur s i s describe d b y o-' , th e angu lar differentia l for m o f th e Rutherfor d cross sectio n (e.g. , Reime r an d Krefting , 1976):

30 MONT

E CARLO MODELING

where a i s the screenin g paramete r discusse d above . A n appropriat e functio n t o select a scattering angl e using a random number can then be obtained from a further application o f Eq. (1.1 ) in the form :

This integration, using the result for <JE given in Eq. (3.1), yields the relation for the scattering angl e (Newbur y an d Myklebust, 1981) :

This equatio n generate s a uniqu e scatterin g angl e i n th e rang e 0 < 4 > < 180° , producing a n angula r distributio n tha t matche s th e on e obtaine d experimentally . Although the full range of angles between 1 and 180 ° is available, th e great majority of scatterin g event s are predicted b y Eq. (3.10) to be low-angle—that is, less than about 10° . Figure 3.3 plots the probability o f obtaining a scattering angle 4 > greate r than some minimum value $ for the particular case of a silicon targe t irradiated at 100 keV. Note tha t whil e there i s onl y a 1 in 10,00 0 chance o f a n electro n bein g scattered by an angle in excess of 110°, more than 50% of all electrons are scattered through at least 1.5° . The electron ca n scatte r freel y t o any point on the base o f the cone show n in Fig. 3.1 . The azimutha l scattering angl e \\i i s then given a s

Figure 3.3. Probabilit y o f elastic scattering angl e exceeding some specified minimu m value .

THE SINGL E SCATTERIN G MODE L 3

1

where RND i s an independent random number selected by the computer. Since , i| /, it cannot be used at lowe r energie s withou t encounterin g problems . I n particular , sinc e / i s o f the order o f 0. 3 t o 0. 6 ke V fo r mos t materials , th e simulatio n o f th e trajector y o f electrons traveling with initial energies of 1 to 1 0 keV (i.e., typical energies encoun tered in a scanning electron microscope ) will be difficult, sinc e a significant part of the trajector y wil l occu r i n th e energ y rang e wher e th e Beth e relatio n canno t b e applied. This problem can be overcome wit h a method due to Rao-Sahib and Wittry (1974). This is a parabolic extrapolation fro m th e tangent to the Bethe curve at the energy E = 6.4 J, where the curve has an inflection, down to E - 0 . Thus for E < 6.4 /, Eq. (3.17) is replaced b y the expression :

THE SINGLE SCATTERING MODEL

35

Figure 3.5. Compariso n of stopping power for copper from Bethe expression, Rao-Sahib and Wittry extrapolation , and Tung e t al. calculation.

This expression provides a convenient extrapolation tha t is well behaved ove r the lo w energ y range , an d s o this approximatio n has bee n widel y used i n Mont e Carlo simulations . I t is not, however, either physically realistic o r accurate. Figure 3.5 compares the rate o f energy loss , (-dE/ds), o r p(dE/dS), fo r copper ( p = 7. 6 g/cm3) ove r the energy range 10 0 eV to 10 0 keV, obtained using the Bethe mode l and the Rao-Sahib-Wittry extrapolation, wit h the results o f detailed computation s by Tun g e t al . (1979) . Th e agreemen t betwee n th e composit e Bethe-Rao-Sahib Wittry profile and the Tung et al. data is satisfactory at energies down to a few kiloelectron volts . However , onc e th e energ y fall s belo w 1 keV, neithe r th e original Bethe curve nor the Rao-Sahib-Wittry extrapolation is close to the Tung et al. value, the error being a factor of two to three times at an energy of a few hundred electron volts. Th e sam e sor t o f resul t i s foun d fo r al l othe r elements , wit h th e error s becoming wors e for high atomi c numbers , sinc e thes e hav e a larger / value [Eq . (3.18)]; consequently th e Bethe expression becomes unusable a t energies a s high as a 4 or 5 keV. We ca n simultaneousl y escap e th e mathematica l singularit y o f th e origina l Bethe expression , eliminat e th e nee d t o us e th e Rao-Sahib-Wittr y extrapolation , and improve the accuracy of the stopping power as compared to the Tung et al. data by using a modified version of the Bethe equation suggested by Joy and Luo (1989). We write the stoppin g power as:

36 MONT

E CARL O MODELIN G

Figure 3.6. Compariso n o f stoppin g power fo r coppe r fro m modifie d Beth e law , wit h dat a from Tun g e t al .

At high energies E> J, this expression converge s t o the original Beth e expressio n of Eq . (3.17 ) bu t th e additio n o f th e extr a ter m 0.8 5 / i n th e numerato r o f the logarithmic term ensures that for all positive values of E the log term evaluates to a positive quantity. As shown in Fig. 3.6 , this modified expression is now a very good fit to the Tung et al. data for all energies above 50 eV. It can be shown that this result is equally applicabl e t o al l other element s an d compounds o f interest, producin g a stopping powe r valu e whic h agree s wel l wit h mor e detaile d calculation s fo r al l energies abov e a few tens of electron volts . This expression [Eq . (3.21)] is therefore used i n this book . The sequence o f operations neede d t o simulate the electron trajector y through the specime n ca n no w b e writte n schematicall y i n a n algorithmi c form . Conven tionally th e electro n i s allowe d t o penetrat e th e firs t ste p lengt h int o th e sampl e before bein g scattered . Thu s the procedure is : Calculate initial entr y dept h int o sampl e [Eq . (3.7) ] then repeat Get startin g energ y E o f the electro n Determine initia l coordinates x, y, z for this ste p Find th e directio n cosine s ex, cy, cz of its motio n relative t o th e axe s Compute the mean free pat h \ fo r this material and energy E [Eq . (3.3) ] Calculate th e ste p lengt h [Eq . (3.7) ] Find th e scatterin g angle s 4> , fy from Eqs . (3.10 ) an d (3.11 )

THE SINGLE SCATTERING MODEL

37

Find finis h coordinate s xn, yn, zn for this ste p [Eqs . (3.12) to (3.15)] Compute final energ y for thi s step E' = E — step*p*(dE/ds) Reset coordinates x = xn, y = yn, z = zn Reset directio n cosines ex — ca, cy = cb, cz = cc Reset energy E *= £ " until the electron leaves the sample or falls below some minimum energy The PASCA L code tha t implement s this outlin e i s se t ou t below , usin g the conventions introduce d in the previous chapter. Note that, by design, the program eliminates pleasing but unessential features like the use of color, or windows, so as to keep the cod e a s clear an d easy to follo w a s possible.

3.4 The single scattering Monte Carlo code Program SingleScatter; {a single scattering Monte Carlo simulation which uses the screened Rutherford cross section] {$N+} {$E+}

{turn on numeric coprocessor] {turn on emulator package]

uses CRT,DOS,GRAPH;

{resources required]

label

exit;

const

two_pi = 6.28318; e_min=0.5;

{2ir} {cutoff energy in keV in bulk case}

var at_num,at_wht,density,inc_energy,mn_ion_pot:extended; al_a,bk_sct,cp,er,ga,lam_a,sp,sg_a,:extended; ca,cb,cc,ex,cy,cz,x,y,z,xn,yn,zn:extended; del_E,m_step,m_t_step,s_en,step:extended; hplot_scale,m_f_p,plot—scale,thick:extended; count,k,num,traj—num:integer; bottom,center,top:integer; thin:Boolean; GraphDriver:Integer; GRAPHMODE:Integer; ErrorCode:Integer; Xasp,Yasp:word; Function power(mantissa,exponent:real):real; {because PASCAL does not have an exponentiation function}

38

MONTE CARLO MODELING

begin if mantissa< = 0 then power:=0 else power :=exp(In(mantissa)*exponent) ; end; Function stop_pwr(energy:real):real; {this computes the stopping power in keV/g/cnt2 using the modified Bethe expression of Eg. (3.21)} var temp:extended; begin if energyot) /mn_ion_pot) ; stop_pwr: =temp*78500*at_num/ (at_wht*energy) ; end; Function lambda(energy:real):real; {computes elastic MFP for single scattering model} var al,ak,sg:real; begin al: =al_a/energy; ak:=al*(l.+al); {giving sg cross-section in cm2 as] sg:=sg_a/(energy*energy*ak); {and lambda in angstroms is} lambda: =lam_a/sg; end; Function yes:Boolean; {reads the keyboard for 'y' or 'Y'} var ch.-char; begin read(ch); if ch in ['Y', 'y'] then yes:=true else yes: =false; end; Procedure get—constants; {computes some constants needed by the program} begin al_a:=power (at_num, 0.67) *3.43E-3; {relativistically correct the beam energy for use up to 500 keV\ er: = (inc_energy+511.0) / ( i n c _ e n e r g y + 1 0 2 2 . 0 ) ; er:=er*er; lam_a:=at_wht/ (density* 6 . OE23 ) ; {lambda in cm}

THE SINGLE SCATTERING MODEL

39

lam_a:=lam_a*1.0E8; {pu t into angstroms} sg_a:=at_num*at_num*12. 56*5.21E-21*er; end;

Procedure set—up—screen; {gets the input data to run this program} begin ClrScr; {erases all previous data from screen} GoToXY(22,1); Writeln ('Single Scattering Monte Carlo Simulation'); {Having set up the screen, now get input data} GoToXY(1,5); Write ('Input beam energy in keV ) ; Readln(Inc_Energy); GoToXY(1,7); Write('Target Atomic Number is'); Readln(at-num); {Calculate J the mean ionization potential mn—ion—pot using the Berger-Selzer analytical fit} mn_ion_pot: = (9 .76*at_num + (58 . 5 /power (at_num, 0 .19 ) ) ) *0 . 001 ; GoToXY(1,9); Write('Target Atomic Weight is'); Readln(At-wht); GoToXY(1,11); Write('Target density in g/cc is'); Readln(Density); GoToXY(40,5); write('Is this a bulk specimen (Y/N)?'); if ye s the n {it' s thick} begin {s o estimate the beam range for graphics t h i c k : = 7 0 0 . 0*power(inc—energy,1.66)/density; if thick/2) , wher e 4 > i s the scatterin g angl e define d i n Fig . 4.2. The value s o f sin(((> ) an d cos(4> ) a fe then obtaine d b y standar d trigonometri c formula s fro m th e valu e o f tan(<j>/2) ,

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Figure 4.4. Trajectorie s i n bulk carbo n an d copper at 20 keV. and th e azimutha l angl e \\i i s foun d fro m Eq . (3.11) . Thes e angle s ar e the n passed t o th e procedure new_coord , identica l to the versio n i n Chap. 3 , which determines th e coordinate s o f th e en d poin t o f thi s step . Th e fina l procedure s reset_next_step, back_scatter , show_traj_num , an d display -backscattering handle the various bookkeeping details for each of the trajectories an d update the screen display . The main body of the program, starting at begin and finishing at end simpl y calls th e procedure s a s the y ar e required , loopin g unti l th e require d numbe r o f trajectories hav e been computed . In th e cente r o f thi s loo p i s th e place , afte r th e new positio n of the electron ha s been determined, where code to compute specific features o f th e electro n interaction (suc h as x-ray s or secondar y electrons ) ca n b e placed. The electron is then be tested to see whether it has been backscattered and , if so, the back_scatter procedur e is called; otherwise, the electron proceeds to the next step . After completion , th e program display s the fractional yiel d of backscat tered electron s an d then terminates.

THE PLURA L SCATTERIN G MODE L 7

5

Figure 4.4. (continued)

4.5 Running the program The program can be ran directly fro m th e disk if this is available by typing PCLMC at th e ">" prompt an d the n hittin g return . Onc e th e require d inpu t dat a (atomi c number, atomi c weight , density , and bea m energ y a s before) hav e bee n provided , the program wil l start running. A comparison o f the speed o f this program wit h the equivalent computatio n fo r a bul k specime n usin g SingleScatte r wil l sho w that PluralScatte r i s 1 0 to 2 0 times faster , becaus e a maximu m of onl y 5 0 steps i s calculated fo r eac h trajectory . Figure 4. 4 show s computed trajector y plot s for 20-ke V electron s i n carbon, copper, silver, and gold to correspond wit h those in Fig. 3.7 . Despit e th e simplifyin g assumption s introduce d b y th e plura l scatterin g model, it is evident that the basic details of the interaction ar e well accounted for . In later chapters, we examine in more detail some verifiable predictions of both models and fin d that , whe n use d wit h prope r precautions , eithe r ca n giv e result s o f goo d accuracy. It should be noted that one consequence o f the model used here is that the shape o f the interactio n volum e is independent o f the actua l inciden t bea m energ y and depends only the material of the target. The form of the interaction volume does, however, depend on the angle of incidence o f the beam, as shown in Fig. 4.5, which plots trajectorie s fo r 30 ° an d 60 ° angle s o f incidenc e i n copper . Not e tha t th e

76 MONT

E CARL O MODELIN G

Figure 4.5. Trajectorie s i n copper a t 20 keV fo r inciden t angle s o f 30 ° and 60° .

maximum extension o f the interaction follows th e general directio n o f the incident beam, s o tha t th e volum e becomes elongate d i n th e directio n o f th e beam . Th e lateral extent of the interaction is therefore quite different i n the plane containing the incident bea m an d in the directio n norma l to this, and this will be eviden t in both imaging an d microanalysis performed under these conditions .

5

THE PRACTICAL APPLICATION OF MONTE CARLO MODELS 5.1 General considerations The remainder of this book is devoted to the application of Monte Carlo models to a variety of problems in electron microscopy an d microanalysis. The programs developed in Chaps. 3 and 4 provide only the framework tha t models the basic details of the electro n interaction . W e will now develo p a variety o f algorithm s that , whe n implemented a s PASCAL procedures and functions, can be used to transform these generic Mont e Carlo model s int o programs customized t o solv e a particular prob lem. Befor e proceedin g t o thes e applications , however , ther e ar e a fe w practica l considerations tha t are wort h discussing.

5.2 Which type of Monte Carlo model should be used? We have developed two types of Monte Carlo model, the singl e scatterin g model , which attempt s t o accoun t fo r ever y elasti c interactio n suffere d b y th e inciden t electron, an d the plural scattering model, which considers only the resultant effect of scattering events occurring within some specified segment of the electron trajectory . These mode l shar e much of the same physics, and so, whe n properly used, can be expected to give comparable results. It is, however, necessary t o decide what constitutes proper usage . An important general property of these models is what we shall call, by analog y wit h photographic film , their granularity. Thi s concep t expresse s the ide a tha t th e Mont e Carl o mode l i s takin g wha t i s i n realit y a continuou s sequence o f scatterin g event s an d modeling i t a s a discret e serie s o f independen t events, just as a piece o f film take s a picture an d breaks it down into fragments th e size of the grains making u p the emulsion. The finer th e grain siz e of the film, th e higher the resolution of the image; and the finer the granularity (i.e., the step size) of the Mont e Carlo simulation , th e better th e qualit y of the mode l generated . I n th e case of the single scatterin g model , th e step size is of the order o f the elastic mean free pat h and thus (depending o n energy) i s between a few nanometer s an d a few tens of nanometers, whil e for the plural scattering model, th e step size is a fractio n of the Bethe range and varies from ten s of nanometers to fractions of a micrometer . In modeling som e effect, i t is therefore necessary to ensure that the granularity 77

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of th e mode l i s sufficien t t o resolv e th e kin d o f effec t tha t i s bein g studied . I t i s obvious that a plural scattering model, which at an energy of 10 0 keV would have a step lengt h o f clos e t o a micron , woul d no t b e suitabl e fo r studie s o f electro n scattering i n a thi n foi l onl y a fe w hundre d angstroms thick—th e dat a woul d b e excessively "grainy." But care mus t also be taken to se e that the granularity is not too small. For example, at low beam energies, the step length in the single scatterin g model is only a few angstroms and so is approaching atomistic dimensions. Caution must be exercised in this condition, because a fundamental assumption of the Monte Carlo method is that the sampl e can be considered t o be a structureless continuum (sometimes calle d a "jellium"), an d this wil l only b e tru e whe n the ste p lengt h i s much greate r tha n eithe r atomi c o r crysta l lattice dimensions . Conclusion s draw n about electro n behavio r unde r thes e extrem e condition s ma y therefore b e i n error , and it would actually be better t o use a more granular model. An appropriate test is to determin e th e critica l dimension s o f th e phenomen a o r experimen t t o b e simu lated an d ensur e tha t th e mode l chose n ha s sufficien t granularit y t o resolv e th e effect. A t the same time we must also ensure that the scale is not so far below that of the problem that excessive time is wasted in computation. For example, th e modeling of backscattered signal s fro m bul k specimens can usually be done with a plural scattering model, because the dimensions of the sample and its features or inclusions are usuall y larg e compare d t o th e ste p size . Bu t i n modelin g secondar y electro n production fro m th e sam e specimen , it may be necessary to use a single scatterin g model, becaus e th e scal e o f th e secondar y emissio n (se t b y th e escap e dept h o f secondary electrons ) is only a few nanometers. Usin g a single scatterin g mode l in the first cas e would probably give the same result, but at the expense o f much extra computing time; while using a plural scattering model in the second case might give a result that is accurate on the scale of the step length (i.e., fractions of a micron) but inaccurate o n the scal e o f a few ten s of angstroms .

5.3 Customizing the generic programs The aim of the subsequent chapters of this book is to develop a library of procedures and functions that , when coupled with one or other of the basic Monte Carlo models, will realisticall y an d accuratel y simulat e th e physica l proces s o f interest . Thes e procedures ca n either be added to the basic programs by typing them in, or the code fragments o r complet e program s ca n b e take n fro m th e disk . I n mos t cases , th e customization requires onl y the addition o f the procedure o f interest to the body of the program together with the declaration at the start of the program o f any additional globa l variables , th e additio n o f a call t o thi s procedur e (o r procedures ) a t th e appropriate poin t within the Monte Carlo loop, and the addition o f any special tests (for example , determinin g th e positio n o f th e electro n relativ e t o som e featur e or surface) that are require to model the specific event of interest. As a practical matter, it is usually most convenient t o modify program s in a three-step process . First, add

PRACTICAL APPLICATIO N 7

9

the procedures an d their associated global variable s to the program an d check that it still compile s correctly . Next , ad d the cod e tha t perform s an y specifi c test s mad e prior to the calling o f the procedure an d again check that the program compiles an d runs. Finally, ad d the call(s) to the procedure itsel f an d see that the program func tions as expected. Breakin g the process int o these steps greatly eases the problem of debugging th e program i n the event of an error, becaus e onl y one change ha s been made t o the program a t each iteration .

5.4 The "all purpose" program An apparentl y irresistibl e temptatio n i n Monte Carl o modelin g i s the urge to construct an all-purpose simulation that can answer any question about any effect fro m a specimen o f arbitrary geometr y and inhomogeneity. The problem wit h producing such an oracle is that the more general a program must become, the more convoluted must be its logic, and the difficulty o f debugging a program rises exponentially with its complexity. It is not just that a general program is longer but that it has built into it a multiplicity o f options (i.e. , i f ....... . .en...........elsee.........statement that ca n (and , perversely, quit e ofte n do ) interac t wit h one anothe r i n unexpected ways. Thus, while a program may function quit e normally and correctly for one set of conditions , anothe r an d apparentl y equall y reasonabl e se t o f condition s ma y cause a malfunction or , worse, a subtl e error i n the outpu t data. The most reasonable advic e is to construct a new simulatio n for each proble m that is to be solved . For example, if we are interested i n the x-ra y production fro m features i n the shap e of either sphere s or cubes, it is best to write one program that tackles the case o f a spherical sampl e an d another separat e progra m that consider s the case of the cube. Since a high fraction of the code will be identical i n both cases, the additional time taken in constructing a second program is small. But the elimination o f complicate d an d messy test s relate d t o th e specime n shap e wil l make th e code easie r t o writ e an d follow , muc h easie r t o debug , an d (ver y likely) faste r i n execution becaus e som e additiona l shortcuts , simplifications , an d optimization s may becom e possible . I n summary , a Mont e Carl o mode l i s mos t efficien t an d effective whe n it is constructed a s a special too l t o solv e a particular problem .

5.5 The applicability of Monte Carlo techniques It ha s bee n sai d tha t "fo r a two-year-ol d chil d wit h a hammer, everythin g i n th e world is a nail." Monte Carlo users can occasionally b e guilty of the same restricte d vision. Whil e th e techniqu e i s o f ver y wid e an d genera l applicability , ther e ar e problems tha t cannot be solved b y this type of approach. Specifically , th e methods discussed her e canno t be applied t o problems tha t violate th e basic assumption s of the models ; fo r example , electro n channelin g an d mos t problem s associate d wit h imaging i n th e transmissio n electro n microscop e canno t b e investigate d becaus e

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they involv e th e crystallin e natur e of a sampl e that the Mont e Carl o simulatio n i s treating a s structureless. No r can Monte Carlo methods be applied to problems tha t rely on effects, suc h as electron diffractio n o r the effect o f electron spin polarization, which ar e no t accounte d fo r i n th e physic s buil t int o th e programs . Finally , th e programs ma y b e of onl y limite d value at very lo w (belo w say 0.5 keV) and very high (abov e 50 0 keV) beam energie s becaus e importan t physica l effects—suc h a s the consequences of relativistic or nuclear interactions—have been omitted from th e physics. Ther e ar e als o occasions—suc h a s scannin g electro n microscop e (SEM ) operation a t low magnifications, where the details o f the electron bea m interaction are less significant than purely geometrical effects—wher e a Monte Carlo approach is simpl y no t a s appropriat e a s a simple r an d faste r analytica l mode l migh t be . Within these boundaries, however, the scope for the use of Monte Carlo simulation techniques i s wid e and , as demonstrated i n the subsequen t chapters, th e program s developed her e ca n b e use d effectivel y t o answe r man y question s abou t t o solv e many commonl y encountere d problem s i n microscopy an d microanalysis.

6

BACKSCATTERED ELECTRONS 6.1 Backscattered electrons In this chapter, we will concentrate on simulations associated with various aspects of backscattered electrons . W e wil l defin e backscattere d electron s a s thos e inciden t electrons tha t are scattered ou t of the target after sufferin g deflection s throug h such an angl e tha t the y leav e th e materia l o n th e sid e b y whic h the y entere d (i.e. , fo r normal incidence , the minimu m scattering angl e require d i s 90°). I n practice, i t is often mor e useful t o defin e backscattere d electron s a s those inciden t electrons that are scattere d i n the target i n suc h a way a s to be collectibl e b y a suitable detecto r placed o n the incident beam side of the specimen. This , of course, implie s tha t the apparent magnitude of the backscattering will be affected b y the size and position of the detector relative to the specimen and incident beam. To avoid confusion, we will separate thes e case s b y callin g th e tota l numbe r o f backscattere d electron s pe r incident electro n th e backscattering yiel d TI and the number per incident electro n a s measured by som e specifie d detecto r the backscattered signal . Since ever y Mont e Carl o mode l track s eac h inciden t electro n i n it s passag e through the target, the determination o f whether or not a given electron is backscattered i s inherent in the simulation. At the same time, the backscattering yiel d TI is a convenient macroscopi c measur e o f th e interactio n o f th e electro n bea m wit h th e target an d form s a usefu l experimenta l tes t o f th e prediction s o f a Mont e Carl o model. Befor e attemptin g t o simulat e detail s o f backscattere d imagin g fro m com plex an d inhomogeneous samples , w e must therefore demonstrat e tha t the models developed i n th e previou s chapter s correctl y matc h experimenta l value s fo r th e backscattering yiel d fro m plana r an d homogeneous materials .

6.2 Testing the Monte Carlo models of backscattering The variatio n o f backscatterin g yiel d wit h atomi c numbe r wa s firs t establishe d nearly a century ag o (Starke, 1898 ; Campbell-Swinton , 1899) . Figur e 6. 1 show s a compilation of some modern backscattering yiel d data plotted agains t atomic number for an incident beam energy of 1 0 keV using results taken fro m Bisho p (1966), Drescher e t al . (1970) , Hunge r an d Kuchle r (1979) , Neuber t an d Rogaschewsk i (1980), Reimer and Tolkamp (1980), and Heinrich (1981). T I is seen to vary consid 81

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Figure 6.1. Experimenta l BSE yield data at 1 0 keV and corresponding Monte Carlo values.

erably acros s the periodic table , increasing fro m abou t 0.06 fo r carbon to about 0.5 for uranium . Th e variation wit h atomic number i s generally smooth, although—a s pointed ou t b y Bisho p (1966)—th e slop e o f th e curve changes discontinuousl y a t about Z = 3 0 and again at about Z = 60 . These changes have been associated wit h the binding energies o f the atomic electrons. However , while the general form of the variation i s evident , i t i s clea r tha t ther e i s significan t scatte r betwee n differen t pieces of data. Although most of the cited authors claim measurement accuracies of better than 5%, variations between comparabl e values are as much as 20% in some cases. Thes e discrepancie s probabl y aris e from th e variety of methods employe d to measure t\ an d i n particula r fro m th e succes s wit h whic h th e effec t o f secondar y electrons ca n be remove d fro m th e data. A complete se t of experimental backscat tered dat a measurement s i s available in Joy (1993) . The backscatterin g coefficien t T\—computed usin g eithe r th e singl e o r th e plural scatterin g Monte Carlo models (1000 trajectories per point)—is seen to be in good agreement wit h the experimental data , either valu e lying within the spread of measured values . Th e singl e scatterin g values , especiall y fo r Z > 40 , ten d t o be slightly on the high side of the best-fit tren d line through the data, but the sense and magnitude of the deviation i s not systematic . The excellent agreemen t fo r either of the model s is , o f course , gratifying , bu t i t i s no t unexpected , sinc e bot h model s contain a parameter that can be selected s o as to match the experimental backscatter ing yields. In the case of the single scattering model, the screening paramete r a can be adjusted t o ensure a good fi t to measured data. The expression fo r a a s given in Eq. (3.2) and used in our model is that suggested by Bishop (1976). Similarly, in the plural scatterin g model , the minimu m impact paramete r p0 [Eqs . (4.3 ) an d (4.4)] , when use d i n th e for m give n in Eq . (4.8) , agai n allow s th e compute d backscatte r data t o b e fitte d t o th e experimenta l values . In bot h models , th e paramete r t o b e

BACKSCATTERED ELECTRON S 8

3

Figure 6.2. Variatio n o f BS yield fro m gold/coppe r solid solution an d corresponding Mont e Carlo simulate d dat a usin g various models .

adjusted is a simple function of the atomic number Z; hence improving the fit to one chosen dat a valu e will usuall y worsen th e fi t a t som e othe r value . The suggeste d values represent th e bes t overal l fi t t o the measure d values a t 1 0 keV. It is also necessary to consider i n somewhat more detail the variation of T| with composition whe n the targe t i s no t a pure elemen t (Herma n an d Reimer, 1984. ) Figure 6.2 plots the variation of the backscattering coefficient for a solid solution of gold i n coppe r (dat a taken fro m Bishop , 1966) . A s the atomi c percentag e o f gol d increases fro m 0 % to 100% , the measured backscattering coefficien t increase s lin early from the expected value for pure copper to the expected value for pure gold. If we have a materia l wit h th e chemica l formul a S^jX , wher e th e Xt ar e element s of th e periodi c tabl e wit h correspondin g atomi c weight s A,- , an d th e ai ar e thei r valences i n the chemical formula (e.g., i n H2O, al - 2 , a2 = 1 , Xl = H, X2 = O, Al = 1 , A2 = 16) , the n w e can defin e a mass concentratio n ci fro m th e formul a Cj = (djAiltpjA^, wher e £,-c, - = 1 . The data shown in Fig. 6. 2 represents a specia l case o f a genera l resul t (Castaing , 1960 ; Heinrich , 1981) , whic h state s tha t th e backscattering coefficien t T| mix for a mixture of elements i s given by the relation :

In simpl e situations , i t i s clearly possibl e t o fin d th e valu e of -n mix b y findin g th e individual values of if y fo r the components, usin g the simulation s alread y described and the summing them using Eq. (6.1). As Fig. 6.2 shows, this provides a very close fit to the experimental data. This procedure is, however, rather tedious if the chemis-

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try o f th e targe t i s comple x (excep t fo r th e cas e o f th e HKLC S model , discusse d below); instead , we migh t try t o obtain th e correc t valu e of T) mix b y makin g som e assumption abou t a n effectiv e o r averag e atomi c numbe r fo r th e compoun d o f interest. Th e simplest suggestio n (Miiller, 1954 ) is to use the mean valu e of atomic number Z mix fo r the compound—i.e.,

The effective atomi c weight Amix is found i n a similar fashion , and the value of the density used is the measured value for the compound. Figure 6. 2 shows the results of employing this assumption and the plural scattering Mont e Carlo model o f Chap. 4 to mode l th e copper-gol d system . The agreemen t betwee n th e experimenta l an d computed dat a i s adequat e t o goo d ove r th e ful l rang e fro m 0 % t o 100 % gold, indicating that the accuracy of this simple approximation is likely to be adequate for most purposes. Alternatively , because electro n stoppin g powers computed from th e Bethe equation are additive, Everhart (1960) suggested that a more physically exact expression for Zmix woul d b e

However, a s show n in Fig. 6.2, application o f this rul e t o th e copper-gol d syste m produces a considerabl e deviatio n betwee n th e predicte d an d experimenta l data , suggesting tha t Eq. (6.2) is a mor e usefu l expression . Simila r result s ar e foun d i n more complex systems. Figure 6.3 plots the experimentally determined backscatter ing coefficien t fo r th e ALpa^^A s syste m a s a functio n o f th e mol e fractio n o f aluminum (Sercel et al., 1989) . Superimposed o n this plot are the computed predic tions fo r th e backscattering, usin g Eqs. (6.2) and (6.3) . I n this example, fo r which the differences betwee n the atomic numbers are relatively small and the composition range i s limited, th e differenc e betwee n th e tw o models i s not a s marked, but i t is clear tha t th e simple r expressio n o f Eq . (6.2 ) is stil l a t leas t a s goo d a s th e mor e complicated expressio n o f Eq . (6.3) . I n th e res t o f thi s volume , therefore , compounds wil l be computed eithe r assumin g the use of Eq. (6.2) or from th e HKLC S procedure, discusse d below. The backscatterin g coefficien t T I i s not constan t wit h energy, a s is often state d in introductor y textbooks , bu t varie s i n a manner tha t depend s o n both th e energ y

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5

Figure 6.3. Variatio n o f B S coefficien t fo r A l i n AlGaA s syste m (Serce l e t al. , 1989 ) and simulated MC data usin g different models . and the atomic number, Figure 6.4 plots data from Hunger and Kuchler (1979) for f] as a function o f Z at 4 and 40 keV. It can be seen that while the general for m o f the variations at the two energies i s similar, the absolute values change significantly. On moving from 4 to 40 keV, the backscattering coefficient of the lightest elements fall s by a factor of up to two times, while that for the heavier elements rise s by as much as 25% . The plots fo r 4 and 40 ke V actuall y cross a t about Z = 40 . The detaile d form of this variation is shown more clearly in Fig. 6.5 , which plots the variation of

Figure 6.4. Variatio n o f B S yiel d wit h energy ,

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Figure 6.5. Variatio n o f B S yiel d wit h energy .

r\ for carbon, silicon, copper, silver , and gold as a function o f beam energy. It is clear that at the lowest beam energies (i.e. , 50 keV ) th e backscatterin g coefficient s fo r ligh t element s ten d t o decreas e b y about 10 % for eac h facto r o f 1 0 increase in bea m energy , while thos e fo r heavie r elements remai n abou t constan t (Antola k an d Williamson , 1985) . Hunge r an d Kiichler (1979) have derived an analytical expression that predicts, fairly accurately , values of T I as a function o f both the atomic number Z and the incident energ y E (i n kilo-electron volts) :

where

Neither of the two Monte Carlo models discussed in the previous chapters can predict thi s type of behavior . I n fact , a plot o f T\ versus energ y usin g eithe r o f ou r models wil l show that, over the energy range 5 to 40 keV, the predicted backscatter ing coefficient is , to within the expected statistical error, independent of energy. (For lower energies , th e value s fro m th e singl e scatterin g mode l ma y vary , but thi s i s because th e computation of trajectories i s terminated a t an arbitrary cut-off energy , which i s a significan t fractio n o f th e inciden t bea m energy) . Ther e ar e severa l

BACKSCATTERED ELECTRON S 8

7

reasons fo r this disagreemen t betwee n experimenta l an d computed data . First , th e parameters in both models that adjust the backscattering values to match experimental data a t some nominal energ y ar e designe d t o hold th e valu e of r\ constan t with beam energy , since , unti l quite recently , low-energy electro n interaction s wer e of little interest ; therefore , th e assumptio n tha t T | wa s constan t wit h energ y wa s a reasonable simplification . Second, a s noted in discussing the data in Fig. 6.1 above, the qualit y of the experimental dat a on backscattering coefficient s i s rather mixed, and th e magnitud e o f the variatio n of -r\ wit h energy for most elements i s less than the sprea d betwee n differen t publishe d value s a t th e sam e energy . Consequently , many workers have preferred to take T| as constant until more accurate experimental values were determined. Finally , an d of more fundamental significance, both models use the screene d Rutherfor d cross sectio n as the basis for the model of electro n scattering. For a wide range of conditions—that is, for elements o f low and medium atomic number, electron energie s greater than 10 keV, and small scattering angles— the Rutherfor d cros s sectio n i s a goo d approximation ; bu t fo r heav y elements , energies below 1 0 keV, and large scattering angles, the Rutherford cross section can be significantl y in error . I t must , instead , b e replace d b y th e Mot t cros s section , which takes into account spin-orbit coupling in the solution of the relativistic Dirac equation (Reime r an d Lodding, 1984) . I n general terms , th e Mot t cros s sectio n i s essential in applications wher e the interaction of interest consist s of a single elastic large-angle scattering event (e.g., in backscattering fro m thi n foils or in calculations of th e energ y spectru m o f backscattere d electrons ) an d i s desirabl e fo r al l low energy applications . But for effect s cause d by electron diffusio n an d plural scatter ing (e.g. , backscatterin g fro m bul k samples, x-ra y production, an d secondary elec tron generation) , th e Rutherfor d cros s sectio n give s rathe r goo d agreemen t wit h experimental data except at low beam energies. The Mott cross section is considered in more detail in the final chapte r of this book, but elsewhere th e Rutherford mode l will b e use d because , unlik e th e Mot t model , whic h require s th e generatio n o f extensive table s o f data, it can be expresse d analytically . With th e increase d interes t i n low-energ y scannin g microscop y an d micro analysis, it would , however, be desirabl e t o try an d modif y th e models develope d earlier to correctly predict the variation of T| with energy so as to obtain som e of the benefits of the Mott cross section without the extra effort tha t is required. This is not readily possibl e for the single scatterin g mode l because of the necessity, discussed in Chap . 3 , of terminating the electrons' trajectorie s a t som e predetermine d cutof f energy, usually 0.5 keV. At low energies, thi s cutoff i s a significant fraction o f the incident energ y an d henc e th e precisio n o f th e computatio n i s doubtful . Fo r th e plural scattering model, however, a variation of Z with E can readily be achieved. As discussed in Chap. 4, the scattering angl e $ can be written in the form [usin g Eqs. (4.5) an d (4.10)]:

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Figure 6.6. B S yiel d vs , ta n ( 0/2). Love et al. (1977) reasoned that since, from the form of the Bethe equation, the variation o f (EQ/E) i s substantially independent of atomic number, the backscattering coefficien t T ) must depend only on the value of tan (cj> 0/2). As shown in Fig. 6.6 , this assumption is correct. Using the code of Chap. 4, we find that the backscattering coefficient t ) depend s o n th e valu e of ta n (4>o/2 ) i n a monotonic fashio n bu t i s t o within statistica l erro r independent o f the atomi c number and density o f th e target and o f th e electro n energy . Fro m a curv e fi t o f th e dat a i n Fig . 6,6 , w e fin d th e relation:

These coefficients diffe r slightl y from thos e originally given by Love e t al . (1977 ) and quoted in Eq. (4.7) because of the effect o f the small angle-scattering correctio n included in our program [Eq . (4.10)]. To obtain the necessary estimat e of TI , we can now us e th e Hunger-Kiichler relation [Eqs . (6.4 ) t o (6.6) ] discussed above . Give n the atomic number of the target and the beam energy E, this gives us the value of TI and hence of tan (4>o/2). This value can now be used in place of the original estimat e [Eqs. (4.6 ) an d (4.8)] , wit h th e advantag e tha t the backscatterin g coefficien t pre dicted by the Monte Carlo program will vary correctly with incident beam energy. It should be clear that this Hunger-Ktichler-Love-Cox-Scott (HKLCS) model is not a radical revisio n o f th e plura l scatterin g schem e bu t simpl y th e replacemen t o f Duncumb's one-parameter fi t to the minimum scattering angle [Eqs. (4.4), (4.8), and (4.9)] with a more comple x fit. To incorporat e thi s modificatio n int o th e plura l scatterin g code , i t i s onl y

BACKSCATTERED ELECTRON S 8

9

necessary to replace one procedure (hence the value of writing the code in a modular fashion). The revised procedure listed below replaces the original procedure of the same name and implements Eqs. (6.4) , (6.5), (6.6), and (6.8). It is stored on the disk as HKLCS.PAS. To replace th e ol d procedure wit h the new one using the TURBO PASCAL editor, g o into EDIT mode and load PSJVIC.PAS; then scroll through the program until the start of the procedure Rutherford—Factor . Type (Control K R) and a smal l windo w wil l appea r askin g fo r a file name . Typ e i n HKLCS(return) . This block of code will then be read in from the disk and be placed at the cursor position. Now delet e th e old procedure an d recompile th e program. Procedure Rutherford—Factor; {computes the Rutherford scattering factor for this incident energy using the Love, Cox, Scott model and the Hunger-Kiichler backscatter equation} var hkbs,hkc,hkl,hkm:extended; begin {compute the Hunger-Kuchler backscattering coefficient}}} htan:=0,1382-0.9211/sqrt(at_num) ; hkl: =l n ( a t _ n u m ); hkc:=0.1904-0.22236*hkl+0.1292*hkl*hkl -0.01491*hkl*hkl*hkl; hkbs : =hkc*power (inc_energy' hkm) ; [now compute the Love, Cox, Scott parameter from Eg.

(6.8)}

rf : =0.016697 + 0. 55108*hkbs-0.96777*hkbs*hkbs+l.8846*hkbs*hkbs*hkbs; r f : =rf*inc_energy; end;

Figure 6. 7 plots the backscattering coefficient , computed using this version of the plural scatterin g program , fo r carbon, silver , and gold a s a function o f inciden t beam energy . Th e variatio n o f T ] wit h energy i s clearl y eviden t belo w 1 0 keV, th e yield fo r carbo n risin g a s th e energ y fall s whil e that for silve r remains essentiall y constant and that for gold falls, ultimately to a value lower tha n that for silver. The predicted value s agre e wel l wit h the dat a plotte d i n Fig. 6.5 excep t a t th e lowes t energies, where the Hunger-Kiichler fit is probably inaccurate. Althoug h this revised model i s not a substitut e for th e mor e accurat e Mot t scatterin g cros s section , i t is often a sufficiently goo d approximatio n to make the use of the more complex Mott model unnecessary . A n additiona l advantag e o f thi s approac h i s that , sinc e a n estimate fo r th e backscatterin g coefficien t i s produced o n th e basi s o f th e atomi c number o f th e targe t fro m th e Hunger-Kiichle r model , w e coul d us e Eq . (6.1)

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Figure 6.7. compute d variatio n o f B S yiel d wit h energy i n HKLCS model .

directly to treat the case of a multicomponent material. Without coding this example in detail, the procedure woul d be a s follows : Get the number n_comp of components in the compound For i = 1 to n_comp Get atomic number Z[i] Get corresponding concentration c[i] Calculate BS coefficient % from Egs. (6.4) through Next i Calculate mean BS coefficient as S^iii Find Rutherford factor from equation 6.8

(6.6)

6.3 Predictions of the Monte Carlo models We can now investigate some of the predictions tha t the Monte Carlo model s make about the properties o f backscattered electrons . Thes e ar e parameters o f the back scattering tha t are not, in either a direct o r a hidden way , built into the models we have developed . 6.3.1 Variation of ^ with angle of incidence The plural scattering models allow us to input the angle of incidence of the electro n beam relative to the surface normal of the specimen. We can therefore calculate how the magnitude of the backscattering coefficient varie s with the specimen tilt . Figure 6.8 show s som e dat a fo r iro n compare d wit h a n experimenta l measuremen t (My klebust e t al., 1976 ) fo r Fe—3.2% Si . In bot h cases th e backscattering coefficien t T](0) at some angle 0 is normalized by the corresponding backscattering coefficien t at normal incidence T|(0) , and th e calculated an d experimental data a s obtained fo r

BACKSCATTERED ELECTRON S 9

1

Figure 6.8. Experimental an d computed variatio n o f B S yield wit h tilt .

an incident energ y of 30 keV. It can be seen that the amount of backscattering rise s rapidly a s the angl e o f incidenc e i s increased, wit h both th e experimenta l an d the Monte Carlo data showing an increase o f around 50% for a tilt angle of 45°, while for a tilt of 60°, the backscattering double s i n magnitude. These number s will vary somewhat with both the incident beam energy an d the atomic number of the target, although th e genera l for m o f th e Ti(0)/^(0 ) variatio n remain s fairl y clos e t o tha t illustrated i n Fig. 6. 8 except fo r the higher atomi c number s ( Z > 50 ) an d for low energies (E < 3 keV). The examination of such phenomena b y running a series of simulations is an excellent wa y of becoming familiar with the use of a Monte Carlo model. Simila r calculation s can, of course, b e performed with the single scatterin g model, but the relative slownes s o f this model make s obtainin g a sufficiently goo d statistical accuracy a rather tedious affair. The actual data obtained ar e close to those of th e plura l scatterin g approximation . 6.3.2 Energy distribution and mean energy of backscattered electrons The energ y distributio n o f th e backscattere d electrons , o r a t leas t thei r averag e energy fo r a given set of conditions, is an important parameter, becaus e the output from a backscattered detecto r depend s o n both th e numbe r o f backscattered elec trons (i.e., the value of -n) and the energy of these electrons. Any measurement o f the backscattered signa l is therefore a convolution o f the backscattering yiel d an d the energy distribution from th e sample, and consequently ther e wil l not be, in general, a simpl e relationship betwee n thi s signa l an d the atomic number of the target. The energy distributio n an d mean energy ca n readily b e found b y modifying th e back -scatter and display_backscattering procedures in the plural scatterin g model. W e simpl y not e o n whic h ste p ( 1 t o 50 ) o f th e trajector y th e electro n i s

92 MONT

E CARLO MODELING

traveling whe n i t i s backscattere d an d ad d 1 to th e & th bo x o f a n arra y bs_e [ ] . Since the energy for all electrons o n the k-th step is the same, i.e., E[k], the average backscattering energ y wil l be :

because th e total number of electrons backscattered i s bk_sct. The modified proce dures below implement these steps and plot the mean-energy value on a thermome ter scal e a s a percentage o f the inciden t bea m energy . Remember: ad d to the va r lis t at the top o f th e program th e entry: bs_e:array[l . . 50] of integer ; Procedure back—scatter ; {handles specia l case of a backscattered electron} begin bk_sct: =bk_sct + l; {ad d one to counter} num: =num+l; {ad d one to total} bs_e[k] :=bs_e[k]+1; {electron backscattered at k-th xyplot (y , z , y n, 99 ) ; {plot BS exit} end;

step}

Procedure display—backscattering; {draws a thermometer to display the backscattering coefficient and another to indicate the mean energy of the backscattered electrons} label var

hang; a,b,c,k:integer; mean—energy:extended;

begin a: =GetMaxX-180; b:=bottom+23; c:=20;

{adjust to suit your screen} {ditto} {ditto}

OutTextXY(a,b-8,"0 . . . . 0.25 . . . 0.5 . . . 0.75"); QutTextXY(a+18,b+5,"BS coefficient" ) ; Line ( a , b, trunc ( a + ( b k _ s e t / t r a j — n u m) * 2 2 0 ) ,b ) ; {dra w it } {now compute the mean backscattering energy value} mean—energy: =0; {initializ e th e value}

BACKSCATTERED ELECTRONS

93

for k:*=l to 5 0 d o

begin {add up number at step k*energy at step k} mean—energy: =mean_energy+E[k] *bs_e[k] ; end; mean—energy : =mean_energy/bk—set; {averaged over total BS] {draw a thermometer to plot up this data value }

O u t T e x t X Y ( c , b - 8 , ' 0 %....................................50%..................................1000%) ;; OutTextXY(c+2,b+5,'% of incident energy'); line (c, b, c + trunc (165* (mean—energy/ inc —energy) ) ,b) ;{{{{draw itn in } hang: if (not keypressed) then goto hang; {freeze display on screen} CloseGraph; {shut down the graphics unit} end;

Figure 6. 9 plot s th e valu e o f mean-energ y a s a functio n o f th e atomi c number for 1 5 keV incident electrons, showing that as Z increases, the value climbs steadily from carbo n to gold. These values are seen to be in good agreement with the experimental data measured by Bishop (1966). From a curve fit of the Monte Carlo data, we can express the value of mean-energy as : mean-energy = £ 0[0.55612 + 3.163*1Q- 3*Z - 2.0666 * IQ-^Z2] (6.10 )

Figure 6.9. Variatio n o f mea n B S energy wit h Z.

94

MONTE CARL O MODELING

Figure 6.10. Compute d energy spectr a o f backscattered electrons .

This variatio n is significan t because backscattere d detector s usuall y respond mor e efficiently t o electrons of higher energy; therefore, as we move through the periodic table, the output from the detector will increase both because of the higher backscattering coefficien t an d becaus e o f th e highe r averag e mean energy . Th e reaso n fo r this variation in mean energy can be seen by plotting the histogram of the bs_e [k ] array, compute d above , whic h measures th e numbe r of backscattered electron s a t each trajector y ste p k. As show n in Fig . 6.10 , whic h plots dat a for E0 = 1 5 keV beam energy, for carbon this histogram is more or less symmetrical about the energy 0.5£0; but for copper, the maximum in the curve has shifted upward in energy, while for gol d th e curv e has n o maximu m but increase s monotonicall y a s th e backscat tered energ y approache s th e bea m energy . Th e for m o f th e energ y distributio n predicted b y th e Mont e Carl o simulatio n i s i n generall y goo d agreemen t wit h experimental dat a (Bishop, 1966 ; Mykelbus t et al., 1976 ) excep t fo r the top end of the distribution clos e to th e inciden t beam energy, where the numbe r of backscat tered electrons is overestimated somewhat . These particular electrons ar e those that are scattere d i n a singl e high-angl e even t clos e t o th e entranc e surface , an d th e scattering o f suc h electron s i s accuratel y described onl y by the Mot t cros s sectio n rather than by the screene d Rutherfor d model use d here. However, sinc e these are only a smal l fractio n o f th e tota l numbe r o f backscattere d electrons , th e erro r i s insignificant. Although we will not pursue it here, this same code fragment ca n also be used to estimate the information depth o f the backscattered electrons—tha t is, the depth

BACKSCATTERED ELECTRONS

95

beneath the surface reached by an electron before it is backscattered. If an electron is backscattered o n the k-th step, the n the maximum possible dept h to which it could have traveled beneath th e surface is k*step, wher e step is the step length in the simulation, whic h i s usuall y one-fiftiet h o f th e Beth e range . A n analysi s o f th e bs_e [k ] distributio n therefore provides an upper bound on the information depth in th e backscattere d image . Typicall y thi s i s abou t 0. 3 o f th e Beth e range . Th e program JustBS on the disk incoporates thi s and the other modification s discussed above to give a detailed mode l of the backscattering effect , includin g an estimate of the surfac e area fro m whic h the electron s ar e emitted . 6.3.3 Variation of backscattering with density In addition t o comparing th e predictions o f our simulations with experimental data , we ca n us e th e Mont e Carl o mode l t o predic t effect s tha t migh t no t b e easil y observable althoug h they migh t also be important. A n example o f such an applica tion i s t o as k th e question : "Ho w doe s th e backscatterin g coefficien t o f a sampl e vary with density?" This question is of particular interest because of attempts to use the backscattered signal as a means o f performing chemica l composition analysis. While suc h a method migh t wor k for a homogeneous material , i t seem s natural to ask whethe r changes i n the densit y o f th e target , cause d perhaps b y differen t pro cessing methods , might not lead to a systematic error. Running the plural scatterin g simulation fo r coppe r a t 1 5 keV, leaving al l o f the physica l parameter s unchange d except for the density an d using 5000 trajectories for each value , produces th e data shown i n Fig. 6.11 . To within th e statistical error of the simulation, it can be seen

Figure 6.11. Compute d variation o f BS yield with sampl e density.

96 MONT

E CARL O MOD

that changing the density over a wide range o f values does not change th e backscattering coefficien t fo r a n otherwis e homogeneou s material . W e als o fin d tha t th e mean energy o f the backscattered electrons , althoug h not plotted, is independent o f the density . While a t first sigh t it might seem tha t this result is counterintuitive, a n examination of the equations in Chaps. 3 and 4, which determine electron scattering , shows tha t the densit y appear s onl y in the stoppin g powe r equation . Consequentl y varying the densit y changes onl y the Beth e rang e o f the electrons , an d this woul d not b e expecte d t o have an y effec t o n t) . If , however, the materia l i s no t homoge neous bu t contain s region s o f differen t density , change s migh t b e expected ; thi s situation i s examine d late r i n this chapter. 6.3.4 Variation of backscattering with thickness A final demonstratio n of the models is to use them to compute how the backscatter ing coefficient migh t vary with the thickness of the target. Clearly, if the specimen is very thin , then the amoun t of backscattering i s small ; while for a sufficientl y larg e thickness a t a given inciden t energy, the backscatterin g coefficient shoul d b e con stant. Our interest her e i s to simulat e the variatio n of i q betwee n these two limitin g conditions and to compare thi s with experimental data. It is immediately clear that, in this case, we cannot use the plural scattering approximation because, as discussed in Chap . 5 , th e granularit y o f thi s mode l i s prese t t o b e one-fiftiet h o f th e Beth e range. When the target is a thin foil, eac h ste p of the plural scattering mode l would be a substantial fraction of the thickness of the foil and the quality of the approximation woul d be very poor. Instead, w e must use the single scatterin g model, becaus e the ste p lengt h her e i s o f th e orde r o f th e elasti c mea n fre e pat h (e.g. , a fe w nanometers at 1 0 keV), and is thus much smaller than the thickness being modeled . Figure 6.12 plots the predicted backscattering coefficien t o f films of carbon, copper , and gold a s a function o f their thickness a t an incident bea m energy o f 1 5 keV. For convenience i n comparison, th e thicknesses i n each cas e hav e been expresse d a s a fraction o f the appropriate Bethe range RB calculated from Eq . (4.1). The backscatter yield varies almost linearly with the specimen thicknes s (Niedrig, 1982 ) right up to the point where the backscattering yield reaches it s "bulk" value. The thickness at which thi s occurs , however , varies significantl y fro m on e materia l t o another , a s would be expected fro m a n examination o f the interaction volume s pictures show n in Fig. 4.4 . I n carbon, th e scatterin g o f the electrons is weak, giving a n interactio n volume tha t hang s lik e a teardro p downwar d from th e entranc e surface . Conse quently th e specime n ha s t o b e almos t 0. 5 RB thic k befor e th e backscatterin g saturates. In the case of gold, the electrons ar e strongly scattered, givin g an interaction volume that is squashed u p against the surface. As a result, the backscatterin g yield reaches it s bulk valu e a t only abou t 0.1 RB. Since th e Bethe range i n gold i s only a small fractio n of that in carbon, thi s show s that the information depth o f the

BACKSCATTERED ELECTRONS

97

BS Yield v s Thickness

Figure 6,12. Variatio n of B S yield wit h specime n thickness .

BS signal, (i.e., th e extent of the volume sampled by the backscattered electrons) is much les s in gold than i n carbon.

6.4 Modeling inhomogeneous materials So far every application o f Monte Carlo modeling w e have considered assume s the specimen to be a infinite half-plan e of uniform composition . I n practice, o f course, samples o f suc h a restricted natur e ar e o f littl e interest , sinc e i n th e real worl d of microscopy our specimens are finite in size; have edges, shape, and surface topography; and sho w wide variations of chemical compositio n fro m on e point to another. Fortunately, Mont e Carl o simulations , whic h trac k th e electro n ste p b y ste p a s it moves through the specimen , are ideally suite d to this type of situation, so we can now start to develop the necessary tools to model such systems. In the simplest case , the sample might consist of a layer of material 1 on top of a substrate of materials 2. If the depth of the interface between these materials occur s at a depth boundary, we can then determin e which material th e electron i s in by testing the value of z .

98

MONTE CARLO MODELING

{the sample consists of two materials i = l and i = 2} i : = l ; {reache s to player first} if z>= boundar y the n i : = 2 ;

The program below implements this example of a thin film on a substrate to illustrate how the original program of Chap. 4 is modified to deal with two materials. In order to save space, functions and procedures that have been discussed already, either in this chapter or earlier chapters, are not listed but simply identified as required. A further generalization to three or more materials, or to the situation where a single scattering model is required, is a simple matter; a program of this type is given in the chapter on x-ray generation. Program BINARY; {this code performs a plural scattering Monte Carlo trajectory simulation for the case where a layer of material A is placed on top of a bulk substrate of some other material B. There is no graphical display in this program} {$N+} {$E+} uses

{turn on numeric coprocessor} {install emulator package}

CRT, DOS;

{resources required}

label

back—scatter,abort;

const

two_pi = 6.28318;

{2 constant}

var

inc_energy,boundary,tilt,s_tilt,c_tilt:extended; nu, sp, cp, ga, an, an__m, an_n: extended; x,y,z,xn,yn,zn,ca,cb,cc,ex,cy,cz,vl,v2,v3,v4:extended E:array[l . . 2,1 . . 51] of extended; at_num:array[1 . . 2] of extended; at_wht:array[1 . . 2] of extended; mn_ion_pot:array[1 . .2] of extended; m_t_step:array[1 . .2] of extended; density:array[1 . .2] of extended; step:array[l . . 2] of extended; rf:array [1 . .2] of extended; bk_sct,num,traj_num,i,k:integer; Function power(mantissa,exponent:real):extended; Function stop_pwr(i:integer;energy:extended):extended; {calculate stopping power for material i using Eg. (3.21)} var temp:extended;

BACKSCATTER

99

begin if energyot[i] ) ; stop_pwr: =7.85E4*at_num[i]*temp/(at_wht[i]*energy) ; end;

Procedure ranged: integer) ; {calculates range assuming Bethe continuous energy loss} var energy,f,fs,bethe—range:extended;: 1,m:integer; begin fs:=0.; for m:=l to 21 do begin

{initialize variable to be sure} {siinpsons rule integration}

energy: = (m-1)*inc_energy720; f : = 1/stop_pwr(i,energy); 1:=2;

if m mod 2 = 0 then 1:=4; if m=l then 1:=1; if m=21 then 1:=1; fs:=fs+l*f;

end; {now use this to find tions}

the range and step length for these condi-

bethe_range: = fs*inc_energy/60 . 0; {i n g/'cm2} m_t_step[i] : =bethe_range/50.0; bethe_range: =bethe_range*10000.0/density[i] ; {i n mi crons} G o T o X Y ( 4 0 * ( i - l ) +1,13) ; {displa y 1 o n LHS, 2 on RHS of screen} w r i t e l n l ' R a n g e in ' i , 'i s bethe_range:4:2 ' ' m i c r o n s ' ) ; s t e p [ i ] : =bethe_range/50 . 0 ; {uni t ste p o f simulation} end; Procedure p r o f i l e ( i : i n t e g e r ) ; {compute 50-ste p energy profile for electron beam} var A l , A 2 , A 3 , A 4 , e m : r e a l ; m:integer; begin

100

MONTE CARLO MODELING

E [ i, 1 ] : =inc_energy ; for m : = 2 t o 5 1 d o begin Al: =m _ t _ s t e p [ i ] * s t o p _ p w r (i , E [i , m - l] ) ; A2 :=m_t_step[i] *stop_pwr (i , E [ i , m - l] - A l / 2) ; A3 : =m _ t _ s t e p [ i ] * s t o p _ p w r ( i , E [ i , m - l ] - A 2 / 2 ); A4:=m_t_step[i]*stop_pwr(i,E[i,m-l]-A3); E[i,m]:=E[i,m-l] -

( A l +2*A 2 +2*A 3 + A 4 J / 6 . ;

end; E[i,51]:=0; {now smooth thes e profiles

out a

little bit}

for m : = 2 t o 5 0 d o begin E [ i , m ] : = ( E [ i , m ] + E [ i , m + l] } /2 . ; end;

end; Procedure Rutherford—Factor(i:integer); {find the screened Rutherford scattering parameter b using the HKLCS method} var hkbs,hkc,dum,hkm:extended; begin hkm: = (0.1382 - 0.9211/sqrt(at_num[i] ) ) ; dum: = In (at_num [ i ] ) ; hkc:=0.1904-0.2236*dum+0 .1292*dum*dum-0.01491*dum*dum*dum; hkbs : =hkc*power (Inc_Energy,hkm) ; {then the scattering factor for material i is} rf [i] : =0.016697 +0.55108*hkbs-0.96777*hkbs*hkbs +1.8846*hkbs*hkbs*hkbs; end; Procedure set—up—screen; {gets necessary input data to run the program} begin ClrScr; {tidy up the display screen}

BACKSCATTERED

101

GoToXY (25,1) ;

Writeln('*A on B MC Simulation in Turbo Pascal*'); {Having set up the screen now get input data} GoToXY(1,5);

Write ('Input beam energy in keV ) ; Readln(Inc—Energy); GoToXY(1,7);

Write('Boundary depth (microns)'); Readln(boundary); GoToXY(1,9);

Write('Tilt angle in degrees'); readln(tilt); s_tilt: =sin(tilt/57 .4) ,{convert degrees to radians] c_tilt:=cos(tilt/57.4) ; ClrScr; {again-to set up for two column input of data} for i:=l to 2 do {get materials data} begin if i = l then {surface layer} begin GoToXY(1,5); writeln('Surface Layer'); end else {we are in the substrate} begin GoToXY (40,5);

writeln('Substrate'); end; [now get the materials information that is needed} GoToXY(40* (i-1) +1,7) ,{LHS of screen for 1 = 1, KHS for 1 = 2}

Write('.. .. ..Atomic Number ig'); Readln(at—num[i]); GoToXY(40* (i-1) +1,9) ;

Write('. . . . . Atomic Weight is'); Readln(At_wht[i]); GoToXY(40* (i-1) +1,11) ; Write ( ' . . . . . density in g/cc is'); Readln(Density[i]); {Calculate th e Berger-Selze r Mean ionization potential inn_ion_pot } mn_ion_pot[i]:=(9,76*at_num[i] + _num[i],0.19)))*0.001;

(58.5/power(a t

102

MONTE CARLO MODELING

{now get the Rutherford factor b using the HKLCS method} Rutherford—factor(i); [now get the range and step length in microns for these conditons} range ( i) ; {and calculate a 50-step energy profile

E[i,k]

using this result]

profile(i); end;

{of

the input

loop}

{get the number of trajectories to be run in this simulation} GoToXY(18,15); write("Number of trajectories required"); readln(traj_num); {and set up the display for output of data}

GoToXYfl,16); writeln GoToXY(1,17); writeln('Number of trajectories'); GoToXY(40,17); writeln('Backscattered fraction'); end;

Procedure init_counters; Procedure reset-coordinates; Procedure p_scatter(i:integer); {calculates scattering angles using plural scattering model of Chap. 4} begin {call the random number generator function} nu:=sqrt(RANDOM) ; nu: = ( ( l / n u ) - 1 . 0) ; an: =n u * r f [ i ] * i n c _ e n e r g y / E [ i , k ]; {and use this to find the scattering angles} sp: = (an+an) / (1 + (an*an ) ) ; c p : = ( l - ( a n * a n ) ) / (1 + (an*an ) ) ;

BACKSCATTERED ELECTRONS

103

{and the azimuthal scattering angle} ga: =two_pi*RANDOM;

end; Procedure new_coord(i:integer);

{calculates new coordiantes xn,yn,zn from x,y,z and scattering angles} begin {the coordinate rotation angles are] if cz = 0 then cz:=0.000001; an_m:=-cx/cz; an_n:=1.0/sqrt(1+(an_m*an_m) ) ;

{save computation time by getting all the transcendentals first] vl: =cos(an_m)*sp; v2:= cos(an_n)*sp; v3 : =cos (ga) ; v4 : =sin (ga) ; {find the new direction cosines} ca =(cx*cp) + (vl*v3) + (cy*v2*v4); cb =(cy*cp) + (v4*(cz*vl - cx*v2)); cc =(cz*cp) + (v2*v3) — (cy*vl*v4);

{and get the new coordinates—using the appropriate step} xn =x + step[i]*ca; yn =y + step[i]*cb; zn =z + step[i]*cc; end;

Procedure**

s thisisthestartofthemainprogram

set_up_screen; init_counters; randomize; {reseed random number

generator}

{

*

104

MONTE CARLO MODELING

}

while num < traj_num do begin reset—coordinates; for k:=l to 50 do begin {first find out where the electron is now i = l or 2. In the simplest case this is done by checking whether or not the electron z coordinate places it below the boundary layer} if z>=boundary then i:=2 else i:=l; {in other situations a different test would be inserted here}

{now allow the electron to be scattered} p_scatter(i) ; new_coord(i); {test for electron position within the sample} if zn< = 0 then begin bk_sct:=bk_sct+l; num:=num+l; goto back—scatter; end else reset_jiext—step; end;

{of the 50-step loop} num:=num+l; {add one to the trajectory total} back—scatter; {end of goto branch for BS} {update the screen displays} GoToXY(25,17); writeln(num); {display trajectory total} GoToXY(65,17); writeln((bk_sct/num):4:3); {display BS coefficient} {to escape from the program press any key} if keypressed then goto abort;

{

*

BACKSCATTERED ELECTRON S 10

}

end;

5

{of the Monte Carlo loop} abort: {escap e from the program} GoToXY(21,21); w r i t e l n l ' . . . h t a t 's al l f o l k s ' ) ; readln; {freeze the display}

end.

6,5 Notes on the program The program starts in the sam e way as for the previous examples b y settin g up the pragmas that select the use of the math coprocessor o r the emulator package. Sinc e this program does not generate any graphical display, the uses declaration include s only th e call s fo r th e function s CR T an d DOS , which ar e required . Th e variabl e declaration sectio n resemble s tha t use d befor e bu t wit h a n importan t difference . Instead o f there bein g a single valu e of , for example, the atomi c numbe r at_wt, there wil l now be one of two possible value s depending o n where within the target the electron i s situated. Therefore the original singl e value of at_wt i s replaced b y an arra y o f at_w t values . I n th e notatio n use d i n PASCAL , w e indicat e thi s by writing at_,w t [ i ] t o represent th e i-th value of at_wt. W e must thus modif y the list of variables in the program to indicate that at_wt, den s i ty, o r any of the other specime n parameter s ca n take two (or more general example s ri) different values. The VAR table woul d then contai n entrie s suc h as: at_wt:array [1 . .4] of extended; at_num:array [1 . .4) of extended; {and so on for every variable

that changes} where the [ . . .] brackets indicat e th e numbe r o f the firs t an d las t member s o f th e array and specify what type of variable it is f Not e that in PASCAL we must specify an actual number of members of the array; this cannot be done in terms of anothe r variable. Having now done this, we must also indicate to the various PROCEDURE S and FUNCTIONS that use these variables whic h of the n values they are to use. We do this by passing a parameter to the procedure, which tells it which particular value to use . In eac h case , wher e i n Chap . 4 w e ha d a single-value d variabl e (e.g. , density), we now have one specific value taken from th e array of possible value s densityD'[. The value of / to be used is passed t o the procedure o r function a t the time that we call it; therefore, instea d o f calling p r o f i le t o calculate th e energy profile E[k], w e cal l profile!)' ] t o tel l i t t o calculat e th e profil e fo r th e /-t h material b y substituting the appropriate value s of stopping_pwr, m_t_ste p etc. On e assumption in this program is that E[l,k] * » E[2,k]', i n other words, that the

106 MONT

E CARL O MODELING

instantaneous energ y of th e electron depend s onl y o n the number of the trajectory step k and not on the actual material. This approximation is, in fact, closel y correct , because—from th e form o f the Bethe equation—the instantaneous energy depend s only on the fraction of the Bethe range traveled and not significantly on the atomic number of the materia l itself. The rest of the program then follow s closel y on the original version given in Chap. 4 except that we now test at the top of the loop which of the two materials the electron i s in , /= ! o r i=2, befor e computin g th e scatterin g angle s an d findin g the new coordinates . Becaus e w e ar e not trying to displa y these dat a visually , the procedures fo r graphics setu p and plotting are missing, and if you run this program, you will notice how much faster it performs than the earlier version . While pictures of th e trajectorie s ar e interestin g an d quit e ofte n helpful , the y ar e ver y time consuming t o produce because o f the large tim e overhea d required i n calculating plotting coordinate s an d drawin g th e scree n displays . Therefor e the y shoul d b e omitted i f not strictl y necessary. Figure 6.1 3 shows an application of this program t o the case o f a thin carbo n film o n to p o f a gold substrat e a t 1 0 keV. Sinc e w e ar e usin g a plural scatterin g model here , th e dat a wil l no t b e reliabl e fo r fil m thicknesse s les s tha n abou t a thousand angstroms (i.e., three or four trajector y steps), but the trend is clear (Hohn et al. , 1976) . A t lo w fil m thicknesses , th e backscatterin g coefficient i s essentiall y that of gold; this then decreases wit h film thickness , eventuall y falling to the value

Figure 6.13. Variatio n of B S yiel d from thi n carbo n fil m o n gol d a t 1 0 keV.

BACKSCATTERED ELECTRONS

107

appropriate fo r bul k carbon . Th e fil m thicknes s a t whic h thi s occur s i s abou t th e same as that for which an unsupported thin film o f carbon attains its bulk scattering value (see Fig. 6.12). This thickness i s an estimate for the "depth of information" in the backscatter imag e at this beam energy (i.e., the maximum distance beneat h th e surface a t whic h w e coul d deduce—fro m informatio n in th e B S image—tha t th e sample was not soli d carbon) . Several othe r applications of this type of multilayer program are considered late r in this book—for example, in the chapters on electron beam induce d charge (EBIC ) an d x-rays. Depending on the problem, it is sometimes adequate to test the end point z n of a trajectory step to see which part of the specimen it is in, but in other cases cases it may b e bette r t o fin d th e midpoin t of th e ste p and test that instead : z_mid: = ( z + z n ) /2 ; if z_mid>= dept h the n i : = 2 ;

The firs t approac h i s quicker , bu t th e secon d make s mor e us e o f th e dat a i n th e simulation an d i s les s likel y t o caus e problem s an d confusio n whe n complicate d geometries are being considered. Car e must also be taken to properly error-tra p the test function; fo r example, if the electron i s backscattered znbotto m the n i : = 3 else i : =1; etc. but this is cumbersome and slow, as these tests have to be made on every step of every trajectory . For a completely arbitrar y geometry, however , suc h as a random steps on a surface, this is the only available way. One alternative is to divide up the entire volume of interest into an array of cubes of a size comparable wit h or smaller than the expected ste p length an d then to appl y the sor t o f test describe d abov e to preassign a value of i to each elemen t o f the arra y an d store it, s o that, give n the coordinates o f the electron, th e correct valu e of / can simpl y be looke d up . This i s

108

MONTE CARLO MODELING

fast and quite general but requires very large arrays of data in all except the simplest cases (Joy, 1989a). In one special case, however, there is a simple and elegant solution to this difficulty. If each of the regions of interest in a material is a closed volume that can be described by an equation of the form f(x, y, z) = 0, then we can immediately determine whether any given point p, q, r is inside, on the surface of, or outside the volume by evaluating/(p, q, r). lff(p, q, r) < 0, then the point is inside the volume; if/(p, q, r) = 0, then the point is on the surface; and iff(p, q, r) > 0, then the point is outside. For example consider the case of a sphere, radius r and centered at the coordinates (0, 0, Zc). We can find whether or not the point (x, y, z) is inside or outside the sphere by a simple PASCAL function, which can be called simply Outside. Function outside(xv,yv,zv:extended):Boolean; {tests whether or not the electron at xv,yv,zv is inside or outside the sphere of radius r centered at 0,0, Zc} var dum:extended;; begin dum:=xv*xv + yv*yv + (zv-Zc)*(zv-zc) — r*r; if dum< = 0 then outside : =false else outside:=true; end;

The start of the Monte Carlo loop in the program above would then be modified to read while num < traj_num do begin reset—coordinates ; for k:=l to 50 do begin {first find out where the electron is now-i=2 inside sphere or i = l outside the sphere}

if outside (x,y,z) then i:=l else i:=2; etc.

remembering that an if .(test) . . then statement is carried out if the test is true (or is a function that evaluates to a positive number). Figure 6.14 shows some data obtained from this version of the program. The

BACKSCATTERED ELECTRON S

109

Figure 6.14. B S yiel d acros s a 0.25-(xm coppe r spher e i n carbo n a t 1 5 keV.

matrix of material wa s carbon, containing a copper spher e 0.2 5 jji m i n radius, and the incident beam energy was 1 5 keV. The program was set to calculate the variation of r\ wit h th e positio n o f th e bea m relativ e t o th e cente r o f th e sphere—which , because of the cylindrical symmetry of the problem, we need only do along a single radius. An additional loop was therefore inserted into the program to move the value of the starting y coordinate o f the beam from 0 to some maximum value y_jnax. i n suitable steps . The data of Fig. 6.1 4 show how the apparent backscatterin g coeffi cient varies with the position of the beam an d the depth of the sphere . W e see that for Zc > 1. 0 (Jim, or about one-third the Bethe range, the sphere is not visible, so the "depth of information" i n this case is about 1 |j,m. As the sphere is brought closer to the surface, it s visibility increases an d its apparent width changes, the full widt h at half maximum of the profile being 0.25 (x m when the sphere is at a depth of 0.7 (x m but 0.4 n, m wide when the sphere is at 0.35 (x m depth. These values for the apparent width of the image are interesting because they indicate the sort of spatial resolution that can be achieved in the backscattered image mode. On a simple basis, we might expect tha t the apparen t siz e would be o f th e orde r o f th e actua l diamete r o f th e sphere plus a term to account fo r the beam interaction volume. However, as we see , the measured size is always less than the actual size. This is because th e mere fact that an electron reaches th e copper sphere doe s not guarantee that it will be backscattered b y it , sinc e i n orde r t o escape , th e electro n mus t firs t trave l throug h a significant amoun t of the carbon matrix. When the sphere i s deep or on the periph-

110

MONTE CARLO MODELING

ery of the interaction volume, the electrons reaching it are already low in energy and have little chance of reaching the surface even if scattered directly toward it. This method of testing the position of the beam relative to a closed surface is certainly a special case, but it is not as restrictive as it might at first appear, because a sphere can be generalized into an ellipse (i.e., the equation becomes of the form ax2 + by2 + cz2 - d), which can then, by an appropriate choice of major and minor axes, be turned into a disk or a plate in any chosen orientation. A search through books on analytical geometry will also reveal a few other closed surfaces (or effectively closed, as in the case of a hyperbola) that might also prove useful. A case of some practical importance occurs when the surface encloses not another material but a vacuum—i.e., when the material contains a void. As can readily be confirmed, this situation cannot be taken care of by expedients—such as representing the void as being a material of zero atomic number and density— because the stopping power and scattering expressions do not behave properly in this limit. Instead, we assume that an electron entering the void will neither be scattered nor deposit significant energy but will simply move in a straight line across the void until it once again leaves. Using the notation from the code given above, the program would be modified to read

while nu m < traj_nu m d o begin reset—coordinates; for k : = l t o 5 0 do begin {fir or 1=1 outside in the matrix" if o u t s i d e ( x , y , z ) the n i : = l else begin {extrapolate previous trajectory step] go_round_agin:{la£>el for next loop if needed} xn: =x + s t e p [ l) *cx; yn: =y+step [1] *cy ; zn:=z+step[l)*cz; if outsid e ( x n , y n , z n ) the n {its gone through} go t o escape d {so leave this loop} else {extend the extrapolation} x: =xn;y: =yn; z : =zn; {reset coordinates} go t o go—round—again ; {and do it again} end;

BACKSCATTERED ELECTRONS

111

escaped: {label for exit from loop] i : = l ; {remember to reset this on exit]

etc.

While th e programmin g style will wi n n o prize s becaus e o f it s us e o f go-t o statements, thi s fragmen t o f cod e correctl y handle s th e void . Th e visibilit y an d apparent siz e o f void s behave s i n a n approximatel y simila r wa y t o tha t fo r th e inclusions discussed above, except that the absence of a scattering material leads to a fall i n the backscattering coefficient rather than to a rise. For an interesting study of contras t fro m void s i n th e scannin g electro n microscop e (SEM ) usin g Mont e Carlo techniques of this type, see Gasper an d Greer (1974) .

6.6 Incorporating detector geometry and efficiency A final topic that must be discussed concerns the difference between the compute d backscattering coefficien t and the actua l signal collected i n the SE M by a suitable backscatter detector . I f the backscatter detecto r were perfect, that is, i f it collected every backscattered electron withi n the 2ir steradians above the sample surface and if ever y electron—regardless o f energy—produce d th e sam e magnitude of output pulse, the signal would be directly proportional to the backscattering coefficient. I n general, this is not the case, and circumstances usually require that the properties of the detector be taken into account when a Monte Carlo simulation is being made. As a rule, the position of the detector must be considered if the specimen to be modeled has anything other than a flat (i.e., planar) surface and, since all practical backscatter detectors hav e an output that varies directl y wit h the energ y o f the electron s collected, modelin g th e efficienc y o f th e detecto r fo r electron s o f differen t energ y is always likely t o be useful . In orde r t o incorporat e th e geometrica l propertie s o f th e detecto r int o th e program, i t i s onl y necessar y t o chec k whethe r o r no t an y give n backscattere d electron i s traveling in the right direction to impinge on the detector. If the detector is assume d t o b e annula r abou t th e inciden t beam—i f th e oute r diameter o f th e detector i s R0 and the distance from th e detector to the sample is D—then, with the coordinate conventio n bein g use d in this book, a n electron wil l be collecte d i f it s direction cosin e c c satisfie s the relation

note that the sign is negative because the positive z-direction is defined to be in the beam direction. Th e code o f Chaps. 4 and 6 would then be modified to read : if zn< = 0 then if cc —0) ) then detector—A—Signal: =detector_A—signal+1 else detector—B_signal: =detector_B_signal + 1; num:=num+l; goto back—scatter; end

where for this split detector w e keep separate track of the electrons collecte d on the two halves A and B (remembering firs t to declare and initialize these variables). The extension o f this cod e to othe r mor e comple x arrangement s is obvious. All backscatte r detector s displa y som e energ y sensitivity . Typically, w e fin d that thei r i s som e minimu m energ y Emin belo w whic h the backscattere d electro n creates n o output signal , whil e abov e Emin a backscatter electro n o f energy E wil l produce a n output signal tha t varies linearl y a s (E—Emin). Th e code for our plural scatter model s woul d now loo k somethin g lik e this : if ((znE_min)) then begin if ( (cc< = -critical angle) and (cb> = 0) then detector—A_signal: =detector_A—signal+ (E[i,k] -E_min) else

detector_B_signal:=detector_£_signal+ ( E [ i , k ] -E-min) ; num:=num+l; goto back—scatter ; end

so that the detected signal now depends on the actual energy of the electron collected. Note that the variables detector_A_signal an d detector_B_signal mus t

BACKSCATTERED ELECTRON S

113

Figure 6.15. BS profile acros s sphere an d corresponding detected signal . now be declared as extended or real rather than integer and that the final result will no longer represent th e backscatter yiel d but instead a scaled representation o f the actua l signal detected . Figure 6.1 5 show s the effec t tha t this kind o f correctio n has. The datum is one of the traces from Fig . 6.14, but now corrected fo r a detector that has an Emin value of 5 keV. The two traces show the variation of the backscattered coefficien t an d th e outpu t signa l fro m th e backscatte r detecto r respectively . While th e form s o f th e profile s ar e essentiall y identical , not e tha t th e effec t o f including the energy sensitivit y of the detector i s to increase the apparent visibility of th e coppe r sphere , sinc e th e "peak-to-background " variatio n o f th e signa l i s increased by about 20%. There is thus no longer a direct proportionality between the backscatter coefficien t an d the detected backscatte r signal. This is of importance if the backscatte r signa l i s being use d fo r chemica l observations , sinc e i t canno t b e assumed tha t th e detecte d signa l an d th e emitte d backscatterin g coefficient—an d hence th e mea n atomi c numbe r of the target—ar e al l simpl y related together. Be cause th e detecto r respond s a s muc h t o change s i n th e energ y o f th e electron s i t receives a s it does to their actua l number, these two effects convolut e together and cannot simpl y be separated. Quantitativ e backscattered imagin g is therefore a diffi cult activity unless care is taken to account for all of these factors (e.g., Sercel et al., 1989).

7

CHARGE COLLECTION MICROSCOPY AND CATHODOLUMINESCENCE 7.1 Introduction Charge collection imagin g in the scanning electron microscope , ofte n know n by the acronym EBI C (electro n beam-induce d current) , has become a widely used tech nique fo r th e characterizatio n o f semiconducto r material s an d device s (Leamy , 1982; Holt and Joy, 1990). While there is a substantial literature on the use of EBIC methods t o measure semiconducto r parameters , suc h a s th e minority carrie r diffu sion length (Leamy , 1982) , the majority o f charge-collected image s ar e interprete d in a purely qualitative manner. Cathodoluminescence (C/L ) imaging of semiconductor materials , whic h i s i n essenc e ver y simila r t o EBIC , ha s similarl y bee n use d mostly i n a picture-taking rathe r tha n a data-producing mode . Th e proble m i s not that there are no good models to explain the image formation but rather that, in order to provide tractable analytical expressions fo r the calculation of contrast, it is invariably necessary t o make significantl y oversimplified assumptions about the interac tion of the electron beam with the specimen. In this chapter, we demonstrate how the Monte Carlo models discusse d earlie r ca n be used to overcome these problems and make EBIC an d C/ L mor e usefu l technique s fo r microcharacterization .

7.2 The principles of EBIC and C/L image formation When an electron beam impinges on a semiconductor, som e of the energy deposite d by the beam is used to promote an electron fro m th e filled valence band , across the band gap , t o th e empt y o r partiall y fille d conductio n ban d (Fig . 7.1) . Sinc e th e valence ban d wa s initially full , th e remova l o f a n electro n leave s behin d a "hole" that has all the physical properties o f an electron bu t carries a positive charge. For each electron promoted acros s th e band gap, one hole is formed, so it is convenient to conside r th e tw o component s togethe r an d tal k o f a n electron-hol e pair . T o generate th e electron-hole pai r requires a n amount of energy esh. Typically, esh is about three times the energy of the band gap (Table 7.1); so, for example, for silicon, eeh is 3.6 eV. If we assume that all of the energy deposited b y the incident bea m of energy E0 is ultimately unavailable for the generation of electron-hole pairs , then the number o f carrier pair s forme d n eh wil l b e 114

CHARGE COLLECTIO N MICROSCOPY/CATHODOLUMINESCENC E 11

5

Figure 7.1. Electron-hole pai r generatio n i n a semiconductor .

That is , a 10-ke V bea m inciden t o n silico n coul d produc e 10,000/3. 6 « 280 0 electron-hole pairs . Because th e electron an d hole hav e opposite charges , they are electrostatically attracted and will tend to drift togethe r through the lattice, maintaining local electrical neutrality. After onl y a few femptoseconds (10~~15 s), the electron will fall back across the band gap and recombine with the hole, giving up, as it does so, some of the energy used to form th e original carrier pair. It is this effect tha t may result i n th e productio n o f C/ L emission , sinc e th e exces s energ y ca n b e carrie d away by an emitted photon. I n any event, within a very short time after the passage of th e incident electron , th e syste m has returned to its origina l state . Since, unlik e a metal, a semiconducto r ha s significan t resistivity, a potential difference ca n be maintained across it, resulting in the generation of an electric fiel d imposed acros s the sampl e (Fig . 7.2A) . A n incident electro n o f energy E0 will , as before, produce some number of carrier pairs neh, but now, because the electrons and holes carry opposite charges, the electrons will tend to move toward the positive end of the sample while the holes will drift towar d the negative end. One result of this is Table 7.1 Electron-hole pair energies

Material C (diamond ) 13. CdS 7.

CdTe 4. GaAs 4.

GaP 7. Ge 3.9 InP 2. PbS 2. Si 3. SiC 9.

Electron-hole pair energy (eV) 1

5

8 6

8 5 2 0 6 0

116

MONTE CARL O MODELING

(a)

(b)

(c)

Figure 7.2 . Thre e configuration s for charge collectio n microscopy .

immediately apparent. Before the electron beam is turned on, the amount of current flowing throug h th e semiconducto r i s ver y smal l o r zero , becaus e ther e ar e n o electrons in the conduction band and hence there is no way to move charge. But with the beam striking the specimen, each electron produces n eh electron-hole pairs , and the presence of these free carriers will permit charge to flow. Thus the electron beam has induce d conductivit y in th e semiconductor . Thi s effec t i s know n a s electro n

CHARGE COLLECTIO N MICROSCOPY/CATHODOLUMINESCENC E 11

7

beam-induced conductivit y (an d s o give s anothe r interpretatio n o f th e acrony m EBIC) or (3-conductivity (Holt and Joy, 1990) . I f the beam is scanned, the resultant conduction signa l ca n be used to for m a n image. In practice , a mor e interestin g procedur e i s t o generat e th e fiel d internall y within the semiconductor. Fo r example, if a semiconductor tha t has been chemically doped to make it a p-type material (i.e., one containing a n excess of holes) is placed in contact with n-type material (i.e., one with an excess of electrons), the n a region is formed around the p-n junction where a potential and hence a field is present (Fig. 7.2B). This field arise s from the attempt of the electrons i n the n-type material to go to th e p-typ e region an d th e hole s i n th e p~typ e materia l to flo w int o th e n-type region. Within a short time, charge fro m uncovere d atomic nuclei produces a fiel d just sufficient t o prevent any further charg e motion. In this region of field, there can be no free charge carriers, so it is called a "depleted region," and it typically extends for a few micrometer s o n eithe r sid e o f th e physica l locatio n o f th e p- n junction (Leamy, 1982) . Alternatively, if a thin metal film is put into atomic contact with the surface o f a semiconductor (Fig . 7.2C) then this "Schottky barrier" again results in the appearanc e o f a depleted regio n extende d downwar d from th e Schottk y layer. Without a n inciden t electro n beam , ther e exists , i n eithe r case , a stati c potentia l between the p and n regions or between the metal and semiconductor; bu t if the pand n-type regions, o r the Schottky barrier and the semiconductor, were connecte d together b y a wire, then n o curren t would flow , becaus e th e fiel d exist s onl y i n a region fro m whic h all of the availabl e charg e carriers have alread y been removed . Now let us place the incident electron beam onto the p-n specimen in either the p- or n-type region but away from th e depleted zone. As before, electron-hole pairs will b e generated, but since the material in which they ar e produced is electrically neutral and has no field across it, they will quickly recombine an d no external effec t will be observed. If, however, the beam is placed in the depleted region, the carriers produced wil l se e the depletio n fiel d an d the carrie r pair s wil l b e separated . Thi s motion o f charge s withi n th e specime n wil l produc e a flo w o f curren t 7 CC i n th e external circuit give n by the relation

since eac h inciden t electro n ca n produce n eh carrie r parirs . This signal , which we will call the charge-collected curren t 7 CC, is seen to be substantially greater than the incident current /B. In the scanning electron microscope (SEM) , the current flowing around the external loop is measured and displayed as function of the beam position to produce th e charge-collected , o r EBIC, signal . Note that a n important practica l consideration i s tha t th e curren t w e wis h t o measur e i s actuall y a short-circui t current. Hence external resistance loa d /?L must be of as low a value as possible or else the ohmic voltage drop across it, which is in the opposite sens e to the potential at the depletion layer , will affec t th e signa l collection process .

118 MONT

E CARL O MODELIN G

In the standard theory of Donolato (1978), two steps are necessary t o compute /cc: first , w e mus t mode l th e generatio n o f carrier s b y th e electro n beam , and , second, we must model the transport and collection o f the beam-generated carriers . Typically the generation o f carrier pairs is described b y a function g(r), i n units of cm~ 3 s~ 1 , modele d a s a three-dimensiona l Gaussia n Distributio n (Fittin g e t al. , 1977) or a sphere of uniform generatio n (Bresse , 1972) , neither of which is a very realistic representatio n o f the beam interactio n wit h the solid . The transport o f the minority componen t o f th e electron-hol e pair s generate d i s a diffusio n proble m described b y the equation:

where p(r) i s the density o f the beam-generated minorit y carriers an d D an d T are, respectively, the minority carrier diffusion coefficien t and lifetime. Given the necessary boundary conditions onp(r) a s defined by the geometry of the problem at hand, Eq. (7.3 ) ca n b e solve d t o giv e p(r) an d th e induce d curren t 7 CC ca n b e foun d b y integrating the normal gradient ofpp over the collection plane. While this descriptionn of th e proble m i s rigorousl y correct , th e drawbac k i s tha t 7 CC canno t b e foun d without knowing p(r), which , in turn, requires a knowledge of g(r), and no realistic model o f g(r) yield s an equation tha t is analytically tractable . However, if we use the Monte Carlo procedures we have developed t o describe the incident bea m interactio n wit h the semiconductor , then , a t any instant alon g a trajectory, g(r) is effectively a point source whose strength is equal to the number of electron-hole pair s generate d i n tha t segmen t o f th e trajectory , whic h fo r th e £ th trajectory step is equal (Akamatsu et al., 1981) to the energy given up (E[k] - E[k + 1]) divide d b y th e energ y t o creat e on e electron-hol e pai r e eh. I f th e trajector y segment is at a distance .s from the collecting junction, then the probability of i|/(s) of these carriers diffusin g t o the collecting junction and producing a charge collecte d current is , (Wittr y an d Kyser , 1964) , fo r a point source:

where L i s the minority carrie r diffusio n lengt h (i.e. , L = vDi). It woul d thu s see m tha t w e coul d comput e 7 CC i n a sequentia l manne r b y summing up the charge-collection contributio n from eac h step of each of the trajectories tha t w e simulat e an d averagin g thi s t o fin d effectiv e charg e collecte d pe r incident electron (Joy , 1986). Although this looks, at first sight, to be quite differen t from actuall y solvin g th e diffusio n proble m o f eq . (7.4 ) an d the n computin g / cc, these two approaches are, in fact, functionally equivalent. A s shown by Possin and Kirkpatrick (1979), w e can generalize th e quantity \\>(s) an d defin e a quantity ij;(r), which describe s th e probabilit y tha t a minorit y carrie r generate d b y a n electro n

CHARGE COLLECTIO N MICROSCOPY/CATHODOLUMINESCENC E 11

9

beam a t r i s collecte d an d thus contribute s t o th e charge-collecte d curren t 7 CC. Donolato (1985,1988) showed that for the geometry of Fig. 7.2C, i|/(r) also satisfies Eq. (7.3) wit h the boundary condition i|;(r ) = 1 at the entrance surfac e (z = 0) . In this case, th e solutio n o f Eq. (7.3 ) reduces t o one dimension an d has th e for m

That is, it is identical with the result of Eq. (7.4), and the charge-collected curren t 7CC is then the su m o f the contributio n o f th e elementar y sources :

where h(z) i s the dept h distribution of the generation. Thu s we can quite generall y calculate the magnitude of 7 CC for a given specimen geometr y an d beam interactio n by first steppin g through the Monte Carlo simulatio n to produce h(z) an d then using Eqs. (7.4 ) o r (7.5 ) an d (7.6 ) t o fin d 7

7.3 Monte Carlo modeling of charge-collection microscopy 7.3.7 The generation function Either th e singl e o r plura l scatterin g model s coul d b e use d a s th e basi s fo r a simulation o f charge-collectio n microscopy , bu t sinc e th e sample s ar e invariabl y bulk, th e plura l scatterin g approac h wil l b e significantl y faste r an d s o wil l b e illustrated here. Th e first tas k is to compute eh, th e number of electron-hole pairs generated alon g each ste p of the trajectory . Working from th e cod e i n chap. 4, w e can easily inser t thi s as shown below. Sinc e i t is interesting t o be able to view the actual spatial distribution o f g(r), we can associate eac h element o f carrier produc tion with the coordinates o f the midpoint (xm, ym, zm) of the trajectory step . Since for norma l incidence th e distribution i s going to be radially symmetrica l about the beam axis, we can store this in an array g(r, zz) where r, the radius from the axis, and zz, th e depth , ar e bot h expresse d i n unit s o f one-fiftiet h o f th e Beth e rang e (i.e., th e ste p length) . D o no t forge t t o ad d t o th e variable s lis t th e extende d quantities xm,ym , zm, eh, th e integers r , z z , an d the array g [ 0 . . . 50 , 0 . . . 5 0 ] o f extended, an d remember t o initialize g [ r, z z ] befor e usin g it. p_scatter; {fin d the scattering angles} new_coord(step); {find where electron goes } {program-sped fie code will go here} {******** and here it is ******** }

120 MONT

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{eh the number of carriers generated is} eh:= ( E [ k ] - E [ k + l] ) * 1 0 0 0 / 3 . 6; (E[] i s in keV\ {assuming that the pair generation energy for silicon is 3.6 eV} xm: = (x+xn) / 2 ; {x coordinate midpoint of step} ym: = ( y + y n ) /2 ; {y coordinate of midpoint} zm:=(z+zn)/2; [z coordinate o£ midpoint] {get radius of midpoint from axis in units of range} r: =round(sqrt (xm*xm+ym*ym ) / s t e p ) ; z z : =r o u n d ( z m / s t e p ); {} if zz > = 0 the n {put thi s carrier contribution into the array at r,zz} g[r,zz]:=g[r,zz]+eh; etc.

The array can be printed out at the end of the run to give the spatial distribution g [ r, z z ] . Becaus e th e dat a hav e bee n store d i n unit s o f a radia l variable , th e volume represented b y successive values of r increases steadily , so, in order to make the values directly comparable, i t is necessary t o divide g [ r , z z] b y ( 2 r+1) — i.e., th e are a betwee n th e rth an d ( r + l ) t h annuli . Figur e 7. 3 show s th e carrie r distribution calculated in this way for silicon, The profile has the familiar "teardrop" shape with a diameter of the order of the range, and with a maximum depth of about 0.75 of the range. It is clear, however, that the distribution of carrier generation is far from uniform , with nearly a quarter of the carriers being produced within a volume

Figure 7.3. Isogeneration contour s for electron-hole pairs i n silicon.

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that i s onl y abou t 0. 2 o f the rang e i n diameter . Th e isogeneratio n contour s ar e in good agreemen t wit h published experimenta l dat a (e.g. , Possi n an d Norton, 1975 ) confirming tha t the physical basis for the mode l i s good . 7.3.2 The gain of a Schottky barrier The specime n configuratio n most normally used for EBI C imaging i s that of Fig . 7.2C, the Schottky barrier geometry; consequently, it is this system that we will use the Monte Carlo method to analyze, although other arrangements such as that using a p-n junction can as easily be studied by obvious modifications to the development below. The Schottk y barrie r consist s o f a thin layer, typically 10 0 to 30 0 A, o f a metal, suc h a s gol d fo r n-typ e silico n o r titaniu m fo r p-typ e silicon , i n intimat e contact with the surface of the semiconductor. Although this film is thin, it may have a significant effect o n the electron beam interaction with the solid, especially at low beam energies, and so must ultimately be included in the simulation. We assume that the Schottk y produce s a depletio n regio n o f dept h zd. T o a firs t approximatio n (Leamy, 1982 )

where fl i s the resistivity (ohm.cm) and V b is the barrier height, (e.g., for Si V b = 0.7 V) plus an y applie d revers e bias . S o for 1 ohm.cm resistivit y silicon , wit h no external applie d bias , zd i s about 0.5 (Jim . The measurable parameter of a Schottky diode in charge-collection mod e is its gain G, which can be define d as:

where 7 CC is the measured short-circui t current collected fo r a given incident beam energy /b. From the discussion above , we can see that G is of the order of n eh, the number of electron-hole pairs generated per incident electron; but it will invariably be lower, because not all of these carriers will ultimately contribute to the signal. Let us consider a single step of a trajectory and calculate the incremental contribution to 7CC. As before, the number of electron-hole pair s e h produce d a t this ste p is eh = (£[*] - E[k + l]/eeh and these we take to be generated at the depth zm = ( z + z«)/2 , the midpoint of the trajectory step . I f zm < zd, the n al l of the carriers produce d ar e separated, s o the incremental charge collected c c i s increased b y eh. I f zm > zd, then the carriers must diffuse bac k to the depleted region before they can be separated and collected.

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The fraction of carriers collected coll_f rac is, from Eq. (7.5),

where L is the minority carrier diffusion length. In general, we want to know how the gain G depends on semiconductor parameters such as the depletion depth zd and diffusion length L. Because the collection of the carriers is totally independent of the generation process, we can conveniently do this by computing just a single set of trajectories but allowing zd and L to take a range of different values and calculating the gain in each case. If we define m values of the depletion depth zd[] and n values of the diffusion length L[], then we need a matrix m*n in size for the current values Icc[]. The code fragment from above would then look like this: p_scatter; new_coord(step);

{find {find

the scattering angles} where electron goes}

{program-specific code will go here} {******** an d here it is ******** } (eh the number of carriers generated is} e h : = ( E [ k ] - E [ k + l ] * 1 0 0 0 / 3 . 6 ; {E[] is in keV} {assuming that the pair-generation energy for silicon is 3.6 eV} zm: = (z + z n ) / 2; [z coordiante of midpoint} {compute the incremental current collected for each value of zd and L} for i : = l t o m do begin for j : = 1 t o n d o begin if zm < =z d [ i] the n {al l the carriers are collected} Ice[i,j] :=Icc[i,j]+eh e Ice [ i , j] : =Icc [ i, j] +eh*exp (- ( zzd[i])/L[j]); end; end; etc. . . .

Figure 7.4 shows the variation of the gain G computed in this way for silicon at 30 keV, as the depletion depth varies from 0.5 to 5 (Jim and as L varies from 0.1 to 5 (xm. When the depletion depth zd or the diffusion length L are small compared to

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Figure 7.4 . Variatio n o f current gain i n a silico n Schottk y diod e a t 3 0 keV .

the electro n rang e (i.e. , abou t 1 0 |xm) , the n th e gai n o f th e diod e i s a sensitiv e function o f thes e parameters ; bu t whe n either o r bot h becom e comparabl e t o th e range, then the gain depends only o n the beam energy. Note that the advantag e of using the same set of trajectories to calculate the gain for each of the variables is not just that of saving computational time. Using the sam e set of random numbers for each piec e o f data ensures that the difference s i n th e gain du e t o a change in th e diffusion lengt h o r depletion depth are not masked because of the statistical variations between one Monte Carlo run and the next. An interesting exercise is to try the effect o n these calculations o f changing the angle of beam incidence (Jakubowicz, 1982; Joy , 1986) . Fo r a given depletio n depth , diffusion length , and beam energy there is a range of angles over which the gain is constant. Since the angle of beam incidence can easily be varied—for example , by rocking the beam or mechanically tilting the sample—this provides a rapid experimental metho d for determining the diffusion lengt h of a diode. In general , w e canno t ignor e th e effec t o n th e inciden t electro n bea m o f th e Schottky barrier metal film. As the electrons pass through this, they lose energy and are scattere d laterally . I n addition , th e effectiv e backscatterin g coefficien t o f th e specimen i s changed . Whil e thes e effect s ar e smal l a t moderat e an d hig h bea m energies (1 5 ke V an d above) , the y ar e quit e significan t a t lowe r bea m energie s because th e meta l film , thoug h thin, i s the n a substantia l fraction o f th e electro n

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range. This situation can be modeled by slightly modifying the program BINARY discussed in Chap. 6. The physics is exactly the same as that described above, except that we only consider electron-hole pair generation in the substrate material (i.e., the semiconductor) and the distance that these have to diffuse before collection is measured to the bottom of the Schottky layer rather than to the surface. It is only necessary to add a few lines of code to BINARY to incorporate these calculations. That is, after the scattering calculation we write {now allow the electron to be scattered] p_scatter(i); new_coord(i); {test for electron position within the sample} if zn< = 0 then begin bk_sct: =bk_sct + l; num:=num+l; goto back—scatter; end else {compute the gain of the Schottky diode—energy of electronhole pair formation is e—eh—i.e., 3.6 eV for silicon etc.} begin e formed} if i = l then eh:=0; {no carriers generated in metal film} if znz d the n ed:=0 {defec t is in neutral material} else begin e d : = 2 * ( 0 . 5 + b i a s ) / t h i c k ; {thic k i s thickness of metal film} ed:=ed*lE4* (1-dd) / z d ; end;

{arid r_crit:=rO*sqrt(5E3/(5E3+ed)); {a

Remember to add these variables to the var list at the start of the program— although, if you do not, the compiler will find them and warn you of their absence. Note that the constant k in Eq. (7.11), which represents the recombination strength of the dislocation, has the units of a length. So in the program, this has been called the "size" rO of the defect, and been assigned a value of 0.3 u-m. This value has been found to give contrast values that are in good agreement with those measured experimentally, and it is also consistent with the expected physical extent of the electrostatic field around a defect in silicon. For other materials or special conditions, such as a decorated defect, the value of rO may have to be adjusted. The presence or absence of the defect does not change the way in which the electron beam interacts with the solid. We can therefore again save time, and also improve our statistics, by simultaneously calculating the signal profile at many points across the dislocation for each trajectory step. If we place the dislocation so that it is parallel to the x axis and at y = 0, then for each trajectory step we can loop through the current calculation shifting the effective position of the beam by a series of values pos [ j ] and calculating the corresponding current i_cc [ j ] . Because the profile is symmetrical about the center of the defect, typically five to ten values in the array pos [ j ] are enough to produce a smooth profile. The chosen values for pos [ j ] can either be input by the user or preset in the program code itself. That is,

C H A

pos[0] : =0;

pos[l] :—0.25; {all values are in micrometers from defect center} pos[2]:=0.5; pos[3] :=1.0; {and so on}

A convenient rule of thumb is to set the maximum value of pos [ j ] to the sum of the electron range plus the diffusion length (i.e., typically in the range 5 to 15 ^m for most materials and usable beam energies) and to choose intermediate values of pos [ j ] with approximately a constant ratio between them (e.g., 0, 0.25, 0.5, 1, 2, 4, 8, 16), so as to give the best definition to the central portion of the profile. Procedure do—defect;

{this calculates the current gain of the Schottky including the effect of the electrically active defect} var begin [by computing number of e_h pairs generated} eh: = (E[i,k]-E[i,k+l])*1000/e_eh;

zval: = (z + zn)/2; {midpoint of trajectory} yval: = (y+yn) 12 ; {midpoint of trajectory} for j:=0 to 10 do

{a loop Shifting the defect position by pos[j]} begin {defect is at depth dd} rl: = (zval-dd)*(zval-dd) + (yval-pos[j])*(yval-pos[j]) ; {Fig. 7.6}

end; {o

r 2 : = r l + 4*dd; {by Pythagoras's theorem—see Fig. 7.6} {now apply Eg. (7.11)} d e l _ I : = ( e x p ( - r l / L ) / r l - e x p ( - r 2 / L ) / r 2 ); del_I:=l-(r_crit*del_I); if zval < =zd the n {al l o f the carriers are collected} i_cc[j] :=i_cc[j] +eh*del_I e l i_cc [ j ] : =i_cc [ j ] +eh*exp(- (zval-zd) / L ) ; e n

f this procedure}

Because computations for ten or so different effective beam positions are made for each step of every trajectory, the program will run more slowly than normal. However, because the data from each of these points come from the same set of random numbers, the precision of the computation is high, and good statistics can be achieved for as few as 400 to 500 electrons. A plot of i_cc [ j ] against pos [ j ] , possibly reflected about the position pos [0] = 0 so as to give a symmetrical profile, will now give the desired profile of the EBIC signal variation across the

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Figure 7.7 . Compute d EBI C profiles acros s a line defect a t various depths i n silicon .

defect. Thi s can be done either by having th e program output th e actual i_cc [ j ] values an d plottin g thes e i n som e othe r progra m o r b y writing th e necessar y fe w lines of code to have the program plot its own output. In either case a great deal can be learned abou t the behavior o f defect image s i n EBIC b y using this program an d trying th e effect s o f variou s set s o f semiconducto r an d defec t parameters . Figure 7. 7 show s the predicted signa l profile s arros s a line defect a t depths of 0.5, 1,1.5 , 2, and 2.5 |xm in silicon. To avoid cluttering the figure, only half profiles are shown . With the physica l parameters assume d here, a resistivity of lOOOft.c m and zero applied bias, the depletion depth is [Eq. (7.7)] about 1 5 |o,m, which is large compared t o bot h th e electro n rang e o f 3 (Ji m an d th e minorit y carrie r diffusio n length o f 1 (jum. Fo r all depths, the profile has the sam e general V-shape d form, but the details of th e profile vary from a narrow, high-contrast di p for a defect just 0.5 |xm below the surface to a broad and rather shallow profile for a defect 2.5 |xm below the surface . These prediction s ar e in excellent agreemen t wit h experimental obser vations. See , fo r example , figur e 3 3 in Leamy , 1982 . I t can als o b e note d tha t the contrast fro m a defec t effectivel y vanishe s i f th e dept h o f th e defec t exceed s th e range of the incident electrons. Thi s is because, as shown in Fig. 7.3, the majority of carriers ar e produce d clos e t o th e surface , an d i t i s eviden t fro m Eq . (7.11 ) tha t carriers generate d fa r from th e defect will contribute little contrast because th e two exponential terms will be about equal and opposite. Fo r good contrast to be observable fro m a defect , th e carriers ' generatio n maximu m mus t b e a t o r clos e t o th e defect. Th e depth at which a dislocation is situated can thus be estimated wit h good

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131

accuracy by noting the beam energy a t which it first become s visibl e and equating the electro n rang e to the depth (Leam y et al., 1976 ; Jo y et al., 1985) . Figure 7. 8 show s th e profile s predicte d fo r th e cas e wher e the defec t i s a t a constant depth , her e 0. 5 jjun , but th e bea m energ y i s varied. A t 5 keV, where th e beam range is also about 0.5 (Jim , the defect is just detectable, a s would be expected from the discussion above. As the energy is raised, both AS, the signal change across the defect, an d S, the total signal collected, increase, so the contrast AS/S also rises and reaches a shallow maximum at about 1 0 keV, at which conditio n th e electro n range, 1. 5 (Jim , i s abou t three time s th e defec t depth . A t stil l highe r energies , th e signal variatio n A S start s t o fal l whil e S slowl y increases , s o th e contras t falls . Simulations fo r a wide variety o f differen t defec t parameter s sho w that a contras t maximum can always be expected durin g a sweep of the incident beam energy and indicate that this maximum occurs whe n the defect is at a depth of between 0.3 and 0.4 times the beam range. The existence o f such a contrast maximum with accelerat ing voltag e wa s firs t predicte d b y Donalat o (1978) , wh o showe d that, for a poin t defect an d a uniform, spherical , generatio n volume, the maximum would occur a t about 0.5 of the beam range. The knowledge that a dislocation a t a given depth does not becom e visibl e unti l th e bea m rang e firs t reache s tha t depth , an d tha t wit h increasing energ y th e contras t the n goe s throug h a maximum (at abou t twic e th e energy at which the defect first becomes visible) and then decays away, provides the

Figure 7.8. Compute d EBI C profiles acros s a line defec t i n silicon a s a function o f bea m energy.

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MONtE CARL O MODELIN G

Figure 7.9. Compute d EBI C profiles acros s a line defect i n silicon a s a function o f diffusio n length.

basis of a method for determining the three dimensional distribution of dislocation s in a semiconductor (Jo y e t al. , 1985) . It will also be expected tha t the minority carrier length L would have a significant effec t o n defec t imagin g in EBIC, since i t is this parameter, togethe r wit h the beam range , tha t set s a physica l scal e fo r th e generatio n o f th e mobil e carriers . Figure 7. 9 show s ho w th e signa l profil e acros s a dislocation , i n thi s cas e 1 (x m below th e surfac e o f 100 0 ft.c m silico n an d viewe d at 1 5 keV, is affecte d b y th e value o f L. A s th e diffusio n lengt h increases , th e shap e o f th e profil e remain s essentially unchanged, but the apparen t size o f the feature increases. However , the full widt h at half maximum contrast of the defect image increases only by a factor of two times as L goes from 0. 5 to 50 (xm, and saturates if L is increased any further. In fact, measurement s fro m simulation s o f thi s typ e sho w clearl y tha t th e spatia l resolution o f th e EBI C imag e i s almos t independen t o f th e diffusio n length . Th e width of the defect image is determined by the beam energy (i.e. , the lateral extent of th e generation volume ) and by the position o f the defec t relativ e to the surface . Consequently, even in materials wit h a long minority carrier diffusio n length , highresolution EBI C imagin g i s possibl e provide d tha t the bea m energ y i s reduce d t o minimize th e interaction volume. Although al l of th e examples give n relate t o EBI C imaging , the extensio n t o C/L profiles is straightforward. All of the physics and program code associated with the generation o f carriers and their interaction with defects remains unchanged, but

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all of the physics concerned with the subsequent collection of carriers is eliminated, In addition , sinc e n o deplete d regio n i s required , ther e i s n o practica l reaso n t o consider th e presence of a Schottky barrier o n the surface. Unless it is necessary t o consider absorptio n of the C/L radiatio n as it leaves the specimen, the computation can be made straightforwardly by equating the resultant C/L signal with i_cc, th e number of carrier pairs determined for each trajectory step. An example of the use of Monte Carlo simulations for both EBIC and C/L imaging is given in Czyzewski and Joy (1989) . The result s discusse d abov e demonstrat e th e powe r o f th e applicatio n o f th e Monte Carlo simulation methods to the analysis of a practical problem such as EBIC imaging. By carrying out a sequence of simulations, i t is rapidly possible t o deter mine what effect differen t experimenta l parameters wil l have on the observed pro file. This , i n turn, makes it possible t o know how best to se t up the microscope t o achieve th e desire d resul t an d allow s a quantitativ e interpretation o f th e image s generated t o be made .

8

SECONDARY ELECTRONS AND IMAGING 8.1 Introduction Secondary electron s (SE) , discovered b y Austin and Starke in 1902 , ar e defined as being thos e electrons emitte d fro m th e specimen tha t have energies betwee n 0 and 50 eV. Because o f thei r lo w energy , secondar y electron s ar e readily deflecte d an d collected b y the application o f an electrostatic o r magnetic field ; a s a consequence , the grea t majorit y o f al l scannin g electro n microscop e (SEM ) image s hav e bee n taken using the SE signal. Secondaries ca n be generated throug h a variety of interactions in the specimen (Wolff , 1954 ; Seiler , 1984) , a typical event being a knock-o n collision i n which the incident electro n impart s some fraction o f its energy t o a fre e electron i n th e specimen . Thi s i s followe d b y a cascad e proces s i n whic h thes e secondaries diffus e throug h the solid, multiplyin g and losing energ y a s they travel, until the y either sink back into the sea of conduction electrons or reach the surfac e with sufficien t energ y to emerge a s true SE. For S E with energies o f a few ten s of electron volts , the inelastic mea n free pat h (MFP) is small, in the range 1 0 to 40 A, so each secondar y typicall y travels only a short distance befor e sharin g som e o f its energy in an inelastic event. However, for most materials, the inelastic MFP reache s a minimum at about 20 to 30 eV; for energies below that value, it increases rapidly, because ther e ar e n o larg e cross-sectio n inelasti c scatterin g event s throug h whe n energy ca n b e transferred . Th e elasti c MF P als o fall s wit h energ y bu t become s approximately constan t a t a few tens o f angstroms for energies belo w abou t 3 0 eV. SE with energies belo w thi s value ar e therefore strongl y elastically scattere d eve n though inelasti c scatterin g i s insignificant . Consequently , a s th e cascad e develop s from som e point below the surface of the irradiated specimen , onl y a finite fractio n will actually reach the surface and escape to be collected. We can thus see that there is a region beneath th e surface—the so-called S E escape depth, perhaps 50 to 15 0 A in extent—beyon d whic h no S E generated ca n reach th e surfac e an d escape . The measur e o f secondary electro n productio n is the S E yield 8 , which is th e total numbe r of SE produced pe r incident electron . Fo r al l materials, 8 varies with incident electron energ y in the same manner as that shown in Fig. 8.1 for silver. The yield is low at high beam energies becaus e most of the secondary productio n occurs too dee p belo w th e surfac e t o escape , bu t i t rise s a s the bea m energ y i s reduced. Eventually a broad pea k i s reached, correspondin g t o the condito n wher e th e inci 134

SECONDARY ELECTRONS AND IMAGING

135

Figure 8.1. Experimenta l S E yield dat a fo r silver .

dent electron range is comparable with the SE escape distance. Typically this occurs for a n energy of 1 to 3 keV. At still lower energies , the yield again falls becaus e of the lower energ y input from th e inciden t beam . Any satisfactor y simulation of S E production must be able to reproduce this yield curve for a given material and set of experimental conditions . In general , there wil l be tw o occasion s whe n S E generated b y the bea m can escape from th e specimen surface : first a s the incident electrons pass down through the escape zone and second a s backscattered electron s agai n pass through this zone on thei r wa y back t o th e surface . While th e S E produced i n these tw o event s ar e identical i n nature, their utilit y for imagin g is quite different , sinc e th e firs t grou p come fro m th e beam impac t point and are therefore capable o f high spatia l resolu tion while the second emerge from an area of the order of the incident electron range and are thus of low spatial resolution. It is thus convenient to denote them separately (Drescher et al., 1970). Those secondaries produced by incident electrons are called SE1, whil e thos e produced b y th e exitin g backscattere d electron s ar e calle d SE2 . The ratio between these two components is also an important quantity to be able to calculate because it is a measure of the likely spatial resolution o f the combined SE signal. In this chapter we discuss three levels o f approach to the problem o f doing a Monte Carl o simulatio n of secondar y electron production : 1. A complete first-principles simulation of all inelastic events leading to secondary production and including a model of the cascade multiplication of the SE and of their diffusio n t o th e surface.

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2. A simplifie d simulatio n i n whic h on e majo r mechanis m fo r S E productio n i s assumed t o be dominant. 3. A parametric model of SE production an d escape that can be added to a standard Monte Carl o mode l fo r inciden t electro n trajectories . The choice of which model to apply is determined by the information required. A first-principle s simulatio n ca n provid e detaile d informatio n abou t an y o f th e parameters of SE production, e.g., energy, angle, and depth distribution of the SE in addition to the computed secondary yield 8. However, a substantial computing effor t is require d becaus e o f th e comple x nature o f th e physica l model s involve d and , realistically, the applicatio n o f this approach i s limited t o a very few elements an d the most basic geometry. The simpler models typically calculate onl y SE yield, but they can provide this information in a short time even for structures of an arbitrary composition, size , and shape. These types of simulations are thus well suited for the computation o f S E images, studie s of charging , an d microanalytica l effects .

8.2 First principles—SE models The secondary emission of SE has been the subject of a sustained body of theoretical research ove r a large number of years since its discovery (Austi n and Starke, 1902). Important pioneer studies were performed by Bethe (1941) and Salow (1940), leading to the development of detailed phenomenological models by Baroody (1950), Jonke r (1952), an d Dekke r (1958) . Followin g Wolf f (1954) , wh o proposed th e use o f the Boltzmann transport equation to describe the process of SE generation, many authors, such as Cailler and Ganachaud (1972), Schou (1980,1988) and Devooght et al. (1987) have use d thi s approach . Th e Mont e Carl o metho d ha s als o bee n widel y applie d following the initial work of Koshikawa and Shimizu (1974). Construction of a Monte Carlo model for SE production involves three separate steps: determining the trajectory of the incident electron, computing the rate of SE generation alon g each portion of this trajectory, and finally calculating the fraction o f SE that escape from th e solid after th e serie s o f cascade processes. The first par t of this procedure i s identical with what we have already consid ered at length, so we will not discuss it again here. Generally, it is sufficient t o use a simple plura l scatterin g metho d t o mode l th e inciden t electro n trajectories . Th e second ste p requires u s to tak e int o accoun t all possible creatio n processe s for SE resulting from th e interaction o f primary electrons an d backscattered electron s wit h free a s well as with bound (i.e., core) electrons. I n addition, th e contribution to SE production fro m th e volum e plasmon deca y shoul d als o b e included . Fo r eac h o f these processes, w e require an excitation cross section. The differential cross section for th e productio n o f S E fro m valenc e an d d-shel l electron s i s (Lu o e t al. , J987 )

SECONDARY ELECTRON S AN D IMAGIN G 13

7

where E is the energy of the incident electron an d £' is the energy transferred to the secondary. The lowest allowed energy for an SE is chosen to be £F + 4 > (where EF is the Ferm i energ y and is the wor k function), s o that the S E can cros s the surfac e potential barrier an d enter the vacuum state. Gryzinski's function (Gryzinski, 1965 ) is employed to describe th e production of SE from th e excitation o f core electrons :

where E-} i s the bindin g energ y o f the cor e electron . For metals, the contribution t o SE production b y the deca y of plasmons must also b e considered. Usin g th e expressions o f Chung and Everhart (1977) gives a n expression for the probability per unit distance of creating SE in the form:

where

and D(E, /zco p, F v), which describes plasmo n deca y by one-electro n transitions , i s

where /zw p is the plasmon energy , E0 is the incident electron energy, a0 is the Bohr radius, an d We i s th e g th Fourier coefficien t o f the lattic e pseudopotentia l fo r the reciprocal lattic e vecto r g. It i s usual (Bruining, 1954 ) t o assum e that the probabilit y Pz o f a secondary escaping fro m som e dept h z below th e inciden t surfac e o f th e soli d i s give n by a function o f the kin d

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(where A i s a constant o f orde r unity ) whic h i s th e "straight-lin e approximation " (Dwyer and Matthew, 1985 ) an d implies tha t the emerging secondarie s ar e unscattered and that any scattering of an SE produces absorption (i.e., only those SE that are not scattere d between their point of generation and the surface can escape). I n fact , these assumptions are not strictly valid (Chung and Everhart, 1977), but the error that they introduce is usually negligible. However , to accurately model th e cascade, it is necessary t o generaliz e Eq . (8.7) . The probability o f a n SE arriving at the surfac e without an y inelastic collisio n i s

because the average escape angl e will be at 45° to the surface. \(E) i s the inelastic mean free pat h for an SE of energy E, whic h for a metal i s given (Sean an d Dench, 1979) b y the formula :

where EF i s the Fermi energy of the metal, and a is the thickness of a monolayer of the meta l i n nanometers . Typicall y \(E) i s o f th e orde r o f a fe w nanometer s fo r energies i n the sub-10 0 eV range . We can now set up the cascade process. From Eq. (8.8), the probability Pz of an SE traveling from z to z' withou t an y inelastic collision s i s

and similarl y the probability o f it traveling from z ' t o z' + Az ' i s

so the probability kP^ tha t the SE has interacted wit h another electron (i.e. , participated i n the cascade) i s

Some SE thus travel to the surface a t a rate governed by the exponential deca y law while others take part in the cascade process an d produce new SE of lower energies . It is convenient to make a distinction between secondaries o f different energ y ranges in th e cascade . Belo w 10 0 eV , th e scatterin g o f th e S E ca n b e assume d t o b e spherically symmetrica l (Koshikawa and Shimizu , 1974) . I f E' i s the energy o f an SE afte r scattering , the n

SECONDARY ELECTRON S AN D IMAGIN G 13

9

where E is the energy before scattering and RND is the usual equidistributed random number. Thu s fo r eac h S E participatin g i n th e cascade , tw o S E appea r afte r a collision wit h energies E' an d E", where

Above 10 0 eV, the scatterin g o f the S E is given by the usua l Rutherford relations, and Eqs. (8.1 ) and (8.2) are used to calculate the rate at which new SE are generated by core electron ionizations an d from valence and d electrons. Th e calculation of the cascade proces s i s carrie d ou t u p t o a sufficientl y larg e dept h fro m th e inciden t surface t o ensure that al l contribution s ar e accounted for. The final step in the analysis is to calculate what fraction of the SE reaching the surface actually pass the energy barrier an d reach the vacuum. The elastic scattering of the SE can be assumed to be isotropic, because at low energies the elastic MFP is only o f the orde r o f atomic dimension s (Samot o an d Shimizu, 1983) , s o all direc tions of motion are equally probable for the internal SE. In order for an SE of energy E to escape, E must be greater than Ep + 4> > where EF i s the Fermi energy and 4 > is the work function. The maximum angle a a t which the SE can approach the surface is determined b y taking the normal componen t o f momentum Pcos a equa l to the value

Thus the escap e probabilit y p(E) fo r a n S E of energy E a t the surfac e i s

Putting al l o f th e piece s describe d abov e togethe r no w give s u s a complet e Monte Carlo description o f the generation, multiplication , diffusion, an d escape of the secondar y electrons fro m a solid. Th e actua l code wil l not be reproduced her e because, o f necessity, it i s length y an d complex . When applie d t o th e proble m o f computing S E propertie s fro m selecte d metals—usuall y Al , Cu , Ag , an d Au — detailed agreemen t wit h experimental dat a i s obtained . Fo r example , as show n in Fig. 8.2 , (adapte d fro m Lu o an d Joy , 1990) , th e compute d normalize d secondar y energy distribution N(E) i s in good agreement wit h the available experimental data . The agreemen t betwee n compute d an d measure d S E yield s i s als o quit e good , although—as was the case for backscattered yiel d data—the spread between differ ent sets of experimental results is often ver y large, making any quantitative assess -

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Figure 8.2. Compariso n o f SE energy spectru m wit h data of Bindi et al . (1980) .

ment of accuracy a difficult matte r (Joy, 1993). It is usual to assume that the angular distribution of emitted secondaries follow s a cosine law (Jonker, 1952), a result that follows directl y fro m th e isotropi c natur e of elastic scatterin g withi n the specime n (Luo and Joy, 1990). However, the detailed simulation shows that when the inelastic scattering that occurs within the cascade is taken into account, there are deviations from th e cosine rule especially fo r the higher-energy SE. Figure 8.3 shows how this deviation appears . Although the effec t i s real, th e fractional number of suc h highenergy S E is small , an d so—fo r mos t practica l purposes—th e cosin e rul e ca n b e taken to apply. The model also calculates how the yield of SE varies as the thickness of the target is increased. Th e SE1 component of the SE emission, that produced by the forward-traveling incident electrons, typically reaches its maximum value for a thickness of only a few nanometers. As the thickness is increased beyon d that value, however, the number of backscattered electrons continues to rise, and so the yield of SE2 secondaries increase s an d the total yield 8 rises. Figure 8.4 shows experimental data fo r copper , illustratin g this effect . Since

we can write

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141

Figure 8.3. Compariso n o f computed angula r distribution with cosine law , Al a t 2 keV.

where i\ i s th e backscatterin g coefficien t fo r th e targe t an d p i s a facto r tha t represent the relative efficiency o f backscattered electrons at generating SE . If SE1 is constant , a s i s th e cas e fo r an y targe t thicke r tha n a fe w nanometers , the n th e variation of 8 with thickness provides an experimental wa y to measure the value of P, sinc e we know that T| varies linearly wit h thickness (se e Chap . 6). This indirec t approach mus t b e use d becaus e w e canno t physicall y distinguis h SE 1 an d SE 2 electrons, sinc e thei r propertie s ar e identical . However , i n a calculation , i t i s a simple matte r to compute the SE 1 and SE2 yield separatel y an d hence to obtain a value for p. The computation shows that p is typically in the range 3 to 6, the value being somewha t higher a t lo w energie s (= = 1 keV ) an d fo r material s wit h highe r atomic numbers. This sor t of value is in fai r agreemen t wit h the availabl e experi mental value s (e.g. , Bronstei n an d Fraiman, 1961 ) an d indicates tha t i n a typical metal or semiconductor for which T\ = 0.3 , les s than 50% of the total SE signal i s being produced b y the incident beam . The use of this model with other than a few familiar metals is difficult, primarily becaus e man y of th e require d parameter s i n th e mode l ar e eithe r unknow n or known onl y imperfectly . Fo r nonmetalli c element s suc h a s semiconductor s an d dielectrics, significan t modification s t o th e mode l woul d b e require d t o mak e i t physically realistic. Work in this area is still proceeding. Thus , while this method is of theoretical importance in studies of the phenomena associated wit h SE emission, in the context of this book the model is of limited value because it cannot be applied to th e real-worl d situatio n i n a n SEM . Fo r thi s kin d o f problem , simpler , les s detailed, bu t mor e pliable, models must be used.

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Figure 8.4. Variatio n o f SE yield wit h sampl e thickness.

8.3 The fast secondary model We ca n simplif y th e complexit y o f th e previou s metho d b y assumin g tha t onl y a single mechanism for the production o f secondary electrons need be considered. As originally suggeste d b y Murat a et al. (1981), a n appropriate assumptio n is that the SE are produced by a knock-on collision i n which an incident electron interact s with a fre e electron . Th e required differentia l cros s section s fo r thi s typ e o f interactio n are poorl y known ; however, thi s typ e o f interactio n ca n b e treate d classicall y b y considering i t as a coulomb interaction. In this case, the cross section per electron is (Evans, 1955 )

where E i s th e inciden t electro n energ y an d flE i s th e energ y o f th e secondar y produced. Sinc e this interaction involve s energy transfer s of [IE an d (1 - fl)E an d since, i n the final state , the incident electro n wil l no longer b e distinguishable, th e two cross sections must be added. For convenience, w e can define the electron with the highest energ y a s the primary; thus fl i s restricted, s o that 0 < fl < 0.5. Th e inelastic scatterin g (Fig . 8.5 ) cause s a deflectio n a o f the primar y electro n i n the laboratory fram e o f reference give n as :

SECONDARY ELECTRON S AN D IMAGIN G

143

Figure 8.5. Geometr y of scattering event to produce a fast secondar y electron.

where t is the kinetic energy of the electron i n units of its rest mass (511 keV). For a small energy transfer, say 500 eV for a 20-keV primary, a i s about 1° ; which is thus of th e sam e genera l orde r o f magnitud e a s th e averag e elasti c scatterin g angle . However, the secondar y electron leave s th e impac t poin t a t a n angl e - y give n by

so that the 500-eV S E leaves at an angle of about 80° to the initial direction of travel of th e primar y electron . Other cros s section s fo r S E productio n b y knock-o n hav e bee n given , fo r example by Mott (1930), Molle r (1931), and Gryzinski (1965). Thes e models diffe r in th e assumption s tha t they mak e an d i n their functiona l form , bu t th e predicte d values o f the cross section s unde r equivalent condition s diffe r relativel y littl e provided tha t the energ y o f the S E is sufficientl y hig h for i t t o b e considere d "free. " Since ther e ar e n o experimenta l dat a clearl y supportin g on e cros s sectio n ove r another, we will use the Evans model [Eq . (8.18) and following], sinc e th e values predicted b y this fall i n the middle of the range of numerical values spanned by the other formulations. 8.3.1 Constructing a fast secondary Monte Carlo model The fas t S E (FSE ) model tha t w e ar e developin g her e i s a doubl e Mont e Carl o simulation, since we track an incident electron , a s usual, but if a secondary electro n is generated , w e freez e th e positio n o f th e primary , trac k th e secondar y unti l i t

144 leaves the specimen or loses it energy, and then resume tracking the incident electron again. The model that we will use is the single scattering model of Chap. 3 because we have to investigate each electron interaction individually to see if it is an elastic or an inelastic event. We will use the program to plot the trajectories of either the incident or the FSE and to calculate the yield of SE from the specimen. Below is the key portion of the original listing of the single scattering program of Chap. 3, but now modified to allow for FSE generation. The changes and additions to the original code are shown in boldface.

* zero—counters; while num < traj_num do begin reset—coordinates; {allow initial entrance of electron} step: =-lambda(s_en)*ln(random); zn:=step; if zn>thick then {this one is transmitted} begin straight—through; goto exit;

end else {plot this position and reset coordinates} begin xyplot(0,0,0,zn); y:=0; z:—zn; end; {now start the single scattering loop} repeat

*

Test_f or_FSE: =RND;

if Test_for_FSE>PBL then {we have generated an FSE}

SECONDARY ELECTRON S AND IMAGIN G 14

5

begin Track_the_FSE; {follow till it escapes or dies) goto reentry; {then go back to main program} end; {otherwise scatter the primary electron in the normal way} s—scatter(s_en); reentry: {FSE program rejoins main loop} step: =-lambda(s_en) *ln(random) ; new_coord(step); [problem-specific code will go here] {decide what happens next} etc.

It ca n b e see n that , a t leas t a t thi s leve l o f detail , th e change s involve d ar e minimal. The initia l entry o f the electro n proceed s a s before, but now—instea d of just allowin g th e electron to be scattered elastically—we test to see whether o r not an inelastic scatterin g event, resulting in the production of a fast SE , has occurred . This is done by seeing if a random number Test_f or_FSE i s greater or less than a variable PEL , whic h is the ratio of the elastic cros s sectio n t o the total scatterin g cross sectio n (i.e. , includin g the inelasti c effect) . The elastic scattering undergone by the incident electron is represented as usual by th e Rutherfor d cross sectio n [Eq . (3.2)] . Thi s cros s sectio n CT E define s a mean free pat h X el give n by th e formula

which represent s th e averag e distanc e tha t an electro n wil l travel between succes sive elasti c scatterin g events . Similarl y w e ca n defin e a n inelasti c MF P X in b y integrating Eq . (8.18) . However , w e not e fro m Eq . (8.18 ) tha t th e cros s sectio n becomes infinite at fl = 0 , so a lower limit fl c must be chosen as a cutoff. Choosin g a finite value of flc no t only avoids the problem of an infinite cross sectio n bu t also has th e effect o f removing from consideratio n very low energy secondarie s which , because of their small MFP, require substantial computer time during the simulation without contributing much to the final result. I t is foun d (Murat a et al., 1981 ) that the choice of Oc does not sensitively affect the results produced, since lowering ft c produces more secondaries bu t each of a lower average energy. So, for convenience,

146 MONT

E CARLO MODELIN G

Table 8.1 Values computed for 20-keV beam incidence Element Carbon 33 Aluminum 28 Silicon 24 Copper 5 Silver 3

X elastic

A. inelastic

8 A 92 9 A 102 0 A 87 4 A 25 6 A 19

7A 0A 1A 0A 8A

a valu e of 0.01 wil l be used here. Equatio n (8.18 ) ca n the n b e integrated ove r th e range flc < d < 0. 5 to give th e total inelastic cros s sectio n cr in an d hence

since the cross section is per electron. X in is the average distance between successive inelastic scattering events. Whereas X el is of the order of 10 0 A for most materials at 20 keV, X. in i s abou t 0.1 (x m a t the sam e energy, s o th e probabilit y o f a n inelasti c collision is small compared to that for an elastic event. Table 8.1 gives some typical values fo r X el and X in. We can define a total scattering MFP—th e average distanc e between scatterin g events o f either type—fro m th e relation

PEL i s the ratio X TAei an d represents the probability tha t a given scatterin g event wil l be elastic. Sinc e A. in > X el, the n PEL is close t o unity an d only a small fraction o f scatterin g event s wil l b e inelastic . Th e typ e o f even t tha t occur s i s determined b y choosin g a random numbe r Test_f or_FSE an d seein g i f thi s is greater tha n PEL . I f i t i s not , then the scatterin g even t wa s elastic , n o FS E wa s generated, an d the program proceeds a s normal. If, however, Text_f or_FSE > PEL, the n an FSE has been produced and the routine Track_the_FSE is called to track the FSE while the tracking o f the incident electro n i s suspended. It is convenient to be able to turn FSE production on and off as required. This can be done in the setu p of the program: GoToXY(some suitable screen coordinates); Write('Include FSE (y/n)?'); if yes then FSE_on:=true else FSE_on: = false;

SECONDARY ELECTRONS AND IMAGING

147

This sets a Boolean variable FSE_on to true if FSE production is required, and to false if FSE are not needed. If FSE_on is true, then PEL is calculated as described above; but if FSE_on is set false, then PEL is set equal to unity, A.T = \el, and the program behaves in exactly the same way as the original single scattering model. The calculation of PEL is carried out in the function lambda (s_en) , which, although it has the same name as before, is now modified to incorporate the inelastic scattering.

Function lambda(energy:real):real; {this now computes both the elastic and inelastic MFP} var al,ak,sg, mfpl,mfp2,mfp3:real; begin {the function} if energy<e_min then {don't allow anything below cutoff energy} energy:=e_min; al:=al_a/energy; ak:=al*(l.+al); {this gives the elastic cross secton sg in cm2 as} sg:=sg_a/(energy*energy*ak); {and hence the elastic mfp in A is} rnfpl: =lam_a/sg; {from Eg. (8.18) the MFP for FSE in A is] mfp2 : =at_wht*energy*energy*2.557(density*at_num); {so the total MFP from Eg. (8.22) is} mfp3 : = ( 1 / m f p i) + ( 1 / m f p 2) ; mfp3:=l/mfp3; {make if FSE_o n the n {compute the ratio constant elastic/total} PEL: = m f p 3 / m f p l else begin PEL:=1; {n o FSEs will be produced} mfp3 : =mf pi; {an d X T = X el} end; {in eithe r cas e th e value returned is] lambda: =mfp3 ; end;

The final part of the operation is to track the FSE that are produced. This involves several steps that can be displayed schematically as

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MONTE CARLO MODELING

Calculate the energy of the FSE Store the coordinates of the incident electron at the point where scattering occurred Find the scattering angles of the FSE and its MFP Find the end point of its first trajectory step Follow FSE through standard single scattering Monte Carlo loop until FSE leaves the specimen or falls below the energy cutoff Calculate scattering angle of incident electron from inelastic event Find energy of incident electron after inelastic event Return to main program

The procedur e Track_the_FS E an d a n associate d functio n FSEmf p carr y ou t these tasks. Function FSEmfp(Energy:real):real; {computes elastic MFP of FSE during its subsequent scattering} var QK,QL,QG:real; begin QL: =power(at_num,0.67)*3.4E-3/energy; QK:=QL*(1+QL); QG: = (at_num*at_num)*9842.II (energy*energy*QK); FSEmfp:=(at_wht*lE8)/(density*QG); end;

Procedure Track—the_FSE; {generates an FSE and then tracks it. This procedure is a complete single scattering MC loop. Note that all variables are local.} label set_up_reentry,FSEloop;

var FSEnergy,eps,sp,cp,ga,FSE_step:real; SI,S2,S3,S4,S5,S6,S7:real; vl,v2,v3,v4,an_m,=an_n:real; deltaE,al,escape:real; begin {increment the counter for FSE production}

FSE_count :=FSE_count +1 ;

{get the energy of the FSE that is produced} eps:=l/ (1000-998*RND) ; FSEnergy:=eps*s_en;

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149

{we now store the coordinates of the primary electron to reuse them later } Sl:=x; S2;=y; S3:=z; S4:=cx; S5:=cy; S6:=cz; S7:=s_en-s_en*eps: {incident energy loss =

FSE energy}

{see i f the FSE exceeds the minimum energy that we want to consider} if FSEnergy<e_mi n the n {it s no t worth tracking} begin escpae: =750*power (FSEnergy, 1 . 6 6) /density;{estimat e range} se_yield: =se_yield+0.5*exp(-z/escape) ; {contributio n to yield } if FSE_o n the n xyplo t ( y ' z 'y + 1' z + 1) ; {plot a dot} goto set_up_reentry ; {an d ou t of here} end; {otherwise ge t th e initial scattering angles for the FSE that are produced} s p : = 2 * ( 1 - e p s ) / ( 2 + eps*FSEnergy/511.0) ; cp:=sqrt(1-sp) ; sp:=sgrt(sp) ;

FSEloop: {single scattering model loops round here} {find the FSE step length} FSE_step:=-FSEmfp(FSEnergy)* In(END) ; ga: =two_pi*RND; {find out where the FSB has gone using usual formula} if cz = 0 then cz: =0.0001; {avoid division by zero} an_m:= (-cx/cz); an_n: =1.0/sqrt(1. + (an_m*an_m) ) ; {save time by getting all the transcendentals first} vl:=an_n*sp; v2 :=an^n*an_m*sp; v3 : =cos (ga) ; v4:=sin(ga) ; {find the new direction cosines} ca:=(cx*cp) + (vl*v3) + (cy*v2*v4); cb: = (cy*cp) + (v4*(cz*vl - cx*v2)); cc:=(cz*cp) + (v2*v3) - (cy*vl*v4);

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MONTE CARLO MODELING

{get the new coordinates} x n : = x + FSE_step*ca ; y n : = y + FSE_step*cb ; z n : = z + FSE_step* c if zn>thic k the n {thi s one is transmitted} begin if FSE_o n the n xyplo t ( y , z , y n , 9 9 9 ) ; goto set—up—reentry ; {get out of this function} end; if zn < = 0 the n {it s an emitted SB] begin if FSE_o n the n xyplo t ( y , z , y n , 9 9 ) ; se_yield:=se_yield+l; goto set—up—reentry ; {ge t out of this function} end; if FSE_o n the n xyplo t (y , z , y n, zn) ; {plot the other case} {find the energy loss of FSB on this step} deltaE:=FSE_step*stop_pwr(FSEnergy)*density*IE-8; {so current FSE energy is} FSEnergy:=FSEnergy-deltaE; if FSEnergy=wh then z_val:=0 else z_val: = (y—val-wh) / tn: {from geometry of Fig. 8.15} if y_val < = tw the n z-val : =height ; end;

The rest o f the progra m i s straightforward and requires n o detailed comment . Th e longest section is the application of Eq. (8.35) to compute the SE yield. The program proceeds by locating the position o f the electron i n one of four possible regions—in the substrate not beneath the structure, in the substrate and underneath the structure , in the walls of the structure, in the central region of the structure—and then computing the possible exi t paths. One small refinement is to consider th e possibility o f the specimen recollecting som e of its own emitted electrons. Ever y emerging BS E (e.g., from the side walls of the feature) is checked to see if it has a component of velocity towards the substrate . If it does, the n it is allowed to reenter th e specimen and the program continue s t o track it until i t is again backscattere d o r comes t o rest . Figure 8.1 7 shows SE and BS profiles for the case of an aluminum structure 1 jjim wide and 1 urn high with walls at +5 ° t o the vertical (i.e. , a n overcut structure) placed on a silicon substrat e and for beam energies of 2,10, and 30 keV. The profiles demonstrate how the "image" of this simple object is affected b y the choice of beam energy. At 2 keV, both the SE and BSE line profiles reveal the edges of the structure clearly. I n the S E image, th e simulatio n show s that the lin e woul d appea r slightl y bright above background, with sharp white edges. Th e BS profile shows the aluminum lin e and the silicon substrat e at about the same brightness, bu t the edges o f the feature ar e marked by bright band s wit h a width of the orde r o f the incident bea m range (i.e., abou t 500 A). At 1 0 keV, the SE profile remains muc h the same, but the

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MONTE CARL O MODELING

Figure 8.17. Compute d SE and BSE signal profiles across an aluminum bar 1 fun wid e and 1 (Jim high, with 5° wall, on a silicon substrate at 2 , 10 , and 30-ke V bea m energy .

edge brightness i s now greater i n magnitude. The BS profile has changed substan tially, however, because the interaction volume of the beam is of the same order of magnitude as the size of the feature. The beam range is now of the order of the width of the feature, so the whole line is bright abov e background with no clearly defined edges/ Note also tha t the background intensit y from th e silico n substrat e dip s as it approaches th e edg e o f th e featur e becaus e o f electro n penetratio n beneat h th e structure. Finally, at 30 keV, the SE and BSE images are similar to those at 1 0 keV, with the SE profile dominated by the edge bright lines and a low contrast in the BSE profile because the thickness of the line is now a small fraction of the electron range.

SECONDARY ELECTRONS AND IMAGING

173

A comparison o f the various line profiles wit h the real cross-sectional geome try of the structure shows that determining th e position of the top and bottom edges from an y of the traces is difficult o r impossible. Hence, althoug h it is easy to image a micron-scale device , "measuring " th e profil e wit h an y accurac y usin g th e S E or BSE signal s i s a muc h mor e challengin g tas k becaus e o f th e constantl y shiftin g relationship betwee n th e real an d apparent edge positions. On e o f the most usefu l applications o f suc h simulation s a s these, therefore , i s t o tr y an d devis e rule s that will allow experimental lin e profiles t o be interpreted in such a way that the width, height, an d wall angl e of the structur e fro m whic h they came ca n be deduced (see , e.g., Poste k an d Joy, 1986) . A comparison of these computed profiles and corresponding experimenta l dat a shows that the qualitative agreement i s excellent, wit h the simulations predicting al l of th e feature s o f th e experimenta l profiles . However , th e leve l o f quantitativ e agreement is not always so encouraging, for a variety of reasons. A key reason is the behavior an d efficiencie s o f th e S E an d BS E detector s use d i n th e experimenta l system. In the computation, it is assumed all secondaries ar e collected. In practice , the detection efficienc y o f the SE detector wil l depend o n where it is relative to the irradiated area , o n the bias applied t o it, and on the presence an d magnitude of any local charging . Thes e consideration s ca n drasticall y modif y th e detai l o f th e S E profile. Unfortunately , computing the efficiency o f the SE detector requires a determination of the electrostatic field distribution s in the SE M specimen chamber , and this is a lengthy task (Suga et al., 1990 ; Bradle y and Joy, 1991 , Czyzewsk i and Joy, 1992). I n addition , i n complex structure s such as parallel array s of closel y space d lines, there is a possibility that emerging BS E will be recollected b y some other part of the structure, generating SE at the point of entry and during the subsequent travel (Kotera et al., 1990) . Detaile d calculations , therefore, requir e testin g for the possi bility o f recollection b y tracing th e path o f th e BS E a s they leave . Expanding thes e line-profil e computation s t o th e simulatio n o f a complet e image is , in principle, straightforward , although the time required t o obtained ade quate statistic s migh t be dauntin g on an y but th e fastes t computers . Nevertheless , with the growing interest in quantitative image interpretation (e.g. , for microdimensional metrology), and for the comparison of images between, say, the SEM and the scanning tunneling microscope (STM) , this is likely t o be an important applicatio n and extensio n o f th e techniques discusse d above .

9

X-RAY PRODUCTION AND MICRO ANALYSIS

9.1 Introduction The process of fluorescent x-ray generation in a sample by electron bea m irradiatio n makes it possible t o perform a chemical microanalysis of small (micrometer siz e or less) region s o f a specimen . However , i n orde r t o obtai n quantitativ e data , i t i s necessary t o b e abl e t o describ e i n considerabl e detai l th e distribution—bot h lat erally and in depth—of the x-ray production. I t was this need that led to the initia l studies of Monte Carlo modeling by Bishop (1966). Although, it many cases, simple analytical model s hav e subsequentl y bee n develope d tha t allo w man y o f th e re quired result s t o b e obtaine d wit h adequat e accuracy , onl y Mont e Carl o method s offer a genera l approac h t o th e microanalysi s o f a n arbitrar y specimen . I n thi s chapter w e examin e x-ra y production , bot h characteristi c an d continuu m (Bremsstrahlung), i n thi n foil s an d i n bul k specimen s usin g th e Mont e Carl o model s previously developed .

9.2 The generation of characteristic x-rays The production of characteristic x-rays along a step of an electron trajectory can be calculated i f th e cros s sectio n fo r x-ra y production i s known . Typically th e Beth e cross sectio n fo r inner-shel l ionizatio n i s used , in the form :

where « s i s th e numbe r o f electron s i n th e shel l o r subshell , Ec i s th e critica l ionization energ y i n kil o electro n volts , U is th e overvoltag e E/EC, bs an d cs ar e constants, an d a the n ha s th e unit s o f ionization s pe r inciden t electron s pe r atom/cm2. Fo r example , fo r th e K shell , bs an d c s ar e 0. 9 an d 0.6 5 respectivel y (Powell, 1976 ) for the overvoltage range 4 < U ^ 25. For the L and M shells, the appropriate constant s an d range o f applicabilit y ar e les s wel l establishe d (Powell , 1976). In some cases, for example, the production of x-rays in a thin foil, the energy of 174

X-RAY PRODUCTIO N AND MICROANALYSI S 17

5

the electrons ma y always be high enough to put U in the correct range ; bu t in the case of bulk specimens, i t is clear that, since the energy of the electron decrease s as it travels through the specimen, ultimately the overvoltage will no longer satisfy th e condition U s4 and the predicted cross section may become inaccurate. Alternativ e cross-section model s ar e available , fo r exampl e Casnat i e t al . (1982) , whic h ar e valid fo r lo w overvoltages , an d thes e shoul d b e considere d fo r th e mos t accurate work; bu t th e familiarit y an d convenienc e o f th e Beth e expressio n ha s le d t o it s nearly universa l use i n all types of conditions, whethe r warranted or not. The numbe r o f x-ray s /s pe r inciden t electro n produce d alon g a ste p o f th e trajectory i s then

where NA i s Avogadro's number (6 X 1023 atoms/mole), A is the atomic weight, p is the density, to is the fluorescent yield, and the step length is in centimeters. Sinc e A, NA, p an d 9 ar e constant , i t i s convenien t t o extrac t thes e quantities , an d th e constants ahead of the functional variables in Eq. (9.1), and evaluate /s in the for m

only restoring the numerical constants whe n an absolute yiel d valu e is required.

9.3 The generation of continuum x-rays In addition t o the characteristic x-ra y signal used for microanalysis, there is also a continuum, or Bremsstrahlung, x-ray signal generate d by th e slowin g down of the electrons in the coulomb field o f an atom. By analogy with Eq. (9.2) , we can write the continuu m yield per inciden t electro n int o a unit steradian 7 CO as

where Q is the continuum cross section. The continuum radiation, unlike the characteristic line , i s anisotropicaly emitted, being peaked abou t the forward direction of travel o f the electron . I n considerin g Bremsstrahlung production i n bulk samples , this effect i s of little consequence because it is averaged ou t by the plural scattering of the electrons; bu t in the case o f thin films, where the incident electron s ar e only scattered throug h smal l angles , th e polarizatio n an d directivit y o f th e continuum must be properly accounte d for. A simple cross section for continuum production can be obtained by combining

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experimental observatio n an d som e theoretica l result s (Fior i e t al. , 1982) . Th e maximum energy that can be given up by an incident electron is equal to its kinetic energy E0, numerically equa l to the beam voltag e V 0; thus the highest energy tha t can appea r i n th e continuu m spectrum i s E0, the so-calle d high-energ y o r DuaneHunt limit. Second , for thin specimens i t is foun d tha t the continuu m energy in an energy interval A £ is about constant (Compton an d Allison, 1935 ) fro m zer o up to the high-energy limit. Thus, the fraction of the total emitted continuum energy Er in the energ y rang e E t o E + A E i s AE/E0. Th e efficienc y o f th e generatio n o f continuum productio n i s define d as th e tota l continuu m energy i n th e rang e fro m near zer o t o E0, generate d b y electron s tha t los e a n amoun t dV o f thei r energ y divided b y th e energ y lost . Kirkpatric k an d Wiedman n (1945 ) showe d fro m th e theory o f Sommerfeld (1931) that this efficiency wa s 2.8 X 10~ 9 Z V0, where V0 is the bea m voltag e i n kilovolt s an d Z i s th e atomi c number . Th e tota l continuu m energy i s thu s

The fractio n o f thi s quantit y i n th e energ y interva l A E the n give s th e numbe r o f photons 7 CO of energ y E; i.e. , th e numbe r of photons i n th e energ y interva l A E is ET.AE/(£ EO). S o

In the Monte Carlo simulation , dV is simply the energy loss occurring alon g a given ste p o f th e trajectory , a quantit y that i s evaluate d eithe r directl y fro m th e stopping power (in the single scattering model) or as the difference E[k] - E[k + 1] in th e plura l scatterin g model , s o Eq . (9.5 ) ca n b e use d t o giv e th e continuu m intensity a t some energ y £ i n the energy interval A£ directly . This simpl e expression produce s quite useful result s in many cases of interest, but i t mus t b e realize d tha t i t i s a drasti c simplification , sinc e i t assume s bot h isotropic emission and uniformity in energy distribution. More accurate and detailed models are available, the most widely used of which is that of Sommerfeld (1931) as modified b y Kirkpatric k an d Wiedman n (1945) . Sommerfeld' s mode l assume s a pure Coulomb field abou t a point nucleus ignoring screenin g effect s an d represents the scattered electrons a s plane waves. Stricly speaking, this theory is valid only for low electro n energies , wher e relativisti c effect s ca n b e neglected . Kirkpatric k an d Wiedmann (1945 ) gav e a n algebrai c fi t t o th e Sommerfel d theor y tha t correctl y accounts fo r th e polarizatio n o f th e Bremsstrahlung an d i s more easil y compute d numerically. The algebra is messy and will not be reproduced here , but a routine for the evaluation of the Kirkpatrick an d Wiedmann expression base d o n that given by Statham (1976 ) will be give n later i n thi s chapter.

X-RAY PRODUCTIO N AN D MICROANALYSI S 17

7

9.4 X-ray production in thin films 9.4.1 Spatial resolution The simples t case of interest is tha t o f determining the spatia l resolution of x-ra y microanalysis. Thi s proble m i s o f importanc e i n analytica l electro n microscop y (AEM), where x-ray analysis is used to determine th e composition o f small precipitates or the composition profile at a boundary. In this kind of situation, the specime n is thin (typically 30 0 to 150 0 A ) and the incident bea m energy i s high (10 0 to 200 keV). A s ca n b e see n b y runnin g th e SS_M C progra m (Chap . 3 ) for thi s typ e of condition, the majority o f electrons pass through the sample unscattered because the specimen thicknes s i s comparabl e t o an d smalle r tha n th e elasti c mea n fre e pat h (MFP) a t this energy. Since , i n addition , th e bea m energ y i s usually much greate r than th e excitation energ y of the x-ra y line o f interest, th e volum e associated wit h x-ray generatio n i s identical wit h that associated wit h the electron scattering , since the ionizatio n cros s sectio n wil l b e essentiall y constant . Sinc e th e x-ra y spatia l resolution is commonly defined as the radius of the volume within which 90% of the emitted x-ray s are generated, i n this approximatio n th e resolution ca n be take n as being th e radiu s withi n whic h 90 % o f th e transmitte d electron s emerg e fro m th e bottom surface of the foil. Incorporation of this into the SSJVIC program is straightforward. If an electron is determined as being transmitted (i.e., zn > thickness of the foil) then the length of the exit vector is first found and the exit radius from the beam axis (x = y = 0 ) i s computed. The botto m surfac e is divide d int o an arra y of 10 0 annular rings, radius [ 0 . . 9 9 ] , o f constant width scale where scale is set to 1 0 A. The exit radius is converted t o give the number of the annular ring (0 to 100) through which the electron left, and one count is added to the total for that ring. {after determining that this electron is leaving} {find length of vector from x,y,z to bottom surface} 11: = (thick-z) /cc {hence the exit coordinates on the bottom surface are} xn: =x+ll*ca; yn: =y+ll*cb; {and the exit radius about the beam axis is} radial:=sqrt((xn*xn)+(yn*yn)); {convert this to an integer identifying the annular ring} r_val:=trunc(radial/scale) ; {and use r—val to index the array and add one count} if r_val=broad then {we have reached 90% at this value of k}} goto test; {so stop counting} end; test: {exit from summing loop} {and display the computed beam broadening as a line of appropriate length} MoveTo(center-trunc(k*scale*plot_scale), bottom+15) ; LineTo(center-trunc(k*scale*plot_scale), bottom+15) ; readln; {freeze the display} Figure 9.1 shows how the beam broadening computed in this way compares with the simple analytical estimate given by Goldstein et al. (1977) for the case of a 100-keV beam incident on a foil of copper. It can be seen that the two models predict the same type of behavior with increasing thickness, even though they give somewhat different values for the broadening. This is not surprising, because the analytical model is estimating the average exit diameter on the assumption that each electron is scattered just once, at the midpoint of the foil, and this figure is not directly related to the 90% definition used in the Monte Carlo case. In addition, once the foil becomes thick enough for plural scattering to be significant, the value estimated by the analytical model is no longer valid. It is clear, in any case, that a single "beam broadening" number cannot faithfully represent the actual experimental situation. While it provides a guide as to the resolution to be expected, it cannot say what effect this resoluton might have on, for example, a measured composition profile. For that kind of problem, it is necessary to use the actual x-ray generation profile of the beam, again on the assumption that this is directly related to the distribution of trajectories of the incident electrons. The program AEM_MC on the disk provides this ability. It is essentially the same single scattering program, the only modification being that the incident electron beam is

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179

Figure 9.1. Bea m broadening in copper at 100 keV using Monte Carlo method and Goldstein et al . (1977) equation.

assumed t o be a Gaussian distribution o f som e given ful l widt h at half maximum height (FWHM) . Basically, the first par t of the procedure i s identical t o that given above, so that the number of electrons travelin g through—and hence the number of x-rays produced in a given annular ring—is determined as before. Imagine now that the beam i s placed a t a distance X fro m some tes t poin t i n the sample. When X i s very large and negative, no x-rays will be produced at the test point, but the number will increas e a s X i s reduced, reac h a maximum when X = 0 , an d then fal l away again as X becomes larg e and positive. A plot of the integrated x-ra y generation in the positive half spac e x > 0 as a function o f the incident beam position X give s a S-shaped profile (fig. 9.2), which rises from 0 % to 100 % of the total x-ray generation. I n a manner analogou s t o tha t use d fo r specifyin g the prob e diamete r o f a n electron beam, the distance travelled in raising the x-ray signal from 10 % to 90% of its maximum value can be called the x-ray spatial resolution. The program plots the profile an d marks on the 10 % and 90% levels so that this number can be measured. Although this value is again a single number, the visual profile provides a guide as to wha t ca n b e expecte d t o happen . Whe n th e specime n i s thin , th e profil e i s dominated by the Gaussian shape of the probe, since electron scattering is small and the profil e i s a standar d erro r functio n (o r "erf" ) curve . A s th e foi l thicknes s increases, however , th e profil e change s shap e an d i s compose d o f tw o distinc t regions, (1) a central region that is still approximately the error function, surrounded by (2) broad tail s as a result o f electron scattering . The effect o f this particular beam profile on an experimental measurement can

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Figure 9.2. Variatio n of integrate d x-ra y intensit y wit h position .

now b e observed . Th e progra m permit s tw o kind s o f situation s t o b e considere d (Fig. 9.3) : th e case i n which two materials A and B are in contact at some interfac e and in which B is diffusing int o A, and the case of an interface region o f some width W o f material B surrounded by material A, again allowing for the diffusion o f B into A. I t i s assume d i n eithe r cas e tha t th e electro n scatterin g power s o f A an d B ar e sufficiently simila r tha t th e for m o f th e electro n bea m profil e i s no t significantl y altered. C(B) X the concentration o f B in A a t some distanc e x fro m th e interface , i s given th e form :

Figure 9.3. Interfac e geometrie s fo r thin-fil m x-ra y program .

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181

Figure 9.4. Compariso n o f compute d an d actua l profile s acros s a 10-A-wid e boundar y i n copper, with L = 3 0 A.

where L is a characteristic diffusio n length . I f L is specified , the n th e for m o f th e composition profile that would be experimentally measured acros s the chosen interface can be found by numerical convolution of the computed beam profile with the 0 '.en for m o f the interface . Becaus e the procedures fo r doing thi s are straightforward, they are not given here, but they can be studied in the source code on the disk. Figure 9. 4 show s ho w the "real " compositio n profile an d the predicte d measure d profile would compare for a 1000-A-thick copper film an d a 25-A beam at 10 0 keV for th e case where the interface region wa s 1 0 A wide (e.g., a grain boundary) and the diffusion lengt h L is 30 A. If experimental data are available from a n analytical electron microscope , th e values of L and W can be obtained by iteratively fitting the computed an d measure d profiles . Th e mode l ca n readil y b e extende d t o othe r geometrical situations . 9.4.2 The effect of fast secondary electrons In the previous chapter, the generation of fast secondar y electrons (FSE ) was incorporated int o a Mont e Carl o program . I t wa s note d tha t thes e FSE , whic h hav e energies u p to half that of the incident beam, tended to travel almost normal to the original directio n o f the inciden t beam. Clearly , i f these FS E ca n generate x-rays , this coul d adversel y affec t th e spatia l resolutio n o f microanalysis . Equatio n (9.3 ) shows tha t the yiel d o f x-ray s by a n electro n o f energ y E i s proportiona l t o l/U, where U is the overvoltage E/Ek. Conside r then the case of oxygen x-rays (Ek = 53 0 eV) being fluoresced by incident 100-keV electrons, and by, say, 2-keV fast second ary electrons . Th e efficienc y o f oxyge n x-ra y ionizatio n b y th e FS E i s 5 0 time s higher tha n th e efficienc y fo r th e primar y electron , s o eve n i f onl y on e primar y

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electron in a hundred generated an FSE, 50% of the measured x-ray yield could be coming from secondaries rather than the incident beam. Since, as we saw, the spatial distribution of the FSE is also quite different from that of the incident beam, it is clear that the effect of including fast secondary production could be significant. To examine this effect, the Monte Carlo model must be modified to include x-ray generation using Eq. (9.3) for some arbitrary x-ray energy. The code for this is straightforward: Procedure generate_x-rays (energy:real;stepsize:real); {computes the x-ray yield using Eq. (9.3) given the electron energy and the critical excitation energy of the x-ray. Fixed constants can be inserted later if required}

var x_ray_yield:real; at_z, a t _ r : i n t e g e r ; begin x_ray_yield: =l n ( e n e r g y / E _ c r i t ) / ( e n e r g y * E _ c r i t) ; x_ray—yield: =x_ray_yieId*stepsize; {assign X-ray production to a box at given radius and depth} a t _ z : = r o u n d ( ( z + zn) / (2*step ) ) ; {depth } at_r: =round(sqrt ( ( x + x n) * ( x + x n) - ( y + y n ) * (y+yn) ) / ( 2 * s t e p ) ) ; {radius } if at_z < = 0 the n at_z : = 0; {t o protect array} if ( (at_z< = 50) an d (at_r < = 50) ) the n {within bounds so put array} x_ray_gen[at_z,at_r] : =x_ray_gen [at—z , at_r] +x_ray—yield;

into

end;

In this example, the computed x-ray yield is deemed to have occurred at a point defined by the midpoints of the trajectory step (x + xri)/2,(y + yri)/2, and at a depth (z 4- zn)/2. The data are stored in an array x_ray_gen [0 . . 5 0 , 0 . . 5 0 ] formed of annuli whose width and depth are defined by the quantity step. The box size can be chosen to be any value, but a convenient choice is often to make step equal to one-fiftieth of the estimated electron range. The variable stepsize passed to the procedure is the length of the trajectory step. If, on this step, the electron is either transmitted through or backscattered from the specimen, the stepsize must be modified accordingly. This can be done exactly—e.g., the actual distance 11 traveled within the specimen for an electron that is backscattered out of the specimen from some point z beneath the surface of the specimen is 11 = z/cc, where cc is the usual direction cosine. Otherwise the stepsize can be approximated putting the stepsize equal to stepsize*RND. The insertion of the procedure into the FSE_MC code is obvious and needs no detailed comment. In either the primary loop, or in the FSE loop, the x-ray genera-

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Table 9.1 Effect of FSE on X-ray resolution Line Used

Mode

IronKa N

o FSE 6 With FSE 6 o FSE 6 With FS E 6 o FSE 6 With FSE 7 o FSE 6 With FSE 8

Silicon K a N Oxygen Ka N Carbon K a N

Resolution

X-ray Yield

0A— 4A + 0A— 5 A +3.1 0A— 5 A +6.0 0A— 5 A +11.2

1% % % %

tion can be computed a s soon as the new coordinates xn,yn,zn have been calculate d and th e subsequen t behavio r o f th e electro n ha s bee n determine d (i.e. , doe s th e electron remai n i n the sampl e o r is it transmitted o r backscattered?). Thu s z n : = z + step*cc ; {get x-ray production now for electron remaining in sample} if E>E_cri t the n {ionization can occur so} generate_x-rays(E,step);

The effec t o f including FS E production i s ofte n quit e significant. Table 9. 1 shows data computed for the effective resolutio n an d total x-ray yield for several differen t elements i n a 500-A thick sampl e o f iron examine d with a 100-ke V beam. It ca n b e see n tha t i n th e simpl e approximatio n wher e secondar y effect s ar e neglected, the spatial resolution is independent of the energy of the x-ray line that is being examined , because only th e volume occupie d by the trajectories o f the inci dent electrons are considered. Bu t when FSE are included, the resolution gets worse because of the lateral motio n of the secondaries. Th e lower the energy of the x-ray, the further a given secondary ca n travel before it no longer has sufficien t energ y to excite th e chose n line , s o a s th e ionizatio n threshol d i s lowered , th e compute d resolution becomes worse. Fo r iron Ka wher e £crit is around 7 keV, the change in resolution is negligible because relatively fe w FSE have high enough energies; bu t for carbo n K a (£ crit = 0.2 8 keV), the relution i s degraded b y som e 40%. Sim ilarly, including FSE production als o changes the predicted yiel d of x-rays from a n element. Th e exten t t o whic h thi s occur s depend s o n th e natur e o f th e matri x i n which the element t o be observed i s sitting. If the matrix is of low density and low atomic number , then th e numbe r o f FSE produced , an d the consequen t chang e i n x-ray yield, will be small; but for a dense, high-atomic-numbe r matrix, the number of FSE produced is high and consequently extra fluorescence of the element occurs. This indicates that care must be used in performing quantitative microanalysis under

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conditions where FSE production is significant. For example, at high beam energies and thus generally hig h overvoltage-ratios, the majority of x-ray production may be coming vi a th e FSE rather than from th e inciden t electron . I n thi s cas e th e spatia l resolution will be significantly worse than predicted b y the simple theory—and, in fact, wil l get worse rather than better a s the beam energy is increased, bu t it will be more o r less independen t o f the foi l thickness—and th e intensit y ratios o f various elements wil l vary in a complex way with composition because changes i n chemistry will cause changes in the number of FSE and hence in the amount of excitation. Microanalysis a t beam energies o f 200 keV o r above is likely to demonstrate these problems. 9.4.3 Peak-to-background ratios A commo n measur e o f th e qualit y o f performanc e o f a n AE M i s th e peak-to background rati o tha t can b e achieve d fo r a give n x-ra y line, typicall y chromium (Williams an d Steel , 1987) . Value s that hav e bee n publishe d i n th e literatur e fo r various microscopes var y by nearly a factor of 10 , and there has also been consider able discussion about how the figure might be expected t o change with the acceler ating energ y o f th e microscope . Th e Mont e Carl o simulatio n can b e modifie d t o address som e o f thes e questions . The simples t approac h i s to us e a Kramer' s la w model for the Bremsstrahlung contribution in the form given in Eqs. (9.4) and (9.5). Although thi s is only an approximation , in particular because it ignores th e aniso tropy of the continuum, it provides a good startin g point for a more detaile d analysis. In order to make use the routine for both electron transparent and bulk samples, it i s convenien t t o calculat e th e dept h distributio n o f th e continuu m signa l i n th e chosen energy window, since this will be needed later on. By analogy with the cf>(pz ) ("phi_ro-z") depth distribution o f characteristic x-ra y production, this can be called bkg_ro_z. The cod e woul d have th e followin g form: Procedure continuum (energy:real;stepsize:real); {this computes the Bremsstrahlung flux per energy step using a Kramer's law model. No account is taken of the anisotropy of radiation. The continuum window is at an energy E—bkg} constant window=0.01; {width of continuum energy window in keV} var bkg_yield:real; del_E:real; begin position: =round(50*z/thick) ; {divide thickness into 50 steps} {now compute the continuum signal} if energy>E_bkg then {the continuum window can be excited} begin del_E: =stepsize*stop_pwr(energy)*density*lE-8; {energ y lost along step}

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bkg_yield:=2 .8E-9* (at_num/E_bkg) *del_E*window; {Kramer's law} end else bkg_yield:= 0; if ( (position>0) an d (position < = 50) ) the n {we are within bounds of array} bkg_ro_z [position] : =bkg_ro_z [position] +bkg_yield; end;

The routine computes the Kramer's law contribution, assuming that the energy window used for measuring the Bremsstrahlung is 10 eV (0.01 keV) wide. The rate of continuum production is proportional to the electron stopping power, so the energy lost along this step of the trajectory must be determined. Note that this is the flux emitted into 4ir steradians. The angular distribution of the continuum is not, however, isotropic, but forward peaked about the beam direction through the thin film, so some account must be taken of this. The corresponding characteristic intensity can be computed by a slight modification of the routine given above, so as to give the depth distribution (phi_ro_z). Procedure generate—x-rays(energy:real; stepsize: real) ; {computes the x-ray yield using Eq. (9.3) given the electron energy and the critical excitation energy of the x-ray. Fixed constants can be inserted later if required} var x_ray—yield:real position:integer; begin if energy>E_crit then {x-rays will be produced} begin x_ray_.yield: = In (energy /E_cr it) / (energy*E_crit) ; x_r ay—yield: =x_ray _yield* steps ize; end else x_ray—yield: =0; {assign x-ray production to a given depth} position: =r o u n d ( 5 0 * z / t h i c k) ; {depth counter} if ( (positon> = 0) an d (position < = 50) ) the n {within bounds of array} phi_ro_z [position] : =x_ray_gen [position] +x_jray—yield ; end;

Again, this is the intensity produced into 4ir steradians. Since we want the peak-tobackground intensity ratio at a given energy, E_Bkg for the continuum can be put equal to E_Crit for the characteristic line. Then adding the lines: {this computes the characteristic and continuum signals} generate—x-rays(s_en,step); continuum(s_en,step);

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to the program a t appropriate point s i n the program (i.e. , whe n the coordinates ar e reset, or when the electron i s backscattered or transmitted) calls the two routines. At the end of the simulation, the depth distributions for the characteristic an d continuum signal s mus t b e summe d u p ove r th e whol e thicknes s o f th e specimen . Sinc e both signals are at the same energy, it is a reasonable approximatio n to assume that any x-ra y absorptio n i n the specime n wil l be the sam e i n both cases an d thus will disappear i n th e ratio . Similarly , the soli d angl e subtende d b y th e detecto r i s th e same for both the continuum and characteristic signal s and also disappears from th e result. Finally , th e missin g constant s (suc h a s Avogadro' s number , density , an d atomic weight ) tha t appea r i n th e origina l equation s bu t no t i n th e cod e mus t b e reinserted s o as to give th e prope r numerica l values. {sum the x-ray contributions} for k : = 0 to 5 0 d o begin the_peak_is:=the_peak_is + phi_ro_z[k] ; the_bkg_is : =the_bkg_is + bkg_ro_ z [ k ] ; end; {put back the missing constants} the_peak_is:=the_peak_is*6.5E-20*1.2*6E23* ( d e n s i t y / at jrfht) *lE-8 ; the _bkg_is:= the_bkg_i s *1E4 ; {so the computed peak-to-background ratio is} p2b_ratio : =the_peak_is/the_bkg_is;

The program "P_to_B" on the disk implements all of this code. Choosing chromiu m as a specimen, the predictions of the program can be tested against published data values (William s an d Steel , 1987 ) fo r th e "Fiori " number—the rati o o f th e inte grated intensit y i n th e C r Ko t lin e t o th e intensit y in a 10-eV-wid e windo w o f th e continuum a t the sam e energy . A t 10 0 keV, the progra m predict s a value between 1300 and 1200, fallin g slightly as the film thickness is increased fro m 250 A to 5000 A. Figur e 9. 5 plot s ho w thi s peak-to-backgroun d (P/B ) rati o varie s wit h inciden t beam energy for a 1000-A-thick film of chromium. At 1 0 keV, the ratio is only about 250, bu t thi s valu e rises rapidl y a s th e energ y i s increase d (an d plura l scatterin g becomes les s pronounced in the sample). Note that the P/B ratio continues to rise for energies abov e 10 0 keV, but only relatively slowly . These value s are in fairly goo d agreement with , althoug h somewha t lower than , both publishe d experimenta l dat a and othe r theoretica l estimate s (e.g. , Fior i e t al., 1982) . Thi s leve l of agreemen t i s only a s goo d a s migh t b e expected , sinc e th e mode l i s no t accountin g fo r th e forward-directed anisotrop y o f the continuum and s o is probably overestimating it s intensity i n th e directio n o f th e detector . Th e simpl e Beth e cros s sectio n fo r th e characteristic signa l i s als o o f uncertain accurac y a t 10 0 keV an d above . To obtai n a mor e accurat e result , i t i s necessar y t o us e a continuu m cros s section, whic h i s no t onl y mor e accurat e bu t whic h explicitl y account s fo r th e

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Figure 9.5. Predicte d P/ B rati o using Kramer's law model.

anisotropy. The most convenient approach is to employ the numerical evaluation and fit of Sommerfeld's Bremsstrahlung cross-section data (1931) made by Kirkpatrick and Wiedmann (1945). The procedure given below uses is a modification by Blackson (private communication) of the routine given by Statham (1976), with the relativistic corrections of Scheer and Zeitter (1955) and Zaluzec (1978). The original, rather cryptic notation of Kirkpatrick and Wiedmann has been retained so as to help in identifying the purpose of the various steps in the code. Procedure continuum (energy:real;stepsize:real); {calculates a relativistically corrected continuum intensity for a 10-eV window at E^bkff using the Kirkpatrick and Wiedmann evaluation of Sommerfeld's cross section. The result is in continuum counts per unit energy width per electron. This version adapted from an original of J. Blackson} var

position:integer; Joe,yl,y2,y3,h,j, m,a,b,c,d,ioverrreal; la,Qr,Qs,EC,EO:real; begin position: =round (50*z/thick); if energy>E_bkg then {the continuum window can be excited} Ec:=E_bkg*1000; {convert from keV to eV} Eo: =energy*1000; {ditto} beta: =1+ (energy/511) ;

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beta: =17 (beta*beta ) ; joe: =Eo/(300*at_num*at_num) , - {K-W scaling {now evaluate the terms in the K-W numerical fit} yl: = exp ( - 2 6 . 9* j o e) , yl:=0.22*(l-(0.39*yl)); y2: 0.06 7 + ( 0 . 0 2 3 / ( j o e + 0 . 7 5 ) ) ; y3:= 0.0025 9 + ( 0 . 0 0 7 7 6 / ( j o e + 0 . 1 1 6 ) ) ;

factor}

h : = ( ( = 0 . 2 1 4 * y l ) + (1.21*y2 ) -y3 ) / ( ( 1 . 4 3 * y l) ( 2 . 4 3 * y 2 ) +y3) ; j : = ( ( l + ( 2 * h ) ) * y 2) - ( (2* (1+h) ) *ye) ; m: = ( 1 + h) * (y3 + j) ; b:=exp(-0.0828*joe) - exp(84.9*joe) ; a : = e x p ( - 0 . 2 2 3 * j o e ) — e x p ( - 5 7 * j o e) ; c:= 1.47* b - 0 . 5 0 7 * a - 0 . 8 3 3 ; d:= 1.7* b - 1.09* a - 0 . 6 2 7 ; lover:=Ec/Eo; {get the components of the continuum in X and Y directions} Ix:=c*(iover-0.135); Ia:=d* ( i o v e r - 0 . 1 3 5 ) * ( i o v e r - 0 . 1 3 5 ); Ix:=(0.252+Ix)-la; Ix: = ( 1 / j o e) *Ix*1.51E-27 ; {X-component} I y : = lover + h ; Iy: = (1/joe) * (-j+ (m/Iy) ) *1.51E-27; {Y-component] {correct for the line of sight of detector to anisotropic Bremsstrahlung} den:=l - beta*cos (TOA) ; den:=den*den*den*den; Qz : =cos(TOA); Qz:=Qz*Qz; Qr:=l-Qz; Qs:=Qz/den; Q z : = (Ix * ( Q r / d e n) ) + (ly* (1+Qs) ) ; {so the net contribution is} bkg_yield:=stepsize*Qz; end {of the if statement} else bkg_yield:= 0; {add this contribution to the histogram} if ( (position> = 0) and(position< = 50) ) then {within array bounds} bkg_ro_z[position] : =bkg-ro_z[position] + bkg —yield; end;

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In orde r t o properl y account for th e anisotrop y o f th e continuum , th e takeoff angle o f the detector must be known. In standar d microanalysis nomenclature , th e takeoff angl e (TOA ) is the angl e between th e line joining the detector t o the beam point and the surface of the specimen. In order to match the Kirkpatrick-Wiedmann notation, this angl e mus t be referenced t o the incident bea m direction . Tha t is , {code to input and properly compute the takeoff angle} GoToXY(40,9); write('Detector take-off angle (deg)') ; readln(TOA); [for compatibility with W-K notation} TOA: = (TOA + 90)757.4; {in radians}

Finally, a s before , th e continuu m an d characteristi c contribution s mus t b e summed, ove r depth, an d the missing constants mus t be reinserted : {sum over depth} for k:=0 to 50 do begin the—peak—is :=the—peak—is + phi_ro_z[k]; the_bkg_is : =the_bkg_is + bkg_ro_z[k] ; end; {put in the missing constants for both terms} the_peak_is : =100*the_peak_is*6 . 02E23*6 . 51E-20* (density/at _wht)*lE-8; the_bkg_is : =100*the_bkg_is*6 . 023E23*lE-8* (density/at _wht)*10*4*3.14159;

Using condition s identica l wit h thos e trie d abov e an d choosin g a detecto r takeoff angl e of 30°, this model predict s a Fiori number at 10 0 keV of about 2400 , again falling by about 5% as the film thicknes s is varied over the range 250 to 5000 A. This number is in excellent agreement with current experimental data. Figure 9.6 show how the peak to local background for a 1000-A chromium film (Fiori number) varies with incident beam energy. The trend is the same that found with the previous model, a rapid rise wit h energy up to 10 0 keV and then a slow increase. Figur e 9. 7 show how the Fiori number at 10 0 keV is changed b y the choice of detector takeof f angle. A s TO A i s varie d fro m 0 (i.e. , lookin g a t glancin g incidenc e acros s th e specimen surface ) t o 60° , th e P/ B rati o increase s b y abou t 50 % becaus e o f th e forward pea k of the continuum, confirming the usual preference fo r a high-takeoff angle detector. Further sophisticatio n i n both th e continuum and characteristic cros s section s could b e use d t o ge t stil l bette r dat a (e.g. , Gra y e t al. , 1983) , i n particula r a t th e highest energies, where relativistic effect s ar e significant. However, for most typical

Figure 9.6. Compute d P/ B rati o vs . energy, using Wiedman-Kirkpatrick model .

Figure 9.7. Compute d P/ B variatio n wit h detector takeoff angle . 190

X-RAY PRODUCTIO N AN D MICROANALYSI S 19

1

operating conditions , the models give n here ar e accurate enough to provide a good guide to what can be expected experimentally .

9.5 X-ray production in bulk samples Extending the models discussed above to the case of a bulk specimen is straightforward, sinc e exactly th e sam e cod e ca n be used . As usual, however, th e spee d o f a bulk simulatio n usin g a single scatterin g mode l ma y be unacceptabl y slow , so the plural scatterin g approac h ma y b e o f mor e practica l use . The foundatio n o f al l quantitative method s fo r bul k microanalysi s i s th e dept h dependenc e o f th e x-ray production, a function usuall y called (|>(pz) , following Castaing (1960), who defined 4>(pz) a s th e x-ra y generation i n a slic e 8(pz ) of th e specimen , o f density p and a t some depth z beneath the surface, normalized by the corresponding x-ray production from a n identical freestandin g slice. Whe n presented i n this way , the 4>(pz ) curves are foun d t o be, t o a good approximation, of a simple generi c an d analytica l for m that is sensibl y dependen t o n the atomi c number o f the elements involved . The computatio n o f 4>(pz ) curve s wa s on e o f th e firs t task s t o whic h Mont e Carlo method s wer e applie d (Bishop , 1966) , an d muc h o f th e developmen t o f quantitation method s ha s relie d o n suc h computation s (e.g. , Curgenve n and Dun cumb, 1971 ; Duncumb, 1992) . A calculation i n th e form specifie d by th e origina l Castaing definitio n is possible, bu t the majorit y of computations have instead normalized the data by the value of c|>(pz) at the surface of the specimen. There is than a constant rati o between th e valu e that woul d be derive d fro m Castaing' s definitio n and the computed (Kpz), this ratio being the quantity 4>(0), which is of the order of 1 + T I wher e T | i s th e backscatterin g coefficien t o f th e bul k sampl e (Merlet , 1992) . The progra m PhiRo Z compute s an d display s (pz ) curv e usin g th e procedur e Generate_x_rays discusse d above and our usual plural scatterin g Monte Carl o model. Since the Bethe range of the incident beam is already divided into 50 steps in this model, these step s are used as the intervals for the 4>(pz ) histogram. A s before, x-ray generatio n i s assigned to the midpoint of the trajectory step. Onc e the energy of incident electrons has fallen below the critical excitation energ y Ecrit for the x-ray line of interest, th e trajectory calculation ca n be aborted without error to save time. However, for illustrative purposes it is sometimes bette r to continue the computation but to identify portion s of the trajectory that are above and below £ crit—for exam ple, b y color-codin g th e tw o portion s o f trajectory . I n thi s wa y th e relationshi p between th e x-ray generation volum e and the beam interaction volum e can readily be seen. In the PHIROZ program on the disk, the approach use d by Curgenven and Duncumb (1971) i n their pioneer wor k is followed. A random weighting algorithm varying with the x-ray ionization cross section determine s whethe r or not a dot will be plotte d a t the midpoin t o f each trajector y ste p fo r whic h th e electro n energ y i s above Eciit. This builds up a display in which dot density i s proportional t o ionization probability. Figure 9.8 shows a typical output from th e program for the genera -

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Figure 9.8. Compute d ()>( p z ) curve fo r F e K a t 2 0 keV .

tion o f F e K a x-ray s at 2 0 keV . 4>(pz ) i s plotted a s a horizontal histogra m on th e same scal e a s the trajectories, s o that the relationship between them can readily b e seen. A s the depth beneath th e surfac e i s increased, c|>(pz ) rises becaus e the ioniza tion cross section [Eq . (9.1)] continues to rise as E falls until E = = 2.5 X £crit. Figure 9.9 shows some typical 4>(pz) curves computed from thi s program for the excitatio n

Figure 9.9. cj>( p z ) curves fo r F e a t variou s energies.

X-RAY PRODUCTIO N AN D MICR O ANALYSIS 19

3

Figure 9.10. Compute d curve s for F e a t 20 keV, S i a t 5 keV , givin g sam e overvoltage .

of iron Ka at 10 , 15, 20, 30, and 50 keV. The systemati c change in the 4>(pz ) curves is evident. As the accelerating voltage is increased the peak magnitude of the profile rises, and moves deeper into the sample but the general shape of the profile remains the same. This is further illustrate d in Fig. 9.10, which shows ()>(pz) profiles for iron Ka a t 20 keV and silicon K at 5 keV, which in both cases represents an overvoltage U = E/Ecrit o f about 3. Note that the two curves are very similar in shape and size, even thoug h the energ y o f th e x-ray s an d th e respectiv e electro n range s diffe r b y nearly a n order o f magnitude, demonstratin g th e valu e o f th e <Xpz ) approac h a s a way o f simplifying th e representation o f x-ray generation. A complete program for quantitative microanalysi s ca n b e constructe d fro m thi s Mont e Carl o approac h (Duncumb, 1992) . Experimental 4>(pz ) profiles are generated b y monitoring th e x-ra y productio n from a thi n fil m o f a "tracer " elemen t embedde d a t differen t depth s withi n th e material of interest (Castaing, 1960), since only in this way can be the specific depth dependence o f generation b e separate d fro m th e total emission . However , because the x-ray line that is being studie d in the tracer is not excited at the same energy as the desired line from th e material, the form of the cf>(pz ) profile will not be identica l with tha t expecte d fro m th e elemen t itself . B y usin g th e PHIRO Z program , thi s problem ca n also b e studied . Figure 9.11 shows two c|>(pz ) profiles, one computed for th e Ka lin e o f a Mn tracer i n iron an d the othe r fo r iro n K a i n iron . Th e Mn profile peaks slightly sooner an d slightly higher than the Fe curve, because the Mn line i s of lower energy . A similar simulatio n for a tracer o f higher energ y tha n the

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Figure 9.11. Compute d cj>( p z ) curves fo r iro n a t 1 0 keV with M n an d F e tracers .

iron would show the opposite situation . Thus, while the tracer method gives a good experimental approximatio n to th e tru e 4>(pz ) profile i t i s not exact . By substitutin g one o f th e continuu m procedure s give n above , th e corre sponding dept h variatio n o f the Bremsstrahlung generatio n ca n als o be studied . In this case the energy of the continuum window to be modeled, previously se t to Ecrit, can b e varie d t o se e ho w th e dept h dependenc e varie s wit h energy . Figur e 9.1 2

Figure 9.12. Dept h variatio n o f continuu m productio n a t 1 , 5, 10 , 15 , and 1 9 keV.

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195

shows a sequence o f "(P Z)" plots obtained i n this way for an iron sample at 20 keV and for continuu m energies o f 1 , 5, 10 , 15, and 1 9 keV obtaine d fro m th e progra m BREMS o n the disk , whic h implements thi s approach . Al l o f the continuum windows are computed simultaneously from th e same set of trajectories, sinc e they are independent events . Note that the shape of the 4>(pz) profile is different fro m that for characteristic radiation s because of the different functiona l for m of the cross section , and this is true even when the energy of the continuum window is the sam e as the energy of the characteristic line . Thus methods that rely on the ratio of the peak (i.e., characteristic line ) to the background (i.e., continuum) at the same energy to correct for x-ra y absorption i n the sampl e (Statha m an d Pawley, 1978 ) will not generall y produce a satisfactory result (Rez and Kanopka, 1984) unless corrected (L u and Joy, 1992) t o accoun t fo r this discrepancy. Bot h Kramer's la w an d the Kirkpatrick and Wiedmann model s fo r continuu m productio n yiel d ver y simila r result s fo r bul k samples because the multiple scattering of the beam averages out the polarization of the continuu m signal. Eithe r ca n thus be used a s desired . The mode l ca n b e furthe r develope d b y computin g th e shap e o f th e Bremsstrahlung a s see n b y a n energ y dispersiv e spectromete r (EDS ) detecto r place d outside of the sample. This requires that the continuum be simulated for a number of energy windows between zero and the incident beam energy; here, 1 9 such regions equally space d ar e used . Eac h o f thes e result s mus t the n b e correcte d fo r x-ra y absorption both in the specimen and in the EDS detector window. The absorption of x-ra from th e point o f production in the for m /(*) = /(O) . expt-^p*] (9.7

)

where p is the densit y an d u , is the mass-absorptio n coefficient. | x depends bot h on the material an d on the energy o f the x-ray passing through it. Tables of values are available (e.g., Goldstein et al., 1992), but since numerical fits through the data have been mad e by Heinrich, a simple routin e can be used to compute the | x values that are needed wit h good accuracy . Procedure abs_corr(photon—energy:real):real; {calculates the mass absorption coefficient using Heinrich's method—the photon energy is in keV and the result is in cm" 2/g} var A,B,C,D,F,dum,dum2,dum3:real;

begin if photon_energy<E_cri t the n [below K-absorption begin A:=1.0; B:=-0.2544711;

edge}

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MONTE CARLO MODELING

C:=4.769245; D:=-10.37878; F:=2.73 end else {w e ar e above the begin

K-edge]

A:=1.0; B:=-0.2322294; C:=4.070053; D:=-6.220746; F: =e x p ( - 0 . 0 0 4 5 5 2 2 * l n ( a t _ n u m ) * l n ( a t _ n u m ) - 0 . 0 0 6 8 5 3 5 * I n ( at _num)+1.070181); end; {compute the value using appropriate constants] dum: =B*ln) at_num) *ln (at_num) +C*l n (at_num) +D ; dum2:=12.398/photon—energy; dum3 : =p o w e r ( d u m 2 , F ); abs_corr:=A*dum3*exp(dum) ; end;

If the TOA of the EDS detector is specified, then for an x-ray generated at depth z beneath the surface, the exit path length is z/sin (TOA). At the end of the simulation, therefore, the continuum "4>(pz)" data is corrected in a stepwise fashion for each of the 19 continuum windows: for i:=l to 19 do begin {by getting energy of the i-th continuum window} photon—energy:=i*inc_energy 720; {get the mass absorption coefficient at this energy} factor: =abs_corr(photon—energy)/sin(TOA/57.4); for j:=0 to 50 do {sum over all layers down from surface} begin {compute exit path length X density X MAC} factor2:=factor*density*j*step*lE-4; factors:=exp(-factor2 ); {Beer's law} {now compute what would be observed outside sample} bremss[i] : =bremss[i] +phi_ro_z [j,i]* factor3 ; end; {j'-loop} end; {i-loop}

Finally, the effect of the detector on the spectrum must be accounted for. Over most of the energy range, the dominant effect is absorption in the beryllium window in front of the Si(Li) diode, so this can be found by using the same routine as before, but replacing the physical constants of the sample with those of the beryllium window:

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Function det_factor(wind_z,wind—thick,wind_density,energy:real}:real ; {com atomic number and density} var f a c t o r , f a c t o r 2 , f a c t o r s :real ; begin at_num: =wind_z , - [for abs—corr routine} factor: =abs_corr(energy) ; {density*path length through window*MAC} factor2: = factor*wind_density*wind_thick*lE-4 ; factors : =e x p ( - f a c t o r 2) ; [Beer's law} det_factor: =factor3; end;

thus we have for i : = l t o 1 9 d o {the 19 continuum windows} begin photon_energy: =i * i n c _ e n e r g y 7 2 0 . 0; {putting in constants for a Be window 6 jju n thick, Z = 4, p = d e t _ e f f i e [ i ] : =d e t _ f a c t o r ( 4 , 6 , 1 . 2 , p h o t o n _ e n e r g y ); {now correct data for this additional absorption} bremss [i] := brems s [i] *det_ef f ic [i] ; end;

1.2}

The final step is then to interpolate data points linearly between the calculated values so as to simulate the whole spectrum. Figure 9.13 shows the result of a calculation for the Bremsstrahlung spectrum of carbon at 20 keV. The dots show the corresponding experimental data, scaled to match at the peak of the continuum. As can be seen, the agreement between the normalized and experimental spectra is excellent, even though the model is simple; although the ratio of absolute predicted and measured intensities is less good. The Bremsstrahlung spectrum is not often considered as being worthy of examination and is usually removed mathematically before any analysis of the characteristic peaks is made. However, with the aid of a model such as we have developed here, the behavior of the continuum can conveniently be studied, and we find that this component of the spectrum can provide much useful information, especially because the Bremsstrahlung covers a wide range of energies. First, the area beneath the continuum after correction for x-ray absorption varies linearly with, the atomic number of the target, so providing a simple check on any qualitative analysis that is performed. Second, the shape of the continuum depends on the absorption experienced by the photons as they travel to the surface. Thus the shape depends on the general shape of the surface—i.e., its topography. This is important in analyzing particles, samples containing edges, or samples with a large degree of surface topography, since the form of the continuum

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Figure 9.13. Compariso n o f experimenta l an d compute d Bremsstrahlung fo r coppe r a t 2 0 keV, 8-|j, m B e window .

can be used both to understand what is happening i n the sample an d then to correct for this . For example, the position of the peak in the Bremsstrahlung depends on the length of the aborption path to the surface. If the effective radiu s of curvature of the surface around the beam entrance point is reduced from infinit y (i.e. , a flat surface ) down to values of a few micrometers, then the peak shifts smoothly . Thus a measure of the surfac e conformation can be obtained from a n observation of the continuum. Similarly, the presence of charging fields in the specimen will systematically distor t the for m o f the Bremsstrahlung profile. The techniques discussed above can readily be extended t o deal with any other geometrical condition . For example, the important case of performing x-ray analysis of a thin film o n a substrate can be modeled by using the procedures give n in Chap. 6 t o comput e th e 4>(P Z) profile s i n th e tw o material s an d the n accountin g fo r th e x-ray absorptio n o f th e substrat e radiatio n i n th e fil m b y usin g th e routin e give n earlier i n this chapter. In general, it is sufficient t o employ a plural scattering routine to comput e th e trajectories, bu t i n an y cas e wher e a region o f interest i s les s tha n about 5 % o f th e electro n range , i t woul d b e bette r t o us e th e alternativ e singl e scattering mode l t o ensure better accuracy .

10

WHAT NEXT IN MONTE CARLO SIMULATIONS? 10.1 Improving the Monte Carlo model The state d ai m o f thi s boo k wa s t o develo p Mont e Carl o simulation s tha t wer e accurate enough to be useful predictive tools but simple enough to be accessible to nonspecialists. Th e previou s chapter s o f th e boo k hav e demonstrate d that , within these limits, a great deal can be done. However, in meeting these goals, approximations an d simplification s hav e ha d t o b e made . Specifically , w e hav e mad e th e assumptions that only elasti c scattering , a s described b y the Rutherfor d cros s sec tion, need be considered an d that energy loss can be treated a s a continuous rather than as a discrete process. Such assumptions are not essential to the construction of a Monte Carl o simulatio n o f electro n scattering , an d s o th e obviou s an d desirabl e move i s t o remov e thes e approximation s fo r thos e case s wher e the y represen t a n unacceptable restriction . An important first ste p is to use the more accurate description of elastic scattering provide d b y th e Mott , o r partia l wav e expansion , cros s section . A s note d i n Chap. 3 , the Rutherfor d cross sectio n work s wel l fo r th e scatterin g o f medium to high-energy electrons by nuclei of low atomic number. However under some conditions, particularly at low beam energies (E < 2 0 keV) and for high-atomic-number elements ( Z > 30) , where the scattering angles ar e large, the Rutherford formula is only an approximation. When the cross section is derived from th e relativistic Dirac equation, takin g int o accoun t spin-orbi t coupling , then th e "Mott" scattering cros s section whic h result s ca n diffe r significantl y fro m th e Rutherfor d values . Suc h discrepancies becom e important, and experimentally detectable, in cases where each electron i s scattered onl y a few times and where the angular distributions of transmitted o r scattere d electron s o f variou s energie s ar e require d t o b e know n accu rately. The reaso n wh y th e Mot t cros s section , despit e it s superiority , ha s no t bee n more widel y use d fo r Mont e Carl o simulation s i s tha t i t i s no t a n analytica l function—i.e., on e that can be evaluated fro m a n equation—but a n array of differ ential cross-sectio n value s representing th e scatterin g probability for electrons o f a particular energ y i n a specifie d direction . Typicall y (Czyzewsk i e t al. , 1990 ) th e Mott cross section must be evaluated at energy steps of 10 0 eV or so from 2 0 eV to 20 keV and for angular increments o f 1 or 2° from 0 to 180° , giving a total of some 199

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2500 sets of numbers for each element. Storing the data for the entire periodic table therefore require s severa l megabytes o f storag e spac e o n disk . Angula r scatterin g probabilities an d total cross section s must then be found b y numerical integrations of thes e dat a arrays. Whil e thes e operation s ar e not intrinsicall y difficult, the y are messy an d ten d t o mak e th e program s tha t emplo y the m loo k rathe r comple x compared t o the normal ru n o f personal compute r software. One approach that has been successfully implemente d to obtain the benefits of the Mott cross section without the complexity is to reduce it to an analytical form by a proces s o f parametrization . Brownin g (1992) showe d tha t th e tota l elasti c Mot t cross sectio n CTM could be writte n as

where u = lo glo(8.£'.Z-i-33) Equation (10.1) is a good description o f the overall trends of the cross section , although i t canno t follo w th e periodicit y o f th e Mot t cros s sections , whic h ar e directly du e t o periodic variation s of the siz e of the atoms . Suc h effects hav e littl e macroscopic effec t o n the scattering o f electrons by a solid, however. This parame trized cross section can be substituted directly for the usual Rutherford cross section in th e singl e scatterin g program . Sinc e th e Mot t cros s sectio n i s typicall y smalle r than the corresponding Rutherfor d value, the elastic mea n free path s that are computed will be larger and the number of scattering event s that must be computed will therefore generall y b e smaller , leadin g t o a useful improvemen t in program speed . It i s als o necessar y fo r th e compute d angular scatterin g distributio n t o b e modified t o match the results of the Mott cross section . A s shown in Fig. 10.1, the cumulative angular scattering distributio n for the Mott model is rather more squar e in for m tha n th e Rutherfor d distribution . Thi s behavio r ca n b e approximate d b y modifying th e form o f the Rutherfor d equation, e.g. , Equation (3.10):

to be

c Parametrized model s are a useful ste p up from th e simplest Rutherford models and should , wit h furthe r work , b e capabl e o f givin g good-qualit y data . I t must , however, be realized tha t such an approach is only an approximation and that under

WHAT NEX T I N MONT E CARL O SIMULATIONS ?

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Figure 10.1. Compariso n o f cumulative scattering probabilities in Mott and screened Rutherford models .

extreme conditions—suc h a s ver y lo w voltage s (< 1 keV ) an d a t hig h scatterin g angles—the approximatio n wil l no t b e accurate . Whe n suc h condition s mus t b e explored, i t i s therefor e necessar y t o g o t o a ful l Mot t model , usin g numerically tabulated differential cross-sectio n data . Man y such programs have been described (e.g., Reimer and Stelter, 1987 ; Kotera et al., 1990 ; Czyzewsk i and Joy, 1989), and tabulated sets of Mott cross sections have been published by several groups (Reimer and Lodding , 1984 ; Czyzewsk i e t al. , 1990) . Suc h program s hav e bee n demon strated to give excellent results for both macroscopic events , such as the variation of backscattering with incident energy, an d microscopic data , such as the angular and energy distribution s o f scattere d an d transmitte d electron s (Reime r an d Krefting , 1976). With the continued increas e i n the memory capacity of personal computers , Mott model s wil l certainl y becom e mor e commonl y applie d becaus e o f th e un doubted improvements that they can provide in some circumstances. However, care must b e take n tha t al l aspect s o f th e Mont e Carl o progra m tha t i s develope d ar e equally advanced. Some published programs that have gone to great lengths to use the Mott cross section have compromised th e quality of their results by using poor electron stopping-powe r formulations or by terminating the simulation at relatively high energies an d relying o n diffusio n model s t o account fo r the rest of the trajec tory. A second important area of research activity has been to construct Monte Carlo simulations in which all types of scattering, elastic and inelastic, are considered and in whic h energ y losses ar e treate d a s discret e rather than continuous event s (e.g. ,

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Shimizu an d Ichimura , 1981) . Suc h model s considerabl y exten d th e boundarie s within whic h Mont e Carl o modelin g ca n b e considere d useful , althoug h a t th e expense of increased complexit y and computation time. The procedure followed is a generalization o f tha t discusse d i n Chap . 7 fo r th e modelin g o f fas t secondar y electrons. I n additio n t o the usual elastic mea n free pat h A E, a total inelasti c mea n free pat h A } is defined , wher e

and th e A J ( y = 1 , 2, etc. ) ar e th e mea n fre e path s for eac h o f th e inelasti c event s considered (Fittin g an d Reinhart , 1985) . Th e tota l mea n fre e pat h i s A- p wher e

and fo r eac h scatterin g even t a rando m numbe r RN D i s compare d wit h th e rati o AT/AE to determine i f the event is elastic or inelastic; then a second rando m number is draw n to decid e wha t type o f inelasti c even t is occurring . Th e usual Bethe la w expression givin g the electron stoppin g power is replaced by the sum of the energy lost in the inelastic event s 2 (A£j) encountered . Sinc e th e mixture of events along each trajector y will be different, th e energy losse s wil l correspondingly vary, leading to range straggling-—th e phenomenon o f a statistical variatio n in electron rang e for a fixed inciden t energy—jus t a s is observed experimentally . A program o f this type is needed i f information about the energy an d angular distribution of electron s is sought , fo r example , to mode l Auge r or energ y los s spectr a (Shimiz u and Ichi mura, 1981) .

10.2 Faster Monte Carlo modeling For many people th e problem wit h Monte Carl o simulation s has been an d remains that it is a sequential procedure. Onc e the model has been constructed, it must be run a large numbe r of times on e after another i n order t o achieve the desired statistica l accuracy befor e th e dat a ca n b e extracted . Althoug h th e spee d o f al l computer s continues t o increase , th e necessit y o f runnin g 500 0 o r mor e trajectorie s pe r dat a point can still represent a major investment in time when perhaps 5 0 or more points must be modeled o r when a large number of different set s of experimental parame ters mus t be compared. Simpl y increasin g compute r speeds stil l furthe r i s a brute force approach that can offer onl y a limited degree of improvement, especially sinc e the ver y fastes t machine s ten d t o b e multiuse r computer s tha t allocat e t o eac h program only some fraction of the available processo r time . For example, whe n the time required for data input and output on a typical time-sharing system is included,

WHAT NEX T I N MONT E CARLO SIMULATIONS ? 20

3

a Cray supercompute r i s less than twic e a s fast i n running th e sam e cod e a s a 33Mhz 486-typ e MS-DO S compute r equippe d wit h a mat h coprocesso r chip . On e solution to this dilemma is to run more than one trajectory at a time—that is, to use parallel computin g techniques . Mont e Carl o method s ar e wel l suite d t o thi s ap proach because th e code is relatively compact and each trajectory is independent of every other . I t i s therefor e straightforwar d to achiev e a larg e gai n i n throughput simply by running one copy o f the program on each of the processors i n a parallel machine. Severa l example s of this have been reporte d (e.g. , Michaels e t al., 1993 ) on machines employin g u p to 12 8 processors. Impressiv e results , combining both good statistics an d many data points, have been demonstrated for situations such as the modeling o f x-ray generation across a complex interface an d for the production of S E imag e simulations . Th e actua l improvemen t i n speed , whil e dramatic , i s usually somewha t les s tha n th e numbe r o f CPU s use d i n parallel , becaus e th e number of computations required for each trajectory is not a constant. One processor may therefore finish all of its allotted simulations while another is still working, and unless th e compute r ca n reorganiz e it s workloa d t o reoccup y al l o f th e availabl e processors, the speed advantage gradually decreases a s more and more CPUs finish their task s an d fal l idle . Nevertheless , wit h computer s tha t hav e mor e an d mor e parallel architecture becoming available, this will be an important avenue of progress in thi s field .

10.3 Alternatives to sequential Monte Carlo modeling A typica l us e o f th e kin d o f Mont e Carl o model s develope d i n thi s boo k i s t o investigate the effect of varying one or more parameters in some experimental setup; for example , seein g ho w changin g th e position of a detector affect s th e for m o f a backscattered electro n signa l profil e o r ho w alterin g th e morpholog y o f a surfac e might vary the x-ray yield. In such cases, even though almost al l of the condition s remain th e same , i t i s stil l necessar y t o reru n th e entir e simulatio n ove r agai n in order t o get the new data, whic h is a wasteful an d time-consuming procedure . A n interesting alternativ e procedure has been described (Desai and Reimer, 1990 ; Wang and Joy , 1991 ; Czyzewsk i an d Joy , 1992) , whic h is a mixture o f a conventional Monte Carl o mode l an d a n alternative physica l representatio n calle d a "diffusio n matrix." Consider th e proble m o f computin g the backseatterin g coefficien t o f a flat , semi-infinite target. Suppose this specimen to be divided up into a larger number of cubes, each of which has the property of emitting electrons at a certain rate and with some know n angula r distribution . Th e measure d yiel d o f backscattere d electron s will resul t fro m th e emission s comin g fro m thos e cube s wit h one face o n the to p surface of the specimen. If the topography, for example, i s now changed—perhaps by placing a groove across the surface—then som e additional cubes will have faces at th e surfac e an d the numbe r o f emitted electron s wil l change .

204 MONT

E CARL O MODELIN G

Figure 10.2. Layou t o f matrix . Th e Mont e Carl o simulatio n find s ho w man y BS E wit h a specified exi t angle pas s throug h a strip of width AS at angle fl t o the horizontal i n each cell.

The strength of each elemental cuboid emitter of electrons i s found b y running a Monte Carlo simulation for the chosen experimental conditions. However, unlike a conventional simulation , th e inciden t bea m i s considere d t o b e emergin g fro m a point insid e th e specime n (Fig . 10.2 ) rathe r tha n impingin g a t a surface . Ever y trajectory i s followe d unti l th e electro n completel y dissipate s it s energy , becaus e there i s now no surfac e fro m whic h to be backscattered. Th e interaction volum e is again divide d int o cubes , bot h abov e an d belo w th e bea m point , an d th e Mont e Carlo is configured so that for the cube X, Y, Z relative to the beam at coordinates (0 , 0, 0), the number and energy of electrons crossin g th e face s o f the cube a t various angles relative to the axes is determined. Th e resultant multidimensional array then completely specifie s th e matri x o f cubes . (Th e nam e diffusion matrix refers t o a n analogous model i n whic h a point diffusio n source—suc h a s a sourc e o f heat—is replaced b y an array of emitters that gives the same resultant properties). Typically the interaction volume is divided into 64 X 64 X 32 cubes, the angular variation into 20 steps, and the energy into 10 steps. As usual, the Monte Carlo simulation must be continued for a sufficient numbe r of trajectories so that each element of the array has adequately goo d statistics . The arra y can be furthe r extende d b y including compo nents representin g th e correspondin g fluxe s o f S E o r x-rays , and som e initia l re search has demonstrate d th e feasibility of considering eve n multicomponent specimens.

WHAT NEX T I N MONT E CARL O SIMULATIONS ? 20

5

Once this has been done, the yield of electrons for any arbitrary surface geometry o r for an y specifie d detecto r positio n relativ e t o the specime n ca n be found b y simply summing the relevant contributions fro m al l the surface cubes. This makes it possible t o almos t instantaneousl y compute , fo r example , th e effec t o f tilt o n th e backscattered yiel d o r th e S E profil e fro m a surfac e o f give n geometry . I n a n implementation o n a Macintos h Ilc i machine , th e tim e t o recomput e a 256-pixe l surface profile for a specified surface geometry was less than 1 sec, so this technique is wel l suite d fo r studie s i n whic h th e compute d profil e i s t o b e interactivel y matched to an experimental profile. For many applications, thi s method has a great deal of promise, since it offers a good balance between speed an d accuracy even on small computers. The diffusion matri x method is, of course, an approximation rather than an exact solution, since it assumes that each cubical element has properties that are fixed an d independent o f its surroundings and position relative to a surface. The method also cannot take account of effects suc h as the recollection, or scattering, of emitted electron s fro m th e sampl e itself , whic h ma y b e significan t i n congeste d geometries. However , i t i s a valuable extensio n o f the standar d Mont e Carl o ap proach and can produce a lot of additional information for little extra computation .

10.4

Conclusions

Simulations will become a n increasingl y important an d effective too l fo r proble m solving i n electro n microscop y an d microanalysis . Wit h furthe r improvement s i n both computers and the software that is run on them, it is not too fanciful t o envisage a "virtual reality" approac h t o electron microscopy i n which fast, interactive , sim ulations ca n b e use d t o tes t alternativ e experimenta l methods , t o investigat e th e effect o f differen t experimenta l parameters , t o optimiz e conditions , an d even produce "images " an d "spectra " withou t th e nee d t o us e expensiv e tim e o n a rea l microscope. Eve n a t a less exalte d level , th e simulation s discusse d i n thi s boo k provide a convenient, flexible , and accurate way of interpreting data and images, a useful resourc e fo r plannin g experiments , an d a satisfyin g teachin g tool .

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INDEX all purpose program , 7 9 angular differentia l Rutherfor d cros s section, 29-3 0 annular detector, 11 2 Archard model , 54-5 5 Auger electron , 3 , 20 2 automatic computin g machines , 5 Avogadro's number , 2 8 axis convention , 10 , 26, 7 2

cascade process , 134 , 138-39 cathodoluminescence, 114 , 132-33 charge collectio n microscopy , 11 4 compiler directives , 46-47 , 71 , 105 compounds atomic numbe r of, 50-51 backscatter yiel d of , 83-84, 106 computer languag e ADA, 1 2 C, 1 2 MODULA-2, 1 2 PASCAL, 12 , 22, 10 5 QUICK PASCAL , 1 3 QUICKBASIC, 1 2 TURBO PASCAL, 12 , 89 computers Macintosh Ilci , 205 main-frame, 6 , 7 parallel processor , 20 3 personal, 7 , 203, 205 conductivity, induced, 116-1 7 coordinate system , 10 , 26 coprocessor chips, 46-4 8 coprocessor, floating-point , 46-47 Cosslet, V . E., 6 Courier typeface , 1 7 csda range , 57 current balance , 16 5 cutoff energy , 34-35, 52-53 , 145-46, 151-52

backscatter yields , 81-86 , 203 discontinuities in , 8 2 variation wit h density, 95-96 variation wit h energy, 85-8 6 variation wit h thickness, 96-9 7 variation wit h tilt , 90-91 backscattered electrons , 4 , 81 energy distributio n of , 91-94, 112-1 3 information depth , 95 , 10 9 line profiles , 171-7 3 mean energ y of , 93-94 beam broadening , 178-7 9 Beer's law, 195 Berger, M., 6 beryllium window , 196 Bethe equation , 26 , 32 modified, 35-36 , 58 Bethe range , 57 , 96, 11 9 steps of , 5 8 BGI files , 2 3 Bishop, H. , 6 bkg_ro_z, 185 Boltzmann transpor t theory , 13 6 Bremsstrahlung, 26, 175-7 7 Buffon, A. , 5 bulk specimen , 56 , 75, 96

debugging, 2 2 defects crystallographic, 125-2 7 EBIC profiles of , 129-3 2 electrically active , 12 5 size of, 12 8 depletion region , 117 , 121-2 2 detector efficiency , 111-12 , 173 , 196-97

calculators, mechanical , 6 carrier pairs , 118-1 9 213

214 detector geometry , 111-12 , 173 diffusion length , 118 , 122-24 , 125-28, 132 diffusion matrix , 203-5 Dirac equation , 19 9 direction cosines , 10 , 26, 31-32, 72, 111-12 drunken walk , 9-12 Duane-Hunt limit, 17 6 Duncumb, P., 6 EBIC, 11 4 EDSAC I I computer, 6 electron lithography , 156-5 7 electrons Auger, 3 , 20 2 backscattered. See backscattere d elec trons energy loss , 2 6 energy los s spectra , 20 2 secondary. See secondar y electron s stopping power, 26 , 32 , 84 tertiary, 15 2 transmitted, 4 valence, 3 electron-hole pair , 114-15 , 121 energy los s continuous model, 26 , 32-36 parabolic model , 34-3 5 energy profile , 58 , 7 3 error (erf ) function , 17 9 E2 energy, 164-6 5 Evans model , 14 2 fast secondar y electrons , 142-4 7 energy of , 15 1 energy depositio n of , 155-5 6 production o f x-rays, 181-8 3 spatial distributio n of , 153-5 5 yield of , 15 4 Fiori number, 186-87 , 18 9 FORTRAN, 6 function generation, 118-2 1 Gryzinski's, 13 7 functions, 19 , 10 5 abs_corr,195 Archard_range,54-55 delay, 21

INDEX

GasDev, 169 GetAspectRatio, 20 GetMaxX, 20 , 48, 73 GetMaxY, 20 , 48 keypressed, 22 lambda, 38 , 47 outside, 108 POWER, 37 , 47 , 62 , 71 , 9 8 stop_pwr, 38 , 47, 62, 71 xyplot, 40, 48 YES, 38 , 47-48 Gaussian distribution , 169 granularity, 77-7 8 graphics drive r routines, 19 , 23 h_scale, 73 holes, 114-1 5 hplot_scale, 41 ICRU tables, 33 impact parameter , 58 , 82-83 inelastic cross section , 14 6 information, dept h of , 107 , 10 9 inhomogeneous materials, 97-10 5 interaction volumes , 52-53, 74-75, 96, 120-21 teardrop shap e of, 96 , 12 0 ionization, 3 energy, critical , 17 4 mean potential of, 32-33, 10 1 / (mean ionization potential), 32-33 , 101 jellium, 7 8 Kahn, H. , 4 keywords, 1 7 begin, 19 const, 18 end, 19 function, 18-19 label, 47 procedure,18-19 program, 17 uses, 17 var, 1 7 knock-on collision , 134 , 142-4 3 Kramer's law , 184

INDEX

Manhattan Project , 5 mass-thickness contrast, 155 matrix, diffusion , 204- 5 mean fre e pat h elastic, 27-29, 202 inelastic, 146 , 20 2 total, 146 , 202 mean ionizatio n potential, 32-33, 51 , 60, 101 metrology, 167 , 17 3 Metropolis, N. , 5 micron markers , 4 9 minicomputers, 7 minority carrie r length . See diffusio n length Monte Carl o method definition of , 4- 5 history of , 5- 7 Monte Carl o simulation , 5 applicability of , 79-80 double, 143-4 4 precision of , 23-24 Mott cros s section , 87 , 199-20 1 nuclear interactions , 8 0 nuclear screening , 27 , 59 numerical integration , 126, 168, 20 0 overcut structure , 170 overvoltage, 174-75 , 18 1 PC. See computer , persona l peak-to-background ratios , 184-91 , 195 photographic emulsion , 51 plasmon decay , 137-3 8 plot—scale, 41 plotting scales , 48-4 9 plural scatterin g model, 56 , 75, 77-78, 124 polymethyl methacrylate (PMMA), 15 7 pragmas, 46-47, 71 , 105 printing, 5 0 procedures, 19 , 105 back_scatter, 43, 49, 69, 71, 74, 92

continuum, 184 , 187-8 8 display-backscattering, 69, 71, 79, 92 do_defect, 127 FSEmfp,148-49

215

generate_x-rays, 182 , 185 , 19 1 get—constants,38, 48 init_counters, 67 , 70, 102, 10 3 initialize, 14 , 19, 21, 40, 49, 66, 70,73 new-coord, 14 , 20, 68, 71, 73, 103 p_scatter, 68 , 71, 73, 102 plot_xy, 15 , 20 prof ile, 58, 65, 72, 99, 105 randomize, 20, 70 range, 57 , 64, 70, 99 reset—coordinates, 15 , 21, 41, 67, 71 , 10 2 reset-next-step, 69, 71, 103 Rutherf ord_Factor, 64, 70, 72, 89, 10 0 s_scatter, 42 , 49 sei_sig,159 set_up_graphics, 41-49 , 66, 70, 73 set-up-screen, 39, 48, 72, 63, 70, 100, 10 3 SetViewPort, 21 show_BS_coeff, 44 , 49 show_traj_num, 44 , 50, 69, 71, 74 stop_pwr, 57, 105 straight-through, 43, 49 Test_for_FSE, 146 Track_the_FSE, 148 , 15 2 transmit—electron, 43, 49 xyplot, 40, 67 pseudopotential, lattice, 137 quadrant detector, 11 2 random number generators, 5-6, 20 random numbers , 4, 20 random sampling , 4, 24 random walk, 9-12, 23-24 range equation, 48, 15 2 Runge-Kutta method, 72 Rutherford cros s section. See screened Rutherford cros s section samples, geometry of, 107- 9 scattering elastic, 3 , 25 inelastic, 3 , 25 small-angle, 61 , 88

216

INDEX

scattering angle , 30 , 59-60, 200- 1 azimuthal, 30 , 6 1 minimum, 59-6 0 Schottky barrier, 117 , 121, 124 gain of , 121-2 4 screen dump , 5 0 screened Rutherfor d cross section , 25, 27 , 87, 19 9 angular differential , 29-3 0 total, 27 , 58-59, 14 5 screening parameter , 27-2 8 secondary electrons , 3 , 134 , 142-47 analytical yiel d model, 15 8 angular distribution , 135 , 140-41 attenuation length , 158-6 1 beta factor , 140-42 , 166-6 7 energy distribution , 135 , 139-40 escape depth , 13 4 escape probability , 152 , 167-6 8 fast. See fas t secondar y electron s generation energy , e, 158-6 1 line profile, 171-7 3 plasmon-generated, 135-3 6 production, 7 8 production b y BSE, 135, 166-67 production b y valence excitations , 135 36 recollection, 17 1 variation wit h thickness, 142 , 154 variation wit h tilt, 161 yield coefficient , 134-35 , 158, 161 SE1, 135 , 140-42, 154 SE2, 135 , 140-42 Simpson's rule , 7 2 single-scattering model , 25 , 75, 77-78 spin-orbit coupling , 19 9 statistical scatter , 2 4 stepsize, 182 stopping power , 26 , 32-36, 157 , 202 straggling, range , 20 2 straight-line approximation , 138 , 154 , 157-58 supercomputer, 20 3

takeoff angl e (TOA) , 189 , 196 thermometer scale , 4 9 thin film , 97-98, 15 4 tilt, effect s of , 72, 75, 123-2 4 topography, 203- 4 total electro n yields , 161-6 3 tracer elements , 193-9 4 Ulam, S. , 5 undercut structure , 170 v_scale, 73 variables, 1 7 Boolean, 17 , 22, 47-48, 14 7 extended, 1 7 global, 17 , 15 2 integer, 1 7 local, 17 , 15 2 longinteger, 1 7 private, 47 real, 1 7 virtual reality , 205 voids, 110-1 1 von Neumann , J., 5 window absorption , 196-9 7 x-rays Bethe ionizatio n cros s section , 17 4 Casnati cross section , 17 5 characteristic, 3 , 17 4 continuum, 175-7 7 continuum depth variation , 194-9 5 continuum, effec t o f curvature , 19 8 continuum, integrate d valu e of, 19 7 continuum polarization , 176 , 187 detector windo w effect s on , 196-9 7 effect o f FSE on , 181-8 3 peak-to-background ratio s for , 184-86, 190 PhiRoZ program fo r computing, 191 94 spatial resolutio n of , 177-79 , 18 3