Modeling and Measurements of the Bidirectional Reflectance of Microrough Silicon Surfaces. A Thesis Presented to The Academic Faculty.

Transcription

1 Modeling and Measuements of the Bidiectional Reflectance of Micoough Silicon Sufaces A Thesis Pesented to The Academic Faculty By Qunzhi Zhu In Patial Fulfillment Of the Requiements fo the Degee Docto of Philosophy in the School of Mechanical Engineeing Geogia Institute of Technology July 2004

2 Modeling and Measuements of the Bidiectional Reflectance of Micoough Silicon Sufaces Appoved by: D. Zhuomin Zhang, Adviso D. J. Robet Mahan D. Andei G. Fedoov D. Dennis W. Hess D. Andew F. Peteson Date Appoved July 9, 2004

3 ACKNOWLEDGEMENT Fist, I would like to take this oppotunity to thank my adviso, D. Zhuomin Zhang, fo his guidance, suppot, and encouagement duing this poject. I appeciate his help in ovecoming the cultue diffeence and his advice on achieving a successful caee. I also thank Pofesso J. Robet Mahan, Pofesso Andei G. Fedoov, Pofesso Dennis W. Hess, and Pofesso Andew F. Peteson fo seving on my thesis eading committee and poviding helpful suggestions and suppot on this poject. I gatefully acknowledge the help povided by D. Zhang s eseach goup. I thank D. Yu-Jiun Shen fo the hands-on expeiences in the instumentation, Ds. Donghai Chen and Yihui Zhou fo helpful advice. I appeciate Ceji Fu fo valuable discussion and sincee fiendship. I enjoyed the collaboations with Yu-Bin Chen and Hyunjin Lee. I also thank Keunhan Pak, Bong Jae Lee, Vinh Khuu, and Dan McComick fo eviewing my papes and helping me impove my English. I thank M. Eic Lambes of Depatment of Mateial Science and Engineeing at Univesity of Floida and M. Joel Pikasky of MIRC at Geogia Institute of Technology fo showing me the wondeful wold of the atomic foce micoscopy and M. Al Ogden of Depatment of Electical and Compute Engineeing at Univesity of Floida fo pepaing coatings on some wafes. Finally, I am indebted to my family, especially my paents, who have suppoted and encouaged me though all these yeas. I also thank my bothe and my siste fo thei suppot. This wok has been suppoted by the National Science Foundation. iii

4 TABLE OF CONTENTS Acknowledgment List of Tables List of Figues List of Symbols Summay iii vi vii x xiii Chapte 1 Intoduction 1 Chapte 2 Liteatue Review Silicon Wafe Manufactuing and Rapid Themal Pocessing Roughness Measuement Theoetical Study of the Bidiectional Reflectance Expeimental Study of the Bidiectional Reflectance 27 Chapte 3 Suface Roughness Chaacteization Roughness Paametes and Functions Roughness Measuement and Data Analysis Roughness Statistics of Silicon Wafes 36 Chapte 4 Bidiectional Reflectance Measuements Thee Axis Automated Scatteomete Chaacteization of TAAS 53 Chapte 5 BRDF Modeling and Measuement fo Silicon Wafes Unification of Slope Models Pedicting BRDF Using the 1-D Slope Distibution Pedicting BRDF Using the 2-D Slope Distibution Suface Unifomity and Batch Repeatability Instument Effects and Revese Pocedue Out-of-plane BRDF and Nomal Emissivity 94 iv

5 Chapte 6 Validity of Hybid Method fo Coated Rough Sufaces Hybid Method Rigoous EM-Wave Solution Numeical Implementation and Validation Citeia Scatteing on Pefectly Conducting Sufaces Simulation on Thin-Film Coated Sufaces 111 Chapte 7 Conclusions and Recommendations 119 Refeences 122 Vita 131 v

6 LIST OF TABLES Table 3.1 Wafe popeties of studied samples 39 Table 3.2 Roughness paametes of studied samples 41 Table 4.1 Components in the combined uncetainty 54 vi

7 LIST OF FIGURES Figue 2.1 Schematic dawing of an AFM 11 Figue 2.2 Geomety fo the definition of BRDF 14 Figue 2.3 Shadowing (a) and masking (b) effects 18 Figue 2.4 Illustation of the specula eflection on a micofacet 19 Figue 2.5 Schematic dawing of eflection fom thin-film coatings on 26 a smooth substate (a) and a ough substate (b). Figue 3.1 Schematic of the nodal netwok fo slope calculation 36 Figue 3.2 Compaison of σ measued with the AFM and the OIP: 38 (a) Sample 1; (b) Sample 2 Figue 3.3 AFM suface images: 40 (a) Sample 3; (b) Sample 4; (c) Sample 5; (d) Sample 6 Figue 3.4 PSD (a) and ACF (b) functions fo Sample 3 42 Figue 3.5 Height distibution functions: (a) Sample 3; (b) Sample 5 43 Figue D slope distibutions of Sample 3 (a) and Sample 5 (b) 43 Figue D slope distibution functions: 48 (a) Sample 3; (b) Sample 4; (c) Sample 5; (d) Sample 6 Figue 4.1 Expeimental setup of the TAAS 50 Figue 4.2 Schematic dawing of the otay stages in the TAAS 50 Figue 4.3 Reflectivity of the smooth side a silicon wafe: 57 (a) λ = 635 nm; (b) λ = 785 nm Figue 4.4 Repeatability of measued BRDFs 58 Figue 5.1 Compaisons of measued BRDFs and pedicted values using 1-D 65 slope along the diagonals: (a) p-polaization; (b) s-polaization Figue 5.2 Compaisons of measued BRDFs and pedicted values 68 using 1-D slope along the ow and column vii

8 Figue 5.3 Compaison of measued and pedicted BRDFs at θ i = 0 71 fo Sample 3: (a) and (b): p-polaization; (c) and (d): s-polaization Figue 5.4 Compaison of measued and pedicted BRDFs at θ i = fo Sample 3: (a) and (b): p-polaization; (c) and (d): s-polaization Figue 5.5 Compaison of measued and pedicted BRDFs at θ i = 0 77 fo Sample 4: (a) and (b): p-polaization; (c) and (d): s-polaization Figue 5.6 Compaison of measued and pedicted BRDFs at θ i = fo Sample 4: (a) and (b): p-polaization; (c) and (d): s-polaization Figue 5.7 Compaison of measued and pedicted BRDFs at θ i = 0 80 fo Sample 5: (a) and (b): p-polaization; (c) and (d): s-polaization Figue 5.8 Compaison of measued and pedicted BRDFs at θ i = fo Sample 5: (a) and (b): p-polaization; (c) and (d): s-polaization Figue 5.9 Compaison of measued and pedicted BRDFs at θ i = 0 83 fo Sample 6: (a) and (b): p-polaization; (c) and (d): s-polaization Figue 5.10 Compaison of measued and pedicted BRDFs at θ i = fo Sample 6: (a) and (b): p-polaization; (c) and (d): s-polaization Figue 5.11 BRDFs at nea-infaed incidence at θ i = 30 fo Sample 3: 86 (a) and (b): p-polaization; (c) and (d): s-polaization Figue 5.12 Effect of wavelength on BRDF modeling: 87 (a) p-polaization; (b) s-polaization Figue 5.13 Suface unifomity tests fo Sample 3 (a) and Sample 6 (b) 88 Figue 5.14 Repeatability test fo wafes in the same batch as Sample 3: 89 (a) θ i = 0 ; (b) θ i = 45 Figue 5.15 Repeatability test fo wafes in the same batch as Sample 4: 89 (a) θ i = 0 ; (b) θ i = 45 Figue 5.16 Pedicted BRDFs using data within an aea of µm 2 : 91 (a) p-polaization; (b) s-polaization Figue 5.17 Effects of sample aea and instument on the coss-sections: 92 (a) coss-section ζ y = 0; (b) coss-section ζ x = ζ y viii

9 Figue 5.18 Compaison of the coss-sections of the 2-D slope distibution: 94 (a) Sample 4; (b) Sample 5 Figue 6.1 Schematic dawing of light scatteing on a thee-laye system 98 Figue 6.2 Validation of the EM-wave solution fo 105 (a) a pefect conducto and (b) a thin-film coating. Figue 6.3 Compaison of simulation esults fo scatteing on a pefectly 108 conducting suface: (a) σ = λ, τ = 5λ; (b) σ = 3λ, τ = 6λ; (c) σ = 2λ, τ = 2λ. Figue 6.4 Effects of film thickness on the validity of the hybid method: 112 (a) σ = 0.2λ, τ = 2λ; (b) σ = 0.5λ, τ = 2λ Figue 6.5 Compaisons of simulation esults fo sufaces with h = 0.1λ: 115 (a) σ = λ, τ = λ; (b) σ = 0.1λ, τ = 4λ Figue 6.6 Validity egion of the hybid method: (a) h = 0.1λ; (b) h = 0.5λ 117 ix

10 LIST OF SYMBOLS C I = instument constant D = distance fom beam cente to detecto d = sampling inteval, o local thickness of film E = y-component of electic field, o spectal emissive powe F = nomal deivative of electic field f = suface oughness wavelength f = bidiectional eflectance distibution function G = Geen function, o geometical attenuation facto g = optical oughness H = y-component of magnetic field h = thickness of film i = 1 k = wave vecto k = wave numbe L = adiance, o nomal deivative of H l = length of a tuncated suface M = numbe of ows in an aay N = numbe of columns in an aay, o numbe of suface nodes n = efactive index, o nomal vecto, o index of suface nodes p = pobability density function R a = aithmetic aveage oughness x

11 = Fesnel eflection coefficient S = shadowing function U = uncetainty V = output of a detecto Geek symbols α = inclination angle of a micofacet, o position of a otay stage β = phase shift, o position of a otay stage δ = infinitesimal numbe, o Konecke delta function ε = elative pemittivity φ = azimuthal angle γ = position of a otay stage η = citeion fo validity κ = extinction coefficient λ = wavelength in vacuum, o efes to spectal value θ = pola angle, o eflection angle, o obsevation angle θ hc = half-cone angle ρ = eflectivity Σ = inteface σ = oot-mean-squae oughness τ = autocoelation length ω = solid angle xi

12 ξ = pofile of inteface, ξ(x) ψ = local incidence angle ζ = slope in x- o y-diections Subscipt d-h = diectional-hemispheical f = film h = efes to hemispheical citeion i = incidence m, n = indexes of an aay, o indexes of suface nodes obs = obsevation = eflection s = substate, o efes to specula citeion xii

13 SUMMARY Bidiectional eflectance is a fundamental adiative popety of ough sufaces. Knowledge of the bidiectional eflectance is cucial to the emissivity modeling and heat tansfe analysis. This thesis concentates on the modeling and measuements of the bidiectional eflectance fo micoough silicon sufaces and on the validity of a hybid method in the modeling of the bidiectional eflectance fo thin-film coated ough sufaces. The suface topogaphy and the bidiectional eflectance distibution function (BRDF) of the ough side of seveal silicon wafes have been extensively chaacteized using an atomic foce micoscope and a lase scatteomete, espectively. The slope distibution calculated fom the suface topogaphic data deviates fom the Gaussian distibution. Both nealy isotopic and stongly anisotopic featues ae obseved in the two-dimensional (2-D) slope distibutions and in the measued BRDF fo moe than one sample. The 2-D slope distibution is used in a geometic-optics based model to pedict the BRDF, which agees easonably well with the measued values. The side peaks in the slope distibution and the subsidiay peaks in the BRDF fo two anisotopic samples ae attibuted to the fomation of {311} planes duing chemical etching. The coelation between the 2-D slope distibution and the BRDF has been developed. A bounday integal method is applied to simulate the bidiectional eflectance of thin-film coatings on ough substates. The oughness of the substate is one dimensional fo simplification. The esult is compaed to that fom a hybid method which uses the geometic optics appoximation to model the oughness effect and the thin-film optics to xiii

14 conside the intefeence due to the coating. The effects of the film thickness and the substate oughness on the validity of the hybid method have been investigated. The validity egime of the hybid method is established fo silicon dioxide films on silicon substates in the visible wavelength ange. The poposed method to chaacteize the micofacet oientation and to pedict the BRDF may be applied to othe anisotopic o non-gaussian ough sufaces. The measued BRDF may be used to model the appaent emissivity of silicon wafes to impove the tempeatue measuement accuacy in semiconducto manufactuing pocesses. The developed validity egime fo the hybid method can be beneficial to futue eseach elated to the modeling fo thin-film coated ough sufaces. xiv

15 CHAPTER 1 INTRODUCTION The bidiectional eflectance distibution function (BRDF) is a fundamental popety of ough sufaces, and knowledge of the BRDF is cucial to the emissivity modeling and heat tansfe analysis (Siegel and Howell, 2002). The study of the BRDF is also impotant to optical engineeing (Bennett and Mattsson, 1999) and object endeing (He et al., 1991). The BRDF of a suface can be pedicted by solving the Maxwell equations if the suface oughness is fully chaacteized. Since the igoous electomagnetic-wave solution geneally equies a huge memoy and a high-speed CPU, this appoach is pactically applicable to one-dimensional (1-D) ough sufaces only, though in some cases, solutions fo two-dimensional (2-D) ough sufaces have been obtained (Saillad and Sentenac, 2001). Thus, it is common to use appoximation methods, such as the Rayleigh-Rice petubation theoy, the Kichhoff appoximation, and the geometic optics appoximation (Beckmann and Spizzichino, 1987; Tang et al., 1997). These appoximations ae only applicable within cetain anges of oughness and wavelength. The Rayleigh-Rice petubation theoy can be used fo elatively smooth sufaces. The Kichhoff appoximation is applicable when the suface pofile is slightly undulating (i.e., without shap cests and deep valleys). The geometic optics appoximation is appopiate to sufaces whose oot-mean-squae (ms) oughness and autocoelation length ae geate than the wavelength of the incident adiation. Recent eseach has found that the geometic optics appoximation can also be used fo ough sufaces whose 1

16 ms oughness and autocoelation length ae compaable to the incidence wavelength (Tang et al., 1997). The geometic optics appoximation can be easily incopoated into a statistical and Monte-Calo method (Tang and Buckius, 2001; Zhou and Zhang, 2003). Thee exists good ageement between the simulation esults employing the geometic optics appoximation and the igoous electomagnetic-wave solution (Tang and Buckius, 1998). Howeve, the simulation based on the geometic optics appoximation equies much less computational esouces and takes much less time than that based on the igoous solution (Tang and Buckius, 1998). Since the BRDF is intinsically dependent on the suface statistics, seveal analytical expessions ae available to appoximately coelate the suface statistics to the BRDF (Toance and Spaow, 1967; Bennett and Mattsson, 1999; Caon et al., 2003). The slope distibution function is a key input in the analytical models based on the geometic optics appoximation. Befoe the invention of the atomic foce micoscope (AFM), the suface pofile was usually measued with a mechanical pofile that scans the suface line-by-line. Theefoe, the estimated 1-D slope distibution function may miss impotant infomation of the suface isotopy. Although some mechanical pofiles can measue ough sufaces with a vetical esolution of a few nanometes, the lateal esolution is usually on the ode of a micomete due to the lage adius of the stylus pobe (Bennett and Mattsson, 1999; Thomas, 1999). On the othe hand, the adius of cuvatue of an AFM pobe tip is in the ange of nm; thus, the AFM can povide detailed infomation of the topogaphy of a small aea on the micoough sufaces with a vetical esolution of sub-nanometes and a lateal esolution aound 10 nm (Wiesendange, 1994). Consequently, it is possible to evaluate the aea statistics fom the 2

17 AFM topogaphy measuement. Although attention has been paid to compae the suface statistics detemined fom the topogaphy measuements to those obtained fom the light scatteing expeiments (Cao et al., 1991; Bawolek et al., 1993; Stove et al., 1998; Nee et al., 2000), little has been done to coelate the aea statistics evaluated fom the AFM topogaphic data to the measued BRDF fo elatively ough sufaces. In geneal, suface oughness is assumed to satisfy the Gaussian statistics in the deivation of the BRDF model and fo the suface geneation in the Monte Calo simulation (Beckmann and Spizzichino, 1987; Tang et al., 1998). Futhemoe, the suface statistics of the 2-D ough suface ae mostly assumed to be isotopic so that the autocoelation function is independent of the diection. Howeve, the Gaussian distibution may miss impotant featues of natual sufaces because this function does not allow any abupt event in the apidly deceasing tails (Guéin, 2002). Vey few papes have been devoted to the BRDF of non-gaussian and anisotopic 2-D ough sufaces. Shen et al. (2001) found that the BRDF models could not pedict the subsidiay peak in the measued BRDF of the ough side of a silicon wafe, although easonable ageement existed between the measuement esults and the model pedictions within a lage angula egion aound the specula diection. This disageement could be caused by the Gaussian distibution applied in the BRDF model. Theefoe, it is impotant to examine the actual suface statistics of the ough side of silicon wafes so that a easonable explanation may be povided fo the occuence of the subsidiay peak. Besides the suface oughness, thin-film coatings on the ough side of silicon wafes can also geatly change the BRDF. Reseaches have applied a hybid method in the modeling of thin-film coatings on ough substates (Tang et al, 1999a). In the hybid 3

18 method, thin-film optics is applied to model the intefeence effect due to the coating, and the ay-tacing algoithm based on the geometic optics appoximation is used to model the scatteing due to the suface oughness. The hybid method is computationally effective since its fomulation is vey simila to that fo a ough suface without thin-film coatings. Some ageement has been obseved between the pedictions using the hybid method and the measuement esults (Tang et al, 1999a). Howeve, thee is a dilemma between the two theoies in the hybid method. The geometic optics appoximation is applicable to ough sufaces while thin-film optics is appopiate fo a laye of coating with pefectly smooth intefaces. Theefoe, it is vey impotant to study the validity of the hybid method so that the advantages of the hybid method may be exploited in the BRDF modeling of thin-film coated ough sufaces. The motivations of this thesis ae to model the BRDF of the ough side of silicon wafes fom the actual suface statistics and to investigate the validity of the hybid method. The topogaphic data measued with an AFM ae analyzed to obtain the suface statistics. The measued suface statistics may be linked to the key paametes in the wafe manufactuing pocesses. The pedicted BRDF fom the measued suface statistics is compaed with the measued BRDF to find the coelation between the suface statistics and the BRDF. The established coelation will help the application of the light scatteing measuement in the suface oughness chaacteization. A easonable explanation fo the subsidiay peak obseved fo the silicon wafe may be povided. Futhemoe, fo the modeling of thin-film coated ough sufaces, the igoous electomagnetic-wave solution is pefomed to study whethe the hybid method is valid. 4

19 The compaisons of simulation esults fom diffeent methods will show the capability and the limitation of the hybid method. The oganization of this thesis is as follows. Chapte 2 povides a eview of the silicon wafe manufactuing, oughness chaacteization instuments, and theoetical and expeimental studies on the BRDF. Chapte 3 pesents the oughness measuement and the statistical analysis on the topogaphic data. The 1-D slope distibution and the 2-D slope distibution ae investigated in detail. Chapte 4 descibes the thee-axis automated scatteomete (TAAS) developed by Shen (2002) and the systematical chaacteization and futhe impovements on the TAAS. Chapte 5 applies a geometic optics model to pedict the BRDF fom the 1-D and the 2-D slope distibution functions. The pedicted esults ae compaed with the expeimental findings. The coelations between the slope distibution and the BRDF ae explained. The instument effects and the invese pocedue ae discussed. Chapte 6 compaes the numeical simulation esults using the igoous electomagnetic-wave solution and the hybid method. The effect of film thickness on the validity of the hybid method is discussed. A validity egime is pesented fo a laye of silicon dioxide on a silicon substate in the visible wavelength egion. Finally, Chapte 7 summaizes the conclusions and the suggested futue wok. 5

20 CHAPTER 2 LITERATURE REVIEW 2.1 Silicon Wafe Manufactuing and Rapid Themal Pocessing Silicon is the pimay cystalline mateial in semiconducto manufactuing industy. A smooth silicon wafe can be poduced by the following pocedue. Fist, polysilicon is poduced fom sand by means of a complex eduction and puification pocess. High puity polycystalline silicon is melted in a cucible. A seed of single cystal silicon is dipped into the melt and pulled out fom the melt gadually. The liquid ises with the seed due to suface tension and cools into a single cystalline ingot. Ends of the ingot ae copped, and the ingot is gound to a unifom diamete with a flat indicating the cystal oientation. Then the ingot is sliced into many silicon wafes. The sliced wafe is mechanically lapped to educe suface oughness due to the saw cut. The lapped wafe is etched by a chemical solution to emove any emaining micocacks and suface damages. The etched wafe is polished to a mio suface. Nomally only one side of the silicon wafe is polished while the othe side emains ough. Finally, the wafe is cleaned by deionized wate and died. It is common to deposit othe mateials on a silicon wafe fo vaious applications. Fo example, the ough silicon wafe can have polysilicon coatings fo potection and insulation. Some coatings on the ough side ae beneficial to attact defects within the silicon wafe duing annealing. Many steps in semiconducto manufactuing equie themal pocessing, such as the gowth of films, annealing and so on (Timans, et al., 2000). Fo example, ion 6

21 implantation is a good method to tune the conductivity of silicon because of its inheent doping contols. The doped wafe has to be themally annealed to estoe the cystal stuctue and educe the stess due to the implanting of heteoatoms (Timans, et al., 2000). The geneal tend of themal pocessing is to educe the pocess tempeatue and duation time as much as possible in ode to estict the motion of atoms though diffusion (Timans, et al., 2000). Taditionally, the annealing pocess is pefomed within a batch funace. Wafes in the batch funace cannot be heated up unifomly because thei edges heat up faste than thei centes (Fan and Qiu, 1998). Futhemoe, the tempeatue diffeence of the wall can affect the quality of the whole batch. Rapid themal pocessing (RTP) is a pomising way to eplace the taditional batch funace method since it povides flexibility to the tempeatue contol. A RTP funace can heat one individual wafe to a specified tempeatue in a shot peiod of time mainly though adiation heat tansfe. It can make all the points on the wafe expeience the same tempeatue-time cycle as defined in the pocess ecipe (Timans, et al., 2000). In many RTP funaces, the tempeatue of the silicon wafe is monitoed by a adiation themomete viewing the ough side of the wafe. Howeve, the detemination of the spectal emissivity of a ough suface is vey difficult. Some eseach has been devoted to the modeling of the spectal hemispheical emissivity fo a ough suface (Demont et al., 1982, Vandenabeele and Maex, 1992; Xu and Stum, 1995; Bhushan et al., 1998; Zhou and Zhang, 2003). Besides the suface oughness, the thin-film coating can significantly change the emissivity. Bidiectional eflectance of coated sufaces can be vey diffeent fom that of the substate (Yeh, 1988; Soell and Gyucsik, 1993). Futhemoe, since the ough side of the wafe and the lowe chambe of the RTP funace 7

22 compose an enclosue, the appaent emissivity should be used to detemine the tempeatue of the wafe (DeWitt el al., 1997; Zhang, 2000). In ode to model the effective emissivity of the wafe, a thoough undestanding of the bidiectional eflectance of the wafe suface is necessay. 2.2 Roughness Measuement Real sufaces all show some extent of oughness. Roughness can be imagined as aspeities (micofacets) on the suface fom a micoscopic view. The summation of the altitudes of the aspeities with espect to the mean plane is zeo. The lateal and vetical scales of the aspeities can be vey lage, as peaks and valleys in a mountain, o can be vey small, such as a small paticle on a mio. In this thesis, the oughness efes to the micooughness. The lateal and vetical dimensions of aspeities on a micoough suface ae in the ode of micometes. The patten of light scatteing can be geatly changed by suface oughness. If a suface is vey smooth, like a mio, most of the incidence light is eflected to the specula diection. If a suface is ough, the scatteed adiation usually exists in the whole hemisphee above the suface. The suface pofile of a deteministic ough suface can be descibed by a function. One of the specific goups of the deteministic suface is the peiodic suface. The scatteed adiation on a peiodic suface can exist only in a finite numbe of diections instead of the whole hemisphee due to diffaction. Howeve, in geneal, the pecise definition equation of the suface pofile is unknown o of little inteest, the shape of the ough suface is descibed by a andom function of space coodinates (Saillad and Sentenac, 2001). Only the andom ough suface is studied in 8

23 this thesis. Since light scatteing is stongly dependent on the suface oughness, it is cucial to study the statistics of eal sufaces. We can oughly sense the suface oughness using the thumbnail and the eye. Both methods ae completely subjective. Vaious types of instuments can be used to map the suface topogaphy. Some instuments ae following the tactile example of the nail, i.e., using a stylus pobe. Some ae mimicking the eye, i.e., using an optical method. The detailed infomation about the oughness instumentation can be found in efeences (Whitehouse, 1997; Bennett and Mattsson, 1999; Thomas, 1999). A shot eview is povided in the following fo some commonly used instuments in the micoelectonics industy and optical engineeing Mechanical Pofile As a stylus is dagged ove a ough suface, it moves up and down when it ides ove peaks and valleys on the suface. The deviation of the stylus fom a efeence (a skid) can be tansfomed to an electical signal and the suface pofile can be detemined fom the signal. The vetical esolution of the stylus pofile is aound 1 nm. The lateal esolution is limited by the adius of the tip. The adius of cuvatue of a tip can be 2 µm, 5 µm, and 10 µm accoding to the ISO standad (Thomas, 1999). The smallest adius of cuvatue of the tip can each 0.2 µm (Bennett and Mattsson, 1999). Consequently, the shotest suface oughness wavelength that can be measued by the mechanical pofile is aound 0.2 µm. One issue fo the mechanical pofile is the load of the stylus. The foce exeted by the stylus may be geate than 1 µn so that it may scatch o even damage the scanned suface. Anothe issue is that most mechanical pofiles can only pefom the 9

24 line scan, although a vey limited numbe of models can pefom the aea scan. If aea pofiling is equied, the scan time will be vey long since the stylus has to aste the whole aea Scanning Pobe Micoscope The atomic foce micoscope (AFM) and the scanning tunneling micoscope (STM) belong to the family of the scanning pobe micoscope (SPM) (Wiesendange, 1994). The name of the SPM comes fom the pobe, like a stylus in phonogaph, asteing a sample. The bith of the SPM gives a mavelous way fo scientists and enginees to view the fine featue in atomic scale. The fist membe of the SPM family is the STM. Although the STM can map the suface featue in mavelous vetical and lateal esolutions, the undelying physics detemine that the STM can wok only fo conductive sufaces. Binning et al. (1996) intoduced the fist atomic foce micoscope. Figue 2.1 shows the pinciple of an AFM. When a shap tip is vey close to a sample, thee is a epulsive foce between the tip and the sample. This foce can cause the bending of a cantileve whee the tip is attached. In the fist AFM, a technique simila to the STM was used to detect the deflection of the cantileve. In the cuent AFMs, an optical method is popula to detect the bending of the cantileve because it induces little noise into the micoscope. Geneally, the tip and the cantileve ae made of silicon, silicon dioxide, o silicon nitide. The adius of the tip can be as small as 10 nm so that the AFM can map the fine featues with an excellent lateal esolution. The vetical esolution of the AFM can achieve 0.1 nm. Because the tip has to aste scan a selected aea, the scan ate of the AFM is usually slow. 10

25 Photodiode Lase Feedback to z-piezo x-y-z Piezo Scanne Cantileve & Tip Sample Figue 2.1 Schematic dawing of an AFM Optical Intefeometic Micoscope The optical intefeometic micoscope (OIM) exploits the wave behavio of light. The main pat of the OIM is an intefeomete. Light fom a souce is split into two pats by a beamsplitte in the intefeomete. A pat of the light is eflected back by the ough suface, and the othe pat is eflected by a efeence suface. Because two lights ae fom the same coheent souce and the optical paths ae diffeent, they can geneate an intefeogam, which epesents the topogaphy of the suface. The intefeogam is pictued by a chaged couple device (CCD) camea. Sophisticated hadwae and softwae ae necessay to extact the suface height infomation fom the intefeogam. The most popula intefeometic techniques ae the phase shift intefeomety (PSI) and the scanning white light intefeomety (SWLI) (Wyant et al., 1986). The light souce in the PSI is a monochomatic souce, and the PSI is applicable fo slightly ough sufaces. The vetical esolution of the PSI is in the sub-nanomete ange. If the suface is modeately ough (the height diffeence between the adjacent peak and valley is geate than a quate of the selected wavelength), the PSI cannot esult in a coect topogaphic 11

26 image fo the suface. Fo elatively ough sufaces, the SWLI is a bette way fo oughness measuement. In the SWLI, the light souce is a wide-band souce instead of a monochomatic souce. The basic pinciple of the SWLI is that the maximum intensity fo white-light finges occus when the optical path diffeence is zeo. When the optical path diffeence is vaied, an intensity envelope is ecoded fo each point on the suface. The height infomation is deduced fom the maximum in the intensity envelope. The vetical esolution of the SWLI is about 1-2 nm, lage than that of the PSI. The OIM has a lage field of view and the image can be obtained in a few seconds. Since the intefeogam is dependent on the phase o the intensity of the eflected light, the suface condition may deteioate the measuement esult if the eflectivity is not unifom ove the scan aea. Futhemoe, the OIM may not be applicable to sufaces with steep aspeities since the eflected light may be not collected by the optical system Othe Pobe Techniques The instuments mentioned in Sec can povide the topogaphic data fo the measued suface. Some othe instuments do not povide the topogaphic data, but povide the statistical infomation of the oughness (Stove, 1995; Bennett and Mattsson, 1999). Total integated scatteing (TIS) and angle-esolved scatteing ae two examples of techniques using the light scatteing method. A evese pocedue is necessay to obtain the oughness paametes fom the available models. A detailed discussion of the egime of suface oughness paametes measuable with the light scatteing method can be found in Vobuge et al. (1993). 12

27 2.3 Theoetical Study of the Bidiectional Reflectance When adiation is eflected by a ough suface, the eflected enegy will be distibuted in the hemisphee, and futhemoe, the distibution of enegy is geneally dependent on the incoming diection. Theefoe, it is necessay to use two diections to descibe the eflection by a ough suface (Nicodemus, 1970; Banes et al., 1998). Bidiectional eflectance is a fundamental adiative popety of ough sufaces in themal science (Bewste, 1992; Modest, 1993; Siegel and Howell, 2002). The study of the bidiectional eflectance is also impotant to othe subjects. In optical engineeing, the bidiectional eflectance is used as a tool to chaacteize the suface oughness. In object endeing, efficient bidiectional eflectance models ae sought to achieve fast and vivid object endeing (Phong, 1975; He et al., 1991). The theoies on the scatteing fom ough sufaces can be found in seveal books and eview papes (Tang et al., 1999b; Saillad and Sentenac, 2001; Tsang et al., 2001; Wanick and Chew, 2001; Zhang et al., 2003). Fo an ideally smooth suface, thee is no scatteing and all the adiation is eflected to the specula diection. Fo an ideally diffuse suface, the eflected adiation in any diection is the same. Howeve, the scatteing on a eal andom suface is neithe the specula eflection no the ideal diffuse eflection. Roughly speaking, the bidiectional eflection may be divided into thee components, a specula spike at the specula diection, a specula lobe aound the specula diection, and a diffuse tem coveing all the eflection angles. In ode to quantitatively descibe the enegy distibution of the scatteed adiation, it is necessay to define the bidiectional eflectance distibution function (BRDF). The geomety to illustate the definition is shown in Figue 2.2. The x- and y-axes ae located in the mean plane of the ough suface, and the z-axis is nomal to 13

28 the mean plane. The BRDF, also called the bidiectional eflectance, is the atio of the eflected adiance (o intensity in most heat tansfe textbooks) to the incident iadiance (Nicodemus, 1970; Banes et al., 1998; Siegel and Howell, 2002), L ( θi, φi, θ, φ ) -1 f ( θ i, φi; θ, φ ) = [s ] (2.1) L ( θ, φ ) cosθ dω i i i i i L i dω i z θ i dω L θ y φ i φ Figue 2.2 Geomety fo the definition of BRDF. x In the above equation, (θ i, φ i ) and (θ, φ ) denote the incoming and scatteing diections, espectively, L i is the incoming adiance, L i cosθ i dω i epesents the incident iadiance (powe pe unit pojected aea), the eflected adiance L is a function of both the incoming and scatteed diections. Since adiance is a spectal popety, BRDF is also a spectal popety. The dependence on the wavelength of the incident adiation λ is not shown in Eq. (2.1) fo the sake of simple notation. Both the theoetical analysis and the expeimental measuement ae mainly devoted to finding the vaiation of the eflected adiance (o eflected powe) with the eflection angle (θ, φ ) when the incidence is fixed at the diection of (θ i, φ i ). 14

29 2.3.1 Rigoous Electomagnetic-Wave Solution Thee ae two main categoies in the igoous EM-wave solution: integal equation methods and diffeential equation methods. The bounday integal method is one example of the integal equation methods. Diffeential equation methods include the finite-diffeence time domain (FDTD) method, the volume finite-element method, and the diffeential method. The integal equation method can deal with homogeneous media suounded by a bounday. The diffeential method can deal with inhomogeneous media as well (Saillad and Sentenac, 2001). The diffeential method equies volumetic meshes; theefoe, the numbe of unknowns may be lage than that equied by a suface mesh (Wanick and Chew, 2001). The bounday integal method is the most common method of the numeical simulation of light scatteing. This method is based on the extinction theoem (Wolf, 1973) and Geen s theoem (Keyszig, 1993). Maadudin et al. (1990) and Sánchez-Gil and Nieto-Vespeinas (1991) applied this method to study the scatteing by onedimensional dielectic ough sufaces. Assuming that light of p-polaization is incident on a one-dimensional ough suface fom vacuum, with the magnetic field H in the y diection, the fomulas govening the light scatteing ae 1 G (, ) H ( ) ( ) = H ( ) + π H ( ) 0 G (, ) d (2.2) 4 n n H i Σ 1 G H = π H s (, ) G s( ) 0 s ( ) s(, ) d (2.3) 4 n n Σ whee = (x, z), z = ξ(x) at the bounday Σ, G is the Geen function, and n is the nomal at the position on the bounday. In Eq. (2.2), and ae in vacuum, while in Eq. (2.3), 15

30 is in vacuum and is in the medium. The subscipts 0 and s stand fo vacuum and the medium, espectively. H 0 ae H s ae linked by the bounday conditions: H ( ) = H ( ) (2.4a) 0 s H0( ) 1 H = s( ) n ε n s (2.4b) The coupled equations (2.2) and (2.3) should be solved simultaneously. The intensity in fa field can be calculated fom the magnetic field H and its nomal deivative H / n at the bounday (Maadudin et al., 1990; Sánchez-Gil and Nieto-Vespeinas, 1991) The bounday integal method has been extensively used to explain the backscatteing on ough sufaces. Maadudin et al. (1990) found that thee exists a citical efactive index fo the occuence of backscatteing, and futhemoe, the citical value becomes smalle with the incease of oughness. This method can also be applied to veify the simulation esults using the Kichhoff appoximation (Chen and Fung, 1988; Thosos, 1988; Sánchez-Gil and Nieto-Vespeinas, 1991) and the geometic optics appoximation (Tang and Buckius, 1997). A lot of wok has been done on scatteing fom the one-dimensional ough suface using the igoous numeical simulation. The simulation fo the two-dimensional ough sufaces based on the igoous appoach is vey computationally intensive, and only a few cases fo pefect conductos and metals ae available (Pak et al., 1995; Johnson et al., 1996). Because of the lage numbe of unknowns in the fomulation fo two-dimensional sufaces, diectly solving the matix equation is not feasible. Consequently, an iteative method should be applied (Wanick and Chew, 2001). In geneal, the igoous appoach equies a lage memoy and a long computation time. 16

31 Theefoe, seveal appoximation methods have been developed to facilitate a fast simulation of light scatteing by ough sufaces Kichhoff s Appoximation In the igoous electomagnetic-wave (EM-wave) appoach, the electomagnetic field at the bounday is unknown. In the Kichhoff appoximation, the field at a cetain point at the bounday is the same as that on a tangential plane passing though the point (Beckmann and Spizzichino, 1987). The total electic field E is the summation of the incidence field E i and the eflected field, i.e., E = ( 1+ ) E i (2.5) whee the local Fesnel eflection coefficient is dependent on the slope at each point. The Kichhoff appoximation is also called the tangent plane appoximation. In geneal, the Kichhoff appoximation is valid fo a ough suface with a adius of cuvatue much lage than the wavelength of the incident adiation λ (Beckmann and Spizzichino, 1987). In othe wods, the Kichhoff appoximation is only applicable to gently undulating sufaces. Unlike the igoous appoach, thee is no need to solve the coupled equations. The applicable egion of the Kichhoff appoximation has been established fo pefectly conducting sufaces (Chen and Fung, 1988; Thosos, 1988; Sánchez-Gil and Nieto-Vespeinas, 1991). It is commonly believed that the Kichhoff appoximation can still give the eliable esult when the ms oughness σ and the autocoelation length τ ae less than o compaable to λ and the atio of σ to τ is less than 0.3 (Tang and Buckius, 1998). Chen and Fung (1988) claimed that the Kichhoff appoximation is eliable at 17

32 small angles of incidence when the atio of τ to λ is less than 0.3 and the atio of σ to λ is less than (a) Shadowing (b) Masking Figue 2.3 Shadowing (a) and Masking (b) effects. Figue 2.3 shows the shadowing and masking effects. Some of suface aspeities ae not illuminated since the incident adiation is blocked. This is efeed to shadowing effect. Similaly, when the adiation is eflected by suface aspeities, it may be not able to leave the ough suface if the eflected adiation is diected to othe suface aspeities. This is efeed to masking effect, o outgoing shadowing. The intecepted beam can bounce back and foth on the ough suface until it finally leaves the ough suface. This is efeed to multiple scatteing. The oiginal Kichhoff appoximation takes into account only the single scatteing (the fist-ode scatteing). Buce and Dainty (1991) incopoated multiple scatteing into the Kichhoff appoximation to model light scatteing by elatively ough sufaces. Sánchez-Gil and Nieto-Vespeinas (1991) demonstated that the valid egion of the Kichhoff appoximation can be even lage fo dielectic sufaces than fo metal sufaces since multiple scatteing on dielectic sufaces is not as significant as that on metal sufaces. 18

33 2.3.3 Geometic Optics Appoximation In the geometic optics appoximation, the ough suface is imaged as a combination of numeous micofacets which ae andomly oiented on the mean plane (Toance and Spaow, 1967; Tang and Buckius, 1998 and 2001). The dimension of the micofacet is much lage than λ. The suface of the micofacet is assumed to be ideally smooth, and the eflection of the adiation on the micofacet obeys Snell s law. Figue 2.4 shows the angula elationships between the incident beam and the eflected beam when the specula eflection takes place on the suface of a micofacet m. The micofacet nomal n bisects the incident beam and the eflected beam. The eflectivity of the micofacet is dependent on the local incidence angle ψ, which is defined by the incident beam and the micofacet nomal. The oientation of the micofacet, i.e., the diection of n, can be defined by an inclination angle α and an azimuthal angle (not shown in Figue 2.4). The oientation of the micofacet can be pesented by its slopes as well. The slopes ae elated to the incoming and scatteing diections by θ i z α n ψ ψ θ m Figue 2.4 Illustation of the specula eflection on a micofacet. 19

34 ζ ζ x y sin θi cosφi + sin θ cosφ = cosθ + cosθ sin θi sin φi + sin θ sin φ = cosθ + cosθ i i (2.6a) (2.6b) The diffaction and intefeence ae ignoed in the geometic optics appoximation. Fom a statistical point of view, the adiation eflected into a finite solid angle is popotional to the eflectivity of the micofacet and the pobability to find the micofacets with coesponding slopes. The shadowing and masking effect may become significant at lage incidence angles and lage eflection angles. The effect due to multiple scatteing may become significant fo vey ough sufaces, and it needs to be included in the modeling. Geneally speaking, the applicable oughness egion of the geometic optics appoximation is σ > λ and τ > λ. Tang et al. (1997) established a validity egion fo onedimensional pefectly conducting ough sufaces. They found that the valid egion can be extended to σcos(θ i )/λ > 0.2 and σ/τ < 2 with easonable accuacy. Although the validity egion fo two-dimensional ough sufaces has not been systematically investigated, Tang et al. (1997) believed that the validity egion fo one-dimensional ough sufaces may apply to two-dimensional ough sufaces. In the simulations using the ay-tacing algoithm based on the geometic optics appoximation, the suface geneation method (Tang et al., 1997; Tang and Buckius, 1998) is vey common. In addition, the micofacet slope method has also been exploed by Zhou et al. (2002) and Pokhoov and Hanssen (2003). The micofacet slope method equies less computation time than the suface geneation method. 20

35 2.3.4 BRDF Models Specula model descibes the bidiectional eflection on an ideally smooth suface. The BRDF is zeo eveywhee except fo at the specula diection (θ i, φ i +180 ), whee ρ ( κ, ) ( n, κ, θi ) δ( θ θ ) δ[ φ ( φ + )] ρ f ( θi, φi, θ, φ ) = i i 180 (2.7) cosθ n, θ i is the eflectivity of the smooth suface, and δ is the Konecke delta function. Diffuse model descibes the bidiectional eflection fom an ideally diffuse suface. The BRDF is independent of the eflection angle, whee ρ ( κ, ) d h n θi ( n, κ, θ ) ρ f θ φ θ φ = d h i ( i, i,, ) (2.8) π, is the diectional-hemisphee eflectance of the diffuse suface. The suface powe spectal density (PSD) function is popula to pesent the oughness statistics. In the golden ule deived based on the Rayleigh-Rice petubation theoy, the BRDF is elated to the PSD function (Stove, 1995; Bennett and Mattsson, 1999), 2 16π f ( θ i, φi, θ, φ ) = Q cos θ cos PSD 4 i θ (2.9) λ whee Q is a facto consideing the eflectance of the suface. One model is deived based on the Kichhoff appoximation and Gaussian oughness statistics (Beckmann and Spizzichino, 1987). The specula component and the off-specula component of the BRDF fo an illuminated suface with a ectangle aea l x l y ae 21

36 f, spe lxly cosθi = exp 2 λ cosθ sinc 2 πl λ y sin θ 2 ( g) sinc x ( sin θ sin θ cosφ ) sin φ πl λ i (2.10a) f, off m= 1 2 πcosθ τ + θ θ θ θ φ = i 1 cos i cos sin i sin cos 2 λ cosθ cosθi (cosθi + cosθ ) m 2 2 g π τ exp! 2 i i m m mλ whee sinc(x) = sin(x)/x, and the optical smoothness is [ 2πσ(cosθ + cos θ ) / λ] 2 = i 2 2 ( sin θ 2sin θ sin θ cosφ + sin θ ) 2 exp ( g) (2.10b) g (2.11) Toance and Spaow (1967) assumed that the distibution function of the inclination angle α of the micofacets is Gaussian and deived a semi-empiical BRDF model, bexp( c α ) f ( θ i, φi, θ, φ ) = ρ( ψ) G + a 4cosθ cosθ i 2 2 (2.12) whee constants b and c define the distibution function of the inclination angle. G is the geometical attenuation facto to include the effect of shadowing. The diffuse tem in the BRDF is epesented by a. All constants have to be fitted by the expeimental esults. Caon et al. (2003) deived the atio of the scatteed adiance to the incident powe flux following the static-phase fomulation (Tsang and Kong, 1980; Kong, 1990). Fo in-plane scatteing, p( ζ x, ζ y ) f ( θi, φi, θ, φ ) = ρ( ψ) (2.13) 4 4cosθ cosθ cos α i whee p is the pobability density function with espect to slopes (ζ x, ζ y ) of micofacets. In the plane of incidence, φ i φ = 0 o 180, α = θ θ i /2, and ψ = (θ i + θ )/2. The y- 22

37 component of the slope has to be zeo so that the eflected beam can fall to the plane of incidence. Tang and Buckius (2001) developed a compehensive statistical model based on the geometic optics appoximation. This model includes contibutions fom both fistode scatteing and multiple scatteing. The BRDF fo the fist-ode scatteing (i.e., the adiation is eflected once by a micofacet befoe leaving the suface) can be pesented as f p( ζ x, ζ y )(1 + ζ x tan θi ) dζ xdζ y θ i, φi, θ, φ ) = ρ( ψ) S( θi ) S( θ ) (2.14) cosθ sin θ dθ dφ ( whee S(θ i ) and S(θ ) ae the shadowing functions (Smith, 1967) fo shadowing and masking effect, espectively. The slope distibution is most commonly modeled as a Gaussian function (Toance and Spaow, 1967; Tang and Buckius, 2001; Caon et al., 2003). The twodimensional pobability density function (PDF) of slopes fo an isotopic suface can be witten as 2 2 p (2.15) 1 ( ) ζ x + ζ y ζ ζ = x, y exp 2 2 2πζms 2ζms whee ζ ms stands fo the ms slope, which is the same in all diections. A simple elation exists fo ough sufaces that satisfy the Gaussian statistics (Beckmann and Spizzichino, 1987): ζ = 2σ / τ ms. Specula model and diffuse model ae fo ideal sufaces, and they have little use in modeling the BRDF fo eal sufaces. The golden ule is based on Rayleigh-Rice s petubation theoy, and theefoe it is only applicable to a suface whose ms oughness is much smalle than λ. Equations (2.10), and (2.13) though (2.15) ae deived eithe 23

38 fom the Kichhoff appoximation o fom the geometic optics appoximation. Theefoe, the validity egion fo these analytical models is constained by the validity egion of the coesponding appoximation methods. Eqs. (2.13) though (2.15) ae all based on the geometic optics appoximation, the only diffeence in the deivations is whethe the distibution function is elated to the inclination angle o to the slopes. Theefoe, it might be possible to unify these equations to one identical fomulation Modeling the Scatteing fom Coated Sufaces The study of scatteing fom thin-film coated sufaces is vey impotant in optical, mateials, and themal engineeing. Many optical components and semiconducto wafes ae coated with thin films accoding to diffeent applications. The BRDF and the emissivity of thin-film coated sufaces can be vey diffeent fom those of the substate. The theoy on the eflection fom a multilaye system with ideally smooth intefaces is well developed (Yeh, 1988). Accoding to thin-film optics, the amplitude of the eflected wave fom a thee-laye system with ideally smooth intefaces is (Bewste, 1991; Siegel and Howell, 2002) f 0, f + f = 1+ 0, f, s f, s exp( j2β) exp( j2β) (2.16) whee 0,f and f,s ae the Fesnel eflection coefficients between ai and film and between film and substate, espectively. The phase shift of wave taveling though the thin film is β = 2πn f hcos( θ) (2.17) λ whee n f is the efactive index of the film, h is the film thickness, and θ is the efaction angle within the film. 24

39 On the contay, the scatteing of adiation by thin-film coatings on a ough suface is extemely difficult to analyze. A coection facto can be added to the Fesnel eflection coefficients to calculate the patially coheent eflectance and tansmittance of a multilaye stuctue with ough intefaces (Filinski, 1972; Mitsas and Siapkas, 1995). The fist-ode vecto petubation theoy (Elson, 1977; Buno et al., 1995) and the Kichhoff appoximation (Lettiei et al., 1991; Icat and Aques, 2000; McKnight, 2001) can be applied to simulate the scatteing fom multilaye systems. Howeve, these methods ae only applicable to eithe vey smooth sufaces o gently undulating sufaces (Elson, 1977; Beckmann and Spizzichino, 1987). The geometic optics appoximation is extensively used in analytical models and Monte Calo simulation. Good ageement has been obseved between the simulation esults employing the geometic optics appoximation and the igoous electomagneticwave solution fo a ough suface without thin-film coatings (Tang et al., 1998), and the simulation is vey computationally effective. Since the simulation based on the igoous electomagnetic-wave solution becomes fomidable if thee ae thin-film coatings on a ough substate, it is wothwhile to exploe othe altenative methods. Some pevious woks (Tang et al., 1999a) assumed that the eflection on the thin-film coating could be well descibed by thin-film optics consideing intefeence effects wheeas the oughness effect can be modeled using the geometic optics appoximation though the ay-tacing method. Afte the eflectivity of the micofacet without thin-film coatings is eplaced by that with thin-film coatings, the analytical models and the developed pogams applying the Monte Calo simulation can be used fo modeling of the BRDF fo coated ough sufaces. The eflectivity of each coated micofacet is detemined fom Eqs. (2.16) and 25

40 (2.17). Figue 2.5 shows the schematic of eflection fom thin-film coatings on a smooth substate and a ough substate. Equation (2.16) is deived fo a thin film on an ideally smooth plane; theefoe, each wave eflected by the ai-film inteface and the one eflected by the film-substate inteface can be accounted fo. Howeve, the micofacet on the ough suface has a finite (usually vey small) aea although the suface of the micofacet is smooth. Because of the small aea of a micofacet, an incident ay on the micofacet can emege fom the neaby micofacet afte it tavels in the film, fist towads the substate and then towads the ai. Heeafte, this will be efeed as the cone effect. Because of the cone effect, the application of Eq. (2.16) fo the coated micofacet may not well descibe the intefeence of a thin film on a ough substate. (a) a smooth substate (b) a ough substate Figue 2.5 Schematic dawing of eflection fom thin-film coatings on a smooth substate (a) and a ough substate (b). Tang et al. (1999a) applied the suface geneation method (SGM) to evaluate the BRDF fo thin-film coated sufaces. Zhou and Zhang (2003) and Lee et al. (2004) used the micofacet slope method (MSM) to model the adiative popeties of thin-film coated opaque o semi-tanspaent sufaces. Some ageement has been demonstated between the modeling esults and the expeimental measuement (Tang et al., 1999a). It indicates 26

41 that the hybid method may be applicable fo some ough sufaces. Theefoe, it is necessay to study the validity of the hybid method so that the advantages of the hybid method may be exploited in the BRDF modeling of thin-film coated ough sufaces. 2.4 Expeimental Study of the Bidiectional Reflectance BRDF Instumentation Expeimental studies ae necessay to veify the pedicted esults fom vaious models and numeical simulations. The instument used to measue the bidiectional eflectance is called a bidiectional eflectomete o scatteomete. Some scatteometes can also measue the bidiectional tansmittance (Pocto and Banes, 1996; Banes et al., 1998). Diffeent types of bidiectional eflectometes ae available fo eseach and industial applications (Zipin, 1966; Andeson et al., 1988; Dolen, 1992; Feng et al., 1993; Roy et al., 1993; Zawoski et al., 1996a; White et al., 1998). Although the sophistication vaies fom instument to instument, the essential components of a scatteomete ae the same: namely, an optical souce, a goniometic table, and a detection and data acquisition system. The wavelengths of the measuements ae usually in the visible and nea-infaed egions due to the difficulty encounteed fo shote o longe wavelengths. A gating monochomato o a coheent lase souce can povide a naow band optical adiation, which is nealy collimated. A spectomete in the detecto assembly could pefom the same function as the monochomato (Feng et al., 1993). Thee is a vaiety of designs of the goniometic table, which manipulates the movements of the detecto, sample, and/o optical souce. If only in-plane measuements (i.e., the eflected light is confined to the plane of incidence) ae needed, two coaxial 27

42 otay stages ae sufficient to vay the incidence and eflection angles. Theefoe, the scatteomete is elatively easy to constuct (Zipin 1966; Roy et al., 1993). The Spectal Ti-function Automated Refeence Reflectomete (STARR) at NIST is a high-accuacy efeence instument fo the in-plane BRDF measuement in the visible and nea-infaed egions (Pocto and Banes, 1996; Banes et al., 1998). A few designs have been ealized fo out-of-plane measuements (Andeson et al., 1988; Dolen, 1992; Feng et al., 1993; Zawoski et al., 1996a; White et al., 1998; Shen et al., 2003). Usually, a flexible hadwae design fo the goniometic table is desied so that it can move the souce, sample, and detecto to diffeent combinations of incidence and viewing angles. In some systems, the souce is stationay while the detecto and the sample holde ae movable (Andeson et al., 1988; Dolen, 1992; Zawoski et al., 1996a; Shen, 2002). The advantage of a fixed souce is that vitually thee is no limit on the size and weight of the souce (Zawoski et al., 1996a). The cicula-tack design is anothe way to otate the souce and detecto aound the sample; howeve, the long-tem stability and eccenticity equiements may be difficult to meet (White et al., 1998). In ode to measue the scatteing and its associated polaization states, an out-of-plane ellipsomety scatteomete has been developed by Geme and Asmail (1999). In this design, the sample can be otated in both its vetical and hoizontal axes BRDF Measuements and Modeling Seveal eseaches measued samples by applying diffeent micoscopic pobe techniques and analyzed the PSD function fo these sufaces. Polished silicon wafes and wafes with polysilicon coatings wee measued, and the suface statistics wee pesented 28

43 in the fom of the PSD function. Vatel et al. (1993) and Mulle et al. (2001) studied the PSDs of polished silicon wafes and polysilicon coated wafes. Dumas et al. (1993) calculated the PSD function of optical-glass sufaces using the AFM topogaphic data and epoted that the esult was satisfactoily compaable to the optical scatteing data ove a lage ange of spatial fequencies. Jahanmi and Wyant (1992) measued vaious samples using an optical pofilomete and a scanning pobe micoscope. They compaed the ms oughness and the aveage oughness fo those samples. Max et al. (1998) measued silicon wafes with vaious anges of oughness using diffeent instuments including the AFM and the intefeometic pofilomete. Howeve, they did not measue the oughness of wafes whose last pocessing stage was lapping. Most of the sufaces studied in the published liteatue have low levels of oughness. Rae eseach has been caied out to study the suface statistics of the backside of silicon wafes, whose ms oughnesses may be as lage as seveal hunded nanometes. The slope distibution function is a key input in the analytical models based on the geometic optics appoximation (Toance and Spaow, 1967; Bennett and Mattsson, 1999; Caon et al., 2003). Befoe the invention of the atomic foce micoscope (AFM), the suface pofile was usually measued with a mechanical pofile that scans the suface line-by-line. Theefoe, the estimated one-dimensional slope distibution function may miss impotant infomation of the suface isotopy. On the othe hand, the AFM can povide detailed infomation of the topogaphy of a small aea on the micoough sufaces with a vetical esolution of sub-nanometes and a lateal esolution aound 10 nm (Wiesendange, 1994). Consequently, it is possible to evaluate the aea statistics fom the AFM topogaphy measuement. Although attention has been paid to compae the 29

44 suface statistics detemined fom the topogaphy measuements to those obtained fom the light scatteing expeiments (Cao et al., 1991; Bawolek et al., 1993; Stove et al., 1998; Nee et al., 2000), little has been done to coelate the aea statistics evaluated fom the AFM topogaphic data to the measued BRDF fo elatively ough sufaces. In geneal, suface oughness is assumed to satisfy the Gaussian statistics in the deivation of the BRDF model and fo the suface geneation in the Monte Calo simulation (Beckmann and Spizzichino, 1987; Tang et al., 1999). Futhemoe, the suface statistics of the two-dimensional ough suface ae mostly assumed to be isotopic so that the autocoelation function is independent of the diection. Vey few papes have been devoted to the BRDF of non-gaussian and anisotopic two-dimensional ough sufaces. Shen et al. (2001) found that the BRDF models could not pedict the subsidiay peak in the measued BRDF of the ough side of a silicon wafe, although easonable ageement existed between the measuement esults and the model pedictions within a lage angula egion aound the specula diection. This disageement could be caused by the Gaussian distibution applied in the BRDF model. Theefoe, it is impotant to examine the actual suface statistics of the ough side of silicon wafes so that a easonable explanation may be povided fo the occuence of the subsidiay peak. The BRDFs of both metal and dielectic sufaces at vaious wavelengths and tempeatues have been epoted. Dolen (1992) investigated the wavelength effect and diectional dependence of adiative popeties fo spacecaft themal contol mateials. Fod et al. (1995) used a Fouie tansfom infaed spectomete to measue BRDF in the nea-infaed and infaed wavelength anges. Thei esults cove a boad spectal ange fom 2.5 to 15 µm with a low angula esolution, which is due to the lage solid angle of 30

45 the instument. Roy et al. (1993) measued the diectional eflectance of seveal dielectic mateials at tempeatues up to 1100 C. The measued BRDF can be inputted into the Monte Calo simulation to impove the accuacy of the modeling. Zawoski et al. (1996b) incopoated the measued BRDF data fo paints in the simulation of the spatial distibution of light though a gap made of two painted plates. A numbe of techniques had to be intoduced to intepolate o extapolate the expeimental data due to the esolution and limitation of the measued BRDF. Reasonable ageement was found between the measued BRDF and the pedicted esult. Zhou et al. (2002) studied the BRDF of a ough silicon wafe and the appaent emissivity of the wafe in a RTP funace. They found that the pedicted BRDFs show simila magnitudes and tends to the measued BRDF. 31

46 CHAPTER 3 SURFACE ROUGHNESS CHARACTERIZATION In most of the published liteatue, the height distibution function and the slope distibution function of andom ough sufaces ae modeled as the Gaussian function. Howeve, the Gaussian distibution may miss impotant featues of natual sufaces because this function does not allow any abupt event in the apidly deceasing tails (Guéin, 2002). Anothe assumption fo the two-dimensional ough suface is that the statistics ae independent of the diection, i.e., the suface is isotopic. Nevetheless, the statistics of eal sufaces can show some extent of anisotopy (Wad, 1992). Since light scatteing is stongly dependent on the suface oughness statistics, it is cucial to undestand the tue statistics of a ough suface. 3.1 Roughness Paametes and Functions Many paametes and functions ae available to descibe the suface oughness quantitatively. The following shows the definition of oughness paametes and functions that will be used in this thesis. Note that these definitions ae applicable fo the onedimensional ough suface. Nevetheless, they may be still valid if the two-dimensional ough suface is consideed as a combination of line sections. The suface topogaphy can be epesented as the vaiation of height with displacement along the sampling diection, z(x). The mean suface is detemined by z 1 l = lim z( x) dx l l (3.1) 0 whee l is the total sampling length. The aithmetic aveage oughness R a is defined by 32

47 R a 1 l = lim z( x) l l 0 z dx (3.2) The oot-mean-squae (ms) oughness σ is defined by σ = 2 [ z( x) z ] dx 1 l lim l l (3.3) 0 R a and σ ae the most commonly used oughness paametes. Howeve, they only povide infomation on the oughness amplitude. The steepness of suface aspeities on two sufaces may be vey diffeent although R a o σ can be the same (Stove, 1995). The ms slope ζ ms can tell the steepness of suface aspeities. The definition of ζ ms is 2 [ ζ( x ] dx 1 l ζms = lim l l ) ζ (3.4) 0 whee ζ(x) is the slope (dz/dx) and ζ is the mean slope. The height distibution function is applied to descibe the faction of suface heights fom a given height to a small incement. The definition of the slope distibution is vey simila to that of the height distibution function. The height distibution function is elated to the beaing aea atio in machine science (Thomas, 1999). The twodimensional slope distibution has been used to descibe the oientation of objects on nano/micomete scales (Inoue et al., 1996; Schleef et al., 1997; Hegeman et al., 1999). The suface pofile can be consideed as a summation of many suface oughness components. Each component is the sine (o cosine) wave with egad to the space coodinate, and the peiod of the wave is defined as the spatial wavelength. The powe spectal density (PSD) function can delineate both the vetical and the spatial extent of the suface oughness components. The definition of the PSD function is given as (Stove, 1995) 33

48 1 l PSD = lim z( x)exp( i2πxf l l 0 x ) dx 2 (3.5) whee f x is the ecipocal of the spatial wavelength. The autocoelation function coelates the deviation fom the mean value with a tanslated vesion by a distance τ (Stove, 1995), 1 1 l ACF( τ) = lim [ z( x) z ][ z( x ) z ]dx 2 L l + τ σ 0 (3.6) The autocoelation length is defined as the value of τ when ACF(τ) is equal to 1/e. The PSD function and the autocoelation function ae a Fouie tansfom pai. The ms oughness can be detemined fom the PSD function, σ = 2 2 fmax fmin PSD( f ) (3.7) x df x Since no instuments can captue suface oughness waves fom zeo fequency to infinite fequency, the integal has to be calculated fom the minimum fequency f min = 1/l to the maximum fequency f max = 1/(2d), whee d is the sampling inteval. 3.2 Roughness Measuement and Data Analysis Seveal silicon wafes have been studied in this thesis. The suface topogaphy of the ough side of these wafes was chaacteized using the state-of-at techniques. We used the Digital Instuments NanoScope scanning pobe micoscope (Multimode and Dimension 3100). The measuements with the AFM wee conducted in the contact mode. The optical intefeometic micoscope (OIM) was also used fo complimentay measuements. We measued some samples with the Veeco WYKO optical pofile (NT 1000 and NT 3300). Measuements wee taken in the vetically scanning mode. 34

49 The suface topogaphic data is mapped into a data aay of size M N, in the two othogonal diections, x and y, espectively. Fist, the mean suface is detemined fo the data aay, and then the ms oughness can be easily calculated. The topogaphic data in the same ow o column in the data aay can be teated as one individual scan. The PSD function and the ACF function can be calculated along eithe the x-diection o the y- diection. The pocedue descibed in the efeence (Stove et al., 1998) is followed to calculate the PSD function fom the pofile data, with the exception that filtes and window functions ae not implemented in ou calculations. Thee ae eithe M o N line scans in one data aay, depending on the diection. The calculated functions ae aveaged among these line scans. The height distibution function is poduced next. Fist, the maximum and minimum heights ae detemined fom the topogaphic data. Then, a finite numbe of bins with equal inteval ae assigned and each data point is checked to find the coesponding bin. The numbe of data points within each bin is divided by the total numbe of data points to obtain the elative fequency. Finally, the elative fequency is nomalized to the pobability density function. The slope distibution function can be poduced similaly. Since both the AFM and the OIM measue the suface topogaphy, suface slope has to be estimated fom the heights of neighboing points. Figue 3.1 shows the schematic of the nodal netwok fo the slope calculation. The one-dimensional slope can be calculated along diffeent diections with espect to the ow diection of the data aay. Fo example, the slope along the x-diection can be calculated fom the data in the same ow, ζ( m, n) = z ( m, n) = z z m + 1, n m, n d (3.8) 35

50 (m, n+1) (m+1, n+1) y x (m, n) (m+1, n) Figue 3.1 Schematic of the nodal netwok fo slope calculation. The slope can also be evaluated along the diagonal, fo instance, ζ( m, n) = z ( m, n) = z m+ 1, n+ 1 2d z m, n (3.9) Howeve, if one assumes that fou neighboing nodes in the data aay fom a suface element, then the oientation of the suface element needs to be pesented as the two-dimensional slopes in the x and y diections, ζ x = dz/dx, ζ y = dz/dy. The fou-node facet may be thought of as two tiangula sufaces with a common side (shown as the dashed line in Figue 3.1). The suface nomals of the two tiangula sufaces can be aveaged to give the mean slope of the fou-node element such that zm+ 1, n zm, n zm+ 1, n+ 1 zm, n+ 1 ζ x( m, n) = + (3.10a) 2d 2d zm, n+ 1 zm, n zm+ 1, n+ 1 zm+ 1, n ζ y ( m, n) = + (3.10b) 2d 2d 3.3 Roughness Statistics of Silicon Wafes In ode to test the unifomity of sufaces and inspect the effect of the scan aea on the measued oughness paametes, we picked up seveal spots on Samples 1 and 2 and 36

51 measued thei suface topogaphies at diffeent scan aeas with the AFM and unde diffeent magnifications with the OIM. Sample 1 is a phosphoous-doped wafe (525 µm thick) and thee is no coating on its backside. Sample 2 has a themal oxide coating on the backside. The thickness of the coating is 140 nm and the thickness of the wafe is 725 µm. The measuement esults fo Samples 1 and 2 ae shown in Figue 3.2. The dashed line denotes the aveaged ms oughness fo the AFM measuements with the same scan aea (Md Nd) and the solid line denotes the aveaged ms oughness fo the OIM measuements at the same magnification. In Figue 3.2a, fom left to ight, fo each goup of data with the same sampling inteval, scan sizes fo the AFM incease fom 5 µm to 40 µm and magnifications fo the OIM decease fom 100X to 10X. When the sampling inteval is less than 200 nm (scan size 150 µm), the deviation of the ms oughness fom the aveage value anges fom 50 nm to 100 nm. When the sampling inteval exceeds 200 nm, the deviation educes to only 30 nm. The degee of deviation elative to the sampling inteval suggests that Sample 1 cannot be statistically analyzed as a unifom suface unless the scan size is geate than 150 µm. Fo the AFM measuements, the ms oughness σ inceases pominently as the scan aea is enlaged. Figue 3.2a illustates that the ate of incease is significant when the scan size inceases fom 5 µm to 20 µm. Since the numbe of sampling points is fixed in these scans, the suface wavelength of the measuable oughness component becomes longe as the sampling inteval inceases. Consequently, the suface fequency of the measuable oughness component becomes lowe. Theefoe, it can be infeed that the highfequency oughness components ae smoothe than the low-fequency oughness 37

52 components fo this sample in the specific egion. When the sampling size appoaches 40 µm, the ms oughness is satuated to a limiting value. At the sampling inteval of 80 nm, which is close to the maximum inteval fo the AFM and the minimum inteval fo the OIM in this measuement, the aveaged σ is 460 nm and 610 nm fo the AFM and the OIM measuements, espectively. Consequently the elative diffeence is about 30% AFM OIP AFM Ave OIP Ave σ (nm) 400 σ (nm) (a) Sample (b) Sample Sampling inteval, d (nm) Sampling inteval, d (nm) Figue 3.2 Compaison of σ measued with the AFM and the OIM: (a) Sample 1; (b) Sample 2. AFM: M = N = 512; OIM: M = 736, N = 480. In Figue 3.2b, fom left to ight, scan sizes fo the AFM incease fom 5 µm to 50 µm, and magnifications fo the OIM decease fom 100X to 10X. The deviation of the AFM measuement is less than 20 nm, and the one fo the OIM measuement is less than 16 nm. Theefoe, Sample 2 is moe unifom compaed to Sample 1. The aveaged σ of the AFM measuement also inceases with the scan size fo Sample 2. Nevetheless, the ate of incease is smalle. Fom Figue 3.2b, it is clea that oughness measued with the OIM is much highe than that with AFM. If the sampling inteval is set at 100 nm in the AFM, the aveaged σ is 148 nm. If the sampling inteval is set at 80 nm in the OIM, the 38

53 aveaged σ is 297 nm. The latte is almost twice as much as the fome. This deviation may be attibuted to the influence of the eflection fom the coating and the substate on the topogaphy measuement. Futhe studies may be needed to systematically investigate the dependence of the oughness paametes on the instuments. The oughness statistics of Samples 3-6 have been systematically chaacteized. Table 3.1 lists the popeties of these wafes. The suface oughness was measued using the AFM in scan aeas of µm 2 and µm 2. Figue 3.3 shows the suface images of Samples 3-6. Thee is no obvious diffeence between the suface images fo these samples. It can be seen that thee ae a lot of micofacets on these sufaces and that the lateal dimension of these micofacets is aound a few micometes. Table 3.1 Wafe popeties of studied samples. Sample Numbe Gowth Method FZ a CZ b FZ CZ Doping Type N P N P Resistivity Range Ω cm a. floating-zone method b. Czochalski method Thickness, µm Table 3.2 lists the oughness paametes calculated fom the topogaphic data within an aea of µm 2. The listed values ae the aveage values and the standad deviations of thee measuements at diffeent positions on each wafe. The ms oughnesses ae between 0.51 µm and 0.61 µm, which ae compaable to the wavelength 39

54 (a) Sample 3 (b) Sample 4 (c) Sample 5 (d) Sample 6 Figue 3.3 AFM suface images: (a) Sample 3; (b) Sample 4; (c) Sample 5; (d) Sample 6. 40

55 of the incident lase beam. Sample 6 is slightly smoothe than the est. The diffeences in the ms oughnesses fo Samples 3, 4, and 5 ae within the standad deviation of the measued σ. The slope is calculated along diffeent diections in the measued data aay. The ms slope ζ ms aveaged along the ow and column diections is vey close to that aveaged along the two diagonals. Theefoe, it is insufficient to detemine whethe o not the suface is isotopic based on the ms slopes. Note that fo a suface that follows the Gaussian statistics, ζ ms = 2 σ/ τ, which is not the case fo the measued samples. Among the measued samples, ζ ms of Sample 6 is the smallest while ζ ms of Sample 4 is 1 the lagest. The aveage inclination angle tan ( ζ ) ms is 15 fo Sample 6 and 22 fo Sample 4. Table 3.2 Roughness paametes of studied samples. Sample Numbe σ, µm 0.578± ± ± ±0.027 ζ ms (along ow/column) 0.334± ± ± ±0.007 ζ ms (along diagonals) 0.326± ± ± ±0.007 τ, µm (along ow/column) 3.219± ± ± ±0.683 Figue 3.4 displays the powe spectal density function (PSD) and the autocoelation function (ACF) fo Sample 3, calculated fom two diffeent scan sizes. The fist data point in the PSD plot is not shown. When the suface fequency f x is highe than 0.05 µm -1, the PSD function calculated fom a scan aea of µm 2 is lage than that fom a scan aea of µm 2. Howeve, the minimum f x is 0.02 µm -1 fo 41

56 the scan aea of µm 2. Accoding to Eq. (3.7), the ms oughnesses ae µm and µm, espectively. The elative diffeence between these values is only 3%. As shown in Figue 3.4b, the diffeence between the ACFs fo diffeent scan sizes is insignificant. The autocoelation lengths ae appoximately 3.1 µm fo the scan aea of µm 2 and 3.2 µm fo the scan aea of µm 2, espectively µm 100 µm µm 100 µm PSD (µm 3 ) ACF(τ) (a) 0.2 (b) Fequency, f x (um -1 ) Tanslation length, τ (µm) Figue 3.4 PSD (a) and ACF (b) functions fo Sample 3. The dotted line in (b) epesents ACF = 1/e. Figue 3.5 shows histogams of height distibution fo Samples 3 and 5. The height distibutions fo Samples 4 and 6 show the same tend with those in Figue 3.5, and theefoe, they ae not shown to avoid the edundancy. The height of the mean suface is zeo and the aea unde each cuve, i.e., the cumulative pobability, is equal to unity. The solid cuve epesents the height distibution (i.e., pobability density) obtained fom the topogaphic data while the dashed cuve is the Gaussian function calculated with a standad deviation equal to the ms oughness σ fo each sample. Some common deviations exist between the measued distibution and the Gaussian. The most 42

57 (a) Sample 3 σ = 0.58 µm (b) Sample 5 σ = 0.61 µm Measued Gaussian Height distibution Height distibution z (µm) z (µm) Figue 3.5 Height distibution functions: (a) Sample 3; (b) Sample 5. Slope distibution, p 1 (ζ x ) (a) Sample 3 ζ ms = AFM data along diagonals AFM data along ow and column Gaussian Slope distibution, p 1 (ζ x ) 2 1 (b) Sample 5 ζ ms = AFM data Gaussian Slope, ζ x Slope, ζ x Figue D slope distibutions of Sample 3 (a) and Sample 5 (b). 43

58 pobable height is shifted towads ight fom the standad Gaussian function that is symmetic at z = 0. The (expeimentally obtained) pobability density is highe than the Gaussian function fo z < 1.2, suggesting that thee ae moe deep valleys in the actual sufaces. A cossove occus within 1.2 < z < 1.0, and then the pobability density is lowe than the Gaussian function until z 0. Aftewads, the pobability density continues to go up and eaches a maximum aound z = 0.2 befoe it goes down and eventually falls below the Gaussian function. Accoding to Bennett and Mattsson (1999), the measued sufaces ae said to be negatively skewed with moe deep valleys and less high peaks than a pefect Gaussian suface. Figue 3.6 shows the one-dimensional (1-D) slope distibutions fo Samples 3 and 5. Fo Sample 3, the calculation shows that the slope distibutions along the ow and the column diections ae vey close to each othe, as ae the slope distibutions along the two diagonals. Howeve, thee is significant diffeence between the fome categoy and the latte categoy. The slope distibutions aveaged within each categoy ae plotted in Figue 3.6a, whee ζ x is used in a boad sense to indicate the slope along the specified diections. The solid line epesents the aveage slope distibution ove the diagonals and the dashed line epesents that ove the ow and column. The slope distibutions show some extent of symmety about ζ x = 0 and the mean slope is almost zeo. The ms slope of the ough suface is appoximately fo both categoies. A Gaussian distibution with a standad deviation of is shown as the dash-dot line. Although the height distibution of this ough suface is close to the Gaussian, the measued slope distibutions deviate significantly fom the Gaussian distibution. The peak at ζ x = 0 in the diagonal slope distibution (solid line) is much highe than that in the Gaussian. 44

59 Futhemoe, instead of deceasing monotonically, thee ae two side peaks at ζ x ±0.47 in the solid line, and the magnitude of these peaks is about one-fouth of that at ζ x = 0. The slope distibution calculated along the ow and column (dashed line) has a lowe peak than that calculated along the diagonals, and the side peaks appea at ζ x ±0.33, close to the cente than those in the solid line. On the othe hand, the slope distibution functions fo Sample 5 ae almost the same no matte whethe the slope is calculated along the ow and column, o along the diagonals. Theefoe, they ae not distinguished in Figue 3.6b. In addition, some deviation can be obseved between the calculated slope distibution and the coesponding Gaussian function. Figue 3.7 plots the two-dimensional (2-D) slope distibutions fo the measued sufaces in the ζ x -ζ y coodinate system. It can be obseved that the slope distibution functions of Samples 3 and 4 ae significantly diffeent fom those of Samples 5 and 6. The slope distibutions fo Samples 3 and 4 ae clealy anisotopic; howeve, the slope distibutions fo Samples 5 and 6 ae nealy isotopic. Because the height distibution of Samples 3 and 4 is close to the Gaussian, it is supising to notice that thee ae side peaks in the slope distibution. As shown in Figue 3.7a and Figue 3.7b, thee ae fou side peaks located at ζ ζ besides the dominant peak located at the cente (ζ x x y = ζ y = 0). The side peaks have simila magnitudes. The magnitude of the side peaks is about one-sixth that of the main peak in Figue 3.7a, and about one-fouth in Figue 3.7b. The side peaks ae located symmetically aound the cental peak, and the coss-sections passing though the planes of ζ x ± ζ y = 0 can nealy bisect these side peaks. A idge along ζ x 0.8 and anothe one along ζ y 0.8 can be obseved fo some samples. These idges ae independent of the otation and measuement spot on the sample. Theefoe, it is 45

60 believed that the idges ae atifacts associated with the geomety of the AFM tip, which is usually a evese pyamid but may not necessaily be symmetic. If the micofacet has a vey lage inclination angle, the AFM tip may not touch the suface of the micofacet, and instead, the AFM may image the shape of the tip. Theefoe, even though thee exist micofacets with slopes ζ x > 0.8 o ζ y < 0.8, they will not contibute to the slope distibution. The effect of the idges in the slope distibution on the pedicted BRDF will be discussed in Chapte 5. The slope distibution functions shown in Figue 3.7c and Figue 3.7d ae vey simila, except that the value at the cente is highe in the latte. Thee is only one dominant peak at the cente in the slope distibutions. The deviation in the coss-sections of the two-dimensional slope distibution at diffeent azimuthal angles is insignificant. The featues shown in the slope distibutions fo Samples 3 and 4 can be elated to the cystal stuctue of silicon. In the coss-section of the 2-D slope distibutions passing though the planes of ζ x ± ζ y = 0 and ζ x + ζ y = 0, the side peaks ae located at ζ x ± = ± If the slope of a micofacet is 0.50, then the coesponding inclination angle is 27. This implies that thee ae a numbe of micofacets that ae tilted aound 27 with espect to the mean suface. By examining the cystalline stuctue of silicon, the side peaks should be associated with the {311} planes, since the angles between any of the fou {311} planes and the (100) plane is 25.2 (Resnik et al., 2000). The pojections of the suface nomals of the {311} planes to the (100) plane is in the diection of <011> and the pojections ae othogonal, coesponding well to the symmetic featues shown in the coss-sections at ζ x ± ζ y = 0. Theefoe, the occuence of the side peaks may be attibuted to the last pocessing of the ough side of the silicon 46

61 wafes. If a smooth single-cystal suface is etched by chemicals, the fomed facet should have defined oientations along the cystalline planes. The position of the fomed facet can be andom o well contolled (Zhao et al., 1996; Resnik et al., 2000). Fo the ough side of the silicon wafes, usually the wet chemical etching is applied to emove micoscopic cacks and suface damages caused by the mechanical pocesses. Vaious chemicals such as hydofluoic, nitic, and acetic acids, and sodium hydoxide ae geneally used. It is not clea how the slope distibution function would look like befoe the chemical etching. The esidual oughness statistics due to the pocesses such as saw cutting and mechanical lapping/gounding might be andom. The chemical etching might have modified some micofacets with a pefeence to the {311} plane, leaving andomly distibuted micofacets with pedominant oientations othe than the (100) plane (coesponding to ζ x = ζ y = 0). On the othe hand, the slope distibution functions fo Samples 5 and 6 do not have side peaks. The eason may be attibuted to the diffeent pocessing conditions such as chemical solution, tempeatue, duation, and so on. 47

62 (a) Sample 3 (b) Sample 4 (c) Sample 5 (d) Sample 6 Figue D slope distibution functions: (a) Sample 3; (b) Sample 4; (c) Sample 5; (d) Sample 6. 48