Recent Developments in the X-ray Reflectivity Analysis

Transcription

1 merican Journal of Psics and pplications 6; 4: 7-49 ttp:// doi:.648/.apa.64. ISSN: Print; ISSN: Online eview rticle ecent Developments in te X-ra eflectivit nalsis Yosiau Fuii Center for Supports to esearc and ducation ctivities Kobe Universit Nada Kobe Japan mail address: To cite tis article: Yosiau Fuii. ecent Developments in te X-ra eflectivit nalsis. merican Journal of Psics and pplications. Vol. 4 No. 6 pp doi:.648/.apa.64. eceived: Februar 8 6; ccepted: Februar 6 6; Publised: Marc 7 6 bstract: X-ra reflectivit X is a powerfull tool for investigations on surface and interface structures of multilaered tin film materials. In te conventional X analsis te X-ra reflectivit as been calculated based on te Parratt formalism accounting for te effect of rougness b te teor of Nevot-Croce conventionall. owever te calculated results ave sown often strange beaviour were interference effects would increase at a roug surface. Te strange result ad its origin in a serious mistae tat te diffuse scattering at te roug interface was not taen into account in te equation. Ten we developed new improved formalism to correct tis mistae. owever te estimated surface and interface rougnesses from te -ra reflectivit measurements did not correspond to te TM image observation results. For deriving more accurate formalism of X we tried to compare te measurements of te surface rougness of te same sample b atomic force microscop FM ig-resolution uterford bacscattering spectroscop BS and X. Te results of analsis sowed tat te effective rougness measured b X migt depend on te angle of incidence. Ten we introduced te effective rougness wit depending on te incidence angle of X-ra. Te new improved X formalism derived more accurate surface and interface rougness wit depending on te sie of coerent X-ras probing area and derived te rougness correlation function and te lateral correlation lengt. In tis review an improved X formalism considering te diffuse scattering and te effective rougness is presented. Te formalism derives an accurate analsis of te -ra reflectivit from a multilaer surface of tin film materials. Kewords: X-ra eflectivit Surface and Interface ougness Multilaered Tin Film Buried Interface. Introduction X-ra scattering spectroscop is a powerful tool for investigations on roug surface and interface structures of multilaered tin film materials [-4] and X-ra reflectometr is used for suc investigations of various materials in man fields. [5 6-4] In man previous studies in X-ra reflectometr te X-ra reflectivit was calculated based on te Parratt formalism [] coupled wit te use of te teor of Nevot and Croce to include rougness. [] owever te calculated results of te X-ra reflectivit done in tis wa often sowed strange results were te amplitude of te oscillation due to te interference effects would increase for a rouger surface. Because te -ra scattering vector in a specular reflectivit measurement is normal to te surface it provides te densit profile solel in te direction perpendicular to surface. Specular reflectivit measurements can ield te magnitude of te average rougness perpendicular to surface and interfaces but cannot give information about te lateral etent of te rougness. In previous studies te effect of rougness on te calculation of te -ra reflectivit onl too into account te effect of te densit canges of te medium in a direction normal to te surface and interface. On te oter and diffuse scattering can provide information about te lateral etent of te rougness. In contrast to previous calculations of te -ra reflectivit in te present analsis we consider te effect of a decrease in te intensit of penetrated -ras due to diffuse scattering at a roug surface and roug interface. In tis review we sow tat te strange result as its origin in a currentl used equation due to a serious mistae in wic te Fresnel transmission coefficient in te reflectivit equation is increased at a roug interface and te increase in te transmission coefficient completel overpowers an decrease in te value of te reflection coefficient because of a lac of consideration of diffuse scattering. Te mistae in Nevot and

2 merican Journal of Psics and pplications 6; 4: Croce s treatment originates in te fact tat te modified Fresnel coefficients were calculated based on te teor wic contains te -ra energ conservation rule at surface and interface. In teir discussion te transmission coefficients were replaced approimatel b te reflection coefficients b te ignoring diffuse scattering term at te roug interface and according to te principle of conservation energ at te roug interface also. Te errors of transmittance witout te modification cannot be ignored. It is meaningless to tr to precisel matc te numerical result based on a wrong calculating formula even to details of te reflectivit profile of te eperimental result. Tus because Nevot and Croce s treatment of te Parratt formalism contains a fundamental mistae regardless of te sie of rougness tis approac needs to be corrected. Ten we developed new improved formalism to correct tis mistae. Te calculated reflectivit obtained b te use of tis accurate reflectivit equation gives a psicall reasonable result and sould enable te structure of buried interfaces to be analed more accuratel. owever te estimated surface and interface rougnesses from te -ra reflectivit measurements did not correspond to te TM image observation results. Ten for deriving more accurate formalism of X we tried to compare te measurements of te surface rougness of te same sample b atomic force microscop FM ig-resolution uterford bacscattering spectroscop BS and X. Te results of analsis sowed tat te effective rougness measured b X migt depend on te angle of incidence. Ten we introduced te effective rougness wit depending on te incidence angle of X-ra. Te new improved X formalism derived more accurate surface and interface rougness wit depending on te sie of coerent X-ras probing area and derived te rougness correlation function and te lateral correlation lengt. In tis review an improved X formalism considering te diffuse scattering and te effective rougness is presented. Te formalism derives an accurate analsis of te -ra reflectivit from a multilaer surface of tin film materials. Tis article is te review article tat summaried te researc articles [33-4] and te later stud.. X-ra eflectivit nalsis In te first subsection we consider te calculation of te -ra reflectivit from a multilaer material b te Parratt formalism [] and in te net subsection te calculation of te -ra reflectivit wen rougness eists in te surface and te interface is considered... X-ra eflectivit from a Multilaer Material wit a Flat Surface and Flat Interface Te intensit of -ras propagating in te surface laers of a material i.e. te electric and magnetic fields can be obtained from Mawell s equations. [9] Te effects of te material on te -ra intensit are caracteried b a comple refractive inde n wic varies wit dept. We divide a material in wic te densit canges continuousl wit dept into N laers wit an inde. Te comple refractive inde of te -t laer is n. Te vacuum is denoted as and n. Te ticness of te -t laer is te ticness of te bottom laer being assumed to be infinite. Te reflectance of an N-laer multilaer sstem can be calculated using te recursive formalism given b Parratt. [] In te following we sow in detail te process of obtaining Parratts epression and furter sow tat tis epression requires conservation of energ at te interface. We go on to sow tat te dispersion of te energ b interface rougness cannot be correctl accounted for Parratts epression. Following tat approac let n be te refractive inde of te -t laer defined as n δ iβ were δ and β are te real and imaginar parts of te refractive inde. Tese optical constants are related to te atomic scattering factor and electron densit of te -t laer material. For -ras of wavelengt λ te optical constants of te -t laer material consisting of N i atoms per unit volume can be epressed as λ re λ re δ f in i β f in i π π i were r e is te classical electron radius and f i and f i are te real and imaginar parts of te atomic scattering factor of te i-t element atom respectivel. We tae te vertical direction to te surface as te ais wit te positive direction pointing towards te bul. Te scattering plane is made te - plane. Te wave vector of te -t laer is related to te refractive inde n of te -t laer b ω c n i const 3 and as tis necessitates tat te -direction components of te wave vector are constant ten te -direction component of te wave vector of te -t laer is n. 4 In te -t laer i.e. in vacuum n π ω. 5 λ c In te -t laer te components of te wave vector are cosθ n cos θ 6 Te electric field of -ra radiation at a glancing angle of incidence θ is epressed as ep[ i r ω ]. 7 t Te incident radiation is usuall decomposed into two

3 9 Yosiau Fuii: ecent Developments in te X-ra eflectivit nalsis geometries to simplif te analsis one wit te incident electric field parallel to te plane of incidence p-polariation and one wit perpendicular to tat plane s-polariation. n arbitrar incident wave can be represented in terms of tese two polariations. Tus and correspond to p-polariation and to s-polariation; tose components of te amplitude s electric vector are epressed as p sinθ s cosθ 8 p Te components of te wave vector of te incident -ras are cosθ sinθ 9 Te electric field of reflected -ra radiation of eit angle θ is epressed as were ep[ i r ω ]. t. Because an -ra is a transverse wave te amplitude and te wave vector are ortogonal as follows. 4 cosθ n cos θ. 5 Te relation of te amplitudes and can be found from te continuit equations of te electric fields for te interface between te -t and -t laers as follows noter relation of te amplitudes and can be found from te continuit equations of te magnetic fields for te interface between te -t and -t laers are sown below 9 From te above equations tese amplitudes are related b te Fresnel coefficient tensor for refraction and te Fresnel coefficient tensor for reflection as follows. ere te Fresnel coefficient tensor for refraction at te interface between te -t and -t laers is given b Figure. eflected and transmitted -ras. We consider te relation of te electric field of -ras incident at a flat surface from vacuum te electric field of -ras propagating in te first laer material te electric field of -ras reflected from te surface eit to vacuum and te electric field of -ras propagating toward to te surface in te first laer material as sown in Figure. Te electric fields in te first laer material below te surface are epressed as were ep[ i r ωt ] t ep[ i r ω ] 3 Te Fresnel coefficient tensor for reflection from te interface between te -t and -t laers is given b 3

4 merican Journal of Psics and pplications 6; 4: ere we consider te reflection from a flat surface of a single laer. Te reflection coefficient is defined as te ratio of te reflected electric field to te incident electric field at te surface of te material. Te reflection coefficient from a single-laer flat surface is equal to te Fresnel coefficient for reflection as te following sows 4 In general wen -ras tat are linearl polaried at an angle χ impinge on te surface at an angle of incidence θ te components of te amplitude s electric vector are epressed as sin θ p p sin χ s cos θ cos χ s p te amplitudes of reflected -ra radiation are epressed as ten 5. 6 sin χ sinθ χ cos. 7 sin χ cosθ Te -ra reflectivit is 8 ten ten n n n n n n n n n n n n 3 n n n n 3 cos χ sin χ 3 Taing an average for χ cos χ sin χ 33 / 34 For te reflectivit in te case of s-polaried -ras incident. 35 χ were sin χ sin θ cos χ sin χ cos θ 9 Net we consider te reflection from a flat surface of a multilaer wit flat interfaces. We consider te electric field - of -ras propagating in te --t laer material te electric field of -ras propagating in te -t laer material and te electric field - of -ras reflected from te -t laer material at - of te interface between te --t laer and -t laers as sown in Figure. Figure. eflection and transmission of -ras in te --t -t and -t laers of a multilaer material.

5 3 Yosiau Fuii: ecent Developments in te X-ra eflectivit nalsis Te electric fields - - at te interface between te --t laer and -t laer and te electric fields below te interface between te --t laer and -t laer are epressed as ep[ i ω ] t ep[ i ω ] t ep[ i ω ] t ep[ i ω ]. 36 t Te electric fields of -ras at te interface between te --t laer and -t laer can be formall epressed as follows were - is te Fresnel coefficient tensor for reflection from te interface between te - and laers and - is te Fresnel coefficient tensor for refraction at te interface between te - and laers. In addition te electric field witin te -t laer varies wit dept as follows ep i 39 ep i. 4 Te amplitudes and at te -t laer are derived from te above equations for te interface between te - and laers as follows ep i ep i 4 ep i 4 Tis relation is epressed b te following matri ep i ep i 43 ere te Fresnel coefficient tensor for refraction at te interface between te --t and -t laers is given b Te Fresnel coefficient tensor for reflection from te interface between te - and laers is given b Te amplitudes - and - of te electric fields - - at te -t laer and te amplitudes and of te electric fields at te -t laer are related b te following equations; ep i ep i 46 For s-polariation te Fresnel coefficients are Ten te relations between te amplitudes - - and at te interface of te --t and -t laers are epressed as follows ep i ep i Te reflection coefficient is defined as te ratio o of te reflected electric field to te incident electric field at te surface of te material and is given b. 5 Te reflection coefficient - of te electric field - to te electric field - at te interface of --t laer and -t laer is - 5 and te ratio - is related to te ratio as follows epi 5 ere from te relation between te Fresnel coefficient for reflection and te Fresnel coefficient for refraction -

6 merican Journal of Psics and pplications 6; 4: we can formulate te following relationsip epi 55 It is reasonable to assume tat no wave will be reflected bac from te substrate so tat 56 N N Ten te -ra reflectivit is simpl 57.. Previous Calculations of X-ra eflectivit Wen ougness ists at te Surface and Interface Wen te surface and interface ave rougness te Fresnel coefficient for reflection is reduced b te rougness. [5-8 -3] Te effect of te rougness was previousl put into te calculation based on te teor of Nevot and Croce. [] Te effect of suc rougness was taen into account onl troug te effect of te canges in densit of te medium in a vertical direction to te surface and interface. Wit te use of relevant rougness parameters lie te root-mean-square rms rougness - of te -t laer te reduced Fresnel reflection coefficient for s-polariation is transformed as sown below - ep - 58 and te -ra reflectivit is calculated using te following equation epi N N. 59 Figure 3 sows te reflectivit from a Gas-covered silicon wafer solid line sows te calculated result in te case of flat surface and flat interface dased line sows te calculated result in te case tat te surface as an rms rougness of 4 nm and dotted line sows te equivalent result wen te surface and interface bot ave an rms rougness of 4 nm. In te latter case te reflectivit curve dots decreases more quicl tan tat in Figure 3. owever te ratio of te oscillation amplitude to te value of te reflectivit does not decrease. It seems unnatural tat te effect of interference does not also decrease at a roug surface and interface because te amount of coerent -ras sould reduce due to diffuse scattering at a roug surface and interface. eflectivit nm nm Figure 3. Calculated reflectivit from a Gas laer wit a ticness of 48 nm on a Si substrate. Te solid curve is for a flat surface and a flat interface. Te dased curve is for a surface rougness of 4 nm and a flat interface wile te dotted curve is for a surface rougness of 4 nm and interface rougness of 4 nm. In te reflectivit curve dased line for a surface rougness of 4 nm and wit a flat interface te ratio of te oscillation amplitude to te sie of te reflectivit near an angle of incidence of.36 is muc larger tan te reflectivit of te flat surface in Figure. It seems ver strange tat te interference effects would increase so muc at a roug surface. eflectivit Figure 4. Calculated reflectivit from a Gas laer wit a ticness of 48 nm on a Si substrate. In te calculation te interface rougness is nm. Tree calculated results are sown for a Gas surface wit an rms rougness of 3.5 nm 4 nm and 4.5 nm. To probe tese effects furter we ten recalculated te reflectivit for surface rougness of 3.5 nm 4 nm and 4.5 nm and wit a flat interface. Tose calculated reflectivit results are sown in Figure 4. Te ratio of te oscillation amplitude to te reflectivit near an angle of incidence of.36 in calculated reflectivit is larger in all cases tan tat of te reflectivit in te case of a flat surface in Figure. For most angles of incidence witin tis range te reflectivit of te surface wit a rougness of 4 nm is near te mean value of te reflectivit of surfaces wit rougnesses of 3.5 nm and 4.5 nm. owever near an angle of incidence of.36 te reflectivit of te surface wit a 4 nm nm 4 nm 4 nm θ deg nm θ deg 3.5nm 4.nm 4.5nm

7 33 Yosiau Fuii: ecent Developments in te X-ra eflectivit nalsis rougness of 4 nm is ver muc attenuated compared to tat same average. It seems ver strange tat te reflectivit of te average rougness as a value quite different from te mean value of te reflectivit of eac rougness because te value of te rougness is not te value of te amplitude of a roug surface but te standard deviation value of various amplitudes of roug surface. eflectivit Figure 5. X-ra reflectivit from a silicon wafer covered wit a tin nm tungsten film calculated b te teor in use prior to tis wor. Solid line sows te case of a flat surface. Dased line sows te case of a surface wit an rms surface rougness of.3 nm. wor. Te ratio of te oscillation amplitude to te value of te reflectivit from a surface wit a rms surface rougness of.3 nm dased line does not decrease near an angle of incidence of.8 but increase. Tis result is strange and not reasonable..3. ffect of ougness on X-ra eflectivit of Multilaer Surface We now consider te above strange result of te -ra reflectivit wic was calculated based on te Parratt formalism [] wit te use of te Nevot and Croce approac to account for rougness. [] In tat calculation te -ra reflectivit is derived using te relation of te reflection coefficient - and as follows epi 6 Figure 5 sows te reflectivit from a tungsten-covered silicon wafer calculated b te teor in use prior to tis epi 6 were te reduced Fresnel reflection coefficient tat taes into account te effect of te rougness is as sown below ep. 6 owever te relationsip between te reflection coefficients - and was originall derived as te following equation ere te following conditional relations between te Fresnel coefficient for reflection and refraction are relevant to te above equation and ten i.e nm nm W nm nm Si... θdeg Te Fresnel coefficients for refraction at te roug interface are derived using te Fresnel reflection coefficient as follows ep > 67 ep. 68 Terefore te Fresnel coefficients for refraction at te roug interface are necessaril larger tan te Fresnel coefficient for refraction at te flat interface. Te resulting increase in te transmission coefficient completel overpowers an decrease in te value of te reflection coefficient. Tese coefficients for refraction obviousl contain a mistae because te penetration of -ras sould decrease at a roug interface because of diffuse scattering.

8 merican Journal of Psics and pplications 6; 4: We propose tat te unnatural results in te previous calculation of te -ra reflectivit originate from te fact tat diffuse scattering was not considered. In fact equation 63 contains te -ra energ conservation rule at te interface as te following identit equation for te Fresnel coefficient. 69 ere we consider te energ flow of te -ra. In electromagnetic radiation te energ flow is equal to te Ponting vector 4 p 7 were µ ε 7 and ε and µ are te dielectric and magnetic permeabilit. Te Ponting vector is terefore 4 4 p ϖµ µ ε µ ε. 7 Ten te Ponting vector tat crosses te interface is S S S p d d d µω µω µω. 73 Te amplitudes - and - of te electric fields - - at te -t laer and amplitudes and of te electric fields at te -t laer are related b te following equations; i i ep ep 74 Wen 75 we can describe te above equation as i i ep ep 76 From te determinant of te refraction matri ep ep i i 77 ten i.e. te -ra energ flow is conserved at te interface. Wen te Fresnel coefficients at te roug interface obes te following equations 8 tese coefficients fulfil -ra energ flow conservation at te interface and so diffuse scattering was not considered at te roug interface. Tis conservation epression sould not appl an longer wen te Fresnel reflection coefficient is replaced b te reduced coefficient wen tere is rougening at te interface. Terefore calculating te reflectivit using tis reduced Fresnel reflection coefficient in equation 6 will incorrectl increase te Fresnel transmission coefficient i.e. <. Te penetration of -ras sould decrease at a roug interface because of diffuse scattering. Terefore te identit equation for te Fresnel coefficients become

9 35 Yosiau Fuii: ecent Developments in te X-ra eflectivit nalsis D <. 83 were D is a decrease due to diffuse scattering. Ten in te calculation of -ra reflectivit wen tere is rougening at te surface or te interface te Fresnel transmission coefficient sould be used for te reduced coefficient. Several teories eist to describe te influence of rougness on -ra scattering. [5-8 -3] Wen te surface and interface are bot roug te Fresnel coefficient for refraction as been derived in several teories. [5-8 -3].4. Te efractive Fresnel Coefficient of a oug Interface Used in Previous eflectivit Calculations Initiall we consider te reduced Fresnel coefficient wic is nown as te Croce-Nevot factor. Wen te position of te interface of -t laer and -t laer o fluctuates verticall as a function of te lateral position because of te interface rougness te relations between te amplitudes and are derived b te use of te Fresnel coefficient tensor for refraction and te Fresnel coefficient tensor for reflection as follows ep i ep i ep i 84 ep i ep i ep i 85 ten ep i ep i ep i ep i ep i ep i 86 ep i ep i ep i ep i ep i ep i 87 ep i ep i ep i ep i 88 were 89 9 ten ep i ep i ep i ep i 9 We tae te average value of te matri over te wole area coerentl illuminated b te incident -ra beam. Tis leads to ep i ep i ep i ep i 9 For Gaussian statistics of standard deviation value g ep π 93 f g f d ep f d π 94

10 merican Journal of Psics and pplications 6; 4: ep i g ep i d ep ep i d ep 95 π ep ep ep ep 96 Terefore ep ep ep ep 4 ep 4 ep 97 Ten te Fresnel reflection coefficients are reduced as follows 98 ep owever te Fresnel refraction coefficients increase as follows. 99 ep ep ep ep 4 4 Te modified Fresnel refraction coefficients corresponds to equation.9 in p. of ol[4] equation 8.4 in p.4 of Daillant [5] and equation.7 in p.9 of Saurai []. owever no one obtained te epression corresponding to. It is peculiar tat and are asmmetrical. It comes to cause a different result if -t laer and -t laers are replaced and calculated. Terefore tis derived sould not be used to calculate te reflectivit from roug surfaces and interfaces. Te derived Fresnel refraction coefficients increase. Tis increase in te transmission coefficient completel overpowers an decrease in te value of te reflection coefficient as te following ep 3 > Moreover if te deformation modulus of is assumed to be te left side of equation eceeds unit and terefore equation is obviousl wrong. In Nevot and Croce s treatment of te Parratt formalism for te reflectivit calculation including surface and interface rougness [] te relations of te Fresnel coefficients between reflection and transmission as quations 63 8 and were not sown. Furtermore te modification of te Fresnel coefficients according to Nevot and Croce as been used for onl surface and interface reflection. owever te modification of te transmission coefficients as an important role wen te rougness of te surface or interface is ig and te effect of diffuse scattering due to tat rougness sould not be ignored as sown in equation 83. Te error in Nevot and Croce s treatment [] originates in te fact tat te modified Fresnel coefficients was calculated based on te Parratt formalism wic contains te -ra energ conservation rule at te surface and interface. In te discussion on pp of Nevot and Croce s [] teir Fresnel coefficients at te roug interface fulfil -ra energ flow conservation at te interface and so diffuse scattering was ignored at te roug interface. In teir discussion te transmission coefficients t and t I were replaced approimatel b te reflection coefficients r and r I b te ignoring diffuse scattering term and according to te principle of conservation energ. Te reflection coefficient r at te roug interface sould be epressed as a function of te reflection coefficient r I and transmission coefficient t I. owever te reflection coefficient r at te roug interface was epressed onl b te reflection coefficient r I wile te transmission coefficient t I ad alread been replaced b te reflection coefficient r I b te ignoring diffuse scattering term in te relationsip based on te principle of te conservation of energ. Tus te reflection coefficient r at

11 37 Yosiau Fuii: ecent Developments in te X-ra eflectivit nalsis te roug interface as equation of p.77 in Nevot and Croce [] ad been epressed wit te reflection coefficient r I onl and tis results in te equation was also sure to include te conservation of energ. Te resulting increase in te transmission coefficient completel overpowers an decrease in te value of te reflection coefficient at te roug interface. Tus because Nevot and Croce s treatment of te Parratt formalism contains a fundamental mistae regardless of te sie of te rougness results using tis approac cannot be correct. Te sie of te modification of te transmission coefficient is one-order smaller tan tat of reflection coefficient but te sie of transmission coefficient is one-order larger tan te reflection coefficient at angles larger tan critical angle. Tus te errors of transmittance witout te modification cannot be ignored. Of course tere are cases were tat Nevot and Croce s treatment can be applied. owever teir metod can be applied onl to te case were tere is no densit distribution cange at all in te direction parallel to te surface on te surface field side and onl wen te scattering vector is normal to te surface. tpical eample of surface medium to wic tis model can be applied is one were onl te densit distribution cange in te vertical direction to te surface eists as caused b diffusion etc. In suc a special case Nevot and Croce s treatment can be applied witout an problem. owever because a general multilaer film alwas as structure in a direction parallel to te surface field side Nevot and Croce s epression fails even wen te rougness is etremel small. Te use of onl Fresnel reflection coefficients b Nevot and Croce is a fundamental mistae tat does not depend on te sie of te rougness..5. Te efractive Fresnel Coefficient of a oug Interface Used in New eflectivit Calculations To proceed we terefore reconsider te derivation of te average value of te matri as te same derivation of qs wen we consider te reduced Fresnel coefficient wic is nown as te Croce-Nevot factor. Wen te position of te interface of te -t laer and -t laer o fluctuates verticall as a function of te lateral position because of te interface rougness te relations between te electric fields are derived b te use of te Fresnel coefficient tensor for refraction and te Fresnel coefficient tensor for reflection as follows were 4 ep[ i ] ep[ i ] ep[ i ] ep[ i ] 5 ten ep[ i ] ep[ i ] ep[ i ] ep[ i ] ep[ i ] ep[ i ]. 6 Ten te amplitudes and are derived as follows ep i ep i ep i ep i ep i ep i. 7 Matri description of te relations is as follows ep i ep i ep i ep i 8 epi ep i ep i ep i 9 ten epi ep i ep i ep i

12 merican Journal of Psics and pplications 6; 4: We tae te average value of tis matri. epi ep i ep i ep i For Gaussian statistics of standard deviation value ep ep ep ep ten te Fresnel reflection coefficients are found as follows ep ep. 3 and te Fresnel refraction coefficients are also produced similarl ep ep 4 4 ep ep < 5 Te modified Fresnel refraction coefficients and of 4 corresponds to equation.5 on p.9 of Saurai[]. Te Fresnel refraction coefficients derived b tis metod are reduced and could be used to calculate te reflectivit from roug surface and interfaces. ccordingl we calculated te reflectivit using tese derived Fresnel refraction coefficients. owever te numerical results of tis calculation did not agree wit te eperimental results wen te angle of incidence smaller tan te total reflection critical angle. In tring to account for te reason for tis disagreement it sould be noticed tat our present approac to constructing te reduced reflection coefficient term does not include an reference to te refractive inde of te medium. Furter -ras tat penetrate an interface reflect from te interface below and penetrate te former interface again witout fail. Terefore te refraction coefficient and sould not be separatel treated..6. New Formula for te eflectivit for oug Multilaer Surface Once again we consider process b wic we derive te average value of te matri. Wen te position of te interface of -t laer and -t laer o fluctuates verticall as a function of te lateral position because of te interface rougness te relations between te amplitudes and are sown b te use of te Fresnel coefficient tensor for refraction and te Fresnel coefficient tensor for reflection as follows ep i ep i ep i ep i 6 ep i ep i ep i ep i ep i ep i 7 ep i ep i ep i ep i ep i ep i ep i ep i ep i ep i ep i ep i ep i ep i ep i ep i 8 9

13 39 Yosiau Fuii: ecent Developments in te X-ra eflectivit nalsis ep i ep i ep i ep i ep i ep i gain we tae te average value of tis matri ep i ep i ep i ep i ep i ep i For Gaussian statistics of standard deviation value te Fresnel reflection coefficient are as follows ep i ep ep i ep ep ep i ep ep ep i ep.. 3 Because -ras tat penetrate an interface reflect from te interface below and penetrate former interface again witout fail it is necessar to treat te refraction coefficients and collectivel. ep i ep i ep i ep i ep i ep 4 Ten te Fresnel coefficients and are reduced as follows ep ep ep ep 5 6 Ten ep ep ep ep 7 ep ep ep ep 8 and

14 merican Journal of Psics and pplications 6; 4: ep ep 4 < 9 Te Fresnel refraction coefficients derived b tis metod are reduced and can be used to calculate te reflectivit from roug surface and interface. Terefore we calculate te reflectivit using tese newl-derived Fresnel coefficients in an accurate reflectivit equation of - and as follows epi 3 eflectivit nm nm 4 nm nm 4 nm 4 nm θ deg Figure 6. New calculated reflectivities from a Gas laer wit a ticness of 48 nm on a Si substrate. Te line is for a flat surface and a flat interface. Te dased curve is for a surface rougness of 4 nm and wit a flat interface wile te dotted curve is for a surface rougness of 4 nm and interface rougness of 4 nm. Based on te above considerations we again calculated te -ra reflectivit for te Gas/Si sstem but now considered te effect of attenuation in te refracted -ras b diffuse scattering resulting from surface rougness. Te results are sown as te dased line in Figure 6 for a surface rougness of 4 nm and flat interface and te dotted line sows te calculated result in te case tat te surface and interface bot ave a rms rougness of 4 nm. Te ratio of te oscillation amplitude to te sie of te reflectivit in te reflectivit curve dot in Figure 6 is smaller tan tat of te reflectivit curve Figure 3. In te reflectivit curve dased line te ver large amplitude of te oscillation near an angle of incidence of.36 in Figure 3 as disappeared in Figure 6. Tese results are now psicall reasonable. ll te strange results seen in Figure 3 ave disappeared in Figure 6. It seems natural tat te effect of interference does decrease at a roug surface and interface because te amount of coerent -ras sould reduce due to diffuse scattering. Figure 7 sows te new calculated reflectivit for surface rougnesses of 3.5 nm 4 nm and 4.5 nm and wit a flat interface. t all angles of incidence te reflectivit of te surface rougness of 4 nm is near te mean value of te reflectivit of te surface rougness of 3.5 nm and te reflectivit of te surface rougness of 4.5 nm. Tis result is psicall reasonable because te value of te rougness is te standard deviation value of various amplitudes of roug surface. owever it was difficult to matc te numerical result of -ra reflectivit to te results of TM observation. eflectivit Figure 7. New calculated reflectivit from a Gas laer wit a ticness of 48 nm on a Si substrate. In te calculation te interface rougness is nm. Tree calculated results for a Gas surface wit rougness of 3.5 nm 4 nm and 4.5 nm are sown. eflectivit Figure 8. X-ra reflectivit from a silicon wafer covered wit a tin nm tungsten film calculated b te new calculation tat considered diffuse scattering. Solid line sows te case of a flat surface. Dased line sows te case of a surface wit a rms surface rougness of.3 nm. Net we again calculated te X-ra reflectivit for te W/Si sstem but now considered te effect of attenuation in te refracted X ras b diffuse scattering resulting from surface rougness. owever te reduced refraction coefficient in prior wor varies. [4-8 -3] Ten about te reduced refraction coefficient reduction as same as reflection coefficient was applied now. Figure 8 sows te calculated results wit te use of improved X-ra reflectivit formalism. In te reflectivit curve from a surface wit an rms surface rougness of.3 nm dased line te amplitude of te nm θ deg 3.5nm 4.nm 4.5nm nm nm W nm nm Si... θdeg

15 4 Yosiau Fuii: ecent Developments in te X-ra eflectivit nalsis oscillation in Figure 5 as reduced in Figure 8. Tese results are now psicall reasonable. Te strange results seen in Figure 5 ave disappeared in Figure 8. It seems natural tat te effect of interference does decrease at a roug surface and interface because te amount of coerent X ras sould reduce due to diffuse scattering. 3. TM Observation and X-ra eflectivit Measurement for Surfaces and Interfaces of Multilaered Tin Film Materials Te surface and interfacial rougness of te same sample of multilaered tin film material was measured b transmission electron microscop TM and compared tem wit tose from -ra reflectivit measurements. Te surface sample for eamination was prepared as follows; a Gas laer was grown on Si b molecular beam epita MB. From TM observations te ticness of te Gas laer was 48 nm te root-mean-square rms rougness of te Gas surface was about.8 nm te rms rougness of te interface between Gas and Si was about.7 nm. Figure 9 sows a cross section image of tis Gas / Si sample observed b TM. X-ra reflectivit measurements were performed using a Cu-Kα -ra beam from an 8 W rotating-anode source. Figure sows te measured reflectivit of -ras wave lengt.54 nm from a Gas laer wit a ticness of 48 nm on a silicon wafer. Te decrease in signal for angles larger tan te total reflection critical angle sows oscillations. Tese oscillations are caused b interference between -ras tat reflect from te surface of Gas laer and tose tat reflect from te interface of te Gas laer and Si substrate. Te caracteristics of tese oscillations reflect te surface rougness and te interface rougness. Te angular resolution in te measurement was. degree. Tis resolution is adequatel smaller tan oscillation period about.4 of X. Ten we compared te measurement data wit calculation witout fitting correction. X formalism sown as - ep - epi N N Figure sows te result dots of a calculation based on tese epressions of te reflectivit of -ras from a Gas laer wit a ticness of 48 nm on Si substrate. Te rms rougness of te interface of Gas and Si was set to.7 nm te value derived from te TM observations. Te rms rougness of te Gas surface was set to.8 nm te value derived from te FM measurements. Te agreement of te calculated and eperimental results in Figure is not good. Figure. Measured -ra reflectivit from a silicon wafer covered wit a tin 48 nm Gas laer. Figure. Calculated dots and measured line reflectivit from a Gas laer wit a ticness of 48 nm on a Si substrate. Te surface rougness is.8 nm and te interfacial rougness is.7 nm. Figure 9. Cross section image of Gas / Si b TM observation. t te first we simulated te X data b conventional Te calculated result suggests te following: if te value of te surface rougness and te interfacial rougness in te calculation would be made larger te calculated result will more closel approac te eperimental result. In te TM observation and FM measurements one alf of te pea to pea value of te interface rougness equates to nm and tat of te Gas surface is 4 nm. We ten recalculated te reflectivit values of tis order for te surface rougness and te interface rougness in te calculation. Tree calculated

16 merican Journal of Psics and pplications 6; 4: results for a rougness of Gas surface of 3.5 nm 4 nm and 4.5 nm wit an interface rougness of nm are sown in Figure. ltoug te calculated results did more closel approac tose from eperiment te still sowed poor agreement. Te ratio of te oscillation amplitude to te value of te reflectivit near an angle of incidence of.36 in te calculated reflectivit for te Gas surface of 4 nm rougness in Figure is larger tan tat of te reflectivit for a small rougness of.8 nm in Figure i. e. near an angle of incidence of.36 interference effects appear to increase te reflectivit in te case of large rougness. It seems ver strange tat interference effects would operate in tis wa. can be written b including an parameters depend on te proposed approimations as ep{ [ C C ] } were parameters C C depend on te proposed approimation. [4-8 -3] Wit te use of te reduced Fresnel reflection coefficient - of q. 3 and te reduced Fresnel transmission coefficient - of q. new accurate reflectivit from a multilaer consisting of N laers wit roug surface and interfaces is sown as N N epi 33 Figure. Calculated dotted dased and tin lines and measured tic line reflectivit from a Gas laer wit a ticness of 48 nm on a Si substrate. In te calculation te interface rougness is. nm. Tree calculated results wit te rougness of Gas surface set at 3.5 nm 4 nm and 4.5 nm are sown. Tis disagreement was mainl caused b te fact tat te diffuse scattering at te roug interface was not correctl taen into account b Nevot and Croce []. For reproducing te result of measurement X te calculated interfacial rougness sould not be.7 nm in conventional X formalism of q. 58. Te result of interfacial rougness b te conventional X formulae sowed large difference wit te TM result and derived wrong structure of surface. Net we sow appling of new improved formalism for tis result of X measurement wit a TM observation. Ten in te calculation of X wen tere is rougening at te surface or te interface te Fresnel transmission coefficient - sould be used for te reduced coefficient. ltoug formula for - is well nown - - ep 3 an accurate analtical formula for - including te effect of te interface rougness is not available. Several teories eist to describe te influence of rougness on X-ra scattering and te Fresnel coefficient for transmission as been derived in several teories. [4-8 -3] Tere are several approimations proposed so far and all tese results In te previous analsis te reduction of transmission coefficient as not be eamined b te oter eperiment. Ten in tis stud we tried to determine te parameters C C in q. eperimentall b comparing te measurements of TM observation results and X. Te X from a Gas laer wit a ticness of 48 nm on Si substrate were surface rougness s was set to 5.5 nm and interface rougness i was set to.7 nm was calculated wit various C and C. fter coosing te parameters C C so tat te calculation result of X accorded wit te eperimental result in tis Gas laer structure C and C. were provided. In Fig. 3 te dased line sows te calculation result of X. Te calculation results reproduce te eperimental results almost well. s suc we could eamine te psicall reasonable reduction of transmission coefficient. In te previous analsis wen it was supposed tat was 4.3 nm and was.7 nm in tis X measurement data [37] different parameters C.5 and C.5 were provided altoug te agreement of te calculation result and te eperimental result was not more good tan tis time result. Tis suggests tat te eperimental X result can be reproduced almost well if appropriate parameters are cosen for different structure lie as even using conventional X formalism. Now we ave got te parameters C and C. but do not get psical grounds of te value of te parameters. It is tougt tat te value of te parameter C C depends on te structure of a parallel direction on te surface in te surface rougness and te interface rougness. Terefore te investigation about man samples will be necessar in future. Te result of interfacial rougness b using te conventional X formulae sowed large difference wit te TM result and derived wrong structure of surface. Wile te result b new improved formalism reproduce te TM result but need appropriate parameters in transmission coefficient. It sows tat new improved X formalism derives more accurate analsis of te X from a multilaer

17 43 Yosiau Fuii: ecent Developments in te X-ra eflectivit nalsis surface but te reduced Fresnel coefficients wit psical grounds in te reflectivit equation are need in furter researc and we continue to discuss te refining tis teor in net section. 4.. FM Observation Te surfaces of sample and sample B were observed b atomic force microscop FM. Figure 4 sow te FM images and te rougness profiles of sample and sample B respectivel. Te r.m.s. rougness s at te area of µm of te SiO surfaces of sample and sample B in Figures 4 a - b were about.7 nm bot and tose at te area of µm in Figures 4 c - d were about.4 nm bot. FM observation sows tat te surface rougness was ardl a cange before and after vapor deposition of te SiO. Figure 3. Solid line sows measured X from a Gas laer wit a ticness of 48 nm on a Si substrate. Dased line sows calculated reflectivit b improved formalism wit te parameters C C. for reproducing measurement X wen is 5.5 nm and is.7 nm. 4. Surface and Interface ougness stimations b X-ra eflectivit wit FM Observation and BS Measurements For deriving more accurate formalism of -ra reflectivit combination of X wit oter analtical tecniques could be useful. We tried to compare te measurements of te surface rougness of te same sample b -ra reflectivit X wit atomic force microscop FM and ig-resolution uterford bacscattering spectroscop BS. X is generall less sensitive to te interface rougness i compared wit te surface rougness s. FM is widel used to measure te surface structures and BS is widel used to measure te interface structures [4]. ltoug BS cannot directl measure te interface rougness it can measure te film ticness and its dispersion t. Wit te elp of te BS measurement X can provide more accurate estimate of te interface rougness i. Fig. 4a. Te FM images and te rougness profiles of sample in te area of µm square. Fig. 4b. Te FM images and te rougness profiles of sample B in te area of µm square. 4.. Sample Preparation Two samples of silicon wafers aving a tin SiO laer were prepared b te following metods. Te sample was prepared b termal oidiing of a Si wafer. Te ticness of te prepared SiO laer is about 5 nm. Te oter sample B was prepared b vacuum deposition of an additional SiO laer of about nm on te sample at room temperature. Te rougness of te SiO /Si interface is epected to be te same as te sample altoug te surface rougness sould be increased after te deposition. Te surface and interface rougnesses of tese samples were measured b X FM and BS. Fig. 4c. Te FM images and te rougness profiles of sample in te area of µm square.

18 merican Journal of Psics and pplications 6; 4: COUNTS arb. units 3 O 4eV e SiO /Si sample []canneling random Si Fig. 4d. Te FM images and te rougness profiles of sample B in te area of µm square BS Measurement Te details of te BS measurement were described elsewere [43 44]. Briefl a beam of 4-eV e ions from a Coccroft Walton tpe accelerator was collimated to mm and to a divergence angle of less tan. b a series of 4-aw slit sstems. Te well-collimated beam was transported to an ultra-ig-vacuum UV scattering camber base pressure 8 Pa via a differential pumping sstem and impinged on a target mounted on a 5-ais precision goniometer. Te tpical beam current was about 3 n. Te e ions scattered from te target at a scattering angle θ 5º were energ analed b a 9º sector tpe magnetic spectrometer and detected b a one-dimensional position sensitive detector D-PSD of mm lengt te energ window was 5% of te central energ. Te energ resolution of te spectrometer was ~.%. Te surface of te sample was cleaned using te ultraviolet/oone cleaning metod before te BS measurement. Figure 5 sows te BS spectra observed at a scattering angle of 5.3º wen 4 ev e ions were incident on te sample. In addition to a random spectrum triangles te [] canneling spectrum circles was also measured to observe te surface and interface structures more precisel. Te angle of incidence of e ions was 54.9 and 5.4 eit angles were θ e 74.8 and 79.3 for te canneling and te random spectra respectivel. In te canneling spectrum tere are two trapeoidal structures at ~ 355 ev and ~ 33 ev. Tese structures correspond to te silicon and ogen signals in te SiO laer. Te widt of te ogen signal is 9.9 ev wic corresponds to a SiO laer of 5. nm. Te surface and interface edges seen at ~ 333 and 35 ev respectivel of te ogen signal were fitted b error functions as is sown b solid curves. Te standard deviation Ω s of te error function for te surface edge was determined to be.47 ev wic is ascribed to te instrumental energ resolution including te energ spread of te incident beam. On te oter and te standard deviation Ω i for te interface edge was.6 ev wic includes effects of te energ loss straggling and te non-uniformit of te SiO laer NGY ev Fig. 5. BS spectra of sample. Te [] canneling and te random spectra are sown b circles and triangles respectivel. COUNTS arb. units 3 O 4eV e SiO /Si sample B []canneling random NGY ev Fig. 6. BS spectra of sample B. Te [] canneling and te random spectra are sown b circles and triangles respectivel. Figure 6 sows te [] canneling and random spectra of te sample B. Te angles of incidence of e ions were 54.4 and 49.9 eit angles were θ e 75.3 and 79.8 for te [] canneling and te random spectra respectivel. Te widt of te ogen signal in te canneling spectrum was.6 ev wic corresponds to a SiO laer of 7.4 nm. Te leading and trailing edges of te ogen signal in te canneling spectrum were fitted b error functions as is sown b solid curves. Te standard deviation of te surface edge was determined to be.47 ev sowing a good agreement wit te sample. Te standard deviation of te interface edge was determined to be. ev. Tis is larger tan tat of te sample indicating tat te non-uniformit of te SiO laer is increased b te deposition of SiO. Te observed Ω i is affected b te instrumental energ resolution wic is given b te observed Ω s and also affected b te energ loss straggling. Te effect of te energ loss straggling Ω str was calculated using te empirical formula given b Yang et al [45]. Te calculated results are.5 and.5 ev for te sample and B respectivel. Te non-uniformit of te SiO laer is estimated b Ω t Ω i Ω str Ω s. Te obtained results are Ω t.4 ev and.5 ev for sample and B respectivel. Using te stopping power of SiO te dispersion t of te SiO laer ticness was derived as.67 nm and.88 nm for Si

19 45 Yosiau Fuii: ecent Developments in te X-ra eflectivit nalsis te sample and B respectivel. Te difference between te sample and B is attributed to te increase of te surface rougness due to te deposition of -nm SiO laer X-ra eflectivit Measurement X-ra reflectivit measurements were performed using a Cu-Kα -ra beam from a 3 W rotating-anode source. Te beam sie of te -ra was about mm perpendicular to te reflection plane.5 mm parallel to te reflection plane. Te results of te -ra reflectivit measured for te sample and B are sown as a function of te angle of incidence θ i b dased curves in Figs. 7 and 8 respectivel. t θ i smaller tan te critical angle for total reflection.º te reflectivit is almost unit. Wit increasing θ i over te critical angle te reflectivit decreases and oscillator structures are seen. Tese oscillations originate from te interference of -ras reflected from te surface and te interface of te SiO /Si. B analsing te θ i dependence of te reflectivit te surface rougness interface rougness and te ticness of te SiO laer can be estimated. Now we sow appling of te following new improved formalism for tis result of X measurement. N N epi - ep ep{ [ C C ] } were parameters C C depend on te proposed approimation. In te present wor we coose C and C wic we believe is te most appropriate approimation. FCTIVITY a s.4 nm Sample eperimental calculation i.4 nm s.6 nm s.5 nm s.4 nm s.6 nm s.5 nm θ: NG OF INCIDNC degrees Fig. 7a. X-ra reflectivit from te sample. Te eperimental result tic dased curve is compared wit te calculated ones for i.4 nm and various s tin curves. FCTIVITY i.4 nm θ: NG OF INCIDNC degrees Fig. 7b. X-ra reflectivit from te sample. Te eperimental result tic dased curve is compared wit te calculated ones for s.5 nm and various i tin curves. FCTIVITY b Sample eperimental calculation s.5 nm i. nm i.4 nm i.84 nm i.84 nm i. nm i.4 nm θ: NG OF INCIDNC degrees Fig. 8a. X-ra reflectivit from te sample B. Te eperimental result tic dased curve is compared wit te calculated ones for s.54 nm and various i tin curves. FCTIVITY Sample B eperimental calculation s.54 nm i. nm i.4 nm i.84 nm i.84 nm i. nm Sample B eperimental calculation i.4 nm s.7 nm s.54 nm s.77 nm s.8 nm s.8 nm s.7 nm s.54 nm s.77 nm.5.5 θ: NG OF INCIDNC degrees Fig. 8b. X-ra reflectivit from te sample B. Te eperimental result tic dased curve is compared wit te calculated ones for i.4 nm and various s tin curves. s was mentioned above te origin of te oscillation is te interference between te -ras reflected from te surface and te interface. Tus te ticness of te SiO laer can be determined from te observed period of te oscillation. Te