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Reflection characterization of anisotropic ultrathin dielectric films on absorbing isotropic substrates
Surface Science 603 (2009) 3227–3233
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Reflection characterization of anisotropic ultrathin dielectric films on absorbing isotropic substrates Peep Adamson Institute of Physics, University of Tartu, Riia 142, Tartu 51014, Estonia
a r t i c l e
i n f o
Article history: Received 21 January 2009 Accepted for publication 6 September 2009 Available online 11 September 2009 Keywords: Semi-empirical models and model calculations Insulating films Ellipsometry
a b s t r a c t The reflection of linearly polarized light from an ultrathin anisotropic dielectric film on isotropic absorbing substrate is investigated analytically in the long-wavelength limit. All analytical results are correlated with the numerical solution of the anisotropic reflection problem on the basis of rigorous electromagnetic theory. Simple analytical approach developed in this work not only gives a physical insight into the reflection problem but also provides a way of estimating the necessary experimental accuracy for optical diagnostics by reflection characteristics. It is shown that obtained expressions are of immediate interest for determining the parameters of anisotropic surface layers. Innovative possibilities for optical diagnostics of anisotropic properties of ultrathin dielectric layers upon absorbing materials are discussed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Nowadays ultrathin dielectric films on absorbing materials have become more important in the technology of electronic devices because the nanoscale dielectric layer forms the basis of field-effect device structure. For instance, high-k Hf and Zr silicates and oxides are extensively studied as leading candidates for a silicon dioxide replacement in the gate-oxide insulating layer in complementary metal-oxide-semiconductor devices [1]. However, the concept of an isotropic surface film in the nanometric region of thicknesses is generally unrealistic because the properties of the film in the direction perpendicular to the surface should differ from those in the direction parallel to the surface [2–5]. In addition, just epitaxially grown high-k oxides have a good interface quality with semiconductor substrate but such crystalline oxides are generally anisotropic [6]. Besides semiconductor technology the anisotropic ultrathin films are of considerable interest in magneto-optics as well because today magneto-optics effects are widely applied in magnetic research [7,8]. For example, the surface magneto-optic Kerr effect has significantly impacted research on magnetic thin films. Therefore, the surface structures, which contain ultrathin anisotropic dielectric layers, are more and more common in nanotechnological industry, and the need to characterize this type of structures in an easy way by fast and noninvasive methods is highly desirable. Optical reflection techniques [9–28] have been successfully employed for investigation of surface films and ultrathin layered structures for a long time because they are fast, noninvasive, and
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inexpensive. For ultrathin surface layers, it is best to apply the differential reflection techniques, which are founded on the direct measurement of the contribution of an ultrathin layer to reflection parameters. The corresponding methods in connection with theoretical analysis are adequately elaborated for isotropic (in)homogeneous ultrathin films [9,13–16,18–23]. However, in the case of anisotropic ultrathin films the differential reflection problem is considered only for several simple cases of anisotropy [24–27]. Notice that a practical implementation of conventional reflection methods for anisotropic thin films is not impossible to realize for diagnostics of ultrathin layers. The reason has to do with the precision of reflectance or ellipsometric measurements for an ultrathin film on massive substrates: instrumental error, as a rule, is greater than the contribution of an ultrathin film to the reflection parameters (the governing factor is the contribution of a substrate). Only surface plasmon resonance technique [29] can be applied to study the optical properties of thin films with relatively small thickness (100 nm), however, the feasibility of this method is limited by the factor that the refractive index of the anisotropic film must be lower than that of the prism. A purpose of this paper, first, is to study the reflection characteristics in the long-wavelength limit for anisotropic dielectric films upon an absorbing, isotropic, and homogeneous substrate. A second aim is to discuss the possibilities of optical reflection diagnostics of ultrathin anisotropic films on the basis of obtained expressions for reflection characteristics where the main challenges is the simultaneous determination of the thickness and refractive index of an ultrathin film. The distinctive feature of this paper is the analytical tackling of the issue. We derive the relatively simple final expressions for reflection characteristics. Notice that the remarkable advantage
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of the analytical approach lies in the fact that this method gives also a physical insight into the reflection problem. We also show the utility of these formulas for handling the problems of optical diagnostics, and it has been just this research that has led to analytical expressions for dielectric constants of ultrathin anisotropic films (it is common knowledge that generally the reflection equations cannot be inverted). Note that in earlier papers the indirect numerical techniques, e.g., the regression analysis was primarily developed for this purpose. But such numerical methods always come across serious difficulties, the main two being the instability and the non-uniqueness of the solution. Moreover, since these inverse problems are essentially ill posed, it is frequently impossible to obtain a solution with the desired accuracy because of the presence of both systematic and random measurement errors. In addition, it is pertinent to note that an analytical algorithm for calculating the parameters of interest directly from the measured data is very fast in comparison, e.g., with regressive type of algorithm. Although the latter is very general by nature, it tends to be computationally slow if more than a few variable parameters are involved. Due to this fact, approximate analytical techniques are used to provide initial guesses at the values of the variable parameters, and regression techniques are then used to fine-tune the desired parameters. The paper is organized as follows. In Section 2 the analytical first-order expressions in a long-wave limit for the reflection characteristics of anisotropic ultrathin films is produced. The third section is concerned with the solution of the inverse problem on the basis of obtained analytical expressions.
e11 e12 e13 e22
e31 e32
3
2
exx
7 6 6 e23 7 5¼A4 0 e33 0
2 2 1 ^ e213 e1 33 Þ cos /s es ð1 ea e33 sin /a Þðd=kÞ;
0
eyy 0
0
3
7 1 0 7 5A ;
ð1Þ
ezz
where A is the orthogonal coordinate rotation matrix [30]. For calculating the reflection characteristics we use the 4 4 matrix method [31] within the framework of a long-wavelength limit. Note that previously the long-wavelength approximation has been used for uniaxially anisotropic films [3,32,33]. In the case of biaxially anisotropic films in the first order with respect to the small parameter d=k the amplitude reflection coefficients rpp and rps (in this paper the first subscript always indicates
ð2Þ
rps iptp ts ðna cos /a Þ1 ½ðe12 e13 e23 e1 33 Þ cos /s ^ s e23 e1 na n 33 sin /a ðd=kÞ:
ð3Þ
For s-polarized incident wave one can obtain that
^ rss r s þ ipt2s ðna cos /a Þ1 ðe22 e223 e1 33 es Þðd=kÞ; 1
ð4Þ
1 13 23 33 Þ cos /s
rsp iptp ts ðna cos /a Þ ½ðe12 e
e e
^ s e23 e1 þ na n 33 sin /a ðd=kÞ;
ð5Þ
2
1=2 where cos /s ¼ ð1 ea ^e1 ; r p ; rs and t p ; t s are the amplis sin /a Þ tude reflection and transmission coefficients, respectively, from bare ðd 0Þ isotropic substrate and are expressed by the standard Fresnel’s formulas
^ s cos /a Þ=ðna cos /s þ n ^ s cos /a Þ; rp ¼ ðna cos /s n ^ ^ rs ¼ ðna cos /a ns cos /s Þ=ðna cos /a þ ns cos /s Þ; ^ s cos /a Þ; tp ¼ 2na cos /a =ðna cos /s þ n
ð7Þ
^ s cos /s Þ: ts ¼ 2na cos /a =ðna cos /a þ n
ð9Þ
ð6Þ ð8Þ
For the reflectances Rpp ¼ jr pp j2 and Rss ¼ jr ss j2 we obtain from Eqs. (2) and (4) the following expressions accurate to the first order in small parameters: 2
Rpp Rp f1 þ 8pK esI na cos /a ½ea ð1 ea e1 33 sin /a Þ ðe11 e
Assuming that all the media are nonmagnetic, we consider the reflection of s- and p-polarized time-harmonic (the complex representation is taken in the form expðixtÞ, where x ¼ 2pc=k, and k is a vacuum wavelength) electromagnetic plane waves in an ambient medium with real isotropic and homogeneous dielectric constant ea n2a from an arbitrarily anisotropic homogeneous dielectric film of thickness d k and with real principal dielectric-tensor components in the crystal-coordinate system exx n2xx ; eyy n2yy ; ezz n2zz , and which is located on a semi-infinite absorbing, isotropic, and homogeneous substrate with com^ 2s ¼ ðnsR þ insI Þ2 . The plex dielectric constant ^es ¼ esR þ iesI n orientations of the crystal axes are described by the Euler angles h; u, and w with respect to a fixed x y z coordinate system (the Cartesian laboratory coordinate system). The laboratory x; y, and z axes are defined as follows. The reflecting surface is the xy plane, and the plane of incidence is the zx plane, with the z axis normal to the surface of the layered medium and directed into it. The incident light beam in the ambient medium makes an angle /a with the z axis. The dielectric tensor of a biaxially anisotropic layer in the x y z coordinate system is given by
6 6 e21 4
rpp rp þ ipt2p ðna cos /a Þ1 ½ðe11
2 1 2 13 33 Þ cos
2. Reflection characteristics
2
the incident light) for p-polarized incident electromagnetic plane wave take the form:
e
/a ðd=kÞg;
ð10Þ
Rss Rs f1 þ 8pK 0 esI na cos /a ðea þ e223 e1 33 e22 Þðd=kÞg;
ð11Þ
where 2
2
K ¼ ½1 2ea a sin /a =½fea ð1 ea a sin /a Þ esR cos2 /a g2 2
þ e2sI ðcos2 /a e2a j^es j2 sin /a Þ2 ; Rp ¼ jr p j2 ; Rs ¼ jr s j2 ; a ¼ esR =j^es j2 ; j^es j2 ¼ e2sR þ e2sI , and K 0 ¼ Kð/a ¼ 0Þ. The reflectances Rps ¼ jrps j2 and Rsp ¼ jrsp j2 are equal to zero in the first order in d=k. The second-order formulas take the form:
Rr p2 ðea cos2 /a Þ1 jtp j2 jts j2 jðe12 e13 e23 e1 33 Þ cos /s 2 2 ^ s e23 e1 þ Pr na n 33 sin /a j ðd=kÞ ;
ð12Þ
where r ¼ sp or ps, and Psp ¼ þ1; Pps ¼ 1. Next we consider the ellipsometric parameters in the longwavelength approximation. Since in anisotropic samples the Jones matrix contains off-diagonal terms, a so-called generalized ellipsometry is designed for anisotropic systems [34–37]. Use of multiple input polarization states enables determination of the off-diagonal elements and thus to collect information regarding the anisotropy from all Jones matrix elements. As in the case with conventional ellipsometry, the absolute values of the Jones matrix elements are not determined. There are three normalized complex ratios, which may be chosen for measurement:
rpp =r ss tan Wpp expðiDpp Þ;
ð13Þ
rps =r ss tan Wps expðiDps Þ;
ð14Þ
rsp =r ss tan Wsp expðiDsp Þ:
ð15Þ
The ratio rpp =r ss is similar to the complex ratio determined by standard ellipsometry. In cases where r ps and r sp vanish, rpp =r ss is identical to the conventional ellipsometric parameters for isotropic systems. For the contributions of an anisotropic multilayer to ellipsometric angles dWpp ¼ Wpp W and dDpp ¼ Dpp D, where W and D are
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the ellipsometric angles of a bare substrate, one can obtain the following formulas:
dWpp 2pna esI cos /a sin 2W
a1 M2 Þðd=kÞ;
ð17Þ 2
2 2 2 1 a1 ¼ sin /a ðea e1 33 1ÞðesR esI Þ þ ðesR ea sin /a Þðe11 e13 e33 Þ 2
2
þ ðea sin /a esR cos2 /a Þðe22 e223 e1 33 Þ ea esR sin /a ; 2
2
1 a 33 Þ
a2 ¼ ea sin /a þ 2esR sin /a ð1 e
e
(a)
ð16Þ
2 dDpp 4pna cos /a A1 1 ða1 M 1 þ esI a2 M 2 Þðd=kÞ; 2
5
2
þ cos /a ðe22 e223 e1 33 Þ
e11 þ e213 e1 33 ;
Relative error, %
A1 1 ða2 M 1
4 3 2 1
2
M 1 ¼ ea esR e2a sin /a ðe2sR e2sI Þ cos2 /a ; 2
M 2 ¼ 2esR cos /a ea ;
0
A1 ¼ M 21 þ e2sI M22 :
2
2 1=2
tan Wr 4pna cos /a ½F r þ Gr tan Dr Gr =F r ;
ð18Þ ð19Þ
;
F r ¼ ½g r Q 1 fr Q 2 =½Q 21 þ Q 22 ðd=kÞ; Gr ¼ ½g r Q 2 þ fr Q 1 =½Q 21 þ Q 22 ðd=kÞ; 1 fr ¼ ½e12 e13 e23 e1 33 Q 3 þ P r e23 e33 na esR sin /a ; 1 g r ¼ f½e12 e13 e23 e1 þ Pr e23 e1 33 ð2Q 3 Þ 33 na sin /a gesI ;
-2 -4 -6 -8
2
Q 1 ¼ Q 3 ðesR ea Þ cos /a e2sI ð2Q 3 Þ1 cos /a þ na ðesR ea Þ sin /a ;
(b)
2
1
Q 2 ¼ ½Q 3 cos /a þ ð2Q 3 Þ ðesR ea Þ cos /a þ na sin /a esI ; 2
Relative error, %
The quantities Wr and Dr can be expressed in the form
2
Q 3 ¼ 21=2 fesR ea sin /a þ ½ðesR ea sin /a Þ2 þ e2sI 1=2 g1=2 : Next we briefly consider the accuracy of approximate formulas obtained above. Since in the case of ultrathin films the experimental error of the overall reflectance Rpp;ss , as a rule, is greater than the contribution of an ultrathin film to Rpp;ss then our interest is in differential quantities, e.g., in the relative change of reflectance DRpp;ss =Rp;s ðRpp;ss Rp;s Þ=Rp;s , which is directly brought on by an ultrathin layer [38–41]. Note that Rr may also be considered as a differential quantity because Rr ¼ 0 if d ¼ 0. The relative errors of DRss;pp =Rs;p and Rr for an anisotropic layer upon different absorbing substrates as functions of d are presented in Fig. 1. As a rough approximation, the error of the first-order equations does not exceed a few percent if d=k 6 102 . However, the accuracy of these equations for a given value of d=k depends on the values of material dielectric constants as well. Because of this, it is difficult to indicate explicitly the value of d=k where the long-wavelength approximation is broken down. In relation to the dependence of the error for DRpp;ss =Rp;s on nsI attention should be drawn to the fact that the first-order formulas for the contributions of anisotropic dielectric layers to the reflectances may be applied only for substrates which have a strong absorption: if the imaginary part of the refractive index of substrate becomes far less than unity, then the error of approximate formulas for DRss;pp =Rs;p increases steeply. This is because DRpp;ss =Rp;s is proportional to nsI , and, therefore, at nsI 1 a small parameter appears as nsI ðd=kÞ which is similar in value to ðd=kÞ2 . Thus in a power series in the small parameter d=k is no longer correct to drop the terms with second powers of d=k. The situation is essentially different for Rps and Rsp where the first terms in a power series in d=k that are not equal to zero prove to be proportional to ðd=kÞ2 . The corresponding approximate Eq. (12) for Rr is inherently a general-purpose formula with respect to nsI (the value of nsI may be anything from zero to infinity), in other words, it works for a transparent substrate ðnsI ¼ 0Þ as well. A comparison between the approximate Eq. (12) and calculations on the basis of exact electromagnetic theory for Rps and Rsp shows that the greater the value of nsI , the greater the relative error of the Eq. (12).
2
4
d, nm
6
8
Fig. 1. Relative errors of approximate formulas for (a) Rps (solid and dash-dotted curve), Rsp (dashed and dotted curve) and for (b) DRpp =Rp (solid and dash-dotted curve), DRss =Rs (dashed and dotted curve) as functions of d for a film with weak anisotropy: nxx ¼ 1:62; nyy ¼ 1:65; nzz ¼ 1:68; h ¼ u ¼ 50 ; w ¼ 10 (solid and dashed curves), and for a film with strong anisotropy: nxx ¼ 1:75; nyy ¼ 1:4; nzz ¼ 1:6; h ¼ u ¼ 50 ; w ¼ 10 (dash-dotted and dotted curves) at na ¼ 1; nsR ¼ 1:5; nsI ¼ 0:2; /a ¼ 40 ; k ¼ 630 nm.
Clearly the practical issue is how the accuracy of approximate expressions depends on the power of film anisotropy. Parallel computations for week and strong anisotropy show that no noticeable difference exists between them (Fig. 1). Finally we briefly dwell on the question how to appreciate the meaning of ‘‘ultrathin film”. A sound basis for this purpose is the ratio of film thickness to wavelength. Thus, if we use optical radiation for measurements, then the thickness of surface layers cannot be in excess of several nanometers. Moreover, it is evident that the advantages of differential methods are most pronounced in the case of surface layers with subnanometer thicknesses, especially for differential reflectance measurements [47]. The reason has to do with the precision of absolute measurements for a system substrate–subnanometer film: instrumental error, as a rule, is greater than the contribution of an ultrathin film to the reflection coefficients (the governing factor is the contribution of a substrate). On the other hand, if we work in the infrared region of wavelengths, then the layer thicknesses can even be close to visual wavelengths, i.e., in principle, the differential methods under discussion can be used in optical diagnostics of conventional thin films as well. Therefore, generally, in the present context the concept of ‘‘ultrathin” should not be taken too literally. 3. Reflection diagnostics Let us consider potential applicability of approximate expressions obtained above for the optical diagnostics of an ultrathin biaxially anisotropic film on absorbing isotropic substrate. From
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the measurements of reflectances ½DRss =Rs ðRss Rs Þ=Rs ; DRpp = Rp ðRpp Rp Þ=Rp ; Rps , and Rsp or ellipsometric parameters ½dWpp ; dDpp ; tan Wr , and tan Dr and on the basis of the corresponding first-order Eqs. (10)–(12) or (16)–(19) one can obtain the values of five independent quantities ðe22 e223 =e33 Þd; ðe11 e213 = 1 e33 Þd; ðe1 33 ea Þd; ðe12 e13 e23 =e33 Þd, and ðe23 =e33 Þd. Note that in this regard the physical situation is closely similar to that in the case of isotropic or uniaxially anisotropic ultrathin surface layers, where, generally, so-called optical invariants emerge out of a d=k expansion of the reflection coefficients in the long-wavelength approximation [3–5,42]. In essence, these quantities are integral invariants and are best suited for optical characterizing of nonuniform interfaces [43–45]. For uniform surface layers one can calculate these invariants as a function of the layer thickness and optical constants. Since in this work we restrict the discussion to the simplest case of uniform anisotropic layers, the implementation of integral invariants to our formalism is meaningless. In the case of a biaxially anisotropic film, generally, we have seven unknown parameters: e11 ; e22 ; e33 ; e12 ; e13 ; e23 (or exx ; eyy ; ezz ; h; u; wÞ, and d. However, it emerges that by the measurements at different angles of incidence we cannot determine the dielectric functions and thickness simultaneously on the basis of the first-order approximation. If we consider the thickness d also as an unknown quantity, then we come up with a homogeneous system of linear equations for unknown quantities. For example, for three unknown x k=d; y e11 e213 =e33 , and z e1 33 the expression (10) gives the following system of equations if we perform three measurements for DRpp =Rp at three different incident angles /ðjÞ a ½j ¼ 1; 2; 3:
2
ð20Þ
and
aD x þ bD ðy ea Þ þ cD ðz e1 a Þ dD ðt ea Þ ¼ 0;
ð23Þ
where
t e22 e223 =e33 aW ¼ A1 ð2pna cos /a sin 2W0 esI Þ1 dW; aD ¼ A1 ð4pna cos /a Þ1 dD; 2
bW ¼ M 1 þ ðes ea sin /a ÞM 2 ; 2
cW ¼ ea ð2es M 1 þ ðe2sR e2sI ÞM2 Þ sin /a ; 2
cD ¼ ea ð2es e2sI M2 ðe2sR e2sI ÞM 1 Þ sin /a ;
2
2
ðjÞ 1 ðjÞ ðjÞ where P j ¼ ½8pna cos /ðjÞ a esI Kð/a Þ ½DRpp ð/a Þ=Rp ð/a Þ. Since this system of equations possesses a solution, its determinant is equal to zero. From this condition follows the relationship between three ðjÞ quantities DRpp ð/ðjÞ a Þ=Rp ð/a Þ:
DRpp ð/ð1Þ 2 ð3Þ 2 ð2Þ ð1Þ 1 2 ð2Þ 2 ð3Þ a Þ ½cos /ð1Þ a Kð/a Þ ðcos /a sin /a cos /a sin /a Þ Rp ð/ð1Þ a Þ þ
DRpp ð/ð2Þ 2 ð1Þ 2 ð3Þ ð2Þ 1 2 ð3Þ 2 ð1Þ a Þ ½cos /ð2Þ a Kð/a Þ ðcos /a sin /a cos /a sin /a Þ Þ Rp ð/ð2Þ a
þ
DRpp ð/ð3Þ 2 ð2Þ ð3Þ 1 2 ð1Þ a Þ ½cos /ð3Þ a Kð/a Þ ðcos /a sin /a Rp ð/ð3Þ a Þ 2
ð22Þ
2
ð3Þ 2 1 P 3 x þ cos2 /ð3Þ a ðy ea Þ þ ea sin /a ðz ea Þ ¼ 0;
ð1Þ cos2 /ð2Þ a sin /a Þ ¼ 0:
aW x þ bW ðy ea Þ þ cW ðz e1 a Þ þ dW ðt ea Þ ¼ 0;
bD ¼ e2sI M2 es M 1 þ ea sin /a M1 ;
2
ð1Þ 2 1 P 1 x þ cos2 /ð1Þ a ðy ea Þ þ ea sin /a ðz ea Þ ¼ 0; ð2Þ 2 1 P 2 x þ cos2 /ð2Þ a ðy ea Þ þ ea sin /a ðz ea Þ ¼ 0;
ðd=kÞ2 or, in other words, the application of exact reflection theory for computation of reflection characteristics is worthwhile, i.e., enables one to determine the thickness and optical constants simultaneously only if the experiment possesses high sensitivity. Note that in the case of an isotropic ultrathin dielectric film on an absorbing substrate (has only two unknown parameters – the refractive index and thickness) one can simply determine these two quantities simultaneously in the framework of the first-order approximation [46]. In addition, by combination of the first-order expressions for complex amplitude reflection coefficient and reflectance one can simultaneously determine all three parameters of an absorbing isotropic film (the real and imaginary parts of refractive index and thickness) as well [47]. Simultaneous determination of the thickness and refractive index of an isotropic ultrathin film on a transparent substrate is also not a particular problem [19]. Similar situation exists in the case of ellipsometric measurements. From the expressions (16) and (17) one can obtain that
ð21Þ
ð1Þ ð2Þ In other words, three quantities DRpp ð/ð1Þ a Þ=Rp ð/a Þ; DRpp ð/a Þ= ð3Þ ð3Þ Þ, and D R ð/ Þ=R ð/ Þ are not independent measurements. Rp ð/ð2Þ pp p a a a If two of these quantities are known, then the remaining one can be determined without knowledge of the parameters of an ultrathin film from the relation (21). Consequently, solely two independent measurements of the differential reflectance DRpp =Rp exist that provide a way of estimating only two unknown parameters of an ultrathin film. If we solve the system of Eq. (20) for the unknown quantities y and z, then the thickness is a freely pre-assigned value, i.e., the optical constants are the immediate functions of the thickness. In other words, a strong parameter correlation between the thickness and optical constants appears in anisotropic ultrathin films and the thickness is inseparable from anisotropic dielectric constants on the basis of the first-order equations for DRpp;ss =Rp;s . This correlation disappears if the equations incorporate also the second-order terms, which are proportional to ðd=kÞ2 . But the effect of such terms arises essentially only when the faithfulness of differential reflectance measurements is notable, almost in order of
dW ¼ ðea sin /a es cos2 /a ÞM 2 cos2 /a M 1 ; 2
dD ¼ ðea sin /a es cos2 /a ÞM 1 þ cos2 /a e2sI M 2 ; dW, and dD are the ellipsometric angles to be measured. Thus, for ultrathin anisotropic films it is possible to carry out only three independent ellipsometric measurements. The fourth ellipsometric parameter can always be calculated via pre-measured three ellipsometric angles without the knowledge of the parameters of an ultrathin film. The corresponding mathematical relationship among four ellipsometric quantities can be obtained from the condition of the determinant of the system of four equations being equal to zero. If the thickness of an anisotropic layer is known, then three quantities e22 e223 =e33 ; e11 e213 =e33 , and e1 33 can simply be determined by reflectance (or ellipsometric) measurements. In this context the expression (11) yields:
e22
e223 ½ðea esR Þ2 þ e2sI DRss ð/a ¼ 0Þ ¼ ea 8pna esI Rs ð/a ¼ 0Þ e33
k ; d
ð24Þ
if /a ¼ 0, and solving the system of Eq. (20) gives:
e11 e1 33
2 2 ð2Þ e213 ðsin /ð1Þ a P 2 sin /a P 1 Þ k ¼ ea þ ; 2 2 ð1Þ d e33 ðsin /ð2Þ a sin /a Þ
2 ð2Þ 1 ðcos2 /ð1Þ a P 2 cos /a P 1 Þ k ¼ þ ; 2 ð1Þ 2 d ea ðcos2 /ð2Þ a cos /a Þea
ð25Þ ð26Þ
where measurements are taken at two different incident angles /ð1Þ a and /ð2Þ a . In what follows we consider the reflectances Rsp and Rps . In this instance it is necessary to measure two quantities: Rps and Rsp both at a fixed incident angle or one reflectance, e.g., Rps at two different angles of incidence. In the former case one can obtain the following system of equations for u e12 e13 e23 =e33 and v e23 =e33 :
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P. Adamson / Surface Science 603 (2009) 3227–3233
au2 þ bv 2 cuv ¼ gps ; au2 þ bv 2 þ cuv ¼ gsp ;
where
ð27Þ
2
^s j where a ¼ j cos /s j ; b ¼ jn
2
2
ea sin /a ; c ¼
^ s 2na sin /a Re½n
gps;sp
cos /s ,
^ s cos /a þ na cos /s j jna cos /a þ n ^ s cos /s j k 2 jn ¼ Rps;sp ; 2 2 d 16p ea cos /a 2
2
0
0 111=2 0 !2 11=2 g þ g g g 4ab B ps C sp B ps sp A C ¼ @ @1 @1 2 AA ; 4b c gps þ gsp ð28Þ
g gps e12 e13 e23 =e33 ¼ sp : 2ce23 =e33
ð29Þ
Clearly in a similar manner one can use general ellipsometric measurements and corresponding expressions (18) and (19) for determining the quantities e12 e13 e23 =e33 and e23 =e33 . Thus, from reflection measurements five quantities e33 ; e23 ; e22 ; e11 e213 =e33 , and e12 e13 e23 =e33 can be determined. All these quantities can be expressed in terms of six desired anisotropic parameters exx ; eyy ; ezz ; h; u, and w. Therefore, we have six unknown parameters for five equations and, consequently, a diagnostics problem can be solved only when one of these parameters is known. By way of example let us consider two specific cases: (i) h ¼ 0 and (ii) u ¼ 0. For h ¼ 0 one can obtain the following system of equations:
ezz ¼ e33 ; exx cos2 ðu þ wÞ þ eyy sin2 ðu þ wÞ ¼ e11 ; exx sin2 ðu þ wÞ þ eyy cos2 ðu þ wÞ ¼ e22 ; ðexx eyy Þ sinðu þ wÞ cosðu þ wÞ ¼ e12 :
ð30Þ
Solving the system (30) gives:
eyy ¼ ðe11 þ e22 ðe11 e22 þ 4e212 Þ1=2 Þ=2; exx ¼ e11 þ e22 eyy ; u þ w ¼ arccosððe11 eyy Þ=ðexx eyy ÞÞ1=2 :
ð31Þ ð32Þ ð33Þ
As may be seen in this instance we can determine only the sum of two angles u and w or, in other words, these angles cannot be separated. In the latter case ðu ¼ 0Þ the following system of equations can be obtained: 2
2
ðexx sin w þ eyy cos2 wÞ sin h þ ezz cos2 h ¼ e33 ; 2
ðexx sin w þ eyy cos2 w ezz Þ sin h cos h ¼ e23 ; 2
2
ðexx sin w þ eyy cos2 wÞ cos2 h þ ezz sin h ¼ e22 ; 2 2 2 1 2 2 yy sin w ð xx yy Þ 33 sin h sin w cos w ¼ 11 13 = 33 ; 1 23 sin hÞð xx yy Þ 33 sin w cos w ¼ 12 13 23 = 33 :
exx cos2 w þ e ðe33 cos h e
2
e
e
e e e
e
e
e e e e
e
e
ð34Þ Solving the system (34) yields:
ezz ¼ ðe22 þ e33 ððe22 e33 Þ2 þ 4e223 Þ1=2 Þ=2; ð35Þ exx ¼ ðe22 þ e33 ezz þ r1 ððe22 þ e33 ezz r1 Þ2 þ 4r22 Þ1=2 Þ=2; ð36Þ
eyy ¼ e22 þ e33 ezz exx þ r1 ; 1=2 e22 ezz h ¼ arccos ; e þ e33 2ezz 22 1=2 r1 eyy ; w ¼ arccos exx eyy
ð37Þ ð38Þ ð39Þ
e213 e13 e23 þ e12 e33 e33
r2 ¼ e12
and the asterisk ð Þ denotes the complex conjugate. Solving the system (27) gives:
e23 e33
r1 ¼ e11
2
e33 sin2 h ; ðe33 cos h e23 sin hÞ2
e13 e23 e33 : e33 e33 cos h e23 sin h
Note that without the slightest difficulty analogous simple expressions can be obtained for exx ; eyy ; ezz ; h, and u if the angle w ¼ 0. Generally the nonlinear system of five equations for five unknowns can be solved with a computer. Finally we consider some difficulties, which can emerge in the design of a spectroscopic experiment. Inasmuch as film thickness is very small, it is evident that a central problem is the quantifiability of reflectance characteristics DRpp;ss =Rp;s ; Rr and ellipsometric parameters. Calculations show that for ultrathin films with nanometric thicknesses on absorbing substrates the film-induced changes in the dominant quantities DRpp;ss ¼ Rpp;ss Rp;s are of sufficiently high values. For the measurements of such quantities the excellent high-precision experimental technique is developed [38– 41]. Analogous is the situation with reference to the ellipsometric parameters dWpp and dDpp . However, the detection of Rr is a serious challenge from an experimental standpoint. For example, if the thickness of an anisotropic layer is several nanometers, then the typical value of Rr is roughly in the range from 108 to 106 . For such measurements the major criterion is the minimum signal that can be detected or, in other words, the signal-to-noise ratio is bound to be quite large values. As is shown in Refs. [13,48–51] by special efforts the measurement of reflectivity of p-polarized light around the Brewster angle in order of 107 is made possible. It must be emphasized that the physical situation for determining Rr is quite analogous to the reflectivity measurement at the Brewster angle. Namely this fact that for a Fresnel interface the reflectance of ppolarized light vanishes at the Brewster angle leads to a high sensitivity to adsorbed layers. In our case, the quantity Rr is also equal to zero if the surface layer is missing ðd ¼ 0Þ, i.e., the reflected signal is free from the s-polarized wave if the incident light has pure p-polarization, and vice versa, the reflected signal is free from the p-polarized wave if the incident light has pure s-polarization. Besides, this circumstance offers good possibilities for the determination of a crucial apparatus parameter-the residual intensity for s-polarized (p-polarized) light if the incident wave has p-polarization (s-polarization). The second critical question is the influence of experimental error of Rr and DRss;pp =Rs;p . For reference we have included a computer simulation for the possible errors of approximate formulas (24)–(29) and (36)–(39) if the incident angles are the following: ð2Þ /ð1Þ a ¼ 0 and /a ¼ 70 . In order to calculate the error of these equations we give certain exact values for all unknown parameters exx ; eyy ; ezz ; h, and w and then calculate by the exact electromagnetic theory the values of DRpp;ss =Rp;s and Rr . Next we use these quantities in the form of DRpp;ss =Rp;s ð1 v pp;ss Þ and Rr ð1 v r Þ {where v pp;ss and v r represent the relative errors of DRpp;ss =Rp;s and Rr , respectively} in Eqs. (24)–(29) for calculating, firstly, e33 ; e23 ; e22 ; e11 e213 =e33 , and e12 e13 e23 =e33 and then for ðcalcÞ ðcalcÞ ðcalcÞ determining exx ; eyy ; ezz ; hðcalcÞ , and wðcalcÞ on the basis of Eqs. (36)–(39). The machine performed computations of the relaðcalcÞ ðcalcÞ ðcalcÞ tive errors ðexx exx Þ=exx ; ðeyy eyy Þ=eyy ; ðezz ezz Þ=ezz ; ðcalcÞ ðcalcÞ Þ=h, and ðw w Þ=w as functions of d=k for different ðh h v pp ; v ss ; v ps , and v sp are plotted in Figs. 2–6. If v pp ¼ v ss ¼ v ps ¼ v sp ¼ 0, then we obtain the pure mathematical error of approximate formulas in hand. Notice that computations show that the uncertainty of Rr is not of first importance (the error of DRpp;ss =Rp;s has a dominant role). Therefore, it may be said that
3232
P. Adamson / Surface Science 603 (2009) 3227–3233
the measurement of Rr at least for ultrathin layers on absorbing substrates with moderate values of the real part of refractive index, for example, glass or noble metal substrates is quite possible. For generalized ellipsometric measurements there is no ideal or standard technique yet devised and this method is still under pro-
gress and development. In Ref. [27], for example, is shown that by generalized ellipsometry sufficiently accurate birefringence measurements are possible on Langmuir–Blodgett films as thin as 10 nm. Notice that considerable attention now focusses on the elaboration of acutely sensitive and highly accurate ellipsometric technique [52] and this circumstance shows promise of extending the measurement range to 1 nm as well.
2
2 1
2. 5
2. 0
0
1
-1
-2
1. 5
Relative error,%
Relative error,%
3
3
-3
1. 0
1 0. 5
0. 0
2 0.001
0.002
0.003
0.004
d/ λ Fig. 2. Relative error of approximate formula (36) for exx as a function of d=k for an anisotropic film with nxx ¼ 2:9; nyy ¼ 2:6; nzz ¼ 2:8; h ¼ 70 ; u ¼ 0 ; w ¼ 30 at na ¼ 1; nsR ¼ 2:0; nsI ¼ 1:0; v pp ¼ v ss ¼ v ps ¼ v sp ¼ 0 (dashed curve), v pp ¼ 0:2% (1) 0.5%, (2) 1%, (3) v ss ¼ 0:3%, (1) 0.5%, (2) 2%, (3) v ps ¼ 10%, (1) 2%, (2 and 3) v sp ¼ 10%, (1) 2%, (2) 1%, (3). Preceding numbers in parentheses are curve labels.
-0.5 0.001
0.003
0.004
d/λ Fig. 4. Relative error of approximate formula (35) for ezz as a function of d=k for an anisotropic film with the same parameters as in Fig. 2.
16
12
3
10
12
8
2
8
Relative error,%
Relative error,%
0.002
6 4
1
2 0
4 0
1
-4
2 -8
-2 -4
3
-1 2
0.001
0.002
0.003
0.004
d/λ Fig. 3. Relative error of approximate formula (37) for eyy as a function of d=k for an anisotropic film with the same parameters as in Fig. 2.
0.001
0.002
0.003
0.004
d/λ Fig. 5. Relative error of approximate formula (38) for h as a function of d=k for an anisotropic film with the same parameters as in Fig. 2.
P. Adamson / Surface Science 603 (2009) 3227–3233
6 5
2
Relative error,%
4 3 3
2 1 1
0 -1 0.001
0.002
0.003
0.004
d/λ Fig. 6. Relative error of approximate formula (39) for w as a function of d=k for an anisotropic film with the same parameters as in Fig. 2.
4. Conclusions Relatively simple analytical approach developed in this work not only gives a physical insight into the reflection problem but also provides a way of estimating the necessary experimental accuracy for optical diagnostics by reflection characteristics. On the basis of the first-order expressions for reflection characteristics one cannot determine simultaneously the thickness and all six unknown components of the dielectric tensor for a biaxially anisotropic ultrathin film on an absorbing substrate because the first order expressions for the reflection parameters at different angles of incidence always generate a homogeneous system of linear equations for which the number of possible solutions is not limited, i.e., for every specified value of thickness the definite values of material constants exist. Furthermore, for a biaxially anisotropic ultrathin film, which thickness is known, we even cannot determine all six unknown anisotropic dielectric constants of film material. Only when one (or more) parameter(s) of anisotropy is known we can determine the remaining parameters on the basis of the first-order expressions for reflection characteristics. References [1] J. Robertson, Rep. Prog. Phys. 69 (2006) 327. [2] T. Yamaguchi, S. Yoshida, A. Kinbara, Thin Solid Films 21 (1974) 173. [3] J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves, Martinus Nijhoff, Dordrecht, 1987.
3233
[4] E.K. Mann, E.A. van der Zeeuw, G.J.M. Koper, P. Schaaf, D. Bedeaux, J. Phys. Chem. 99 (1995) 790. [5] D. Bedeaux, J. Vlieger, Optical Properties of Surfaces, Imperial College Press, London, 2004. [6] G.-M. Rignanese, X. Gonze, G. Jun, K. Cho, A. Pasquarello, Phys. Rev. B 69 (2004) 184301. [7] S. Visnovsky, Czech. J. Phys. 48 (1998) 1083. [8] S.Q. Qiu, S.D. Bader, J. Magn. Magn. Mater. 200 (1999) 664. [9] J.P.E. McIntyre, D.E. Aspnes, Surf. Sci. 24 (1971) 417. [10] W.K. Paik, J. O’M. Bockris, Surf. Sci. 28 (1971) 61. [11] R.C. O’Handley, Surf. Sci. 46 (1974) 24. [12] B.D. Cahan, Surf. Sci. 56 (1976) 354. [13] P. Schaaf, P. Dejardin, A. Schmitt, Langmuir 3 (1987) 1131. [14] I. Kamiya, D.E. Aspnes, L.T. Florez, J.P. Harbison, Phys. Rev. B 46 (1992) 15894. [15] H. Wormeester, D.J. Wentink, P.L. de Boeij, C.M.J. Wijers, A. van Silfhout, Phys. Rev. B 47 (1993) 12663. [16] E.A. van der Zeeuw, L.M.C. Sagis, G.J.M. Koper, E.K. Mann, M.T. Haarmans, D. Bedeaux, J. Chem. Phys. 105 (1996) 1646. [17] L. Bleckmann, O. Hunderi, W. Richter, E. Wold, Surf. Sci. 351 (1996) 277. [18] A. Rosental, P. Adamson, A. Gerst, A. Niilisk, Appl. Surf. Sci. 107 (1996) 178. [19] P. Adamson, J. Opt. Soc. Am. B 21 (2004) 645. [20] P. Adamson, Thin Solid Films 492 (2005) 221. [21] P. Adamson, Surf. Sci. 600 (2006) 735. [22] J.P. Landry, J. Gray, M.K. O’Toole, X.D. Zhu, Opt. Lett. 31 (2006) 531. [23] M. Lublow, H.J. Lewerenz, Surf. Sci. 602 (2008) 1677. [24] K. Naegele, W.J. Plieth, Surf. Sci. 61 (1976) 504. [25] M.K. Kelly, S. Zollner, M. Cardona, Surf. Sci. 285 (1993) 282. [26] K. Hingerl, D.E. Aspnes, I. Kamiya, Surf. Sci. 287/288 (1993) 686. [27] B. Lecourt, D. Blaudez, J.M. Turlet, J. Opt. Soc. Am. A 15 (1998) 2769. [28] H.G. Tompkins, W.A. McGahan, Spectroscopic Ellipsometry and Reflectometry, John Wiley & Sons, New York, 1999. [29] Y.J. Jen, C.H. Hsieh, T.S. Lo, Opt. Commun. 244 (2005) 269. [30] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, 1965. [31] R.M.A. Azzam, N.M. Bashara, Ellipsometry and Polarized Light, North-Holland, Amsterdam, 1977. [32] F. Abelés, Thin Solid Films 34 (1976) 291. [33] D. Beaglehole, Physica B&C 100 (1980) 163. [34] R.M.A. Azzam, N.M. Bashara, J. Opt. Soc. Am. 62 (1972) 1521. [35] D.J. De Smet, Surf. Sci. 56 (1976) 293. [36] M. Schubert, B. Rheinländer, J.A. Woollam, B. Johs, C.M. Herzinger, J. Opt. Soc. Am. A 13 (1996) 875. [37] M. Schubert, Thin Solid Films 313–314 (1998) 323. [38] P. Chiaradia, A. Cricenti, S. Selci, G. Chiarotti, Phys. Rev. Lett. 52 (1984) 1145. [39] S. Selci, F. Ciccacci, G. Chiarotti, J. Vac. Sci. Technol. A 5 (1987) 327. [40] P.E. Wierenga, M.J. Sparnaay, A. Van Silfhout, Surf. Sci. 99 (1980) 59. [41] E.G. Keim, A. van Silfhout, Surf. Sci. 152 (1985) 1096. [42] D. Bedeaux, G.J.M. Koper, E.A. van der Zeeuw, J. Vlieger, M. Wind, Physica A 207 (1994) 285. [43] E.K. Mann, L. Heinrich, J.C. Voegel, P. Schaaf, J. Chem. Phys. 105 (1996) 6082. [44] R.C. van Duijvenbode, G.J.M. Koper, J. Phys. Chem. B 104 (2000) 9878. [45] C. Picart, G. Ladam, B. Senger, J.C. Voegel, P. Schaaf, F.J.G. Cuisinier, C. Gergely, J. Chem. Phys. 115 (2001) 1086. [46] P. Adamson, Opt. Spectrosc. 86 (1999) 408. [47] I.K. Kim, D.E. Aspnes, J. Appl. Phys. 101 (2007) 033109. [48] G.J.M. Koper, P. Schaaf, Europhys. Lett. 22 (1993) 543. [49] L. Heinrich, E.K. Mann, J.C. Voegel, G.J.M. Koper, P. Schaaf, Langmuir 12 (1996) 4857. [50] R.C. van Duijvenbode, G.J.M. Koper, M.R. Böhmer, Langmuir 16 (2000) 7713. [51] R.C. van Duijvenbode, E.A. van der Zeeuw, G.J.M. Koper, Rev. Sci. Instrum. 72 (2001) 2407. [52] M. Gilliot, A. En Naciri, L. Johann, J.P. Stoquert, J.J. Grob, D. Muller, Phys. Rev. B 76 (2007) 045424.
Contents lists available at ScienceDirect
Surface Science journal homepage: www.elsevier.com/locate/susc
Reflection characterization of anisotropic ultrathin dielectric films on absorbing isotropic substrates Peep Adamson Institute of Physics, University of Tartu, Riia 142, Tartu 51014, Estonia
a r t i c l e
i n f o
Article history: Received 21 January 2009 Accepted for publication 6 September 2009 Available online 11 September 2009 Keywords: Semi-empirical models and model calculations Insulating films Ellipsometry
a b s t r a c t The reflection of linearly polarized light from an ultrathin anisotropic dielectric film on isotropic absorbing substrate is investigated analytically in the long-wavelength limit. All analytical results are correlated with the numerical solution of the anisotropic reflection problem on the basis of rigorous electromagnetic theory. Simple analytical approach developed in this work not only gives a physical insight into the reflection problem but also provides a way of estimating the necessary experimental accuracy for optical diagnostics by reflection characteristics. It is shown that obtained expressions are of immediate interest for determining the parameters of anisotropic surface layers. Innovative possibilities for optical diagnostics of anisotropic properties of ultrathin dielectric layers upon absorbing materials are discussed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Nowadays ultrathin dielectric films on absorbing materials have become more important in the technology of electronic devices because the nanoscale dielectric layer forms the basis of field-effect device structure. For instance, high-k Hf and Zr silicates and oxides are extensively studied as leading candidates for a silicon dioxide replacement in the gate-oxide insulating layer in complementary metal-oxide-semiconductor devices [1]. However, the concept of an isotropic surface film in the nanometric region of thicknesses is generally unrealistic because the properties of the film in the direction perpendicular to the surface should differ from those in the direction parallel to the surface [2–5]. In addition, just epitaxially grown high-k oxides have a good interface quality with semiconductor substrate but such crystalline oxides are generally anisotropic [6]. Besides semiconductor technology the anisotropic ultrathin films are of considerable interest in magneto-optics as well because today magneto-optics effects are widely applied in magnetic research [7,8]. For example, the surface magneto-optic Kerr effect has significantly impacted research on magnetic thin films. Therefore, the surface structures, which contain ultrathin anisotropic dielectric layers, are more and more common in nanotechnological industry, and the need to characterize this type of structures in an easy way by fast and noninvasive methods is highly desirable. Optical reflection techniques [9–28] have been successfully employed for investigation of surface films and ultrathin layered structures for a long time because they are fast, noninvasive, and
E-mail address: [email protected] 0039-6028/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2009.09.007
inexpensive. For ultrathin surface layers, it is best to apply the differential reflection techniques, which are founded on the direct measurement of the contribution of an ultrathin layer to reflection parameters. The corresponding methods in connection with theoretical analysis are adequately elaborated for isotropic (in)homogeneous ultrathin films [9,13–16,18–23]. However, in the case of anisotropic ultrathin films the differential reflection problem is considered only for several simple cases of anisotropy [24–27]. Notice that a practical implementation of conventional reflection methods for anisotropic thin films is not impossible to realize for diagnostics of ultrathin layers. The reason has to do with the precision of reflectance or ellipsometric measurements for an ultrathin film on massive substrates: instrumental error, as a rule, is greater than the contribution of an ultrathin film to the reflection parameters (the governing factor is the contribution of a substrate). Only surface plasmon resonance technique [29] can be applied to study the optical properties of thin films with relatively small thickness (100 nm), however, the feasibility of this method is limited by the factor that the refractive index of the anisotropic film must be lower than that of the prism. A purpose of this paper, first, is to study the reflection characteristics in the long-wavelength limit for anisotropic dielectric films upon an absorbing, isotropic, and homogeneous substrate. A second aim is to discuss the possibilities of optical reflection diagnostics of ultrathin anisotropic films on the basis of obtained expressions for reflection characteristics where the main challenges is the simultaneous determination of the thickness and refractive index of an ultrathin film. The distinctive feature of this paper is the analytical tackling of the issue. We derive the relatively simple final expressions for reflection characteristics. Notice that the remarkable advantage
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P. Adamson / Surface Science 603 (2009) 3227–3233
of the analytical approach lies in the fact that this method gives also a physical insight into the reflection problem. We also show the utility of these formulas for handling the problems of optical diagnostics, and it has been just this research that has led to analytical expressions for dielectric constants of ultrathin anisotropic films (it is common knowledge that generally the reflection equations cannot be inverted). Note that in earlier papers the indirect numerical techniques, e.g., the regression analysis was primarily developed for this purpose. But such numerical methods always come across serious difficulties, the main two being the instability and the non-uniqueness of the solution. Moreover, since these inverse problems are essentially ill posed, it is frequently impossible to obtain a solution with the desired accuracy because of the presence of both systematic and random measurement errors. In addition, it is pertinent to note that an analytical algorithm for calculating the parameters of interest directly from the measured data is very fast in comparison, e.g., with regressive type of algorithm. Although the latter is very general by nature, it tends to be computationally slow if more than a few variable parameters are involved. Due to this fact, approximate analytical techniques are used to provide initial guesses at the values of the variable parameters, and regression techniques are then used to fine-tune the desired parameters. The paper is organized as follows. In Section 2 the analytical first-order expressions in a long-wave limit for the reflection characteristics of anisotropic ultrathin films is produced. The third section is concerned with the solution of the inverse problem on the basis of obtained analytical expressions.
e11 e12 e13 e22
e31 e32
3
2
exx
7 6 6 e23 7 5¼A4 0 e33 0
2 2 1 ^ e213 e1 33 Þ cos /s es ð1 ea e33 sin /a Þðd=kÞ;
0
eyy 0
0
3
7 1 0 7 5A ;
ð1Þ
ezz
where A is the orthogonal coordinate rotation matrix [30]. For calculating the reflection characteristics we use the 4 4 matrix method [31] within the framework of a long-wavelength limit. Note that previously the long-wavelength approximation has been used for uniaxially anisotropic films [3,32,33]. In the case of biaxially anisotropic films in the first order with respect to the small parameter d=k the amplitude reflection coefficients rpp and rps (in this paper the first subscript always indicates
ð2Þ
rps iptp ts ðna cos /a Þ1 ½ðe12 e13 e23 e1 33 Þ cos /s ^ s e23 e1 na n 33 sin /a ðd=kÞ:
ð3Þ
For s-polarized incident wave one can obtain that
^ rss r s þ ipt2s ðna cos /a Þ1 ðe22 e223 e1 33 es Þðd=kÞ; 1
ð4Þ
1 13 23 33 Þ cos /s
rsp iptp ts ðna cos /a Þ ½ðe12 e
e e
^ s e23 e1 þ na n 33 sin /a ðd=kÞ;
ð5Þ
2
1=2 where cos /s ¼ ð1 ea ^e1 ; r p ; rs and t p ; t s are the amplis sin /a Þ tude reflection and transmission coefficients, respectively, from bare ðd 0Þ isotropic substrate and are expressed by the standard Fresnel’s formulas
^ s cos /a Þ=ðna cos /s þ n ^ s cos /a Þ; rp ¼ ðna cos /s n ^ ^ rs ¼ ðna cos /a ns cos /s Þ=ðna cos /a þ ns cos /s Þ; ^ s cos /a Þ; tp ¼ 2na cos /a =ðna cos /s þ n
ð7Þ
^ s cos /s Þ: ts ¼ 2na cos /a =ðna cos /a þ n
ð9Þ
ð6Þ ð8Þ
For the reflectances Rpp ¼ jr pp j2 and Rss ¼ jr ss j2 we obtain from Eqs. (2) and (4) the following expressions accurate to the first order in small parameters: 2
Rpp Rp f1 þ 8pK esI na cos /a ½ea ð1 ea e1 33 sin /a Þ ðe11 e
Assuming that all the media are nonmagnetic, we consider the reflection of s- and p-polarized time-harmonic (the complex representation is taken in the form expðixtÞ, where x ¼ 2pc=k, and k is a vacuum wavelength) electromagnetic plane waves in an ambient medium with real isotropic and homogeneous dielectric constant ea n2a from an arbitrarily anisotropic homogeneous dielectric film of thickness d k and with real principal dielectric-tensor components in the crystal-coordinate system exx n2xx ; eyy n2yy ; ezz n2zz , and which is located on a semi-infinite absorbing, isotropic, and homogeneous substrate with com^ 2s ¼ ðnsR þ insI Þ2 . The plex dielectric constant ^es ¼ esR þ iesI n orientations of the crystal axes are described by the Euler angles h; u, and w with respect to a fixed x y z coordinate system (the Cartesian laboratory coordinate system). The laboratory x; y, and z axes are defined as follows. The reflecting surface is the xy plane, and the plane of incidence is the zx plane, with the z axis normal to the surface of the layered medium and directed into it. The incident light beam in the ambient medium makes an angle /a with the z axis. The dielectric tensor of a biaxially anisotropic layer in the x y z coordinate system is given by
6 6 e21 4
rpp rp þ ipt2p ðna cos /a Þ1 ½ðe11
2 1 2 13 33 Þ cos
2. Reflection characteristics
2
the incident light) for p-polarized incident electromagnetic plane wave take the form:
e
/a ðd=kÞg;
ð10Þ
Rss Rs f1 þ 8pK 0 esI na cos /a ðea þ e223 e1 33 e22 Þðd=kÞg;
ð11Þ
where 2
2
K ¼ ½1 2ea a sin /a =½fea ð1 ea a sin /a Þ esR cos2 /a g2 2
þ e2sI ðcos2 /a e2a j^es j2 sin /a Þ2 ; Rp ¼ jr p j2 ; Rs ¼ jr s j2 ; a ¼ esR =j^es j2 ; j^es j2 ¼ e2sR þ e2sI , and K 0 ¼ Kð/a ¼ 0Þ. The reflectances Rps ¼ jrps j2 and Rsp ¼ jrsp j2 are equal to zero in the first order in d=k. The second-order formulas take the form:
Rr p2 ðea cos2 /a Þ1 jtp j2 jts j2 jðe12 e13 e23 e1 33 Þ cos /s 2 2 ^ s e23 e1 þ Pr na n 33 sin /a j ðd=kÞ ;
ð12Þ
where r ¼ sp or ps, and Psp ¼ þ1; Pps ¼ 1. Next we consider the ellipsometric parameters in the longwavelength approximation. Since in anisotropic samples the Jones matrix contains off-diagonal terms, a so-called generalized ellipsometry is designed for anisotropic systems [34–37]. Use of multiple input polarization states enables determination of the off-diagonal elements and thus to collect information regarding the anisotropy from all Jones matrix elements. As in the case with conventional ellipsometry, the absolute values of the Jones matrix elements are not determined. There are three normalized complex ratios, which may be chosen for measurement:
rpp =r ss tan Wpp expðiDpp Þ;
ð13Þ
rps =r ss tan Wps expðiDps Þ;
ð14Þ
rsp =r ss tan Wsp expðiDsp Þ:
ð15Þ
The ratio rpp =r ss is similar to the complex ratio determined by standard ellipsometry. In cases where r ps and r sp vanish, rpp =r ss is identical to the conventional ellipsometric parameters for isotropic systems. For the contributions of an anisotropic multilayer to ellipsometric angles dWpp ¼ Wpp W and dDpp ¼ Dpp D, where W and D are
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P. Adamson / Surface Science 603 (2009) 3227–3233
the ellipsometric angles of a bare substrate, one can obtain the following formulas:
dWpp 2pna esI cos /a sin 2W
a1 M2 Þðd=kÞ;
ð17Þ 2
2 2 2 1 a1 ¼ sin /a ðea e1 33 1ÞðesR esI Þ þ ðesR ea sin /a Þðe11 e13 e33 Þ 2
2
þ ðea sin /a esR cos2 /a Þðe22 e223 e1 33 Þ ea esR sin /a ; 2
2
1 a 33 Þ
a2 ¼ ea sin /a þ 2esR sin /a ð1 e
e
(a)
ð16Þ
2 dDpp 4pna cos /a A1 1 ða1 M 1 þ esI a2 M 2 Þðd=kÞ; 2
5
2
þ cos /a ðe22 e223 e1 33 Þ
e11 þ e213 e1 33 ;
Relative error, %
A1 1 ða2 M 1
4 3 2 1
2
M 1 ¼ ea esR e2a sin /a ðe2sR e2sI Þ cos2 /a ; 2
M 2 ¼ 2esR cos /a ea ;
0
A1 ¼ M 21 þ e2sI M22 :
2
2 1=2
tan Wr 4pna cos /a ½F r þ Gr tan Dr Gr =F r ;
ð18Þ ð19Þ
;
F r ¼ ½g r Q 1 fr Q 2 =½Q 21 þ Q 22 ðd=kÞ; Gr ¼ ½g r Q 2 þ fr Q 1 =½Q 21 þ Q 22 ðd=kÞ; 1 fr ¼ ½e12 e13 e23 e1 33 Q 3 þ P r e23 e33 na esR sin /a ; 1 g r ¼ f½e12 e13 e23 e1 þ Pr e23 e1 33 ð2Q 3 Þ 33 na sin /a gesI ;
-2 -4 -6 -8
2
Q 1 ¼ Q 3 ðesR ea Þ cos /a e2sI ð2Q 3 Þ1 cos /a þ na ðesR ea Þ sin /a ;
(b)
2
1
Q 2 ¼ ½Q 3 cos /a þ ð2Q 3 Þ ðesR ea Þ cos /a þ na sin /a esI ; 2
Relative error, %
The quantities Wr and Dr can be expressed in the form
2
Q 3 ¼ 21=2 fesR ea sin /a þ ½ðesR ea sin /a Þ2 þ e2sI 1=2 g1=2 : Next we briefly consider the accuracy of approximate formulas obtained above. Since in the case of ultrathin films the experimental error of the overall reflectance Rpp;ss , as a rule, is greater than the contribution of an ultrathin film to Rpp;ss then our interest is in differential quantities, e.g., in the relative change of reflectance DRpp;ss =Rp;s ðRpp;ss Rp;s Þ=Rp;s , which is directly brought on by an ultrathin layer [38–41]. Note that Rr may also be considered as a differential quantity because Rr ¼ 0 if d ¼ 0. The relative errors of DRss;pp =Rs;p and Rr for an anisotropic layer upon different absorbing substrates as functions of d are presented in Fig. 1. As a rough approximation, the error of the first-order equations does not exceed a few percent if d=k 6 102 . However, the accuracy of these equations for a given value of d=k depends on the values of material dielectric constants as well. Because of this, it is difficult to indicate explicitly the value of d=k where the long-wavelength approximation is broken down. In relation to the dependence of the error for DRpp;ss =Rp;s on nsI attention should be drawn to the fact that the first-order formulas for the contributions of anisotropic dielectric layers to the reflectances may be applied only for substrates which have a strong absorption: if the imaginary part of the refractive index of substrate becomes far less than unity, then the error of approximate formulas for DRss;pp =Rs;p increases steeply. This is because DRpp;ss =Rp;s is proportional to nsI , and, therefore, at nsI 1 a small parameter appears as nsI ðd=kÞ which is similar in value to ðd=kÞ2 . Thus in a power series in the small parameter d=k is no longer correct to drop the terms with second powers of d=k. The situation is essentially different for Rps and Rsp where the first terms in a power series in d=k that are not equal to zero prove to be proportional to ðd=kÞ2 . The corresponding approximate Eq. (12) for Rr is inherently a general-purpose formula with respect to nsI (the value of nsI may be anything from zero to infinity), in other words, it works for a transparent substrate ðnsI ¼ 0Þ as well. A comparison between the approximate Eq. (12) and calculations on the basis of exact electromagnetic theory for Rps and Rsp shows that the greater the value of nsI , the greater the relative error of the Eq. (12).
2
4
d, nm
6
8
Fig. 1. Relative errors of approximate formulas for (a) Rps (solid and dash-dotted curve), Rsp (dashed and dotted curve) and for (b) DRpp =Rp (solid and dash-dotted curve), DRss =Rs (dashed and dotted curve) as functions of d for a film with weak anisotropy: nxx ¼ 1:62; nyy ¼ 1:65; nzz ¼ 1:68; h ¼ u ¼ 50 ; w ¼ 10 (solid and dashed curves), and for a film with strong anisotropy: nxx ¼ 1:75; nyy ¼ 1:4; nzz ¼ 1:6; h ¼ u ¼ 50 ; w ¼ 10 (dash-dotted and dotted curves) at na ¼ 1; nsR ¼ 1:5; nsI ¼ 0:2; /a ¼ 40 ; k ¼ 630 nm.
Clearly the practical issue is how the accuracy of approximate expressions depends on the power of film anisotropy. Parallel computations for week and strong anisotropy show that no noticeable difference exists between them (Fig. 1). Finally we briefly dwell on the question how to appreciate the meaning of ‘‘ultrathin film”. A sound basis for this purpose is the ratio of film thickness to wavelength. Thus, if we use optical radiation for measurements, then the thickness of surface layers cannot be in excess of several nanometers. Moreover, it is evident that the advantages of differential methods are most pronounced in the case of surface layers with subnanometer thicknesses, especially for differential reflectance measurements [47]. The reason has to do with the precision of absolute measurements for a system substrate–subnanometer film: instrumental error, as a rule, is greater than the contribution of an ultrathin film to the reflection coefficients (the governing factor is the contribution of a substrate). On the other hand, if we work in the infrared region of wavelengths, then the layer thicknesses can even be close to visual wavelengths, i.e., in principle, the differential methods under discussion can be used in optical diagnostics of conventional thin films as well. Therefore, generally, in the present context the concept of ‘‘ultrathin” should not be taken too literally. 3. Reflection diagnostics Let us consider potential applicability of approximate expressions obtained above for the optical diagnostics of an ultrathin biaxially anisotropic film on absorbing isotropic substrate. From
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P. Adamson / Surface Science 603 (2009) 3227–3233
the measurements of reflectances ½DRss =Rs ðRss Rs Þ=Rs ; DRpp = Rp ðRpp Rp Þ=Rp ; Rps , and Rsp or ellipsometric parameters ½dWpp ; dDpp ; tan Wr , and tan Dr and on the basis of the corresponding first-order Eqs. (10)–(12) or (16)–(19) one can obtain the values of five independent quantities ðe22 e223 =e33 Þd; ðe11 e213 = 1 e33 Þd; ðe1 33 ea Þd; ðe12 e13 e23 =e33 Þd, and ðe23 =e33 Þd. Note that in this regard the physical situation is closely similar to that in the case of isotropic or uniaxially anisotropic ultrathin surface layers, where, generally, so-called optical invariants emerge out of a d=k expansion of the reflection coefficients in the long-wavelength approximation [3–5,42]. In essence, these quantities are integral invariants and are best suited for optical characterizing of nonuniform interfaces [43–45]. For uniform surface layers one can calculate these invariants as a function of the layer thickness and optical constants. Since in this work we restrict the discussion to the simplest case of uniform anisotropic layers, the implementation of integral invariants to our formalism is meaningless. In the case of a biaxially anisotropic film, generally, we have seven unknown parameters: e11 ; e22 ; e33 ; e12 ; e13 ; e23 (or exx ; eyy ; ezz ; h; u; wÞ, and d. However, it emerges that by the measurements at different angles of incidence we cannot determine the dielectric functions and thickness simultaneously on the basis of the first-order approximation. If we consider the thickness d also as an unknown quantity, then we come up with a homogeneous system of linear equations for unknown quantities. For example, for three unknown x k=d; y e11 e213 =e33 , and z e1 33 the expression (10) gives the following system of equations if we perform three measurements for DRpp =Rp at three different incident angles /ðjÞ a ½j ¼ 1; 2; 3:
2
ð20Þ
and
aD x þ bD ðy ea Þ þ cD ðz e1 a Þ dD ðt ea Þ ¼ 0;
ð23Þ
where
t e22 e223 =e33 aW ¼ A1 ð2pna cos /a sin 2W0 esI Þ1 dW; aD ¼ A1 ð4pna cos /a Þ1 dD; 2
bW ¼ M 1 þ ðes ea sin /a ÞM 2 ; 2
cW ¼ ea ð2es M 1 þ ðe2sR e2sI ÞM2 Þ sin /a ; 2
cD ¼ ea ð2es e2sI M2 ðe2sR e2sI ÞM 1 Þ sin /a ;
2
2
ðjÞ 1 ðjÞ ðjÞ where P j ¼ ½8pna cos /ðjÞ a esI Kð/a Þ ½DRpp ð/a Þ=Rp ð/a Þ. Since this system of equations possesses a solution, its determinant is equal to zero. From this condition follows the relationship between three ðjÞ quantities DRpp ð/ðjÞ a Þ=Rp ð/a Þ:
DRpp ð/ð1Þ 2 ð3Þ 2 ð2Þ ð1Þ 1 2 ð2Þ 2 ð3Þ a Þ ½cos /ð1Þ a Kð/a Þ ðcos /a sin /a cos /a sin /a Þ Rp ð/ð1Þ a Þ þ
DRpp ð/ð2Þ 2 ð1Þ 2 ð3Þ ð2Þ 1 2 ð3Þ 2 ð1Þ a Þ ½cos /ð2Þ a Kð/a Þ ðcos /a sin /a cos /a sin /a Þ Þ Rp ð/ð2Þ a
þ
DRpp ð/ð3Þ 2 ð2Þ ð3Þ 1 2 ð1Þ a Þ ½cos /ð3Þ a Kð/a Þ ðcos /a sin /a Rp ð/ð3Þ a Þ 2
ð22Þ
2
ð3Þ 2 1 P 3 x þ cos2 /ð3Þ a ðy ea Þ þ ea sin /a ðz ea Þ ¼ 0;
ð1Þ cos2 /ð2Þ a sin /a Þ ¼ 0:
aW x þ bW ðy ea Þ þ cW ðz e1 a Þ þ dW ðt ea Þ ¼ 0;
bD ¼ e2sI M2 es M 1 þ ea sin /a M1 ;
2
ð1Þ 2 1 P 1 x þ cos2 /ð1Þ a ðy ea Þ þ ea sin /a ðz ea Þ ¼ 0; ð2Þ 2 1 P 2 x þ cos2 /ð2Þ a ðy ea Þ þ ea sin /a ðz ea Þ ¼ 0;
ðd=kÞ2 or, in other words, the application of exact reflection theory for computation of reflection characteristics is worthwhile, i.e., enables one to determine the thickness and optical constants simultaneously only if the experiment possesses high sensitivity. Note that in the case of an isotropic ultrathin dielectric film on an absorbing substrate (has only two unknown parameters – the refractive index and thickness) one can simply determine these two quantities simultaneously in the framework of the first-order approximation [46]. In addition, by combination of the first-order expressions for complex amplitude reflection coefficient and reflectance one can simultaneously determine all three parameters of an absorbing isotropic film (the real and imaginary parts of refractive index and thickness) as well [47]. Simultaneous determination of the thickness and refractive index of an isotropic ultrathin film on a transparent substrate is also not a particular problem [19]. Similar situation exists in the case of ellipsometric measurements. From the expressions (16) and (17) one can obtain that
ð21Þ
ð1Þ ð2Þ In other words, three quantities DRpp ð/ð1Þ a Þ=Rp ð/a Þ; DRpp ð/a Þ= ð3Þ ð3Þ Þ, and D R ð/ Þ=R ð/ Þ are not independent measurements. Rp ð/ð2Þ pp p a a a If two of these quantities are known, then the remaining one can be determined without knowledge of the parameters of an ultrathin film from the relation (21). Consequently, solely two independent measurements of the differential reflectance DRpp =Rp exist that provide a way of estimating only two unknown parameters of an ultrathin film. If we solve the system of Eq. (20) for the unknown quantities y and z, then the thickness is a freely pre-assigned value, i.e., the optical constants are the immediate functions of the thickness. In other words, a strong parameter correlation between the thickness and optical constants appears in anisotropic ultrathin films and the thickness is inseparable from anisotropic dielectric constants on the basis of the first-order equations for DRpp;ss =Rp;s . This correlation disappears if the equations incorporate also the second-order terms, which are proportional to ðd=kÞ2 . But the effect of such terms arises essentially only when the faithfulness of differential reflectance measurements is notable, almost in order of
dW ¼ ðea sin /a es cos2 /a ÞM 2 cos2 /a M 1 ; 2
dD ¼ ðea sin /a es cos2 /a ÞM 1 þ cos2 /a e2sI M 2 ; dW, and dD are the ellipsometric angles to be measured. Thus, for ultrathin anisotropic films it is possible to carry out only three independent ellipsometric measurements. The fourth ellipsometric parameter can always be calculated via pre-measured three ellipsometric angles without the knowledge of the parameters of an ultrathin film. The corresponding mathematical relationship among four ellipsometric quantities can be obtained from the condition of the determinant of the system of four equations being equal to zero. If the thickness of an anisotropic layer is known, then three quantities e22 e223 =e33 ; e11 e213 =e33 , and e1 33 can simply be determined by reflectance (or ellipsometric) measurements. In this context the expression (11) yields:
e22
e223 ½ðea esR Þ2 þ e2sI DRss ð/a ¼ 0Þ ¼ ea 8pna esI Rs ð/a ¼ 0Þ e33
k ; d
ð24Þ
if /a ¼ 0, and solving the system of Eq. (20) gives:
e11 e1 33
2 2 ð2Þ e213 ðsin /ð1Þ a P 2 sin /a P 1 Þ k ¼ ea þ ; 2 2 ð1Þ d e33 ðsin /ð2Þ a sin /a Þ
2 ð2Þ 1 ðcos2 /ð1Þ a P 2 cos /a P 1 Þ k ¼ þ ; 2 ð1Þ 2 d ea ðcos2 /ð2Þ a cos /a Þea
ð25Þ ð26Þ
where measurements are taken at two different incident angles /ð1Þ a and /ð2Þ a . In what follows we consider the reflectances Rsp and Rps . In this instance it is necessary to measure two quantities: Rps and Rsp both at a fixed incident angle or one reflectance, e.g., Rps at two different angles of incidence. In the former case one can obtain the following system of equations for u e12 e13 e23 =e33 and v e23 =e33 :
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P. Adamson / Surface Science 603 (2009) 3227–3233
au2 þ bv 2 cuv ¼ gps ; au2 þ bv 2 þ cuv ¼ gsp ;
where
ð27Þ
2
^s j where a ¼ j cos /s j ; b ¼ jn
2
2
ea sin /a ; c ¼
^ s 2na sin /a Re½n
gps;sp
cos /s ,
^ s cos /a þ na cos /s j jna cos /a þ n ^ s cos /s j k 2 jn ¼ Rps;sp ; 2 2 d 16p ea cos /a 2
2
0
0 111=2 0 !2 11=2 g þ g g g 4ab B ps C sp B ps sp A C ¼ @ @1 @1 2 AA ; 4b c gps þ gsp ð28Þ
g gps e12 e13 e23 =e33 ¼ sp : 2ce23 =e33
ð29Þ
Clearly in a similar manner one can use general ellipsometric measurements and corresponding expressions (18) and (19) for determining the quantities e12 e13 e23 =e33 and e23 =e33 . Thus, from reflection measurements five quantities e33 ; e23 ; e22 ; e11 e213 =e33 , and e12 e13 e23 =e33 can be determined. All these quantities can be expressed in terms of six desired anisotropic parameters exx ; eyy ; ezz ; h; u, and w. Therefore, we have six unknown parameters for five equations and, consequently, a diagnostics problem can be solved only when one of these parameters is known. By way of example let us consider two specific cases: (i) h ¼ 0 and (ii) u ¼ 0. For h ¼ 0 one can obtain the following system of equations:
ezz ¼ e33 ; exx cos2 ðu þ wÞ þ eyy sin2 ðu þ wÞ ¼ e11 ; exx sin2 ðu þ wÞ þ eyy cos2 ðu þ wÞ ¼ e22 ; ðexx eyy Þ sinðu þ wÞ cosðu þ wÞ ¼ e12 :
ð30Þ
Solving the system (30) gives:
eyy ¼ ðe11 þ e22 ðe11 e22 þ 4e212 Þ1=2 Þ=2; exx ¼ e11 þ e22 eyy ; u þ w ¼ arccosððe11 eyy Þ=ðexx eyy ÞÞ1=2 :
ð31Þ ð32Þ ð33Þ
As may be seen in this instance we can determine only the sum of two angles u and w or, in other words, these angles cannot be separated. In the latter case ðu ¼ 0Þ the following system of equations can be obtained: 2
2
ðexx sin w þ eyy cos2 wÞ sin h þ ezz cos2 h ¼ e33 ; 2
ðexx sin w þ eyy cos2 w ezz Þ sin h cos h ¼ e23 ; 2
2
ðexx sin w þ eyy cos2 wÞ cos2 h þ ezz sin h ¼ e22 ; 2 2 2 1 2 2 yy sin w ð xx yy Þ 33 sin h sin w cos w ¼ 11 13 = 33 ; 1 23 sin hÞð xx yy Þ 33 sin w cos w ¼ 12 13 23 = 33 :
exx cos2 w þ e ðe33 cos h e
2
e
e
e e e
e
e
e e e e
e
e
ð34Þ Solving the system (34) yields:
ezz ¼ ðe22 þ e33 ððe22 e33 Þ2 þ 4e223 Þ1=2 Þ=2; ð35Þ exx ¼ ðe22 þ e33 ezz þ r1 ððe22 þ e33 ezz r1 Þ2 þ 4r22 Þ1=2 Þ=2; ð36Þ
eyy ¼ e22 þ e33 ezz exx þ r1 ; 1=2 e22 ezz h ¼ arccos ; e þ e33 2ezz 22 1=2 r1 eyy ; w ¼ arccos exx eyy
ð37Þ ð38Þ ð39Þ
e213 e13 e23 þ e12 e33 e33
r2 ¼ e12
and the asterisk ð Þ denotes the complex conjugate. Solving the system (27) gives:
e23 e33
r1 ¼ e11
2
e33 sin2 h ; ðe33 cos h e23 sin hÞ2
e13 e23 e33 : e33 e33 cos h e23 sin h
Note that without the slightest difficulty analogous simple expressions can be obtained for exx ; eyy ; ezz ; h, and u if the angle w ¼ 0. Generally the nonlinear system of five equations for five unknowns can be solved with a computer. Finally we consider some difficulties, which can emerge in the design of a spectroscopic experiment. Inasmuch as film thickness is very small, it is evident that a central problem is the quantifiability of reflectance characteristics DRpp;ss =Rp;s ; Rr and ellipsometric parameters. Calculations show that for ultrathin films with nanometric thicknesses on absorbing substrates the film-induced changes in the dominant quantities DRpp;ss ¼ Rpp;ss Rp;s are of sufficiently high values. For the measurements of such quantities the excellent high-precision experimental technique is developed [38– 41]. Analogous is the situation with reference to the ellipsometric parameters dWpp and dDpp . However, the detection of Rr is a serious challenge from an experimental standpoint. For example, if the thickness of an anisotropic layer is several nanometers, then the typical value of Rr is roughly in the range from 108 to 106 . For such measurements the major criterion is the minimum signal that can be detected or, in other words, the signal-to-noise ratio is bound to be quite large values. As is shown in Refs. [13,48–51] by special efforts the measurement of reflectivity of p-polarized light around the Brewster angle in order of 107 is made possible. It must be emphasized that the physical situation for determining Rr is quite analogous to the reflectivity measurement at the Brewster angle. Namely this fact that for a Fresnel interface the reflectance of ppolarized light vanishes at the Brewster angle leads to a high sensitivity to adsorbed layers. In our case, the quantity Rr is also equal to zero if the surface layer is missing ðd ¼ 0Þ, i.e., the reflected signal is free from the s-polarized wave if the incident light has pure p-polarization, and vice versa, the reflected signal is free from the p-polarized wave if the incident light has pure s-polarization. Besides, this circumstance offers good possibilities for the determination of a crucial apparatus parameter-the residual intensity for s-polarized (p-polarized) light if the incident wave has p-polarization (s-polarization). The second critical question is the influence of experimental error of Rr and DRss;pp =Rs;p . For reference we have included a computer simulation for the possible errors of approximate formulas (24)–(29) and (36)–(39) if the incident angles are the following: ð2Þ /ð1Þ a ¼ 0 and /a ¼ 70 . In order to calculate the error of these equations we give certain exact values for all unknown parameters exx ; eyy ; ezz ; h, and w and then calculate by the exact electromagnetic theory the values of DRpp;ss =Rp;s and Rr . Next we use these quantities in the form of DRpp;ss =Rp;s ð1 v pp;ss Þ and Rr ð1 v r Þ {where v pp;ss and v r represent the relative errors of DRpp;ss =Rp;s and Rr , respectively} in Eqs. (24)–(29) for calculating, firstly, e33 ; e23 ; e22 ; e11 e213 =e33 , and e12 e13 e23 =e33 and then for ðcalcÞ ðcalcÞ ðcalcÞ determining exx ; eyy ; ezz ; hðcalcÞ , and wðcalcÞ on the basis of Eqs. (36)–(39). The machine performed computations of the relaðcalcÞ ðcalcÞ ðcalcÞ tive errors ðexx exx Þ=exx ; ðeyy eyy Þ=eyy ; ðezz ezz Þ=ezz ; ðcalcÞ ðcalcÞ Þ=h, and ðw w Þ=w as functions of d=k for different ðh h v pp ; v ss ; v ps , and v sp are plotted in Figs. 2–6. If v pp ¼ v ss ¼ v ps ¼ v sp ¼ 0, then we obtain the pure mathematical error of approximate formulas in hand. Notice that computations show that the uncertainty of Rr is not of first importance (the error of DRpp;ss =Rp;s has a dominant role). Therefore, it may be said that
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P. Adamson / Surface Science 603 (2009) 3227–3233
the measurement of Rr at least for ultrathin layers on absorbing substrates with moderate values of the real part of refractive index, for example, glass or noble metal substrates is quite possible. For generalized ellipsometric measurements there is no ideal or standard technique yet devised and this method is still under pro-
gress and development. In Ref. [27], for example, is shown that by generalized ellipsometry sufficiently accurate birefringence measurements are possible on Langmuir–Blodgett films as thin as 10 nm. Notice that considerable attention now focusses on the elaboration of acutely sensitive and highly accurate ellipsometric technique [52] and this circumstance shows promise of extending the measurement range to 1 nm as well.
2
2 1
2. 5
2. 0
0
1
-1
-2
1. 5
Relative error,%
Relative error,%
3
3
-3
1. 0
1 0. 5
0. 0
2 0.001
0.002
0.003
0.004
d/ λ Fig. 2. Relative error of approximate formula (36) for exx as a function of d=k for an anisotropic film with nxx ¼ 2:9; nyy ¼ 2:6; nzz ¼ 2:8; h ¼ 70 ; u ¼ 0 ; w ¼ 30 at na ¼ 1; nsR ¼ 2:0; nsI ¼ 1:0; v pp ¼ v ss ¼ v ps ¼ v sp ¼ 0 (dashed curve), v pp ¼ 0:2% (1) 0.5%, (2) 1%, (3) v ss ¼ 0:3%, (1) 0.5%, (2) 2%, (3) v ps ¼ 10%, (1) 2%, (2 and 3) v sp ¼ 10%, (1) 2%, (2) 1%, (3). Preceding numbers in parentheses are curve labels.
-0.5 0.001
0.003
0.004
d/λ Fig. 4. Relative error of approximate formula (35) for ezz as a function of d=k for an anisotropic film with the same parameters as in Fig. 2.
16
12
3
10
12
8
2
8
Relative error,%
Relative error,%
0.002
6 4
1
2 0
4 0
1
-4
2 -8
-2 -4
3
-1 2
0.001
0.002
0.003
0.004
d/λ Fig. 3. Relative error of approximate formula (37) for eyy as a function of d=k for an anisotropic film with the same parameters as in Fig. 2.
0.001
0.002
0.003
0.004
d/λ Fig. 5. Relative error of approximate formula (38) for h as a function of d=k for an anisotropic film with the same parameters as in Fig. 2.
P. Adamson / Surface Science 603 (2009) 3227–3233
6 5
2
Relative error,%
4 3 3
2 1 1
0 -1 0.001
0.002
0.003
0.004
d/λ Fig. 6. Relative error of approximate formula (39) for w as a function of d=k for an anisotropic film with the same parameters as in Fig. 2.
4. Conclusions Relatively simple analytical approach developed in this work not only gives a physical insight into the reflection problem but also provides a way of estimating the necessary experimental accuracy for optical diagnostics by reflection characteristics. On the basis of the first-order expressions for reflection characteristics one cannot determine simultaneously the thickness and all six unknown components of the dielectric tensor for a biaxially anisotropic ultrathin film on an absorbing substrate because the first order expressions for the reflection parameters at different angles of incidence always generate a homogeneous system of linear equations for which the number of possible solutions is not limited, i.e., for every specified value of thickness the definite values of material constants exist. Furthermore, for a biaxially anisotropic ultrathin film, which thickness is known, we even cannot determine all six unknown anisotropic dielectric constants of film material. Only when one (or more) parameter(s) of anisotropy is known we can determine the remaining parameters on the basis of the first-order expressions for reflection characteristics. References [1] J. Robertson, Rep. Prog. Phys. 69 (2006) 327. [2] T. Yamaguchi, S. Yoshida, A. Kinbara, Thin Solid Films 21 (1974) 173. [3] J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves, Martinus Nijhoff, Dordrecht, 1987.
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