752 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Peep Adamson

Reflection of light in a long-wavelengthapproximation from an N-layer

system of inhomogeneous dielectricfilms and optical diagnostics of

ultrathin layers. I. Absorbing substrate

Peep Adamson

Institute of Physics, University of Tartu, Riia 142, Tartu 51014, Estonia

Received July 23, 2002; revised manuscript received November 8, 2002

The reflection of s- and p-polarized light from an N-layer system of inhomogeneous ultrathin dielectric filmsupon absorbing homogeneous substrates is investigated. The first-order approximate expressions for differ-ential reflectance and changes in the ellipsometric parameters that are caused by a multilayer system are ob-tained in the long-wavelength limit. The possibilities of using these formulas for resolving the inverse prob-lem for inhomogeneous ultrathin films are discussed. A number of novel options are developed forsimultaneously determining the dielectric constant and thickness of a homogeneous ultrathin film by differ-ential reflectance and ellipsometric measurements. © 2003 Optical Society of America

OCIS codes: 260.2110, 240.0310, 310.6860, 260.2130, 120.5700, 120.4290.

1. INTRODUCTIONOptical methods, particularly reflection methods(ellipsometry1,2 and various photometric techniques3),have been successfully employed for a long time for inves-tigation of thin films because they are fast, inexpensive,and noninvasive. Furthermore, ellipsometry is charac-terized by high sensitivity, and photometric methods havethe advantage of simplicity of experimental setup. Atpresent, the diagnostics of ultrathin films and multilayersis attracting a high level of interest, especially in modernmaterials technology.4 In the case of ultrathin layers, itis best to apply the differential methods5–7 that arefounded on direct measurement of the contribution of anultrathin layer to a reflection coefficient or an ellipsomet-ric angle.

From a theoretical standpoint, ultrathin layers have anessential advantage, namely, thickness d is much lessthan an optical wavelength l, which enables the reflectionproblem to be addressed analytically within the frame-work of a long-wavelength approximation, i.e., by the per-turbation theory. This advantage is significant owing tothe complexity of the exact solution of the reflection prob-lem for inhomogeneous films; in such a case the analyticaltreatment is known only for a few types ofinhomogeneity.8 Generally, the solution for an arbitraryrefractive-index profile can be found solely by numericalmethods.9

The use of numerical techniques to resolve the inverseproblem, i.e., to determine the parameters of layeredstructures from reflection characteristics, is rather com-plicated even for homogeneous films. The reason is thatthe relevant equations are transcendental and highlynonlinear and have, as a rule, many physically meaning-ful solutions. Moreover, because the inverse problem is

0740-3224/2003/040752-08$15.00 ©

essentially ill posed, often this problem may not have asolution at all (or it is impossible to obtain a solution withthe desired accuracy) because of the presence of both sys-tematic and random measurement errors.

It is worthwhile stressing that in the long-wavelengthapproximation a thin-film structure with a strongly ab-sorbing substrate provides exactly the conditions in whichthe contribution of an ultrathin dielectric film (d/l ! 1)to the reflection characteristics of light can be expressedin terms of up to first order of smallness in d/l. There-fore, in the issue under consideration mathematical rela-tionships take a relatively simple form (they are linearwith respect to film thickness), which is important fortackling an inverse problem.

A purpose of this paper, first, is to investigate furtherthe change of reflectance and ellipsometric parameters inthe long-wave limit if an N-layer system of nonuniform di-electric films is deposited upon an absorbing homoge-neous substrate. A second aim is to study the possibili-ties for determining the parameters of ultrathin films onthe basis of first order in d/l approximate expressions forreflection characteristics.

The paper is organized as follows: In Section 2 a de-tailed analysis of the angular dependence of differentialreflectance caused by ultrathin nonuniform layers is car-ried out. This is done because the literature is restrictedonly to derivation of the relevant basic formulas, butknowledge of the angular dependence is essential to theability to perform optical diagnostics through differentialreflectance. In Section 3 new approximate first-order for-mulas in the long-wave limit for the changes in the ellip-sometric angles produced by an N-layer system of nonuni-form films are obtained. The fourth section is concernedwith the solution of the inverse problem on the basis of

2003 Optical Society of America

Peep Adamson Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. B 753

first-order formulas obtained previously. A number ofnovel straightforward expressions for the unambiguousdetermination of the dielectric constants of uniform lay-ers are established. It is shown that simultaneously de-termining the thickness and the average material param-eters of ultrathin nonuniform layers is in principleimpossible. Notice that all the approximate analyticalresults obtained are correlated with the exact computersolution of the reflection problem for a multilayer systemof nonuniform films.

2. DIFFERENTIAL REFLECTANCEWe assume that all media are isotropic and nonmagnetic.We analyze the reflection of s- and p-polarized time-harmonic [the complex representation is taken in theform exp(2ivt), where v 5 2pc/l] electromagnetic planewaves in an ambient medium (a) of real dielectric con-stant «a [ na

2 from a plane-parallel layer medium con-sisting of a semi-infinite absorbing substrate (s) with com-plex (∧ denotes the complex quantity) dielectric constant«s 5 «s 1 ijs [ (ns 1 iks)

2 and N inhomogeneous dielec-tric layers with thickness di ! l (i 5 1,...,N) and real di-electric constant « i(z) [ ni

2(z) that vary only in the di-rection perpendicular to the layers (along the z axis).Ultrathin dielectric layers are considered phenomenologi-cally within the framework of macroscopic electrodynam-ics by use of the concept of a local dielectric constant.Therefore the thickness of the layers must be muchgreater than the spatial dimension on which the refrac-tive index is formed (thickness > a few nanometers).

The issue of calculating the contribution of ultrathinlayers to the reflection characteristics in the first order ind/l has long been studied. Within the scope of macro-scopic electrodynamics three formally different methodscan be used for this purpose: the standard matrixmethod for stratified media10 (applied in this paper), themethod of effective (integral) boundary conditions,1,11 andthe method of Green’s function.12 In a macroscopic con-text none of the methods outlined above is more generalor rigorous, whereas all three methods furnish exactly thesame result; i.e., in this approach they are actually onequal terms. But it must be emphasized that the Green’sfunction method is more general in the sense that it canalso be used in the microscopic theory of surface layers.13

Let us consider, in long-wavelength approximation tofirst order in di /l, the differential reflectance (the rela-tive change of the reflectance caused by a multilayer sys-tem of ultrathin inhomogeneous films) (DRN /R0)(s, p)

[ @RN(s, p) 2 R0

(s, p)#/R0(s, p) , where RN

(s, p) and R0(s, p) are

the reflectance of s- or p-polarized light for an N-layer sys-tem and a bare substrate (di [ 0), respectively. Fors-polarized light one can obtain

S DRN

R0D ~s !

' 8pna cos waDjs (i51

N

~«a 2 « ti!~di /l!,

« ti 5 di21 E

0

di

« i~z !dz,

D 5 @~«s 2 «a!2 1 js2#21, (1)

in which wa is the angle of incidence; and forp-polarization,

S DRN

R0D ~ p !

' 8pna cos wajsK (i51

N

~«a 2 « ti!@1 2 sin2 wa~1

1 «a« ti21pi!#~di /l!,

K 5 @1 2 2«aa sin2 wa#/$@«a~1 2 «aa sin2 wa!

2 «s cos2 wa#2 1 js2@cos2 wa

2 «a2«s

21a sin2 wa#2%,

«ni21 5 di

21 E0

di

« i21~z !dz,

pi 5 ~1 2 «a /«ni!/~1 2 «a /« ti!,

a 5 «s /u«su2, u«su2 5 «s2 1 js

2. (2)

A discussion of physical interpretation for the averagedmaterial parameters « ti and «ni is found, e.g., in Ref. 14.

In what follows, we consider the angular dependence ofdifferential reflectance in a long-wavelength approxima-tion. From formula (1) one can make the conclusion thatfor s-polarized light the angular dependence is trivial@(DRN /R0)(s) ; cos wa#. Hence there is no way to deter-mine the optical parameters of ultrathin dielectric filmsupon absorbing substrates with s polarization only. Atthe same time, for p-polarized light a wealth of informa-tion can be extracted from expression (2). First, the signof the contribution of each dielectric layer to reflectivity@DRi

( p)# depends on angle wa in the following way: If« ti . «a , wa , wBi , where

wBi 5 arctan@« ti /~«api!#1/2, (3)

and «s@1 1 (js /«s)2# . 2«a sin2 wa , we have DRi

( p) , 0.Moreover, the quantity DRi

( p) is negative when any two ofthese conditions are not fulfilled. But when only one ofthem or all three conditions are not fulfilled, we haveDRi

( p) . 0. Thus the angular dependence of the quantity(DR1 /R0)( p) for one layer (N 5 1) always has at leastone zero point for angle of incidence wB1 . If the condition2«aa sin2 wa 5 1 is fulfilled, we have also a second zeropoint, and DR1

( p) changes sign the second time when wavaries from 0 to 90°. However, one should bear in mindthat zero points are determined approximately (within anaccuracy of d1 /l). First-order formula (2) is inapplicablein the neighborhood of these points, where one shouldtake into account terms of higher (second) order in the ex-pansion in a small parameter. The angles for whichDR1

( p) exactly vanishes @DR1( p) [ 0# are slightly different

from the values obtained by approximate formula (3).Next, for p-polarized light the pseudo-Brewster angle,

where the absolute value of (DRN /R0)( p) can reach amaximum, appears to be of particular interest. If g[ js /u«s 2 «au ! 1, a condition that in practice, e.g., canbe fulfilled for semiconductor substrates in air («s @ «a5 1), the pseudo-Brewster angle differs slightly fromangle wBs 5 arctan(ns /na). From formula (2) it followsthat if g ! 1 then for wa 5 wBs

754 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Peep Adamson

S DRN

R0D

Bs

~ p !

' 8p«s2@js~«s 2 «a!~«s 1 «a!1/2#21

3 (i51

N

« ti21~« ti 2 «a!~«spi 2 « ti!~di /l!.

(4)

For a single layer, one can obtain from formulas (4) and(1) the relation

~DR1 /R0!Bs~ p !/@~DR1 /R0!~s !~ wa 5 wBs!#

' «s2~« t1 2 «sp1!~«s 2 «a!/~«a« t1js

2!. (5)

Consequently, the advantage of using angle wBs forp-polarized light increases with decreasing js and in-creasing «s . However, it is fruitless to use p-polarizedlight incident at angle wBs for « t1 → p1«s , where(DR1 /R0)Bs

( p) → 0 in the first order in d1 /l; in actuality, itis different from zero but represents a very small quantity@;(d1 /l)2#.

In the research reported in this paper the analyticalcalculations are supported by the exact solution of the in-homogeneous problem. Note that to eliminate errors inthe computer calculations of the reflectance of inhomoge-neous layers we utilize two completely different algo-rithms. The first is the classic method in which an inho-mogeneous layer was represented by a large set ofhomogeneous layers, and the second is the direct numeri-cal integration of the differential wave equation.9 Thedependences of the relative errors v 5 @(DR1 /R0)ex

(s, p)

2 (DR1 /R0)(s, p)#/(DR1 /R0)ex(s, p) [where (DR1 /R0)ex

(s, p)

was obtained by exact numerical solution of the problemand (DR1 /R0)(s, p) was calculated by approximate formu-las] on wavelength l and on angle of incidence wa are pre-sented in Fig. 1; that on ns and ks , in Fig. 2. The angu-lar dependence of differential reflectance is presented in

Fig. 1. Relative errors of approximate formulas (1) (dashedcurves) and (2) (solid, dashed–dotted, and dashed–dot-dottedcurves) as functions (a) of wavelength l if wa 5 45° and (b) ofangle of incidence wa if l 5 1000 for a single film with d1 5 1(curves 1 and 2), a three-film system with d1 5 d2 5 d3 5 1 (3),and a two-film system with d1 5 d2 5 1 (4) at na 5 1; ns 5 4;ks 5 1; n01 5 3 (1), 1.5 (2), 1 (3), 4 (4); nd1 5 3 (1), 4 (2), 1.5 (3,4); n02 5 nd1 (3, 4); nd2 5 5 (3), 4 (4); n03 5 nd2 (3); and nd35 2 (3). Profiles n1(z) are described by Eq. (6) with g 5 1 (2),2 (3), 0.25 (4); profiles n2(z) are described by Eq. (7) with g 5 1(3) 4 (4); and profiles n3(z) are described by Eq. (8) (3). Thequantities l and di are measured in arbitrary common units.

Fig. 3. The various dependences are plotted for inhomo-geneous layers with different distributions of refractiveindex n1(z). In specific calculations the refractive indexof an inhomogeneous layer is described by one of the fol-lowing functions:

Fig. 2. Relative errors of approximate formulas (1) (dashedcurves) and (2) (solid curves) as functions (a) of ks if ns 5 2 and(b) of ns if ks 5 2 for a single film with d1 /l 5 1023 (curve 1),2 3 1023 (4); a three-film system with d1 /l 5 d2 /l 5 d3 /l5 1023 (2); and a two-film system with d1 /l 5 d2 /l 5 23 1023 (3) at wa 5 75°; na 5 1; n01 5 1 (3), 1.5 (2), 3 (4), 4 (1);nd1 5 1.5 (1), 2.5 (2), 3 (4), 4 (3); n02 5 nd1 (2, 3); nd2 5 2 (3), 4(2); n03 5 nd2 (2); and nd3 5 1.5 (2). Profiles n1(z) are de-scribed by Eq. (6) with g 5 4 (1), 3 (2), 1 (3); n2(z) by Eq. (7) withg 5 0.5 (2), and by Eq. (8) (3), and n3(z) by Eq. (12) (2).

Fig. 3. (a) Differential reflectivities (DR1 /R0)(s) (dotted curves)and (DR1 /R0)( p) (solid, dashed–dotted, and dashed–dot-dottedcurves) and (b) the relation (DR1 /R0)( p)/(DR1 /R0)(s) (solid,dashed–dotted, and dashed–dot-dotted curves) as functions ofangle of incidence wa for a single inhomogeneous layer with a lin-ear profile [Eq. (6)] if g 5 1, na 5 1, d1 /l 5 2 3 1023, n015 1.5, nd1 5 ns 5 4, ks 5 0.5 (solid and dotted curves), ks 5 1(dashed–dotted curves), and ks 5 3 (dashed–dot-dotted curves).Dashed curves correspond to the calculation by approximate for-mulas (1) and (2).

Peep Adamson Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. B 755

ni~z ! 5 n0i 1 ~ndi 2 n0i!~z/di!g, (6)

ni~z ! 5 n0indi@ndig 2 ~ndi

g 2 n0ig !~z/di!#

21/g, (7)

ni~z ! 5 n0i~ndi /n0i!z/di

[ n0i exp@ln~ndi /n0i!~z/di!#, (8)

where n0i and ndi are values of the refractive index atz 5 0 and z 5 di , respectively, and g Þ 0 is a certainreal number.

3. CONTRIBUTION TO ELLIPSOMETRICANGLESThe contribution of a multilayer to ellipsometric angles inlong-wavelength approximation dC 5 C 2 C0 and dD5 D 2 D0 can be calculated from the classic relation2

@ rN( p) / rN

(s)# @ r0( p) / r0

(s)#21 5 (tan C / tan C0) exp @i(D 2 D0)#' (1 1 2dC/sin 2C0)exp(idD) 5 1 1 aC 1 iaD , where C, D

and C0 , D0 are the ellipsometric angles and rN( s) and r0

( s)

( s 5 s, p) are the amplitude reflection coefficients of amultilayer system and a bare substrate, respectively.Hence

dC 5 ~aC/2!sin 2C0 , dD ' tan~dD! ' aD , (9)

and to first order with respect to the small parametersdi /l on the basis of formula

Fig. 4. Relative errors of approximate Eqs. (11) (solid anddashed curves) and (12) (dashed–dotted, dashed–dot-dotted, anddotted curves) as functions (a) of wavelength l and (b) of angle ofincidence wa for na 5 1, ns 5 4, and ks 5 0.5. (a) wa 5 50°,d1 5 3, n01 5 1.5, nd1 5 4 (solid, dashed, dashed–dot-dotted,and dotted curves) and nd1 5 1.5 (dashed–dotted curve). Pro-files n1(z) are described by Eq. (6) with g 5 1 (solid and dashed–dot-dotted curves) and g 5 0.5 (dashed and dotted curves). Thequantities l and d1 are measured in arbitrary common units.(b) A single film with d1 /l 5 2 3 1023 (solid and dashed–dot-dotted curves), a three-film system (dotted curve) with d1 /l5 d2 /l 5 d3 /l 5 1023, and a two-film system (dashed–dottedcurve) with d1 /l 5 1023 and d2 /l 5 2 3 1023; n01 5 1.5; nd15 4 (solid, dashed–dot-dotted, and dotted curves), and nd15 1.5 (dashed–dotted curve); n02 5 nd1 ; nd2 5 1.5 (dottedcurve) and nd2 5 4 (dashed–dotted curve); n03 5 nd2 and nd35 2.5 (dotted curve). Profiles n1(z) are described by Eq. (6)with g 5 1 (solid and dashed–dot-dotted curves) and g 5 2 (dot-ted curve), n2(z) by Eq. (7) with g 5 0.5 (dotted curve) and g5 4 (dashed–dotted curve), and n3(z) by Eq. (8) (dotted curve).

rN~ s! ' r0

~ s!(1 1 i4pPa~ s!$@Pa

~ s!#2 2 @Ps~ s!#2%21

3 (i51

N

$ci~ s! 2 @Ps

~ s!#2bi~ s!%~di /l!), (10)

where Pa(s) 5 na cos wa , Ps

(s) 5 ( «s)1/2 cos ws , Pa

( p)

5 na /cos wa , Ps( p) 5 ( «s)

1/2/cos ws , cos ws 5 (12 «a«s

21 sin2 wa)1/2, ci(s) 5 « ti 2 «a sin2 wa , ci

( p) 5 « ti ,bi

( p) 5 1 2 «a«ni21 sin2 wa , and bi

(s) 5 1, we obtain

aC 5 4pnajs cos wa sin2 waA121 (

i51

N

~« ti 2 «a!

3 « ti21~a2iM1 2 a1iM2!~di /l!, (11)

aD 5 4pna cos wa sin2 waA121 (

i51

N

~« ti 2 «a!

3 « ti21~a1iM1 1 js

2a2iM2!~di /l!, (12)

where a1i 5 «s« ti 2 pi(«s2 2 js

2), a2i 5 2«spi 2 « ti ,M1 5 «a«s 2 «a

2 sin2 wa 2 («s2 2 js

2)cos2 wa , M2 5 2«scos2 wa 2 «a , and A1 5 M1

2 1 js2M2

2.It should be pointed out that Eq. (12) works at arbi-

trary values of js . If js 5 0 we get

Fig. 5. Relative errors of approximate Eqs. (11) (dashed, short-dashed, and dotted curves) and (12) (solid, dashed–dotted, anddashed–dot-dotted curves) as functions (a) of ks if ns 5 2.5 and(b) of ns if ks 5 1 for a single film (solid and dashed curves) withd1 /l 5 3 3 1023, a two-film system (dashed–dotted and short-dashed curves) with d1 /l 5 2 3 1023 and d2 /l 5 1023, and athree-film system (dashed–dot-dotted and dotted curves) withd1 /l 5 d2 /l 5 d3 /l 5 1023; wa 5 45°; na 5 1; n01 5 1.5(solid, dashed, dashed–dot-dotted, and dotted curves), and n015 2.5 (dashed–dotted and short-dashed curves); nd1 5 1.5(dashed–dotted and short-dashed curves), nd1 5 3 (solid anddashed curves), and nd1 5 4 (dashed–dot-dotted and dottedcurves); n02 5 nd1 ; nd2 5 1.5 (dashed–dot-dotted and dottedcurves) and nd2 5 2.5 (dashed–dotted and short-dashed curves);n03 5 nd3 5 1.5 (dashed–dot-dotted and dotted curves). Pro-files n1(z) are described by Eq. (6) with g 5 3 (dashed–dot-dotted and dotted curves), by Eq. (7) with g 5 1 (dashed–dottedand short-dashed curves), and by Eq. (8) (solid and dashedcurves), n2(z) by Eq. (7) with g 5 1 (dashed–dot-dotted and dot-ted curves) and by Eq. (8) (dashed–dotted and short-dashedcurves), and n3(z) by Eq. (8) (dashed–dotted and dotted curves).

756 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Peep Adamson

tan~dD! 5 4pna«s cos wa sin2 wa@~«s 2 «a!~«s cos2 wa

2 «a sin2 wa!#21 (i51

N

~«a 1 «s 2 « ti

2 «a«s«ni21!~di /l!. (13)

Equation (13) presents a generalization of the well-knownDrude approximate formula for one inhomogeneous di-electric layer upon a transparent substrate1 to themultilayer system of inhomogeneous dielectric films.But it must be emphasized that Eq. (11) makes sensemerely when js @ di /l because aC } js . In practicethis means that Eq. (11) may be applied only for sub-strates with strong absorption; e.g., for semiconductorsthe energy of a quantum of light must be greater than thematerial energy gap.

The dependences of relative errors of approximate for-mulas (9) on wavelength l and angle of incidence wa areshown in Fig. 4, and those on optical constants of sub-strate ns and ks in Fig. 5. For one (N 5 1) homogeneouslayer (« t1 5 «n1 [ «1 [ n1

2), Eq. (12) gives exactly thesame values for dD as the previously derived formula.15

However, the first-order expression in d1 /l for dC that iscommonly found in the literature5,15 is in error. Further-more, it is also in error when js ! «s and js is not muchgreater than d1 /l. For example, if wa 5 45°, na 5 1,n1 5 1.5, and ns 5 4 1 i0.5, then the relative errors for

Fig. 6. Ellipsometric quantities dD and dC as functions of angleof incidence wa for na 5 1, d1 /l 5 2 3 1023; ns 5 4 (solid,dashed–dotted, and short-dashed curves), 1.5 (dashed anddashed–dot-dotted curves); ks 5 0.1 (dashed–dot-dotted curves),0.5 (solid curve), 1 (dashed–dotted curve), 3 (dashed curve), 6(short-dashed curves); n01 5 1 (dashed and dashed–dot-dottedcurves), 1.5 (the other curves); nd1 5 1.5 (dashed and dashed–dot-dotted curves), 4 (the other curves). Profiles n1(z) are de-scribed by Eq. (6) with g 5 1. Dotted curves correspond to thecalculation by approximate formulas (9).

dC of approximate expression (6) of Ref. 15 and our for-mula are 2873.4% and 2.919% for d1 /l 5 1023 and2902.4% and 0.0303% for d1 /l 5 1025, respectively.

Finally, we note several facts with respect to the angu-lar dependence of differential ellipsometric quantitieswhen N 5 1 and «a 5 1 (for an actual situation). Ifjs 5 0, then the angular dependence of dD is governed bythe expression cos wa sin2 wa /(«s cos2 wa 2 «a sin2 wa).From that expression we can conclude that in the regionswa , wBs and wa . wBs the sign of dD is always different:if «sp1 . « t1 , then dD . 0 for wa , wBs and dD , 0 forwa . wBs ; if «sp1 , « t1 then dD , 0 for wa , wBs anddD . 0 for wa . wBs . The behavior of dD( wa) is practi-cally the same when js ! «s , but tan@dD( wBs)# is now notequal to infinity (Fig. 6). If js @ «s and « t1 , 2«sp1 ,then dD . 0 for any wa . For quantity dC it may benoted that the greater ks is, the greater the maximum ofudC( wa)u is. One can see from Fig. 7 the dependence ofdC and dD on ks .

4. OPTICAL DIAGNOSTICSA. Homogeneous Ultrathin FilmIn the homogeneous case a particularly useful attribute ofthe first-order expressions is that they are invertible, per-mitting, for example, direct calculation of the parametersfor dielectric films upon absorbing substrates. We con-sider here the instance of one (N 5 1) homogeneous layer(« t1 5 «n1 [ «1 and p1 5 1). Then approximate expres-sions (1) and (2) permit a simple determination of the re-fractive index of an ultrathin layer. First, one can mea-sure for a certain oblique angle of incidence wa thequantity t1 5 (DR1 /R0)( p)/(DR1 /R0)(s). From formulas(1) and (2) it then follows that

«1 ' «a~K/D !sin2 wa@~K/D !cos2 wa 2 t1#21. (14)

Second, one can use (DR1 /R0)( p) for normal incidenceand oblique (substantially different from normal) inci-dence and to define «1 from the formula

«1 ' «a~K/D !sin2 wa cos wa@~K/D !cos3 wa 2 t2#21,(15)

Fig. 7. Ellipsometric quantities dD (solid, dashed–dotted, anddashed–dot-dotted curves) and dC (dashed, short-dashed, anddotted curves) as functions of ks for na 5 1, wa 5 50°, d1 /l5 1023, n01 5 1.5, nd1 5 3.5, ns 5 1.5 (solid and dashed curves)and ns 5 3.5 (the other curves). Profiles n1(z) are described byEq. (6) with g 5 5 (dashed–dotted and short-dashed curves) andg 5 1 (the other curves).

Peep Adamson Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. B 757

where t2 5 (DR1 /R0)wa

( p)/(DR1 /R0)wa50( p) .

The advantage of these methods is that they provide anunambiguous determination of «1 . The dependences ofrelative errors of expression (14) on d1 /l and of formula(15) on ks are depicted in Figs. 8 and 9, respectively.

Next, we examine the possibility of determining the di-electric constant of an ultrathin film by using the pure el-lipsometric method or a combination of ellipsometric andreflectivity measurements. For ellipsometry a conve-nient quantity is the ratio dC/dD (or vice versa). On con-dition that dD and dC are measured at the same angle ofincidence wa , from formulas (9) we obtain

«1 ' ~2«sf1 1 «s2 2 js

2!/~ f1 1 «s!, (16)

where f1 5 @M1 2 2M2js sin21 2C0(dC/dD)#/@M21 2M1(js sin 2C0)

21(dC/dD)#.One can obtain similar unambiguous expressions for «1

by combining the changes in the ellipsometric angles andthe differential reflectance of s- or p-polarized light, i.e.,the ratio dD/(DR1 /R0)( s) or dC/(DR1 /R0)( s). Becauseat oblique angles (DR1 /R0)( p) @ (DR1 /R0)(s), in practice

Fig. 8. Relative error of approximate formula (14) as a functionof d1 /l for wa 5 60°; na 5 1; ns 5 4 (curve 1), 1.5 (2); ks5 0.5; n1 5 1.5 (1), 2 (2); m 5 0 (solid curves), 5% (dashedcurves); m is the relative error of t1 .

Fig. 9. Relative error of approximate formula (15) as a functionof ks for wa 5 55°, d1 /l 5 2 3 1023; na 5 1; ns 5 4 (solid,dashed–dotted, and dashed–dot-dotted curves), 1.5 (dashed,short-dashed, and dotted curves); n1 5 1.5 (solid, dashed–dotted,and dashed–dot-dotted), 2 (dashed, short-dashed, and dottedcurves); h 5 0 (solid and dashed curves), 5% (short-dashed anddashed–dotted curves), 25% (dashed–dot-dotted and dottedcurves). h is the relative error of t2 .

p-polarized light can be used more successfully. UsingEqs. (2) and (9), we obtain the following approximate for-mulas:

«1@ ' @M1~«s2 2 js

2! 2 2M2«sjs2 1 f2«a#/

~M1«s 2 M2js2 1 f2 cotan2 wa!,

f2 5 2jsKA1@dD/~DR1 /R0!~ p !#, (17)

«1 ' @2M1«s 1 M2~«s2 2 js

2! 2 f3«a#/

~M1 1 M2«s 2 f3 cotan2 wa!,

f3 5 4KA1 sin21 2C0@dC/~DR1 /R0!~ p !# (18)

if differential reflectance and ellipsometric quantities aredetermined at the same incident angle wa . The depen-dences of the relative error in approximate expression(17) are shown in Figs. 10–12. Note that the use of dif-ferential reflectance in combination with ellipsometric pa-rameters as an aid in optical diagnostics of ultrathin lay-ers was proposed long ago.16–18 However, in thosepapers, only numerical techniques, which suffer from sig-

Fig. 10. Relative errors of approximate formula (17) as func-tions (a) of d1 /l if wa 5 40° and (b) of wa if d1 /l 5 2 3 1023 forns 5 4 (solid, dashed, dashed–dot-dotted, and short-dashedcurves), 1.5 (dashed-dotted and dotted curves); ks 5 2 (short-dashed curve), 0.5 (the other curves); n1 5 1.5 (solid, dashed,and short-dashed curves), 2 (dashed–dotted and dotted curves),3.5 (dashed–dot-dotted curves); g 5 0 (solid, dashed–dotted, anddashed–dot-dotted curves), 5% (dashed and dotted curves). g isthe relative error dD/(DR1 /R0)( p).

Fig. 11. Relative error of approximate formula (17) as a functionof ks for na 5 1; wa 5 55°; d1 /l 5 4 3 1023; ns 5 4 (curve 1),1.5 (2, 3, 4); n1 5 1.5 (1), 2 (2), 2.5 (3), 3 (4); m 5 0 (solid curves),25% (dashed curve 1), 5% (dashed curve 2). m is the relativeerror dD/(DR1 /R0)( p).

758 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Peep Adamson

nificant shortcomings as we outlined in Section 1, wereemployed for solution of inverse problem.

B. Inhomogeneous Ultrathin FilmFor an inhomogeneous ultrathin film we have three un-known parameters: « t1 , «n1 , and d1 . Determination ofthese parameters requires at least three independentmeasurements. However, from approximate expressions(1), (2), and (9) it follows that to first order in d1 /l thequantities dD, dC, and (DR1 /R0)( p,s) are not independentcharacteristics of the ultrathin film. If two of thesequantities are known, then the remaining two can easilybe determined without knowledge of the parameters of anultrathin film (disregarding whether the film to be stud-ied is homogeneous or inhomogeneous). For example,among dD, (DR1 /R0)(s), and (DR1 /R0)( p) the followingrelation exists:

«a sin2 wa~ p !$A2~DR1 /R0!~s !

1 2 cos wa~s !DA1js@cos wa

~D! sin2 wa~D!#21dD%

5 A3$@D cos wa~s !/K cos wa

~ p !#~DR1 /R0!~ p !

2 cos2 wa~ p !~DR1 /R0!~s !%, (19)

and among dD, dC, and (DR1 /R0)(s) there is the followingrelationship:

A4$2 cos wa~s !@cos wa

~D! sin2 wa~D!#21DA1jsdD

1 A2~DR1 /R0!~s !%

5 A3$4 cos wa~s !@cos wa

~C! sin2 wa~C! sin 2C0#21DA1dC

2 A5~DR1 /R0!~s !%, (20)

where

A2 5 M1«s 2 js2M2 ,

A3 5 2«sjs2M2 2 ~«s

2 2 js2!M1 ,

A4 5 2«sM1 1 ~«s2 2 js

2!M2 ,

A5 5 M1 1 «sM2 ,

Fig. 12. Relative error of approximate formula (17) as a func-tion of ns for na 5 1; wa 5 55°; d1 /l 5 4 3 1023; ks 5 0.5(solid and dashed curves), 1 (dashed–dotted curve), 2 (dashed–dot-dotted curve); n1 5 1.5 (solid and dashed curves), 3 (dashed–dotted and dashed–dot-dotted curves); m 5 5% (dashed curve), 0(the other curves).

wa(s) and wa

( p) are the incident angles for (DR1 /R0)(s) and(DR1 /R0)( p), respectively, and wa

(D) and wa(C) are the inci-

dent angles for measurements of ellipsometric quantitiesdD and dC. Analogous relationships can be obtained forthe other two ternary groups: dC, (DR1 /R0)(s),(DR1 /R0)( p) and dD, dC, (DR1 /R0)( p). Therefore, on thebasis of approximate formulas (1), (2), and (9) we cannotsimultaneously determine all three parameters « t1 , «n1 ,and d1 of an inhomogeneous layer from differential reflec-tance and ellipsometric measurements. If the thicknessof the inhomogeneous layer is known, then the quantities« t1 and «n1 can easily be determined, for example, by theformulas

« t1 ' «a 1 ~A4c1 2 A3c2!~A2A4 1 A3A5!21, (21)

«n121 ' «a

21@1 2 ~A5c1 1 A2c2!~A2A4 1 A3A5!21#,

(22)where

c1 5 A1@4pna cos wa~D! sin2 wa

~D!#21~l/d1!dD,

c2 5 A1@2pna cos wa~C! sin2 wa

~C! sin 2C0js#21~l/d1!dC.

The fact that the angle that yields DR1( p) 5 0 for an in-

homogeneous layer is different (in the case of strong inho-mogeneity it may be substantially different) from the cor-responding angle for a homogeneous layer @wB15 arctan(n1 /na)# can be used in practice to estimate thedegree of homogeneity of ultrathin films. In the case ofweak inhomogeneity («ni ' « ti), formula (3) enables oneto estimate the average dielectric constant « ti of an inho-mogeneous layer. Note that for a homogeneous film thefact that the reflectivity for p-polarized light remains un-changed for wa 5 wB1 is typical not only for an ultrathinfilm but in all cases: For a transparent homogeneousfilm of arbitrary thickness the change in reflectivity forwa 5 wB1 5 arctan(n1 /na) exactly vanishes.

The system in which one can provide a rather continu-ous change of «a offers one more way to elucidate thequestion whether an ultrathin layer is inhomogeneous.It is precisely homogeneous layers that give DRi

( p) → 0for «a → « i for oblique incidence of p-polarized radiation;i.e., DRi

( p) has a zero value, but for inhomogeneity in anultrathin layer the dependence of DRi

( p) on «a never van-ishes [for « ti 5 «a we have Li(« ti 2 «a) 5 («a2 «ni)«a«ni

21 sin2 wa].

5. CONCLUSIONSThe first-order formulas for the contribution of dielectriclayers to reflectance and ellipsometric angle C may be ap-plied only for substrates with «s @ di /l; i.e., for ultrathindielectric films and optical wavelengths the substratesmust have a strong absorption (for semiconductor sub-strates the energy of a quantum of light must be muchgreater than the material energy gap). But the corre-sponding first-order expression for the change caused byan ultrathin layer in ellipsometric angle D works at arbi-trary values of js .

The most useful property of the first-order expressionsthat were obtained for the differential reflectance and thechanges of ellipsometric angles is that they are simply in-

Peep Adamson Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. B 759

vertible, permitting direct calculation of the refractive in-dex and the thickness of a uniform ultrathin film on thebasis of differential measurements. But there is no wayto determine simultaneously all three parameters « t1 ,«n1 , and d1 of a nonuniform film because the quantitiesdD, dC, and (DR1 /R0)( p,s) are not independent character-istics: If two of these quantities are known, then the re-maining two can easily be determined without knowledgeof the parameters of the ultrathin film.

Finally, it is important to keep in mind that all conclu-sions and formulas presented here can be applied not onlyfor ultrathin films; in infrared and microwave wavelengthregions these formulas work also for thick films (only con-dition di /l ! 1 must be fulfilled).

The author’s e-mail address is peep@fi.tartu.ee.

REFERENCES1. P. Drude, Lehrbuch der Optik (Hirzel, Leipzig, Germany,

1912).2. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Po-

larized Light (North-Holland, Amsterdam, 1977).3. L. Ward, The Optical Constants of Bulk Materials and

Films (IOP Publishing, Bristol, UK, 1988).4. G. Bauer and W. Richter, eds., Optical Characterization of

Epitaxial Semiconductor Layers (Springer-Verlag, Berlin,1996).

5. A. N. Saxena, ‘‘Changes in the phase and amplitude of po-larized light reflected from a film-covered surface and theirrelations with the film thickness,’’ J. Opt. Soc. Am. 55,1061–1067 (1965).

6. J. P. E. McIntyre and D. E. Aspnes, ‘‘Differential reflectionspectroscopy of very thin surface films,’’ Surf. Sci. 24, 417–434 (1971).

7. P. V. Adamson, ‘‘Differential reflection spectroscopy of sur-face layers on thick transparent substrates with normallyincident light,’’ Opt. Spectrosc. 80, 459–468 (1996).

8. G. Tyras, Radiation and Propagation of ElectromagneticWaves (Academic, New York, 1969).

9. B. Sheldon, J. S. Haggerty, and A. G. Emslie, ‘‘Exact com-putation of the reflectance of a surface layer of arbitraryrefractive-index profile and an approximate solution of theinverse problem,’’ J. Opt. Soc. Am. 72, 1049–1055 (1982).

10. F. Abeles, ‘‘Optical properties of thin absorbing films,’’ J.Opt. Soc. Am. 47, 473–482 (1957).

11. W. J. Plieth and K. Naegele, ‘‘Uber die Bestimmung der Op-tischen Konstanten Dunnster Oberflachenschichten unddas Problem der Schichtdicke,’’ Surf. Sci. 64, 484–496(1977).

12. J. Lekner, Theory of Reflection of Electromagnetic and Par-ticle Waves (Nijhoff, Dordrecht, The Netherlands, 1987).

13. A. Bagchi, R. G. Barrera, and A. K. Rajagopal, ‘‘Perturba-tive approach to the calculation of the electric field near ametal surface,’’ Phys. Rev. B 20, 4824–4838 (1979).

14. S. A. Tretyakov and A. H. Sihvola, ‘‘On the homogenizationof isotropic layers,’’ IEEE Trans. Antennas Propag. 48,1858–1861 (2000).

15. R. J. Archer and G. W. Gobeli, ‘‘Measurement of oxygen ad-sorption on silicon by ellipsometry,’’ J. Phys. Chem. Solids26, 343–351 (1965).

16. W.-K. Paik and J. O’M. Bockris, ‘‘Exact ellipsometric mea-surement of thickness and optical properties of a thin light-absorbing film without auxiliary measurements,’’ Surf. Sci.28, 61–68 (1971).

17. R. C. O’Handley, ‘‘Obtaining three surface-film parametersfrom two ellipsometric measurements,’’ Surf. Sci. 46, 24–42(1974).

18. B. D. Cahan, ‘‘Implications of three parameter solutions tothe three-layer model,’’ Surf. Sci. 56, 354–372 (1976).