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Ellipsometry of anisotropic (sub)nanometric dielectricfilms on absorbing materialsPeep Adamsona

a Institute of Physics, University of Tartu, Riia 142, Tartu 51014, EstoniaPublished online: 26 Feb 2014.

To cite this article: Peep Adamson (2014) Ellipsometry of anisotropic (sub)nanometric dielectric films on absorbing materials,Journal of Modern Optics, 61:5, 424-434, DOI: 10.1080/09500340.2014.890255

To link to this article: http://dx.doi.org/10.1080/09500340.2014.890255

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Ellipsometry of anisotropic (sub)nanometric dielectric films on absorbing materials

Peep Adamson*

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

(Received 24 October 2013; accepted 29 January 2014)

A new ellipsometric method was developed for determining optical parameters of ultrathin uniaxially anisotropicdielectric films. It is based on the phase conversion measurements of polarized reflected light at different incident angles.The elaborated technique possesses very high sensitivity and is successfully applicable even for sub-nanometric layersbecause the phase changes for such layers on absorbing substrates are generally markedly greater than the measurementerror. Another interesting facet of this method lies in the fact that the traditional model-based regression analysis is notused for data handling. The inversion problem is resolved on the basis of an original analytical approach, which has noneed of initial guesses for the desired parameters. The presented method is tested using a numerical simulation.

Keywords: ultrathin dielectric layer; anisotropic film; ellipsometry; optical diagnostics

1. Introduction

An appropriate technique for determining the opticalparameters of ultrathin films (thickness d much less thanoptical wavelength k) is, obviously, the ellipsometricmethod [1–7]. This is because ellipsometry is character-ized by high sensitivity, is nondestructive, noninvasive,and can successfully be performed in real-time for inves-tigating the dynamics of surfaces and nanostructures.The ellipsometric method is now well established in afairly wide spectral region – from vacuum UV tomid-IR. The bands that appear in an infrared spectrumcontain the highly individual fingerprint of a materialand, consequently, infrared ellipsometry can also beemployed to characterize surface layers in the low nano-meter range, i.e. thinner than 1/1000th of the wavelength[8–13]. More recently, the spectral range of ellipsometricstudies has also been extended to the far-infrared range[14–19].

Furthermore, ultrathin dielectric films are currentlythe object of much concentrated attention in micro- andnanoelectronics, and for that reason an optical probe withhigh sensitivity to measure the parameters of such filmsis highly desirable [20]. However, this brings up twoserious problems. The first lies in the fact that ellipsome-try is an indirect method, i.e. the desired parameters mustbe inferred from measured ellipsometric characteristicsby solution of an inversion problem, for which, as a rule,the widely accepted model-based regression analysis isused. Unfortunately, in the case of ultrathin films, themodel-based regression analysis, where unknownparameters result from fitting the experimental data forellipsometric quantities (prevalently frequency spectrum)

to the computed values of these quantities on the basisof a presumed optical model of the structure under study,is characterized by a strong correlation between opticalconstants and film thickness, i.e. by this method it is notpossible to determine the thickness and optical constantsof an ultrathin (thickness <10 nm) film simultaneously[3]. Because of this, a different approach is used fordetermining the unknown parameters of ultrathin films[21–24]. Namely, in the case of ultrathin films it isappropriate to separate the contributions of an ultrathinfilm to the ellipsometric angles Δ and Ψ from the corre-sponding contributions Δ0 and Ψ0 of the bare substrate,i.e. to use the differential quantities δΔ = Δ – Δ0 andδΨ = Ψ – Ψ0 for determining the parameters of anultrathin film. The quantities δΔ and δΨ are expressibleanalytically as a power series in the small parameterd=k. Such an approach gives us relatively simple analyti-cal equations which offer a clearer view of how themeasured ellipsometric quantities depend on the incidentangle and unknown film parameters, which in turnmakes it possible to analyze the equations in the light ofthe resolution of the inverse problem by multi-anglemeasurements, which produce the effect of enhancinginformation content.

A second critical problem is directly related to theaccuracy of ellipsometric measurements. If δΨ or δΔbecomes smaller than experimental error, then the use ofthis quantity in data processing fails. Generally, it hasbeen known that ellipsometric angle δΨ is unsuitable fortreatment of sub-nanometric films (or nanometric films ininfrared spectral range) because nowadays a properlyaligned ellipsometer with high-quality optics is capable

*Email: peep.adamson@ut.ee

© 2014 Taylor & Francis

Journal of Modern Optics, 2014Vol. 61, No. 5, 424–434, http://dx.doi.org/10.1080/09500340.2014.890255

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of precision only of about several hundredths of degreesin Δ and Ψ. For clarity, in this paper, the correspondingcalculations are also presented for anisotropic films(Section 2) and it is shown that δΨ is that quantity whichat first becomes smaller than experimental error ifd=k\10�3. Therefore, in this case, we can use only onemeasurable quantity, i.e. Δ. Naturally, this brings up thequestion: What parameters of ultrathin films can wedetermine solely on the basis of Δ? In general, one evi-dent advantage of ellipsometry, by comparison withintensity-based measurements, definitely lies in the factthat it involves simultaneous measurement of two inde-pendent quantities that contain much more informationabout a sample than just the one measurable quantity inphotometry. For example, in a recently published paper[24], it was shown that by applying a new mathematicalapproach for solving the inverse problem it is possibleon the basis of ellipsometric parameters Ψ and Δ todetermine the thickness and all optical constants of auniaxially anisotropic dielectric layer on an absorbingisotropic substrate simultaneously, i.e. without the usualcorrelation between the thickness and optical constants.However, in light of the issues mentioned above,this method is no longer valid in the case of asub-nanometric film in the visible region or nanometricfilms in the infrared spectral range, where δΨ is still notattainable by present-day ellipsometric techniques.

A purpose of this paper is to analyze the inverseissue for uniaxially anisotropic ultrathin films onisotropic absorbing substrates by a technique which isbased only on the phase conversion measurements, i.e.by using nothing else but Δ.

The paper is organized as follows. In Section 2, theformulas for ellipsometric angles of uniaxially aniso-tropic transparent ultrathin films on absorbing isotropicsubstrates are derived. The third section is concernedwith the solution of the ellipsometric inverse problem forsuch films.

2. Ellipsometric angles for nanometric films

Let an ultrathin (d � k) uniaxially anisotropicnon-absorbing dielectric film be located upon a semi-infinite isotropic homogeneous and absorbing substratewith complex dielectric constant es ¼ esR þ i esI �n2s ¼ ðnsR þ i nsI Þ2. The dielectric tensor for a uniaxiallyanisotropic transparent material in the x y z coordinatesystem is given by

e11 e12 e13e21 e22 e23e31 e32 e33

24

35 ¼ A

eo 0 00 eo 00 0 ee

24

35A�1; (1)

where A is the coordinate rotation matrix [25,26].Therefore,

e11 ¼ eo cos2 uþ ðeo cos2 hþ ee sin2 hÞ sin2 u; (2)

e12 ¼ e21 ¼ ðeo � eeÞ sin2 h sinu cos/; (3)

e22 ¼ eo sin2 uþ ðeo cos2 hþ ee sin2 hÞ cos2 u; (4)

e13 ¼ e31 ¼ ðee � eoÞ sin h cos h sinu; (5)

e23 ¼ e32 ¼ ðeo � eeÞ sin h cos h cosu; (6)

e33 ¼ eo sin2 hþ ee cos2 h; (7)

where eo � n2o and ee � n2e are the real principal dielec-tric-tensor components in the crystal-coordinate system,and θ and φ are the Euler angles with respect to a fixedx y z coordinate system (the Cartesian laboratory coordi-nate system). The laboratory x; y; and z axes are definedas follows. The reflecting surface is the xy plane, and theplane of incidence is the zx plane, with the z-axis normalto the surface of the layered medium and directed into it.The incident linearly polarized time-harmonic (thecomplex representation is taken in the form exp ( –iωt),where x ¼ 2p c=k, and k is a vacuum wavelength) elec-tromagnetic plane wave in a transparent ambient mediumwith isotropic and homogeneous dielectric constantea � n2a makes an angle ϕa with the z axis. Assume thatall the media are nonmagnetic.

Since in anisotropic systems the Jones matrixcontains off-diagonal terms, a so-called generalizedellipsometry [5,6,27–31] is designed for anisotropicsystems. There are three normalized complex ratios,which may be chosen for the measurement in thereflection mode ellipsometry, for example,

rpp=rss � tanWpp expðiDppÞ; (8)

rps=rss � tanWps expðiDpsÞ; (9)

rsp=rss � tanWsp expðiDspÞ; (10)

where the first subscript indicates the incident light andrpp ; rps ; rsp, and rss are the amplitude reflectioncoefficients.

In the following, we examine the major practicalissue, which arises in connection with ellipsometricquantities δΔ and δΨ. Namely, on the one hand, it isapparent that these quantities are rather small whend=k � 1. On the other hand, ellipsometers determineellipsometric angles always with a certain precision. Thisbrings up the question: What ellipsometric angles are atall measurable for nanometric or sub-nanometric films?

Figures 1–3 show the differential ellipsometric anglesdDpp ¼ Dpp � D0, dWpp ¼ Wpp �W0, Dr, and Wr (Δ0and Ψ0 are the ellipsometric angles for the bare substrate;Wr ¼ Dr ¼ 0 if d = 0; σ = ps or sp) as functions of d=kfor some different values of refractive indexes ofanisotropic films and substrates. The ellipsometric anglesdDpp, dWpp, Dr, and Wr in Figures 1–3 are computed by

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the rigorous electromagnetic theory for anisotropiclayered systems on the basis of Equations (8)–(10) (therelevant computational technique is outlined, e.g., in[1]). Note that the ambient refractive index na = 1 inFigures 1–12. The machine-performed computationsdemonstrate that for ultrathin films with d=k < 10−3 themeasurable ellipsometric quantities are only dDpp andDr, the others (dWpp and Ψσ) are smaller than themeasurement error (currently front-ranking ellipsometersare capable of precision of about several hundredths ofdegrees in Δ and Ψ ). In other words, in this case, theellipsometric angles which are bound up with thephase changes in the reflection process are chieflyapplicable for ultrathin films.

A complete solution of the reflection problem for ahomogeneous anisotropic uniaxial film is discussed in[32], where analytical expressions for the reflection coef-

ficients rpp; rps; rsp, and rss are also obtained in the thinlayer case (d=k � 1). On the basis of these formulas, inthe special case that the substrate is absorbing, for thequantity dDpp ¼ Dpp � D0 one can obtain the followingapproximate formula:

dDpp � 4p na cos/aða1M1 þ e2sI a2M2Þ

M21 þ e2sIM

22

d

k; (11)

where

a1 ¼ sin2 /aðeae�133 � 1Þðe2sR � e2sI Þ

þ ðesR � ea sin2 /aÞ ðe11 � e213e

�133 Þ

þ ðea sin2 /a � esR cos2 /aÞ ðe22 � e223e

�133 Þ

� eaesR sin2 /a;

a2 ¼ ea sin2 /a þ 2esR sin

2 /a

� ð1� eae�133 Þ þ cos2 /a � ðe22 � e223e

�133 Þ � e11

þ e213e�133 ;

M1 ¼ eaesR � e2a sin2 /a � ðe2sR � e2sI Þ

� cos2 /a;

M2 ¼ 2esR cos2 /a � ea:

Figure 1. Differential ellipsometric angle, dDpp ¼ Dpp � D0,as a function of d=k for an anisotropic film withno ¼ 1:7 ; ne ¼ 1:8, and θ = φ = 40º on an absorbing isotropicsubstrate with ns ¼ 1:5þ i 0:6 at ϕa = 50º (dotted curve) andfor an anisotropic film with no ¼ 2:8 ; ne ¼ 2:6, θ = 60º, andφ = 30º on different substrates with ns ¼ 3:9þ i 0:5 (solidcurve), ns ¼ 2:5þ i 0:1 (dash-dotted curve), and ns ¼ 0:3þ i 4(dash-dot-dotted curve) at ϕa = 70º.

0.0002 0.0004 0.0006 0.00080.00

0.01

0.02

0.03

0.04

20

30

40

50

60

70

80

2

2

1

1

(b)

ps,

sp,

deg.

d /

2

2

1

1

(a)

ps,

sp,

deg.

Figure 3. (a) Ellipsometric angles Dps (solid curves) and Dsp

(dashed curves) and (b) ellipsometric angles Wps (solid curves)and Wsp (dashed curves) as functions of d=k for an anisotropicfilm with no ¼ 2:8 ; ne ¼ 2:6, θ = 60º, and φ = 30º on differentsubstrates with ns ¼ 3:9þ i 0:5 (1) and ns ¼ 1:5þ i (2) atϕa = 70º. Preceding numbers in parentheses are curve labels.

Figure 2. Differential ellipsometric angle dWpp ¼ Wpp �W0

as a function of d=k. All the parameters and designations arethe same as in Figure 1.

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The quantity Dr (r ¼ ps or sp) can be expressed in theform:

tanDr � frQ1 þ grQ2

frQ2 � grQ1; (12)

where

fr ¼ ½e12 � e13e23e�133 �Q3 þ Pre23e

�133 na sin/a esR;

gr ¼ e12 � e13e23e�133

� � ð2Q3Þ�1 þ Pre23e�133 na sin/a

n oesI ;

Q1 ¼ Q3ðesR � eaÞ cos/a � e2sI ð2Q3Þ�1 cos/a

þ naðesR � eaÞ sin2 /a;

Q2 ¼ ½Q3 cos/a þ ð2Q3Þ�1ðesR � eaÞ cos/a þ na sin2 /a�esI ;

Q3 ¼ 2�1=2 esR � ea sin2 /a þ ½ðesR � ea sin

2 /aÞ2 þ e2sI �1=2n o1=2

;

in which Pps = –1 and Psp = +1. It is notable that forultrathin films on absorbing substrates the quantity Dr isindependent of film thickness and radiation wavelength.On the other hand, such drastic changes in phase atinterfaces are not rare in the process of reflection ofelectromagnetic waves from layered structures. Mathe-matically, it means that in the present context (in the firstorder) rps; sp=rss is a complex quantity which real andimaginary parts both proportional to d=k, so that theargument (phase) of this complex quantity, i.e. the ratioof the imaginary part to the real part, does not dependon d=k at all. As illustrated (in Figure 3), the computa-tions on the basis of exact electromagnetic theory alsoshow a very weak dependency of Δσ on d=k for ultrathinfilms (a detectable dependency on d=k appears onlywhen d=k� 10�2).

In relation to measurement of Dr it is pertinent tonote that intensity Iσ = RσI0 (where I0 is the intensity ofthe incident light) is usually very weak because Rr � 1,if d=k � 1 (Figure 4). The reflectance Rps (Rsp) shows

what fraction of the incident p-polarized (s-polarized)light intensity transforms into s-polarized (p-polarized)reflected light intensity. The corresponding analytical for-mulas for Rσ are derived in [33]. One can see from theseequations that Rσ ~ ðd=kÞ2. That is why the reflectedlight intensity into orthogonal polarization in the case ofultrathin films is very small. In order for the measure-ment of such intensity, the signal-to-noise ratio must be>1/Rσ. Thus, the signal-to-noise ratio must be very highin the course of the measurement of the ellipsometricquantity Dr. Note that the ellipsometric quantities dWpp

and Ψσ become measurable, in general, if d=k ≥ 10−2,perchance also then if d=k ≥ 10−3 (depends on materialsparameters).

It is also pertinent to note that standard ellipsometersused widely nowadays do not allow direct measurementof the quantity Δσ. Commonly, ellipsometric angles ofanisotropic objects are determined from a regressionanalysis which is applied over a large set of measuredintensity data at multiple settings of the polarizer andcompensator azimuths, and the phase retardation [5,6]. Itis clear that in the case of low values of reflected lightintensity this technique is not generally the most appro-priate for determining Δσ. Therefore, the development ofnovel experimental techniques that allow direct measure-ment of the quantity Δσ of an anisotropic ultrathin film ishighly topical.

3. Analytical inverse relationships

In the case of a uniaxially anisotropic layer, generally,we have five unknown parameters: eo; ee, h; u; and d.We examine here the question of what can be obtainedfrom the measurement of dDpp or Dr and show how afresh mathematical approach opens up new possibilitiesfor determining the optical parameters of an anisotropicultrathin layer. As already shown in [24], expressly thedependence of ellipsometric quantities on the incidenceangle via the relatively simple approximate equationsplays a dramatic role in solving the ellipsometric inverseproblem.

By applying the measurements of dDpp at differentincident angles and Equation (11), it can be shown that thestrongly nonlinear inverse problem rearranges to the solu-tion of a simple linear system of equations. Namely, asfollows from Equation (11), on the basis of dDpp we canobtain the following quantities: ðe11 � e213e

�133 � eaÞ d � x,

ðe�133 � e�1

a Þ d � y, and ðe22 � e223e�133 � eaÞ d � z (it is

suggested, certainly, that optical constants of a substrateand an ambient medium are known). Indeed, taking themeasurements of dDpp ð/aÞ at three different incidentangles /ð1Þ

a , /ð2Þa , and /ð3Þ

a , we can simply create the fol-lowing system of equations for determining the unknownquantities x; y; and z:

1 2 3 4

1E-6

1E-5

1

2

2

nsI

Rps

, R

sp

Figure 4. Reflectances Rps (solid curves) and Rsp (dashedcurves) as functions of nsI for an anisotropic film with d ¼ 4 nmno ¼ 3:6 ; ne ¼ 3:8, θ = 60º, and φ = 40º at two differentvalues of nsR = 1.5 (1) and 4 (2) if ϕa = 60º and k ¼ 1000 nm.Preceding numbers in parentheses are curve labels.

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bð1ÞD xþ cð1ÞD yþ dð1ÞD z ¼ að1ÞD ;

bð2ÞD xþ cð2ÞD yþ dð2ÞD z ¼ að2ÞD ;

bð3ÞD xþ cð3ÞD yþ dð3ÞD z ¼ að3ÞD ; ð13Þwhere

aðiÞD ¼ dDpp ð/ðiÞa Þ ðM ðiÞ

1 Þ2 þ e2sI ðM ðiÞ2 Þ2

4p na cos/ðiÞa

" #k;

dDpp ð/ðiÞa Þ ¼ Dpp ð/ðiÞ

a Þ � D0 ð/ðiÞa Þ; and i ¼ 1; 2; 3.

Expressions for bðiÞD , cðiÞD , dðiÞD , M ðiÞ1 , and M ðiÞ

2 are availablein [24]. Therefore,

x � ðe11 � e213e�133 � eaÞ d ¼ Dx=D; (14)

y � ðe�133 � e�1

a Þ d ¼ Dy=D; (15)

z � ðe22 � e223e�133 � eaÞ d ¼ Dz=D; (16)

where D, Dx, Dy, and Dz are the determinants of theCramer’s rule:

D ¼bð1ÞD cð1ÞD dð1ÞD

bð2ÞD cð2ÞD dð2ÞD

bð3ÞD cð3ÞD dð3ÞD

���������

���������; Dx ¼

að1ÞD cð1ÞD dð1ÞD

að2ÞD cð2ÞD dð2ÞD

að3ÞD cð3ÞD dð3ÞD

���������

���������;

Dy ¼bð1ÞD að1ÞD dð1ÞD

bð2ÞD að2ÞD dð2ÞD

bð3ÞD að3ÞD dð3ÞD

���������

���������; Dz ¼

bð1ÞD cð1ÞD að1ÞD

bð2ÞD cð2ÞD að2ÞD

bð3ÞD cð3ÞD að3ÞD

���������

���������:

Furthermore, from the measurement of Δσ at theincident angle /ðrÞ

a one can obtain:

e13 � e12e33e�123

¼ Prna sin/a½esRQ1 þ esIQ2 þ tanDrðesIQ1 � esRQ2Þ�Q1Q3 þ esIQ2ð2Q3Þ�1 � tanDrðQ2Q3 � esIQ1ð2Q3Þ�1Þ � b:

ð17ÞIt is apparent that on the basis of four independent

measurable quantities, i.e. ðe11 � e213e�133 � eaÞ d,

ðe�133 � e�1

a Þ d, ðe22 � e223e�133 � eaÞ d, and ðe13 � e12e33e�1

23 Þ ,one can determine only four unknown quantities from allfive desired parameters eo; ee, h; u; and d.

First we consider the case where the thickness d of auniaxially anisotropic ultrathin film is known. Then onthe basis of Equations (14)–(17) and (2)–(7) one canobtain the following set of equations for the fourunknown quantities eo; ee, h, and φ:

2eo þ ðee � eoÞ sin2 h� sin2 h cos2 hðee � eoÞ2 ðc3 þ e�1a Þ

¼ c1 þ c2 þ 2ea; ð18Þ

cos 2u ½ðee � eoÞ sin2 h� sin2 h cos2 hðee � eoÞ2� ðc3 þ e�1

a Þ� ¼ c1 � c2; ð19Þ

ðeo sin2 hþ ee cos2 hÞ�1 ¼ c3 þ e�1

a ; (20)

eo sinu tan h ¼ �b; (21)

where c1 ¼ Dx=ðDdÞ, c2 ¼ Dy=ðDdÞ, and c3 ¼ Dz=ðDdÞ.Equations (18) and (19) yield:

u ¼ arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

21� c1 � c2

P1 � 2eo

� �s����������; (22)

where P1 = γ1 + γ2 + 2ɛa. Equation (20) gives:

h ¼ arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2 � eeeo � ee

r��������; (23)

where P2 ¼ ðc3 þ e�1a Þ�1. Next, by using Equations (22)

and (23), we obtain from Equation (21) that

ee ¼ P2 � b2

e20

ðP1 � 2eoÞ ðeo � P2Þðc2 � eo þ eaÞ : (24)

Finally, by substituting Equations (24) and (23) intoEquation (18) we obtain the following cubic equationfor ɛo:

e3o þ m1e2o þ m2eo þ m3 ¼ 0; (25)

where

m1 ¼ c1 � c22

� P1 � b2

P2;

m2 ¼ b2 1þ P1

2P2

� þ P1ðc2 þ eaÞ

2;

m3 ¼ � b2P1

2:

Note that for two special cases, i.e. φ = 0 (θ ≠ 0) andθ = 0, the formula for ɛo takes the form:

eo ¼ c2 þ ea; (26)

and for ɛe:

ee ¼ P2c1 þ eac2 þ ea

� ; (27)

if φ = 0 and θ ≠ 0, and

ee ¼ P2; (28)

if θ = 0.Secondly, we consider a situation where one parame-

ter of anisotropy is known, for example, φ (φ ≠ 0), andthe desired parameters are eo; ee, h; and d. In this casethe quantities eo; ee, and h can be determined from thefollowing system of equations:

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e11 � e213e�133 � ea ¼ c12ðe�1

33 � e�1a Þ;

e22 � e223e�133 � ea ¼ c32ðe�1

33 � e�1a Þ;

e13 � e12e33e�123 ¼ b; ð29Þ

where γ12 = Dx/Dy and γ32 = Dz/Dy. Solving the systemof Equations (29) gives for ɛo the ensuing cubic equation:

a0e3o þ a1e

2o þ a2eo þ a3 ¼ 0; (30)

where

a0 ¼ 4e2aAðB� CAÞ;

a1 ¼ 2ea½Bð1� 4e2aAÞ � 2CAð1� 2e2aAÞ�;

a2 ¼ C½2e2aAð3� 2e2aAÞ � 1� � 2e2aBð1� 2e2aAÞ;

a3 ¼ Ceað1� 2e2aAÞ;in which

A � cos 2u

2ðc12 cos2 u� c32 sin2 uÞ ;

B � 1� ðc12 þ c32ÞA;

C � ðb= sinuÞ2;and for ɛe and θ the following expressions:

ee ¼ ea � Ce�2o feo½1þ 2eaAðeo � eaÞ� � eag

1þ 2eaAðeo � eaÞ ; (31)

tan h ¼ � b = ðeo sinuÞ: (32)

The special case φ = 0, where ɛ12 = ɛ13 = 0 and,therefore, β = 0, has only three independent Equations(14)–(16) for four unknown quantities, eo; ee, h and d.Because of this the simultaneous determination of theoptical constants and thickness is not possible or, inother words, correlation-free measurements cannot beperformed.

If we know θ (θ ≠ 0) and eo; ee, φ, and d areunknown, then for the desired parameters eo; ee, and φwe obtain:

e4o þ b1e3o þ b2e

2o þ b3eo þ b4 ¼ 0; (33)

where

b1 ¼ c12=ea � ea;

b2 ¼ ðc32 � ð1þ tan2 hÞc12 � 2b2Þ= tan2 h;

b3 ¼ ðb2ð2ea � ðc12 þ c32Þ=eaÞ � eaðc32 � c12ÞÞ= tan2 h;

b4 ¼ b2ðc12 þ c32Þ= tan2 h;

ee ¼ eo þ eacos2 h

1

1þ t� eo

� ; (34)

in which

t ¼ eaðeo � eaÞ ðe2o tan2 h� 2b2Þe2oc12 tan

2 h� b2ðc12 þ c32Þ;

and

sinu ¼ � b = ðeo tan hÞ: (35)

The special case θ = 0, where ɛ12 = ɛ13 = ɛ23 = 0,ɛ11 = ɛ22 = ɛo, and ɛ33 = ɛe, has only two equations,

ðeo � eaÞ d ¼ Dx

D(36)

and

ðe�1e � e�1

a Þ d ¼ Dy

D; (37)

for three unknown parameters, eo; ee, and d. Therefore,in this simple case of anisotropy, correlation-freemeasurements are also impossible (the material constantsɛo and ɛe can be determined only if the thickness isknown in advance).

Note that if the material parameters (eo; ee, θ, φ) aredetermined, then the thickness d can simply becalculated on the basis of Equations (14)–(16), i.e.

d ¼ Dy

Dðe�133 � e�1

a Þ : (38)

For reference, we have included a computer simula-tion for the evaluation of possible errors of the obtainedequations. Computer simulations offer a clearer view ofhow the analytical formulas work because such anapproach makes it possible to analyze more complicatedsituations than we can create in real experiments. Inorder to calculate the error of approximate expressions,we give certain exact values for all unknown parameters,ɛo, ɛe, θ, φ, and d, and then calculate by the exact elec-tromagnetic theory the values of δΔpp and Δσ. Next weuse these quantities in the form of δΔppð1� vð ppÞD Þ andDrð1� vðrÞD Þ, where vð ppÞD and vðrÞD represent the relativeinstrumental errors of δΔpp and Δσ, respectively, inapproximate Equations (22)–(25), (30)–(35), and (38) forcalculating by these equations the unknown quantitieseðcalcÞo , eðcalcÞe , θ(calc), φ(calc), and d (calc). Note that inFigures 5–11, the incident angles /ð1Þ

a ¼ 50, /ð2Þa ¼ 60,

and /ð3Þa ¼ 70. The machine computations of the rela-

tive errors ðeo � eðcalcÞo Þ=eo, ðee � eðcalcÞe Þ=ee, (θ – θ(calc))/θ, and (φ – φ(calc))/φ, where eðcalcÞo , eðcalcÞe , θ(calc), φ(calc)

are calculated by Equations (25), (24), (23), and (22),respectively, as functions of d=k for vð ppÞD = vðrÞD = 0are plotted in Figures 5 and 6 (if vð ppÞD =vð psÞD = 0, thenwe obtain the pure mathematical error of the approxi-mate formulas that has nothing to do with the errors

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vð ppÞD and vð psÞD of δΔpp and Δps that can occur in the mea-surements of these quantities). These results demonstratethe fact that in the case, where the refractive index of asubstrate is greater than the refractive index of an ultra-thin film, Equations (22) and (23) for determining φ andθ have reasonable error only for small optical mismatchbetween the film and the substrate (Figure 5(b)). In theopposite instance (the refractive index of an ultrathinfilm is greater than the refractive index of a substrate),these equations do not work if d=k� 10�4and theEquations (24) and (25) for determining ɛe and ɛo workonly for relatively small refractive index jump betweenthe film and the substrate (Figure 6).

Figures 7 and 8 illustrate the role of the instrumentalerrors vð ppÞD and vð psÞD . Note that the sign of these errors is,of course, meaningless. However, in this paper, in placeof practical measurements we use the computer simula-tions where we must decide whether given (‘measured’)incorrect ellipsometric angles, which we use in the solu-tion of the inverse problems, are greater or smaller thaninitially computed exact ellipsometric angles. Besides,theoretically it is also interesting to examine how the

0.0002 0.0003 0.0004 0.0005 0.0006

-12

-8

-4

0

4

8

12

-2

0

2

4

6

8

10

1

1

(b)

Rel

ativ

e er

ror o

f a

nd

, %

d /

2

2

1

2

2

(a)

Rel

ativ

e er

ror o

f o a

nd

e, %

Figure 5. Relative errors of (a) ɛo (solid curves) and ɛe (dashedcurves) and (b) θ (solid curves) and φ (dashed curves)determined by Equations (27), (26), (25), and (24), respectively,as functions of d=k for anisotropic films with no ¼ 2.7,ne ¼ 2.6, θ = φ = 40º(1) and no ¼ 1.7, ne ¼ 1.6, θ = φ = 40º(2)

on substrates with ns ¼ 3:9þ i 1:5 if vð ppÞD ¼ vð psÞD ¼ 0.Preceding numbers in parentheses are curve labels.

0.0001 0.0002 0.0003 0.0004 0.0005

-12

-8

-4

0

4

8

1

23

1

2

3

d /

Rel

ativ

e er

ror o

f o a

nd

e, %

Figure 6. Relative errors of ɛo (solid curves) and ɛe (dashedcurves) determined by Equations (27) and (26), respectively, asfunctions of d=k for anisotropic films with no ¼ 1.7, ne ¼ 1.8(1); no ¼ 2.7, ne ¼ 2.8 (2); no ¼ 3.1, ne ¼ 3.2 (3); θ = 70º andφ = 30º (1–3) on substrates with ns ¼ 1:5þ i if vð ppÞD ¼ vð psÞD ¼0. Preceding numbers in parentheses are curve labels.

0.0002 0.0004 0.0006 0.0008

-2

0

2

4

6

8

10

-8

-6

-4

-2

0

2

3

2

1

(b)

Rel

ativ

e er

ror o

f , %

d /

3

2

1

(a)

Rel

ativ

e er

ror o

f , %

Figure 7. Relative errors of (a) θ and (b) φ determined byEquations (25) and (24), respectively, as functions of d=k foran anisotropic film with no ¼ 2.7, ne ¼ 2.6, and θ = φ = 40º

on a substrate with ns ¼ 3:5þ i if vð ppÞD = 0 (1), 0.3% (2),

−0.3% (3) and vð psÞD = 0 (solid curves), 15% (dashed curves),and −15% (dotted curves). Preceding numbers in parenthesesare curve labels.

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results depend on the sign of the errors. Certainly,making an estimate of the errors of desired parameters ofanisotropic films under study, strictly speaking, we mustperform the calculations for errors with positive andnegative signs and then out of these obtained values forerrors we can choose the greatest number.

It must be emphasized that the error of Equations(22)–(25) is strongly dependent on vð ppÞD , but weakly onvðrÞD : if the quantity vð ppÞD can be varied only within a fewdecimals of percent (Figure 7), then vðrÞD is subject towide variations (within several percent (Figure 8)).Another moment lies in the fact that the influence of themathematical error and instrumental error can possessesdifferent signs that can involve a decrease of the totalerror of approximate equations around the certain valueof d=k (Figure 7).

Figure 9 shows the mathematical error of Equations(30)–(32) and (38) for the case where the angle φ isknown and desired parameters are eo; ee, h and d. Asdistinct from the foregoing Equations (22)–(25) (the casewhere d is known), the accuracy of Equations (30)–(32)

-8 -4 0 4 8

-2

0

2

4

6

8

10

Rel

ativ

e er

ror o

f o,

e, , a

nd

, %

v (ps), %

Figure 8. Relative errors of ɛo (solid curve), ɛe (dashedcurve), θ (dash-dotted curve), and φ (dotted curve) determinedby Equations (27), (26), (25), and (24), respectively, as

functions of vð psÞD if vð ppÞD = 0 for an anisotropic ultrathin filmwith d=k ¼ 5� 10�4, no ¼ 1.5, ne ¼ 1.6, θ = 50º, and φ = 70º

on a substrate with ns ¼ 2:6þ i 0:5.

500 1000 1500 2000

-15

-10

-5

0

5

10

15

-10

-5

0

5

10

15

(b)

Rel

ativ

e er

ror o

f o,

e, , a

nd d

, %

Wavelength, nm

(a)

Rel

ativ

e er

ror o

f o,

e, , a

nd d

, %

Figure 9. Relative errors of ɛo (dash-dotted curves), ɛe(dashed curves), θ (solid curves), and d (dotted curves)determined by Equations (32), (33), (34), and (40), respec-tively, as functions of k for an anisotropic film (a) withd ¼ 0:4 nm, no ¼ 1.7, ne ¼ 1.8, θ = 70º, and φ = 60º on asubstrate with ns ¼ 4þ i 2 and (b) with d ¼ 0:4 nm, no ¼ 3.7,ne ¼ 3.8, θ = 70º, and φ = 60º on a substrate withns ¼ 1:5þ i 2 if vð ppÞD = vð psÞD = 0.

-4 -2 0 2 4

-2

0

2

4

-6

-4

-2

0

2

(b)

Rel

ativ

e er

ror o

f d, %

v (pp), %

(a)

Rel

ativ

e er

ror o

f , %

Figure 10. Relative errors of (a) θ and (b) d determined byEquations (34) and (40) for the case where φ is known asfunctions of vð ppÞD for an anisotropic ultrathin film withd=k ¼ 10�3, no ¼ 3.5, ne ¼ 3.6, θ = 20º, and φ = 60º on asubstrate with ns ¼ 2:5þ i 0:5 if vð psÞD = 0 (solid curves), 2%(dash-dotted curve), −2% (dash-dot-dotted curve), 5% (dashedcurves), and −5% (dotted curves).

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and (38) does not depend significantly on opticalmismatch between the film and the substrate and theseformulas work similarly well when the refractive indexof a film is greater or smaller than the refractive index ofa substrate.

However, a more interesting feature of Equations(30)–(32) lies in the fact that the error of these equationsdoes not depend on the instrumental error vð ppÞD atall (Figure 10(a)), if we suppose that the systematic errorat three different incident angles /ð1Þ

a , /ð2Þa , and /ð3Þ

a arepractically equal (this is quite realistic if we take intoconsideration that we can select comparatively close-setincident angles). The reason is that Equations (30)–(32)depend on relations Dx/Dy and Dz/Dy. Note that randomerrors (noise) can be suppressed by adjustment of the sig-nal-to-noise ratio. At the same time, Equation (38) fordetermining d depends on vð ppÞD (Figure 10(b)) because itis governed by the quantity Dy/D. It is also significant thatthe dependence of Equations (30)–(32) and (38) on vðrÞD isgenerally not strong (Figure 11), however, it is markedly

controlled by material parameters, especially for θ(e.g. the solid line in Figure 11(b)).

In relation to mathematical error of approximate for-mulas, it is well to bear in mind that there can be existcertain values of nsR and nsI (Figure 12(b)) where Dps→0 (certainly, this would hold for Dsp as well). It is clearthat the first-order approximate formula (Equation (12))does not work in this case; consequently, deducedapproximate expressions for determining unknownparameters ɛo, ɛe, θ, φ, and d also lose their meaning.

In conclusion on the accuracy of the obtainedapproximate formulas, we can state the following. Ofcourse, the quantity that has a dramatic effect on theaccuracy of the approximate formulas is d=k, whichmust be � 1 (in fact, all previous analyses started fromthis assumption). Broadly speaking, it may be concludedthat if we have anisotropic films with d=k~10−4

(mid- and far-infrared regions for nanometric layers),then the elaborated theory gives the best fit. On the otherhand, the answer to the question how small must thequantity d=k be depends also on the material parameters.As a consequence, if we have anisotropic films withthicknesses of several nanometers, then for radiation with

-8 -4 0 4 8

-25

-20

-15

-10

-5

0

5

10

-6

-4

-2

0

2

4

6

8

(b)

Rel

ativ

e er

ror o

f , %

v (ps), %

(a)

Rel

ativ

e er

ror o

f o,

%

Figure 11. Relative errors of (a) ɛo and (b) θ determined byEquations (32) and (34), respectively, as functions of vð psÞD foran anisotropic film with d ¼ 1 nm, no ¼ 3:5 ; ne ¼ 3:6, θ = 20º,and φ = 60º on a substrate with ns ¼ 2:5þ i 0:5 (dotted curves)and for anisotropic films with d ¼ 0:8 nm, no ¼ 2:3 ; ne ¼ 2:2,θ = 20º, and φ = 20º (solid curves); d ¼ 1:2 nm, no ¼ 2:6 ;ne ¼ 2:7, θ = 60º, and φ = 10º (dashed curves); d ¼ 1:5 nm,no ¼ 3:3 ; ne ¼ 3:4, θ = 20º, and φ = 70º (dash-dotted curves)on substrates with ns ¼ 3:9þ i at k ¼ 1000 nm.

1 2 3 4

-60

-40

-20

0

20

40

60

1.3

1.4

1.5

1.6

1.7

1.8

(b)

ps,

deg.

nsI

(a)

pp ,

deg

.

Figure 12. (a) Differential ellipsometric anglesdDpp ¼ Dpp � D0 and (b) Dps as functions of nsI for an ultrathinanisotropic film with d=k ¼ 10�3, no ¼ 2:5 ; ne ¼ 2:55,θ = 60º, and φ = 40º if nsR = 1.5 (solid curves), 2.6 (dashedcurves), and 4 (dotted curves) at ϕa = 70º.

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k~ a few micrometers (d=k~10−3) we need to analyzeeach situation with specific material parametersindividually in order to make the best use of theobtained formulas (the computations make it clear thatfor certain values of material parameters the approximateexpressions work sufficiently well for d=k~10−3). It ispertinent to note that nowadays considerable study isalso being devoted to the possibilities of optical diagnos-tics for ultrathin films less than 1 nm: thick [34] becausethis is realistic in the context of real-time control ofdeposition, the identification of physisorbed and chemi-sorbed layers on the monolayer scale, and applicationsinvolving preparation of next-generation electronic andoptoelectronic devices. Thus, the obtained approximateformulas are of practical interest in the near-IR and evenin the visible spectrum as well. It is also worth bearingin mind that if we consider anisotropic films withd=k~10−2, then we cannot say with confidence thatapproximate formulas are not applicable: the machinecomputations show that for certain material parametersthese formulas even work, although with greater error. Inpractice, however, such results are of interest as suitablestarting points for numerical iterative methods, where theproper choice of the initial data plays a crucial role.

Further, one important point to remember is that inthis paper we consider an optical model where all inter-faces are physically ideally flat (the surface roughnesshas not been taken into consideration). This simplesituation is also actual because at present very uniformultrathin films with smooth interfaces can be grown, e.g.by means of atomic layer deposition [35,36] or molecu-lar beam epitaxy [37,38]. It should be mentioned that theinclusion of interface roughness is not a simple problemand therefore anisotropic ultrathin films with roughboundaries should be considered further in their ownright.

However, in general terms, one can state thefollowing. If the boundaries of an ultrathin film can berepresented by microscopic roughness, i.e. these arecomposed of the random mixtures of void and solidmaterial or two different solid materials (the interfacebetween the film and substrate), so that the characteristicsizes – the heights and lateral dimensions of the homo-geneous regions (irregularities) – are much smaller thanthe radiation wavelength, one can replace the boundariesby fictitious ultrathin films with suitable effective opticalconstants and thicknesses [6]. These fictitious films,which boundaries that are ideally flat, are described byoptical constants calculated using the formulas derivedwithin the effective medium approximation (EMA). Forexample, for isotropic materials the most popular is theBruggeman formula where the effective medium itselfacts as a host medium [39,40]. Thus, in the case wherethe interfaces are microscopically rough, a one-layermodel becomes a two- or three-layer system which can

be analyzed within the traditional transfer matrixformalism.

Notice that for ultrathin films the basic condition ofusing the EMA at the ellipsometric studies of roughinterfaces is finely fulfilled for the heights of irregulari-ties but not always for their lateral dimensions. If therough surface contains the spatial wavelengths compara-ble with the wavelength of incident light, then use couldbe made of the Rayleigh–Rice scattering theory [41,42],which, in addition, gives an insight into what is theinfluence of surface morphology on the optical responseof surfaces [43]. Although it is evident that for the opti-cal characterization of the rough surfaces this theory ismore advantageous than the EMA from the general pointof view, then the practical applications of the Rayleigh–Rice theory, unfortunately, are connected with compli-cated numerical calculations, particularly in the solutionof the inversion problem.

Finally, it might be worth pointing out that thealternative reflection technique for optical diagnostics ofanisotropic ultrathin films is based on reflectancemeasurements [44]. In principle, it is possible to inte-grate ellipsometric and reflectance measurements toobtain a complete solution of the optical properties of ananisotropic ultrathin film. For example, we can use fordetermining ðe22 � e223e

�133 � eaÞ d, ðe11 � e213e

�133 � eaÞ d,

and ðe�133 � e�1

a Þ d the measurements of ellipsometric dif-ferential angle dDpp at three different incident angles/ð1Þa , /ð2Þ

a , and /ð3Þa , and instead of ellipsometric angle Δσ

we can use the reflectances Rps and Rsp. As follows fromcorresponding analytical expressions for these reflec-tances [44], the quantities ðe12 � e13e23e�1

33 Þ d andðe23e�1

33 Þ d can be determined by the use of Rps and Rsp.Thus, from such combined optical measurements fivequantities ðe22 � e223e

�133 � eaÞ d, ðe11 � e213e

�133 � eaÞ d,

ðe�133 � e�1

a Þ d, ðe12 � e13e23e�133 Þ d, and ðe23e�1

33 Þ d can bedetermined. On the basis of these five independentquantities, all five unknown parameters, i.e. eo; ee, h; u;and d, can be found. However, the detection of Rσ forultrathin films is also a serious challenge from anexperimental standpoint.

4. Conclusions

The long-wavelength approximation makes it possible toelaborate a relative simple analytical method fordetermining the parameters of an ultrathin uniaxiallyanisotropic transparent film on an isotropic absorbingsubstrate only from phase information-based ellipsomet-ric data. It must be emphasized that a key capability ofsuch an analytical approximation is to clear up thequestion of when we can decouple the usual correla-tions between the optical constants and the thickness ofan ultrathin anisotropic film. The latter feature iscurrently critical in optics of nanoscale films because the

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standard regression methods for determining the parame-ters of ultrathin films on the basis of ellipsometricmeasurements are characterized by a strong correlationbetween film thickness and dielectric response. The elab-orated simple mathematical model enables us also toestimate the accuracy of ellipsometric measurements,which is needed for operation in the field of nanoscalefilms.

AcknowledgmentsThis research was supported by Estonian Research Council(Project IUT2-24) and Estonian Centre of Excellence inResearch (Project TK117).

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