Pseudodielectric functions of uniaxial materials in certain symmetry directions

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468 J. Opt. Soc. Am. A/Vol. 23, No. 2 /February 2006 G. E. Jellison Jr. and J. S. Baba

Pseudodielectric functions of uniaxial materials incertain symmetry directions

G. E. Jellison, Jr.

Condensed Matter Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6030

J. S. Baba

Engineering Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6030

Received April 28, 2005; revised June 28, 2005; accepted July 8, 2005

The pseudodielectric function is often used to represent ellipsometric data and corresponds to the actual di-electric functions of materials when there is no surface overlayer and the material is isotropic. If a uniaxialmaterial is oriented such that the optic axis is in the plane of incidence or is perpendicular to the plane ofincidence, then the cross-polarization terms are zero and appropriate pseudodielectric functions can be deter-mined from the ellipsometry data. We calculate the pseudodielectric functions for uniaxial crystals in threeprimary symmetry directions: (1) the optic axis is perpendicular to the plane of incidence, (2) the optic axis isin the plane of the sample surface and parallel to the plane of incidence, and (3) the optic axis is in the planeof the sample surface and perpendicular to the plane of incidence. These results are expanded in terms of thedifference in the ordinary and extraordinary dielectric functions and compared with the approximation ofAspnes [J. Opt. Soc. Am. 70, 1275 (1980)]. Comparisons are made with experimental results on oriented crys-tals of rutile �TiO2�, and a simple procedure is presented to determine the complex dielectric function fromstandard ellipsometry techniques.

OCIS codes: 120.2130, 160.1190, 260.2130.

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. INTRODUCTIONhe optical functions of many uniaxial and biaxial mate-ials are quite complicated in that the complex dielectricunctions of the material can depend significantly on theolarization of the probing light beam. Below the band-ap, these differences usually manifest themselves as dif-erences in the spectroscopic refractive indices, each ofhich tends to behave in a traditional, dispersive way asescribed by the Sellmeier approximation.Above the band edge, the complex dielectric function

an be considerably more complicated. Critical points inhe optical spectra can be quite different for light polar-zed along the optic axis compared with those for light po-arized perpendicular to the optic axis. Since the 1960’s,hese spectra and critical points have been obtained usingormal-incidence reflection measurements. During theast five to ten years, however, it is more common to usepectroscopic ellipsometry. For isotropic samples, stan-ard spectroscopic ellipsometry measurements can besed to determine unambiguously the optical functions ofaterials, once the effects of surface roughness and/or

urface interface layers have been taken into account.For anisotropic samples, generalized ellipsometry tech-

iques must be used, since considerably more informations needed to determine the optical functions of the mate-ial. For an arbitrary direction of the optic axis with re-pect to the sample surface, the s- and p-polarizationtates are no longer eigenstates of reflection, and theample itself can mix s- and p-polarized light (cross polar-zation). If this cross polarization is measured, then the

easured values can be used to determine the opticalunctions of uniaxial crystals.

If a uniaxial crystal is oriented with the optic axis ei-her in the plane of incidence or perpendicular to thelane of incidence, then the cross-polarization reflectionoefficients are 0. In this case, standard ellipsometricechniques can be used to measure the pseudodielectricunction (PDF) of the material in that configuration (thats, the optical functions of the material, where it is as-umed that the surface overlayer vanishes).

According to the work of Aspnes,1 the PDF of a materials a close approximation to the projection of the dielectricunction tensor along the direction defined by the inter-ection of the plane of incidence and the surface. In Ref. 1,our approximations are assumed: (1) one of the principalxes is perpendicular to the surface, (2) the dielectricunction itself is large ��10�, (3) the differences betweenhe principal values of the dielectric function are smallnough to be treated to first order, and (4) the angle of in-idence is large.

Physically, it is relatively easy to see why the dielectricunction tensor component along the intersection of thelane of incidence and the sample surface plane is theost important: For large dielectric functions, Snell’s law

ays that the penetrating light beam will be nearly nor-al into the material and the associated electric field will

e primarily in the plane of the sample surface, diminish-ng the influence of the component perpendicular to theample surface. Furthermore, for large angles of inci-ence, the reflection coefficient for light polarized perpen-icular to the plane of incidence rs is normally mucharger than the reflection coefficient for light polarizedarallel to the plane of incidence rp, making the ellipso-etric ratio �=r /r much more sensitive to the compo-
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G. E. Jellison Jr. and J. S. Baba Vol. 23, No. 2 /February 2006 /J. Opt. Soc. Am. A 469

ent of the dielectric tensor in the plane of incidence. TheDF, being derived from �, will then depend primarily on

his same component of the dielectric tensor.In this paper, the PDFs of uniaxial crystals oriented in

hree principal symmetry directions will be considered:1) the optic axis is perpendicular to the sample surfaceOA�surf�, (2) the optic axis is parallel to the sample sur-ace and in the plane of incidence �OA�surf, �POI�, and (3)he optic axis is parallel to the sample surface and per-endicular to the plane of incidence �OA�surf, �POI�. Ex-ressions will be presented to second order in the differ-nces between the ordinary and the extraordinaryielectric functions, where it can be seen directly how theDFs deviate from the actual dielectric functions as theielectric functions get small or as the dielectric functionifference becomes large. To quantify these results, weill present generalized ellipsometry data from single-

rystal rutile �TiO2�. For determining the actual opticalunctions of rutile, generalized ellipsometry measure-ents are made with the optic axis in the plane of the

ample, oriented at ±45° with respect to the plane of in-idence. Measurements are also made in the three princi-al symmetry directions (OA�surf; OA�surf, �POI; andA�surf, �POI), which are then compared with the theo-

etically predicted values and with the measured complexielectric functions. All of these measurements are alsoade at several angles of incidence to explore this depen-

ence.

. THEORYssume that the sample can be approximated as a simpleir–material structure and that the material is uniaxialith complex dielectric functions �o for light polarizederpendicular to the optic axis and �e for light polarizedarallel to the optic axis. The incident light beam makesn angle � with respect to the surface normal, and we willse the shorthand notation

� = sin��, �1a�

� = cos��. �1b�

For standard ellipsometric angles (�=60° to 75°), �0.866–0.966; �=0.5–0.259, and the squares ranges �2

0.75–0.933 and �2=0.25–0.067.] We will also assign

�o = �� − �2�1/2, �1c�

�� = �e − �o. �1d�

e will explore three high-symmetry cases (OA�surf;A�surf, �POI; and OA�surf, �POI) discussed Section 1.he calculations of the complex reflection coefficientsave been in the literature for decades (see, for example,efs. 2–5), but we will start with the complex reflectionoefficients given by Filippov et al.2

Once the complex reflection coefficients rp (for light po-arized in the plane of incidence) and rs (for light polar-zed perpendicular to the plane of incidence) are calcu-ated, the complex reflection ratio �=rp /rs is calculated,rom which we can determine the PDF:

�� = �2�1 +�2

�2

�1 − ��2

�1 + ��2� . �2�

his expression is somewhat deceiving, since the PDF ishown to be explicitly a function of the angle of incidence.owever, we know that in the limit of isotropy (that is,hen ��=0) ��=� and the dependence on the angle of in-

idence is eliminated. In the discussion that follows, theole of the angle of incidence will be explicitly stated.

. Optic Axis Is Perpendicular to the Plane of IncidenceOA�surf…or this case, let us define

�e = ��o��e − �2�/�e1/2. �3�

he complex reflection coefficients rp and rs are given by

rp =�e� − �e

�o� + �e, �4a�

rs =� − �o

� + �o, �4b�

nd the cross-polarization reflection coefficients are zero.he PDF [Eq. (2)] becomes

�� = �2�1 + �2 �o�o − �e

�o�2 − �o�e�2� . �5�

he PDF shown in Eq. (5) is explicitly expressed as aunction of �2 and therefore the angle of incidence. If theirefringence is small, we can expand Eq. (5) in terms of�:

�� o −��

�o − 1+

��2

4�o��o − 1�2

4�o2 − �o − 3�o�2 + �2

�o − �2 , �6�

here we have truncated the expansion to second order in�. Similarly, if we assume that the dielectric functions �ond �e are large, we can use a Taylor expansion of �o ande (to second order in �2 /�o and �2 /�e, ignoring the quan-ities �6 /16�o

3 and �6 /16�c3 and higher orders) to obtain

�� o + �2�2

�1 + ��2 − �2�2 , �7�

here

� =��

2�e��o − 1�. �8�

ccording to the Aspnes approximation for OA�surf, ��o for small �� and large �o and �e . This is certainly

onsistent with expressions (6) and (7). However, expres-ions (6) and (7) are considerably less restrictive, requir-ng either small �� / �� −1��2 [expression (6)] or small
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3� [expression (7)]. Interestingly, bothxpressions (6) and (7) show that the angle of incidenceoes not enter into the first-order expressions but onlynto the second-order terms and then as a minor contri-ution. Therefore, variable angle-of-incidence measure-ents are unlikely to yield additional information for the

ase of uniaxial materials with the optic axis perpendicu-ar to the plane of incidence, unless � is large.

. Optic Axis in the Plane of the Sample Surface and inhe Plane of Incidence „OA¸surf, ¸POI…n this case, the quantity �e is defined as

�e = ��e��o − �2�/�o1/2 = �ave�o/�o, �9�

here �ave= ��o�e�1/2. The complex reflection coefficientsre given by

rp =�ave� − �o

�ave� + �o, �10a�

rs =� − �o

� + �o. �10b�

he PDF can be calculated using Eq. (2) to obtain

�� = �2�1 + �2�o2 �ave − 1

�ave�2 − �o

2�2� . �11�

s before, we can expand Eq. (11) to second order in �� tobtain

�� o + ��o − �2�

�2��o − 1�+ ��2

�3�o�2 + �2��o − �2�

4�o�4��o − 1�2 ;

�12a�

o convergent expansion was found under the assumptionf large �o and large �c.

If it can be assumed that the angle of incidence is largesuch that �2�1) and that �o is large (the Aspnes approxi-ation), then the first-order term becomes �� /�2. Apply-

ng this to the first-order term of expression (12a), we findhat the PDF becomes

�� o + ��e − �o�/�2 = ��e − �o�2�/�2. �12b�

learly, the extraordinary dielectric function is the domi-ant term in the PDF at normal ellipsometric angles, buthe dependence on the angle of incidence remains, even torst order. As a result, variable angle-of-incidence ellip-ometry does give valuable additional information for thisrientation of the crystal.

. Optic Axis in the Plane of the Sample Surface anderpendicular to the Plane of IncidenceOA¸surf, �POI…n this case, the quantity �e is defined as

�e = ��e − �21/2, �13�

nd the complex reflection coefficients are given by

rp =�o� − �o

�o� + �o, �14a�

rs =� − �e

� + �e. �14b�

s before, the PDF can be calculated using Eq. (2) to ob-ain

�� = �2�1 + �2 �o�e − �o

�o�2 − �e�o�2� . �15�

quation (15) can be expanded to second order in �� tobtain

�� o − ��o − �2 − �o�2�

�2��o − 1�+ ��2

�3�o − 4�2��o − �2 − �o�2�

4�4��o − 1�2��o − �2�.

�16a�

s with the case described in Subsection 2.B, no conver-ent expansion was found under the assumption of largeo and large �e. Looking now only at the first-order term,ssuming large angles of incidence and large values of �othe Aspnes approximation), we can set the �o−�2 term too−1, yielding

�� o + ��1

�2��o − 1�. �16b�

his is similar to the case where the optic axis is in theample surface plane and in the plane of incidenceOA�surf, �POI, Subsection 2.B), but now the �� term isivided by ��o−1�, making this term much less dependentn the angle of incidence. Once again, the Aspnes approxi-ation is valid in that ��o for large angles of incidence

nd large values of �o, but the PDF can become decidedlyifferent from �o if either of these conditions are violated.he first-order term does depend on the angle of inci-ence, but the 1/ ��o−1� term decreases this dependence.ariable angles of incidence may or may not give addi-ional information.

. EXPERIMENTAL VERIFICATION ANDISCUSSION: RUTILE. General Methodhe material rutile �TiO2� was chosen for experimentalerification, since it has a bandgap well within the rangef most spectroscopic ellipsometers ��3.6 eV�, it is highlynisotropic ( �n �0.3 below the band edge), properly ori-nted and polished crystals are readily available, and sev-ral measurements of the optical functions have beenade that agree once the data are corrected for surface

oughness effects.6–9 Generalized spectroscopic ellipsom-try measurements were made using the two-modulatoreneralized ellipsometer (2-MGE),10,11 which has beentilized in previous measurements of the optical functionsf rutile.6,7

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Epitaxial-ready samples of the optic axis perpendicularo the surface and the optic axis in the plane of theample were obtained from Commercial Crystal Labora-ories, Inc. (Naples, Florida). The 2-MGE measurementsere made from 220 to 860 nm (5.64 to 1.44 eV) at severalngles of incidence. For the sample with the optic axis inhe plane of the sample surface, measurements wereade with the optic axis in the plane of incidence, perpen-

icular to the plane of incidence, and at ±45° with respecto the plane of incidence. Additional 2-MGE measure-ents were made of a different sample with the optic axis

erpendicular to the plane of incidence at several anglesf incidence.

The analysis of the experimental results were per-ormed in several ways. The raw data are presented inhe � representation, where

� =rpp

rss= tan��ei� =

C + iS

1 + N, �17a�

�ps =rps

rss= tan�ps�ei�ps =

Cps + iSps

1 + N, �17b�

�sp =rsp

rss= tan�sp�ei�sp =

Csp + iSsp

1 + N, �17c�

here the sp and ps terms are the cross-polarizationerms and are 0 for situations where the optic axis is inhe plane of incidence or perpendicular to the plane of in-idence. Alternatively, the raw data can also be expressedn terms of the PDF [see Eq. (2)] for cases where the cross-olarization terms are zero.All spectra taken showed a small nonzero imaginary

art below the direct band edge, which is due to a thinurface overlayer between the ambient (air) and the ma-erial underneath. One common way of removing the ef-ects of this overlayer is to treat the near-surface region ofhe sample as air/surface roughness/material.12,13 The op-ical functions of the surface roughness are modeled usinghe Bruggeman effective-medium approximation,14 con-isting of 50% voids and 50% material. For finding thehickness of the surface roughness layer, the spectra be-ow the band edge �3.0 eV� are fit, where the opticalunctions of the material are modeled using a Sellmeierispersion relation. In this case, the resultant surfaceoughness thickness of the sample with the optic axis inhe plane of the sample varied from 1.6 to 1.9 nm for allhe measurements made, and the surface roughnesshickness for the sample with the optic axis perpendicularo the plane of the sample surface varied from 2.0 to 2.2m. Once this thickness is determined, the raw data cane converted into complex dielectric functions using stan-ard calculation procedures.6,12,13 For cases where the op-ic axis is ±45° with respect to the plane of incidence, theesultant values are just the ordinary and extraordinaryielectric functions of the material. For cases where theptic axis is in the plane of incidence or perpendicular tohe plane of incidence, the resultant values are interme-iate between the uncorrected PDFs and the real dielec-ric functions, in that we removed the effects of the sur-ace overlayer and the angle of incidence to first order.

. Optic Axis Is in the Plane of the Sample, ±45° to thelane of Incidencehis is just the case that has been used before (Refs. 6nd 7) to determine the optical functions of rutile. Figure

ig. 1. Raw generalized ellipsometry data, expressed in the �epresentation, taken on rutile with the optic axis in the plane ofhe sample surface, oriented at +45° with respect to the plane ofncidence.

ig. 2. Optical functions of rutile determined from generalizedllipsometry measurements with the optic axis of the crystal ori-nted in the plane of the sample surface at +45° with respect tohe plane of incidence. The data have been corrected for a surfaceverlayer. The difference between � and � is also plotted.
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shows one of the experimental spectra for the sampletudied here, where the angle of incidence �=65° , �45°. If the sample were positioned such that �=−45°, theremains the same, but �ps��= +45° �=−�ps��=−45° �, etc.

see Ref. 6). These measurements were performed athree angles of incidence (nominally 60°, 65°, and 70°) athe two orientations of � �±45° �, and the results were av-raged to determine the complex dielectric functions,hown in Fig. 2. There was a small difference in the opti-al functions determined at the same angle of incidence athe two orientations of �; this was due to the imprecisionf determining the position of the optic axis with respecto the plane of incidence and was corrected by averagingver the two orientations of �. There was no discernableifference between the averaged dielectric functions athe three angles of incidence.

It will be noted that there is a discrepancy between theielectric functions shown here and those reported in Ref.above the band edge. This is due primarily to the sur-

ace roughness corrections required in Ref. 6, where theurface overlayer thickness was 5.5 nm (compared with.6–2.0 nm for this work). The reduced surface overlayerhickness required for this data makes them more accu-ate and more in line with the results of Ref. 9.

. Optic Axis Is Perpendicular to the Plane of IncidenceOA�surf…igure 3 shows the raw spectroscopic ellipsometry data inhe � representation taken on a rutile crystal with the op-ic axis perpendicular to the sample surface. As can beeen, there is a significant dependence of the raw data onhe angle of incidence. However, when the raw data areonverted into the PDF using Eq. (2), there is no depen-ence on the angle of incidence, within the limits of error.

ig. 3. Raw ellipsometry data, expressed in terms of real andmaginary parts of the complex reflection ratio � from a rutilerystal with the optic axis perpendicular to the surface plane,aken at several different angles of incidence.

his confirms the results of the calculations shown inubsection 2.A.: Ellipsometric measurements made onamples with the optic axis perpendicular to the sampleurface at variable angles of incidence are very insensi-ive to angles of incidence, when expressed in terms of theDF.Figure 4 shows several plots of the complex dielectric

unction for rutile with the optic axis perpendicular to theample surface. The dashed curves show the raw data (at4.92° angle of incidence) expressed in terms of the PDF.he narrow curves show the same data corrected for aurface overlayer, as described in Subsection 3.A. The or-inary dielectric function data (shown by the heavy solidurves in Fig. 4) are taken from Fig. 2, and the dottedurves show the PDF data determined from Eq. (5) usinghe optical functions shown in Fig. 2.

Clearly, the simple overlayer correction to the PDFata is nearly sufficient to determine the actual ordinaryielectric function of rutile, which is consistent with theork of Aspnes. Moreover, the corrected data are very

lose to the calculated spectra from Eq. (5). In order toompare the Taylor expansion approximations shown inxpressions (6) and (7) with the exact expression shown inq. (5), a figure of merit (FOM) was defined:

FOM =1

N�� PDF − PDFapprox��PDF − PDFapprox�*,

�18�

here N is the total number of points, PDF is the PDFaken from Eq. (5), PDFapprox is taken from expression

ig. 4. Several plots of the complex dielectric function of a rutilerystal with the optic axis perpendicular to the sample surfaceOA�surf�. Dashed curves show the raw ellipsometric data, ex-ressed in terms of the PDF, taken at �=64.92°. Narrow solidurves show the raw data, corrected as described in Subsection.A. This to be compared with the ordinary dielectric functionheavy solid curves), also shown in Fig. 2. The calculated PDFusing Eq. (5)] is shown by the dotted curves.

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6) or (7), and the * symbolizes the complex-conjugate op-ration. With Eq. (18) and expression (6) as the approxi-ation, the resulting FOM=3.3�10−3, whereas a FOM3.3�10−6 is obtained if expression (7) is used as the ap-roximation. Therefore, expression (6) is a good approxi-ation, but expression (7) is a much better approximation

or the case of rutile in this energy range.

. Optic Axis in the Plane of the Sample Surface and inhe Plane of Incidence „OA¸surf, ¸POI…igure 5 shows the PDF spectra obtained for rutile for theptic axis in the plane of the sample surface and in thelane of incidence at two different angles of incidence. Asan be seen, there is a significant difference between theDFs at different angles of incidence for the optic axis inhe plane of incidence. The largest difference occurs for�2� near 4.2 eV, but significant differences occur through-ut the spectrum. This is consistent with the conclusioneached in Subsection 2.B. that the dielectric function willepend on the angle of incidence in the first order.Figure 6 shows the PDF obtained when the simple cor-

ection procedure described in Subsection 3.A, which re-oves the dependence on the surface overlayer and the

ngle of incidence to first order, is applied. This is com-ared with the PDF calculated from Eq. (11) and with thectual extraordinary dielectric function determined fromhe �= ±45° spectra and shown in Fig. 2. Clearly, the cor-ected PDF agrees well with the calculation using Eq.11), but neither corresponds to the actual dielectric func-ion, particularly near 4.2 eV. As can be seen from the ��pectra shown in Fig. 2, this is precisely the spectral re-ion where the �� is the largest and where the angle-of-ncidence dependence matters the most.

ig. 5. PDF raw data (uncorrected for surface overlayer) for theptic axis in the plane of the sample, parallel and perpendicularo the plane of incidence �OA�surf, �POI�, taken for differentngles of incidence. The data are shown for two different anglesf incidence.

The calculated PDF [Eq. (11)] can also be comparedith the approximation given in expression (12a) using

he FOM given in Eq. (18), where it is found that FOM5�10−4; clearly, third-order terms are not significant for

he PDFs of rutile in this energy range.

. Optic Axis in the Plane of the Sample Surface anderpendicular to the Plane of IncidenceOA¸surf, �POI…igure 5 also shows the PDF spectra obtained for rutileith the optic axis in the plane of the sample surface anderpendicular to the plane of incidence at two differentngles of incidence. As can be seen, there is some differ-nce between the PDFs at different angles of incidence,ut the difference is much smaller than the case wherehe optic axis is in the plane of incidence. This is consis-ent with the conclusion reached in Subsection 2.C thathe dielectric function will not depend significantly on thengle of incidence for this configuration.Figure 7 shows the PDF obtained when the simple cor-

ection procedure described in Subsection 3.A is applied.his is compared with the PDF calculated from Eq. (15)nd with the actual ordinary dielectric function deter-ined from the �= ±45° spectra and shown in Fig. 2.learly, the corrected PDF agrees well with the calcula-

ion using Eq. (15), but both PDFs differ from the ordi-ary dielectric function, above �3.5 eV, although the dis-greement is not as severe as with the case where theptic axis is perpendicular to the plane of incidence (seeubsection 3.D).

ig. 6. Several plots of the complex dielectric function of a rutilerystal with the optic axis parallel to the sample surface and par-llel to the plane of incidence. The PDF data, corrected for theverlayer and angle of incidence as described in Subsection 3.A,re shown by the dotted curves. The PDF calculated using Eq.11) is shown by the thin solid curves. This is compared with thectual extraordinary dielectric function in the thick solid curves.

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The calculated PDF [Eq. (15)] can also be comparedith the approximation given in expression (16a) using

he FOM given in Eq. (18), where it is found that FOM0.011. Of the four approximations considered in this pa-er, this case compares the least favorably with the actualDF and indicates that third-order terms may be impor-ant.

. Alternate Method to Calculate Dielectric Functionsubsections 3.D and 3.E show that the PDF determined

or the cases where OA�surf, �POI and OA�surf, �POIre approximations of the extraordinary and ordinary di-lectric functions (respectively) of the material when theurface layer is mathematically removed. This suggests aimple way of getting reasonably accurate values of thectual dielectric functions using standard ellipsometryechniques (that is, where cross polarization is not mea-ured). First of all, the PDFs are determined using mea-urements at two orientations of the optic axis:A�surf, �POI and OA�surf, �POI. The surface over-

ayer is mathematically removed. These experimentalalues are then used as the left-hand side of Eqs. (11) and15), which are then solved to determine �o and �e usingteration.

Figure 8 shows the result of this procedure for the dataiscussed in this paper and indicates that the two inde-endent determinations of the actual dielectric function ofutile are quite similar. Although the spectra are nearlydentical, there are small differences between the complexielectric functions determined using the two methods

ig. 7. Several plots of the complex dielectric function of a rutilerystal with the optic axis parallel to the sample surface and per-endicular to the plane of incidence �OA�surf, �POI�. The PDFata, corrected for the overlayer and angle of incidence as de-cribed in the text, are shown by the dotted curves. The PDF cal-ulated using Eq. (15) is shown by the thin solid curves. This isompared with the actual ordinary dielectric function in the thickolid curves.

hat are outside the error limits. The generalized ellip-ometry method usually results in smaller errors and isherefore the preferred method.

. CONCLUSIONShe pseudodielectric function is calculated for uniaxialrystals in several orientations of the optic axis where theross-polarization terms are zero: (1) the optic axis per-endicular to the sample surface plane �OA�surf�, (2) theptic axis in the sample surface plane, parallel to thelane of incidence �OA�surf, �POI�, and (3) the optic axisn the sample surface plane, perpendicular to the plane ofncidence �OA�surf, �POI�. In their exact form, these ex-ressions depend directly on the angle of incidence. How-ver, all expressions can also be expressed in a convergentaylor expansion in terms of the difference of the dielec-ric function ��=�e−�o, where it is assumed that �� ismall.

These expressions are then compared with the Aspnespproximation1 where it is assumed that the angle of in-idence � is large enough that sin2��1 and that theagnitude of the dielectric functions is large. In this

imit, the PDFs of cases 1 and 3 approximate �o, whereasase 2 approximates �e, and these calculations confirmhat approximation. However, significant deviations canccur if any of the Aspnes approximation limits is not sat-sfied.

ig. 8. Complex dielectric functions of rutile determined usingwo different techniques. The dark solid curves represent theata from Fig. 2 and are determined by utilizing the generalizedllipsometry data for the optic axis in the plane of the sample,riented at �= ±45° with respect to the plane of incidence. Theotted curves are the complex dielectric functions determined us-ng the normal ellipsometry data for the optic axis in the plane ofhe sample, oriented parallel and perpendicular to the plane ofncidence, utilizing the procedure outlined in Subsection 3.F.

sotfaipm1pp(accn

dtftwHitopt

ARuNso0ntd

j

R

1

1

1

1

1


For examining these calculations in detail, spectro-copic generalized ellipsometric measurements are madef single-crystal rutile. If the optic axis is perpendicular tohe sample surface (case 1 above), then the PDF, correctedor any surface overlayer and the angle of incidence, givesreasonably good approximation to �o. If the optic axis is

n the sample surface plane and perpendicular to thelane of incidence (case 3 above), the PDF also approxi-ates �o, although the deviations are more than for case

. If the optic axis is in the sample surface plane and isarallel to the plane of incidence, then the Aspnes ap-roximation states that the PDF should approximate �ecase 2). Athough this is the case in the limit of the Aspnespproximation, it results in some significant errors for thease of rutile. In particular, it is found that the �� coeffi-ient depends directly on 1/sin2� �, giving this PDF a sig-ificant dependence on the angle of incidence.Therefore, the PDF of uniaxial crystals can be used to

etermine an approximation of the actual dielectric func-ions for light polarized parallel to the optic axis ��e� andor light polarized perpendicular to the optic axis ��o�, buthis approximation can be subject to significant errorshen the Aspnes approximation criteria are violated.owever, a simple iterative procedure can be devised us-

ng the PDF data, corrected for the surface overlayer, forhe two orientations of the crystal (optic axis in the planef the sample surface, parallel and perpendicular to thelane of incidence) that results in very accurate values ofhe actual dielectric functions of rutile.

CKNOWLEDGMENTesearch sponsored by the U.S. Department of Energynder contract DE-AC05-00OR22725 with the Oak Ridgeational Laboratory, managed by UT-Battelle, LLC. The

ubmitted manuscript has been authored by a contractorf the U.S. Government under contract No. DE-AC05-0OR22725. Accordingly, the U.S. Government retains aonexclusive, royalty-free license to publish or reproducehe published form of this contribution, or allow others to
o so, for U.S. Government purposes.
Corresponding author G. E. Jellison’s e-mail address [email protected].

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6. G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner,“Measurement of the optical functions of uniaxial materialsby two-modulator generalized ellipsometry: rutile �TiO2�,”Opt. Lett. 22, 1808–1810 (1997).

7. G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B. S. Jeong,and D. P. Norton, “Spectroscopic ellipsometry of thin filmand bulk anatase �TiO2�,” J. Appl. Phys. 93, 9537–9541(2003).

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9. T. E. Tiwald and M. Schubert, “Measurement of rutile TiO2dielectric tensor from 0.148 to 33 �m using generalizedellipsometry,” in Proc. SPIE 4103, 19–29 (2000).

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Pseudodielectric functions of uniaxial materials in certain symmetry directions