Optical constants such as refractive index, n, andextinction coefficient, k, are commonly calculatedfrom optical transmission and reflection spectra ofthin films. An accurate determination of the value ofthe wavelength-dependent optical constants is veryuseful because it gives fundamental informationabout the optical bandgap, defect levels, oscillatorenergy, oscillator strength, etc. Practically speaking,optical constants are essential for the design andmodeling of optical devices and optical coatings.
Most of the published literature does not considerthe wedge and nonuniformity in thickness of the filmsthat occur in many practical instances. The presenceof thickness variations can drastically change thetransmission spectra, and thus the formulas used for
uniform smooth films are no longer valid because theinterference structure of the spectra becomes quitedistorted. In general as thickness variations increase,the visibility of the interference fringes is decreased;i.e., the transmission maxima decrease, the transmis-sion minima increase, and the envelope of the spec-trum shrinks. The determination of optical constantsby using conventional methods results in an overes-timate of the thickness and an erroneous refractive-index dispersion, especially in the short-wavelengthregion of the spectrum.1
Various methods have been employed for deter-mining the properties of thin films, particularly thosethat are smooth but have thickness variations. Thatis, the film’s thickness variation is not sufficientlylarge or of short enough period to cause significantoptical scattering or to disrupt optical interferencewithin the film. These methods can be classified intotwo general categories:
1. Virtual calculation of transmission (or reflec-tion) spectra based on spectral envelopes2
2. Fitting (optimization) methods using parame-terized dispersion relations3–5
Spectral envelope methods determine film parame-ters from functions describing the locus of interfer-
J. Baek ([email protected]) and M. F. Becker are with theDepartment of Electrical and Computer Engineering, D. Kovar iswith the Department of Mechanical Engineering, and J. W. Keto iswith the Department of Physics, Texas Materials Institute, TheUniversity of Texas at Austin, Austin, Texas 78712.
Received 5 May 2005; revised 26 August 2005; accepted 29 Au-gust 2005.
ence maxima and minima in the transmittance orreflectance spectrum. On the other hand, optimiza-tion methods use the full spectral-interference pat-tern to fit an exact, but parameterized, model to thedata.
Marquez et al.6,7 developed and applied a physi-cally appealing procedure that was first proposedby Swanepoel1 to calculate the average thickness,thickness variation, refractive index, and extinctioncoefficient of wedge-shaped films. (From now on werefer to this method as the wedge-shaped envelopemethod.) Unfortunately, this method suffers from acritical disadvantage: It assumes that the film istransparent in the long-wavelength region, which isnot true in many cases. One would expect that theerror resulting from this assumption would increaseas the extinction coefficient of the film increases. Thisassumption is verified in detail in this paper.
Petrich and Stenzel5 developed a fitting techniquethat is applicable to the much more complex problemof fitting thin-film parameters when scattering anddisruption of optical resonance are significant owingto short-range surface roughness. Although this tech-nique could perhaps be applied to smooth, wedged,nonuniform films, as discussed in this paper, it wouldnot be the optimum technique owing to its complexityand the increased likelihood of finding local minimainstead of the global minimum owing to the largernumber of parameters to be determined.
In this paper a novel, physically attractive, andsimple numerical method that considers nonunifor-mity in the thickness of the film is presented. Themethod is based on Swanepoel’s calculation of wedgedfilm transmission through a finite aperture, but it uti-lizes a fitting method so that it can be applied accu-rately to absorptive and wedged films. Our optimumparameter extraction (OPE) method allows the calcu-lation of film thickness, thickness variation, refractiveindex, and extinction coefficient from only a transmis-sion spectrum. We also obtain optical constants byusing the wedge-shaped envelope method and the con-ventional envelope method2 for comparison and dis-cuss the differences between the two. We compareresults for a suite of test cases and for actual AlN thinfilms fabricated in our laboratory. The AlN thin filmswere deposited on sapphire (0001) substrates, usingPLD (pulsed laser deposition) at different tempera-tures and laser fluences. Our aim is to determine theeffects of substrate temperature, background gas pres-sure, and laser fluence on the properties of the films.
2. Theoretical Background of the Optimum ParameterExtraction Method
We start by considering a homogeneous thin film ofuniform thickness, d, refractive index, n, and extinc-tion coefficient, k (complex refractive index, n � n� ik). The film is on a transparent substrate withrefractive index, s. The substrate is considered to beperfectly smooth but thick enough so that the sur-faces are not perfectly parallel and that interferenceeffects due to the substrate are negligible. The systemis surrounded by air with a refractive index n0 � 1.
Taking all the multiple reflections at the threeinterfaces into account, we follow the derivation ofSwanepoel,2 of the rigorous expression for the opticaltransmission, T, for normal incidence that isgiven by
In the last line of Eqs. (2), � is referred to as the filmtransmittance since it is the internal transmittanceof the film.8 It should be noted that these formulasare valid only for smooth films with uniform thick-ness.
The case now considered is shown schematically in
Fig. 1. (a) Absorbing thin film with a variation in thickness on athick, finite transparent substrate; (b) triangular surface irregu-larities; (c) rectangular surface irregularities; (d) sinusoidal sur-face irregularities.
Fig. 1(a). It is assumed that the film is wedged so thatthe thickness varies linearly over the long dimension(x direction) of the illuminated area so that the thick-ness can be expressed in the following form:
d � d� � d, (3)
where
� 2x�L ��1 � � 1�. (4)
Here d is the local deviation in thickness from theaverage thickness d� and x is the position across theaperture as shown in Fig. 1(a). The OPE method wasderived to be applied to films with a one-dimensional(1-D) linear variation in d as shown in Fig. 1(a). Weshow later that the OPE method can be extended tocases in which there is a transverse linear thick-ness variation or in which several two-dimensional(2-D) irregularities occur over the illuminated area.Figures 1(b)–1(d) show examples of triangular, rect-angular, and sinusoidal surface irregularities, re-spectively. The meaning of r is indicated in thefigures. More random types of surface irregularitycould be approximated by any of the three formsshown in Figs. 1(b)–1(d). Recently, Gonzalez-Lealet al.9 applied their wedge-shaped envelope method tolow-loss films with 2-D surface irregularity parame-terized by a single d and no wedge.
The transmission Td at a specific wavelength forthe case of 1-D nonuniform thickness can be obtainedby integrating Eq. (1) over d, n, and k. However, thisis prohibitively difficult analytically, and a simplify-ing assumption is to use an average value of n and kover the range of integration over thickness. Thisapproximation is excellent provided d �� d� . Thusthe expression for the transmission Td is
Td �1
�2 � �1�
�1
�2 A�
B � C� � D�2 d�, (5)
where �1 � �d� � d��� and �2 � �d� � d���.The integral can easily be converted mathemati-
cally to the following simpler form that parameter-izes the nonuniformity by using the nondimensionalparameter �:
Td �12 �
�1
1A�
B � C� � D�2 d, (6)
where d � d� � d and �1 � � 1 over the lengthof the measurement beam L.
We now extend this concept to the case of simulta-neous 2-D wedge and thickness variation to arrive ata case that will be close to the worst case for real-world samples. In this case the thickness can be ex-pressed in the following parameterized form:
d � d� � dx � �dy � rx � �ry, (7)
where �1 � , �, , � � 1; and dx and dy are thevariations in film thickness due to the wedge in thex and y directions, respectively. Similarly, rx, ry arethe other thickness variations. The integral expres-sion for the optical transmission of this wedged andnonuniform film becomes
Td �12
12
12
12
� ��1
1 ��1
1 ��1
1 ��1
1A�
B � C� � D�2 dd�d d�.
(8)
Equation (8) is valuable when the illumination is a2-D rectangular area rather than 1-D line, and itincludes the effects of both a 2-D wedge and othervariations simultaneously. In this paper the illumi-nation area is confined to 1-D by putting a long, nar-row aperture in front of the light source so that wehave to consider only 1-D nonuniformity. In Subsec-tion 3.B we evaluate the error introduced by thisassumption. Usually thin films have a sufficientlysmall thickness nonuniformity so that the wedgedominates. Our numerical fitting of the films’ opticalconstants is conducted under the assumption of sucha 1-D wedged smooth film.
The reliability of fitting methods for determiningfilm properties depends mainly on the dispersionmodel that is employed; often this is a more or lessempirical choice. Fitting methods and dispersionequations are used in several commercial opticalthin-film design and analysis software packages suchas TFCalc10 and Film Wizard.11 The dispersion equa-tions that are generally used are the Cauchyequation,12 the Sellmeier equation,13 the Foroughi–Bloomer equation,14 and the Drude equation.15 Any ofthese dispersion equations can give very good resultsfor a large number of materials over quite a widerange of wavelengths. A particular dispersion modelis applicable if there is a good fit between the exper-imental transmission (or reflection) spectrum and theone calculated by using that equation. Certainly, allof these dispersion equations are slowly varying func-tions of wavelength, and they are very unlikely toproduce a good fit over a wide wavelength range if thevalues of n and k are incorrect. This is due to the factthat n, k, and the film thickness directly determinethe form of the transmission (or reflection) spectrum.From the point of view of the fitting routine, a simplerdispersion expression with fewer adjustable param-eters is more efficient at and effective for fitting mea-sured data. To accelerate the fitting routine and toavoid finding local minima instead of the global min-imum, we propose a packing-density model for refrac-tive index. In this model the refractive index has onefitted parameter, the packing density, p. That is,
where nref��� is the reference refractive index thatwas previously measured or tabulated for a similarthin-film or bulk material. In other words, n��� isfree to increase or decrease but not to change itsfrequency-dependent shape. Provided that we knowthe film material and that the change of the film’srefractive-index dispersion due to experimental fac-tors is not large, this packing-density model is veryaccurate. This technique is particularly effective forthin, strongly absorbing, or severely wedged films,since the OPE method can give a unique solution nomatter how small or few the interference fringes arein the transmission spectrum. Conventional modelstend to suffer from multiple solutions in these cases.
The second dispersion parameter is the extinctioncoefficient. We chose the exponential form
k��� � A exp�B���, (10)
where A and B are fitted constants. This exponentialmodel is the simplest among several common modelsand is very effective because a large number of thin-film materials show an exponential UV absorptionedge16–19 and are widely adopted in other thin-filmmodels. There are two additional unknown parame-ters needed for determining film transmission: aver-age thickness, d� , and nonuniformity, d. Implicit inthis definition is the length of the measured area, L,or alternatively a wedge angle, � � tan�1�2d�L�, forthe film surface [see Eqs. (3) and (4) and Fig. 1(a)].Using this set of parameters, one can determine thetransmission of a 1-D wedge-shaped thin film fromEq. (6).
Our OPE method minimizes the difference be-tween the measured spectrum and the simulated onewith optimized values of dispersion constants, thick-ness, and nonuniformity. This difference or cost func-tion is defined as
Cost �� 1m �
j�1
m
��Tex, j � Tfit,j�Wj�21�2
, (11)
where m is the number of optimization targets(spectral measurement positions); Tex, j and Tfit, j arethe experimental and fitted transmission values atthe jth wavelength, respectively; and W is a weight-ing factor for each wavelength. Cost is the quantitythat the optimization process attempts to minimize.To minimize the cost, we utilized the sequential qua-dratic programming (SQP) method20,21 implementedin MATLAB. In the SQP method a QP subproblem issolved for each iteration. Optimization variables mayall be varied simultaneously, but this may cause con-vergence to multiple local minima and make it diffi-cult to find the global minimum (best solution). Thisproblem is partially alleviated by reducing thenumber of optimization variables by employing thepacking-density model for the refractive index andthe two-variable exponential model for the extinctioncoefficient. To further reduce the likelihood of con-verging to a local minimum, the use of appropriate
initial values that are practical and reasonable isessential. Constraints can also be used to limit theparameters found by optimization to save time andto avoid unreasonable values. Although uniformweighting is normally used, we can use other appro-priate weighting factors in the cost. For example,when only the optical information at the long-wavelength region is required, larger weighting fac-tors can be used for the long-wavelength terms inthe cost. This gives better agreement between ex-perimental data and simulated data in the long-wavelength region. Care must be exercised, however,when weighting factors are used in this way becausedisagreement can increase in the short-wavelengthregion, depending on the dispersion model used.
In summary, the OPE method has the followingadvantages over other methods:
1. The ranges of the refractive index and extinc-tion coefficient that may be computed are larger thanany other envelope methods.
2. The packing-density model works well regard-less of the number of interference fringes in the trans-mission spectrum.
3. The method is robust with respect to the signal-to-noise ratio of the data.
The OPE method also has its limitations; one has toknow (or estimate) the dispersion of the refractiveindex for the film material before the fitting proce-dure is started when the packing-density model isused. Usually it is sufficient to know n��� for thecompound in either the bulk or the thin-film form. Wediscuss an alternative dispersion model in Subsec-tion 4.B if n��� is totally unknown. If the dispersionmodel is not appropriate, then the disagreement be-tween the measured (or simulated) transmissionspectrum and the fitted one is usually quite large.
3. Model Films
Model films are useful in testing film-parameter com-putation techniques, such as the OPE method, sincethe transmission (or reflection) spectra can be calcu-lated exactly and the exact starting film parametersare known so that they can be compared with theextracted parameters for error analysis. A muchwider range of test cases is also possible since thefilms do not have to be actually fabricated. Themethod can be tested for convergence to the correctsolution and for its absolute accuracy. Finally, theadditional uncertainty of making a physical measure-ment of the fabricated film parameters is eliminated.For this investigation, we have studied both modelfilms (this section) and actual deposited AlN films(Section 4) to test the OPE method.
In Subsection 3.A we use model films to show howa large error can result when the wedge-shaped en-velope method is applied to absorptive films with 1-Dwedge. We also apply our OPE method for compari-son. In Subsection 3.B we investigate how much erroroccurs when the OPE method is used for films with2-D nonuniformity but only 1-D nonuniformity in
film thickness is fitted. Owing to a finite illuminationwidth, transmission spectra exhibit the full effects of2-D nonuniformity. In principle, we can obtain moreaccurate information by considering all these effectsin the OPE fitting procedure. However, there is apractical difficulty in considering all of them simul-taneously because the cost function [Eq. (11)] willhave many more local minima around the global min-imum, thus making it impossible or extremely timeconsuming to converge to the global minimum withour algorithm. We demonstrate that the reduction ofthe OPE method to 1D for practical apertures is agood approximation, even when the films exhibitsome 2-D nonuniformity.
A. Limitations of the Wedge-Shaped Envelope Method
To check the reliability of the wedge-shaped envelopemethod for absorbing films, we generated various sim-ulated transmission spectra with 1-D nonuniformity inthickness from Eq. (6) by using realistic values of therefractive index, extinction coefficient, thickness, andnonuniformity. Both the wedge-shaped envelopemethod cited in Refs. 22–25 and the OPE method werethen used to extract optical properties from the simu-lated transmission spectra, and these values werecompared with the true values.
The following is a brief description of the wedge-shaped envelope method. Analytical expressions forthe envelopes around the interference maxima andminima of the transmission spectrum were derived bysolving Eq. (5) with the assumption that the absorp-tion was approximately zero in the long-wavelengthregion. The optical properties were obtained by solvingthose equations given the experimental envelopes. Up-per and lower experimental envelopes were computa-tionally derived from the measured transmissionspectrum by using the algorithm suggested by Mc-Clain et al.26
For these 1-D test cases, the refractive index of thefilms was assumed to be that of bulk AlN as reportedby the manufacturers.27 The packing-density modelwith p � 1 was used with this form of n���. Theextinction coefficient, k, was assumed to be constantover the range of wavelengths from 200 to 1200 nm.Thickness nonuniformity, d, was assumed to be
40 nm for all cases. The results computed by thewedge-shaped envelope method for two film thick-nesses and four different extinction values are shownin Table 1. Figure 2 plots these errors in thickness,d� , and nonuniformity, d, versus film transmittance,� � exp���d�, for a 2000 nm thick film at a wave-length of 633 nm. Note that film transmittance variesinversely with both extinction coefficient and absorp-tion coefficient, �.
These data clearly show that the error increasesas the extinction coefficient increases (film transmit-tance decreases). The computed value of the filmthickness is always less than the true value, thusleading to an overestimate of the refractive index.The error in d is even more sensitive to the extinc-tion coefficient. When using the wedge-shaped enve-lope method, one obtains errors of less than 5% in d�only when the film transmittance is greater than
Table 1. Error (%) for the Index of Refraction, n, Thickness, d� , and Nonuniformity, �d, Due to the Nonzero Extinction Coefficient Determined forSimulated Spectra by Using the Wedge-Shaped Envelope Methoda
� at 633 nm 0.998 0.990 0.980 1.000 0.996 0.980 0.961Error (%) of n 1.5 2.9 4.4 0.0 1.1 5.0 11.5Error (%) of d �1.5 �2.8 �4.2 �0.1 �1.1 �4.8 �17.4Error (%) of �d 5.5 8.1 12.7 1.1 5.8 18.9 31.8
aThe values used to simulate the spectra were the packing density, p � 1, and the thickness variation, �d � 40 nm. The extinctioncoefficient, k, was constant versus the wavelength for all films in the table. The film transmittance [� � exp(��d)] at a wavelength of633 nm was calculated for each film.
Fig. 2. Plot of the error of thickness, d� , and nonuniformity, �d,versus the film transmittance [� � exp(��d)] at 633 nm, using thewedge-shaped envelope method. The simulated film had a thick-ness of 2000 nm and �d � 40 nm. The dotted lines are a guide forthe eye.
0.98. Therefore large errors are possible when thewedge-shaped envelope method is applied to absorb-ing films. Indeed, some extinction coefficients re-ported previously for various arsenic sulfide filmswere negative for some wavelengths, and transmis-sion values were larger than the substrate transmis-sion when the wedge-shaped envelope method wasused.22,23 These results are clearly unphysical. Evenfor less extreme cases, this method leads to an over-estimate of the refractive index and an underesti-mate of the film thickness. For all these same cases,the OPE method correctly determined the values ofthe parameters of the model films (with �0.1% error).
B. Model Films with Two-Dimensional Nonuniformity
Although transmission spectra are often acquired byusing a finite aperture width, one might worry aboutthe errors caused by this technique. We devised sev-eral simulations to ascertain how large these effectsare on the extracted optical parameters and to checkfor convergence and absolute accuracy of the OPEmethod. First, to determine the effect of linear thick-ness variations transverse to the aperture axis and ofother thickness nonuniformity �r� on transmissionspectra, we simulated the transmission spectra withthe standard AlN refractive index �p � 1�, a constantextinction coefficient over the whole wavelengthrange �k � 1 � 10�3�, and a 2000 nm thick film.Transmission was calculated by using Eq. (8) whilesimultaneously including a 2-D wedge and nonuni-formity. The four cases generated were T�dx � dy
The simulated transmission spectra for these casesare shown in Fig. 3. In Fig. 3(b) the wavelength rangeof 355–375 nm is magnified to show the effects on theinterference ripples. As we expected, the envelopefor T�dx � 40� is severely decreased compared withT�dx � dy � rx � 0� [Fig. 3(a)], whereas T�dx
� 40�, T�dx � 40, dy � 20�, and T�dx � 40, dy
� 20, rx � 10� are indistinguishable in Fig. 3(a).Figure 3(b) shows that the envelopes for the caseswith transverse thickness variation and nonunifor-mity are further reduced. From these test cases, itappears that envelope distortion depends mainly onthe magnitude of the wedge parallel to the apertureand less on transverse variations. This severe changein the transmission spectra envelopes explains whythe conventional envelope method can fail for wedge-shaped films. Furthermore, it gives an indication thatnot considering transverse thickness variations in afitting method can still give good estimates of filmproperties. We note that the first two of these caseshad no wedge and a 1-D wedge, respectively. Theerror in the derived parameters when the OPEmethod was applied to these cases was again lessthan 0.1%. The second two cases had 2-D nonunifor-mity; their error analysis is included in the next para-graph, which considers the effect of 2-D wedge andnonuniformity on the OPE method.
To check the capability of the 1-D OPE method to
derive optical constants of absorbing samples thathave some transverse nonuniformity in the spec-trometer aperture, we generated several simulatedtransmission spectra by using the refractive index,extinction coefficient, thickness, wedge, and nonuni-formity values shown in Table 2 for eight cases for2000 nm thick films. Again, the refractive index of allfilms was assumed to be that of bulk AlN �p � 1�, andthe extinction coefficient was assumed to be constant�k � 1 � 10�3� over the entire wavelength range.Transverse nonuniformity �ry� was assumed to bezero, but the cases included longitudinal nonunifor-mity (in addition to wedge) and transverse wedge.
Fig. 3. (a) Simulated transmission spectra for an absorbing filmwith uniform thickness and perfect smoothness compared withfilms with T(�dx � 40), T(�dx � 40, �dy � 20), and T(�dx � 40, �dy
� 20, �rx � 10). At this scale T(�dx � 40), T(�dx � 40, �dy � 20),and T(�dx � 40, �dy � 20, �rx � 10) are indistinguishable from oneanother. (b) Magnified section of (a) over the wavelength range355–375 nm.
The 1-D OPE method that considers only longitudi-nal wedge was applied to extract the optical con-stants, and we compared these values with the actualvalues input into the transmission spectrum calcula-tion. The fitted parameters and error percentages forthe eight cases are shown in Table 2. From this datait is apparent that the errors for refractive index andfilm thickness are less than 1%, even for the cases inwhich the deviation transverse to the aperture is aslarge as the thickness variation longitudinal to themeasurement aperture. Errors for k were less than5%, but errors for d were larger. The effects of trans-verse thickness variation and longitudinal nonuni-formity were interpreted by the OPE method, whenrestricted to a 1-D wedge, as an increase in the valueof dx. The transverse thickness makes by far thelargest contribution to this error, but it generatesan error of only 8.2% in dx when dy � dx�2 �20 nm.
As Table 2 indicates, the errors caused by assumingonly a 1-D wedge and no other thickness nonunifor-mity are less than 0.7% and 0.4% in refractive indexand thickness, respectively, even in the most extremecase shown. The largest errors result when the thick-ness variation is equal in both dimensions. However,even in this case, T�dx � 40, dy � 40�, the OPEmethod accurately predicts n, k, and d� . We concludethat one can apply the OPE method, consideringonly 1-D nonuniformity, to determine n, k, and d�accurately from spectra obtained with rectangularaperture illumination without regard to samplewedge or orientation. Furthermore, if the transversevariation within the illumination aperture is lessthan 25% of the longitudinal variation, then dx canbe determined accurately as well.
4. Experimental Results
AlN films were grown by using PLD28 with the aimof studying film quality and properties for differentdeposition conditions, such as substrate temperature,background gas pressure, and ablation laser fluence.The OPE method was used to derive film parametersbased on experimentally measured transmissionspectra of the films. In addition, several samples weresacrificed to make direct physical measurements thatwould confirm the accuracy of the OPE method whenapplied to extract film thickness and thickness vari-ation. From these measurements, we are able to con-clude that the OPE method is applicable to andaccurate for the determination of parameters forpractical thin films.
A. Deposition of AlN Thin Films
The AlN films were deposited inside a UHV systemevacuated by a turbomolecular pump to a base pres-sure of 1 � 10�8 Torr �1.3 � 10�6 Pa�, using PLD froman AlN target. A schematic diagram of the PLD systemis shown in Fig. 4. A Lumonics PM-848 KrF excimerlaser, � � 248 nm, pulsed repetition frequency � 30Hz, and � � 14 ns, was used to ablate the AlN target.The laser output was rectangular, 30 mm � 10 mm,
and we used an eccentric rotating 32 cm focal-lengthlens to direct each pulse to a fresh location on a1.5 mm � 3 mm oval so as to achieve as consistentablation as possible. The target was a rectangular,stoichiometric AlN slab with a purity of 99.99%, fab-ricated by Sputtering Materials Inc. by using thepressure-assisted densification process. The targetwas irradiated by the laser through a UV fused-silicawindow at an angle of 60° to the target surface normal.The ablated material was then allowed to condenseonto the heated substrate that was parallel to the tar-get surface at a distance of 5.1 cm. Commercial sap-phire (0001) substrates were first degreased withacetone and methanol in an ultrasonic bath and werethen heated to 900 °C in vacuum for 45 min beforedeposition. The depositions were carried out at differ-ent substrate temperatures varying from 100 °C to800 °C for 25 or 50 min. Ultrahigh-purity nitro-gen was introduced into the PLD chamber at 4.5� 10�4 Torr �6.0 � 10�2 Pa�during deposition. Theeccentric rotation of the focusing lens resulted in afactor of 3 variation in fluence around the oval path onthe target �1–3 J�cm2 and 4–12 J�cm2 to achieve low-fluence and high-fluence sputtering regimes, respec-tively). Furthermore, the fluence at the sputteringtarget decreased slowly owing to deposition on thechamber window. For all the deposition runs, the win-dow was cleaned after a fluence drop of 25%–30%. Thethicknesses of the deposited films ranged from 1500 to2300 nm for a high laser fluence and from 400 to600 nm for a low fluence. The films had nonuniformityin thickness but minimal short-period surface rough-ness; these characteristics were verified by mechanicalprofilometry, atomic force microscopy (AFM), andcross-section scanning electron microscopy (SEM).
Optical transmission spectra of the films were ac-quired over the 200 to 1000 nm spectral region�1.2 to 6.2 eV� by UV–VIS spectroscopy. To mask thesample and define the illumination region, we con-structed a 4 mm � 1 mm aperture so that only 1-Dnonuniformity was considered.
B. Optical Properties of AlN Thin Films
We show here the result of applying our OPE methodto these films. A subset was chosen for detailed anal-
Fig. 4. Diagram of the UHV system used for AlN film depositionwith PLD.
ysis of how the various methods for extracting theoptical parameters from the transmission spectraperformed. Complete characterization of the films’optical and physical properties versus the depositionconditions will be reported in a future paper. Therefractive index, n���, extinction coefficient, k���, av-erage thickness, d� , and thickness variation, d, of ourfilms were obtained by using the OPE method. Wealso applied the wedge-shaped envelope method andthe conventional envelope method for comparison.The upper and lower envelopes of the spectra wereacquired by using McClain’s algorithm, as noted pre-viously.
Typical results are presented in Figs. 5 and 6 forrepresentative films deposited at high and low laserfluence, respectively. The measured spectral trans-mittance and the transmittance calculated from theparameters derived by using the OPE method for thetwo films selected are shown in Figs. 5(a) and 6(a).The fitted curves agree exactly with the experimentaldata at long wavelengths and deviate only slightly atshort wavelengths. Two possible reasons may be pos-tulated for this deviation: (1) the transverse wedge
and thickness variation in the illuminated area (ex-plained in Subsection 3.B and (2) the single-parameter, packing-density dispersion model forrefractive index did not have enough parameters tofit the short-wavelength region �300–400 nm�. Tocheck the accuracy of our dispersion model, we alsoapplied our method by substituting the Sellmeierequation13 to model refractive-index dispersion. Inthis case the derived parameters are virtually un-changed, and the difference between the measuredand the fitted transmission spectra at short wave-lengths is also similar. From this we conclude thatthe OPE method is robust and that the small spectraldifference is due to the thickness variation in thetransverse direction in addition to the 1-D wedge.
Figures 5(b), 5(c), 6(b), and 6(c) show the refractiveindex (b) and the extinction coefficient (c) versus thewavelength derived by using all three methods: theOPE method, the wedge-shaped envelope method,and the conventional envelope method. Extracteddata for the entire set of films at a wavelength of633 nm are shown in Table 3.
First, we consider which of these parameters ex-
Fig. 5. Optical properties of an AlN film deposited at high fluence and at a 500 °C substrate temperature: (a) transmittance, (b) refractiveindex, (c) extinction coefficient. Notation in the legends: exp, experimental measurement; fitted, the OPE method; wedge, wedge-shapedenvelope method; convention, conventional envelope method.
tracted by the OPE method can be verified by phys-ical measurements. Film thickness and thicknessvariation are the most amenable to direct measure-ment, so several types of measurements were made todocument the film thickness and thickness nonuni-formity. To bracket the film thickness, d� , a mechan-ical profilometer was used to measure thickness stepsnear both edges of each film. For all films, these mea-surements bracketed the OPE thickness and wedgevalues. A second measurement was made by scribingand fracturing a film along the center axis of theoptical transmission aperture. A number of cross-section SEM micrographs were recorded along thefracture surface, and both short-range and long-range thickness variations were determined. A rep-resentative cross-section SEM micrograph is shownin Fig. 7 for the AlN film deposited at high fluenceand at a 100 °C substrate temperature. One immedi-ately observes that the film is smooth on a microme-ter scale. Based on this and the AFM measurementsof root-mean-square (RMS) surface roughness (allfilms �3 nm as shown in Table 3), we conclude thatthe films do not have sufficient surface roughness to
significantly scatter light or disrupt the optical inter-ference. The long-range thickness variation of thisfilm is plotted in Fig. 8. The 4 mm long aperture iscentered at 40.4 mm ��1 mm�. The inset in Fig. 8shows this region of the aperture and a linear least-squares fit to the data. This fit results in a centerthickness of 2273 nm ��32 nm� and a thickness vari-ation, d, of 65 nm ��10 nm�. The correspondingOPE derived parameters, d� and d, are 2264 and73 nm, respectively. These are in agreement with thedirect measurements and are well within their accu-racy.
Next we consider the parameters extracted by thewedge-shaped envelope method. For both the high-fluence and low-fluence cases, this method results invalues of thickness, d� , that are 3% lower and invalues of refractive index, n, that are correspondingly3% higher than those for the OPE method. Consid-ering that the film transmittance is between 0.98and 0.99 for both cases, this agrees with our previousresult for model films, which indicated that thewedge-shaped envelope method gives errors of less
Fig. 6. Optical properties of an AlN film deposited at low fluence and at a 300 °C substrate temperature: (a) transmittance, (b) refractiveindex, (c) extinction coefficient. Notation in the legends: exp, experimental measurement; fitted, the OPE method; wedge, wedge-shapedenvelope method; convention, conventional envelope method.
than 5%. The film transmittance is still large enoughfor the extinction coefficient to be accurately esti-mated by this method. The extinction coefficient isfound to be negative over most of the wavelengthregion and diverges further at short wavelengths.The thickness variation, d, has larger and moreerratic errors. For the film deposited at high fluencewhere the wedge is significant, the error in estimat-ing d is �25% for the wedge-shaped envelopemethod. For the film deposited at low fluence, thesmaller d� contributes to a greatly increased error ford.
Finally, fitting the film parameters by using theconventional envelope method results in thicknesserrors of �35% and �2% compared with the OPEmethod for the films deposited at high- and low-fluence, respectively. For the two films considered,the variation in thickness for the high-fluence film ismuch greater than that for the low-fluence film. Theconventional envelope method is not expected to becapable of accounting for wedged samples properly. Itis relatively more successful in estimating the extinc-
tion coefficient; however, it consistently predicts thatthe shrinkage of the transmission spectrum envelopeis caused by absorption rather than by thickness vari-ation, which can also reduce the transmission spec-trum envelope. The result is an overestimate of theextinction coefficient, as shown in Figs. 5(c) and 6(c).In general, when the nonuniformity is small, say lessthan 20 nm, the conventional envelope method givesresults similar to those of the OPE method.
The following is a summary of the relation of filmparameters to deposition conditions. The refractiveindex does not show any specific trend with substratetemperature. But low laser fluence always producesfilms with a higher refractive index. The smallestextinction coefficient is 4.55 � 10�4 and is obtained athigh laser fluence and at a 300 °C substrate temper-ature. In general low substrate temperatures producesmaller extinction coefficients. Increased film thick-
Fig. 7. Representative cross-section SEM micrograph of an AlNfilm deposited at high fluence and at a 100 °C substrate tempera-ture.
Fig. 8. Thickness profile along the optical spectrometer slit axis ofa fractured AlN film deposited at high fluence and at a 100 °Csubstrate temperature. The inset shows the profile in the region ofthe 4 mm long spectrometer aperture centered at 40.4 mm as wellas the linear regression fit to that data.
Table 3. Properties of AlN Films Analyzed by the OPE Method (and by the Wedge-Shaped Method and Conventional Method for Selected Cases)a
ness naturally leads to larger nonuniformity that inturn results in reduced transmission spectra enve-lopes. The direct bandgaps of the AlN films werederived by fitting the derived absorption coefficient tothe relation �h� � A�h� � Eg�1�2 and are shown inTable 3. High-temperature and low-fluence deposi-tion give the largest bandgap energy, which is close tothat of bulk AlN �6.2 eV�. This might imply that highsubstrate temperature and low laser fluence (lowdeposition rate) favor fewer defects in the film crystalstructure. These aspects of film structure and prop-erties will be addressed in a future publication.
5. Summary
We have described a new method for extracting prop-erties of wedged and absorbing thin films; includingrefractive index, extinction coefficient, and thickness,based only on a measurement of the transmission(or reflection) spectrum. This OPE method utilizes acombination of numerical optimization; efficient mod-els of dispersive film parameters; and film trans-mission equations derived for wedged, nonuniformlythick, and absorbing films (with 2-D nonuniformity)measured through a rectangular aperture.
We also introduced a one-parameter, packing-density model for refractive index that requires mea-surement of the refractive-index dispersion of thesame or a similar material. The Sellmeier equationwas also shown to work, but with the disadvantage ofhaving more parameters to fit.
The OPE method was found to accurately predictthe values of optical constants for test cases thatproduced large errors when determined by using pre-vious methods. We showed that the OPE methodcould incorporate only 1-D nonuniformity in thick-ness and still converge rapidly and predict the valuesof optical constants accurately when a rectangularillumination area was used to record the transmis-sion spectrum. It is also possible to include both 2-Dwedge and nonuniformity in thickness if a stochasticalgorithm, such as a genetic algorithm, were to beemployed. Usually a hybrid algorithm that combinesa stochastic algorithm with a deterministic algorithmis employed when the cost function has numerouslocal minima around global minimum. In this case itwould take longer to converge.
A number of AlN films were also tested to comparethe performances of the various methods. Using ourOPE method allowed us to determine the opticalproperties of AlN thin films with accuracy, whereasthe presence of absorption and thickness nonunifor-mity in the films resulted in large errors when othermethods were used.
We gratefully acknowledge support from the Na-tional Science Foundation through Nanoscale Inter-disciplinary Research Teams (NIRT) grant DMI-0304031 for this research.
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