Optical scattering from oxidized metals.verification for oxidized copper
2: Model
Mikael Bergkvist, Arne Roos, Carl G. Ribbing, Jean M. Bennett, and Lars Mattsson
A model for the calculation of diffuse reflectance spectra for oxidized metals is applied to thermally oxidized
copper films and compared with experiments. The model calculations reproduce the spectral structureobserved in the experiments. It is demonstrated that the air-oxide interface roughness dominates thescattering for wavelengths shorter than the absorption threshold of the oxide, and the oxide-metal interfaceroughness dominates for longer wavelengths. Using the model for fitting calculations the rms roughness
values for the two interfaces are determined independently. The roughness values agree with the results from
high accuracy stylus profiling of the oxide front surface as well as stylus profiling and total integrated
scattering of He-Ne light from the bare metal surface, obtained after etching away the oxide. The goodoverall agreement between the calculated and experimental diffuse reflectance spectra, as well as the rms
roughness values determined with different techniques, confirms the applicability of the model.
1. Introduction
In this work we shall test a model1 for the calculationof diffuse light scattering from an oxidized metal. Themodel is based on the scalar scattering theory2 andpredicts the diffuse reflectance from the double layerusing the rms roughness values of the air-oxide andoxide-metal interfaces a1, a2 and the optical constantsof the oxide and metal. In the derivation of the ex-pression for the diffuse intensity reflectance the time-averaged square of the electric field of the light wave iscalculated. The field is normalized vs the incomingintensity but also with respect to the effect of the oxideand metal, respectively. In this way, the high scatter-ing power of the metal surface is taken into account,which makes the present model superior to a previoussimpler model. 3
The roughness patterns of the two interfaces areassumed completely uncorrelated (see Sec. III), and,therefore, the total intensities of scattered light, RD1and RD2, from the two interfaces are added as scalars.
Jean Bennett and Lars Mattsson are with Institute of OpticalResearch, S-100 44 Stockholm, Sweden; the other authors are with
Uppsala University, Department of Technology, Solid State Phys-ics, Box 534, S-751 21 Uppsala, Sweden.
Each of the two contributions can, according to Ref. 2,be written
RD. = 1 - exp[-(4roi/X)21RSj",=o, (1)
where Rs is the specular reflectance obtained from theconventional Fresnel formalism. It should be ob-served that X should be the actual wavelength in themedium in front of the scattering interface. UsingE(X,z) as the normalized electric field intensity at wave-length X and position z, the diffuse reflectance is ob-tained in closed form1 :
RD(X) = E(X,d) R + (Xl0) RD exp(-ad).e(oxide) D1
e(metal) (2)
With the film thickness d, z = -d and z = 0 representthe air-oxide and oxide-metal interfaces, respectively,and e(oxide) and e(metal) denote the function e calcu-lated for the interfaces of a semi-infinite oxide and asemi-infinite metal. a is the absorption coefficient47rk/X for the oxide. The factor exp(-cad) thus takesabsorption in the film into account. No scatteringfrom the bulk of the film is assumed.
In the model study, diffuse reflectance spectra werecalculated for hypothetical nondispersive materials:a dielectric layer on a highly and a moderately reflect-ing metal. The results qualitatively agree with experi-mental spectra4 6 ; i.e., the calculated spectra exhibitstrong spectral variation, which is oppositely phased tothe interference variation in the specular reflectancecurves. Based on this partial success, it was suggestedthat Eq. (2) could be used to fit calculated diffusereflectance spectra of oxide covered metals to experi-
mental curves using the a values as fitting parameters.The oxide thickness d is determined independentlyfrom interference fringes in the specular spectrum.
We shall demonstrate the degree of quantitativeagreement between the calculated and experimentalspectra for Cu2O grown on copper. This particularsystem was selected for several reasons. The amountof oxidation induced roughness is unusually large inthis case, giving rise to scattering levels up to 30-35%from an originally smooth evaporated copper film.Second, the oxide film can conveniently be removed byetching, leaving the bare copper surface7 with itsroughness accessible for independent measurementswith other techniques. We shall, therefore, be in aposition to compare the a values used as parameters inour calculations with experimental values obtained bystylus profiling of the oxide and the metal surfaces andtotal integrated scattering (TIS) from the bare metalsurfaces. This feature is an important advantagecompared with the recent application of the model tothe system SiO2/Si.8 Another characteristic feature ofthe Cu2O/Cu system is the strong optical absorption inthe oxide for wavelengths shorter than 500 nm. Forreasons that will be made clear later, this simplifies thefitting procedure.
In the next section we give details of the three ex-perimental techniques used. In the following sectionswe then demonstrate the sensitivity of the fitting pro-cedure and finally compare the calculated roughnessvalues with the directly measured results.
11. Experimental
A. Sample Preparation
The four copper films, -300 nm thick, were pre-pared on microscope slides by thermal evaporationfrom a tungsten boat in a conventional diffusionpumped system with a base pressure in the 10-6-mbarrange. The source material was 99.99% pure electro-
Table 1. Data on Sample Preparation and Surface Roughness rms-Valuesas Obtained by Different Techniquesa
a MC, model calculation; TS, Talystep; TIS, total integrated scat-tering.
lytic copper, and the glass slides were ultrasonicallycleaned in a detergent solution, rinsed in distilled wa-ter, and finally dried in a jet of filtered nitrogen gas.
The thickness of the metal films, as prepared, weredetermined by surface profiling as described below.The smoothness of the films was measured by TIS anda stylus profilometer (see Table I). An example of asurface profile of a copper film is shown in Fig. 4. Therms roughness of 0.76 nm for this film is very small,which explains the low scattering level (0.03% as mea-sured by TIS) for visible and near IR light, which isbelow the 0.1% level of detection for diffuse reflectancein the spectrophotometer.
Next the samples were thermally oxidized at 1900Cin an ordinary oven. The thickness of the oxide layerwas controlled by varying the oxidation time in the 15-120-min range. It is known that a minimum oxidethickness is required to develop sufficiently rough in-terfaces for noticeable diffuse reflectance to occur.6After oxidation the reflectance and stylus measure-ments were repeated.
Finally, the oxide layers were removed by etching in10% hydrochloric acid. Etching times of -10 s aresufficient to remove the oxide, leaving the bare rough-ened metal surface which was carefully rinsed in dis-tilled water. The extent to which this roughness isidentical to that of the oxide-metal interface is open todiscussion, but it has been established that it is mainlycreated by the original oxidation, not the etching.7Information about the four films is summarized inTable I along with the results of the measurements andcalculations.
B. Integrating Sphere Measurements
The reflectance spectra were recorded in a Beckman5240 UV-VIS spectrophotometer equipped with a dou-ble beam integrating sphere. The sphere detectorreadings either characterize the total reflectance of thesample or, if a specular exit port is opened, the diffusereflectance. A barium sulfate plate, the same materialas the inside coating of the sphere, was used as areference. This reference in turn was calibrated withNBS total reflectance standards.9 The reflectanceproperties of the sphere coating limit the useful spec-tral range of these measurements to the 0.3-2.5-gimwavelength region.
C. Talystep Measurements
Surface roughness was measured on the four initialcopper films, the oxidized copper films, and the filmsafter the oxide had been removed. The instrumentused for the measurements was a Talystep surfaceprofiling instrument 0 at the Surface Evaluation Lab-oratory, Institute of Optical Research, Stockholm. Itwas interfaced to a Hewlett Packard 330 computerusing an HP 3478A multimeter as a digitizer. The 0.3-,gm radius stylusl was used with a 1-mg loading. Six50-,um long profiles were made at different places oneach film to give a good statistical sampling of the filmsurface. The data point spacing was 34 nm, making a
total of 1488 points/profile. Height and slope distri-bution functions, autocovariance functions, and powerspectral density functions were calculated from thesurface profile data.
Because of the small stylus radius, 0.3 gm, and 1-mgloading, one would expect that the large force/unit areaon the copper and copper oxide films would plasticallydeform the surface and give too small roughness val-ues. To verify whether this was indeed the case, thefilms were also profiled with a 3-gm radius stylus and1-mg loading, giving an -100 times smaller force/unitarea. The roughnesses measured with the 3-,gm radiusstylus were up to 20% smaller, indicating that theprimary effect was of the stylus radius rather than toolarge a loading with the 0.3-gm radius stylus. Otherexperiments on a very soft evaporated aluminum filmshowed quite different results. The film roughnessmeasured with the 3-gm radius stylus with 1-mg load-ing was 3.6 nm rms, while the roughness value in-creased to 4.0 nm rms when the loading was decreasedto 0.2 mg. The roughness measured with the 0.3-gmradius stylus with 1-mg loading was only 2.1 nm rms,smaller than the value measured with the 3-gm radiusstylus, while the roughness increased to 5.8 nm rmswhen the loading decreased to 0.1 mg. Clearly in thiscase the 1-mg stylus loading was far too much for the0.3-gm radius stylus profiling the aluminum film,while the difference between the rms roughnesses of5.8 and 4.0 nm was caused by the difference in thestylus radii.'2
The data point spacing of 34 nm (0.034 gim) wasconsiderably smaller than the lateral resolution of the0.3-,um radius stylus. There is some confusion aboutdefining lateral resolution. Spectroscopists have tra-ditionally said that spectral lines were resolved if theycould see a tiny dip between two peaks. The so-calledRayleigh criterion defines this dip as 0.81 of the maxi-mum value,'3 while the more stringent Abbe criteriondefines the dip was 0.98 of the maximum value.'4 Us-ing this criterion for the Talystep stylus, structureseparated by -0.1-0.2,gm could be seen in the profiles.The definition of lateral resolution used previously forprofiling instruments is that it is the separation ofsurface features that could be profiled perfectly withno distortion caused by the stylus radius.'5"16 Whenthis criterion is applied to a sinusoidal surface of am-plitude a being profiled by a stylus of radius r, thelateral resolution dmin is15
dmin = 27rar
Using a value of 200 A for a (400 A peak to valley, arealistic value for the measured data for both the oxi-dized copper films and the rough copper after the oxidehad been removed), the calculated value for dmin is 0.5gim, considerably larger than the measured just-re-solved value mentioned above. Between these twolimits, the surface structure can be measured withvarying amounts of distortion as the separation be-tween surface features becomes smaller.'6 The mea-sured correlation lengths of the film structure were in
the 0.1-0.3-gm range for both the oxidized copperfilms and the rough copper after the oxide had beenremoved.
The roughness values measured for the copper andcopper oxide films were found to be independent ofprofile length for profiles from 50 to 200 gm long. Ifthe profile length had been limited to 15 gm to matchthe bandwidth of the TIS instrument, more profileswould have been required to obtain good statisticalsampling of the surface. It would require 67-15-gmlong profiles to equal a distance of 1 mm, the diameterof the illuminated area in the TIS instrument.
Film thicknesses were also measured for the fourevaporated copper films, the oxidized copper (copperoxide plus the remaining metallic copper), and thecopper remaining after the copper oxide had been re-moved. In this way, the thicknesses of the oxide filmscould be determined. The stylus for these measure-ments had a 15-gm radius and was used with 1-mgloading. Although good thickness measurementswere obtained in most cases, the step made by a wire onthe surface of the microscope slide as the copper wasevaporated was not always sharp. The copper filmthickness of sample 1 graded from the end of the mi-croscope slide to the center, where the scattering mea-surements were made. Thus a better oxide film thick-ness value was obtained from the model fittingprocedure. In most cases the measured and calculatedoxide film thicknesses agreed very well, as seen inTable I.
D. Total Integrated Scattering Measurements
Total integrated scattering (TIS) was measured us-ing a low noise instrument' 7 based on a hemisphericalmirror and a He-Ne laser (632.8-nm wavelength) witha spot size of 1 mm. The instrument fulfills the ASTMF-1048 standard for measurement of the effective sur-face roughness.'8 In the diffuse reflectance mode, itcollects scattered light in the angular range from 2.3 to80°. This angular range corresponds to scattering
from correlated surface features of 0.7-15-gm surfacespatial wavelengths. The Talystep data cover spatialwavelengths of 0.2-50 gm and would consequently beexpected to yield higher roughness values. In general,this is the case, but for samples with localized scatter-ing sites, like pits or particulates, TIS measures a high-er scattering level giving rms values higher than thosefrom the Talystep. The TIS instrument was calibrat-ed with a white reference surface made of Teflon mi-crospheres as described in ASTM F-1048. The instru-ment was capable of measuring scattering levels as lowas 0.2 part per million, corresponding to a surfaceroughness of 0.02 nm rms on silicon wafers.
TIS is defined as the ratio between the diffuse reflec-tance RD and the total reflectance, Rs + RD. Rs wasmeasured by putting the 2.3- X 2.3-mm2 TIS detectorin the specularly reflected beam 200 mm from thesample. The scalar scattering theory2 gives the rela-tion between TIS and rms surface roughness:
where a is the rms roughness and X the wavelength.RD was obtained for each of twenty points separated by1 mm along a line close to the center of the sample,yielding TIS and rms roughness values. The averageand standard deviations of these values were also cal-culated.
Ill. Results and Discussion
A. Fitting of RD Spectra
In this section we demonstrate the possibility ofusing Eq. (2) in a fitting procedure to obtain a valuesfor the air-oxide and oxide-metal interfaces. In par-ticular, we want to emphasize the advantage of havinga partly absorbing oxide film. The fact that Cu2Oabsorbs5 strongly for wavelengths shorter than 500 nm(extinction coefficient k > 0.35) implies that virtuallyall scattering at shorter wavelengths must come fromthe front surface of the oxide. A reliable a value can,therefore, be determined from the behavior in theshort wavelength part of the spectra without consider-ing the a2 value. In Fig. 1, we illustrate the sensitivityof this procedure for sample 4, i.e., the one having thethickest oxide film. In this figure the experimentalcurve is compared to a set of calculated diffuse reflec-tance spectra on an expanded scale. The calculatedcurves differ only in terms of their a, values as indicat-ed. At wavelengths shorter than 400 nm, the a2value has no influence and is, therefore, set equal tozero. In that region we can, therefore, make a singleparameter match. The agreement between the curvesillustrates how well the a, value can be determined. Itappears that a sensitivity of -0.1 nm can be reachedwith a conventional integrating sphere and spectro-photometer. In Fig. 2 an example of the correspond-ing procedure for determining the a2 value is shown forsample 3, using a fixed a, value obtained as explainedabove. Figure 2 includes the full 0-100% scale, and thespecular reflectance spectra are, therefore, included.In this case it is advantageous to consider a spectralregion where the oxide is transparent (longer wave-lengths), so as to make the scattering from the highlyreflecting metal dominant. The fit between the calcu-lated and experimental diffuse reflectance spectra inFig. 2 is not as good as in Fig. 1, but as seen from theindicated parameter values one can reasonably claim asensitivity better than 0.5 nm. The lack of agreementis more pronounced in the specular reflectance spectra.The most likely explanation for the discrepancies inFig. 2 is uncertainties in the optical constants of copperand copper oxide. The double layer calculations inthis spectral region are very sensitive to small changesin the extinctiion coefficient of copper oxide.
In Fig. 3 we illustrate a total fit of the diffuse reflec-tance curves. The overall behavior is well reproducedby the model. In particular, one observes the spectraloscillations in the calculated diffuse reflectance, whichis an essential feature of the experiment to be ex-
expLU3 z
.22 -100 calch _U.
0.25 0.5 1 2 4WAVELENGTH (um)
Fig. 1. Diffuse reflectance spectra illustrating the fitting of a, forsample 4. In the short wavelength region the -2-value is unimpor-tant and is set = 0. Notice the expanded reflectance scale: ---
in the cacuaios- -- , 2 a, 7. nm; - - - ,a - 6.5 nm
100 I
~~~~8--*-,0 =' 7._ 7
LU U60-
exp
LU40 -
nouce ------- inteIta d h calceons
20 / RD
0 I
0.25 0.5 1 2 4WAVELENGTH Cujm)
Fig. 2. Reflectance spectra as inFig.1, illustrating the fitting of theo2-Value for sample 3. The experimental and calculated specularspectra are also shown. a, 6.8 nm, obtained as in Fig. 1, was usedin the calculations. -- - = 5.5 nm; - - -, lih = 6.5 nm;
plained by the model. The use of dispersion in Eq. (2)leads to realistic results. An unexpected observationis that the experimental curves exhibit more pro-nounced minima in the IR than do the calculated ones.At present we do not have a complete explanation forthis phenomenon. One possible explanation is thatthe angular distribution of scattered light is differentfor maxima and minima. Different fractions of thescattering from the back surface would then betrapped owing to total internal reflection. This lossmechanism is not included in the model. Anotherlimitation of the model is the assumption that theroughness of the two interfaces is uncorrelated. Theseeffects are included in other more detailed models (seereferences in Ref. 1), and it would be of interest tocompare results obtained from these models with ours.The good overall agreement between calculated andexperimental results in Fig. 3, however, confirms theapplicability of Eq. (2) to predict light scattering froman oxidized metal. A more quantitative check is re-quired to fully validate the model. This is the subjectof Sec. III.B, where we compare the parameter values
Fig. 3. Experimental and calculated diffuse reflectance spectra forthe four samples as indicated. The r-values used in the calculations
are given in Table I.
a and a2 obtained from the fitting calculations withmeasured values.
B. Talystep and TIS Results
The film roughnesses as measured by the Talystepsurface profiling instrument and total integrated scat-tering values at X = 632.8 nm are given in Table I.Examples of surface profiles for an evaporated copperfilm, oxidized copper, and copper after oxide removalare shown in Fig. 4. Note that there is a large differ-ence in roughness between the initial evaporated cop-per film and the copper surface after the oxide hasbeen removed. As the copper oxidizes, it is well knownthat the growth rate is higher at grain boundaries, thuscreating a large roughness where none was presentoriginally. From this result, it is clear that it is notpossible to use the roughness of the initial copper filmas the roughness of the copper-copper oxide interface.The roughness of the copper oxide is that of interface 1,and the copper with oxide removed corresponds to theinterface between the copper oxide and the copper, i.e.,interface 2 in our formalism. Note that in this case theroughness of interface 2 is larger than for interface 1,but in other cases the reverse is true, as seen in Table I.If the difference in x- and y-scales in Fig. 4 is consid-ered, it is also evident that in all cases the roughness
Fig. 4. Talystep profiles of the various interfaces as indicated.The a-values obtained from the analysis of the profile are indicated.
Notice the differences in x- and y-scales.
profiles are shallow, with slopes that are typically 5 X10-2
TIS could not be used to determine the roughness ofthe oxide film, because it was transparent at X = 632.8nm. Interference between the outer and inner inter-faces strongly affected the diffuse scattering as shownin Fig. 3. The copper oxide could have been coveredwith an opaque reflecting material such as silver oraluminum, but then it would have been more difficultto remove the copper oxide.
The height and slope distribution functions calcu-lated for the films showed that all were Gaussian. Theautocovariance lengths obtained were in the 0.1-0.2-gm range. They could possibly be shorter, since this is
close to the lateral resolution of the stylus. From thepresent measurements we could not verify that theroughness between the interfaces was uncorrelated,but it is most likely to be the case. If the two interfaceswere highly correlated one would expect the diffusereflectance to vary in phase with the specular interfer-ence. This has been verified by combined calculationsand experiments for dielectric Ta 2O5 films on glass.19For this system, the dielectric film roughness closelyfollows that of the substrate.
studies of interfaces with potential importance for cor-rosion and mechanical or microelectronic failure stud-ies.
8 10
8 10
Fig. 5. Comparison of a-values obtained with the three differenttechniques as discussed in the text. (a) a, (model calculation) vs a,(Talystep) (b) a2 (model calculation) and 02 (TIS) vs 02 (Talystep).
In Figs. 5(a) and (b) the a values determined fromthe three different methods are compared. In Fig.5(a) the a, values obtained from our model calculationsare plotted against the values determined by Talystepmeasurements, and in Fig. 5(b) the -2 values obtainedboth by TIS and model calculations are plotted againstthe Talystep values. The diagonal is a guideline forthe eye showing perfect agreement between the differ-ent methods. The agreement between the values de-termined by our model calculation and direct measure-ments is clearly demonstrated.
In conclusion, we find good agreement between avalues determined from several independent tech-niques. The agreement demonstrated in Table I andFig. 5 gives credibility to the model, which is essential-ly a combination of scalar scattering theory, interfer-ence formalism, and a normalization of the electricfield intensity. At present the model has only beenused for calculating diffuse reflectance spectra for oxi-dized metals and determining interface roughness val-ues. It opens new possibilities for nondestructive
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The work reported here has been sponsored by TheSwedish Board for Technical Developent (STU), Thestay in Sweden for one of the authors (J.B.)-on leavefrom the Michelson Laboratory of the U.S. NavalWeapons Center-was financed by the Swedish Na-tional Science Foundation.
