Determination of the extinction coefficient of dielectricthin films from spectrophotometric measurements
Jean-Pierre Borgogno and Emile Pelletier
Data obtained from reflectance and transmittance measurements are used to determine the extinctioncoefficient. We show that accuracy is limited by the substrate quality and interface roughnesses of the layer.
1. Introduction
Characterization of a material in thin film form isoften achieved by spectrophotometric methods. Forthat, we consider a sample made of a single layer of thematerial under study, deposited on a transparent sub-strate of perfectly known refractive index, and mea-surement of its optical properties is carried out over awide spectral range. We obtain in this way a set ofvalues for reflection Rexp(X) and transmission Texp(X).If the layer is roughly homogeneous, the classical mod-el of an isotropic layer with plane and parallel surfacescorrectly accounts for the observed optical properties,with the following parameters characterizing the layer:thickness e; refractive index n(X); extinction coeffi-cient k(X).
These parameters are deduced from the measuredvalues Rexp(X) and Texp(X); this problem, generallysolved by successive approximations methods, hasbeen studied by numerous authors.'
In the case of inhomogeneous layers, we have pro-posed a model with an index gradient; that is, therefractive index varies linearly as a function of thick-ness; the single parameter n(X) is then replaced by thecouple [n(X), An/n(X)] in which n is the mean indexvalue in the layer and An represents the whole indexvariation in the layer thickness; An/n(X) thus charac-terizes the inhomogeneity degree. Usually, we consid-er that the variation of An is negative in the case where
The authors are with Ecole Nationale Superieure de Physique deMarseille, Laboratoire d'Optique des Surfaces et des CouchesMinces, CNRS U. A. 1120, Domaine Universitaire St. Jerome, 13397Marseille CEDEX 13, France.
the index decreases from the substrate toward air andpositive in the opposite case.
An extension of the methods of index determina-tion 2 allows us to determine the values of all theseparameters. This characterization technique hasbeen systematically applied to a great number of layersgenerally obtained by vacuum evaporation or otherdeposition methods rather classical now.3 These lay-ers are usually made of dielectric materials which aretransparent and of good quality for optical applica-tions.
All the experimental results show that it is interest-ing to take account of this inhomogeneity degree be-cause, due to this additional parameter, a very goodagreement is generally obtained between the opticalproperties measured and those given by a calculationusing this layer model. Some examples of results havebeen published.4
In most cases, absorption losses are extremely weak,and we take no account of the value of the extinctioncoefficient k(X) because we are at the limit of sensitiv-ity of the measurement method, the precision on theRexp(X) and Texp(X) values being unfortunately insuffi-cient. This justifies the fact that, in the model of theinhomogeneous layer, we take no acount of a possiblevariation of k(X) in the thickness of the layer.
However, in optical applications, it is extremely im-portant to have a precise determination of k(X), even ifthese values are very low. (In practice, they neverexceed some 10-4.) This is why different methodshave been developed for measuring these losses: mi-crocalorimetric methods5 and more recently phototh-ermal methods.6 7
In this paper we deal with the problem of determin-ing k(X) from the values Rexp(X) and Texp(X). A deter-mination of k(X) is expected for a correct calibration ofthe photothermal method.7 We first give a detaileddescription of the experimental technique used andthen the results obtained concerning the values ofextinction coefficients of materials in thin film form.
Fig. 1. TiO 2 layer. Measured values of the reflection R(X) andtransmission T(X) coefficients of the TiO 2 layer. k(X) is the value of
the extinction coefficient calculated from these measurements.
11. Analysis of the Experimental Results
A. Complex Refractive Index
A detailed study of the experimental results at ourdisposal owing to the above mentioned method2 withparticular attention to the extinction coefficient k(X)leads to the following remarks:
The curve k(X) generally has a high level noise. Forsome materials, titanium oxide, for example, the ab-sorption band is in the neighborhood of the spectralrange involved in the measurements, and the disper-sion law k(X) can be detected rather distinctively.
Practically, the values of k(X) must be >10-3 in thewhole spectral range to be able to give a dispersion lawafter smoothing of the whole numerical values relatedto each measurement wavelength Xi.
Nevertheless it is not exceptional to obtain a set ofvalues k() which exhibits a curious modulation withwavelength. Figure 1 is a good display of one of theseresults. With different abscissa scales, we have repre-sented for this titanium oxide sample the measuredvalues of the reflection coefficient Rexp(X) and thetransmission coefficient Texp(X). This layer is practi-cally homogeneous: n/n(X) is near zero in the wholespectral range. Its thickness e is equal to 397 nm, andits optical constants are given by the following Cauchydevelopments:
The refractive index can be written as n(X) = A + B!X2 + C/X4 with
A = 2.264,
B = 2.62 X 10+4 nm2 ,
C = 5.47 X 10+9 nm4 .
The extinction coefficient k(X) = A' + B'/X2 + C'/X4,
A' = 7.4 x 10-5 ,
B' -5.4 X 10+1 nm2 ,
C' = 3.0 X 10+7 nm4 .
Obviously, k(X) presents an apparent modulationwith wavelength, which is a priori abnormal. Thismodulation is approximately in phase with the oscilla-
0.015 .
0.010
0.005
0.000 -
400.
Measured R and T1.0
0.8
T ~~0.6
rT R .0.4
0.2
"11 P A 19-1 /\ 0.0600. 700. 800. 900. l00.
Wavelength (nm)
Fig. 2. TiO2 layer. From the measurements of the reflection R(X)and transmission T(X) coefficients we have plotted the variations of
1 - R(X) - T(X) vs the wavelength X.
tions of R(X). Thus we tried to explain the origin ofthis phenomenon, which is the consequence of a defectin the characterization technique. This leads us toanalyze in a more detailed way the principle of themethod and to search for the possible sources of error.This will be made in two steps: the first concernsmeasurement of Rexp(X) and Texp(X) and the second theprocessing of the numerical data that leads to k(X).Between these two steps, we obviously get some infor-mation on absorption losses A(X) of the studied sam-ple, and it is convenient to start this study with adetailed analysis of absorption A(X).
B. Study of 1- Rexp - Texp
The curve of Fig. 2 is a direct consequence of Fig. 1,since we have plotted for each measurement wave-length the value Aexp(Xi) by simply writing
Aexp(Xi) = 1 - Rep(Xi) -Tep(Xi)-
On the curve A(X) plotted with each of the measure-ment wavelengths, for the short wavelengths, the be-ginning of the absorption band can be seen well; on theother hand, for long wavelengths (>800 nm), the valueof A(X) is extremely low and practically lost in thenoise.
Nevertheless, it is necessay to note that, in this ex-perimental result, A(X) does not show any oscillationwith wavelength in phase with the oscillation of T(X).This is not in agreement with the elementary theory.Indeed, we can show8 that (1 - R - T)/T as a functionof X can be put under the form of a limited develop-ment if k is low:
1 - R(X)- T(X) = kT(X)[(Q + 1)(F
+ (1 - Q) sink]/2n% + ke(X),
with e(X) leading toward 0 with X and with Q = n/n2and 1b = 47rnlel/X. no is the substrate refractive index;n2 and e2 are, respectively, the refractive index andthickness of the layer supposed to be homogeneous.
Consequently, 1 - R(X) - T(X) is proportional to kand T(X). This last proportionality justifies the pres-ence of a modulation similar to that of T(X). This
result is confirmed by a simple simulation calculation:Fig. 3 gives the values of the reflection R(X) and trans-mission T(X) factors together with the difference A(X)= 1 - R(X) - T(X) calculated from the values obtainedfor the titanium oxide layer described above; the sub-strate is made of BK7; the surrounding medium is air.
This calculation shows that 1 - R - T should bemodulated with wavelength with minimum values cor-responding to those of T(X). Thus, to explain theexperimental result given on Fig. 2, we must examinein detail two essential questions. The first concernsthe problems of calibration of the apparatus used tomeasure R and T. The second one is related to theequation of the energy balance where it is written thatabsorption is equal to 1 - (R + T) while totally neglect-ing diffusion losses. We successively examine thesetwo problems.
Ill. Measurement of the Reflection and TransmissionFactors
A. Principle
It is important to note here that we have access toabsolute values of layer reflection and transmissionfactors by measuring flux ratios. Indeed, to determineRexp for each wavelength, we must measure successive-ly the flux 4'R, and 4RO, which are, respectively, the fluxreflected by the layer and substrate (Fig. 4). In thecase of a prismatic shaped substrate [Fig. 4(a)], para-sitic reflections on the back surface are not disturbing.We have
Rexp = CRTexp =- TO,
Difference i-R-T
0.010
0.005
0.000 400.
Calculated R and T,1.0
0.8
0.6
0.4
0.2
500. 600. 700. 800. 900. 1000.Wavelength (nm)
Fig.3. Calculated optical properties R(X), T(X) and 1-R(X) - T(X)of a single TiO2 layer with thickness e 2 = 397 nm, refractive indexn2(X), and extinction coefficient k2 (X). The values of n2(X) and k2 (X)have been calculated from the measurements of Fig. 1, and theresulting Cauchy expansions are given in the text. We can noticethat the modulation of the theoretical curve 1 - R(X) - T(G) is
different from that given by the measurements (Fig. 2).
RRo i
n
(1)
with
° (1 - n )2 T = 1-RO,(1+ n )2 (2)
no being the substrate bulk index.For a substrate with parallel surfaces, the formula-
tion takes account of the back surface multiple reflec-tions [Fig. 4(b)]. We can easily show that
Rexp 1 + R-2R0OR (1 + RO) - RO T
exp (1 + Ro)O RQT R°,(1 + R,)' -RO '
with OR = (PRj/Ro and OT = 4%MPTo-A correct calibration of Rexp and Texp directly de-
pends on Ro, which implies good knowledge of therefractive index of the nonabsorbing and perfectly pol-ished substrate. In general, we can admit that theindex of bulk materials as glasses, fused silica, etc arewell known. However, polishing problems or cleanli-ness can give rise to some difficulties.
Numerous authors have studied the problems relat-ed to surface polishing or cleaning on various types ofsubstrate. A transition layer has been displayed, thethickness e and index n, of which cal be roughlydetermined by ellipsometry, 9 scattering, 1 0 or x-ray
(a)
n-..
//~ / 70~
](b)
Fig. 4. Principle of the measurement of reflection and transmissioncoefficients. Measurements of reflected and transmitted flux areperformed on the bare substrate and the substrate coated by thelayer under study. Knowledge of the substrate refractive indexenables us to calibrate these reflectance and transmittance measure-ments: (a) with a prismatic substrate, this measurement is notaltered by the back face of the substrate; (b) using a substrate withparallel faces, we must calculate the influence of the back face of the
substrate.
study.11 This layer has proved as disturbing9 for anellipsometric determination of index and thickness ofa layer in thin film form. Thus it is not surprising tohave the same problem in methods using spectropho-
Fig. 5. In the case of a transition layer (nl,el) on the substrate, thereflectance measurement R is altered because of an incorrect calibra-
tion of the bare substrate.
R. T. Rexp and TeXp values1.0
0.0 -ext
0.6 T'
0.4 . -1R
0.204 ti
0.0400. 500. 600. 700. 800. 900. 1000.
Wavelength (nm)
(a)
Aexp tl Apparent value of k
tometric measurements with the best possible accura-cy.
B. Influence of a Transition Layer in the Substrate
For this study, we have made a simulation reproduc-ing the exact conditions of spectrophotometric mea-surements. Notations are given in Fig. 5; we consider alayer of thickness e2 = 397 nm of a material of refrac-tive index (n2,k2), with the values of Cauchy expan-sions given above.
Figure 4 shows the perfect model of the sample used:the calculation of reflection Ro and transmission Tofactors involves a perfect substrate of index no.
Figure 5 shows what we actually measure when thereis a transition layer (thickness el and index nO) at thesubstrate (index no) surface.
For our simulation calculation, we suppose thatthere is a transition layer 60 nm thick, the refractiveindex of which is n1 = no + 0.01.
We have then all the elements necessary to simulatea measurement. We measure the reflection factor R'oon the substrate (with a transition layer) and R', whichis the reflection factor when the titanium oxide layer ison the substrate. Optical constants (n2,k2) and layerthickness e2 are the same as those already used.
The ratio R'/R' thus represents the ratio of the mea-sured flux, and we will make a calibration error bywriting that the reflection factor is given by the expres-sion (1) given above with Ro of Eq. (2), which corre-sponds to a bulk substrate of index no.
The simulation calculation leads us to write that theexperimental result concerning the reflection factor ofa TiO2 layer with a transition layer is
Rexpatj = R'RoIRo.
We can calculate TexptI. It is interesting to comparethese values with those we should have actually ob-tained with the assumption of a perfect measurement,that is, the values calculated by taking account of thetransition layer R' and T'.
On Fig. 6 we have represented the reflection factorsR' and Rexp,tu together with the transmission factors T'and Texpti. We can note that the transition layer hasno significant influence on the transmission factors,while it modifies the reflection factor, especially at the
0.025
0.020
0.015
0.0o0
0.005
0.000100. 500. 600. 700.
0.004
0 003
0.002
0. 001
0.000800. 900. 1000.Wavelength (nm)
(b)
Fig. 6. Influence of a substrate transition layer. The calibrationerror on the flux reflected and transmitted by the bare substratecauses the measurements to be wrong. Instead of T' and R' valuesthat correspond to the optical properties of a TiO2 layer (withthickness e2 = 397 nm, refractive index n2, extinction coefficient k2 ,
with the Cauchy expansion given in the text), we find slightly differ-ent values Texp,tj and Rexpti (a). Calculation gives an apparentabsorption Aexp,tI, and consequently it appears a modulation of thecoefficient k(X) vs the wavelength (b). In this example, we assumedthat the BK7 substrate had a transition layer with thickness el = 60nm and refractive index ni slightly higher than that no of the sub-
strate (ni = no + 0.01).
maximum values; for the given example, this differ-ence is -0.02. A good display of this influence is givenby the curve representing Aexpti(X), which is very dif-ferent from that represented in Fig. 3. In particular,we can note that there is also a modulation, but, unlikethe preceding case, the minimum values correspondhere to the maximum values of the transmission factor.
C. Influence of Scattering
Until now, we have totally neglected the fact thatpart of the light could be scattered. In fact, we havewritten A = 1 - R - T, which implies that all the lossesare light absorbed by the layer. Practically, even if thesubstrate is well polished, the scattered light cannot becompletely neglected. Above all, we must examinewith great attention the TiO2 layer which can scatter agreat quantity of light. In our laboratory we have thepossibility of studying in detail this phenomenon withimportant theoretical and experimental means.12
Fig. 7. Influence of scattering in the case of uncorrelated surfaces(a = 0). We assume here that the roughness spectrum of surfacesdefects can be approximated with the following parameters: g = 1.5nm, Lg = 200 nm, '5k = 3 nm, Le = 2000 nm. Calculation of the opticalproperties R8,o and To of a TiO2 layer (with thickness e2 = 397 nmand complex refractive index n2 ,k2 , with the Cauchy expansion aregiven in the text) have been performed taking into account the globalscattering SG,O- (a) the variations of global scattering SG,O andtransmission coefficient To vs wavelength. (b) the variations vswavelength of the extinction coefficient k,o calculated from thepreceding optical properties Ro and TO. We can compare thesevalues of k 0,o(X) with that corresponding to the Cauchy expansion.
Owing to simulation calculation, we again considerthe possible consequences of scattering losses to see towhat extent they can perturb our determination ofextinction coefficients.
For these calculations, we had at our disposal somerecent results obtained in our laboratory, in particular,by Amra and Grezes-Besset who study scattering frommultilayers. The aim is to determine theoretically theimportance of scattering losses, taking account of thedependence of these losses with wavelength.
Amral2 gives the description of a rough surface byusing an analytical function to represent the roughnessspectrum of the surface defects. It is the Hankeltransform (HT) of the sum of a Gaussian rg and expo-nential re functions: y = HT(r) with r = rg + re,where Fg(T) = 62 exp(-r2/L2) and re(T) = e exp(-r/Lg). For the substrate, we have defined the followingparameters: the Gaussian function represents theshort spatial period defects with g = 1.5 nm and Lg =200 nm. The exponential one concerns the large peri-
ods, and we have supposed that e = 3 nm and Le =2000 nm. The TiO2 layer of 397-nm thickness withn(X) and k(X) given by the Cauchy developments pre-viously used is then deposited on this substrate. Tocalculate the scattering we must also know the rough-ness of the interface film-air. We assume that it isidentical to that of the substrate. In a classical way,we consider two extreme cases: a = 1 case, the twosubstrate-layer and layer-air interfaces are perfectlycorrelated. The opposite case corresponds to the a = 0value for which the two interfaces are completely de-correlated.
Knowing the autocorrelation function, we can calcu-late for well defined illumination conditions the law ofspatial distribution of the light scattered in the wholespace. We consider here the case of normal incidencewith nonpolarized light. By integrating in the wholehalf-space, which corresponds to the front surface ofthe substrate, we have access to the global scatteringSG,r, and for the complementary half-space, we obtainSG,b- 13
In that way, if the TiO2 layer is rough, instead of thetheoretical values R and T which correspond to theresults of calculations in the assumption of perfectlyplane surfaces, we must observe lower values of thereflection and transmission factors which are Rg,,, andTs,a, the subscript s showing that the calculation takesaccount of scattering. Practically, as a first approxi-mation, we have Rsao = R - SG,r,a and TS,a! = T - SG,b,awith a = 0 or a = 1 according to the correlation state.
Thus the scattering losses add with absorptionlosses. To conclude this study, we must consider thevalues Rs,ag and T,a to calculate the apparent value ofthe extinction coefficient ka for the two classical val-ues of a.
Figure 7 concerns the case of decorrelated interfaces(a = 0), Fig. 8 that of correlated interfaces (a = 1). Inthe two cases, the apparent extinction coefficient ks,a isperceptibly modulated with wavelength. In Fig. 7(b),on the value of the k,o vs wavelength, a modulationappears in phase with the transmission factor Ts, ofthe layer. In the same figure, we plotted the value ofthe coefficient k(X) given by the Cauchy development,which was used to simulate the absorption losses in theTiO2 layer together with all the scattering losses SG,O =SG,r,0 + SGbo as a function of wavelength [Fig. 7(a)], forthe concerned correlation coefficient. We find thatk5,o(X) is always lower than k(X), the apparent value ofk8,o differing from k mainly for the wavelengths forwhich the transmission factor is maximum.
It is interesting to observe in Fig. 8(b) (which corre-sponds to a = 1) that, if we still have k, 1(X) greaterthan k(X), the modulation on the apparent value of theextinction coefficient is now practically in phase withthe reflection factor: the difference k, 1(X) - k(X) ismaximum for the wavelengths for which R, 1 shows amaximum.
IV. Conclusion
If we want to obtain the values of layer extinctioncoefficients from spectrophotometrical values, we are
tance absorption losses. Our simulation calculations show0 ° that this situation leads to a modulation of the appar-
0.6 ent values of k(X). The concordance of maximums ofk(X) with those of R(X) or T(X) essentially depends on
0.6 the correlation state between the two interfaces of thelayer. Obviously, we have no a priori information on
0.4 this subject because the intercorrelation law depends,for each studied sample, on the precise preparation
0.2 conditions of the substrate and layer.If we really want to develop this spectrophotometri-
0.0 cal technique to determine the extinction coefficients°°. of the materials deposited in thin films, it is important
to take all the useful precautions in the choice of thesubstrate, and it is necessary to complete experiments
lance by scattering measurements.l. We know that it poses difficult problems. We must
at least make a choice among the samples to keep only0.8 those for which scattering losses are negligible with
respect to absorption losses. In these conditions it can.0.6 be possible to determine correctly the extinction coef-
ficients of materials in thin film form..0.4
.0.2
- -------- ----- -I .700. 800. 900. 1000.
Wavelength (nm)
This paper is based on one presented at the FourthTopical Meeting on Optical Interference Coatings, 12-15 Apr. 1988, in Tucson.
(b)
Fig. 8. Influence of scattering in the case of perfectly correlatedsurfaces (a = 1). We performed the same calculations as those inFig. 7 with the same roughness spectrum. The global scattering SGlis in phase with R 8,,(X). It appears on the calculated value of theextinction coefficient k 8, 1(X) a modulation was not present on k(X).
limited by measurement precision. Practically, it isdifficult to determine k if it is lower than some 10-4.In fact, optical thin layers of classical quality exhibitextinction coefficients of this order of magnitude. Forlayers of thickness of -400 nm, losses do not exceedsome thousandths. The analysis of numerous experi-mental results has shown that k(X) often shows a mod-ulation with wavelength, which could be in phase withthe maximum of R(X). In this work we wanted toexamine this problem in detail to explain it. This ledus to a precise study of the principle of measurementsof transmission and reflection factors. Poor knowl-edge of the substrate refractive index is sufficient toexplain an apparently anomalous result on A(X) andk(X). In particular, the quality of the interface can bebad, and the presence of a transition layer at the sub-strate surface perturbs the flux calibration, and thespectrophotometrical measurements can be wrong.Consequently the determination of k(X) is not correct.
At this degree of precision, we cannot neglect theinfluence of scattering from surface roughnesses.Theoretical studies allow a prevision of the law ofvariation of scattering vs wavelength. We have takena simple numerical example to illustrate the type ofresults we can expect when in the energy balance thescattering losses are of the same order of magnitude as
References
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NASA continued from page 2853
the edges of the wires by looking for signals above and below athreshold value and uses the locations of the edges to compute thedistances between the wires.
This work was done by Thomas J. Magner, Richard D. Barney,William L. Eichhorn, and Henry P. Sampler of Goddard SpaceFlight Center. Refer to GSC-13117.
Flexible polymer-filled metallic conductorsA procedure has been developed to make materials that are both
flexible and reasonably good electrical conductors. Heretofore,electrical conductors for power and signal use have been manufac-tured from pure metals and metal alloys because these materialspossess the requisite electrical conductivity. Unfortunately, therepeated flexure of these metals in ordinary handling producesstress fatigue and, ultimately, mechanical failure. Polymers thathave much better flexure durability have been tried, but they areinadequate because of their very low electrical conductivity. Fur-thermore, because they have low thermal decomposition tempera-tures, the polymers cannot be heated to a liquid state and mixed withmolten metals to produce metal organic materials that might haveimproved stress fatigue properties.
In the new manufacturing procedure, a metal or a polymer sheetsubstrate is cleaned with a beam of energetic inert gas ions to removeadsorbed gases and contaminants from its surface. After this clean-ing, the substrate is coated by cosputter deposition of both a conduc-tive metal and a flexible polymer. The substrate is then removed byeither a mechanical or a chemical dissolution technique, and theresulting flexible metal/polymer conductor can then be bonded atlow temperature to conductor surface contacts. Material thus pro-duced exhibits both adequate electrical conductivity to convey pow-
er or signals and a flexibility that is superior to that of conventionalmetal conductors.
To demonstrate the beneficial properties of such a material, amixture of 76-vol % gold and 24-vol % polytetrafluoroethylene 8000A thick was codeposited by ion beam sputtering on a Kapton polyi-mide substrate 12.7 um thick. The resulting laminate was capableof being bent (with the coating in tension) on a radius of curvature of0.17 mm without either cracking or crazing the conductive coating.However, a 100% gold coating of the same thickness was found tocraze at a radius of curvature of 1.6 mm. Furthermore, the additionof the 24% of polymeric constituent raised the resistivity to only 3.7times that of the pure gold.
This work was done by Bruce A. Banks and Diane M. Swec ofLewis Research Center. Inquiries concerning rights for the com-mercial use of this invention should be addressed to the PatentCounsel, G. E. Shook, Lewis Research Center, Mail Code 301-6,21000 Brookpark Road, Cleveland, OH 44135. Refer to LEW-14161.
Functional microspheresA new process forms beads of polyglutaraldehyde directly from a
solution. Thus far, beads (microspheres) of 0.5-1.0-,am diameterwith fluorescent or magnetic properties have been made. They areuseful in biology, clinical chemistry, and biochemistry since theyreadily attach to red blood cells when combined with suitable anti-bodies. The fluorescent or magnetic properties allow such markedcells to be traced and identified.
Previously, glutaraldehyde was polymerized in an aqueous solu-tion. In the new process, a surfactant is added to the aqueoussolution, and microspheres are formed. The diameters are uniformand controllable, depending on the glutaraldehyde concentration,the emulsifier concentration, and the pH (see Fig. 9). In this pro-
1.
1.
Fig. 9. Size of microspheres is controlled by vary-ing the concentration of aldehyde or surfactant orby altering the pH. The microsphere diameterdecreases as the surfactant concentration is in-creased (left), decreases for more basic solutions(center), and increases as the aldehyde concentra-
