245
Langmuir 1986, 2, 245-241
Resolution of Uniaxial Optical Anisotropy in Thin Films Jennifer A. Bardwell? and Michael J. Dignam* Department of Chemistry, Uniuersity of Toronto, Toronto, Ontario, Canada M5S 1A1 Received October 2, 1985. I n Final Form: December 5, 1985 The various methods employed in attempting, in part or in whole, to resolve the optical constants of uniaxial thin films into components transverse and parallel to the optic axis are reviewed and previous errors pointed out. A correct procedure for the complete resolution of the problem is developed which can be based on either reflection or transmission spectroscopy combined with the Kramers-Kronig (KK) transformation. The method requires only that the films be thin in comparison with the wavelength of light and is applicable on a routine basis using a Fourier transform (FT) spectrometer and its fast FT algorithm to effect the KK transformation.
Introduction Determination of the optical constants for a uniaxial f i i much thinner than the wavelength of light, with its optic axis normal to the surface, has been a problem of interest to many workers. Langmuir-Blodgett mono- and multilayer films have been a particular focus of this interest. In the infrared they have been studied by two major techniques; neither, however, results in the desired two sets of spectra, one for the direction transverse to the optic axis (the t direction) the other for the direction normal to the surface (n direction). The first major technique' involves a comparison between the normal incidence transmission spectrum and the grazing incidence external reflection spectrum. The former samples only the transverse components of the optical constants, while the latter primarily samples the normal components. This approach has been extensively used by a group at IBM to study fatty acid monolayers.2 The second approach uses attenuated total internal reflection spectroscopy (ATR) with polarized light at a In this case, light polarized single angle of in the s direction (perpendicular to the incident plane) samples the transverse optical constants, while light polarized in the p direction (parallel to the incident plane) samples a combination of both the normal and transverse optical constants. Interpretation of these spectra is based on the effective thickness formulas of H a r r i ~ k derived ,~ by using the assumptions that the film is much thinner than the wavelength of light, that its refractive index is constant and isotropic over the wavelength range, and that the absorption coefficient, k, is everywhere less than 0.1. The direction of the transition dipole moment is determined from the ratio of the absorbance with light polarized in the s direction to that in the p direction. Although the data of Haller and Rice5 and Maoz and Sagiv6 have been interpreted using the correct equations, unfortunately some workers3!*have used equations identical with early, erroneous formulas published by Harrick.8 Thus their conclusions regarding molecular alignment are incorrect. In addition, their experimental data seem to be subject to substantial base-line problems, as regions where the film does not absorb do not show 100% reflectance. This error is likely due to imperfectly matched ATR accessories placed in the sample and reference beams. A third approach is due to Plieth and NaegelegJo and forms the basis of this paper. It has to date been attempted only for anodically formed, strongly absorbing thin uniaxial films. Correctly applied, it requires only (i) 'Present address: National Research Council of Canada, Metallic Corrosion and Oxidation Section, Montreal Road, Ottawa, Ontario, Canada K1A OR6.
the films be thin compared to the wavelength of light and (ii) knowledge of the film thickness and refractive indices a t a frequency where the film is transparent. This new approach needs no assumptions with respect to the strength of the absorption or whether the experiment is performed using external or internal reflection techniques. Furthermore, it provides complete information on the two sets of optical constants within the limits of accuracy of the Kramers-Kronig transformation. Finally, as developed here, it can be applied easily and routinely employing a Fourier transform spectrometer and its software. The equations to be used were developed by Dignam et a1.l1 and are shown below. In
(P1p3/F^13)u
= (1/2) In R,
+ ioR, = B F ~ , ( B ) (1)
where FR,(u)
u = s or p,
= - i 4 ~ d t ~ ' / ~ ( cao)s[ l - X,-
Yu(Xt - X,)]
Y p = (cot2 a - ~1/Z3)-', Y,= 0, X,= :,/€I i3/€1
1 -€'/Zn
-1
x, = - 1'
~
1-
tl/i3
The symbols are defined as follows: i12, is the complex Fresnel reflection coefficient for the film covered interface, P,I is the complex Fresnel coefficient for the bare interface, R, is the measured intensity ratio for reflected light for the film covered to bare interface for u-polarized light ( u = s or p), OR, is the phase shift difference between the covered and bare surfaces for u-polarized light, d is the film thickness in centimeters, i = (-1)1/2, 3 is the frequency in wavenumbers, a is the angle of incidence, and tl and i, are the dielectric constants for the incident and substrate media. In general ii = fi2 = (n - ik)2 where n is the refractive index and k the absorption index. For external reflection, i3is complex valued if the substrate is absorbing. For internal reflection el > 1 while e3 is generally unity. (1) Greenler. R. G. J. Chem. Phvs.. 1966. 44. 310. (2) See, for example: Rabolt, F.; Burns, F. C.; Schlotter, N. E.; Sawlen, J. D. J. Chem. Phys. 1983, 78, 946. (3) Takenaka, T.; Harada, K.; Matsumoto, M. J . Colloid Interface Sci. 1979, 73, 569. (4) Takeda, F.; Matsumoto, M.; Takenaka, T.; Fujiyoshi, Y. J . CIdloid Interface Sci. 1981, 84, 220. (5) Haller, G. L.; Rice, R. W. J . Phys. Chem. 1970, 74, 4386. (6) Maoz, R.; Sagiv, J. J. Colloid Interface Sci. 1984, 100, 465. (7) Harrick, N. J.; du Pre, K. F. Appl. Opt. 1966, 5, 1739. (8) Harrick, N. J. J. O p t . SOC.Am. 1965, 55, 851. (9) Plieth, W. J.; Naegele, K. Surf. Sci. 1975, 50, 53. (10)Naeeele. K.: Plieth. W. J. Surf. Sci. 1976, 61, 504. (11) DigLam; Mi J.; Moskovits, M.; Stobie, R. W. J . Chem. SOC., Faraday Trans. 2 1971, 67, 3306.
0743-7463/86/ 2402-0245$01.50/0 0 1986 American Chemical Society
246 Langmuir, Vol. 2, No. 2, 1986
Bardwell and Dignam
Finally, 2, is the transverse component and Z,, the normal component of the film's dielectric constant. With the assumptions k, < 0.1% k, < 0.1, and n, = n, = a constant, these equations reduce to the effective thickness formulas of Harrick7 for ATR. Plieth and NaegelegJoused the Kramers-Kronig (KK) technique12 to obtain the phase angles O'R, and elRpfrom the experimental spectra of (1/2) In R, and (1/2) In R, and then solved eq 1 for n,, k,, n,, and k , assuming OR, = @'R,. The original KK technique employs a mathematical transformation of normal incidence reflectance data over the entire spectral range to obtain the correct phase shift, OR, at each frequency.13 With these two sets of data it is then possible to calculate n and k for a semiinfinite isotropic substrate. This technique can be routinely applied to data for bulk isotropic materials obtained either at normal incidence14 or by ATR,15 by using the existing software of a Fourier transform infrared spectrometer (FTIR) to perform the KK transformation. However, applied to ATR, a correction must be made to the KK transform of (1/2) In R, @'R, in order to obtain the correct phase angle, In this paper we show that eq 1 in conjunction with the KK transformation can be used to calculate f i n and fit but that the unmodified values of YR,and O'Rp obtained directly from the transformation, as used by Plieth and Naegele,lo do not yield the correct values of the optical constants. A correction to @'R, is required to obtain OR,.
Theory The KK relationship between the reflectance, R, and the phase, OR, is shown below:
Here P indicates that the Cauchy principle value of the following integral is to be taken. A mathematically equivalent, Fourier transform (FT) relationship is'"'' OR($
L-In R(ijJ
= - 2 L m d t sin(27root)
COS
(2ai+t) d31 (3)
and may be evaluated rapidly and routinely by using the software of a commercial Fourier transform spectrometer.'* The KK and FT equations are applicable to any complex valued function, of which In r^ is an example, which describes the response of a real system to a periodic, harmonic stimulus, under the following restrictive condition~:'~ 1. The function when regarded as a function of a complex frequency, 5, = 3' - iV,must be analytic in the lower half plane where if' > 0. 2. The function must be square integrable on the lower half plane. Plieth and Naegelegshowed that ijFR,(p) satisfies the first condition except for at most one singularity for s-polarized light and at most two for p-polarized light. All of these singularities can occur only on the imaginary axis, and one can correct for the result of the nonanalyticity by the addition of a Blashke product term to the transformed phase.1° However, while they did examine the exact re(12) Landau, L. D.; Lifschitz, E. M. "Electrodynamics of Continuous Media"; Pergamon: London, 1960. (13)Cardona, M. In "Optical Properties of Solids"; Nudelman, S., Mitra, S. S.,Eds.; Plenum Press: New York, 1969;p 137. (14)Bardwell, J. A.; Dignam, M. J. Anal. Chim. Acta 1985,172, 101. (15)Bardwell, J. A.; Dignam, M. J. Anal. Chim. Acta, in press. (16)Sceats, M. G.;Morris, G. C. Phys. Status Solidi A 1972,14, 643. Knight, B. W. J . Opt. SOC.Am., 1973,63,1238. (17)Peterson, C.W.;
flection equation against the second condition, they failed to examine whether or not DFR (ij) satisfies it. In fact, it does not, since it is unbounded as Z, goes to infinity. The KK or FT relationships apply rigorously to data that are known over the entire frequency range from 3 = 0 to infinite frequency. In any real experiment, the reflectance spectrum is only known over a limited frequency range. Indeed, the meaning of the reflectance at very high frequencies becomes questionable. Because of the local character of the integral in eq 1,however, the relationship is a good approximation when applied to a limited frequency range, provided that the reflectance is not changing rapidly at the ends of the spectral range. If it is, corrections to the calculated phase are required.ls In order to avoid having to consider such corrections, we treat a hypothetical case in which the absorption bands for all of the phases are well confined within the experimental frequency range, [D,, cU], so that to a satisfactory approximation
FR,(D) = FR,($ in [O,fiil = F R , ( z , ~ )in [iju,m]
(4)
and fte[F~~(P,)] = Re[FR (81)] = 0, where FR,(P)is given by eq 1. For such a case dR"(p), defined according to GR,(p) =
3[FR,,(D) - FR,(Du)I
(5)
is both square integrable and analytic (whenever FR,(3) is analytic) and furthermore its real value matched the experimental function, i.e.: Re[GR,(P)]= PR~[FR,(D)]- PR~[FR,(V,)] = (1/2) In R,(D) (6) on making use of eq 1 and 4. Thus, the KK transform of (1/2) in R,(D) will yield elR, = Im[G~"(fi)] so that @'R, = DIm[FR,(3)] - DIm[FR,(D,)] = OR, - DIm[FR,(i',)] (7) where OR, is the desired phase angle appearing in eq 1. Thus, the desired phase, OR,, is obtained from a graph of the transformed phase, O'R,, by adding the following straight line through the origin, of slope Im[FRu(~,)]: A ~ R ="
- O'R, = 5Im[F~,(8,)]
(8)
The slope of the line correction to O I R , can be calculated from a knowledge of the film refractive index a t a frequency where the film is transparent. In the event that one or more of the dielectric functions is changing with frequency a t a significant rate a t D~ and/or D, a further correction must be applied and will have the same form as that used in connection with conventional normal incidence reflectance from a semiinfinite dielectric.'* An alternative approach to the fact that DFR,(z,) is not square-integrable is to divide eq 1 by D and hence KK transform (l/fi)(l/2) In R, to obtain the correct phase for FRu(ij). While this approach is attractive theoretically, we have found that in practice it does not work well since the function to be KK-transformed is heavily weighted to low frequencies where no data are available, a point noted by us earlier in connection with ATR of bulk phases.18 Finally we note that transmission measurements made on thin films supported on wedges (to eliminate multiple reflected rays at the detector) can be handled in exactly the same way as the reflection measurements. Thus in place of eq 1for the Fresnel reflection coefficient PIz3 and i l 3 , we have for the corresponding transmission coefficients l123 and f13 the f o l l o ~ i n g : ~ ~ (18) Bardwell, J. A,; Dignam, M. J. J . Chem. Phys. 1985,83, 5468.
Langmuir, Vol. 2, No. 2, 1986 241
Uniaxial Optical Anisotropy Resolution In
(&/i13)u
= (1/2) In Tu+ ioT, = PF~,(P)
where
%
FT,(v) = - i 2 d [ a X , a = (e3 - el
b=
(€1/63)(t3
+ bY,(X, - X,)]
sin2 a)lJ2-
- el sin2 a)1/2-
i
40.05
95
cos a cos
cy
with Tubeing the measured intensity ratio for transmitted light for the film covered to bare interface for u-polarized light (u = s or p ) with the other terms defined as in eq 1. Clearly the phase correction takes the same form as for reflection, with FT,(p) simply replacing FR,(3). For transparent substrates, the same information is in principle available from reflection and transmission measurements, the choice being a matter of convenience and relative sensitivity.
I
0
3600 WAVENUMBER
5403
(cm-')
Figure 1. Top spectrum: synthetic internal reflection spectrum for p-polarized light calculated from eq 1. Curve b correct phase spectrum corresponding to (1/2) In R,, calculated using eq 1. Curve a: phase spectrum resulting from the KK transformation of (1/2) In R,.
tained within the spectral range. For examples where the absorption of the film is not close to zero a t the ends of the spectral range, an error will remain if only a straight line correction is made to the phase. This remaining error is a basic limitation of the KK or FT technique, though a correction for it is possible as noted earlier.
Model Calculations To illustrate the application of the modified KK technique to thin films, model calculations were made. Starting with spectra of n,, k,, n,, and k, calculated by using harmonic oscillator functions as previously described,ls synthetic reflectance spectra were calculated by using eq 1. A number of such spectra were calculated for various values of ii,, ii,, d, cy, el, and E,. One example, shown in Figure 1, corresponds to attenuated total internal reflection for light polarized in the p direction at an angle of incidence of 4 5 O . The small central peak in R corresponds to an absorption in the normal direction, the remaining two to the transverse direction. The values el = 16, e3 = 1, and d = 2 X lo4 cm were chosen for this calculation. Figure l a shows the phase resulting from the uncorrected FT transformation of (1/2) In R,. This transformation procedure has been carried out on a Nicolet 8000 series FTIR spectrometer with a 7199 computer as previously described.14 Figure l b shows the phase calculated by using eq 1and the known optical constants. The difference between curves b and a is indeed a straight line of slope Im[F, (P,)] as predicted, where 3, = 7899 cm-l, the upper limig of the integration range. Spectra were also calculated for s-polarized light, for both internal and external reflection; in the latter case the substrate was chosen to be absorbing (E3 complex valued). Again the difference between the phase from eq 1and from the FT transformation gave a straight line of slope Im[ F R , ( ~ Jwith ] 3, = 7899 cm-'. In addition, when the transformation was restricted to the range 400 cm-l IP I4000 cm-', the slopes were almost the same as those obtained over the range, 0 IP I7899 cm-'. In these examples, the complex refractive indices of the film were chosen so that the absorption bands were effectively con(19) Bardwell, J. A.; Dignam, M. J. In "FTIR Characterization of Polymers"; Ishida, H., Ed.; Plenum: New York, in press.
I800
Summary and Conclusions I t appears that the full quantitative resolution of the optical constants for a thin uniaxial film with its optic axis normal to the surface can be effected on a routine basis using a F T spectrometer either in reflection or transmission mode. In general, all that is required is knowledge of the refractive indices of the film at some frequency and of the film thickness, which must also be small compared to the wavelength of the radiation in the film medium. Specifically, the condition 27rdlEt - (Elet/eu) sin2
