D. DEN ENGELSEN
3390 the phosphorescence observed throughout the induction period can be due to the presence of many more terminal hydroxylated framework Fe3+(A13+) ionic species in the gel which do not phosphoresce. The final crystals, however, having a very much higher ratio of bulk to surface framework sites would therefore have a much more intense phosphorescence signal. Similarly, the 754-cm-’ (CH3)&+ Raman band in the solid phase during the induction period is explicable as (CH3)4N+ions bound to the negatively charged AI framework sites of the amorphous gel. On transfor-
mation of the amorphous material to zeolite crystals the (CH&N+ ions become trapped in the cages of the crystals. Summarizing, it can be said that our experiments are compatible with a zeolite crystallization mechanism occurring in the solid phase where some amorphous laminae are produced upon gel. formation. During the induction period further laminar species can be formed, destroyed, and modified until the first zeolite crystals are formed, after which autocatalysis of crystal growth produces an “a~alanche”of zeolite crystals.
Transmission Ellipsometry and Polarization Spectrometry of Thin Layers
Philips Research Laboratories, Eindhoven, Netherlands
(Received M a y 19,1972)
Publication costs assisted by Philips Research Laboratories
A theoretical description is given of the transmission ellipsometry and polarization spectrometry (dichroic spectra) of thin films deposited on a transparent substrate. It is shown that transmission ellipsometry is a sensitive method of measuring the anisotropy of the optical constants. The dichroism of films thinner than the wavelength of the light used to make the measurements depends strongly on their thickness, whereas for thick films the dichroism is almost independent of the thickness. The theory about transmission ellipsometry and dichroic spectra is illustrated by Langmuir-Blodgett layers.
Introduction Ellipsometric and spectroscopic methods are a t present widely used for studying thin films. Most applications deal with isotropic thin layers on which a vast literature exists, including several well-known barndbool~s.~--~ Thus far, little attention has been paid to the transmittance and reflectance of anisotropic thin films. Anisotropy may refer to either the components of the dielectric tensor or the componente of the conductivity tensor. The absence of much information or knowledge about this subject is caused by the difficulties inherent in preparing thin anisotropic layers. For instance, evaporation techniques seldom yield anisotropic systems. An exception is formed by polymer sheets containing rod-shaped dye molecules. Stretching of such sheets gives rise to a more or less well-defined orientation of the dye molecules with their long axes in the stretch d i r e ~ t i o n . ~ However, nonrodlike molecules can also be oriented.5 The dichroic. spectra of these anisotropic systems can he interpreted in terms of the degree of orientation of thc dye molecules.6 Another class of essentially anisotropic layers BS formed by the so-called LangmuirBlodgett (LB) l a y e r ~ . ~ ,The s birefringence in nonThe Journal of Phystcal Chemistry, Vol. 76, N o . 23, 1972
absorbing layers of barium stearate has been measured by Langmuir and Blodgett by comparing the respective transmittances of the p (parallel) wave and s (“senkrecht”) wave.R The anisotropy in the optical constants of both nonabsorbing and absorbing LB layers can also be determined with (reflection) ellipsometry.g,10 I n this paper I shall present a theoretical description of the dichroic spectra of uniaxially anisotropic layers (1) A . Vasicek, “Optics of Thin Films,” North-Holland, Amsterdam,
1960. (2) 0. S. Heavens, “Optical Properties of Thin Solid Films,” Butter-
worths, London, 1955. (3) 0. S. Heavens, “Thin Film Physics,” Methuen, London, 1970, Chapter 6. (4) (a) E. W. Thulstrup, J. Michl, and J. H. Eggers, J . P h u s . Chem., 74, 3868 (1970); (b) H. Inoue, et al., Ber. Bunsenges P h y s . Chem., 75,441 (1971). (5) This was kindly pointed out t o me by the reviewer of the manuscript. (6) Y. Tanizaki, Btdl. Chem. SOC.J a p . , 38,1798 (1965). (7) K. B. Blodgett, J. Rmer. Chem. SOC.,57, 1007 (1935). (8) K. B. Blodgett and I. Langmuir, P h y s . Reo., 51,964 (1937). (9) D . den Engelsen, J . Opt. Soc. Amer., 61, 1460 (1971). (10) E. P. Honig, J. H . Th. Hengst, and D. den Engelsen, J . Colloid Interface Sci., in press.
TRANSMISSION ELLTPSOMETRY OF THIN LAYERS
h, .A,
3391 considered. Hence, for small beam diameters the fivephase system may be split in two three-phase systems. If tl, and r l , are, respectively, Fresnel’s coefficients of transmission and reflection for the p wave at the air-film interface, and t Z p and r2, are the corresponding coefficients at the film-glass interface, then the total transmission coefficient T , becomes1n2
I \\
(2) Figure 1. Anisotropic film on glass: Y axis normal to the plane of the drawing; X,Zplane is plane of incidence.
with the optic axis either in or normal to the plane of incidence. The range of film thicknesses considered varies between aero and the wavelength of the incident light. Furthermore, a description will be given of transmission ellipsometry which is a useful technique for determining the optical constants of anisotropic layers. The theoretical considerations are illustrated by some absorbing and nonabsorbing LB layers. With these layers the degree of anisotropy in the indices of absorption can easily be varied by using mixed layers of dyes and fatty acid^^^*'^
Theory The reflectance of an anisotropic film deposited on an isotropic substrate has becn described p r e v i o ~ s l y . ~ Again, only uniaxial anisotropy will be considered with the optic axis coinciding with one of the Cartesian axes of Figure 1. First, we will deal with the case of 2 axis as the optic axis. Latcr on, simple transformations will be given to deal with systems having their optic axis in the direction of either the X or the Y axis. Thus, the principal indices of refraction of the film with respect to the coordinate system of Figure 1are
Gz f i x ==
&
=
1
nZ(l-
iKZ)
nx(1 -
~Kx)
(1)
If the film is nonabsorbing, then the indices of absorp, zero. In the next section n j ( j = x, tion, K= and K ~ are y, or x ) will stand lor both absorbing and nonabsorbing films. It will be assumed that the substrate is an isotropic transparent medium (glass, fused quartz), Figure 1 has becn made in accordance with the experimental conditions, because with the Langmuir-Blodgctt dipping technique both sides of the slide are equally covered by a “built-up” film. The equations derived hereafter refer to this case, but thc transformation to a singly covered slide is trivial. When a monochromatic light beam falls on the covered part of the slide, the beam passes four phase boundaries successively. Since the slides are thick enough, internal reflections within the slide need not be
where the superscripts (u) and (1) refer to the upper and the lower surfaces of the slide (Figure I). The transmission coefficient T , of the s wave is similar to eq 2 by changing subscript p’s into S’S. The phase differences p, and p, are
pp = 2ndn, cos (ppX-’ and ps = 2ndn, cos
pJ-‘
(3)
where n is the refractive index, cp is the angle of refraction, d is the thickness of the film, and X is the wavelength of the light under vacuum. Only in isotropic media or when using normally incident light are pp and (p, equal. When the film absorbs light, then the refractive angles (p, and (pb are complex quantities too.”,’* Using the equations of Snell and Fresnel. n,, n,. pp, and cps can be expressed in terms of the optical constants nx nd n, of the film, the angle of incidence 91 and the refractive index nl of the outer medium (air).“I2 Thus
pp = 2ndn,X,/(n,X) and p3 = 2ndXs/X
(4)
where s = nl sin p1, and X, = d n Z 2- s2 and X , = d n X 2- s2. As a consequence of the preparation technique of the LB layers, the film thicknesses on both = PP(’). sides of the slide are equal; therefore For a singly covered slide the second factor on the right-hand side of ( 2 ) becomes simply Fresnel’s coefficient of transmission for the glass-air interface. Fresnel’s coefficients of reflection at the successive phase boundaries have been given beforcB The reflection coefficients at the upper and lower interface of the slide are related by ~
~
~
=( -y u )m k ( l )
where m is 1 or 2 and IC stands for p or s. We further haveg
(5) and
(11) H.Bother, et al., Mol. Crust., 2, 199 (1967) (12) M.Born and E. Wolf, “Principles of Optics,” Pergnmon Press, Elmsford, N. Y., 1965. The Journal of Phusical Chemistry, Vel. 7 6 , N o . I S , 197%
D. DEN ENGELSEN
3392 /a
I
I
30
I
4100
I 1
20
\
$\,a
+O
\
\
\ \
10
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I m n
-0'
Loo
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I
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I
4
a
-1 0
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I
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I
I I
-20
-30
+ Idegrees I
J) [ degrees 1
Figure 2. $ us. A plot of the transmitted wave for nonabsorbing film: ( o ~= 60°,X 500 nm, n8 = 1.54. Full lines refer t o isotropic films with n = 1.4, 1.5, and 1.7, respectively. Curve a: n, = 1.5, nz = 1.7. Curve b: nZ 1.7, nS = 1.5. Inserted numbers indicate film thickness in r m .
=
Figure 3. $ us. A plot of the reflected wave for a nonabsorbing film. Since the curve Cor n = 1.4 almost coincides with curve a, a curve for n = 1.45 has been drawn. Other data as in Figure 2.
about 200 nm. For the sake of comparison the corresponding curves for the case of reflection ellipsometry have been plotted in Figures 3 and 3, using the equations of ref 9. At reflection the ellipsometric effect is induced by the upper film only, because multiple reflections of the slide are neglected. It is shown in Figures 2 and 4 that a thin isotropic film on both sides of a transparent slide gives rise to relatively small ellipsometric effects in the transmitted 16nln8(cos pr cos ( ~ ~ ) n , ~ n , ~-X 2i@p ,~e wave. The reason for this is that the exponential 2i T, = (7 ) [ ( n ~ ncQs , pi nix,) x (pS p,) in eq 9 is always zero for the isotropic case. (nzn, cos pa n3X,)(1 r1pr2pe-2i@p)12 Hence, the periodicity of the ellipsometric effect as a function of the film thickness stems solely from the T, becomes ratio (1 r 1 , ~ ~ ~ e - ~ ~ P 9 ) /rlprZpe-zi~p), (1 which is close T, = to unity for the used limited range of values for the re16nlndcos v1 cos ( P ~ ) X ~ ~ ~ - ~ ~ P ~ fractive indices of film and substrate. I n the case of anisotropy in a nonabsorbing film (n, f [(nl cos p1 X,)(na cos pa Xd(1 r 1 ~ r ~ ~ e - ~ ~ @dielectric 8)]~ n,) 2i@, - P,) is purely imaginary, which leads to a relatively large effect in A only. This is shown for (8) instance by the dashed line of Figure 2 . On the other I n analogy with reflection ellipsometry we define $ and hand, anisotropy in the indices of absorption (i.e., K, A with # K ~ causes ) fairly large variations in $, as can be seen from Figure 4. Principally, this effect stems from T, (tan $)ezA = - the real part of exp[2i(Ps - &)I. The fairly sharp TS angle in curve b of Figure 4 is purely accidental; inn,n,Xpei(@5-~p)(nl cos p1 .Xs) X sertion of other values for the optical constants of (n3 cos 9 3 X s )(1 r1sr2Re- 2 2 P s ) the film can yield a spiral as curve b of Figure 2 . (9) Xs(n,flz cos vi n*X,) x Finally, Figure 6 shows the amplification of the ellipso(n,n, cos ( ~ 3 naX,)(l rlprzpe-2z@p) metric effect, by increasing the refractive index of the substrate. In Figures 2 , 4, and G, A has been plotted us. $ at various values of the film thickness d between zero and It can be seen from Figures 2-5 that transmission
As shown in Figure 1, p3 refers to the direction of the wave normal in the glass slide with refractive index n3. Corresponding equations for tlp(l) t2,(l) and t l B ( l ) tzs(') are readily derived from ( 5 ) and (G) by replacing nl and (01 by n3 and pa. Making use of ( 5 ) and (6) the total transniission coefficient T,, defined in eq 2 , can be re written
+
+
+
I -
+
+
+
i
+
I
+
+
+
+
+ +
The Journal of Physical Chemistry, VoZ. 7 6 , N o . 83,197.9
]
+
TRANSMISSION ELLIPSOMETRY OF THINLAYERS
42 5
AS
50
L75
I
I
52 5
55
3393
$[degrees1
Figure 4. rl. us. A plot of the transmitted wave for an absorbing film. Full lines refer t o isotropic films with K = 0.05 and 0.1, respectively. Curve a: K % = 0, K$ = 0.1. Curve b: K~ = 0.1, K~ = 0. For all curves n, = n, = 1.5. Other data as in Figure 2 .
50 -
n
$i2 5
-
cn
W 'u
Q
0-
1
3
5
7
9
11
13
$[degrees1
Figure 5 . ic. us. A plot of the reJlected wave for absorbing film. Same conditions as in Figure 4.
ellipsometry is not a very practical technique for determining the optical constants of an isotropic film; reflection ellipsometry under the same conditions is better. I n the case of anisotropy of either the dielectric or the conductivity tensor (or both), transmission ellipsometry can be very useful for determining the optical constants of the film along the principal axes. Transniission ellipsometry is indeed a very sensitive method t o measure the birefringence of thin films. Comparing Figures 2 and 3, it can be seen that although curve a of the latter figure refers t o a bire-
p nz[vp(l lim D -= d+m
D,
+ +
w2) vsnz(1
p p ( b - dl/ 1
Kz')
(16)
The Journal of Physical Chemistry, Vol. 76, No. 23, 1972
D. DEN ENGELSEN
3394
O9
4-
05
I t
I0
1-
46
I 1
0 ,
1
,
I
SL
52
50
L0
300
FILM THICKNESS !NM I
40
I..-,
200
100
0
Figure 8. Dichroic ratio D,/D, us. film thickness for K $ = 0 and K. = 0.005, 0.05, and 0,5. Other data as in Figure 7 .
1 [degrees]
Figure 6. i us. A plot of the transmitted wave for a birefringent nonabsorbing film on different substrates. Refractive index of the film: nz = 1.5, nz = 1.6. Full line: na = 1.8. Dashed line: n3 = 1.45. Other data as in Figure 2 ,
01 0
I -
100 200 FILM THICKNESS [ N M I
300
Figure 9. Semilog plot of dichroic ratio R,/D, us. film thickness. For curve a: K$ = 0.1, K~ = Q.003. Other curves: K~ = 0, K~ = 0.01, 0.1, and 1. Other data as in Figure 7 . I
0
I
100 200 FILM T H I C K N E S S I N M I
1
300
Figure 7 . Dichroic ratio DJD,us. film thickness for K* = K~ and ?az = nz = 1.52. (nl = 45", A 500 nm, n3 = 1.5. Absorption index K is 0.0033, 0.033, and 0.33.
K~ K~
e 0.15, then D , / D s is constant when d > A. If > 0.16, then the limiting value of D,/Ds is reached
at a smaller film thickness (see Figures 7 and 8). Figure 9 shows the opposite anisotropy in the absorption index: K ,
