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Determination of thickness, refractive indices, optical anisotropy of, and stresses in SiO2 films on silicon wafers
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15 May 1995
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d
OPTICS COMMUNICATIONS
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ELSEWJER
Optics Communications 117 (1995) l-7
Determination of thickness, refractive indices, optical anisotropy of, and stresses in Si02 films on silicon wafers W. Lukosz, P. Pliska Optics Laboratory,
Swiss Federal Institute of Technology Ziirich, 8093 Ztirich, Switzerland
Received 13 January 1995
Abstract In a previous paper we demonstrated a simple yet accurate method for determining the thickness and refractive index of thin SiOZ films on silicon substrates, the only equipment required being an Abbe refractometer. The films were assumed to be isotropic. In the present communication we show that the method described, used with s- and p-polarized light, also permits to determine the optical anisotropy of the films which are optically uniaxial with their optic axis oriented normally to the film. The measured ordinary and extraordinary refractive indices of 2-10 km thick SiO, lilms are reported. The anisotropy is induced photoelastically by compressive stresses in the SiOZ films. From the measured anisotropy the stresses are determined to be UII= -290 MPa.
1. Introduction
In a previous paper [ l] we demonstrated a simple yet accurate new method for the determination of the thickness and refractive index of thin silicon dioxide (SiO*) films on silicon (Si) wavers. The methods which uses a commercially available Abbe refractometer without any modification can also be used for any other transparent low-index thin film on an absorbing high-index substrate. Thin SiOZ films have important applications in microelectronics, micromechanics, and in ‘integrated optics on silicon’, where a-few-micrometer-thick SiO* films are used as buffer layers between the waveguides and the absorbing Si [ 21. We use such waveguides in our work on acoustical [3] and biochemical [ 41 integrated optical sensors. In our previous paper [ 11, the SiOZ films were assumed to be optically isotropic. In the present communication: 0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD10030-4018(95)00116-6
(i) We show that also the optical anisotropy of the SiOZ films, i.e., their ordinary and extraordinary refractive indices no and neo. and the thickness d can be determined; the films are assumed to be optically uniaxial with their optic axis normal to the films. (ii) We report measurements of the refractive indices no and n,, of thermally grown SiOa films of tbicknesses d = 2-10 p.m. These results were obtained with a computerized opto-electronic measurement set-up which has a higher resolution than the visual observation with au Abbe refractometer. The physical method used remains basically the same: we determine the angular positions Nz’ and iVcp) of the minima of the reflectances R(“) and Rep)ofyhe SiO,/Si wafer illuminated through a prism P with monochromatic s- or p-polarized light; from ~2) and N,$) the optical parameters Q, near and d of the SiO, film are calculated. (iii) We explain of the observed anisotropy by the photoelastic effect and determine the compressive stresses (+,Iin the SiO, films.
2
W. Lukosz. P. Pliska / Optics Communications 117 (1995) I-7
2. Theory The SiOJSi wafer is laid with a droplet of immersion liquid I on a prism P - with the SiO, film facing the prism - as shown in Fig. 1. It is illuminated with sor p-polarized monochromatic light of vacuum wavelength A through the prism P. In the reflectances R (s,p)(IV) minima occur at certain values N pp) of the angle variable N=np
sin c+ =nr sin cur,
(1)
where np and n, are the refractive indices of the prism P and the immersion liquid I, respectively, and CY,and cu,are the angles of incidence in the two media P and I (seeFig. l).Thereflectanceminimam=1,2, . . ..are seen as dark lines in the eyepiece of the Abbe refractometer and their positions Nz’ and N$‘) are read visually, while in the new set-up described below, they are determined opto-electronically. The reflectance minima correspond to resonances of the optical field inside the film; they occur if A~~22k,d+S,(k,)+Ssi(k,)=27m2,
(2)
where A@ is the total phase shift of a plane wave in the film for one zig-zag reflection between the interfaces SiO,Si and SiO,-I, respectively, k, and k, are the xand z-components of the wave vector of the plane wave in the film (x and z being the coordinates parallel and perpendicular to the film, respectively), Sj( k,) are the phase shifts upon reflection of a plane wave incident
from the SiOZ film onto the medium j=I or j= Si, respectively, and fi = 0, 1, . . . . The phase shifts 8jj(s,p)(k,) can be calculated from the Fresnel reflection coefficients. For reflection at the nonabsorbing denser medium I we have 6I(‘) ((w) = - R- at all angles CX, and Sip) ( CX)= - rr at =arctan(n,ln). For reflection at the cw> aBm~, absorbing Si, we have for nearly grazing incidence in very good approximation 8gp’ = - r. This result holds independent of polarization and practically independent of the imaginary part of the complex refractive index of Si, i.e., of its absorption. Inserting S, + Ssi = - 27~ for both S- and p-polarization into Eq. (2)) we obtain 2kzd=2?mz,
(3)
wherem=fi+1=1,2, ... . For a monochromatic plane wave of (angular) frequency w and wave vector k (k,, ky = 0, k,) propagating in the anisotropic film, arelationship between the wavevector components k, and k, follows from Maxwell’s equations and the constitutive relations D, = rz,,nzE, and D, = e&&E, (where ee is the permittivity of vacuum) between the displacement D and the electric field E. For s-polarization, we have (k,ln,)*+
(kzln,J2=k2,
(4)
while for p-polarization (kxlrQ2+
(kzln,)2=k2,
(5)
where k= w/c= 2rrlh. We use k, = M and resolve Eqs. (4) and (5)) respectively, for k, which we insert into Eq. (3). Thus we obtain the final results [N@)]2=n2 o - (mh/2d)2 m
(6)
[N$q2=&
(7)
e. - (n,Jn,)2(mA/2d)2.
Eqs. (6) and (7) give the angular positions NC’ and N 2,” of the resonances or reflectance minima in s- and p-polarized light, respectively.
Fig. 1. Schematic of the basic configuration. P, prism (of Abbe refractometer or of the new set-up, respectively) with refractive index n,; I, immersion liquid of refractive index lzl closely matching that of the prism P, i.e., nl = np> n,, n,,; SiOz, film of thickness d and of ordinary and extraordinary refractive indices n, and n,, respectively; Si, silicon wafer; cu,, rr,, a, angles of incidence in the media P, I, and SiO,, respectively.
3. Determination SiO, film
of the optical parameters
of the
The method works as follows: firstly, the thickness refractive index n,, are determined from relation (6) with the values N 2’ of d of the film and its ordinary
W. Lukosz, P. Pliska /Optics Communications 117 (1995) l-7
3
two or more reflectance minima m = 1,2, . . . measured in s-polarized light as in the previous paper [ 1] . Then, the extraordinary refractive index neo is determined from the measured values N,$) and N,$‘) in the same order m, from the relation
( 11) and ( 12)) the refractive-index be expressed as
12, =n,N$“/N:‘,
where
(8)
which follows from Eqs. (6) and (7). Eq. (8) shows that the ratio N $“lN A,S’is the same for any order m. The reflectance minima are sharpest for rn = 1 and become broader with increasing order m. Therefore, the lowest orders m are to be preferred for the determination of the ratio q,,/n, from Eq. (8).
4. Stress-induced
(PI1
hl-*=~-*+a712~,,
+Pl*)q,
+Pl*el
+PllEl
9
Y
(9) ( 10)
where n is the isotropic refractive index in the strainand stress-free medium, and pr, and p12 are the strainoptic coefficients. The strains et, and el and stresses a, = u,,~ = (T,, and a, = U* in the medium are related as follows: E,, =E-‘[(lel =E-‘[
v)(T,, - vu11 ,
(11)
+cr,] ,
(12)
-2va,,
-2_n-
-2 rho
where E is Young’s modulus and v Poisson’s ratio of the film. In the Si02 film only (isotropic) stresses (T,, parallel to the surface occur, while CT~=O; note that o,, > 0 describes a tensile stress and (T,,< 0 a compressive stress. By combining Eqs. (9) and ( 10) with Eqs.
--12
2= (C,
+c*>(+,,
-2vPpl21
C,=E-‘E(l-
(13)
,
)
-*=2c*a,,
C, =E-‘[p,,
( 14)
7
(15)
VIII
vh,-
(16)
are the photoelastic
(or elasto-optic) constants. Eqs. ( 13) and ( 14) enable us to simultaneously determine the values of (+,, and n from the measured values of no and n,,: ~II
anisotropy
It is well-known that stresses are virtually always present in thin films [5], particularly in SiOz films grown by thermal oxidation of Si [ 61. However, in the thin-film literature, stress-induced optical anisotropy seems to have attracted little or no attention. According to the general theory of the photoelastic effect (see, for instance, Ref. [7] ) in an isotropic medium an optical anisotropy can be caused by the strains E,= E,,~= E,, and em= lL. (The x- and y-axes lie in the plane of the thin film; the z-axis is normal to the film. The suffixes II and _L mean parallel and normal to the film, respectively.) The ordinary and extraordinary refractive indices no and n,,, respectively, are (%)-*=n-*+
no
changes can also
=
(no
-2
n=non,,{(C,
-&;*)l(G
(17)
-C2),
-C,)/[n~(C,
+C,) -2n&C2]}“*. (18)
In the following discussion of Eqs. (13)-( 18), we use the definitions An,=n,--n, and the approximations no*-n-*= -2Aq,ln3 and ncY2 -n-*= - 2An,,ln3. Eq. ( 17) can be written as gII =2Anl[n3(C1
-C,)]
,
(19)
where An = neo - n, denotes the stress-induced birefringence or anisotropy. With the experimental value of An, the stress u,, can be determined from Eq. (19) even if the absolute values of no and n,, are not exactly known; the value of n need only be known approximately. From Eqs. (13) and (14) we obtain for the ratio of the changes of the extraordinary and ordinary refractive indices An,lAn,=2/(1+C1/C2),
(20)
the ratio Cr / C2 is independent of the value of E, and depends only on vand the ratiop,, /p12. Eq. (20) shows that the ratio &nJ An, is independent of the stress oil. Fused bulk silica has the elastic constants [ 81 E = 76 GPa, v=O.164, and the strain-optic coefficients [9] pI1 -0.121, p12=0.270; its refractive index at the wavelength h= 645 nm is n = 1.456672 [lo]. With the values these data for SiOp we obtain C,=0.427X10-12Pa-‘,C2=2.71X10-12Pa-’;furthermore, Cr - C,= -2.28 X lo-‘* Pa-’ and C,/ C,=O.lSS. We conclude from Eq. (20) that An,/ An,, = 1.728. Anisotropy is often described as
W. Lz&&, P. Pliska /Optics Communications I I7 (1995) 1-7
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An = Co,, with the stress-optic coefficient 2)n3( Cl -C,) = - 3.53 X lo-‘*Pa-‘.
5. Experimental
C= (11
set-up
The anisotropy of the SiOZ film can be observed with an Abbe refractometer used without any modification in visual observation as in our previous paper [ 11. The positions N 2’ and N 2,“’, respectively, of the dark resonance lines visible in s- and p-polarized light can be read from the refractive-index scale visible in the eyepiece of the commercial Abbe refractometer; the resolution of that refractometer was about aNti,= 2 X 10p4. A higher resolution is desirable, because the anisotropy of the SiO, films is AnEn,-n,= 1 X 10e3. Therefore, we built the setup schematically shown in Fig. 2, which is essentially an Abbe refractometer with an opto-electronic readout. The prism P is a BK7 glass prism (nr = 1..5147), the immersion liquid I is benzyl alcohol (C,HsO; n, = 1.5361). Only a very small droplet of immersion liquid should be used, because the thickness of the immersion layer I has to be spatially uniform and not wedge-like (the actual thickness value does not influence the determination of the Si02 film parameters). With a light-emitting diode (LED, type HSMH-T400 from Hewlett Packard; median wavelength h = 645 nm, bandwidth Ah = 25 nm) and a-few-mm2-large spot on the Si02/Si wafer is illuminated, under an angle of incidence c+ = 69” with a convergent beam of angular width ICYr = 12”. With a rotatable polarizer (Pol) s- or
p-polarization is selected. The reflected light intensity is measured in the focal plane of a lens L3 (of focal length f3 = 60 mm) with a linear diode array (type CCDl8 1 from Loral Fairchild Imaging Sensors; 2592 diodes, pixel separation 10 pm). The data are digitized by an analog-to-digital converter (ADC) and transferred to a personal computer (PC). The positions NcP) of the minima (m= 1,2, . ..) of the reflectance R (s*p)(N) are determined. The positions on the diode array have to be translated into the N values; the system was calibrated by measuring the critical angles of total internal reflection at a fused-silica (n = 1.456672, [lo]) and an LiF (n=1.391007, [ll]) plate. The accuracy of the measured Nzp) values is 6N= 1 X 10-4.ThedifferenceAN,=N$) -N$’ can, however, be determined with a much higher accuracy of S(AN) = 1 x 10-5.
6. Experimental
The samples were SiO, films of thicknesses d= 29.5 p,m on (100) 3i wafers; the films had been thermally grown by wet oxidation at temperatures of T= 1000-l 100°C and afterwards cooled down to room temperature To = 20°C with a rate of a few degrees per minute. With the new opto-electronic set-up we measured the positions N As’ and N $” of the reflectance minima in s- and p-polarized light of wavelength h = 645 nm (for an example, see Fig. 3), and determined the film parameters n,, n,,, and d. For all Si02 films we found that Ng’ > Ng’ . From Eq. (8)) we
1500
Fig. 2. Measurement set-up. LED, light-emitting diode;,Pol, polarizer (rotatable) ; L,, L, La, lenses of focal lengths fi = 30 mm, fi = 40 mm, fs = 60 mm; P, BK7 prism; I, immersion liquid (benzyl alcohol); CCD, diode array; ADC, analog/digital converter; PC, personal computer. cur= 69”, median angle of incidence; G+= 12’, angular width of convergent beam.
results
1600
1700
1800
1900
2000
Pixel number
Fig. 3. Experimental angular-reflectance spectra, i.e., R”’ versus pixel number of CCD. Reflectance minima m = 1,2 p,m thick SiOz film on Si (sample no. 3). The shift of about between the minima of different polarizations (and same is a clear indication for the anisotropy of the film.
and Rep’ for a 3.5 25 pixels order m)
W. Lukosz, P. Pliska / Optics Communications I I7 (I 995) l-7
conclude that neo > IZ,,,and from Eq. ( 17) that u,, < 0, i.e., the stresses are compressive. The refractive indices *, and neo of the SiOZ film are greater than that (n = 1.456672 [ lo] ) of fused bulk silica. The detailed results are given in Table 1. The thickness values d agree very well with the results d’ of an independent film-thickness measurement with a mechanical stylus instrument (Alpha-Step 200 from Tencor Instruments). For the latter measurement part of the SiO, film is removed by etching; the stylus measures the step height between the surface of the SiOZ film and the Si substrate. The sample wafers had different dotations and were oxidized under somewhat different conditions. Only the 3.5 pm thick SiOZ film (sample no. 3) had been grown on a wafer polished on both sides; we use SiOZ films of this thickness in our experiments on acoustical and biochemical integrated-optical sensors [3,4]. Table 2 contains values of the compressive stresses ( -a,,) calculated from the anisotropy An = 12, - n, with Eq. ( 19). The anisotropy is approximately~=(0.8-1.1)X10-3.Thestresseslieinthe interval -o,, = 215 - 316 MPa, which is in good agreement with the values -o,, =240 - 320 MPa reported by Ref. [ 121. Part of the stresses in the SiOZ film are caused by (slowly) cooling the oxidized wafer from the oxidation temperature of T= 1000-l 100°C down to room temperature of To= 20°C due to the different thermalexpansion coefficients cU,i, and Lysi. The thick Si substrate contracts more than a free SiOZ film would. Consequently, a strain ell = 1%iO*
-
(21)
asil CT- To)
5
Table 2 Compressive stresses ( - o,, ) in the SiOz films determined from the optical anisotropies An, and measured isotropic refractive indices n for the same samples as in Table 1. The stress values ( - (r,, ) are compared with the values ( - u{, ) measured with the bending-plate method of Stoney; the measured n values agree well with the value n = 1.456672 for fused bulk silica. Sample no.
--l1 (MPa)
- (T{, (MPa)
n
1 2 3 4 5 6 7 8 9 Average
215 288 282 271 291 316 316 316 294 288
270 332 326 351 321 348 401 351 367 341
1.45716 1.45693 1.45673 1.45694 1.45712 1.45663 1.45652 1.45640 1.45686 1.45681
develops, where the (Y’Sare the median values of the thermal-expansion coefficients in this temperature range. Thus, according to Eq. (ll), the stress (T,,= E( 1 - V) -‘e,, develops. With asio2 = 0.54 X 10-6K-’ [13],cusi=3.97X10-6K-’ [14],andwith the elastic constants [ 81 of silica E =76 GPa, V= 0.164, we calculate the thermal-stress value (T,,= - 320 MPa. There are residual stresses which depend on the manner of growth of the film and on the stress relief by tempering (i.e., on the cooling rate) [ 15,161. In order to determine the internal stresses (+,, in the SiO, films by an independent measurement, we used the well-known ‘bending-plate’ method [ 5,171. We measured the radius of curvature R of the wafer inter-
Table 1 Measured thicknesses d, ordinary and extraordinary refractive indices n, and n,, respectively, and anisotropies An = n, - n, of SiOz films on ( 100) Si wafers. For masons of comparison, the thicknesses d’ measured with a mechanical stylus instrument are listed. Sample no.
d (pm)
d’ (pm)
n0
Q0
An (10-a)
1 2 3 4 5 6 7 8 9 Average
2.008 3.051 3.524 4.054 5.003 5.608 6.994 7.809 9.460
2.010 3.055 3.530 4.085 5.005 5.565 6.955 7.795 9.430 -
1.45820 1.45833 1.45810 1.45826 1.45853 1.45816 1.45805 1.45793 1.45829 1.45821
1.45896 1.45935 1.45910 1.45922 1.45956 1.45928 1.45917 1.45905 1.45933 1.45922
0.76 1.02 1.00 0.96 1.03 1.12 1.12 1.12 1.04 1.01
W. Lukosz, P. Pliska / Optics Communications I I7 (1995) I-7
6
ferometrically in a Michelson interferometer after the SiOZ film on the back side of the wafer had been removed by etching. (In the optical measurements on the prism, the wafers being oxidized on both sides are symmetrically and, consequently, flat.) The ‘Stoney formula’ [5,17] gives the connection between R and g;, (the apostrophe distinguishes the stress values determined by the bending-plate method from those calculated from the anisohopy An) : a;, =EsiD2/[6dR(1-
vsi)] ,
(22)
where E,i is Young’s modulus and Vsi Poisson’s ratio of the Si substrate, D the substrate thickness and d the film thickness. For a compressive stress u,, < 0, the film tends to expand, and this leads to a convex curvature (i.e., R < 0) seen from the side of the film. For the ( 100) wafer the stress is along the [ 1 lo] direction and the contraction along the [ ii01 direction; therefore, we use the values E,i = 170.7 GPa, Vsi= 0.057 [ 181. The experimental results for a;, calculated from Eq. (22) are compared from the anisotropy
with the stresses u,, determined in Table 2. The 1CT;,1-values are
systematically between lO-30% higher than the 1ull Ivalues; this discrepancy seems to indicate that the elasto-optic constants (C, - C,) or C of the SiO* films are somewhat different from the values for fused bulk silica. However, (T,, and ai, are correlated; a linear regression analysis yielded a correlation coefficient of r=0.81. The refractive indices n calculated from Eq. ( 18) are also listed in Table 2. They agree well with the value n = 1.456672 [ lo] for fused bulk silica. A complete agreement is not expected because the refractive indices of the films can slightly be influenced, e.g., by the dopants in the Si wafers. In Table 3 we test the prediction of Eq. (8) that the ratio N m (PI/N 2) of the measured positions N $’ and N,$) of the reflectance minima of the same order m is the same a 9.5 ym m = l-9 satisfied,
for any order m = 1,2, . . . . In the example of thick SiO, film (sample no. 9) where orders were measured, the prediction is quite well with
Nm (p)/N n, (‘) = n,/n,
= 1.0007 .
Table 3 Experimentally measured ratios NzP’/Nc’ of the positions of the reflectance minima of the same order m in the example of a 9.5 pm thick SiOl film (sample no. 9) for m = 1-9. Test of the theoretical prediction that N$‘)/N!$ =n,/n,= 1.00071. ?n
N,$P’INc’
1 2 3 4 5 6 7 8 9 Average
1.00072 1.00070 1.00069 1.00068 1.00070 1.00071 1.00068 1.00068 1.00076 1.00070
7. Conclusions We developed a method to determine the thickness d, the refractive indices n, and n,,, and the anisotropy An=&,n, of thermally grown SiOz films on Si wafers. All investigated SiOa films were optically anisotropic and uniaxial with their optic oriented normally to the film. The observed anisotropy An can be explained by the compressive stress g,, in the film which influences n, and neo via the photoelastic effect. Consequently, the measurement of An offers the possibility to determine (T,,. From the measured anisotropies of An= (0.8- 1.1) X 10e3, we determined the stresses to be ( -a,,) = 125-316 MPa. These stress values are lO-30% lower than those determined independently by the bending-plate method, possibly because the elasto-optic constants of the films are not well known; nevertheless, the results of the two methods are correlated. The new method, suitable for film thickness of d = 215 p,m, is non-destructive. A disadvantage is that an immersion liquid has to be used to establish optical contact between the SiOJSi wafer and the prism; to obtain reliable results, it must be ensured that the immersion layer be spatially uniform. The method described can obviously be applied to any other lowindex and non-absorbing thin film on a high-index substrate. Acknowledgements We thank E. Hausammann and G. Natterer for the mechanical construction and the electronics, respec-
W. Lukosz, P. Pliska /Optics
tively, of the measuring set-up. We also thank A.C. Amrein and other students who helped developing and testing the set-up. We are indebted to Q. Lia from the Swiss Federal Institute of Technology Zurich and B.J. Curtis from Paul Scherrer Institute Zurich for supplying some of the samples with various film thicknesses.
References
Communications I I7 (I 99.5) I-7
I
]7] R. Guenter, Modem Optics (Wiley, New York, 1990) Ch. 14. [S] W. Primak and D. Post, J. Appl. Phys. 30 (1959)
779.
[9] M. Gottlieb, in: CRC Handbook of Laser Science and Technology, Vol. IV, Part 2, ed. M.J. Weber (CRC Press, Boca Raton, 1986) Ch. 2.3. [ 101 I.H. Malitson, J. Opt. Sot. Am. 55 (1965)
1205.
111 I J.S. Browder, S.S. Ballard and P. Klocek, in: Handbook of Infrared Optical Materials, ed. P. Klocek (Marcel Dekker, New York, 1991) Ch. 6. [ 121 R.J. Jaccodine and W.A. Schlegel, J. Appl. Phys. 37 (1966) 2429.
[I] W. Lukosz. and P. Pliska, Optics Comm. 85 (1991) 381. [2] H. Nishihara, M. Hamna and T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1985) Ch. 6.7. [3] P. Pliska and W. Lukosz, Sensors and Actuators A 41-42 (1994) 93. [4] Ch. Stamm and W. Lukosz, Sensors and Actuators B 18-19 (1994) 183. [5] M. Ohring, The Material Science of Thin Films (Academic Press, London, 1992) Ch. 9. [6] L.E. Katz, in: VLSI-Technology 2nd Ed., ed. S.M. Sze (McGraw-Hill, New York, 1988) Ch. 3.5.3.
[13] Optical Silica (Electra-Quartz,
1987) p. 4.
[14] T. Soma and H.-Matsuo Kagaya, in: Properties of Silicon (INSPEC, EMIS Datareviews Series No. 4, London and New York, 1988) Ch. 1.12. [ 151 M. Jarosz, L. Kocsanyi and J. Giber, Appl. Surface Science 14 (1982-83)
122.
[ 161 E.P. EerNisse, Appl. Phys. L&t. 30 (1977) 290; 35 (1979) 1171 G.G. Stoney, Proc. Roy. Sot. London, A 82 (1909)
8.
172.
1181 S. Johansson, Micromechanical Properties of Silicon (Acta Universitatis Upsaliensis 146, Dissertation, 1988) Ch. 4.2.
c&J
15 May 1995
mm
d
OPTICS COMMUNICATIONS
__
ELSEWJER
Optics Communications 117 (1995) l-7
Determination of thickness, refractive indices, optical anisotropy of, and stresses in Si02 films on silicon wafers W. Lukosz, P. Pliska Optics Laboratory,
Swiss Federal Institute of Technology Ziirich, 8093 Ztirich, Switzerland
Received 13 January 1995
Abstract In a previous paper we demonstrated a simple yet accurate method for determining the thickness and refractive index of thin SiOZ films on silicon substrates, the only equipment required being an Abbe refractometer. The films were assumed to be isotropic. In the present communication we show that the method described, used with s- and p-polarized light, also permits to determine the optical anisotropy of the films which are optically uniaxial with their optic axis oriented normally to the film. The measured ordinary and extraordinary refractive indices of 2-10 km thick SiO, lilms are reported. The anisotropy is induced photoelastically by compressive stresses in the SiOZ films. From the measured anisotropy the stresses are determined to be UII= -290 MPa.
1. Introduction
In a previous paper [ l] we demonstrated a simple yet accurate new method for the determination of the thickness and refractive index of thin silicon dioxide (SiO*) films on silicon (Si) wavers. The methods which uses a commercially available Abbe refractometer without any modification can also be used for any other transparent low-index thin film on an absorbing high-index substrate. Thin SiOZ films have important applications in microelectronics, micromechanics, and in ‘integrated optics on silicon’, where a-few-micrometer-thick SiO* films are used as buffer layers between the waveguides and the absorbing Si [ 21. We use such waveguides in our work on acoustical [3] and biochemical [ 41 integrated optical sensors. In our previous paper [ 11, the SiOZ films were assumed to be optically isotropic. In the present communication: 0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD10030-4018(95)00116-6
(i) We show that also the optical anisotropy of the SiOZ films, i.e., their ordinary and extraordinary refractive indices no and neo. and the thickness d can be determined; the films are assumed to be optically uniaxial with their optic axis normal to the films. (ii) We report measurements of the refractive indices no and n,, of thermally grown SiOa films of tbicknesses d = 2-10 p.m. These results were obtained with a computerized opto-electronic measurement set-up which has a higher resolution than the visual observation with au Abbe refractometer. The physical method used remains basically the same: we determine the angular positions Nz’ and iVcp) of the minima of the reflectances R(“) and Rep)ofyhe SiO,/Si wafer illuminated through a prism P with monochromatic s- or p-polarized light; from ~2) and N,$) the optical parameters Q, near and d of the SiO, film are calculated. (iii) We explain of the observed anisotropy by the photoelastic effect and determine the compressive stresses (+,Iin the SiO, films.
2
W. Lukosz. P. Pliska / Optics Communications 117 (1995) I-7
2. Theory The SiOJSi wafer is laid with a droplet of immersion liquid I on a prism P - with the SiO, film facing the prism - as shown in Fig. 1. It is illuminated with sor p-polarized monochromatic light of vacuum wavelength A through the prism P. In the reflectances R (s,p)(IV) minima occur at certain values N pp) of the angle variable N=np
sin c+ =nr sin cur,
(1)
where np and n, are the refractive indices of the prism P and the immersion liquid I, respectively, and CY,and cu,are the angles of incidence in the two media P and I (seeFig. l).Thereflectanceminimam=1,2, . . ..are seen as dark lines in the eyepiece of the Abbe refractometer and their positions Nz’ and N$‘) are read visually, while in the new set-up described below, they are determined opto-electronically. The reflectance minima correspond to resonances of the optical field inside the film; they occur if A~~22k,d+S,(k,)+Ssi(k,)=27m2,
(2)
where A@ is the total phase shift of a plane wave in the film for one zig-zag reflection between the interfaces SiO,Si and SiO,-I, respectively, k, and k, are the xand z-components of the wave vector of the plane wave in the film (x and z being the coordinates parallel and perpendicular to the film, respectively), Sj( k,) are the phase shifts upon reflection of a plane wave incident
from the SiOZ film onto the medium j=I or j= Si, respectively, and fi = 0, 1, . . . . The phase shifts 8jj(s,p)(k,) can be calculated from the Fresnel reflection coefficients. For reflection at the nonabsorbing denser medium I we have 6I(‘) ((w) = - R- at all angles CX, and Sip) ( CX)= - rr at =arctan(n,ln). For reflection at the cw> aBm~, absorbing Si, we have for nearly grazing incidence in very good approximation 8gp’ = - r. This result holds independent of polarization and practically independent of the imaginary part of the complex refractive index of Si, i.e., of its absorption. Inserting S, + Ssi = - 27~ for both S- and p-polarization into Eq. (2)) we obtain 2kzd=2?mz,
(3)
wherem=fi+1=1,2, ... . For a monochromatic plane wave of (angular) frequency w and wave vector k (k,, ky = 0, k,) propagating in the anisotropic film, arelationship between the wavevector components k, and k, follows from Maxwell’s equations and the constitutive relations D, = rz,,nzE, and D, = e&&E, (where ee is the permittivity of vacuum) between the displacement D and the electric field E. For s-polarization, we have (k,ln,)*+
(kzln,J2=k2,
(4)
while for p-polarization (kxlrQ2+
(kzln,)2=k2,
(5)
where k= w/c= 2rrlh. We use k, = M and resolve Eqs. (4) and (5)) respectively, for k, which we insert into Eq. (3). Thus we obtain the final results [N@)]2=n2 o - (mh/2d)2 m
(6)
[N$q2=&
(7)
e. - (n,Jn,)2(mA/2d)2.
Eqs. (6) and (7) give the angular positions NC’ and N 2,” of the resonances or reflectance minima in s- and p-polarized light, respectively.
Fig. 1. Schematic of the basic configuration. P, prism (of Abbe refractometer or of the new set-up, respectively) with refractive index n,; I, immersion liquid of refractive index lzl closely matching that of the prism P, i.e., nl = np> n,, n,,; SiOz, film of thickness d and of ordinary and extraordinary refractive indices n, and n,, respectively; Si, silicon wafer; cu,, rr,, a, angles of incidence in the media P, I, and SiO,, respectively.
3. Determination SiO, film
of the optical parameters
of the
The method works as follows: firstly, the thickness refractive index n,, are determined from relation (6) with the values N 2’ of d of the film and its ordinary
W. Lukosz, P. Pliska /Optics Communications 117 (1995) l-7
3
two or more reflectance minima m = 1,2, . . . measured in s-polarized light as in the previous paper [ 1] . Then, the extraordinary refractive index neo is determined from the measured values N,$) and N,$‘) in the same order m, from the relation
( 11) and ( 12)) the refractive-index be expressed as
12, =n,N$“/N:‘,
where
(8)
which follows from Eqs. (6) and (7). Eq. (8) shows that the ratio N $“lN A,S’is the same for any order m. The reflectance minima are sharpest for rn = 1 and become broader with increasing order m. Therefore, the lowest orders m are to be preferred for the determination of the ratio q,,/n, from Eq. (8).
4. Stress-induced
(PI1
hl-*=~-*+a712~,,
+Pl*)q,
+Pl*el
+PllEl
9
Y
(9) ( 10)
where n is the isotropic refractive index in the strainand stress-free medium, and pr, and p12 are the strainoptic coefficients. The strains et, and el and stresses a, = u,,~ = (T,, and a, = U* in the medium are related as follows: E,, =E-‘[(lel =E-‘[
v)(T,, - vu11 ,
(11)
+cr,] ,
(12)
-2va,,
-2_n-
-2 rho
where E is Young’s modulus and v Poisson’s ratio of the film. In the Si02 film only (isotropic) stresses (T,, parallel to the surface occur, while CT~=O; note that o,, > 0 describes a tensile stress and (T,,< 0 a compressive stress. By combining Eqs. (9) and ( 10) with Eqs.
--12
2= (C,
+c*>(+,,
-2vPpl21
C,=E-‘E(l-
(13)
,
)
-*=2c*a,,
C, =E-‘[p,,
( 14)
7
(15)
VIII
vh,-
(16)
are the photoelastic
(or elasto-optic) constants. Eqs. ( 13) and ( 14) enable us to simultaneously determine the values of (+,, and n from the measured values of no and n,,: ~II
anisotropy
It is well-known that stresses are virtually always present in thin films [5], particularly in SiOz films grown by thermal oxidation of Si [ 61. However, in the thin-film literature, stress-induced optical anisotropy seems to have attracted little or no attention. According to the general theory of the photoelastic effect (see, for instance, Ref. [7] ) in an isotropic medium an optical anisotropy can be caused by the strains E,= E,,~= E,, and em= lL. (The x- and y-axes lie in the plane of the thin film; the z-axis is normal to the film. The suffixes II and _L mean parallel and normal to the film, respectively.) The ordinary and extraordinary refractive indices no and n,,, respectively, are (%)-*=n-*+
no
changes can also
=
(no
-2
n=non,,{(C,
-&;*)l(G
(17)
-C2),
-C,)/[n~(C,
+C,) -2n&C2]}“*. (18)
In the following discussion of Eqs. (13)-( 18), we use the definitions An,=n,--n, and the approximations no*-n-*= -2Aq,ln3 and ncY2 -n-*= - 2An,,ln3. Eq. ( 17) can be written as gII =2Anl[n3(C1
-C,)]
,
(19)
where An = neo - n, denotes the stress-induced birefringence or anisotropy. With the experimental value of An, the stress u,, can be determined from Eq. (19) even if the absolute values of no and n,, are not exactly known; the value of n need only be known approximately. From Eqs. (13) and (14) we obtain for the ratio of the changes of the extraordinary and ordinary refractive indices An,lAn,=2/(1+C1/C2),
(20)
the ratio Cr / C2 is independent of the value of E, and depends only on vand the ratiop,, /p12. Eq. (20) shows that the ratio &nJ An, is independent of the stress oil. Fused bulk silica has the elastic constants [ 81 E = 76 GPa, v=O.164, and the strain-optic coefficients [9] pI1 -0.121, p12=0.270; its refractive index at the wavelength h= 645 nm is n = 1.456672 [lo]. With the values these data for SiOp we obtain C,=0.427X10-12Pa-‘,C2=2.71X10-12Pa-’;furthermore, Cr - C,= -2.28 X lo-‘* Pa-’ and C,/ C,=O.lSS. We conclude from Eq. (20) that An,/ An,, = 1.728. Anisotropy is often described as
W. Lz&&, P. Pliska /Optics Communications I I7 (1995) 1-7
4
An = Co,, with the stress-optic coefficient 2)n3( Cl -C,) = - 3.53 X lo-‘*Pa-‘.
5. Experimental
C= (11
set-up
The anisotropy of the SiOZ film can be observed with an Abbe refractometer used without any modification in visual observation as in our previous paper [ 11. The positions N 2’ and N 2,“’, respectively, of the dark resonance lines visible in s- and p-polarized light can be read from the refractive-index scale visible in the eyepiece of the commercial Abbe refractometer; the resolution of that refractometer was about aNti,= 2 X 10p4. A higher resolution is desirable, because the anisotropy of the SiO, films is AnEn,-n,= 1 X 10e3. Therefore, we built the setup schematically shown in Fig. 2, which is essentially an Abbe refractometer with an opto-electronic readout. The prism P is a BK7 glass prism (nr = 1..5147), the immersion liquid I is benzyl alcohol (C,HsO; n, = 1.5361). Only a very small droplet of immersion liquid should be used, because the thickness of the immersion layer I has to be spatially uniform and not wedge-like (the actual thickness value does not influence the determination of the Si02 film parameters). With a light-emitting diode (LED, type HSMH-T400 from Hewlett Packard; median wavelength h = 645 nm, bandwidth Ah = 25 nm) and a-few-mm2-large spot on the Si02/Si wafer is illuminated, under an angle of incidence c+ = 69” with a convergent beam of angular width ICYr = 12”. With a rotatable polarizer (Pol) s- or
p-polarization is selected. The reflected light intensity is measured in the focal plane of a lens L3 (of focal length f3 = 60 mm) with a linear diode array (type CCDl8 1 from Loral Fairchild Imaging Sensors; 2592 diodes, pixel separation 10 pm). The data are digitized by an analog-to-digital converter (ADC) and transferred to a personal computer (PC). The positions NcP) of the minima (m= 1,2, . ..) of the reflectance R (s*p)(N) are determined. The positions on the diode array have to be translated into the N values; the system was calibrated by measuring the critical angles of total internal reflection at a fused-silica (n = 1.456672, [lo]) and an LiF (n=1.391007, [ll]) plate. The accuracy of the measured Nzp) values is 6N= 1 X 10-4.ThedifferenceAN,=N$) -N$’ can, however, be determined with a much higher accuracy of S(AN) = 1 x 10-5.
6. Experimental
The samples were SiO, films of thicknesses d= 29.5 p,m on (100) 3i wafers; the films had been thermally grown by wet oxidation at temperatures of T= 1000-l 100°C and afterwards cooled down to room temperature To = 20°C with a rate of a few degrees per minute. With the new opto-electronic set-up we measured the positions N As’ and N $” of the reflectance minima in s- and p-polarized light of wavelength h = 645 nm (for an example, see Fig. 3), and determined the film parameters n,, n,,, and d. For all Si02 films we found that Ng’ > Ng’ . From Eq. (8)) we
1500
Fig. 2. Measurement set-up. LED, light-emitting diode;,Pol, polarizer (rotatable) ; L,, L, La, lenses of focal lengths fi = 30 mm, fi = 40 mm, fs = 60 mm; P, BK7 prism; I, immersion liquid (benzyl alcohol); CCD, diode array; ADC, analog/digital converter; PC, personal computer. cur= 69”, median angle of incidence; G+= 12’, angular width of convergent beam.
results
1600
1700
1800
1900
2000
Pixel number
Fig. 3. Experimental angular-reflectance spectra, i.e., R”’ versus pixel number of CCD. Reflectance minima m = 1,2 p,m thick SiOz film on Si (sample no. 3). The shift of about between the minima of different polarizations (and same is a clear indication for the anisotropy of the film.
and Rep’ for a 3.5 25 pixels order m)
W. Lukosz, P. Pliska / Optics Communications I I7 (I 995) l-7
conclude that neo > IZ,,,and from Eq. ( 17) that u,, < 0, i.e., the stresses are compressive. The refractive indices *, and neo of the SiOZ film are greater than that (n = 1.456672 [ lo] ) of fused bulk silica. The detailed results are given in Table 1. The thickness values d agree very well with the results d’ of an independent film-thickness measurement with a mechanical stylus instrument (Alpha-Step 200 from Tencor Instruments). For the latter measurement part of the SiO, film is removed by etching; the stylus measures the step height between the surface of the SiOZ film and the Si substrate. The sample wafers had different dotations and were oxidized under somewhat different conditions. Only the 3.5 pm thick SiOZ film (sample no. 3) had been grown on a wafer polished on both sides; we use SiOZ films of this thickness in our experiments on acoustical and biochemical integrated-optical sensors [3,4]. Table 2 contains values of the compressive stresses ( -a,,) calculated from the anisotropy An = 12, - n, with Eq. ( 19). The anisotropy is approximately~=(0.8-1.1)X10-3.Thestresseslieinthe interval -o,, = 215 - 316 MPa, which is in good agreement with the values -o,, =240 - 320 MPa reported by Ref. [ 121. Part of the stresses in the SiOZ film are caused by (slowly) cooling the oxidized wafer from the oxidation temperature of T= 1000-l 100°C down to room temperature of To= 20°C due to the different thermalexpansion coefficients cU,i, and Lysi. The thick Si substrate contracts more than a free SiOZ film would. Consequently, a strain ell = 1%iO*
-
(21)
asil CT- To)
5
Table 2 Compressive stresses ( - o,, ) in the SiOz films determined from the optical anisotropies An, and measured isotropic refractive indices n for the same samples as in Table 1. The stress values ( - (r,, ) are compared with the values ( - u{, ) measured with the bending-plate method of Stoney; the measured n values agree well with the value n = 1.456672 for fused bulk silica. Sample no.
--l1 (MPa)
- (T{, (MPa)
n
1 2 3 4 5 6 7 8 9 Average
215 288 282 271 291 316 316 316 294 288
270 332 326 351 321 348 401 351 367 341
1.45716 1.45693 1.45673 1.45694 1.45712 1.45663 1.45652 1.45640 1.45686 1.45681
develops, where the (Y’Sare the median values of the thermal-expansion coefficients in this temperature range. Thus, according to Eq. (ll), the stress (T,,= E( 1 - V) -‘e,, develops. With asio2 = 0.54 X 10-6K-’ [13],cusi=3.97X10-6K-’ [14],andwith the elastic constants [ 81 of silica E =76 GPa, V= 0.164, we calculate the thermal-stress value (T,,= - 320 MPa. There are residual stresses which depend on the manner of growth of the film and on the stress relief by tempering (i.e., on the cooling rate) [ 15,161. In order to determine the internal stresses (+,, in the SiO, films by an independent measurement, we used the well-known ‘bending-plate’ method [ 5,171. We measured the radius of curvature R of the wafer inter-
Table 1 Measured thicknesses d, ordinary and extraordinary refractive indices n, and n,, respectively, and anisotropies An = n, - n, of SiOz films on ( 100) Si wafers. For masons of comparison, the thicknesses d’ measured with a mechanical stylus instrument are listed. Sample no.
d (pm)
d’ (pm)
n0
Q0
An (10-a)
1 2 3 4 5 6 7 8 9 Average
2.008 3.051 3.524 4.054 5.003 5.608 6.994 7.809 9.460
2.010 3.055 3.530 4.085 5.005 5.565 6.955 7.795 9.430 -
1.45820 1.45833 1.45810 1.45826 1.45853 1.45816 1.45805 1.45793 1.45829 1.45821
1.45896 1.45935 1.45910 1.45922 1.45956 1.45928 1.45917 1.45905 1.45933 1.45922
0.76 1.02 1.00 0.96 1.03 1.12 1.12 1.12 1.04 1.01
W. Lukosz, P. Pliska / Optics Communications I I7 (1995) I-7
6
ferometrically in a Michelson interferometer after the SiOZ film on the back side of the wafer had been removed by etching. (In the optical measurements on the prism, the wafers being oxidized on both sides are symmetrically and, consequently, flat.) The ‘Stoney formula’ [5,17] gives the connection between R and g;, (the apostrophe distinguishes the stress values determined by the bending-plate method from those calculated from the anisohopy An) : a;, =EsiD2/[6dR(1-
vsi)] ,
(22)
where E,i is Young’s modulus and Vsi Poisson’s ratio of the Si substrate, D the substrate thickness and d the film thickness. For a compressive stress u,, < 0, the film tends to expand, and this leads to a convex curvature (i.e., R < 0) seen from the side of the film. For the ( 100) wafer the stress is along the [ 1 lo] direction and the contraction along the [ ii01 direction; therefore, we use the values E,i = 170.7 GPa, Vsi= 0.057 [ 181. The experimental results for a;, calculated from Eq. (22) are compared from the anisotropy
with the stresses u,, determined in Table 2. The 1CT;,1-values are
systematically between lO-30% higher than the 1ull Ivalues; this discrepancy seems to indicate that the elasto-optic constants (C, - C,) or C of the SiO* films are somewhat different from the values for fused bulk silica. However, (T,, and ai, are correlated; a linear regression analysis yielded a correlation coefficient of r=0.81. The refractive indices n calculated from Eq. ( 18) are also listed in Table 2. They agree well with the value n = 1.456672 [ lo] for fused bulk silica. A complete agreement is not expected because the refractive indices of the films can slightly be influenced, e.g., by the dopants in the Si wafers. In Table 3 we test the prediction of Eq. (8) that the ratio N m (PI/N 2) of the measured positions N $’ and N,$) of the reflectance minima of the same order m is the same a 9.5 ym m = l-9 satisfied,
for any order m = 1,2, . . . . In the example of thick SiO, film (sample no. 9) where orders were measured, the prediction is quite well with
Nm (p)/N n, (‘) = n,/n,
= 1.0007 .
Table 3 Experimentally measured ratios NzP’/Nc’ of the positions of the reflectance minima of the same order m in the example of a 9.5 pm thick SiOl film (sample no. 9) for m = 1-9. Test of the theoretical prediction that N$‘)/N!$ =n,/n,= 1.00071. ?n
N,$P’INc’
1 2 3 4 5 6 7 8 9 Average
1.00072 1.00070 1.00069 1.00068 1.00070 1.00071 1.00068 1.00068 1.00076 1.00070
7. Conclusions We developed a method to determine the thickness d, the refractive indices n, and n,,, and the anisotropy An=&,n, of thermally grown SiOz films on Si wafers. All investigated SiOa films were optically anisotropic and uniaxial with their optic oriented normally to the film. The observed anisotropy An can be explained by the compressive stress g,, in the film which influences n, and neo via the photoelastic effect. Consequently, the measurement of An offers the possibility to determine (T,,. From the measured anisotropies of An= (0.8- 1.1) X 10e3, we determined the stresses to be ( -a,,) = 125-316 MPa. These stress values are lO-30% lower than those determined independently by the bending-plate method, possibly because the elasto-optic constants of the films are not well known; nevertheless, the results of the two methods are correlated. The new method, suitable for film thickness of d = 215 p,m, is non-destructive. A disadvantage is that an immersion liquid has to be used to establish optical contact between the SiOJSi wafer and the prism; to obtain reliable results, it must be ensured that the immersion layer be spatially uniform. The method described can obviously be applied to any other lowindex and non-absorbing thin film on a high-index substrate. Acknowledgements We thank E. Hausammann and G. Natterer for the mechanical construction and the electronics, respec-
W. Lukosz, P. Pliska /Optics
tively, of the measuring set-up. We also thank A.C. Amrein and other students who helped developing and testing the set-up. We are indebted to Q. Lia from the Swiss Federal Institute of Technology Zurich and B.J. Curtis from Paul Scherrer Institute Zurich for supplying some of the samples with various film thicknesses.
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Communications I I7 (I 99.5) I-7
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