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Determination of thickness and refractive index of SiO2 films on silicon wafers using an Abbe refractometer
OPTICS COM MUNICATIONS
Optics Communications 85 ( 1991 ) 381-384 North-Holland
Determination of thickness and refractive index of silicon wafers using an Abbe refractometer
SiO2 films on
W. L u k o s z a n d P. P l i s k a Optics Laboratory. Swtss Federal lnstttute of Technolog), Ziirich. 8093 Ziirich, Switzerland Received 14 May 1991
We have demonstrated a simple yet accurate method determining the thickness and refractive index of SiO2 films on silicon substrates. The method requires only an Abbe refractometer which is used without any modification.
1. Introduction
Silicon dioxide (SIO2) films on silicon (Si) wafers have m a n y applications in microelectronics [ 1 ]. In the field of integrated optics, interest is growing for waveguides on Si wafers. Because of the absorption of Si at visible and near infrared wavelengths, a few micrometer thick buffer layers of SiO2 are required between the waveguides and the Si substrates [2]. We had grown such SiO2 buffer layers by thermal oxidation; our object was to determine the thicknesses d and refractive index n of the SiO2 films. Instruments for that purpose are commercially available but have the following disadvantages: (i) Ellipsometers ( m e a s u r i n g at one fixed angle of incidence and at one fixed wavelength 2) are widely used for measurements of very thin film,s with thicknesses d~< 100-300 rim, while for greater thicknesses they are not suitable because of the periodicity of the reflection coefficients with ( n d / 2 ) cos o~, where o~ is thc angle of incidence in the film, and because it is generally difficult to determine both n and d so that a value of n has to be assumed in order to determine d. (ii) Spectro-photometric m e a s u r e m e n t s of thc reflectance at near normal incidence versus wavelength 2 are suitable for d e t e r m i n a t i o n of the optical thickness (rid). However, for the d e t e r m i n a t i o n of the geometrical thickness d the n value has to be known,
2. The new method
The object of this c o m m u n i c a t i o n is to reporl on a simple method for d e t e r m i n i n g both n and d which requires only a commercially available Abbe refractometer (for visual observation). The refractometer is used without any modification. The method works as follows: the Si wafer with an SiO2 film is laid upside down with a droplet of immersion liquid of high refractive index rh> n on the prism P of an Abbe refractometer (fig. 1). It is illuminated in reflection from the side of the prism P with diffuse monochromatic light, or alternatively with white light. In the eyepiece of the i n s t r u m e n t we see a n u m b e r of sharp dark lines in a bright field (fig. 2). The po-
Fig. 1. Schematic of the configuration. P, prism of Abbe refractometer of high refractive index rip; I, immersion liquid of high refractive index nt; SIO2, film of refractive index n and thickness d to be determined: Si, silicon wafer; c~p,cq, c~, angles of incidence in the media P, 1, and SiO2, respectively.
0030-4018/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
381
Volume 85. number 5.6
OPTICS COMMIJNICATIONS 1.00
m-4 i
0
m=2
1.45
1.46 I
Fig. 2. Schematic diagram of dark resonance lines observed visually in the eyepiece of an Abbe refractometer. Also visible is a scale as well known from the conventional use of the Abbe refractometer, from which the N values of the resonance lines are read.
sitions N,, of the dark lines n u m b e r e d m = 1,2 .... are read on the scale which is also visible in the cyepiece. From the readings N,,, of at least thc two dark lines r n = 1 and 2, the values of n and d are determined from a simple formula given below. As a m o n o c h r o m a t i c light source we used a spectral lamp, such as a sodium lamp ( N a - D line with 2D= 589.3 n m ) or alternatively a laser, in particular a HeNe laser ( 2 = 6 3 2 . 8 n m ) . I f a laser is used, rotation of a diffusor between the laser and the prism P eliminates the laser speckle in the field of view and improves the visibility of the dark lines. Alternatively white light of, for example, a halogen lamp can be used. T h e n the resonance lines m = 1, 2 .... can be made nearly achromatic by adjusting the compensator C. The Abbe refractometer (Zeiss type B2) had a prism P of refractive index n p = 1.740 (at 2t)= 589.3 n m ) . This prism has a refractive index measuring range of 1 . 3 0 < N < 1.71. The i m m e r s i o n liquid was methylene iodide (n~= 1.748). The use of an immersion liquid of lower refractive index n~ is not reco m m e n d e d , since the contrast of resonance lines is reduced. The film I of immersion liquid has to be thin so that it can be assumed to be approximately plan-parallel. The refractive index n~ does not enter into the d e t e r m i n a t i o n o f n and d, nor does the complex refractive index ns~ of the Si wafer.
382
[ I
dL 1.06pm :
~ I
i
t
~
I
i
,
,
i
I
i
1.35 1.40 1.45 anglevariable N
rrl-1
1.44
0.75 0.50 0.25
m - 3
.43
1 October 1991
Fig. 3. Calculated reflectances R "~ for s-polarized light of two SiO2 films (n= 1.46, d= 1.06/am and d=3.38 lam, respectively, at 2-589.3 nm) between a silicon wafer (with refractive index ns,= 3.97 + i0.024 ) and an immersion liquid (nx= 1.748 ) versus variable N-z-_n~ sin at = nf, sin ctv.
3. Theory Fig. 3 shows the calculated reflectances R ~ for spolarized light of two SiO2 films of different thicknesscs d between the immersion liquid I and the Si wafer versus the variable
N=_n~, sin ap=n~ sin a~.
(1)
wherc av is the angle of incidence of the incident light in the prism P, and at is the corresponding angle in the m e d i u m I. According to the law of refraction, for N~ n, the transmitted wave in the SiO2 film is an evanescent wave; this is the regime of total internal reflection where R~'°)= 1 for s- and p-polarization. The reading on the scale visible in the eyepiece of the Abbe refractometer directly givcs the value of N at a certain wavelength, normally at the Na-D line 2D= 589.3 nm; for other wavelengths ). the reading has to be corrected by the known dispersion of the prism P. In the conventional use of the Abbe refractometer [3 ]wherc the refractive index n of a bulk solid (or liquid) m e d i u m is measured, a sharp borderline between the dark and bright fields corrcsponding to the angle of total internal refraction is observed; its location read on the scale directly gives the value of the refractive index n. The calculated reflectances R
Volume 85, n u m b e r 5,6
OPTICS COMMUNICATIONS
onances o f the field inside the SiO2 film; they occur if
A~/27r=2(nd/A) cos o~ + ( 2 n ) - ' [6,(o~)+as,(O~) ] = f i t ,
(2)
where ritz=0, 1.... is the order o f the interferences and d~'°)(o~) are the phase shifts upon reflection o f a plane wave incident from the SiO2 film onto the med i u m j = l and j = S i , respectively. The phase shifts a~.P~(~) can be calculated from Fresnel coefficients. For reflection at the nonabsorbing denser med i u m I we have 6 [ ~ ( o z ) = - n at all angles o~, and d[P~(o~)= - z r at o~>C~Br~,~,=arctan(nx/n). For reflection at the absorbing silicon, we have for nearly grazing incidence, i.e. for N ~ n , or o~ , 9 0 °, in very good a p p r o x i m a t i o n d~'P~ ~ - n . This result holds independent o f polarization and practically independent o f the imaginary part o f the complex refractive index o f silicon, i.e., o f its absorption. Inserting d j + ~ s , = - 2 z t for both s- and p-polarization into eq. ( 2 ) , we obtain
2(d/A)(n2-N~,,)'/2=m ,
(3)
where m - fit + 1 = 1, 2 ..... The physical reasons for the sharpness o f the resonance lines are: ( i ) the SiO2 film is a low-index film which is sandwiched between two high-index m e d i a (the refractive index o f Si is ns,= 3 . 9 7 + i 0 . 0 2 4 at )to [4] ); (ii) the resonances occur at N,, values which arc not much smaller than n, corresponding to angles of incidence a a p p r o a c h i n g 90 °. But for grazing incidence the Fresnel reflection coefficients tend to r"-°~(o~) , 1. Therefore. the plane waves in the low index film are efficiently reflected back and forth between the two interfaces and multiple beam interferences occur. The system is analogous to a FabryPerot resonator with a high finesse. Eq. ( 3 ) can be rewritten in the form
N~, =n 2 - (mA/2d) 2 .
(4)
Consequently, from the values N,, and Nm for two orders rn and m ' (where m, m ' = 1, 2 .... ) we obtain
d/2= ½[(m2-m'2)/(NA,,c - N ~,) ] '/2 ,
(5)
n=[(m2N~,.--m'ANA,,)/(mA--m'2)]]/2.
(6)
In cases where m o r e than two resonance lines can be
1 October 1991 m
1 2 3 4 5 2.2 E , , , , ~e'~~j.~d..3.38tJm ~ j- . 1.45 °'E 2.0 i--'~ ~ 1.40 E 1.8 1.35 Z i, ,'~, fdl,~1,.,06~7, , , ,1 Z 1.30 1.6
0
10
20
30
m2
Fig. 4. E x p e r i m e n t a l results for two different SiO2 films. Shown are the m e a s u r e d values N 2,, (in s-polarized light) versus m 2 for the observed resonance lines m = I and 2 for the t h i n n e r film and m = 1..... 5 for the thicker film. F r o m a linear regression analysis we o b t a i n e d the results d = 1.06 tam and n = 1.4603. and d = 3.38 lam and n = 1.4624, respectively.
observed, we used an evaluation as shown in fig. 4 plotting N ~,, versus m 2.
4. Accuracy and measuring range With the Abbe refractometer the N values can be visually read with an accuracy of 6 N z 2 × 10-4. From cqs. (4) and ( 5 ) we derive that the relative error with which d can be d e t e r m i n e d is
8d/d= x/-2 ( 2 d / A ) A n 6 N / I m A - m' 21 .
(7)
F r o m eq. ( 6 ) it follows that the error 6n in the refractive index n is - independent o f d/2 -
~ n = S N ( m 4 + m '4) ' / 2 / [ m a - m ' A I .
(8)
In an example where d/A= 6 and n = 1.46, we find, considering the special case m = 1 and m ' = 2, that 6d/d=2% and 6 n = 3 × 10 -4. For a 3.38 lam thick SiO2 film we observed five resonance lines. In this case, the evaluation, i.e., the linear regression analysis (see fig. 4), gave the correlation coefficient Ir1=0.999975. Thicknesses d d e t e r m i n e d at different wavelengths agreed within 6d/d= 1.3%. In the above given theory we assumed that the SiO2 films are homogeneous and isotropic. The theory showed that the positions N,,, o f the resonance lines m = 1,2 .... practically coincide for s- and p-polarized light; the calculated differences N~,,~ - N ~ # ~ are at least one o r d e r o f magnitude smaller than the reso383
Volume 85, number 5,6
OPTICS C O M M U N I C A T I O N S
lution ~ N ~ 2 X I 0 -4 o f the refractometer. But actually we observed that the resonance lines show a p o l a r i z a t i o n - d e p e n d e n t splitting somewhat larger than 8N. Possibly the SiO2 films are slightly anisotropic. We discuss the range o f film thicknesses d thai can bc measured. Thc lower limit dmi, is d e t e r m i n e d by thc lower limit Nmm o f the measuring range o f the Abbe refractometer. To d e t e r m i n e both n and d, at least the two resonance lines m = 1 and m = 2 have to be observed. U n d e r this condition we derive from eq. ( 4 ) that d , , , , , / ) . = ( n 2 - N ~ i , ) -~/'-. With n = 1.46 and N m , , = l . 3 1 we have d , m , / 2 , ~ l . 6 , and thus dm,,~ 1 p.m at 2 = 0 . 6 p.m. If only the resonance line m = 1 is observed, the thickness d can only be determined under the a s s u m p t i o n of an n value: the lower thickness limit is then d, li,/)t ,~ 0.8, or dram -~ 0.5 p.m at 2 = 0 . 6 p.m. For greater thicknesses d the resonancc lines move closer together. Therefore, the upper limit d,,,~ is d e t e r m i n e d by the resolution 5A'~ 2 × 1 0 - 4 o f the refractometer. F r o m eq. ( 4 ) it follows that d,~,~/;,.-~ ( ~ n SN) 1/2 .~ 36, and d .... ~ 22 p.m at ) . = 0 . 6 p.m.
5. Dispersion of SiO2 film With m e a s u r e m e n t s in white light, the value o f the dispersion On~02 o f the SiO2 film's refractive index n can be d e t e r m i n e d as follows: by a suitable setting C o f the c o m p e n s a t o r the resonance line m = 1 becomes nearly achromatic. With the value C read on the compensator, from a table supplied by the manufacturer o f the refractometer the refractive index diffcrence between the F r a u n h o f e r lines F and C ( 2 v = 4 8 6 . 1 nm, 2 c = 6 5 6 . 3 nm; A A = ; t ~ - - 2 c ) is determined. In our case the reading C yields the difference A N - N ( 2 v ) - N ( 2 c ) , i.e., a value for the dispersion A N / A 2 . F r o m eq. ( 4 ) it follows that
N ON~02= n 0 n / 0 2 - 2 ( m / 2 d ) 2 .
(9 )
In the regime o f normal dispersion we have On/O2 < 0 and the second term in eq. ( 9 ) is an i m p o r t a n t cor-
384
1 October 1991
rection o f the same sign. Eq. ( 9 ) shows that the resonance lines o f higher o r d e r m show higher dispersion. F r o m eqs. ( 4 ) and ( 9 ) we derive that 0 n / 0 2 = [ 1 - ( m 2 / 2 n d ) 2 ] ~/'- ON,,,/ O).
+ (2/n)(m/2d)2.
(10)
With the measured values o f n, d, and AN, the dispcrsion o f the refractive index o f the SiO2 film can be calculated from eq. (10) as O n / O 2 ~ A n / A 2 , where An-n(2v)-n(2c). O u r m e a s u r e m e n t s gave a typical value o f A n / A ) . = - 4 . 2 × 10 -5 nm ~; for fused silica the analogously defined dispersion value is - 4 . 0 ; 4 10 -5 nm - I [5].
6. Conclusion We d e m o n s t r a t e d a simple m e t h o d to determine both the thickness and refractive index o f SiO2 films on silicon substrates. The method is suitable for thicknesses d in the range of d/;t.~ 1.6-3.6, i.e., for d ~ 1-22 p.m at 2 = 0 . 6 p.m. The attractive feature o f the method is that it is not only simple but also very' accurate; only an Abbc refractometer is required. A disadvantage is that it is not a contactless method: contact with the prism using an immersion liquid is required. The m e t h o d described can obviously be applied to any other low-index and non-absorbing thin film on a high-index substrate.
References [ 1 ] L.E. Katz, in: VLSI Technology, cd. S.M. Sze (McGraw-Hill, New York, 1983 ) ch. 3. [ 2 ] H. Nishihara, M. Haruna and T. Suhara, Optical integrated circuits (McGraw-Hill, New York, 1985 ) ch. 6.7. [ 3 ] G.E. Fishier, in: Applied optics and engineering, Vol. IV, part l, ed. R. Kingslake (Academic Press, New York and London, 1967) pp. 370ff. [4] D.E. Aspnes, in: Properties of silicon (INSPEC, EMIS Datarevicws Series No. 4, 1988) ch. 2.6. [ 5 ] l.H. Malitson, J. Opt. Soc. Am. 55 ( 1965 ) 1205.
Optics Communications 85 ( 1991 ) 381-384 North-Holland
Determination of thickness and refractive index of silicon wafers using an Abbe refractometer
SiO2 films on
W. L u k o s z a n d P. P l i s k a Optics Laboratory. Swtss Federal lnstttute of Technolog), Ziirich. 8093 Ziirich, Switzerland Received 14 May 1991
We have demonstrated a simple yet accurate method determining the thickness and refractive index of SiO2 films on silicon substrates. The method requires only an Abbe refractometer which is used without any modification.
1. Introduction
Silicon dioxide (SIO2) films on silicon (Si) wafers have m a n y applications in microelectronics [ 1 ]. In the field of integrated optics, interest is growing for waveguides on Si wafers. Because of the absorption of Si at visible and near infrared wavelengths, a few micrometer thick buffer layers of SiO2 are required between the waveguides and the Si substrates [2]. We had grown such SiO2 buffer layers by thermal oxidation; our object was to determine the thicknesses d and refractive index n of the SiO2 films. Instruments for that purpose are commercially available but have the following disadvantages: (i) Ellipsometers ( m e a s u r i n g at one fixed angle of incidence and at one fixed wavelength 2) are widely used for measurements of very thin film,s with thicknesses d~< 100-300 rim, while for greater thicknesses they are not suitable because of the periodicity of the reflection coefficients with ( n d / 2 ) cos o~, where o~ is thc angle of incidence in the film, and because it is generally difficult to determine both n and d so that a value of n has to be assumed in order to determine d. (ii) Spectro-photometric m e a s u r e m e n t s of thc reflectance at near normal incidence versus wavelength 2 are suitable for d e t e r m i n a t i o n of the optical thickness (rid). However, for the d e t e r m i n a t i o n of the geometrical thickness d the n value has to be known,
2. The new method
The object of this c o m m u n i c a t i o n is to reporl on a simple method for d e t e r m i n i n g both n and d which requires only a commercially available Abbe refractometer (for visual observation). The refractometer is used without any modification. The method works as follows: the Si wafer with an SiO2 film is laid upside down with a droplet of immersion liquid of high refractive index rh> n on the prism P of an Abbe refractometer (fig. 1). It is illuminated in reflection from the side of the prism P with diffuse monochromatic light, or alternatively with white light. In the eyepiece of the i n s t r u m e n t we see a n u m b e r of sharp dark lines in a bright field (fig. 2). The po-
Fig. 1. Schematic of the configuration. P, prism of Abbe refractometer of high refractive index rip; I, immersion liquid of high refractive index nt; SIO2, film of refractive index n and thickness d to be determined: Si, silicon wafer; c~p,cq, c~, angles of incidence in the media P, 1, and SiO2, respectively.
0030-4018/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
381
Volume 85. number 5.6
OPTICS COMMIJNICATIONS 1.00
m-4 i
0
m=2
1.45
1.46 I
Fig. 2. Schematic diagram of dark resonance lines observed visually in the eyepiece of an Abbe refractometer. Also visible is a scale as well known from the conventional use of the Abbe refractometer, from which the N values of the resonance lines are read.
sitions N,, of the dark lines n u m b e r e d m = 1,2 .... are read on the scale which is also visible in the cyepiece. From the readings N,,, of at least thc two dark lines r n = 1 and 2, the values of n and d are determined from a simple formula given below. As a m o n o c h r o m a t i c light source we used a spectral lamp, such as a sodium lamp ( N a - D line with 2D= 589.3 n m ) or alternatively a laser, in particular a HeNe laser ( 2 = 6 3 2 . 8 n m ) . I f a laser is used, rotation of a diffusor between the laser and the prism P eliminates the laser speckle in the field of view and improves the visibility of the dark lines. Alternatively white light of, for example, a halogen lamp can be used. T h e n the resonance lines m = 1, 2 .... can be made nearly achromatic by adjusting the compensator C. The Abbe refractometer (Zeiss type B2) had a prism P of refractive index n p = 1.740 (at 2t)= 589.3 n m ) . This prism has a refractive index measuring range of 1 . 3 0 < N < 1.71. The i m m e r s i o n liquid was methylene iodide (n~= 1.748). The use of an immersion liquid of lower refractive index n~ is not reco m m e n d e d , since the contrast of resonance lines is reduced. The film I of immersion liquid has to be thin so that it can be assumed to be approximately plan-parallel. The refractive index n~ does not enter into the d e t e r m i n a t i o n o f n and d, nor does the complex refractive index ns~ of the Si wafer.
382
[ I
dL 1.06pm :
~ I
i
t
~
I
i
,
,
i
I
i
1.35 1.40 1.45 anglevariable N
rrl-1
1.44
0.75 0.50 0.25
m - 3
.43
1 October 1991
Fig. 3. Calculated reflectances R "~ for s-polarized light of two SiO2 films (n= 1.46, d= 1.06/am and d=3.38 lam, respectively, at 2-589.3 nm) between a silicon wafer (with refractive index ns,= 3.97 + i0.024 ) and an immersion liquid (nx= 1.748 ) versus variable N-z-_n~ sin at = nf, sin ctv.
3. Theory Fig. 3 shows the calculated reflectances R ~ for spolarized light of two SiO2 films of different thicknesscs d between the immersion liquid I and the Si wafer versus the variable
N=_n~, sin ap=n~ sin a~.
(1)
wherc av is the angle of incidence of the incident light in the prism P, and at is the corresponding angle in the m e d i u m I. According to the law of refraction, for N~
Volume 85, n u m b e r 5,6
OPTICS COMMUNICATIONS
onances o f the field inside the SiO2 film; they occur if
A~/27r=2(nd/A) cos o~ + ( 2 n ) - ' [6,(o~)+as,(O~) ] = f i t ,
(2)
where ritz=0, 1.... is the order o f the interferences and d~'°)(o~) are the phase shifts upon reflection o f a plane wave incident from the SiO2 film onto the med i u m j = l and j = S i , respectively. The phase shifts a~.P~(~) can be calculated from Fresnel coefficients. For reflection at the nonabsorbing denser med i u m I we have 6 [ ~ ( o z ) = - n at all angles o~, and d[P~(o~)= - z r at o~>C~Br~,~,=arctan(nx/n). For reflection at the absorbing silicon, we have for nearly grazing incidence, i.e. for N ~ n , or o~ , 9 0 °, in very good a p p r o x i m a t i o n d~'P~ ~ - n . This result holds independent o f polarization and practically independent o f the imaginary part o f the complex refractive index o f silicon, i.e., o f its absorption. Inserting d j + ~ s , = - 2 z t for both s- and p-polarization into eq. ( 2 ) , we obtain
2(d/A)(n2-N~,,)'/2=m ,
(3)
where m - fit + 1 = 1, 2 ..... The physical reasons for the sharpness o f the resonance lines are: ( i ) the SiO2 film is a low-index film which is sandwiched between two high-index m e d i a (the refractive index o f Si is ns,= 3 . 9 7 + i 0 . 0 2 4 at )to [4] ); (ii) the resonances occur at N,, values which arc not much smaller than n, corresponding to angles of incidence a a p p r o a c h i n g 90 °. But for grazing incidence the Fresnel reflection coefficients tend to r"-°~(o~) , 1. Therefore. the plane waves in the low index film are efficiently reflected back and forth between the two interfaces and multiple beam interferences occur. The system is analogous to a FabryPerot resonator with a high finesse. Eq. ( 3 ) can be rewritten in the form
N~, =n 2 - (mA/2d) 2 .
(4)
Consequently, from the values N,, and Nm for two orders rn and m ' (where m, m ' = 1, 2 .... ) we obtain
d/2= ½[(m2-m'2)/(NA,,c - N ~,) ] '/2 ,
(5)
n=[(m2N~,.--m'ANA,,)/(mA--m'2)]]/2.
(6)
In cases where m o r e than two resonance lines can be
1 October 1991 m
1 2 3 4 5 2.2 E , , , , ~e'~~j.~d..3.38tJm ~ j- . 1.45 °'E 2.0 i--'~ ~ 1.40 E 1.8 1.35 Z i, ,'~, fdl,~1,.,06~7, , , ,1 Z 1.30 1.6
0
10
20
30
m2
Fig. 4. E x p e r i m e n t a l results for two different SiO2 films. Shown are the m e a s u r e d values N 2,, (in s-polarized light) versus m 2 for the observed resonance lines m = I and 2 for the t h i n n e r film and m = 1..... 5 for the thicker film. F r o m a linear regression analysis we o b t a i n e d the results d = 1.06 tam and n = 1.4603. and d = 3.38 lam and n = 1.4624, respectively.
observed, we used an evaluation as shown in fig. 4 plotting N ~,, versus m 2.
4. Accuracy and measuring range With the Abbe refractometer the N values can be visually read with an accuracy of 6 N z 2 × 10-4. From cqs. (4) and ( 5 ) we derive that the relative error with which d can be d e t e r m i n e d is
8d/d= x/-2 ( 2 d / A ) A n 6 N / I m A - m' 21 .
(7)
F r o m eq. ( 6 ) it follows that the error 6n in the refractive index n is - independent o f d/2 -
~ n = S N ( m 4 + m '4) ' / 2 / [ m a - m ' A I .
(8)
In an example where d/A= 6 and n = 1.46, we find, considering the special case m = 1 and m ' = 2, that 6d/d=2% and 6 n = 3 × 10 -4. For a 3.38 lam thick SiO2 film we observed five resonance lines. In this case, the evaluation, i.e., the linear regression analysis (see fig. 4), gave the correlation coefficient Ir1=0.999975. Thicknesses d d e t e r m i n e d at different wavelengths agreed within 6d/d= 1.3%. In the above given theory we assumed that the SiO2 films are homogeneous and isotropic. The theory showed that the positions N,,, o f the resonance lines m = 1,2 .... practically coincide for s- and p-polarized light; the calculated differences N~,,~ - N ~ # ~ are at least one o r d e r o f magnitude smaller than the reso383
Volume 85, number 5,6
OPTICS C O M M U N I C A T I O N S
lution ~ N ~ 2 X I 0 -4 o f the refractometer. But actually we observed that the resonance lines show a p o l a r i z a t i o n - d e p e n d e n t splitting somewhat larger than 8N. Possibly the SiO2 films are slightly anisotropic. We discuss the range o f film thicknesses d thai can bc measured. Thc lower limit dmi, is d e t e r m i n e d by thc lower limit Nmm o f the measuring range o f the Abbe refractometer. To d e t e r m i n e both n and d, at least the two resonance lines m = 1 and m = 2 have to be observed. U n d e r this condition we derive from eq. ( 4 ) that d , , , , , / ) . = ( n 2 - N ~ i , ) -~/'-. With n = 1.46 and N m , , = l . 3 1 we have d , m , / 2 , ~ l . 6 , and thus dm,,~ 1 p.m at 2 = 0 . 6 p.m. If only the resonance line m = 1 is observed, the thickness d can only be determined under the a s s u m p t i o n of an n value: the lower thickness limit is then d, li,/)t ,~ 0.8, or dram -~ 0.5 p.m at 2 = 0 . 6 p.m. For greater thicknesses d the resonancc lines move closer together. Therefore, the upper limit d,,,~ is d e t e r m i n e d by the resolution 5A'~ 2 × 1 0 - 4 o f the refractometer. F r o m eq. ( 4 ) it follows that d,~,~/;,.-~ ( ~ n SN) 1/2 .~ 36, and d .... ~ 22 p.m at ) . = 0 . 6 p.m.
5. Dispersion of SiO2 film With m e a s u r e m e n t s in white light, the value o f the dispersion On~02 o f the SiO2 film's refractive index n can be d e t e r m i n e d as follows: by a suitable setting C o f the c o m p e n s a t o r the resonance line m = 1 becomes nearly achromatic. With the value C read on the compensator, from a table supplied by the manufacturer o f the refractometer the refractive index diffcrence between the F r a u n h o f e r lines F and C ( 2 v = 4 8 6 . 1 nm, 2 c = 6 5 6 . 3 nm; A A = ; t ~ - - 2 c ) is determined. In our case the reading C yields the difference A N - N ( 2 v ) - N ( 2 c ) , i.e., a value for the dispersion A N / A 2 . F r o m eq. ( 4 ) it follows that
N ON~02= n 0 n / 0 2 - 2 ( m / 2 d ) 2 .
(9 )
In the regime o f normal dispersion we have On/O2 < 0 and the second term in eq. ( 9 ) is an i m p o r t a n t cor-
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1 October 1991
rection o f the same sign. Eq. ( 9 ) shows that the resonance lines o f higher o r d e r m show higher dispersion. F r o m eqs. ( 4 ) and ( 9 ) we derive that 0 n / 0 2 = [ 1 - ( m 2 / 2 n d ) 2 ] ~/'- ON,,,/ O).
+ (2/n)(m/2d)2.
(10)
With the measured values o f n, d, and AN, the dispcrsion o f the refractive index o f the SiO2 film can be calculated from eq. (10) as O n / O 2 ~ A n / A 2 , where An-n(2v)-n(2c). O u r m e a s u r e m e n t s gave a typical value o f A n / A ) . = - 4 . 2 × 10 -5 nm ~; for fused silica the analogously defined dispersion value is - 4 . 0 ; 4 10 -5 nm - I [5].
6. Conclusion We d e m o n s t r a t e d a simple m e t h o d to determine both the thickness and refractive index o f SiO2 films on silicon substrates. The method is suitable for thicknesses d in the range of d/;t.~ 1.6-3.6, i.e., for d ~ 1-22 p.m at 2 = 0 . 6 p.m. The attractive feature o f the method is that it is not only simple but also very' accurate; only an Abbc refractometer is required. A disadvantage is that it is not a contactless method: contact with the prism using an immersion liquid is required. The m e t h o d described can obviously be applied to any other low-index and non-absorbing thin film on a high-index substrate.
References [ 1 ] L.E. Katz, in: VLSI Technology, cd. S.M. Sze (McGraw-Hill, New York, 1983 ) ch. 3. [ 2 ] H. Nishihara, M. Haruna and T. Suhara, Optical integrated circuits (McGraw-Hill, New York, 1985 ) ch. 6.7. [ 3 ] G.E. Fishier, in: Applied optics and engineering, Vol. IV, part l, ed. R. Kingslake (Academic Press, New York and London, 1967) pp. 370ff. [4] D.E. Aspnes, in: Properties of silicon (INSPEC, EMIS Datarevicws Series No. 4, 1988) ch. 2.6. [ 5 ] l.H. Malitson, J. Opt. Soc. Am. 55 ( 1965 ) 1205.