Measurement of thin films using fourier amplitude
View Patent ↗Thin-film thickness and refractive index are measured using the Fourier amplitude of a broadband interferometric spectrum. Due to the smooth nature of the Fourier amplitude as a function of wavelength, as compared to the fast varying Fourier phase conventionally used to measure thickness, increased stability and repeatability of measurement are achieved. As a result, measurements of ultra-thin films with thickness below 100 nm are possible with reliable results.
1. A method for determining a property of a thin film placed over a substrate, comprising the following steps:
producing a test correlogram of said thin film over the substrate;
generating a test linear transform of the test correlogram and extracting a test transform amplitude from the test linear transform;
deriving a test-beam amplitude component of said test transform amplitude by removing a reference amplitude function, wherein the reference amplitude function comprises the product of the amplitude components contributed by the light source and the reference beam; and
finding a value of said property of the thin film by comparing said test-beam amplitude component to a model test-beam amplitude function.
2. The method of claim 1 , wherein said reference amplitude function is obtained by calibration using a reference substrate having a known reflectivity over a predetermined waveband.
3. The method of claim 2 , wherein said calibration comprises the steps of:
generating a calibration correlogram with a broadband light source included within said predetermined waveband;
generating a calibration linear transform of the calibration correlogram;
extracting a calibration transform amplitude from the calibration linear transform; and
establishing said reference amplitude function from said calibration transform amplitude and said known reflectivity of the reference substrate, said reference amplitude function representing light-source and reference-beam amplitude components of the calibration transform amplitude.
4. The method of claim 3 , wherein said known reflectivity of the reference substrate is substantially constant over said predetermined waveband.
5. The method of claim 3 , wherein said reference amplitude function is established by dividing said calibration transform amplitude by a spectral amplitude of a test beam reflected from said reference substrate during said calibration steps.
6. The method of claim 1 , wherein said test-beam amplitude component is calculated by dividing the test transform amplitude by said reference amplitude function.
7. The method of claim 1 , wherein said finding step is carried out by defining an error function equal to a difference between said test-beam amplitude component and said model test-beam amplitude function, and by minimizing said error function with respect to said property of the thin film.
8. The method of claim 1 , wherein said model test-beam amplitude function is derived from a theoretical expression modeling a reflection from said thin film over the substrate.
9. The method of claim 8 , wherein said theoretical expression is
ρ
(
k
,
d
)
=
r
01
+
r
12
exp
(
-
ⅈ2δ
)
1
+
r
01
r
12
exp
(
-
ⅈ2δ
)
=
b
′
(
k
,
d
)
exp
[
ⅈϕ
(
k
,
d
,
θ
)
]
,
where ρ(k,d) is a coefficient of total reflection, r 01 is a Fresnel coefficient of reflection at an incident-medium to film boundary, r 12 is a Fresnel coefficient of reflection at a film to substrate boundary, δ=2 knd cos θ is a phase difference between consecutive reflections off the film, n is an index of refraction of the film at wavenumber k, θ is a refractive angle in the film, d is a film thickness, and b′(k,d) is said theoretical test-beam amplitude function.
10. The method of claim 1 , wherein said reference amplitude function is derived from an optical model.
11. The method of claim 1 , wherein said linear transform is a Fourier transform.
12. The method of claim 3 , wherein said linear transform is a Fourier transform.
13. The method of claim 1 , wherein said property is a thickness of the thin film.
14. The method of claim 3 , wherein said property is a thickness of the thin film.
15. The method of claim 1 , wherein said property is an index of refraction of the thin film.
16. The method of claim 3 , wherein said property is an index of refraction of the thin film.
17. The method of claim 3 , wherein said linear transform is a Fourier transform; said property is a thickness of the thin film; said reference amplitude function is established by dividing said calibration transform amplitude by a spectral amplitude of a test beam reflected from said reference substrate during the calibration steps; said test-beam amplitude component is calculated by dividing the test transform amplitude by said reference amplitude function; said finding step is carried out by defining an error function equal to a difference between said test-beam amplitude component and said theoretical test-beam amplitude function, and by minimizing said error function with respect to said thickness of the thin film; and said model test-beam amplitude function is derived from a theoretical expression modeling a reflection from said thin film over the substrate.
18. The method of claim 3 , wherein said linear transform is a Fourier transform; said property is an index of refraction of the thin film; said reference amplitude function is established by dividing said calibration transform amplitude by a spectral amplitude of a test beam reflected from said reference substrate during the calibration steps; said test-beam amplitude component is calculated by dividing the test transform amplitude by said reference amplitude function; said finding step is carried out by defining an error function equal to a difference between said test-beam amplitude component and said theoretical test-beam amplitude function, and by minimizing said error function with respect to said index of refraction of the thin film; and said model test-beam amplitude function is derived from a theoretical expression modeling a reflection from said thin film over the substrate.
19. The method of claim 13 , further including the step of determining a profile of said substrate using said thickness of the thin film.
20. The method of claim 19 , wherein said step of determining a profile of the substrate includes utilizing phase information extracted from said test correlogram.
21. The method of claim 1 , wherein said thin film includes multiple layers, said property is a thickness of each layer, and said thickness of each layer is determined using a corresponding model test-beam amplitude function.
22. The method of claim 1 , wherein said steps are repeated using light of different polarization to identify a bi-refringent thin film.