Abstract

A method is proposed for processing ellipsometric data, leading to strict decoupling of the system describing the optical response of a transparent film deposited on a thick opaque substrate. The method is very efficient and enables a thorough semianalytical discussion of the errors involved in the measurements. In particular, resonant errors as well as optimum incidence angles are systematically derived from the formalism.

© 1985 Optical Society of America

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Corrections

D. Chariot and A. Maruani, "Ellipsometric data processing: an efficient method and an analysis of the relative errors; erratum," Appl. Opt. 26, 1167-1167 (1987)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-26-7-1167

References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4.
  2. A. R. Reinberg, “Ellipsometer Data Analysis with a Small Programmable Desk Calculator,” Appl. Opt. 11, 1273 (1972).
    [CrossRef] [PubMed]
  3. Y. Yoriume, “Method for Numerical Inversion of Ellipsometry Equation for Transparent Film,” J. Opt. Soc. Am. 73, 888 (1983).
    [CrossRef]
  4. M. E. Pedinoff, O. M. Stafsudd, “Multiple Angle Ellipsometric Analysis of Surface Layers and Surface Layer Contaminants,” Appl. Opt. 21, 518 (1982).
    [CrossRef] [PubMed]

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Figures (12)

Fig. 1
Fig. 1

Geometry of the model.

Fig. 2
Fig. 2

Plot of G vs n1 for a fixed sample thickness (d1 = 1000 Å) for different angles of incidence. (ϕ0 lies between 60 and 85°.)

Fig. 3
Fig. 3

Plot of ϕ0 optimum ( ϕ ˜ 0 ) vs d1.

Fig. 4
Fig. 4

Plot of G vs n1 for a fixed angle of incidence (ϕ0 = 74°) for different sample thicknesses (100 Å ≤ d1 ≤ 400 Å).

Fig. 5
Fig. 5

Plot of G vs n1 for a fixed angle of incidence (ϕ1 = 74°) for different sample thicknesses (600 Å ≤ d0 ≤ 1200 Å).

Fig. 6
Fig. 6

Relative errors kn and kd in percent per 0.01° vs the angle of incidence ϕ0 for a SiO2 film of 50 Å capped on a Si substrate.

Fig. 7
Fig. 7

Relative errors kn and kd in percent per 0.01° vs the angle of incidence ϕ0 for a SiO2 film of 1000 Å capped on a Si substrate.

Fig. 8
Fig. 8

Relative resonant errors of n and d at a critical angle when d = Dϕ (see text).

Fig. 9
Fig. 9

Relative errors in percent per 0.01° in the calculated parameters n1 and d1 for a GaAs oxide film (n1 ≃ 2.32) capped on a GaAs substrate (n2 = 3.841–i0.2157). The thickness of the film is 50 Å.

Fig. 10
Fig. 10

Relative errors in percent per 0.01° in the calculated parameters n1 and d1 for a GaAs. Optical parameters as in Fig. 9. The thickness of the film is 1000 Å.

Fig. 11
Fig. 11

Relative error in the calculated parameter d1 of a SiO2 thin film of a known index (n1 = 1.46). Note the peak around the Brewster angle.

Fig. 12
Fig. 12

Relative error in d1 for a SiO2 thick film (≃500 Å) of a known film index (n1 = 1.46). The peak at the Brewster angle has disappeared.

Tables (1)

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Table I Calculation of the Refractive Index (n1) and the Thickness (d1) of a Transparent Oxide Film Deposited on a Thick GaAs Substrate

Equations (37)

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ρ = r p r s = ( tan ψ ) · exp i Δ
f i ( n 1 , d 1 ) = 0 i = 1 , 2 ,
r p = r 01 ( p ) + r 12 ( p ) · X 1 + r 01 ( p ) · r 12 ( p ) X ;
r s = r 01 ( s ) + r 12 ( s ) · X 1 + r 01 ( s ) r 12 ( s ) · X ,
r i j ( p ) = ( τ i τ j τ i + τ j ) · ( 1 τ i τ j 1 + τ i τ j ) ;
r i j ( s ) = τ i τ j τ i + τ j ,
τ i = n i cos ϕ i n 0 sin ϕ 0 = cot ϕ i ,
1 + τ i 2 1 + τ j 2 = n i 2 n j 2
X = exp [ 4 i π d 1 λ ( n 1 2 n 0 2 sin 2 ϕ 0 ) 1 / 2 ]
X = exp ( 2 i π d 1 D ϕ )
ρ = ρ 0
a ( n 1 ) + b ( n 1 ) · X + c ( n 1 ) · X 2 = 0 ,
a ( n 1 ) = r 01 ( p ) ρ 001 ( s ) ;
b ( n 1 ) = r 12 ( p ) ρ 012 ( s ) + r 01 ( p ) r 01 ( s ) · [ r 12 ( s ) ρ 012 ( p ) ] ,
c ( n 1 ) = r 12 ( p ) · r 12 ( s ) · [ r 01 ( s ) ρ 001 ( p ) ] .
| X | = 1 .
X * = X 1 ,
a * ( n 1 ) · X 2 + b * ( n 1 ) · X + c * ( n 1 ) = 0 .
g ( a , b , c ) = ( | a | 2 | c | 2 ) 2 | a * b c b * | 2 = 0 = G ( n 1 ) ,
G ( 1 + ) = c 2 ( c > 0 ) ,
G ( n 2 ) > 0 ,
X 0 = | a | 2 | c | 2 c b * a * b
d 1 = i D ϕ 2 π ln X 0 .
D ϕ = 2830 Å
Δ n 1 = i | n 1 x i | · Δ x i ,
Δ d 1 = i | d 1 n 1 · n 1 x i + d 1 x i | · Δ x i ,
Δ d 1 = i | δ d 1 δ x i | · Δ x i .
δ G = 0 = i G x i Δ x i + G n 1 · Δ n 1
n 1 x i = ( G x i ) / ( G n 1 ) .
( d f d x i ) x i = f ( x i + Δ x i ) f ( x i ) Δ x i .
k n = 1 n 1 ( | n 1 ϕ 0 | + | n 1 ψ | + | n 1 Δ | ) ,
k d = 1 d 1 ( | δ d 1 δ ϕ 0 | + | δ d 1 δ ψ | + | δ d 1 δ Δ | ) ,
δ d 1 δ x i = d 1 δ n 1 · n 1 x i + d 1 x i
d 1 = k D ϕ = k · λ 2 n 1 2 n 0 2 sin 2 ϕ 0 , k N .
k λ 2 n 1 d 1 k · λ 2 · n 1 2 n 0 2 ,
ϕ min = arcsin ( n 1 2 n 0 2 2 d 1 λ n 0 2 ) 1 / 2 .
k d = | d 1 ϕ 0 | Δ ϕ 0 + | d 1 ψ | Δ ψ + | d 1 Δ | · Δ ( Δ )

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