Abstract

We develop and solve the equations of multiple-wavelength ellipsometry for thin uniaxial nonabsorbing films. We show that the values of the thickness and the real refractive indices (parallel and perpendicular to the optical axis) of the thin film can be obtained by three measurements, at three different wavelengths, of the phase difference (between the filmed and bare substrate phase changes caused by reflection), provided that the refractive indices of both the incident medium and the substrate are significantly dispersive. We also show, in the case of a dispersive thin film, how to determine the indices of refraction in terms of wavelength up to any desired accuracy by making the appropriate number of measurements of the phase difference at different wavelengths.

© 1986 Optical Society of America

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References

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  1. G. T. Ayoub, N. M. Bashara, J. Opt. Soc. Am. 68, 978–983 (1978).
    [CrossRef]
  2. M. J. Dignam, M. Moskovits, R. W. Stobie, Trans. Faraday Soc. 67, 3306–3317 (1971).
    [CrossRef]
  3. F. B. Hildebrand, Methods of Applied Mathematics (Prentice-Hall, Englewood Cliffs, N.J., 1952).
  4. J. M. Ziman, Principles of the Theory of Solids (Cambridge U. Press, Cambridge, 1964), pp. 223–229.
  5. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, pp. 374–378.
  6. D. E. Aspnes, “Optical characterization techniques for semiconductor technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 276, 188–195 (1981).

1981 (1)

D. E. Aspnes, “Optical characterization techniques for semiconductor technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 276, 188–195 (1981).

1978 (1)

1971 (1)

M. J. Dignam, M. Moskovits, R. W. Stobie, Trans. Faraday Soc. 67, 3306–3317 (1971).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, “Optical characterization techniques for semiconductor technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 276, 188–195 (1981).

Ayoub, G. T.

Bashara, N. M.

Dignam, M. J.

M. J. Dignam, M. Moskovits, R. W. Stobie, Trans. Faraday Soc. 67, 3306–3317 (1971).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, pp. 374–378.

Hildebrand, F. B.

F. B. Hildebrand, Methods of Applied Mathematics (Prentice-Hall, Englewood Cliffs, N.J., 1952).

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, pp. 374–378.

Moskovits, M.

M. J. Dignam, M. Moskovits, R. W. Stobie, Trans. Faraday Soc. 67, 3306–3317 (1971).
[CrossRef]

Stobie, R. W.

M. J. Dignam, M. Moskovits, R. W. Stobie, Trans. Faraday Soc. 67, 3306–3317 (1971).
[CrossRef]

Ziman, J. M.

J. M. Ziman, Principles of the Theory of Solids (Cambridge U. Press, Cambridge, 1964), pp. 223–229.

J. Opt. Soc. Am. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

D. E. Aspnes, “Optical characterization techniques for semiconductor technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 276, 188–195 (1981).

Trans. Faraday Soc. (1)

M. J. Dignam, M. Moskovits, R. W. Stobie, Trans. Faraday Soc. 67, 3306–3317 (1971).
[CrossRef]

Other (3)

F. B. Hildebrand, Methods of Applied Mathematics (Prentice-Hall, Englewood Cliffs, N.J., 1952).

J. M. Ziman, Principles of the Theory of Solids (Cambridge U. Press, Cambridge, 1964), pp. 223–229.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, pp. 374–378.

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Figures (1)

Fig. 1
Fig. 1

Relationship of parameters of Eqs. (1) to the experimental setup.

Tables (1)

Tables Icon

Table 1 Multiple-Wavelength Ellipsometric Data Related to the Experiment of Ayoub and Basharaa

Equations (114)

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δ Δ = Δ Δ ¯ = ( 4 π d λ ) [ n 1 sin φ tan φ 1 ( n 1 n 3 ) 2 tan 2 φ ] × [ n n 3 2 n 1 2 n 3 2 ] [ 1 n 1 2 α 3 n 2 ] ,
α 3 = ( n 2 n 2 ) ( n 2 n 3 2 n 2 n 3 2 ) ,
δ Δ = ( 4 π λ ) d [ n 1 sin φ tan φ 1 ( n 1 n 3 ) 2 tan 2 φ ] × [ n 2 n 3 2 n 1 2 n 3 2 n 1 2 n 2 ( n 2 n 3 2 n 1 2 n 3 2 ) ]
d [ n 2 + n 1 2 n 3 2 ( 1 n 2 ) ( n 1 2 + n 3 2 ) ] = f ( δ Δ , λ , φ , n 1 , n 3 ) ,
f ( δ Δ , λ , φ , n 1 , n 3 ) = δ Δ ( λ 4 π ) ( n 1 2 n 3 2 ) × [ 1 ( n 1 n 3 ) 2 tan 2 φ n 1 sin φ tan φ ]
x = d n 2 , y = d n 2 , z = d
x + n 1 2 n 3 2 y ( n 1 2 + n 3 2 ) z = f .
a i = n 1 i 2 , b i = n 3 i 2 , f i = f ( δ Δ i , λ i , φ i , n 1 i , n 3 i ) .
x + a i b i y ( a i + b i ) z = f i , i = 1 , 2 , 3.
D = | 1 a 1 b 1 ( a 1 + b 1 ) 1 a 2 b 2 ( a 2 + b 2 ) 1 a 3 b 3 ( a 3 + b 3 ) |
D = [ a 1 a 2 ( b 1 b 2 ) + a 2 a 3 ( b 2 b 3 ) + a 3 a 1 ( b 3 b 1 ) + b 1 b 2 ( a 1 a 2 ) + b 2 b 3 ( a 2 a 3 ) + b 3 b 1 ( a 3 a 1 ) ] .
a 1 = a 2 = a 3 and b 1 = b 2 = b 3 ,
a 1 = a 2 = a 3 or b 1 = b 2 = b 3 .
a 1 a 2 a 3 a 1 and b 1 b 2 b 3 b 1 ,
r = [ x y z ] and f = [ f 1 f 2 f 3 ]
M = [ 1 a 1 b 1 ( a 1 + b 1 ) 1 a 2 b 2 ( a 2 + b 2 ) 1 a 3 b 3 ( a 3 + b 3 ) ] .
Mr = f .
r = M 1 f ,
M 1 = 1 D [ a 2 a 3 ( b 2 b 3 ) + b 2 b 3 ( a 2 a 3 ) a 3 a 1 ( b 3 b 1 ) + b 3 b 1 ( a 3 a 1 ) a 1 a 2 ( b 1 b 2 ) + b 1 b 2 ( a 1 a 2 ) ( a 2 a 3 ) + ( b 2 b 3 ) ( a 3 a 1 ) + ( b 3 b 1 ) ( a 1 a 2 ) + ( b 1 b 2 ) ( a 2 b 2 a 3 b 3 ) ( a 3 b 3 a 1 b 1 ) ( a 1 b 1 a 2 b 2 ) ] .
c i j = a i a j ( b i b j ) + b i b j ( a i a j ) .
D = ( c 12 + c 23 + c 31 ) ,
x = c 12 f 3 + c 23 f 1 + c 31 f 2 c 12 + c 23 + c 31 ,
y = ( a 1 + b 1 ) ( f 3 f 2 ) + ( a 2 + b 2 ) ( f 1 f 3 ) + ( a 3 + b 3 ) ( f 2 f 1 ) c 12 + c 23 + c 31 ,
z = a 1 b 1 ( f 3 f 2 ) + a 2 b 2 ( f 1 f 3 ) + a 3 b 3 ( f 2 f 1 ) c 12 + c 23 + c 31 .
n = x / z , n = z / y , d = z .
n = α 0 + α 1 ( λ λ 0 ) + α 2 ( λ λ 0 ) 2 + = l = 0 α l ( λ λ 0 ) l
n = β 0 + β 1 ( λ λ 0 ) + β 2 ( λ λ 0 ) 2 + = l = 0 β l ( λ λ 0 ) l .
x = x 0 + x 1 ( λ λ 0 ) + x 2 ( λ λ 0 ) 2 + = l = 0 ( λ λ 0 ) l l
y = y 0 + y 1 ( λ λ 0 ) + y 2 ( λ λ 0 ) 2 + = l = 0 y l ( λ λ 0 ) l .
l = o k x l ( λ λ 0 ) l + n 1 2 n 3 2 l = 0 k y l ( λ λ 0 ) l ( n 1 2 + n 3 2 ) z = f .
l = 0 k x l ( λ i λ 0 ) l + a i b i l = o k y l ( λ i λ 0 ) l ( a i + b i ) z = f i , i = 1 , 2 , , 2 k + 3 ,
x = x 0 + x 1 ( λ i λ 0 ) , y = y 0 + y 1 ( λ i λ 0 ) ,
x 0 + x 1 ( λ i λ 0 ) + a i b i [ y 0 + y 1 ( λ i λ 0 ) ] ( a i + b i ) z = f i , i = 1 , 2 , , 5.
r = [ x 0 x 1 y 0 y 1 z ] , f = [ f 1 f 2 f 3 f 4 f 5 ] ,
M = [ 1 λ 1 λ 0 a 1 b 1 a 2 b 1 ( λ 1 λ 0 ) ( a 1 + b 1 ) 1 λ 2 λ 0 a 2 b 2 a 2 b 2 ( λ 2 λ 0 ) ( a 2 + b 2 ) 1 λ 3 λ 0 a 3 b 3 a 3 b 3 ( λ 3 λ 0 ) ( a 3 + b 3 ) 1 λ 4 λ 0 a 4 b 4 a 4 b 4 ( λ 4 λ 0 ) ( a 4 + b 4 ) 1 λ 5 λ 0 a 5 b 5 a 5 b 5 ( λ 5 λ 0 ) ( a 5 + b 5 ) ] ,
r = M 1 f ,
M i j = ( λ i λ 0 ) j 1 , j = 1 , 2 , , k + 1 ,
M i j = a i b i ( λ i λ 0 ) j k 2 , j = k + 2 , , 2 k + 2 ,
M i j = ( a i + b i ) , for j = 2 k + 3 ,
r i + 1 = x i , r i + k + 2 = y i , r 2 k + 3 = z , i = 0 , 1 , 2 , , k .
M i 1 = 1 , M i , k + 2 = a i b i , M i , 2 k + 3 = ( a i + b i ) .
M i 1 = a M i 1 = a , M i , k + 2 = 1 a M i , k + 2 = b i , M i , 2 k + 3 = M i , 2 k + 3 + M i , k + 2 = a .
x = d n 2 = d [ j = 0 α j ( λ λ 0 ) j ] [ m = o α m ( λ λ 0 ) m ] = d j = 0 m = 0 α j α m ( λ λ 0 ) j + m .
x = d l = 0 [ ( λ λ 0 ) l j = 0 l α j α l j ] .
x l = d j = 0 l α j α l j , l = 0 , 1 , 2 , , k .
x 0 = d α 0 2 ,
x 1 = d ( α 0 α 1 + α 1 α 0 ) ,
α 0 = x 0 / z , α 1 = 1 2 x 1 x 0 x 0 z ,
n = x 0 / z + 1 2 x 1 x 0 x 0 z ( λ λ 0 ) + .
d = y n 2 = [ l = 0 y l ( λ λ 0 ) l ] [ j = 0 β j ( λ λ 0 ) j ] × [ m = 0 β m ( λ λ 0 ) m ] = l = 0 j = 0 m = 0 y l β j β m ( λ λ 0 ) l + j + m .
d = p = 0 [ ( λ λ 0 ) p ( l = 0 p j = 0 p l y l β j β p l j ) ] .
l = 0 p j = 0 p l y l β j β p l j = { d for p = 0 0 for p = 1 , 2 , .
y 0 β 0 2 = d
y 0 ( β 0 β 1 + β 1 β 0 ) + y 1 β 0 2 = 0.
β 0 = z / y 0 and β 1 = 1 2 y 1 y 0 z / y 0 .
n = z / y 0 1 2 y 1 y 0 z / y 0 ( λ λ 0 ) + .
j = 0 m ( m 3 j ) x m j α j = 0 , m = 1 , 2 , , k ,
α 0 = x 0 / z .
α m = 1 2 m x 0 j = 0 m 1 ( m 3 j ) x m j α j , m = 1 , 2 , , k ,
j = 1 m ( 3 j m m α 0 ) x m j α j = x m , m = 1 , 2 , , k .
α = [ α 1 α 2 . . . α k ] , x = [ x 1 x 2 . . . x k ]
A m j = { ( 3 j m m α 0 ) x m j for j m 0 for j > m .
α = A 1 x .
ξ m = j = 0 m β j β m j , m = 1 , 2 , , k ,
l = 0 p y p l ξ l = { z for p = 0 0 for p = 1 , 2 , , k ,
j = 0 m ( m 3 j ) ξ m j β j = 0 , m = 1 , 2 , , k ,
β 0 = ξ 0 .
ξ = Q 1 y ,
ξ = [ ξ 1 ξ 2 . . . ξ k ] , y = [ y 1 y 2 . . . y k ] ,
Q m j = { y m j ξ 0 , j m 0 , j > m .
ξ p = 1 y 0 l = 0 p 1 y p l ξ l , p = 1 , 2 , , k ,
ξ = Z / y 0 .
β = B 1 ξ ,
β = [ β 1 β 2 . . . β k ] ,
B m j = { 3 j m m β 0 ξ m j , j m 0 , j > m .
β m = 1 2 m ξ 0 j = 0 m 1 ( m 3 j ) ξ m j β j , m = 1 , 2 , , k ,
x = 48.254 , y = 9.075 , z = 22.00.
h j i = [ 0.2327 0.0152 + 0.1845 0.0587 0.0358 + 0.0749 0.1212 0.0282 + 0.1144 ] ,
H j = ( 0.0634 , 0.0196 , 0.0350 ) .
δ n = 0.5 δ ( δ Δ ) , δ n = δ ( δ Δ ) , δ d = 113 δ ( δ Δ ) ,
δ n 3 i = δ n 1 i = δ φ i = δ λ i = 0 , δ ( δ Δ i ) 0.
δ a i = δ b i = δ c i j = δ D = 0 and δ f i = f i δ ( δ Δ i ) δ Δ i .
δ x = c 12 δ f 3 + c 23 δ f 1 + c 31 δ f 2 D ,
δ y = ( a 1 + b 1 ) ( δ f 2 δ f 3 ) + ( a 2 + b 2 ) ( δ f 3 δ f 1 ) + ( a 3 + b 3 ) ( δ f 1 δ f 2 ) D ,
δ z = a 1 b 1 ( δ f 2 δ f 3 ) + a 2 b 2 ( δ f 3 δ f 1 ) + a 3 b 3 ( δ f 1 δ f 2 ) D ,
δ n = 1 2 [ δ x x δ z z ] ,
δ n = 1 2 [ δ z z δ y y ] ,
δ d = δ z .
δ x = 1 D i = 1 3 h 1 i δ ( δ Δ i ) , h 1 i = f i δ Δ i i j k c j k ,
δ y = 1 D i = 1 3 h 2 i δ ( δ Δ i ) , h 2 i = f i δ Δ i i j k [ ( a j + b j ) ( a k + b k ) ] ,
δ z = 1 D i = 1 3 h 3 i δ ( δ Δ i ) , h 3 i = f i δ Δ i i j k ( a j b j a k b k ) ,
i j k = { + 1 if ( i , j , k ) is an even permutation of ( 1 , 2 , 3 ) 1 if ( i , j , k ) is an odd permutation of ( 1 , 2 , 3 ) 0 otherwise .
δ ( δ Δ 1 ) = δ ( δ Δ 2 ) = δ ( δ Δ 3 ) = δ ( δ Δ ) .
H j = i = 1 3 h j i ,
δ n = 1 2 D ( H 1 x H 3 z ) δ ( δ Δ ) ,
δ n = 1 2 D ( H 3 z H 2 y ) δ ( δ Δ ) ,
δ d = H 3 d δ ( δ Δ ) .
x l = d i = 0 l α i α l i , l = 0 , 1 , 2 , , k
j = 0 m ( m 3 j ) x m j α j = 0 , m = 1 , 2 , , k ,
α 0 = x 0 / d .
S = 1 d j = 0 m ( m 3 j ) x m j α j .
S = i = 0 m α i j = 0 m i ( m 3 j ) α j α m j i .
S = 1 d j = 0 m ( 3 j 2 m ) α m j x j .
S = i = 0 m α i j = 0 m i ( 3 j + 3 i 2 m ) α j α m j i .
2 S = i = 0 m α i ( 3 i m ) j = 0 m i α j α m j i = 1 d i = 0 m ( 3 i m ) α i x m i = S ;
S = 0.
g = λ δ Δ 4 π d 1 sin φ tan φ
g ( 1 n 1 2 tan 2 φ n 3 2 ) ( n 1 2 n 3 2 ) n 1 × [ ( n 2 n 3 2 ) + n 1 2 ( n 3 2 n 2 1 ) ] = 0
l = 0 4 γ l n 1 l = 0 ,
γ 0 = g n 3 2 , γ 1 = ( n 2 n 3 2 ) , γ 2 = g ( 1 + tan 2 φ ) , γ 3 = ( n 3 2 n 2 1 ) , γ 4 = g tan 2 φ n 3 2 .
g ( n 3 2 n 1 2 tan 2 φ ) ( n 1 2 n 3 2 ) n 1 n 3 4 × ( n 1 2 n 2 1 ) n 1 n 3 2 ( n 2 n 1 2 ) = 0
ρ 4 n 3 4 + ρ 2 n 3 2 + ρ 0 = 0 ,
ρ 0 = g n 1 4 tan 2 φ , ρ 2 = n 1 ( n 2 n 1 2 ) g n 1 2 ( 1 + tan 2 φ ) , ρ 4 = g + n 1 ( n 1 2 n 2 1 ) .
n 1 2 = ρ 2 ± ( ρ 2 2 4 ρ 0 ρ 4 ) 1 / 2 2 ρ 0 ,

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