Abstract

A straightforward approach for estimation of thickness (d), real (ε1) and imaginary parts (ε2) of the complex permittivity of very thin films from spectrophotometric measurements is presented. The uncertainties in ε1, ε2, and d due to methodical error and the uncertainties in the measured quantities are investigated. It is shown that the influence of these factors is considerable when ε1, ε2, and d are obtained simultaneously for each wavelength. The accuracy of ε1, ε2, and d is significantly increased if the value of d is evaluated first, its value is kept constant over the whole spectral region, and then ε1 and ε2 are calculated for each wave length.

© 2008 Optical Society of America

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  1. I. Chambouleyron, J. M. Martínez, A. C. Morettin, and M. Mulato, “Retrieval of optical constants and thickness of thin films from transmission spectra,” Appl. Opt. 36, 8238-8247 (1997).
    [CrossRef]
  2. D. Pekker and L. Pekker, “A method for determining thickness and optical constants of absorbing thin films,” Thin Solid Films 425, 203-209 (2003).
    [CrossRef]
  3. X. Sun, R. Hong, H. Hou, Z. Fan, and J. Shao, “Optical properties of silver thin films deposited by magnetron sputtering with different thicknesses,” Chin. Opt. Lett. 4, 366-369(2006).
  4. W. McGahan, B. Johs, and J. Woollam, “Techniques for ellipsometric measurements of thickness and optical constants of thin absorbing films,” Thin Solid Films 234, 443-446 (1993).
    [CrossRef]
  5. E. G. Birgin, I. E. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
    [CrossRef]
  6. E. Bondar, Yu. Kulyupin, and N. Popovich, “The inverse problem of the phenomenological theory of the optical properties of thin films,” Thin Solid Films 55, 201-209 (1978).
    [CrossRef]
  7. H. Wolter, “Zur optik dünner metallfilme,” Z. Phys. 105, 269-308 (1937).
    [CrossRef]
  8. F. Abeles, “Facteurs de reflexion et de transmission des coudhes metalliques tres minces: Methode nouvell pour determiner leurs indeces et leurs epaisseurs,” Rev. Opt., Theor. Instrum. 32, 257-268 (1953).
  9. M. Born and E. Wolf, Principles of Optics (Pergamon, 1983).
  10. B. Hristov, P. Gushterova, and P. Sharlandjiev, “Analytical solution of the inverse optical problem for very thin films,” J. Optoelectron. Adv. Mater. 9, 217-220 (2007).
  11. P. Gushterova, P. Sharlandjiev, and K. Petkov, “Optical response of very thin As-Se films,” Proc. SPIE 6252, 6252OT(2006).

2007

B. Hristov, P. Gushterova, and P. Sharlandjiev, “Analytical solution of the inverse optical problem for very thin films,” J. Optoelectron. Adv. Mater. 9, 217-220 (2007).

2006

P. Gushterova, P. Sharlandjiev, and K. Petkov, “Optical response of very thin As-Se films,” Proc. SPIE 6252, 6252OT(2006).

X. Sun, R. Hong, H. Hou, Z. Fan, and J. Shao, “Optical properties of silver thin films deposited by magnetron sputtering with different thicknesses,” Chin. Opt. Lett. 4, 366-369(2006).

2003

D. Pekker and L. Pekker, “A method for determining thickness and optical constants of absorbing thin films,” Thin Solid Films 425, 203-209 (2003).
[CrossRef]

E. G. Birgin, I. E. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
[CrossRef]

1997

1993

W. McGahan, B. Johs, and J. Woollam, “Techniques for ellipsometric measurements of thickness and optical constants of thin absorbing films,” Thin Solid Films 234, 443-446 (1993).
[CrossRef]

1978

E. Bondar, Yu. Kulyupin, and N. Popovich, “The inverse problem of the phenomenological theory of the optical properties of thin films,” Thin Solid Films 55, 201-209 (1978).
[CrossRef]

1953

F. Abeles, “Facteurs de reflexion et de transmission des coudhes metalliques tres minces: Methode nouvell pour determiner leurs indeces et leurs epaisseurs,” Rev. Opt., Theor. Instrum. 32, 257-268 (1953).

1937

H. Wolter, “Zur optik dünner metallfilme,” Z. Phys. 105, 269-308 (1937).
[CrossRef]

Abeles, F.

F. Abeles, “Facteurs de reflexion et de transmission des coudhes metalliques tres minces: Methode nouvell pour determiner leurs indeces et leurs epaisseurs,” Rev. Opt., Theor. Instrum. 32, 257-268 (1953).

Birgin, E. G.

E. G. Birgin, I. E. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
[CrossRef]

Bondar, E.

E. Bondar, Yu. Kulyupin, and N. Popovich, “The inverse problem of the phenomenological theory of the optical properties of thin films,” Thin Solid Films 55, 201-209 (1978).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1983).

Chambouleyron, I.

Chambouleyron, I. E.

E. G. Birgin, I. E. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
[CrossRef]

Fan, Z.

Gushterova, P.

B. Hristov, P. Gushterova, and P. Sharlandjiev, “Analytical solution of the inverse optical problem for very thin films,” J. Optoelectron. Adv. Mater. 9, 217-220 (2007).

P. Gushterova, P. Sharlandjiev, and K. Petkov, “Optical response of very thin As-Se films,” Proc. SPIE 6252, 6252OT(2006).

Hong, R.

Hou, H.

Hristov, B.

B. Hristov, P. Gushterova, and P. Sharlandjiev, “Analytical solution of the inverse optical problem for very thin films,” J. Optoelectron. Adv. Mater. 9, 217-220 (2007).

Johs, B.

W. McGahan, B. Johs, and J. Woollam, “Techniques for ellipsometric measurements of thickness and optical constants of thin absorbing films,” Thin Solid Films 234, 443-446 (1993).
[CrossRef]

Kulyupin, Yu.

E. Bondar, Yu. Kulyupin, and N. Popovich, “The inverse problem of the phenomenological theory of the optical properties of thin films,” Thin Solid Films 55, 201-209 (1978).
[CrossRef]

Martínez, J. M.

E. G. Birgin, I. E. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
[CrossRef]

I. Chambouleyron, J. M. Martínez, A. C. Morettin, and M. Mulato, “Retrieval of optical constants and thickness of thin films from transmission spectra,” Appl. Opt. 36, 8238-8247 (1997).
[CrossRef]

McGahan, W.

W. McGahan, B. Johs, and J. Woollam, “Techniques for ellipsometric measurements of thickness and optical constants of thin absorbing films,” Thin Solid Films 234, 443-446 (1993).
[CrossRef]

Morettin, A. C.

Mulato, M.

Pekker, D.

D. Pekker and L. Pekker, “A method for determining thickness and optical constants of absorbing thin films,” Thin Solid Films 425, 203-209 (2003).
[CrossRef]

Pekker, L.

D. Pekker and L. Pekker, “A method for determining thickness and optical constants of absorbing thin films,” Thin Solid Films 425, 203-209 (2003).
[CrossRef]

Petkov, K.

P. Gushterova, P. Sharlandjiev, and K. Petkov, “Optical response of very thin As-Se films,” Proc. SPIE 6252, 6252OT(2006).

Popovich, N.

E. Bondar, Yu. Kulyupin, and N. Popovich, “The inverse problem of the phenomenological theory of the optical properties of thin films,” Thin Solid Films 55, 201-209 (1978).
[CrossRef]

Shao, J.

Sharlandjiev, P.

B. Hristov, P. Gushterova, and P. Sharlandjiev, “Analytical solution of the inverse optical problem for very thin films,” J. Optoelectron. Adv. Mater. 9, 217-220 (2007).

P. Gushterova, P. Sharlandjiev, and K. Petkov, “Optical response of very thin As-Se films,” Proc. SPIE 6252, 6252OT(2006).

Sun, X.

Ventura, S. D.

E. G. Birgin, I. E. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1983).

Wolter, H.

H. Wolter, “Zur optik dünner metallfilme,” Z. Phys. 105, 269-308 (1937).
[CrossRef]

Woollam, J.

W. McGahan, B. Johs, and J. Woollam, “Techniques for ellipsometric measurements of thickness and optical constants of thin absorbing films,” Thin Solid Films 234, 443-446 (1993).
[CrossRef]

Appl. Numer. Math.

E. G. Birgin, I. E. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
[CrossRef]

Appl. Opt.

Chin. Opt. Lett.

J. Optoelectron. Adv. Mater.

B. Hristov, P. Gushterova, and P. Sharlandjiev, “Analytical solution of the inverse optical problem for very thin films,” J. Optoelectron. Adv. Mater. 9, 217-220 (2007).

Proc. SPIE

P. Gushterova, P. Sharlandjiev, and K. Petkov, “Optical response of very thin As-Se films,” Proc. SPIE 6252, 6252OT(2006).

Rev. Opt., Theor. Instrum.

F. Abeles, “Facteurs de reflexion et de transmission des coudhes metalliques tres minces: Methode nouvell pour determiner leurs indeces et leurs epaisseurs,” Rev. Opt., Theor. Instrum. 32, 257-268 (1953).

Thin Solid Films

E. Bondar, Yu. Kulyupin, and N. Popovich, “The inverse problem of the phenomenological theory of the optical properties of thin films,” Thin Solid Films 55, 201-209 (1978).
[CrossRef]

D. Pekker and L. Pekker, “A method for determining thickness and optical constants of absorbing thin films,” Thin Solid Films 425, 203-209 (2003).
[CrossRef]

W. McGahan, B. Johs, and J. Woollam, “Techniques for ellipsometric measurements of thickness and optical constants of thin absorbing films,” Thin Solid Films 234, 443-446 (1993).
[CrossRef]

Z. Phys.

H. Wolter, “Zur optik dünner metallfilme,” Z. Phys. 105, 269-308 (1937).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics (Pergamon, 1983).

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Figures (10)

Fig. 1
Fig. 1

Dispersion of the model ε 1 ( λ ) (—), ε 2 ( λ ) (– – –), and ε s ( λ ) (…).

Fig. 2
Fig. 2

Dispersion of Δ d / d (—), Δ ε 1 / ε 1 (– – –), Δ ε 2 / ε 2 (…), calculated with d 1 * and Δ d / d (–□–), Δ ε 2 / ε 2 (–○–), calculated with d 2 * .

Fig. 3
Fig. 3

Dispersion of Δ d / d : d = 15 nm (—), d = 20 nm (– – –), and d = 25 nm (…).

Fig. 4
Fig. 4

Dispersion of Δ ε 1 / ε 1 : d = 15 nm (—), d = 20 nm (– – –), and d = 25 nm (…).

Fig. 5
Fig. 5

Dispersion of Δ ε 2 / ε 2 : d = 15 nm (—), d = 20 nm (– – –), and d = 25 nm (…).

Fig. 6
Fig. 6

Dispersion of the calculated d, ε 1 , and ε 2 , obtained considering the uncertainties of T f , R f , and R f (gray curves) and without taking into account the uncertainties of T f , R f , and R f (black curves).

Fig. 7
Fig. 7

Dispersion of Δ d / d , calculated considering the uncertainties of T f , R f , and R f (gray curves) and without taking into account the uncertainties of T f , R f , and R f (black curves) for three film thicknesses: d = 15 nm , d = 20 nm , and d = 25 nm .

Fig. 8
Fig. 8

Dispersion of Δ ε 1 / ε 1 , calculated considering the uncertainties of T f , R f , and R f (gray curves) and without taking into account the uncertainties of T f , R f , and R f (black curves) for three film thicknesses: d = 15 nm , d = 20 nm , and d = 25 nm .

Fig. 9
Fig. 9

Dispersion of Δ ε 2 / ε 2 , calculated considering the uncertainties of T f , R f , and R f (gray curves) and without taking into account the uncertainties of T f , R f , and R f (black curves) for three film thicknesses: d = 15 nm , d = 20 nm , and d = 25 nm .

Fig. 10
Fig. 10

Dispersion of Δ ε 1 / ε 1 and Δ ε 2 / ε 2 , calculated considering the uncertainties in T f and R f (gray curves) and without taking into account uncertainties in T f , R f , and R f (black curves).

Equations (17)

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| m 11 m 12 m 21 m 22 | = | cos ψ 1 ( ε ) 1 / 2 sin ψ ( ε ) 1 / 2 sin ψ cos ψ | ,
1 + R f T f = ( 0.5 ε 2 2 + ε 1 2 ε 2 2 ε 1 + 0.5 ε 2 2 n s 2 ε 1 n s 2 + ε 1 2 n s 2 ε 1 3 ) ω 4 d 4 6 n s + ( ε 2 n s n s ε 2 ε 1 ) ω 3 d 3 3 n s + [ ( 1 ε 1 ) ( n s 2 ε 1 ) + ε 2 2 ] ω 2 d 2 2 n s + 2 ω d ε 2 n s + ( n s 2 + 1 ) 2 n s ,
1 R f T f = ε 2 2 ω 4 d 4 6 + ( ε 2 n s 2 ε 2 ε 1 ) ω 3 d 3 3 n s + ω d ε 2 + n s n s ,
1 R f T f = ε 2 2 ω 4 d 4 6 + ( ε 2 ε 2 ε 1 ) ω 3 d 3 3 + ω d ε 2 + 1 ,
C 1 ε 1 + C 0 = 0 ,
D 2 ε 2 2 + D 1 ε 2 + D 0 = 0 ,
A 3 ε 1 3 + A 2 ε 1 2 + A 1 ε 1 + A 0 = 0 ;
C 1 ε 1 + C 0 = 0 ;
D 2 ε 2 2 + D 1 ε 2 + D 0 = 0 ,
S A C = | A 3 A 2 A 1 A 0 C 1 C 0 0 0 0 C 1 C 0 0 0 0 C 1 C 0 | .
E 5 ε 2 5 + E 4 ε 2 4 + E 3 ε 2 3 + E 2 ε 2 2 + E 1 ε 2 + E 0 = 0 ,
S E D = | E 5 E 4 E 3 E 2 E 1 E 0 0 0 E 5 E 4 E 3 E 2 E 1 E 0 D 2 D 1 D 0 0 0 0 0 0 D 2 D 1 D 0 0 0 0 0 0 D 2 D 1 D 0 0 0 0 0 0 D 2 D 1 D 0 0 0 0 0 0 D 2 D 1 D 0 | .
F 5 U 5 + F 4 U 4 + F 3 U 3 + F 2 U 2 + F 1 U + F 0 = 0 ,
A 3 ε 1 3 + A 2 ε 1 2 + A 1 ε 1 + A 0 = 0 ,
B 1 ε 1 + B 0 = 0 ,
S A B = | A 3 A 2 A 1 A 0 B 1 B 0 0 0 0 B 1 B 0 0 0 0 B 1 B 0 | .
I 6 ε 2 6 + I 5 ε 2 5 + I 4 ε 2 4 + I 3 ε 2 3 + I 2 ε 2 2 + I 1 ε 2 + I 0 = 0 ,

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