Abstract

A comparative analysis is made of the errors in deriving the optical parameters (n, refractive index; k, absorption coefficient; d, film thickness) of thin films from spectrophotometric measurements at normal light incidence. The errors in determining n, k, and d by the (TR f R b), (TR f R m), (TR b R m), (TR f), (TR m), and T(k = 0) methods are compared. It is shown that they are applicable to optical constants of thin films in the n > 1.5, k < 4.5, and d/λ = (0.02–0.3) range, and their combinations make possible the determination of n and k to an accuracy of better than ±4%. To derive the optical constants in a wide spectral range with high accuracy and isolate the correct physical solutions reliably, one should apply all methods, using the relevant solutions with the lowest errors, as shown in this research, when determining the optical constants of As2S3 and Sb2Se3 films.

© 1998 Optical Society of America

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References

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  1. V. Panayotov, I. Konstantinov, “Determination of thin film optical parameters from photometric measurements: an algebraic solution for the (T, Rf, Rb) method,” Appl. Opt. 30, 2795–2800 (1991).
    [CrossRef] [PubMed]
  2. V. Panayotov, I. Konstantinov, “Algebraic determination of thin-film optical constants from photometric (T, Rf, Rm) and (T, Rb, Rm) measurements,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 1070–1079 (1994).
  3. V. Panayotov, I. Konstantinov, “Determination of thin film optical parameters from photometric measurements: algebraic solutions for the (T, Rf, Rm) and (T, Rb, Rm) methods,” Bulg. Chem. Commun. 26, 612–624 (1993).
  4. A. Hjortsberg, “Determination of optical constants of absorbing materials using transmission and reflection of thin films on partially metallized substrates: analysis of the new (T, Rm) technique,” Appl. Opt. 20, 1254–1262 (1981).
    [CrossRef] [PubMed]
  5. C. Nagendra, G. Thutupalli, “Determination of optical properties of absorbing materials: a generalized scheme,” Appl. Opt. 22, 587–591 (1983).
    [CrossRef] [PubMed]
  6. J. Pozo, L. Diaz, “Method for determination of optical constants of thin films: dependence on experimental uncertainties,” Appl. Opt. 31, 4474–4482 (1992).
    [CrossRef] [PubMed]
  7. W. Hansen, “Optical characterization of thin films: theory,” J. Opt. Soc. Am. 63, 793–802 (1973).
    [CrossRef]
  8. H. Liddell, Computer-Aided Techniques for Design of Multilayer Filters (Hilger, Bristol, UK, 1981), Chap. 1, p. 19.
  9. T. Yasuda, D. E. Aspnes, “Optical-standard surfaces of single-crystal silicon for calibrating ellipsometers and reflectometers,” Appl. Opt. 33, 7435–7438 (1994).
    [CrossRef] [PubMed]
  10. D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
    [CrossRef]
  11. M. Sachatchieva, B. Mednikarov, “Sensitivity of the evaporated photoresist—As2S3,” J. Inf. Rec. 23, 337–346 (1996).
  12. D. Dimitrov, D. Tzocheva, D. Kovacheva, “Calorimetric study of amorphous SbSe thin films,” Thin Solid Films (to be published).
  13. F. Abeles, M. Theye, “Methode de calcul’ des constantes optiques des couches minces absorbantes à partir de mesures de reflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
    [CrossRef]

1996

M. Sachatchieva, B. Mednikarov, “Sensitivity of the evaporated photoresist—As2S3,” J. Inf. Rec. 23, 337–346 (1996).

1994

1993

V. Panayotov, I. Konstantinov, “Determination of thin film optical parameters from photometric measurements: algebraic solutions for the (T, Rf, Rm) and (T, Rb, Rm) methods,” Bulg. Chem. Commun. 26, 612–624 (1993).

1992

1991

1983

C. Nagendra, G. Thutupalli, “Determination of optical properties of absorbing materials: a generalized scheme,” Appl. Opt. 22, 587–591 (1983).
[CrossRef] [PubMed]

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

1981

1973

1966

F. Abeles, M. Theye, “Methode de calcul’ des constantes optiques des couches minces absorbantes à partir de mesures de reflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Abeles, F.

F. Abeles, M. Theye, “Methode de calcul’ des constantes optiques des couches minces absorbantes à partir de mesures de reflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Aspnes, D. E.

T. Yasuda, D. E. Aspnes, “Optical-standard surfaces of single-crystal silicon for calibrating ellipsometers and reflectometers,” Appl. Opt. 33, 7435–7438 (1994).
[CrossRef] [PubMed]

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Diaz, L.

Dimitrov, D.

D. Dimitrov, D. Tzocheva, D. Kovacheva, “Calorimetric study of amorphous SbSe thin films,” Thin Solid Films (to be published).

Hansen, W.

Hjortsberg, A.

Konstantinov, I.

V. Panayotov, I. Konstantinov, “Determination of thin film optical parameters from photometric measurements: algebraic solutions for the (T, Rf, Rm) and (T, Rb, Rm) methods,” Bulg. Chem. Commun. 26, 612–624 (1993).

V. Panayotov, I. Konstantinov, “Determination of thin film optical parameters from photometric measurements: an algebraic solution for the (T, Rf, Rb) method,” Appl. Opt. 30, 2795–2800 (1991).
[CrossRef] [PubMed]

V. Panayotov, I. Konstantinov, “Algebraic determination of thin-film optical constants from photometric (T, Rf, Rm) and (T, Rb, Rm) measurements,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 1070–1079 (1994).

Kovacheva, D.

D. Dimitrov, D. Tzocheva, D. Kovacheva, “Calorimetric study of amorphous SbSe thin films,” Thin Solid Films (to be published).

Liddell, H.

H. Liddell, Computer-Aided Techniques for Design of Multilayer Filters (Hilger, Bristol, UK, 1981), Chap. 1, p. 19.

Mednikarov, B.

M. Sachatchieva, B. Mednikarov, “Sensitivity of the evaporated photoresist—As2S3,” J. Inf. Rec. 23, 337–346 (1996).

Nagendra, C.

Panayotov, V.

V. Panayotov, I. Konstantinov, “Determination of thin film optical parameters from photometric measurements: algebraic solutions for the (T, Rf, Rm) and (T, Rb, Rm) methods,” Bulg. Chem. Commun. 26, 612–624 (1993).

V. Panayotov, I. Konstantinov, “Determination of thin film optical parameters from photometric measurements: an algebraic solution for the (T, Rf, Rb) method,” Appl. Opt. 30, 2795–2800 (1991).
[CrossRef] [PubMed]

V. Panayotov, I. Konstantinov, “Algebraic determination of thin-film optical constants from photometric (T, Rf, Rm) and (T, Rb, Rm) measurements,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 1070–1079 (1994).

Pozo, J.

Sachatchieva, M.

M. Sachatchieva, B. Mednikarov, “Sensitivity of the evaporated photoresist—As2S3,” J. Inf. Rec. 23, 337–346 (1996).

Studna, A. A.

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Theye, M.

F. Abeles, M. Theye, “Methode de calcul’ des constantes optiques des couches minces absorbantes à partir de mesures de reflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Thutupalli, G.

Tzocheva, D.

D. Dimitrov, D. Tzocheva, D. Kovacheva, “Calorimetric study of amorphous SbSe thin films,” Thin Solid Films (to be published).

Yasuda, T.

Appl. Opt.

Bulg. Chem. Commun.

V. Panayotov, I. Konstantinov, “Determination of thin film optical parameters from photometric measurements: algebraic solutions for the (T, Rf, Rm) and (T, Rb, Rm) methods,” Bulg. Chem. Commun. 26, 612–624 (1993).

J. Inf. Rec.

M. Sachatchieva, B. Mednikarov, “Sensitivity of the evaporated photoresist—As2S3,” J. Inf. Rec. 23, 337–346 (1996).

J. Opt. Soc. Am.

Phys. Rev. B

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Surf. Sci.

F. Abeles, M. Theye, “Methode de calcul’ des constantes optiques des couches minces absorbantes à partir de mesures de reflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Other

D. Dimitrov, D. Tzocheva, D. Kovacheva, “Calorimetric study of amorphous SbSe thin films,” Thin Solid Films (to be published).

V. Panayotov, I. Konstantinov, “Algebraic determination of thin-film optical constants from photometric (T, Rf, Rm) and (T, Rb, Rm) measurements,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 1070–1079 (1994).

H. Liddell, Computer-Aided Techniques for Design of Multilayer Filters (Hilger, Bristol, UK, 1981), Chap. 1, p. 19.

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Figures (9)

Fig. 1
Fig. 1

Regions in the nk plane, defined by terms of the first column of the error matrix (6) with values E ij ≤ 1 at d/λ = 0.1: (a) The influence of inaccuracies in the measurement of T, R f , and R m in the determination of n. (b) The influence of inaccuracies in n s , n m , and k m on the determination of n. The terms of k m have values of ≤0.1 over the whole nk plane, and only the contours with values of 0.1 are plotted.

Fig. 2
Fig. 2

Regions in the nk plane, defined by the relative error in d ≤ 4% at various values of d/λ.

Fig. 3
Fig. 3

Regions in the nk plane, defined by the relative error in n and k ≤ 4%, at various values of d/λ: (a), (b), (TR f R m ); (c), (d), (T R f ); and (e), (f), (TR m ) method.

Fig. 4
Fig. 4

T(k = 0) method: solid curves, contours of the relative error in n with the indicated values, plotted in the nd/λ plane; dashed curves, contours plotted for (∂T/∂n) = 0, showing the areas of uncertainty of the method in extracting n.

Fig. 5
Fig. 5

Solutions for film thickness d by the (TR f R m ) method and absolute error in d for the correct values [d(λ) ≈ const]: (a), (b), As2S3 films, d = 58 ± 1.6 nm; and (c), (d), Sb2Se3 films, d = 78 ± 0.9 nm.

Fig. 6
Fig. 6

Dispersion curves of n and k and their absolute errors for As2S3 (d = 58 ± 1.6 nm), obtained by the three designated methods.

Fig. 7
Fig. 7

Dispersion curves of n and k and their absolute errors for Sb2Se3 (d = 78 ± 0.9 nm), obtained by the three designated methods.

Fig. 8
Fig. 8

Dispersion curves of n and the absolute errors, obtained by the T(k = 0) method for As2S3, d = 58 ± 1.6 nm and Sb2Se3, d = 78 ± 0.9 nm.

Fig. 9
Fig. 9

Dispersion curves of n and k and the corresponding absolute errors, assembled from the spectral ranges with maximum accuracy of the used methods for (a), As2S3, d = 58 ± 1.6 nm; and (b) Sb2Se3, d = 78 ± 0.9 nm.

Tables (1)

Tables Icon

Table 1 Spectrophotometric Quantities L , ℜ f , ℜ b , and ℜ m of As2S3 and Sb2Se3 Samples

Equations (13)

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Δ n = n T Δ T + n R Δ R + n R m Δ R m + n n s Δ n s + n n m Δ n m + n k m Δ k m , Δ k = k T Δ T + k R Δ R + k R m Δ R m + k n s Δ n s + k n m Δ n m + k k m Δ k m , Δ d = d T Δ T + d R Δ R + d R m Δ R m + d n s Δ n s + d n m Δ n m + d k m Δ k m .
Δ T = T n Δ n + T k Δ k + T d Δ d + T n s Δ n s , Δ R = R n Δ n + R k Δ k + R d Δ d + R n s Δ n s , Δ R m = R m n Δ n + R m k Δ k + R m d Δ d + R m n m Δ n m + R m k m Δ k m .
Δ T Δ R Δ R m = M 1   *   Δ n Δ k Δ d + M 2   *   Δ n s Δ n m Δ k m .
Δ n Δ k Δ d = | M 1 - 1 |   *   Δ T Δ R Δ R m + | M 1 - 1   *   M 2 |   *   Δ n s Δ n m Δ k m .
Δ n n = | E 11 |   Δ T T + | E 21 |   Δ R R + | E 31 |   Δ R m R m + | E 41 |   Δ n s n s + | E 51 |   Δ n m n m + | E 61 |   Δ k m k m ,
E = T n n T T k k T T d / λ d / λ T R n n R R k k R R d / λ d / λ R R m n n R m R m k k R m R m d / λ d / λ R m n s n n n s n s k k n s n s d / λ d / λ n s n m n n n m n m k k n m n m d / λ d / λ n m k m n n k m k m k k k m k m d / λ d / λ k m .
Δ T = T n Δ n + T d Δ d + T n s Δ n s .
Δ n = | E 1 | Δ T + | E 2 | Δ d + | E 3 | Δ n s , E 1 = 1 T n ,   E 2 = T d T n , E 3 = T n s T n .
Δ n n = T n E 1 Δ T T + d n E 2 Δ d d + n s n E 3 Δ n s n s .
T = L R s - 1 1 + b R s - 2 R s , R f = f - L 2 R s 1 + b R s - 2 R s ,     R s = n s - 1 n s + 1 2 , R b = b - R s 1 + b R s - 2 R s ,
Δ T = T L Δ L + T b Δ b + T R s Δ R s , Δ R f = R f L Δ L + R f f Δ f + R f b Δ b + R f R s Δ R s , Δ R b = R b b Δ b + R b R s Δ R s .
n s = 1 + A 1 - B / λ 2 1 / 2 ,   A = 1.26 ,   B = 98.94 .
k m = 1 / A + B λ 2 + C λ 3 ,   A = 5.86 , B = - 2.3 × 10 - 4 ,   C = 5.2 × 10 - 7 .

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