Abstract

The analytical dependence of σn and σk on related experimental uncertainties when conventional reflectance–transmittance methods for the determination of optical constants of thin films are used has been found. Two kinds of singularity appear. These are responsible for the loss of solution in these methods. From the properties of the derivatives of n and k with respect to the measured thickness, a new method has been developed without loss of solution. Different derivative behaviors for physical and nonphysical solutions were found. The film thickness is also determined by this method with an accuracy better than 0.2%. The method has been applied to thin films of amorphous germanium.

© 1992 Optical Society of America

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References

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  1. F. Abelès, M. L. Thèye, “Méthode de calcul des constantes optiques des couches minces absorbantes à partir de mesures de réflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
    [CrossRef]
  2. J. M. Bennett, M. J. Booty, “Computational method for determining n and k for a thin film from the measured reflectance, transmittance, and film thickness,” Appl. Opt. 5, 41–43 (1966).
    [CrossRef] [PubMed]
  3. P.-O. Nilsson, “Determination of optical constants for intensity measurements at normal incidence,” Appl. Opt. 7, 435–442 (1968).
    [CrossRef] [PubMed]
  4. S. G. Tomlin, “Optical reflection and transmission formulae for thin films,” J. Phys. D 2, 1667–1671 (1968).
    [CrossRef]
  5. G. Lubberts, B. C. Burkey, F. Moser, E. A. Trabka, “Optical properties of phosphorus-doped polycrystalline silicon layers,” J. Appl. Phys. 52, 6870–6878 (1981).
    [CrossRef]
  6. M. García-Castaneda, H. Sańchez-Machet, “An iterative and consistent method for the complex refraction index calculation of absorbent thin films,” Thin Solid Films 176, 69–72 (1989).
    [CrossRef]
  7. R. C. McPhedran, L. C. Botten, D. R. McKenzie, R. P. Netterfield, “Unambiguous determination of optical constants of absorbing films by reflectance and transmittance measurements,” Appl. Opt. 23, 1197–1205 (1984).
    [CrossRef] [PubMed]
  8. C. L. Nagendra, G. K. M. Thutupalli, “Determination of optical properties of absorbing materials: a generalized scheme,” Appl. Opt. 22, 587–591 (1983).
    [CrossRef] [PubMed]
  9. J. E. Nestell, R. W. Christy, “Derivation of optical constants of metals from thin-film measurements as oblique incidence,” Appl. Opt. 11, 643–651 (1972).
    [CrossRef] [PubMed]
  10. J. P. Borgogno, B. Lazarides, P. Roche, “An improved method for the determination of the extinction coefficient of thin film materials,” Thin Solids Films 102, 209–220 (1983).
    [CrossRef]
  11. E. Elizalde, F. Rueda, “On the determination of the optical constants n(λ) and k(λ) of thin supported films,” Thin Solid Films 122, 45–57 (1984).
    [CrossRef]
  12. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4, pp. 74–77.
  13. G. A. N. Connell, R. J. Temkim, W. Paul, “Amorphous germanium III. Optical properties,” Adv. Phys. 22, 643–665 (1973).
    [CrossRef]
  14. E. D. Palik, W. R. Hunter, “Lithium fluoride (LiF),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, London, 1985), pp. 680–693.
  15. D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, London, 1985), pp. 555–569.

1989

M. García-Castaneda, H. Sańchez-Machet, “An iterative and consistent method for the complex refraction index calculation of absorbent thin films,” Thin Solid Films 176, 69–72 (1989).
[CrossRef]

1984

1983

J. P. Borgogno, B. Lazarides, P. Roche, “An improved method for the determination of the extinction coefficient of thin film materials,” Thin Solids Films 102, 209–220 (1983).
[CrossRef]

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

1981

G. Lubberts, B. C. Burkey, F. Moser, E. A. Trabka, “Optical properties of phosphorus-doped polycrystalline silicon layers,” J. Appl. Phys. 52, 6870–6878 (1981).
[CrossRef]

1973

G. A. N. Connell, R. J. Temkim, W. Paul, “Amorphous germanium III. Optical properties,” Adv. Phys. 22, 643–665 (1973).
[CrossRef]

1972

1968

P.-O. Nilsson, “Determination of optical constants for intensity measurements at normal incidence,” Appl. Opt. 7, 435–442 (1968).
[CrossRef] [PubMed]

S. G. Tomlin, “Optical reflection and transmission formulae for thin films,” J. Phys. D 2, 1667–1671 (1968).
[CrossRef]

1966

F. Abelès, M. L. Thèye, “Méthode de calcul des constantes optiques des couches minces absorbantes à partir de mesures de réflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

J. M. Bennett, M. J. Booty, “Computational method for determining n and k for a thin film from the measured reflectance, transmittance, and film thickness,” Appl. Opt. 5, 41–43 (1966).
[CrossRef] [PubMed]

Abelès, F.

F. Abelès, M. L. Thèye, “Méthode de calcul des constantes optiques des couches minces absorbantes à partir de mesures de réflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Bennett, J. M.

Booty, M. J.

Borgogno, J. P.

J. P. Borgogno, B. Lazarides, P. Roche, “An improved method for the determination of the extinction coefficient of thin film materials,” Thin Solids Films 102, 209–220 (1983).
[CrossRef]

Botten, L. C.

Burkey, B. C.

G. Lubberts, B. C. Burkey, F. Moser, E. A. Trabka, “Optical properties of phosphorus-doped polycrystalline silicon layers,” J. Appl. Phys. 52, 6870–6878 (1981).
[CrossRef]

Christy, R. W.

Connell, G. A. N.

G. A. N. Connell, R. J. Temkim, W. Paul, “Amorphous germanium III. Optical properties,” Adv. Phys. 22, 643–665 (1973).
[CrossRef]

Edwards, D. F.

D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, London, 1985), pp. 555–569.

Elizalde, E.

E. Elizalde, F. Rueda, “On the determination of the optical constants n(λ) and k(λ) of thin supported films,” Thin Solid Films 122, 45–57 (1984).
[CrossRef]

García-Castaneda, M.

M. García-Castaneda, H. Sańchez-Machet, “An iterative and consistent method for the complex refraction index calculation of absorbent thin films,” Thin Solid Films 176, 69–72 (1989).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4, pp. 74–77.

Hunter, W. R.

E. D. Palik, W. R. Hunter, “Lithium fluoride (LiF),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, London, 1985), pp. 680–693.

Lazarides, B.

J. P. Borgogno, B. Lazarides, P. Roche, “An improved method for the determination of the extinction coefficient of thin film materials,” Thin Solids Films 102, 209–220 (1983).
[CrossRef]

Lubberts, G.

G. Lubberts, B. C. Burkey, F. Moser, E. A. Trabka, “Optical properties of phosphorus-doped polycrystalline silicon layers,” J. Appl. Phys. 52, 6870–6878 (1981).
[CrossRef]

McKenzie, D. R.

McPhedran, R. C.

Moser, F.

G. Lubberts, B. C. Burkey, F. Moser, E. A. Trabka, “Optical properties of phosphorus-doped polycrystalline silicon layers,” J. Appl. Phys. 52, 6870–6878 (1981).
[CrossRef]

Nagendra, C. L.

Nestell, J. E.

Netterfield, R. P.

Nilsson, P.-O.

Palik, E. D.

E. D. Palik, W. R. Hunter, “Lithium fluoride (LiF),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, London, 1985), pp. 680–693.

Paul, W.

G. A. N. Connell, R. J. Temkim, W. Paul, “Amorphous germanium III. Optical properties,” Adv. Phys. 22, 643–665 (1973).
[CrossRef]

Roche, P.

J. P. Borgogno, B. Lazarides, P. Roche, “An improved method for the determination of the extinction coefficient of thin film materials,” Thin Solids Films 102, 209–220 (1983).
[CrossRef]

Rueda, F.

E. Elizalde, F. Rueda, “On the determination of the optical constants n(λ) and k(λ) of thin supported films,” Thin Solid Films 122, 45–57 (1984).
[CrossRef]

Sanchez-Machet, H.

M. García-Castaneda, H. Sańchez-Machet, “An iterative and consistent method for the complex refraction index calculation of absorbent thin films,” Thin Solid Films 176, 69–72 (1989).
[CrossRef]

Temkim, R. J.

G. A. N. Connell, R. J. Temkim, W. Paul, “Amorphous germanium III. Optical properties,” Adv. Phys. 22, 643–665 (1973).
[CrossRef]

Thèye, M. L.

F. Abelès, M. L. Thèye, “Méthode de calcul des constantes optiques des couches minces absorbantes à partir de mesures de réflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Thutupalli, G. K. M.

Tomlin, S. G.

S. G. Tomlin, “Optical reflection and transmission formulae for thin films,” J. Phys. D 2, 1667–1671 (1968).
[CrossRef]

Trabka, E. A.

G. Lubberts, B. C. Burkey, F. Moser, E. A. Trabka, “Optical properties of phosphorus-doped polycrystalline silicon layers,” J. Appl. Phys. 52, 6870–6878 (1981).
[CrossRef]

Adv. Phys.

G. A. N. Connell, R. J. Temkim, W. Paul, “Amorphous germanium III. Optical properties,” Adv. Phys. 22, 643–665 (1973).
[CrossRef]

Appl. Opt.

J. Appl. Phys.

G. Lubberts, B. C. Burkey, F. Moser, E. A. Trabka, “Optical properties of phosphorus-doped polycrystalline silicon layers,” J. Appl. Phys. 52, 6870–6878 (1981).
[CrossRef]

J. Phys. D

S. G. Tomlin, “Optical reflection and transmission formulae for thin films,” J. Phys. D 2, 1667–1671 (1968).
[CrossRef]

Surf. Sci.

F. Abelès, M. L. Thèye, “Méthode de calcul des constantes optiques des couches minces absorbantes à partir de mesures de réflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Thin Solid Films

M. García-Castaneda, H. Sańchez-Machet, “An iterative and consistent method for the complex refraction index calculation of absorbent thin films,” Thin Solid Films 176, 69–72 (1989).
[CrossRef]

E. Elizalde, F. Rueda, “On the determination of the optical constants n(λ) and k(λ) of thin supported films,” Thin Solid Films 122, 45–57 (1984).
[CrossRef]

Thin Solids Films

J. P. Borgogno, B. Lazarides, P. Roche, “An improved method for the determination of the extinction coefficient of thin film materials,” Thin Solids Films 102, 209–220 (1983).
[CrossRef]

Other

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4, pp. 74–77.

E. D. Palik, W. R. Hunter, “Lithium fluoride (LiF),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, London, 1985), pp. 680–693.

D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, London, 1985), pp. 555–569.

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Figures (11)

Fig. 1
Fig. 1

Denominator of Eqs. (3) and (4) versus λ for a simulated 100-nm-thick a-Ge film: CP, critical point; IP, interferential point.

Fig. 2
Fig. 2

Plot of n and k versus λ for the same film as in Fig. 1 but using three different values of tm (99, 100, and 101 nm) in the analysis: IP/2, points that verify Eq. (6) for half-integer values of m.

Fig. 3
Fig. 3

[∂n(λ)/∂tm]tm versus λ for a simulated 100-nm-thick a-Ge film for three different values of tm: 99, 100, and 101 nm.

Fig. 4
Fig. 4

[∂n(λ)/∂tm]tm =d versus λ for a simulated 300-nm-thick LiF film.

Fig. 5
Fig. 5

[∂n(λ)/∂tm]tm=d versus λ for a simulated 100-nm-thick c-Si film.

Fig. 6
Fig. 6

Plot of σn/n versus λ for the same film as in Fig. 1.

Fig. 7
Fig. 7

Contribution to σn of TexpnT), RexpnR), and tmnt) versus λ for a simulated 100-nm-thick a-Ge film.

Fig. 8
Fig. 8

Contribution to σk of TexpkT), RexpkR), and tmkt) versus λ for a simulated 100-nm-thick a-Ge film.

Fig. 9
Fig. 9

Plot of n and k (circles), n ± σn and k ± σk (continuous curve) versus λ for the a-Ge film B. The refined thickness was 111.9 nm.

Fig. 10
Fig. 10

Plot of n and k (circles), n ± σn and k ± σk (continuous curve) versus λ for the a-Ge film A. The refined thickness was 57.4 nm.

Fig. 11
Fig. 11

Plot of n and k (circles), n ± σn and k ± σk (continuous curve) versus λ for the a-Ge film C. The refined thickness was 602 nm.

Equations (20)

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T th ( n , k , d , λ ) - T exp ( λ ) = 0 , R th ( n , k , d , λ ) - R exp ( λ ) = 0.
( T th n ) λ d n + ( T th k ) λ d k + ( T th t m ) λ d t m - d T exp = 0 , ( R th n ) λ d n + ( R th k ) λ d k + ( R th t m ) λ d t m - d R exp = 0
d n = ( R th k ) λ d T exp - ( T th k ) λ d R exp + [ ( T th k ) λ ( R th t m ) λ - ( T th t m ) λ ( R th k ) λ ] d t m ( T th n ) λ ( R th k ) λ - ( T th k ) λ ( R th n ) λ ,
d k = - ( R th n ) λ d T exp + ( T th n ) λ d R exp + [ ( T th t m ) λ ( R th n ) λ - ( T th n ) λ ( R th t m ) λ ] d t m ( T th n ) λ ( R th k ) λ - ( T th k ) λ ( R th n ) λ .
d n = d T exp - ( T th t m ) λ d t m ( T th n ) λ ,
2 n d = m λ ,
tan ( 2 π n d / λ ) = - 2 π d λ n 5 - ( n 2 2 + 1 ) n 3 + n 2 2 - n n 4 - n 2
[ n ( λ ) t m ] λ = ( T th k ) λ ( R th t m ) λ - ( T th t m ) λ ( R th k ) λ ( T th n ) λ ( R th k ) λ - ( T th k ) λ ( R th n ) λ ,
[ k ( λ ) t m ] λ = ( T th t m ) λ ( R th n ) λ - ( T th n ) λ ( R th t m ) λ ( T th n ) λ ( R th k ) λ - ( T th k ) λ ( R th n ) λ .
[ n ( λ ) t m ] λ = ( T th t m ) λ ( T th n ) λ .
[ n ( λ ) t m ] t m = d = - n d ;             k = 0.
σ n 2 = ( R th k ) λ 2 σ T 2 + ( T th k ) λ 2 σ R 2 + [ ( T th k ) λ ( R th t m ) λ - ( T th t m ) λ ( R th k ) λ ] 2 σ t 2 [ ( T th n ) λ ( R th k ) λ - ( T th k ) λ ( R th n ) λ ] 2 ,
σ k 2 = ( R th n ) λ 2 σ T 2 + ( T th n ) λ 2 σ R 2 + [ ( T th t m ) λ ( R th n ) λ - ( T th n ) λ ( R th t m ) λ ] 2 σ t 2 [ ( T th n ) λ ( R th k ) λ - ( T th k ) λ ( R th n ) λ ] 2 .
T th = C a 1 exp ( x ) + b 1 exp ( - x ) + A 1 cos φ + B 1 sin φ ,
R th = a 2 exp ( x ) + b 2 exp ( - x ) + A 2 cos φ + B 2 sin φ a 1 exp ( x ) + b 1 exp ( - x ) + A 1 cos φ + B 1 sin φ ,
a 1 = [ ( n + n 0 ) 2 + k 2 ] [ ( n + n 2 ) 2 + k 2 ] , b 1 = [ ( n - n 0 ) 2 + k 2 ] [ ( n - n 2 ) 2 + k 2 ] , a 2 = [ ( n - n 0 ) 2 + k 2 ] [ ( n + n 2 ) 2 + k 2 ] , b 2 = [ ( n + n 0 ) 2 + k 2 ] [ ( n - n 2 ) 2 + k 2 ] , A 1 = - 2 [ ( n 2 - n 0 2 + k 2 ) ( n 2 - n 2 2 + k 2 ) - 4 n 0 n 2 k 2 ] , B 1 = 4 k [ n 2 ( n 2 - n 0 2 + k 2 ) + n 0 ( n 2 - n 2 2 + k 2 ) ] , A 2 = - 2 [ ( n 2 - n 0 2 + k 2 ) ( n 2 - n 2 2 + k 2 ) + 4 n 0 n 2 k 2 ] , B 2 = 4 k [ n 2 ( n 2 - n 0 2 + k 2 ) - n 0 ( n 2 - n 2 2 + k 2 ) ] , C = 16 n 0 n 2 ( n 2 + k 2 ) , φ = 4 π n t m / λ , x = 4 π k t m / λ .
( 1 / T th ) n = ( B 1 cos φ - A 1 sin φ ) 4 π t m / λ C + ( a 1 / n ) exp ( x ) + ( b 1 / n ) exp ( - x ) + ( A 1 / n ) cos φ + ( B 1 / n ) sin φ C - a 1 exp ( x ) + b 1 exp ( - x ) + A 1 cos φ + B 1 sin φ C 2 C n ,
( 1 / T th ) k = [ a 1 exp ( x ) - b 1 exp ( - x ) ] 4 π t m / λ C + ( a 1 / k ) exp ( x ) + ( b 1 / k ) exp ( - x ) + ( A 1 / k ) cos φ + ( B 1 / k ) sin φ C - a 1 exp ( x ) + b 1 exp ( - x ) + A 1 cos φ + B 1 sin φ C 2 C k ,
( 1 / T th ) t m = 4 π λ [ a 1 exp ( x ) - b 1 exp ( - x ) ] k + ( B 1 cos φ - A 1 sin φ ) n C .
T th θ = - ( 1 / T th ) θ T th 2 , R th θ = [ ( R th / T th ) θ - R th ( 1 / T th ) θ ] T th ,

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