Abstract

Formulas are given for the partial derivatives of the complex index of refraction ñ with respect to the ellipsometric angles Δ and ψ, the angle of incidence φ, and the thickness of a surface film d. The formulas are very useful in a study of the propagation to ñ of systematic errors in data obtained with an ellipsometer and are easily adapted to either machine or hand calculation.

© 1971 Optical Society of America

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References

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  1. R. J. Archer, Manual of Ellipsometry (Gaertner Scientific Corp., Chicago).
  2. G. Hass, J. E. Waylonis, J. Opt. Soc. Amer. 51, 719 (1961).
    [CrossRef]
  3. K. H. Zaininger, A. G. Revesz, RCA Rev. 85, (1964).
  4. F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Nat. Bur. Stand. Phys. Chem. 67A, 363 (1963).
  5. P. H. Smith, Surface Sci. 16, 34 (1969).
    [CrossRef]
  6. H. G. Jerrard, Surface Sci. 16, 67 (1969).
    [CrossRef]
  7. Ref. 1, p. 11.
  8. L. Ward, A. Nag, Brit. J. Appl. Phys. 18, 277 (1967).
    [CrossRef]
  9. A. Vasicek, in Ellipsometry in the Measurement of Surfaces and Thin Films, NBS Misc. Pub. 256 (U.S. GPO, Washington, D.C., 1964), p. 25.
  10. Ref. 1, p. 16.
  11. R. W. Fane, W. E. J. Neal, J. Opt. Soc. Amer. 60, 790 (1970).
    [CrossRef]

1970 (1)

R. W. Fane, W. E. J. Neal, J. Opt. Soc. Amer. 60, 790 (1970).
[CrossRef]

1969 (2)

P. H. Smith, Surface Sci. 16, 34 (1969).
[CrossRef]

H. G. Jerrard, Surface Sci. 16, 67 (1969).
[CrossRef]

1967 (1)

L. Ward, A. Nag, Brit. J. Appl. Phys. 18, 277 (1967).
[CrossRef]

1964 (1)

K. H. Zaininger, A. G. Revesz, RCA Rev. 85, (1964).

1963 (1)

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Nat. Bur. Stand. Phys. Chem. 67A, 363 (1963).

1961 (1)

G. Hass, J. E. Waylonis, J. Opt. Soc. Amer. 51, 719 (1961).
[CrossRef]

Archer, R. J.

R. J. Archer, Manual of Ellipsometry (Gaertner Scientific Corp., Chicago).

Fane, R. W.

R. W. Fane, W. E. J. Neal, J. Opt. Soc. Amer. 60, 790 (1970).
[CrossRef]

Hass, G.

G. Hass, J. E. Waylonis, J. Opt. Soc. Amer. 51, 719 (1961).
[CrossRef]

Jerrard, H. G.

H. G. Jerrard, Surface Sci. 16, 67 (1969).
[CrossRef]

McCrackin, F. L.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Nat. Bur. Stand. Phys. Chem. 67A, 363 (1963).

Nag, A.

L. Ward, A. Nag, Brit. J. Appl. Phys. 18, 277 (1967).
[CrossRef]

Neal, W. E. J.

R. W. Fane, W. E. J. Neal, J. Opt. Soc. Amer. 60, 790 (1970).
[CrossRef]

Passaglia, E.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Nat. Bur. Stand. Phys. Chem. 67A, 363 (1963).

Revesz, A. G.

K. H. Zaininger, A. G. Revesz, RCA Rev. 85, (1964).

Smith, P. H.

P. H. Smith, Surface Sci. 16, 34 (1969).
[CrossRef]

Steinberg, H. L.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Nat. Bur. Stand. Phys. Chem. 67A, 363 (1963).

Stromberg, R. R.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Nat. Bur. Stand. Phys. Chem. 67A, 363 (1963).

Vasicek, A.

A. Vasicek, in Ellipsometry in the Measurement of Surfaces and Thin Films, NBS Misc. Pub. 256 (U.S. GPO, Washington, D.C., 1964), p. 25.

Ward, L.

L. Ward, A. Nag, Brit. J. Appl. Phys. 18, 277 (1967).
[CrossRef]

Waylonis, J. E.

G. Hass, J. E. Waylonis, J. Opt. Soc. Amer. 51, 719 (1961).
[CrossRef]

Zaininger, K. H.

K. H. Zaininger, A. G. Revesz, RCA Rev. 85, (1964).

Brit. J. Appl. Phys. (1)

L. Ward, A. Nag, Brit. J. Appl. Phys. 18, 277 (1967).
[CrossRef]

J. Opt. Soc. Amer. (2)

G. Hass, J. E. Waylonis, J. Opt. Soc. Amer. 51, 719 (1961).
[CrossRef]

R. W. Fane, W. E. J. Neal, J. Opt. Soc. Amer. 60, 790 (1970).
[CrossRef]

J. Res. Nat. Bur. Stand. Phys. Chem. (1)

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Nat. Bur. Stand. Phys. Chem. 67A, 363 (1963).

RCA Rev. (1)

K. H. Zaininger, A. G. Revesz, RCA Rev. 85, (1964).

Surface Sci. (2)

P. H. Smith, Surface Sci. 16, 34 (1969).
[CrossRef]

H. G. Jerrard, Surface Sci. 16, 67 (1969).
[CrossRef]

Other (4)

Ref. 1, p. 11.

A. Vasicek, in Ellipsometry in the Measurement of Surfaces and Thin Films, NBS Misc. Pub. 256 (U.S. GPO, Washington, D.C., 1964), p. 25.

Ref. 1, p. 16.

R. J. Archer, Manual of Ellipsometry (Gaertner Scientific Corp., Chicago).

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Figures (3)

Fig. 1
Fig. 1

Values of the partial derivatives of the refractive index n of aluminum (ñ = 1.45 − i6.12).

Fig. 2
Fig. 2

Values of the partial derivatives of the extinction index k of aluminum.

Fig. 3
Fig. 3

Values of the partial derivatives of the absorption index κ of aluminum.

Tables (2)

Tables Icon

Table I Comparison of Pseudoconstant Calculations and Derivative Calculations for Aluminum Oxide (ñ = 1.635 − i0.0) on Aluminum (ñ = 1.45 − i6.12)

Tables Icon

Table II The Quantities (∂n/∂Δ)/(∂Δ/∂d) and (∂n/∂ψ)/(∂ψ/∂d) and Their Sum for Aluminum Oxide on Aluminum

Equations (27)

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n ˜ = n ˜ 0 + δ n ˜ ,
δ n ˜ = ( n ˜ / Δ ) δ Δ + ( n ˜ / ψ ) δ ψ + ( n ˜ / φ ) δ φ ,
n ˜ = n - i k
n ˜ = n ( 1 - i κ ) ,
n 2 - k 2 = t 2 ( cos 2 2 ψ - sin 2 2 ψ sin 2 Δ ) ( 1 + sin 2 ψ cos Δ ) 2 + sin 2 φ ,
2 n k = t 2 sin 4 ψ sin Δ ( 1 + sin 2 ψ cos Δ ) 2 ,
t = sin φ tan φ .
A = n 2 - k 2 ,
B = 2 n k .
( n + i k ) 2 = A + i B ,
n + i k = ( A + i B ) 1 2 .
d n = n d A + k d B 2 ( n 2 + k 2 ) ,
d k = n d B - k d A 2 ( n 2 + k 2 ) .
d κ = κ ( A d B - B d A ) B ( n 2 + k 2 ) .
A Δ = B cos 2 ψ ( - cos Δ sin 2 ψ + C · D ) ,
A ψ = - 2 B [ ( 1 + sin 2 Δ ) sin Δ + C · D cot Δ sin 2 ψ ] ,
A φ = 4 C ( 1 + cos 2 φ ) sin 2 φ + sin 2 φ ,
B Δ = B ( cos Δ + B · D t 2 cos 2 ψ ) ,
B ψ = 4 B ( cot 4 ψ - cos 2 ψ cos Δ D ) ,
B φ = 4 B ( 1 + cos 2 φ ) sin 2 φ ,
C = A - sin 2 φ ,
D = 1 + sin 2 ψ cos Δ .
d ψ ~ d A
d Δ ~ 2 d P
d ψ ~ d P
d Δ ~ 2 d A
n ˜ d = n ˜ Δ Δ d + n ψ ψ d .

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