Abstract

The theoretical precision attainable from a number of types of ideal null and photometric ellipsometers is investigated quantitatively. Photometric and modulated null ellipsometer systems are shown to be approximately comparable with respect to precision when operating under shot-noise limited conditions. Differences are due principally to intrinsic detector noise levels, which are more significant in null designs. The equations derived can be used to estimate practical measurement limits or to choose components to achieve a specific objective.

© 1975 Optical Society of America

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References

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  1. D. E. Aspnes, J. Opt. Soc. Am. 64, 639 (1974).
    [CrossRef]
  2. See, for instance, the following sources: Ellipsometry in the Measurement of Surfaces and Thin Solid Films, E. Passaglia, R. R. Stromberg, J. Kruger, Eds. (U.S. Nat. Bur. Stand., Washington, D.C., 1964) Misc. Pub. 256;A. C. Hall, N. M. Bashara, A. B. Buckman, Eds., Proceedings of the Symposium on Recent Developments in Ellipsometry, Surf. Sci. 16, (1969);review papers by R. H. Muller, J. Krueger, in Advances in Electrochemistry and Electrochemical Engineering, R. H. Muller, Ed. (Academic, New York, 1973), Vol. 9.
  3. D. E. Aspnes, Phys. Rev. Lett. 28, 168 (1972);in Proceedings of the Twelfth International Conference on the Physics of Semiconductors, M. Pilkuhn, Ed. (Springer-Verlag, Berlin, 1974), p. 1197.
    [CrossRef]
  4. RCA Photomultiplier Manual (RCA Corporation, Harrison, N.J., 1970).
  5. This form supposes each event is counted independently, which requires continuous integration or digital sampling at a sufficiently rapid rate to follow the highest frequency component in ṅ(t).
  6. For an ideal digital system, T equals the total data acquisition time. For an ideal filter with bandwidth Δf, T = 1/(2Δf). We assume sufficient averaging so that the contribution to the fluctuation term arising from terminating a partially completed detection cycle is negligible.
  7. D. J. Scholtens, J. F. Kleibeuker, J. Kommandeur, Rev. Sci. Instrum. 44, 153 (1973).
    [CrossRef]
  8. R. H. Muller, Surf. Sci. 16, 14 (1969).
    [CrossRef]
  9. Values used here: uncooled near ir-sensitive photomultiplier, S1 response: Inep = 6 × 10−13 W, η0 = 0.006, ℏω0 = 3.3 eV, Δf0 = 1 sec−1; visible-uv photomultiplier, S20 response: Inep = 4 × 10−16 W, η0 = 0.29, ℏω0 = 3.3 eV, Δf0 = 1 sec−1. These values pertain to models 9684B and 9558QB photomultiplier detectors manufactured by EMI Electronics, Ltd., Hayes, Middelsex, England.
  10. M. Billardon, Ann. Phys. 7, 233 (1962).
  11. A. B. Winterbottom, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger, Eds. (U.S. Nat. Bur. Stand., Washington, D.C., 1964), Misc. Pub. 256, p. 97.
  12. H. G. Jerrard, Surf. Sci. 16, 137 (1969).
    [CrossRef]
  13. I. Willmans, Surf. Sci. 16, 147 (1969).
    [CrossRef]
  14. H. J. Mathieu, D. E. McClure, R. H. Muller, Rev. Sci. Instrum. 45, 798 (1974).
    [CrossRef]
  15. H. Takasaki, J. Opt. Soc. Am. 51, 463 (1961);J. Opt. Soc. Am. 56, 557 (1966);Appl.Opt. 5, 759 (1966).
    [CrossRef]
  16. C. V. Kent, J. Lawson, J. Opt. Soc. Am. 27, 117 (1937).
    [CrossRef]
  17. B. D. Cahan, R. F. Spanier, Surf. Sci. 16, 166 (1969);B. D. Cahan, J. Horkans, E. Yeager, Surf. Sci. 37, 559 (1973).
    [CrossRef]
  18. R. Greef, Rev. Sci. Instrum. 41, 532 (1970).
    [CrossRef]
  19. D. E. Aspnes, Opt. Commun. 8, 222 (1973);D. E. Aspnes, A. A. Studna, Appl. Opt. 14, 220 (1975).
    [CrossRef] [PubMed]
  20. P. S. Hauge, F. H. Dill, IBM J. Res. Dev. 17, 472 (1973);Y. J. van der Meulen, N. C. Hien, J. Opt. Soc. Am. 64, 804 (1974).
    [CrossRef]
  21. J. F. Archard, P. L. Clegg, A. M. Taylor, Proc. Phys. Soc. (London) 65B, 758 (1952).
  22. T. E. Faber, N. V. Smith, J. Opt. Soc. Am. 58, 102 (1968).
    [CrossRef]
  23. S. N. Jasperson, S. E. Schnatterly, Rev. Sci. Instrum. 40, 761 (1969);S. N. Jasperson, D. K. Burge, R. C. O'Handley, Surf. Sci. 37, 548 (1973);J. I. Treu, A. B. Callendar, S. E. Schnatterly, Rev. Sci. Instrum. 44, 793 (1973).
    [CrossRef]
  24. F. W. J. Olver, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds. (U.S. Nat. Bur. Stand., Washington, D.C., 1964), Appl. Math. Ser., Vol. 55, p. 361.
  25. J. L. Ord, B. L. Wills, Appl. Opt. 6, 1673 (1967);J. L. Ord, Surf. Sci. 16, 155 (1969).
    [CrossRef] [PubMed]
  26. H. F. Hazebroeck, A. A. Holscher, J. Phys. E6, 822 (1973).

1974 (2)

H. J. Mathieu, D. E. McClure, R. H. Muller, Rev. Sci. Instrum. 45, 798 (1974).
[CrossRef]

D. E. Aspnes, J. Opt. Soc. Am. 64, 639 (1974).
[CrossRef]

1973 (4)

D. E. Aspnes, Opt. Commun. 8, 222 (1973);D. E. Aspnes, A. A. Studna, Appl. Opt. 14, 220 (1975).
[CrossRef] [PubMed]

P. S. Hauge, F. H. Dill, IBM J. Res. Dev. 17, 472 (1973);Y. J. van der Meulen, N. C. Hien, J. Opt. Soc. Am. 64, 804 (1974).
[CrossRef]

H. F. Hazebroeck, A. A. Holscher, J. Phys. E6, 822 (1973).

D. J. Scholtens, J. F. Kleibeuker, J. Kommandeur, Rev. Sci. Instrum. 44, 153 (1973).
[CrossRef]

1972 (1)

D. E. Aspnes, Phys. Rev. Lett. 28, 168 (1972);in Proceedings of the Twelfth International Conference on the Physics of Semiconductors, M. Pilkuhn, Ed. (Springer-Verlag, Berlin, 1974), p. 1197.
[CrossRef]

1970 (1)

R. Greef, Rev. Sci. Instrum. 41, 532 (1970).
[CrossRef]

1969 (5)

B. D. Cahan, R. F. Spanier, Surf. Sci. 16, 166 (1969);B. D. Cahan, J. Horkans, E. Yeager, Surf. Sci. 37, 559 (1973).
[CrossRef]

R. H. Muller, Surf. Sci. 16, 14 (1969).
[CrossRef]

H. G. Jerrard, Surf. Sci. 16, 137 (1969).
[CrossRef]

I. Willmans, Surf. Sci. 16, 147 (1969).
[CrossRef]

S. N. Jasperson, S. E. Schnatterly, Rev. Sci. Instrum. 40, 761 (1969);S. N. Jasperson, D. K. Burge, R. C. O'Handley, Surf. Sci. 37, 548 (1973);J. I. Treu, A. B. Callendar, S. E. Schnatterly, Rev. Sci. Instrum. 44, 793 (1973).
[CrossRef]

1968 (1)

1967 (1)

1962 (1)

M. Billardon, Ann. Phys. 7, 233 (1962).

1961 (1)

1952 (1)

J. F. Archard, P. L. Clegg, A. M. Taylor, Proc. Phys. Soc. (London) 65B, 758 (1952).

1937 (1)

Archard, J. F.

J. F. Archard, P. L. Clegg, A. M. Taylor, Proc. Phys. Soc. (London) 65B, 758 (1952).

Aspnes, D. E.

D. E. Aspnes, J. Opt. Soc. Am. 64, 639 (1974).
[CrossRef]

D. E. Aspnes, Opt. Commun. 8, 222 (1973);D. E. Aspnes, A. A. Studna, Appl. Opt. 14, 220 (1975).
[CrossRef] [PubMed]

D. E. Aspnes, Phys. Rev. Lett. 28, 168 (1972);in Proceedings of the Twelfth International Conference on the Physics of Semiconductors, M. Pilkuhn, Ed. (Springer-Verlag, Berlin, 1974), p. 1197.
[CrossRef]

Billardon, M.

M. Billardon, Ann. Phys. 7, 233 (1962).

Cahan, B. D.

B. D. Cahan, R. F. Spanier, Surf. Sci. 16, 166 (1969);B. D. Cahan, J. Horkans, E. Yeager, Surf. Sci. 37, 559 (1973).
[CrossRef]

Clegg, P. L.

J. F. Archard, P. L. Clegg, A. M. Taylor, Proc. Phys. Soc. (London) 65B, 758 (1952).

Dill, F. H.

P. S. Hauge, F. H. Dill, IBM J. Res. Dev. 17, 472 (1973);Y. J. van der Meulen, N. C. Hien, J. Opt. Soc. Am. 64, 804 (1974).
[CrossRef]

Faber, T. E.

Greef, R.

R. Greef, Rev. Sci. Instrum. 41, 532 (1970).
[CrossRef]

Hauge, P. S.

P. S. Hauge, F. H. Dill, IBM J. Res. Dev. 17, 472 (1973);Y. J. van der Meulen, N. C. Hien, J. Opt. Soc. Am. 64, 804 (1974).
[CrossRef]

Hazebroeck, H. F.

H. F. Hazebroeck, A. A. Holscher, J. Phys. E6, 822 (1973).

Holscher, A. A.

H. F. Hazebroeck, A. A. Holscher, J. Phys. E6, 822 (1973).

Jasperson, S. N.

S. N. Jasperson, S. E. Schnatterly, Rev. Sci. Instrum. 40, 761 (1969);S. N. Jasperson, D. K. Burge, R. C. O'Handley, Surf. Sci. 37, 548 (1973);J. I. Treu, A. B. Callendar, S. E. Schnatterly, Rev. Sci. Instrum. 44, 793 (1973).
[CrossRef]

Jerrard, H. G.

H. G. Jerrard, Surf. Sci. 16, 137 (1969).
[CrossRef]

Kent, C. V.

Kleibeuker, J. F.

D. J. Scholtens, J. F. Kleibeuker, J. Kommandeur, Rev. Sci. Instrum. 44, 153 (1973).
[CrossRef]

Kommandeur, J.

D. J. Scholtens, J. F. Kleibeuker, J. Kommandeur, Rev. Sci. Instrum. 44, 153 (1973).
[CrossRef]

Lawson, J.

Mathieu, H. J.

H. J. Mathieu, D. E. McClure, R. H. Muller, Rev. Sci. Instrum. 45, 798 (1974).
[CrossRef]

McClure, D. E.

H. J. Mathieu, D. E. McClure, R. H. Muller, Rev. Sci. Instrum. 45, 798 (1974).
[CrossRef]

Muller, R. H.

H. J. Mathieu, D. E. McClure, R. H. Muller, Rev. Sci. Instrum. 45, 798 (1974).
[CrossRef]

R. H. Muller, Surf. Sci. 16, 14 (1969).
[CrossRef]

Olver, F. W. J.

F. W. J. Olver, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds. (U.S. Nat. Bur. Stand., Washington, D.C., 1964), Appl. Math. Ser., Vol. 55, p. 361.

Ord, J. L.

Schnatterly, S. E.

S. N. Jasperson, S. E. Schnatterly, Rev. Sci. Instrum. 40, 761 (1969);S. N. Jasperson, D. K. Burge, R. C. O'Handley, Surf. Sci. 37, 548 (1973);J. I. Treu, A. B. Callendar, S. E. Schnatterly, Rev. Sci. Instrum. 44, 793 (1973).
[CrossRef]

Scholtens, D. J.

D. J. Scholtens, J. F. Kleibeuker, J. Kommandeur, Rev. Sci. Instrum. 44, 153 (1973).
[CrossRef]

Smith, N. V.

Spanier, R. F.

B. D. Cahan, R. F. Spanier, Surf. Sci. 16, 166 (1969);B. D. Cahan, J. Horkans, E. Yeager, Surf. Sci. 37, 559 (1973).
[CrossRef]

Takasaki, H.

Taylor, A. M.

J. F. Archard, P. L. Clegg, A. M. Taylor, Proc. Phys. Soc. (London) 65B, 758 (1952).

Willmans, I.

I. Willmans, Surf. Sci. 16, 147 (1969).
[CrossRef]

Wills, B. L.

Winterbottom, A. B.

A. B. Winterbottom, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger, Eds. (U.S. Nat. Bur. Stand., Washington, D.C., 1964), Misc. Pub. 256, p. 97.

Ann. Phys. (1)

M. Billardon, Ann. Phys. 7, 233 (1962).

Appl. Opt. (1)

IBM J. Res. Dev. (1)

P. S. Hauge, F. H. Dill, IBM J. Res. Dev. 17, 472 (1973);Y. J. van der Meulen, N. C. Hien, J. Opt. Soc. Am. 64, 804 (1974).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Phys. (1)

H. F. Hazebroeck, A. A. Holscher, J. Phys. E6, 822 (1973).

Opt. Commun. (1)

D. E. Aspnes, Opt. Commun. 8, 222 (1973);D. E. Aspnes, A. A. Studna, Appl. Opt. 14, 220 (1975).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

D. E. Aspnes, Phys. Rev. Lett. 28, 168 (1972);in Proceedings of the Twelfth International Conference on the Physics of Semiconductors, M. Pilkuhn, Ed. (Springer-Verlag, Berlin, 1974), p. 1197.
[CrossRef]

Proc. Phys. Soc. (London) (1)

J. F. Archard, P. L. Clegg, A. M. Taylor, Proc. Phys. Soc. (London) 65B, 758 (1952).

Rev. Sci. Instrum. (4)

S. N. Jasperson, S. E. Schnatterly, Rev. Sci. Instrum. 40, 761 (1969);S. N. Jasperson, D. K. Burge, R. C. O'Handley, Surf. Sci. 37, 548 (1973);J. I. Treu, A. B. Callendar, S. E. Schnatterly, Rev. Sci. Instrum. 44, 793 (1973).
[CrossRef]

H. J. Mathieu, D. E. McClure, R. H. Muller, Rev. Sci. Instrum. 45, 798 (1974).
[CrossRef]

D. J. Scholtens, J. F. Kleibeuker, J. Kommandeur, Rev. Sci. Instrum. 44, 153 (1973).
[CrossRef]

R. Greef, Rev. Sci. Instrum. 41, 532 (1970).
[CrossRef]

Surf. Sci. (4)

R. H. Muller, Surf. Sci. 16, 14 (1969).
[CrossRef]

B. D. Cahan, R. F. Spanier, Surf. Sci. 16, 166 (1969);B. D. Cahan, J. Horkans, E. Yeager, Surf. Sci. 37, 559 (1973).
[CrossRef]

H. G. Jerrard, Surf. Sci. 16, 137 (1969).
[CrossRef]

I. Willmans, Surf. Sci. 16, 147 (1969).
[CrossRef]

Other (7)

F. W. J. Olver, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds. (U.S. Nat. Bur. Stand., Washington, D.C., 1964), Appl. Math. Ser., Vol. 55, p. 361.

Values used here: uncooled near ir-sensitive photomultiplier, S1 response: Inep = 6 × 10−13 W, η0 = 0.006, ℏω0 = 3.3 eV, Δf0 = 1 sec−1; visible-uv photomultiplier, S20 response: Inep = 4 × 10−16 W, η0 = 0.29, ℏω0 = 3.3 eV, Δf0 = 1 sec−1. These values pertain to models 9684B and 9558QB photomultiplier detectors manufactured by EMI Electronics, Ltd., Hayes, Middelsex, England.

A. B. Winterbottom, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger, Eds. (U.S. Nat. Bur. Stand., Washington, D.C., 1964), Misc. Pub. 256, p. 97.

RCA Photomultiplier Manual (RCA Corporation, Harrison, N.J., 1970).

This form supposes each event is counted independently, which requires continuous integration or digital sampling at a sufficiently rapid rate to follow the highest frequency component in ṅ(t).

For an ideal digital system, T equals the total data acquisition time. For an ideal filter with bandwidth Δf, T = 1/(2Δf). We assume sufficient averaging so that the contribution to the fluctuation term arising from terminating a partially completed detection cycle is negligible.

See, for instance, the following sources: Ellipsometry in the Measurement of Surfaces and Thin Solid Films, E. Passaglia, R. R. Stromberg, J. Kruger, Eds. (U.S. Nat. Bur. Stand., Washington, D.C., 1964) Misc. Pub. 256;A. C. Hall, N. M. Bashara, A. B. Buckman, Eds., Proceedings of the Symposium on Recent Developments in Ellipsometry, Surf. Sci. 16, (1969);review papers by R. H. Muller, J. Krueger, in Advances in Electrochemistry and Electrochemical Engineering, R. H. Muller, Ed. (Academic, New York, 1973), Vol. 9.

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Figures (1)

Fig. 1
Fig. 1

Theoretical attainable precision for a null ellipsometer (NE), a modulated null ellipsometer (MNE), and a representative photometric system, a rotating-analyzer ellipsometer (RAE). The curves are calculated from Eqs. (15), (16), and (23), respectively, using the parameter values ℏω = 2 eV, η = 0.3, T = 1 sec, γ = 1 × 10−5, and ΔA = 0.01 = 0.57°. The numbers on the curves give the values of log10(dT), where dT is the total number of events due to detector dark current and stray light registered during the measurement interval, assumed here to be 1 sec.

Equations (37)

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i = η I ( t ) / ( ω ) ,
d = [ η 0 I nep / ( ω 0 ) ] 2 / ( 2 Δ f 0 ) + s ,
1 = γ η I 0 / ( ω ) ,
δ n ν = ( n ν ) 1 / 2 ,
n ν = t ν t ν + Δ t ( t ) dt .
N s = 0 T ( t ) f ( t ) dt ,
δ N s = [ 0 T ( t ) f 2 ( t ) dt ] 1 / 2 ,
N s δ N s .
I ( t ) = I 0 + I c cos ( ω t ) + I s sin ( ω t ) ,
| I c | / I 0 2 N 1 / 2 1.41 N 1 / 2 ,
N = η I 0 T / ( ω )
= i T .
I c / I 0 ( π / 2 ) N 1 / 2 1.57 N 1 / 2 .
r p / r s = tan ψ exp ( i Δ ) ,
I ( t ) = [ I 0 ( | r p | 2 + | r s | 2 ) / 2 ] × 1 2 [ 1 cos 2 ψ cos 2 A sin 2 ψ sin 2 A sin ( Δ 2 P ) ] ,
δ ψ = δ A ;
δ Δ = 2 δ P ;
I I 0 ( δ A 2 + δ P 2 ) .
2 δ ψ = δ Δ = 2 δ A = 2 δ P = 2 [ ( δ A 2 + δ P 2 + γ + d / i ) / N ] 1 / 4 ,
2 δ ψ = δ Δ = 2 δ A = 2 δ P = ( 5 / 2 ) 1 / 2 { [ 1 + 4 ( γ + d / n i ) / ( 5 Δ A 2 ) ] / N } 1 / 2 .
Δ A ( γ + d / i ) 1 / 2
= [ γ + d ω / ( η I 0 ) ] 1 / 2 .
I = 1 4 I 0 { | r p | 2 ( 1 sin Δ θ P ) ( 1 + cos 2 A cos Δ θ A ) + | r s | 2 ( 1 + sin Δ θ P ) ( 1 cos 2 A cos Δ θ A ) 2 | r s | | r p | cos Δ θ p [ sin ( Δ 2 P ) sin 2 A cos Δ θ A + cos ( Δ 2 P ) sin Δ θ A ] } ,
I I 0 ( δ A 2 + δ P 2 + Δ θ 2 / 2 + δ A · Δ θ + δ P · Δ θ ) ,
2 δ ψ = δ Δ = 2 δ A = 2 δ P = ( 32 / 5 ) 1 / 2 { [ 1 + 16 ( γ + d / i ) / ( 5 Δ θ ) 2 ] / N } 1 / 2 ,
I s ( t ) = 1 2 I 0 ( | r s | 2 sin 2 P + | r p | 2 cos 2 P ) × ( 1 + α cos 2 A + β sin 2 A ) ,
α + i β = ( tan 2 ψ tan 2 P + 2 i tan ψ tan P cos Δ ) / ( tan 2 ψ + tan 2 P ) .
2 δ ψ = δ Δ = δ α = δ β = 2 [ ( 1 + γ + 2 d / i ) / N ] 1 / 2 .
2 δ ψ = δ Δ = 2 δ P = 2 δ Δ = 2 2 [ ( 1 + γ + 4 d / i ) / N ] 1 / 2 .
I ( t ) = [ I 0 2 ( | r p | 2 + | r s | 2 ) ] × 1 2 [ 1 + α cos ( Δ θ sin ω t ) + β sin ( Δ θ sin ω t ) ] ,
α + i β = cos 2 ψ + i sin 2 ψ sin Δ .
cos ( Δ θ sin ω t ) = J 0 ( Δ θ ) + 2 J 2 ( Δ θ ) cos ( 2 ω t ) + . . . ,
sin ( Δ θ sin ω t ) = 2 J 1 ( Δ θ ) sin ω t + 2 J 3 ( Δ θ ) sin ( 3 ω t ) + . . . ,
δ α = 2 δ ψ = [ 1 / J 1 ( Δ θ 0 ) ] [ ( 1 + γ + 2 d / i ) / N ] 1 / 2 ,
δ β = δ Δ = [ 1 / J 2 ( Δ θ 0 ) ] [ ( 1 + γ + 2 d / i ) / N ] 1 / 2 ,
Δ A > ( γ + d / i ) 1 / 2 ,
d < i / 2 .

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