Abstract

Single-pass polarizer–surface-analyzer null ellipsometry (PSA-NE) can be used to characterize film–substrate systems, provided that the film thickness lies within one of a set of permissible-thickness bands (PTB). For a transparent film on a transparent or absorbing substrate, the PTB structure consists of a small number of finite-bandwidth bands followed by a continuum band that extends from a film thickness of about half the wavelength of light to infinity. We show that this band structure is a direct consequence of the periodicity of the ellipsometric function ρ (the ratio Rp/Rs of the complex amplitude-reflection coefficients for the p and s polarizations) with film thickness. The PTB for the SiO2-Si film–substrate system at He–Ne laser and mercury spectral lines are calculated. The angles of incidence for PSA-NE on a film-substrate system with known film thickness are easily predicted with the help of a graphical construction in the angle of incidence-vs-thickness ϕd plane. PSA-NE is generally applicable to the determination of both film thickness and optical properties of a film-substrate system. The procedure for its application to the special, but important, case of film-thickness measurement alone, when the optical properties are known, is given and is checked experimentally by the determination of the oxide-film thickness on Si wafers. In an automated form, PSA-NE can be a serious competitor for interferometric reflectance methods.

© 1975 Optical Society of America

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References

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  1. R. J. Archer, J. Opt. Soc. Am. 52, 970 (1962); D. A. Holmes and D. L. Feucht, J. Opt. Soc. Am. 57, 466 (1967); W. G. Oldham, J. Opt. Soc. Am. 47, 617 (1967); P. H. Smith, Surf. Sci. 16, 34 (1969); F. L. McCrackin, J. Opt. Soc. Am. 60, 57 (1970); J. A. Johnson and N. M. Bashara, J. Opt. Soc. Am. 60, 221 (1970); R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
    [Crossref] [PubMed]
  2. H. M. O’Bryan, J. Opt. Soc. Am. 26, 122 (1936). In this paper, O’Bryan mentions the similarity of his idea to that of Brewster, who over a century earlier [Philos. Trans. 69, 133 (1830)] observed that light multiply reflected between a pair of parallel mirrors can be extinguished between two Nicol prisms. In this case, the phase shift per reflection is π/M, so that for M reflections, the cummulative shift is (π/M) × M= π.
    [Crossref]
  3. M. Yamamoto, Opt. Commun. 10, 200 (1974).
    [Crossref]
  4. R. M. A. Azzam, A. -R. M. Zaghloul, and N. M. Bashara, J. Opt. Soc. Am. 65, 252 (1975).
    [Crossref]
  5. The existence of PTB has also been encountered in the design of film–substrate reflection retarders; A. -R. M. Zaghloul, R. M. A. Azzam, and N. M. Bashara, J. Opt. Soc. Am. 65, 1043 (1975).
    [Crossref]
  6. See Eq. (13) and Fig. 3 of Ref. 4.
  7. s or p suppression refers to the total extinction of the s or p components of the incident light upon reflection and identifies the condition when the film–substrate system acts as a reflection polarizer.
  8. The vertical translation of a base band by a multiple of the thickness period preserves the range of ψ(high or low).
  9. Note the difference of notation between dl1m and dmldl1m indicates the lower edge of a translated band, whereas dml is the lower edge of a higher band. A translated band may or may not constitute a higher band, dependent on the overlapping of all translated bands.
  10. Elliipsometric Tables of the Si-SiO2 System for Mercury and He–Ne Laser Spectral Lines, edited by G. Gergely (Akademiai Kiado, Budapest, 1971).
  11. W. A. Pliskin, in Progress in Analytic Chemistry; Vol. 2, edited by E. M. Murt and W. G. Guldner (Plenum, New York, 1969), pp. 1–34.
  12. B. D. Cahan and R. F. Spanier, Surf. Sci. 16, 166 (1969).
    [Crossref]
  13. D. E. Aspnes, Opt. Commun. 8, 222 (1973); D. E. Aspnes and A. A. Studna, Appl. Opt. 14, 220 (1975).
    [Crossref] [PubMed]
  14. P. S. Hauge and F. H. Dill, IBM J. Res. Devel. 17, 472 (1973).
    [Crossref]
  15. Y. J. van der Meulen and N. C. Hien, J. Opt. Soc. Am. 64, 804 (1974).
    [Crossref]
  16. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 64, 1459 (1974).
    [Crossref]
  17. An important parameter for the precise determination of the null by the PSA ellipsometer is the value of the derivative ∂Δ/∂ϕ at the nulling angle ϕ. This derivative determines how rapidly the reflected light becomes elliptically polarized as the angle of incidence is offset from its null position. An initial investigation shows a tendency for high values of ∂Δ/∂ϕ at null, in most cases, leading to very precise null definition.

1975 (2)

1974 (3)

1973 (2)

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

P. S. Hauge and F. H. Dill, IBM J. Res. Devel. 17, 472 (1973).
[Crossref]

1969 (1)

B. D. Cahan and R. F. Spanier, Surf. Sci. 16, 166 (1969).
[Crossref]

1962 (1)

1936 (1)

Archer, R. J.

Aspnes, D. E.

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

Azzam, R. M. A.

Bashara, N. M.

Cahan, B. D.

B. D. Cahan and R. F. Spanier, Surf. Sci. 16, 166 (1969).
[Crossref]

Dill, F. H.

P. S. Hauge and F. H. Dill, IBM J. Res. Devel. 17, 472 (1973).
[Crossref]

Hauge, P. S.

P. S. Hauge and F. H. Dill, IBM J. Res. Devel. 17, 472 (1973).
[Crossref]

Hien, N. C.

O’Bryan, H. M.

Pliskin, W. A.

W. A. Pliskin, in Progress in Analytic Chemistry; Vol. 2, edited by E. M. Murt and W. G. Guldner (Plenum, New York, 1969), pp. 1–34.

Spanier, R. F.

B. D. Cahan and R. F. Spanier, Surf. Sci. 16, 166 (1969).
[Crossref]

van der Meulen, Y. J.

Yamamoto, M.

M. Yamamoto, Opt. Commun. 10, 200 (1974).
[Crossref]

Zaghloul, A. -R. M.

IBM J. Res. Devel. (1)

P. S. Hauge and F. H. Dill, IBM J. Res. Devel. 17, 472 (1973).
[Crossref]

J. Opt. Soc. Am. (6)

Opt. Commun. (2)

M. Yamamoto, Opt. Commun. 10, 200 (1974).
[Crossref]

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

Surf. Sci. (1)

B. D. Cahan and R. F. Spanier, Surf. Sci. 16, 166 (1969).
[Crossref]

Other (7)

An important parameter for the precise determination of the null by the PSA ellipsometer is the value of the derivative ∂Δ/∂ϕ at the nulling angle ϕ. This derivative determines how rapidly the reflected light becomes elliptically polarized as the angle of incidence is offset from its null position. An initial investigation shows a tendency for high values of ∂Δ/∂ϕ at null, in most cases, leading to very precise null definition.

See Eq. (13) and Fig. 3 of Ref. 4.

s or p suppression refers to the total extinction of the s or p components of the incident light upon reflection and identifies the condition when the film–substrate system acts as a reflection polarizer.

The vertical translation of a base band by a multiple of the thickness period preserves the range of ψ(high or low).

Note the difference of notation between dl1m and dmldl1m indicates the lower edge of a translated band, whereas dml is the lower edge of a higher band. A translated band may or may not constitute a higher band, dependent on the overlapping of all translated bands.

Elliipsometric Tables of the Si-SiO2 System for Mercury and He–Ne Laser Spectral Lines, edited by G. Gergely (Akademiai Kiado, Budapest, 1971).

W. A. Pliskin, in Progress in Analytic Chemistry; Vol. 2, edited by E. M. Murt and W. G. Guldner (Plenum, New York, 1969), pp. 1–34.

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Figures (9)

FIG. 1
FIG. 1

Polarizer (P)–surface (S)–analyzer (A) ellipsometer. In the nulling mode, one of the azimuths P or A is fixed and the other, together with the angle of incidence ϕ, are adjusted to extinguish the light transmitted by the PSA optical train.

FIG. 2
FIG. 2

Left. Monotonic Δ -vs-ϕ curves for a bare substrate (Si; continuous line) and a substrate (Si) covered by a very thin film (SiO2; 20 Å; dashed line) (at λ = 6328 Å). Middle. Oscillatory Δ-vs-ϕ curve for a film–substrate (SiO2-Si) system with film thickness (d = 1200 Å) in a PTB (at λ = 6328 Å). Right. Monotonic Δ-vs-ϕ curve for a film–substrate (SiO2-Si) system with a film thickness (d = 2167 Å) that corresponds to a band edge (at λ = 6328 Å).

FIG. 3
FIG. 3

Mapping between the complex plane of the ellipsometric function ρ (left) and the plane of its two real arguments; the angle of incidence ϕ and the film thickness d (right). Points that are the images of one another are identified by the same letter. The real axis of the complex ρ plane is mapped into a closed contour in the ϕd plane, NSGPN. The origin of the PTB for PSA-NE is explained with the help of this figure.

FIG. 4
FIG. 4

The cases of (1) separate (left), and (2) overlapping (right) permissible base bands. In case (2) (right), the two base bands merge together to form one permissible band.

FIG. 5
FIG. 5

The LPB and HPB, dL and dH, respectively, for the SiO2-Si system at λ = 6328 Å. dL and dH are the loci of points (ϕ, d) in the reduced zone (0 ≤ d< Dϕ, 0 ≤ ϕ ≤ 90°) that lead to Δ = 0 or π. ψ = 0 at P and 90° at S. corresponding to p- and s-suppression, respectively. Notice that dLDp and d H 1 2 D ϕ. This can be traced to the small extinction coefficient of Si at λ = 6328 Å. dL = Dϕ, d H = 1 2 D ϕ hold exactly for an all-transparent ambient-film-substrate system.

FIG. 6
FIG. 6

Schematic of the structure of the PTB for PSA-NE on the SiO2-Si system at λ = 6328 Å.

FIG. 7
FIG. 7

The RTC for three choices of the film thickness. Left, 0 < d < D0; middle, D0 < d < D90; right: d > mD90, where m is an integer.

FIG. 8
FIG. 8

Construction for the determination of the angles of incidence at which PSA-NE is possible on a film–substrate system with a known film thickness.

FIG. 9
FIG. 9

The high- and low-psi quadrants of the analyzer transmission axis when the polarizer is set at an azimuth P = 45°.

Tables (3)

Tables Icon

TABLE I Angle of incidence ϕ and reduced film thickness d for a SiO2-Si film–substrate system at λ = 6328 Å that lead to Δ = 0 or π (thus allowing PSA-NE) for different values of ψ in the range 45° < ψ ≤ 90° (HPB). Dϕ is the film-thickness period and p and s are the p and s reflectances.

Tables Icon

TABLE II Same as in Table I but for ψ in the range 0 ≤ ψ < 45°. (LPB).

Tables Icon

TABLE III The PTB for PSA-NE on the SiO2-Si film–substrate system at selected mercury spectral lines.

Equations (28)

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d i l < d < d i u ,             i = 1 , 2 , 3 , , L ,
d i l < d < d i u ,             i = 1 , 2 , 3 , , L - 1 ,
d > d L l ,
ρ = R p / R s
0 < d H < d L < D ϕ .
m D ϕ < d < ( m + 1 ) D ϕ ,             m = 0 , 1 , 2 , 3 , .
ρ = tan ψ e j Δ ,
ρ H > 1 ,             45° < ψ H 90° ,
ρ L < 1 ,             0 ψ L < 45° ,
0 d < D ϕ ,             0 ϕ 90° ,
Δ d G = d L - d H ,
Δ d G > 0 ,             0 ϕ 90° ,
case 1 ,             d L 0 > d H 90 ,
case 2 ,             d L 0 < d H 90 .
d H 0 d H d H 90 ,             d L 0 d L d L 90 ,
d 1 l = d H 0 , d 1 u = d H 90 ; d 2 l = d L 0 , d 2 u = d L 90 .
d 1 l = d H 0 ,             d 1 u = d L 90 .
d l 1 m = d 1 l + m D 0 ,             d u 1 m = d 1 u + m D 90 .
d l 2 n = d 2 l + n D 0 ,             d u 2 n = d 2 u + n D 90 .
1083 < d < 1487 Å , 2167 < d < 2974 Å , d > 3250 Å .
d - k D ϕ = 0 ,             k = m ,             m - 1 ,             m - 2 , .
d = d r i + k D ϕ r i , i = 1 , 2 , 3 , , k = { M , M - 1 , M - 1 , M - 2 , M - 2 , M , M , M - 1 , M - 1 , M - 2 , M - 2 , .
d = ( d 1 D 1 - d 2 D 2 ) / ( 1 D 1 - 1 D 2 ) .
d = d 1 + D 1 [ D 2 - ( d 2 - d 1 ) ] ( D 2 - D 1 ) .
tan ψ = tan P tan A ,
tan ψ = tan A ,
ψ = π - A , when Δ = 0 ψ = A , when Δ = π .
0 < A < 45° , 135° < A < 180° , when 0 < ψ < 45° , 45° < A < 135° , when 45° < ψ < 90° .