Abstract

When the film thickness is considered as a parameter, a system composed of a transparent film on an absorbing substrate (in a transparent ambient) is characterized by a range of principal angle ϕ¯minϕ¯ϕ¯max over which the associated principal azimuth ψ¯ varies between 0° and 90° (i.e., 0°ψ¯90°) and the reflection phase difference Δ assumes either one of the two values: +π/2 or −π/2. We determine the principal angle ϕ¯(d) and principal azimuth ψ¯(d) as functions of film thickness d for the vacuum-SiO2-Si system at several wavelengths as a concrete example. When the film thickness exceeds a certain minimum value, more than one principal angle becomes possible, as can be predicted by a simple graphical construction. We apply the results to principal-angle ellipsometry (PAE) of film-substrate systems; the relationship between ϕ¯ and ψ¯ during film growth is particularly interesting.

© 1977 Optical Society of America

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  1. These definitions of the principal angle and the principal azimuth can be extended to include the case of an arbitrary transparent ambient.We should also mention that the value of Δ = + π/2, instead of −π/2, is based on the choice of conventions discussed in the paper by R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surface Sci. 16, 14–33 (1969)[also in Proceedings of the Symposium on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North-Holland, Amsterdam, (1969)].
    [Crossref]
  2. See, for example, K. Kinosita and M. Yamamoto, “Principal-Angle-of-Incidence Ellipsometry,” Surface Sci. 56, 64–75 (1976)[also in Proceedings of the Third International Conference on Ellipsometry, edited by N. M. Bashara and R. M. A. Azzam (North-Holland, Amsterdam, 1976)].
    [Crossref]
  3. P is measured from the plane of incidence, positive in a counterclockwise sense when looking into the beam.
  4. C. V. Kent and J. Lawson, “A photoelectric method for the determination of the parameters of elliptically polarized light,” J. Opt. Soc. Am. 27, 117–144 (1937).
    [Crossref]
  5. H. M. O’Bryan, “The Optical Constants of Several Metals in Vacuum,” J. Opt. Soc. Am. 26, 122–127 (1936).
    [Crossref]
  6. At this uv spectral line of mercury, the refractive indices of SiO2 and Si are assumed to be 1.5 and (1.67-j 3.59), respectively.[Ellipsometric Tables of the Si-SiO2 System for Mercury and He-Ne Laser Spectral Lines, edited by G. Gergely (Akademiai Kiado, Budapest, 1971).]
  7. R. M. A. Azzam, A.-R. M. Zaghloul, and N. M. Bashara, “Ellipsometric function of a film-substrate system: Applications to the design of reflection-type optical devices and to ellipsometry,” J. Opt. Soc. Am. 65, 252–260 (1975).
    [Crossref]
  8. See Eq. (13) and Fig. 3 of Ref. 7 and also Fig. 3(b) of this paper.
  9. A11 CAIC’s between 66.3° and 84.7° that appear in Fig. 2 intersect the imaginary axis of the complex p plane at four points. It is clear, however, that two and three points of intersection (and tangency) will occur, e.g., at angles of incidence between 66.3° and 75°.
  10. A.-R. M. Zaghloul, R. M. A. Azzam, and N. M. Bashara, “Design of film-substrate single-reflection retarders,” J. Opt. Soc. Am. 65, 1043–1049 (1975).
    [Crossref]
  11. R. M. A. Azzam, A.-R. M. Zaghloul, and N. M. Bashara, “Design of film-substrate single-reflection linear partial polarizers,” J. Opt. Soc. Am. 65, 1472–1474 (1975).
    [Crossref]
  12. R. M. A. Azzam, A.-R. M. Zaghloul, and N. M. Bashara, “Polarizer-surface-analyzer null ellipsometry for film-substrate systems,” J. Opt. Soc. Am. 65, 1464–1471 (1975).
    [Crossref]
  13. At these five wavelengths, the refractive indices of SiO2 are 1.48, 1.475, 1.47, 1.46, and 1.46, and those of Si are (5.06-j 3.04), (6. 63-j 2.74), (5. 63-j 0.29), (4. 83-j 0.116), and (3.85-j 0.02), respectively (after Gergely, Ref. 6).
  14. Exact symmetry of ϕ¯(d) and ψ¯(d) around the line d= ds occurs in the limit of zero absorption in the substrate.
  15. The reduced-thickness curve (RTC) is obtained by subtracting from the ordinate of each point on the line d= const the proper multiple of the thickness period Dϕ that is required to bring that point vertically down below the Dϕ boundary curve of the reduced-thickness zone (RTZ) (see the discussion in Ref. 12).
  16. The equations that give d when m1= m2 and m1= m2+ 1 are the same as Eqs. (18a) and (18b), respectively, in Ref. 12.
  17. See, for example, the method discussed in Sec. IV of Ref. 7.
  18. See, for example, M. M. Ibrahim and N. M. Bashara, “Parameter correlation and computational considerations in multiple-angle ellipsometry,” J. Opt. Soc. Am. 61, 1622–1629 (1971).
    [Crossref]
  19. This contour is readily derived from Fig. 4 by plotting vs for different values of d.
  20. It is interesting to observe that when the film-substrate system acts as p or s reflection polarizer, such a condition is detected experimentally by the extinction of the reflected beam (the sample under measurement and the polarizer of the ellipsometer now operate as a pair of crossed polarizers). The null in both O’Bryan and Kent and Lawson’s ellipsometers becomes due to the reflected beam being extinguished, and not because it is circularly polarized.
  21. Furthermore, the coincident branches also become exactly symmetrical around the ψ¯=45° line.

1976 (1)

See, for example, K. Kinosita and M. Yamamoto, “Principal-Angle-of-Incidence Ellipsometry,” Surface Sci. 56, 64–75 (1976)[also in Proceedings of the Third International Conference on Ellipsometry, edited by N. M. Bashara and R. M. A. Azzam (North-Holland, Amsterdam, 1976)].
[Crossref]

1975 (4)

1971 (1)

1969 (1)

These definitions of the principal angle and the principal azimuth can be extended to include the case of an arbitrary transparent ambient.We should also mention that the value of Δ = + π/2, instead of −π/2, is based on the choice of conventions discussed in the paper by R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surface Sci. 16, 14–33 (1969)[also in Proceedings of the Symposium on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North-Holland, Amsterdam, (1969)].
[Crossref]

1937 (1)

1936 (1)

Azzam, R. M. A.

Bashara, N. M.

Ibrahim, M. M.

Kent, C. V.

Kinosita, K.

See, for example, K. Kinosita and M. Yamamoto, “Principal-Angle-of-Incidence Ellipsometry,” Surface Sci. 56, 64–75 (1976)[also in Proceedings of the Third International Conference on Ellipsometry, edited by N. M. Bashara and R. M. A. Azzam (North-Holland, Amsterdam, 1976)].
[Crossref]

Lawson, J.

Muller, R. H.

These definitions of the principal angle and the principal azimuth can be extended to include the case of an arbitrary transparent ambient.We should also mention that the value of Δ = + π/2, instead of −π/2, is based on the choice of conventions discussed in the paper by R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surface Sci. 16, 14–33 (1969)[also in Proceedings of the Symposium on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North-Holland, Amsterdam, (1969)].
[Crossref]

O’Bryan, H. M.

Yamamoto, M.

See, for example, K. Kinosita and M. Yamamoto, “Principal-Angle-of-Incidence Ellipsometry,” Surface Sci. 56, 64–75 (1976)[also in Proceedings of the Third International Conference on Ellipsometry, edited by N. M. Bashara and R. M. A. Azzam (North-Holland, Amsterdam, 1976)].
[Crossref]

Zaghloul, A.-R. M.

J. Opt. Soc. Am. (7)

Surface Sci. (2)

These definitions of the principal angle and the principal azimuth can be extended to include the case of an arbitrary transparent ambient.We should also mention that the value of Δ = + π/2, instead of −π/2, is based on the choice of conventions discussed in the paper by R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surface Sci. 16, 14–33 (1969)[also in Proceedings of the Symposium on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North-Holland, Amsterdam, (1969)].
[Crossref]

See, for example, K. Kinosita and M. Yamamoto, “Principal-Angle-of-Incidence Ellipsometry,” Surface Sci. 56, 64–75 (1976)[also in Proceedings of the Third International Conference on Ellipsometry, edited by N. M. Bashara and R. M. A. Azzam (North-Holland, Amsterdam, 1976)].
[Crossref]

Other (12)

P is measured from the plane of incidence, positive in a counterclockwise sense when looking into the beam.

See Eq. (13) and Fig. 3 of Ref. 7 and also Fig. 3(b) of this paper.

A11 CAIC’s between 66.3° and 84.7° that appear in Fig. 2 intersect the imaginary axis of the complex p plane at four points. It is clear, however, that two and three points of intersection (and tangency) will occur, e.g., at angles of incidence between 66.3° and 75°.

At this uv spectral line of mercury, the refractive indices of SiO2 and Si are assumed to be 1.5 and (1.67-j 3.59), respectively.[Ellipsometric Tables of the Si-SiO2 System for Mercury and He-Ne Laser Spectral Lines, edited by G. Gergely (Akademiai Kiado, Budapest, 1971).]

At these five wavelengths, the refractive indices of SiO2 are 1.48, 1.475, 1.47, 1.46, and 1.46, and those of Si are (5.06-j 3.04), (6. 63-j 2.74), (5. 63-j 0.29), (4. 83-j 0.116), and (3.85-j 0.02), respectively (after Gergely, Ref. 6).

Exact symmetry of ϕ¯(d) and ψ¯(d) around the line d= ds occurs in the limit of zero absorption in the substrate.

The reduced-thickness curve (RTC) is obtained by subtracting from the ordinate of each point on the line d= const the proper multiple of the thickness period Dϕ that is required to bring that point vertically down below the Dϕ boundary curve of the reduced-thickness zone (RTZ) (see the discussion in Ref. 12).

The equations that give d when m1= m2 and m1= m2+ 1 are the same as Eqs. (18a) and (18b), respectively, in Ref. 12.

See, for example, the method discussed in Sec. IV of Ref. 7.

This contour is readily derived from Fig. 4 by plotting vs for different values of d.

It is interesting to observe that when the film-substrate system acts as p or s reflection polarizer, such a condition is detected experimentally by the extinction of the reflected beam (the sample under measurement and the polarizer of the ellipsometer now operate as a pair of crossed polarizers). The null in both O’Bryan and Kent and Lawson’s ellipsometers becomes due to the reflected beam being extinguished, and not because it is circularly polarized.

Furthermore, the coincident branches also become exactly symmetrical around the ψ¯=45° line.

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Figures (7)

FIG. 1
FIG. 1

The constant-angle-of-incidence contour (CAIC) at ϕ =75° for the vacuum-SiO2-Si system at λ = 2537 Å intersects the imaginary axis of the complex ρ plane at four points I1, I2, I3, and I4. Consequently, ϕ = 75° is a principal angle for this system at four discrete values of film thickness in each film-thickness period. The point B on the curve represents the bare substrate (d = 0) and the arrow indicates the direction of increase of film thickness d.

FIG. 2
FIG. 2

A family of CAIC’s in the complex p plane for the vacuum-SiO2-Si system,at λ = 2537 Å. The angle of incidence in degrees is marked by each contour. The range of principal angle is specified by the limiting contours at 66.3° and 84.7° that touch the imaginary axis, but otherwise lies entirely to its left and entirely to its right, respectively.

FIG. 3
FIG. 3

Mapping of the imaginary axis of the complex ρ plane (a) into the real plane of the angle of incidence ϕ and film thickness d (b). Points that are images of one another are denoted by the same letter. B represents the bare substrate, and P and S represent zero ρ and s reflectances for the system, respectively. In Fig. 3 (b) the image BSPB’ of the imaginary axis is closed by the vertical transition shown by tiie dashed line between B′ and B. The film thickness at B′ equals the film-thickness period D ϕ ¯ B evaluated at the principal angle ϕ ¯ B of the bare substrate. The film-thickness period Dϕ defines the upper boundary of the reduced-thickness zone (RTZ) in the ϕd plane. In Fig. 3(b) the image of the positive imaginary axis (Δ = + π/2) is represented by PB’BS, while that of the negative imaginary axis (Δ = −π/2) is represented by SP. We assume the vacuum-SiO2-Si system at λ = 2537 Å.

FIG. 4
FIG. 4

The principal angle ϕ ¯ and the principal azimuth in degrees as functions of film thickness d in Å for the vacuum-SiO2-Si system at λ = 2537 Å. The points B, S, and P represent the bare substrate (d = 0), the s-suppressing polarizer (d = ds), and the p-suppressing polarizer (d = dp), respectively.

FIG. 5
FIG. 5

Same as in Fig. 4 for five other different wavelengths: (a) 3341 Å, (b) 3650 Å, (c) 4046 Å, (d) 4358 Å, and (e) 6328 Å.

FIG. 6
FIG. 6

Graphical construction for the determination of all principal angles of incidence that correspond to a given value of film thickness. The reduced-thickness curve (RTC) for d = 10λ intersects the principal-angle curve (PAC) at three points 1, 2, and 3, leading to three principal angles of incidence. The vacuum-SiO2-Si system at λ = 2537 Å is assumed.

FIG. 7
FIG. 7

The principal angle ϕ ¯ is plotted against the principal azimuth ψ ¯ to represent the growth of SiO2 film on Si as observed by principal-angle ellipsometry (PAE) at λ = 2537 Å. B represents the bare substrate (d = 0) and the arrow indicates the direction of increasing film thickness. The marks correspond to 50 Å increments. P and S are the points at which the film-substrate system acts as p - and s-suppressing reflection polarizer, respectively; R+ and R represent the points at which the system operates as a quarter-wave retarder with the fast axis parallel (p) and perpendicular (s) to the plane of incidence, respectively. The bottom and top branches of the contour are the images of the positive (Δ = + π/2) and negative (Δ = − π/2) branches of the imaginary axis of the complex ρ plane [Fig. 3(a)], respectively.