Abstract

An analytical solution for the determination of the substrate refractive index of a single-layered system from ellipsometric measurements is presented. It is shown that the above ellipsometric inverse problem is reduced to the finding of the roots of a third-degree polynomial. A unique approximate solution in the case of a thin covering layer is also presented.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
    [CrossRef]
  3. R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (France) C10, 67–70 (1983).
  4. R. M. A. Azzam, “Transmission ellipsometry on transparent unbacked or embedded thin films with application to soap films in air,” Appl. Opt. 30, 2801–2806 (1991).
    [CrossRef] [PubMed]
  5. J. Lekner, “Analytic inversion of ellipsometric data for an unsupported nonabsorbing uniform layer,” J. Opt. Soc. Am. A 7, 1875–1877 (1990).
    [CrossRef]
  6. J. Lekner, “Inversion of reflection ellipsometric data,” Appl. Opt. 33, 5159–5165 (1994).
    [CrossRef] [PubMed]
  7. J. P. Drolet, S. C. Russev, M. I. Boyanov, R. M. Leblanc, “Polynomial inversion of the single transparent layer problem in ellipsometry,” J. Opt. Soc. Am. A 11, 3284–3292 (1994).
    [CrossRef]
  8. S. C. Russev, I. Mircheva, J.-P. Drolet, D. Ducharme, R. M. Leblanc, “Polynomial solution for two thicknesses of a multilayer system from a single ellipsometric measurement,” J. Opt. Soc. Am. A 13, 152–157 (1996).
    [CrossRef]
  9. I. Ohlidal, F. Lukes, “Analysis of semiconductor surfaces with very thin native oxide layers by combined immersion and multiple angle of incidence ellipsometry,” Appl. Surf. Sci. 5, 259–273 (1989).
  10. F. Lukes, “Ellipsometry of silicon with natural surface film at 632.8 nm,” Phys. Status Solidi A 93, 223–230 (1986).
    [CrossRef]
  11. J. P. Moy, “Immersion ellipsometry,” Appl. Opt. 20, 3821–3822 (1986).
  12. H. R. Philipp, “Influence of oxide layers on the determination of optical properties of silicon,” J. Appl. Phys. 43, 2835–2839 (1972).
    [CrossRef]
  13. A. F. Antippa, R. M. Leblanc, D. Ducharme, “Multiple-wavelength ellipsometry in thin uniaxial nonabsorbing films,” J. Appl. Phys. 43, 2835–2839 (1972).
  14. D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films89, 249–262 (1982).
    [CrossRef]
  15. W. S. Dorn, D. D. McCracken, Numerical Methods with fortranIV: Case Studies (Wiley, New York, 1972).

1996

1994

1991

S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
[CrossRef]

R. M. A. Azzam, “Transmission ellipsometry on transparent unbacked or embedded thin films with application to soap films in air,” Appl. Opt. 30, 2801–2806 (1991).
[CrossRef] [PubMed]

1990

1989

I. Ohlidal, F. Lukes, “Analysis of semiconductor surfaces with very thin native oxide layers by combined immersion and multiple angle of incidence ellipsometry,” Appl. Surf. Sci. 5, 259–273 (1989).

1986

F. Lukes, “Ellipsometry of silicon with natural surface film at 632.8 nm,” Phys. Status Solidi A 93, 223–230 (1986).
[CrossRef]

J. P. Moy, “Immersion ellipsometry,” Appl. Opt. 20, 3821–3822 (1986).

1983

R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (France) C10, 67–70 (1983).

1972

H. R. Philipp, “Influence of oxide layers on the determination of optical properties of silicon,” J. Appl. Phys. 43, 2835–2839 (1972).
[CrossRef]

A. F. Antippa, R. M. Leblanc, D. Ducharme, “Multiple-wavelength ellipsometry in thin uniaxial nonabsorbing films,” J. Appl. Phys. 43, 2835–2839 (1972).

Antippa, A. F.

A. F. Antippa, R. M. Leblanc, D. Ducharme, “Multiple-wavelength ellipsometry in thin uniaxial nonabsorbing films,” J. Appl. Phys. 43, 2835–2839 (1972).

Aspnes, D. E.

D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films89, 249–262 (1982).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, “Transmission ellipsometry on transparent unbacked or embedded thin films with application to soap films in air,” Appl. Opt. 30, 2801–2806 (1991).
[CrossRef] [PubMed]

R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (France) C10, 67–70 (1983).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Boyanov, M. I.

Dorn, W. S.

W. S. Dorn, D. D. McCracken, Numerical Methods with fortranIV: Case Studies (Wiley, New York, 1972).

Drolet, J. P.

Drolet, J.-P.

Ducharme, D.

S. C. Russev, I. Mircheva, J.-P. Drolet, D. Ducharme, R. M. Leblanc, “Polynomial solution for two thicknesses of a multilayer system from a single ellipsometric measurement,” J. Opt. Soc. Am. A 13, 152–157 (1996).
[CrossRef]

A. F. Antippa, R. M. Leblanc, D. Ducharme, “Multiple-wavelength ellipsometry in thin uniaxial nonabsorbing films,” J. Appl. Phys. 43, 2835–2839 (1972).

Georgieva, D. D.

S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
[CrossRef]

Leblanc, R. M.

Lekner, J.

Lukes, F.

I. Ohlidal, F. Lukes, “Analysis of semiconductor surfaces with very thin native oxide layers by combined immersion and multiple angle of incidence ellipsometry,” Appl. Surf. Sci. 5, 259–273 (1989).

F. Lukes, “Ellipsometry of silicon with natural surface film at 632.8 nm,” Phys. Status Solidi A 93, 223–230 (1986).
[CrossRef]

McCracken, D. D.

W. S. Dorn, D. D. McCracken, Numerical Methods with fortranIV: Case Studies (Wiley, New York, 1972).

Mircheva, I.

Moy, J. P.

Ohlidal, I.

I. Ohlidal, F. Lukes, “Analysis of semiconductor surfaces with very thin native oxide layers by combined immersion and multiple angle of incidence ellipsometry,” Appl. Surf. Sci. 5, 259–273 (1989).

Philipp, H. R.

H. R. Philipp, “Influence of oxide layers on the determination of optical properties of silicon,” J. Appl. Phys. 43, 2835–2839 (1972).
[CrossRef]

Russev, S. C.

Appl. Opt.

Appl. Surf. Sci.

I. Ohlidal, F. Lukes, “Analysis of semiconductor surfaces with very thin native oxide layers by combined immersion and multiple angle of incidence ellipsometry,” Appl. Surf. Sci. 5, 259–273 (1989).

J. Appl. Phys.

H. R. Philipp, “Influence of oxide layers on the determination of optical properties of silicon,” J. Appl. Phys. 43, 2835–2839 (1972).
[CrossRef]

A. F. Antippa, R. M. Leblanc, D. Ducharme, “Multiple-wavelength ellipsometry in thin uniaxial nonabsorbing films,” J. Appl. Phys. 43, 2835–2839 (1972).

J. Mod. Opt.

S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. (France)

R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (France) C10, 67–70 (1983).

Phys. Status Solidi A

F. Lukes, “Ellipsometry of silicon with natural surface film at 632.8 nm,” Phys. Status Solidi A 93, 223–230 (1986).
[CrossRef]

Other

D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films89, 249–262 (1982).
[CrossRef]

W. S. Dorn, D. D. McCracken, Numerical Methods with fortranIV: Case Studies (Wiley, New York, 1972).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

Real and imaginary parts of the substrate refractive index for the air–SiO2(n=1.46)Si(n=3.88-i*0.018) system as computed by using the zero approximation with Eq. (28) (n20, layer presence neglected), and the first approximation n2 with Eq. (27). The ellipsometric data are simulated for an angle of incidence of 70° and wavelength 632.8 nm. The dashed curve represents exact values.

Equations (45)

Equations on this page are rendered with MathJax. Learn more.

ρ=Rp/Rs.
Rp=r01p+r12pX1+r01pr12pX,
Rs=r01s+r12sX1+r01sr12sX,
r01p=n1 cos ϕ0-n0 cos ϕ1n1 cos ϕ0+n0 cos ϕ1,
r01s=n0 cos ϕ0-n1 cos ϕ1n0 cos ϕ0+n1 cos ϕ1,
r12p=n2 cos ϕ1-n1 cos ϕ2n2 cos ϕ1+n1 cos ϕ2,
r12s=n1 cos ϕ1-n2 cos ϕ2n1 cos ϕ1+n2 cos ϕ2,
X=exp-i4πd1λn1 cos ϕ1.
n0 sin ϕ0=n1 sin ϕ1=n2 sin ϕ2.
ρ12=r12p/r12s.
n2=n1 sin ϕ11+1-ρ121+ρ122 tan2 ϕ11/2.
n2 cos ϕ2=n1sin2 ϕ1cos ϕ11-ρ121+ρ12.
r12s=ρ12+cos 2ϕ11+ρ12 cos 2ϕ1.
r12p=ρ12ρ12+cos 2ϕ11+ρ12 cos 2ϕ1.
aρ123+bρ122+cρ12+d=0.
a=(z2X+c1z1)X,
b=2c1z2X2+(c12z1+c1z4+z1)X+c12z3,
c=c12z2X2+(c12z4+c1z1+z4)X+2c1z3,
d=c1z4X+z3,
z1=1-ρr01pr01s,
z2=r01s-ρr01p,
z3=r01p-ρr01s,
z4=r01pr01s-ρ,
c1=cos 2ϕ1=1-2(n0/n1)2 sin2 ϕ0,
u2s=n2 cos ϕ2n0=(n22/n02-sin2 ϕ0)1/2
p=n02[(r01p+X)(1-r01sX)-ρ(r01s-X)×(1+r01pX)],
q=n0u1s[(r01p+X)(1+r01sX)-ρ(r01s+X)×(1+r01pX)],
r=n12[(r01p-X)(1+r01sX)-ρ(r01s+X)×(1-r01pX)],
s=n0u1p[(r01p-X)(1-r01sX)-ρ(r01s-X)×(1-r01pX)],
u1s=n1 cos ϕ1,
u1p=n1/cos ϕ1,
pu2s3+(q+s)u2s2+(r+p sin2 ϕ0)u2s+q sin2 ϕ0=0.
n2=n0(u2s+sin2 ϕ0)1/2.
e323+e222+e12+e0=0,
2=n22/n02,
e3=p2,
e2=2pr-p2 sin2 ϕ0-(q+s)2,
e1=r2-2[pr-s(q+s)]sin2 ϕ0,
e0=-(r2+s2 sin2 ϕ0)sin2 ϕ0.
d1/λ1.
δx=-k=0nδakx0k/P(x0).
20=n02 sin2 ϕ01+1-ρ1+ρ2 tan2 ϕ0.
δ20=-d1k=0320k ekd1d1=0P3d1d1=0,=20.
n2=n201+2πid1λ(n202-n02 sin2 ϕ0)1/2×(n12-n02)(n12-n202)n12(n02-n202),
n20=20=n0 sin ϕ01+1-ρ1+ρ2 tan2 ϕ01/2

Metrics