Abstract

Reflections from the back surface of a transparent substrate influence the evaluation of optical constants of thin films from ellipsometric measurements. If the thickness of the substrate is large compared with the coherence length of the light, the relative phase between the p and s mode, which commonly is measured by ellipsometry, cannot be defined properly. We show how the reflections from the back surface of the substrate are taken into account in ellipsometric measurements by calculating the intensities of reflections for arbitrary angles of polarization. Applications of the new method, such as transmittance ellipsometry, ellipsometry at the back surface of the substrate, and the determination of the optical constants at the substrate–layer interface, are compared with measurements.

© 1997 Optical Society of America

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References

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  1. A. Röseler, Infrared Spectroscopic Ellipsometry (Akademie Verlag, Berlin, 1990).
  2. A. Röseler, “Problem of polarization degree in spectroscopic photometric ellipsometry (polarimetry),” J. Opt. Soc. Am. A 9, 1124–1131 (1992).
    [Crossref]
  3. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  4. K. B. Ozanyan, O. Hunderi, “Spectroscopic transmission ellipsometry studies of semiconductor heterostructures,” Physica Status Solidi A 207, 420–426 (1994).
  5. F. Abelès, “Recherches sur la propagation des ondes électromagnétique sinussoïdales dans les milieux stratifiés. Application aux couches minces.” Ann. Phys. 5, 596–640 (1950).
  6. G. E. Jellison, “Data analysis for spectroscopic ellipsometry,” Thin Solid Films 234, 416–422 (1993).
    [Crossref]
  7. I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Verlag Harri Deutsch, Thun, Switzerland1985), pp. 55–59.
  8. A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960), p. 140.
  9. A. Gombert, M. Köhl, U. Weimar, “Broadband spectroscopic ellipsometry based on a Fourier transform spectrometer,” Thin Solid Films 234, 352–355 (1993).
    [Crossref]
  10. D. F. Edwards, in Handbook of Optical Constants of SolidsE. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 547–569.
  11. P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, (12), 4370–4379 (1972).
    [Crossref]

1994 (1)

K. B. Ozanyan, O. Hunderi, “Spectroscopic transmission ellipsometry studies of semiconductor heterostructures,” Physica Status Solidi A 207, 420–426 (1994).

1993 (2)

G. E. Jellison, “Data analysis for spectroscopic ellipsometry,” Thin Solid Films 234, 416–422 (1993).
[Crossref]

A. Gombert, M. Köhl, U. Weimar, “Broadband spectroscopic ellipsometry based on a Fourier transform spectrometer,” Thin Solid Films 234, 352–355 (1993).
[Crossref]

1992 (1)

1972 (1)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, (12), 4370–4379 (1972).
[Crossref]

1950 (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétique sinussoïdales dans les milieux stratifiés. Application aux couches minces.” Ann. Phys. 5, 596–640 (1950).

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnétique sinussoïdales dans les milieux stratifiés. Application aux couches minces.” Ann. Phys. 5, 596–640 (1950).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Bronstein, I. N.

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Verlag Harri Deutsch, Thun, Switzerland1985), pp. 55–59.

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, (12), 4370–4379 (1972).
[Crossref]

Edwards, D. F.

D. F. Edwards, in Handbook of Optical Constants of SolidsE. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 547–569.

Gombert, A.

A. Gombert, M. Köhl, U. Weimar, “Broadband spectroscopic ellipsometry based on a Fourier transform spectrometer,” Thin Solid Films 234, 352–355 (1993).
[Crossref]

Hunderi, O.

K. B. Ozanyan, O. Hunderi, “Spectroscopic transmission ellipsometry studies of semiconductor heterostructures,” Physica Status Solidi A 207, 420–426 (1994).

Jellison, G. E.

G. E. Jellison, “Data analysis for spectroscopic ellipsometry,” Thin Solid Films 234, 416–422 (1993).
[Crossref]

Johnson, P. B.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, (12), 4370–4379 (1972).
[Crossref]

Köhl, M.

A. Gombert, M. Köhl, U. Weimar, “Broadband spectroscopic ellipsometry based on a Fourier transform spectrometer,” Thin Solid Films 234, 352–355 (1993).
[Crossref]

Ozanyan, K. B.

K. B. Ozanyan, O. Hunderi, “Spectroscopic transmission ellipsometry studies of semiconductor heterostructures,” Physica Status Solidi A 207, 420–426 (1994).

Röseler, A.

Semendjajew, K. A.

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Verlag Harri Deutsch, Thun, Switzerland1985), pp. 55–59.

Vasicek, A.

A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960), p. 140.

Weimar, U.

A. Gombert, M. Köhl, U. Weimar, “Broadband spectroscopic ellipsometry based on a Fourier transform spectrometer,” Thin Solid Films 234, 352–355 (1993).
[Crossref]

Ann. Phys. (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétique sinussoïdales dans les milieux stratifiés. Application aux couches minces.” Ann. Phys. 5, 596–640 (1950).

J. Opt. Soc. Am. A (1)

Phys. Rev. B (1)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, (12), 4370–4379 (1972).
[Crossref]

Physica Status Solidi A (1)

K. B. Ozanyan, O. Hunderi, “Spectroscopic transmission ellipsometry studies of semiconductor heterostructures,” Physica Status Solidi A 207, 420–426 (1994).

Thin Solid Films (2)

G. E. Jellison, “Data analysis for spectroscopic ellipsometry,” Thin Solid Films 234, 416–422 (1993).
[Crossref]

A. Gombert, M. Köhl, U. Weimar, “Broadband spectroscopic ellipsometry based on a Fourier transform spectrometer,” Thin Solid Films 234, 352–355 (1993).
[Crossref]

Other (5)

D. F. Edwards, in Handbook of Optical Constants of SolidsE. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 547–569.

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Verlag Harri Deutsch, Thun, Switzerland1985), pp. 55–59.

A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960), p. 140.

A. Röseler, Infrared Spectroscopic Ellipsometry (Akademie Verlag, Berlin, 1990).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

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Figures (7)

Fig. 1
Fig. 1

Model of the sample geometry under consideration and conventions for the indices used. Media 1 and 2 are usually air, the coherent systems 1 and 2 could be thin oxide layers. The thickness of the substrate is assumed to be large compared to the coherence length of the light.

Fig. 2
Fig. 2

α, β, and γ reflectance data for a silicon substrate with 3-nm oxide coatings on both surfaces at an angle of incidence of 70°: —, measured for a substrate with a thickness of 0.5 mm; ––––, calculated for a substrate with a thickness of 0.5 mm; …., calculated for an infinitely thick substrate.

Fig. 3
Fig. 3

Degree of polarization P=α2+β2+γ2 for the silicon substrate. The reflected light is totally polarized (P = 1) in the opaque region and only partly polarized (P < 1) in the transparent region: ——, calculated; ––––, measured.

Fig. 4
Fig. 4

α, β, and γ for transmittance ellipsometry for a 0.5-mm-thick silicon substrate: ——, measured; ––––, calculated.

Fig. 5
Fig. 5

Comparison between the reflectance ellipsometric coefficients for a silicon substrate in which the back surface is coated with a 150-nm-thick dielectric layer and an uncoated substrate. Two different k values, k1 = 10-4 and k2 = 10-5, were used for λ > 1.1 µm. For the coated back surface the value of k affects the ellipsometric coefficients much more than for the uncoated back surface: ——, coated back surface k = 10-5; ––––, coated back surface k = 10-4; -.-.-., uncoated back surface k = 10-5; …., uncoated back surface k = 10-4.

Fig. 6
Fig. 6

Comparison between n and k values: ——, values from Ref. 11; …., from measurements from the gold–air interface; ––––, from measurements from the gold–glass interface.

Fig. 7
Fig. 7

Ellipsometric coefficients α, β, and γ and polarization P. (a) reflectance at the gold–air interface, (b) reflectance at the gold–glass interface.

Equations (23)

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α=cos 2ψ,  β=sin 2ψ cos Δ,  γ=sin 2ψ sin Δ,
rprs=rprs×expiΔ,  tan ψ=rprs.
α=Rs-RpRs+Rp,
β=2Rs×RpRs+Rpcos Δ.
cos Δ=rsrp*+rs*rp2RsRp
Rϕ=rs cos ϕ+rp sin ϕ2=Rs cos2 ϕ+Rp sin2 ϕ+2RsRp cos Δ sin ϕ cos ϕ.
β=2R45°R0°+R90°-1.
rs,p=r1,S+t1,StS,1-r1,SrS,1rS,2 expiδn+δk1-rS,2rS,1 expiδn+δks,p,
δn=4πnsdλ cos θs,  δk=-4πksdλ cos θs,  ns=the refractive index of the substrate,  ks=the extinction coefficient of the substrate,  d=the thickness of the substrate,  λ=the vacuum wavelength of the incident light,  θs=the complex angle of incidence in the substrate as given by Snell’s law, and  ϕ=the polarization angle.
Rϕ=rp sin ϕ+rs cos ϕ2=Rp sin2 ϕ+Rs cos2 ϕ+2Rersrp*sin ϕ cos ϕ,
rsrp*=12π02πrsrp*dδn.
rsrp*=a+b1 exp+iδn+b2 exp-iδnc+d1 exp+iδn+d2 exp-iδn
=a+b1+b2cosδn+ib1-b2sinδnc+d1+d2cosδn+id1-d2sinδn,
a=r1,Ssr1,Sp*+usup* exp2δk,  b2=r1,Ssup*expδk,  b1=r1,Sp*us expδk,  c=1+rS,1rS,2srS,1rS,2p* exp2δk,  d2=-rS,1rS,2p* expδk,  d1=-rS,1rS,2s expδk,  us,p=t1,StS,1-r1,SrS,1rS,2s,p.
rsrp*=a+2b1d2+b2d14d1d2cosδn+φ+2ib1d2-b2d14d1d2sinδn+φc+4d1d2 cosδn+φ,
φ=cos-1d1+d24d1d2.
12π02πrsrp*dδn=ac2-4d1d2+2b1d2+b2d1c2-4d1d2c2-4d1d2-c4d1d2.
rsrp*=a-b1d2+b2d1c2-4d1d2.
rsrp*=r1,Ssr1,Sp*+t1,StS,1St1,StS,1p*-r1,SrS,1sr1,SrS,1p*rS,2srS,2p* exp2δk1-rS,1rS,2srS,1rS,2p* exp2δk.
γ=2Rγ45°R0°+R90°-1,
Rγϕ=Rp sin2 ϕ+Rs cos2 ϕ+2Rersirp*sin ϕ cos ϕ=Rp sin2 ϕ+Rs cos2 ϕ+2Imrsrp*sin ϕ cos ϕ.
Tϕ=Tp sin2 ϕ+Ts cos2 ϕ+2 sin ϕ cos ϕ×Ret1StS2 exp12δkst1StS2 exp12δkp*1-rS2rS1 expδksrS2rS1 expδkp*.
P=α2+β2+γ2.

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