Applicability of the effective medium approximation in the ellipsometry of randomly micro-rough solid surfaces

Yuanbin Liu; Jun Qiu; Linhua Liu

doi:10.1364/OE.26.016560

Abstract

The effective medium approximation (EMA) has been widely applied to model the effect of a solid sample with surface roughness in spectroscopic ellipsometry (SE). There are two specific cases to utilize the EMA model. One is utilizing the EMA model to perform the inversion of the optical constants of solid samples from the SE measurements. Another is utilizing the EMA model to estimate the thickness of the rough layer at solid surface from the SE measurements under the condition in which the optical constants of samples are known. For the first case, the thickness of the rough layer is usually assumed to be the root mean square (rms) roughness as measured by atomic force microscopy (AFM). We theoretically investigate the error of the EMA model to estimate optical constants for different surface morphologies and materials. Because the EMA model only accounts for the height irregularities of rough surfaces but neglects the effect of the lateral irregularities on electromagnetic scattering from rough surfaces, it is difficult to obtain high-precision results for optical constants. Moreover, the inversion error of optical constants by using the EMA model is difficult to evaluate. In the second case, the thickness of the rough layer is estimated by using the EMA model from the SE measurements, called the EMA model roughness. We show that the EMA model roughness generally has a deviation from the rms roughness as measured by AFM. Some correlated relationships are established between the EMA model roughness and the morphological parameters of rough surfaces. It is found that these relationships have similar forms but not identical coefficients for different materials. The results from this work may facilitate a better understanding and utilization for the EMA model in SE.

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References

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Article Order
Year
Author
Publication

M. Losurdo and K. Hingerl, Ellipsometry at the Nanoscale (Springer, 2013).
V. Sirtori, L. Magagnin, E. Saglia, and P. L. Cavallotti, “Calculation model of rough gold optical constants,” Surf. Sci. 554(2–3), 119–124 (2004).
[Crossref]
C. A. Fenstermaker and F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
[Crossref]
J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, 1995).
D. Bergström, J. Powell, and A. F. H. Kaplan, “A ray-tracing analysis of the absorption of light by smooth and rough metal surfaces,” J. Appl. Phys. 101(11), 113504 (2007).
[Crossref]
R. W. Collins, I. An, H. Fujiwara, J. C. Lee, Y. W. Lu, J. Y. Koh, and P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313–314, 18–32 (1998).
[Crossref]
J. Koh, Y. W. Lu, C. R. Wronski, Y. L. Kuang, R. W. Collins, T. T. Tsong, and Y. E. Strausser, “Correlation of real time spectroellipsometry and atomic force microscopy measurements of surface roughness on amorphous semiconductor thin films,” Appl. Phys. Lett. 69(9), 1297–1299 (1996).
[Crossref]
P. Petrik, L. P. Biro, M. Fried, T. Lohner, R. Berger, C. Schneider, J. Gyulai, and H. Ryssel, “Comparative study of surface roughness measured on polysilicon using spectroscopic ellipsometry and atomic force microscopy,” Thin Solid Films 315(1–2), 186–191 (1998).
[Crossref]
H. Fujiwara, M. Kondo, and A. Matsuda, “Real-time spectroscopic ellipsometry studies of the nucleation and grain growth processes in microcrystalline silicon thin films,” Phys. Rev. B 63(11), 115306 (2001).
[Crossref]
D. Franta and I. Ohlídal, “Comparison of effective medium approximation and Rayleigh–Rice theory concerning ellipsometric characterization of rough surfaces,” Opt. Commun. 248(4–6), 459–467 (2005).
[Crossref]
A. Yanguas-Gil, B. A. Sperling, and J. R. Abelson, “Theory of light scattering from self-affine surfaces: Relationship between surface morphology and effective medium roughness,” Phys. Rev. B 84(8), 085402 (2011).
[Crossref]
J. Qiu, D. F. Ran, Y. B. Liu, and L. H. Liu, “Investigation of ellipsometric parameters of 2D microrough surfaces by FDTD,” Appl. Opt. 55(20), 5423–5431 (2016).
[Crossref] [PubMed]
I. Ohlídal, J. Vohánka, M. Čermák, and D. Franta, “Optical characterization of randomly microrough surfaces covered with very thin overlayers using effective medium approximation and Rayleigh–Rice theory,” Appl. Surf. Sci. 419, 942–956 (2017).
[Crossref]
H. Arwin and D. E. Aspnes, “Determination of optical properties of thin organic films by spectroellipsometry,” Thin Solid Films 138(2), 195–207 (1986).
[Crossref]
H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (John Wiley & Sons, 2007).
E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

2017 (1)

I. Ohlídal, J. Vohánka, M. Čermák, and D. Franta, “Optical characterization of randomly microrough surfaces covered with very thin overlayers using effective medium approximation and Rayleigh–Rice theory,” Appl. Surf. Sci. 419, 942–956 (2017).
[Crossref]

2016 (1)

J. Qiu, D. F. Ran, Y. B. Liu, and L. H. Liu, “Investigation of ellipsometric parameters of 2D microrough surfaces by FDTD,” Appl. Opt. 55(20), 5423–5431 (2016).
[Crossref] [PubMed]

2011 (1)

A. Yanguas-Gil, B. A. Sperling, and J. R. Abelson, “Theory of light scattering from self-affine surfaces: Relationship between surface morphology and effective medium roughness,” Phys. Rev. B 84(8), 085402 (2011).
[Crossref]

2007 (1)

D. Bergström, J. Powell, and A. F. H. Kaplan, “A ray-tracing analysis of the absorption of light by smooth and rough metal surfaces,” J. Appl. Phys. 101(11), 113504 (2007).
[Crossref]

2005 (1)

D. Franta and I. Ohlídal, “Comparison of effective medium approximation and Rayleigh–Rice theory concerning ellipsometric characterization of rough surfaces,” Opt. Commun. 248(4–6), 459–467 (2005).
[Crossref]

2004 (1)

V. Sirtori, L. Magagnin, E. Saglia, and P. L. Cavallotti, “Calculation model of rough gold optical constants,” Surf. Sci. 554(2–3), 119–124 (2004).
[Crossref]

2001 (1)

H. Fujiwara, M. Kondo, and A. Matsuda, “Real-time spectroscopic ellipsometry studies of the nucleation and grain growth processes in microcrystalline silicon thin films,” Phys. Rev. B 63(11), 115306 (2001).
[Crossref]

1998 (2)

P. Petrik, L. P. Biro, M. Fried, T. Lohner, R. Berger, C. Schneider, J. Gyulai, and H. Ryssel, “Comparative study of surface roughness measured on polysilicon using spectroscopic ellipsometry and atomic force microscopy,” Thin Solid Films 315(1–2), 186–191 (1998).
[Crossref]

R. W. Collins, I. An, H. Fujiwara, J. C. Lee, Y. W. Lu, J. Y. Koh, and P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313–314, 18–32 (1998).
[Crossref]

1996 (1)

J. Koh, Y. W. Lu, C. R. Wronski, Y. L. Kuang, R. W. Collins, T. T. Tsong, and Y. E. Strausser, “Correlation of real time spectroellipsometry and atomic force microscopy measurements of surface roughness on amorphous semiconductor thin films,” Appl. Phys. Lett. 69(9), 1297–1299 (1996).
[Crossref]

1986 (1)

H. Arwin and D. E. Aspnes, “Determination of optical properties of thin organic films by spectroellipsometry,” Thin Solid Films 138(2), 195–207 (1986).
[Crossref]

1969 (1)

C. A. Fenstermaker and F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
[Crossref]

Abelson, J. R.

A. Yanguas-Gil, B. A. Sperling, and J. R. Abelson, “Theory of light scattering from self-affine surfaces: Relationship between surface morphology and effective medium roughness,” Phys. Rev. B 84(8), 085402 (2011).
[Crossref]

An, I.

R. W. Collins, I. An, H. Fujiwara, J. C. Lee, Y. W. Lu, J. Y. Koh, and P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313–314, 18–32 (1998).
[Crossref]

Arwin, H.

H. Arwin and D. E. Aspnes, “Determination of optical properties of thin organic films by spectroellipsometry,” Thin Solid Films 138(2), 195–207 (1986).
[Crossref]

Aspnes, D. E.

H. Arwin and D. E. Aspnes, “Determination of optical properties of thin organic films by spectroellipsometry,” Thin Solid Films 138(2), 195–207 (1986).
[Crossref]

Berger, R.

P. Petrik, L. P. Biro, M. Fried, T. Lohner, R. Berger, C. Schneider, J. Gyulai, and H. Ryssel, “Comparative study of surface roughness measured on polysilicon using spectroscopic ellipsometry and atomic force microscopy,” Thin Solid Films 315(1–2), 186–191 (1998).
[Crossref]

Bergström, D.

D. Bergström, J. Powell, and A. F. H. Kaplan, “A ray-tracing analysis of the absorption of light by smooth and rough metal surfaces,” J. Appl. Phys. 101(11), 113504 (2007).
[Crossref]

Biro, L. P.

P. Petrik, L. P. Biro, M. Fried, T. Lohner, R. Berger, C. Schneider, J. Gyulai, and H. Ryssel, “Comparative study of surface roughness measured on polysilicon using spectroscopic ellipsometry and atomic force microscopy,” Thin Solid Films 315(1–2), 186–191 (1998).
[Crossref]

Cavallotti, P. L.

V. Sirtori, L. Magagnin, E. Saglia, and P. L. Cavallotti, “Calculation model of rough gold optical constants,” Surf. Sci. 554(2–3), 119–124 (2004).
[Crossref]

Cermák, M.

I. Ohlídal, J. Vohánka, M. Čermák, and D. Franta, “Optical characterization of randomly microrough surfaces covered with very thin overlayers using effective medium approximation and Rayleigh–Rice theory,” Appl. Surf. Sci. 419, 942–956 (2017).
[Crossref]

Collins, R. W.

R. W. Collins, I. An, H. Fujiwara, J. C. Lee, Y. W. Lu, J. Y. Koh, and P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313–314, 18–32 (1998).
[Crossref]

J. Koh, Y. W. Lu, C. R. Wronski, Y. L. Kuang, R. W. Collins, T. T. Tsong, and Y. E. Strausser, “Correlation of real time spectroellipsometry and atomic force microscopy measurements of surface roughness on amorphous semiconductor thin films,” Appl. Phys. Lett. 69(9), 1297–1299 (1996).
[Crossref]

Fenstermaker, C. A.

C. A. Fenstermaker and F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
[Crossref]

Franta, D.

I. Ohlídal, J. Vohánka, M. Čermák, and D. Franta, “Optical characterization of randomly microrough surfaces covered with very thin overlayers using effective medium approximation and Rayleigh–Rice theory,” Appl. Surf. Sci. 419, 942–956 (2017).
[Crossref]

D. Franta and I. Ohlídal, “Comparison of effective medium approximation and Rayleigh–Rice theory concerning ellipsometric characterization of rough surfaces,” Opt. Commun. 248(4–6), 459–467 (2005).
[Crossref]

Fried, M.

P. Petrik, L. P. Biro, M. Fried, T. Lohner, R. Berger, C. Schneider, J. Gyulai, and H. Ryssel, “Comparative study of surface roughness measured on polysilicon using spectroscopic ellipsometry and atomic force microscopy,” Thin Solid Films 315(1–2), 186–191 (1998).
[Crossref]

Fujiwara, H.

H. Fujiwara, M. Kondo, and A. Matsuda, “Real-time spectroscopic ellipsometry studies of the nucleation and grain growth processes in microcrystalline silicon thin films,” Phys. Rev. B 63(11), 115306 (2001).
[Crossref]

R. W. Collins, I. An, H. Fujiwara, J. C. Lee, Y. W. Lu, J. Y. Koh, and P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313–314, 18–32 (1998).
[Crossref]

Gyulai, J.

P. Petrik, L. P. Biro, M. Fried, T. Lohner, R. Berger, C. Schneider, J. Gyulai, and H. Ryssel, “Comparative study of surface roughness measured on polysilicon using spectroscopic ellipsometry and atomic force microscopy,” Thin Solid Films 315(1–2), 186–191 (1998).
[Crossref]

Kaplan, A. F. H.

D. Bergström, J. Powell, and A. F. H. Kaplan, “A ray-tracing analysis of the absorption of light by smooth and rough metal surfaces,” J. Appl. Phys. 101(11), 113504 (2007).
[Crossref]

Koh, J.

J. Koh, Y. W. Lu, C. R. Wronski, Y. L. Kuang, R. W. Collins, T. T. Tsong, and Y. E. Strausser, “Correlation of real time spectroellipsometry and atomic force microscopy measurements of surface roughness on amorphous semiconductor thin films,” Appl. Phys. Lett. 69(9), 1297–1299 (1996).
[Crossref]

Koh, J. Y.

R. W. Collins, I. An, H. Fujiwara, J. C. Lee, Y. W. Lu, J. Y. Koh, and P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313–314, 18–32 (1998).
[Crossref]

Kondo, M.

H. Fujiwara, M. Kondo, and A. Matsuda, “Real-time spectroscopic ellipsometry studies of the nucleation and grain growth processes in microcrystalline silicon thin films,” Phys. Rev. B 63(11), 115306 (2001).
[Crossref]

Kuang, Y. L.

J. Koh, Y. W. Lu, C. R. Wronski, Y. L. Kuang, R. W. Collins, T. T. Tsong, and Y. E. Strausser, “Correlation of real time spectroellipsometry and atomic force microscopy measurements of surface roughness on amorphous semiconductor thin films,” Appl. Phys. Lett. 69(9), 1297–1299 (1996).
[Crossref]

Lee, J. C.

R. W. Collins, I. An, H. Fujiwara, J. C. Lee, Y. W. Lu, J. Y. Koh, and P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313–314, 18–32 (1998).
[Crossref]

Liu, L. H.

J. Qiu, D. F. Ran, Y. B. Liu, and L. H. Liu, “Investigation of ellipsometric parameters of 2D microrough surfaces by FDTD,” Appl. Opt. 55(20), 5423–5431 (2016).
[Crossref] [PubMed]

Liu, Y. B.

J. Qiu, D. F. Ran, Y. B. Liu, and L. H. Liu, “Investigation of ellipsometric parameters of 2D microrough surfaces by FDTD,” Appl. Opt. 55(20), 5423–5431 (2016).
[Crossref] [PubMed]

Lohner, T.

P. Petrik, L. P. Biro, M. Fried, T. Lohner, R. Berger, C. Schneider, J. Gyulai, and H. Ryssel, “Comparative study of surface roughness measured on polysilicon using spectroscopic ellipsometry and atomic force microscopy,” Thin Solid Films 315(1–2), 186–191 (1998).
[Crossref]

Lu, Y. W.

R. W. Collins, I. An, H. Fujiwara, J. C. Lee, Y. W. Lu, J. Y. Koh, and P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313–314, 18–32 (1998).
[Crossref]

J. Koh, Y. W. Lu, C. R. Wronski, Y. L. Kuang, R. W. Collins, T. T. Tsong, and Y. E. Strausser, “Correlation of real time spectroellipsometry and atomic force microscopy measurements of surface roughness on amorphous semiconductor thin films,” Appl. Phys. Lett. 69(9), 1297–1299 (1996).
[Crossref]

Magagnin, L.

V. Sirtori, L. Magagnin, E. Saglia, and P. L. Cavallotti, “Calculation model of rough gold optical constants,” Surf. Sci. 554(2–3), 119–124 (2004).
[Crossref]

Matsuda, A.

H. Fujiwara, M. Kondo, and A. Matsuda, “Real-time spectroscopic ellipsometry studies of the nucleation and grain growth processes in microcrystalline silicon thin films,” Phys. Rev. B 63(11), 115306 (2001).
[Crossref]

McCrackin, F. L.

C. A. Fenstermaker and F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
[Crossref]

Ohlídal, I.

I. Ohlídal, J. Vohánka, M. Čermák, and D. Franta, “Optical characterization of randomly microrough surfaces covered with very thin overlayers using effective medium approximation and Rayleigh–Rice theory,” Appl. Surf. Sci. 419, 942–956 (2017).
[Crossref]

D. Franta and I. Ohlídal, “Comparison of effective medium approximation and Rayleigh–Rice theory concerning ellipsometric characterization of rough surfaces,” Opt. Commun. 248(4–6), 459–467 (2005).
[Crossref]

Petrik, P.

P. Petrik, L. P. Biro, M. Fried, T. Lohner, R. Berger, C. Schneider, J. Gyulai, and H. Ryssel, “Comparative study of surface roughness measured on polysilicon using spectroscopic ellipsometry and atomic force microscopy,” Thin Solid Films 315(1–2), 186–191 (1998).
[Crossref]

Powell, J.

D. Bergström, J. Powell, and A. F. H. Kaplan, “A ray-tracing analysis of the absorption of light by smooth and rough metal surfaces,” J. Appl. Phys. 101(11), 113504 (2007).
[Crossref]

Qiu, J.

J. Qiu, D. F. Ran, Y. B. Liu, and L. H. Liu, “Investigation of ellipsometric parameters of 2D microrough surfaces by FDTD,” Appl. Opt. 55(20), 5423–5431 (2016).
[Crossref] [PubMed]

Ran, D. F.

J. Qiu, D. F. Ran, Y. B. Liu, and L. H. Liu, “Investigation of ellipsometric parameters of 2D microrough surfaces by FDTD,” Appl. Opt. 55(20), 5423–5431 (2016).
[Crossref] [PubMed]

Rovira, P. I.

R. W. Collins, I. An, H. Fujiwara, J. C. Lee, Y. W. Lu, J. Y. Koh, and P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313–314, 18–32 (1998).
[Crossref]

Ryssel, H.

P. Petrik, L. P. Biro, M. Fried, T. Lohner, R. Berger, C. Schneider, J. Gyulai, and H. Ryssel, “Comparative study of surface roughness measured on polysilicon using spectroscopic ellipsometry and atomic force microscopy,” Thin Solid Films 315(1–2), 186–191 (1998).
[Crossref]

Saglia, E.

V. Sirtori, L. Magagnin, E. Saglia, and P. L. Cavallotti, “Calculation model of rough gold optical constants,” Surf. Sci. 554(2–3), 119–124 (2004).
[Crossref]

Schneider, C.

P. Petrik, L. P. Biro, M. Fried, T. Lohner, R. Berger, C. Schneider, J. Gyulai, and H. Ryssel, “Comparative study of surface roughness measured on polysilicon using spectroscopic ellipsometry and atomic force microscopy,” Thin Solid Films 315(1–2), 186–191 (1998).
[Crossref]

Sirtori, V.

V. Sirtori, L. Magagnin, E. Saglia, and P. L. Cavallotti, “Calculation model of rough gold optical constants,” Surf. Sci. 554(2–3), 119–124 (2004).
[Crossref]

Sperling, B. A.

A. Yanguas-Gil, B. A. Sperling, and J. R. Abelson, “Theory of light scattering from self-affine surfaces: Relationship between surface morphology and effective medium roughness,” Phys. Rev. B 84(8), 085402 (2011).
[Crossref]

Strausser, Y. E.

J. Koh, Y. W. Lu, C. R. Wronski, Y. L. Kuang, R. W. Collins, T. T. Tsong, and Y. E. Strausser, “Correlation of real time spectroellipsometry and atomic force microscopy measurements of surface roughness on amorphous semiconductor thin films,” Appl. Phys. Lett. 69(9), 1297–1299 (1996).
[Crossref]

Tsong, T. T.

J. Koh, Y. W. Lu, C. R. Wronski, Y. L. Kuang, R. W. Collins, T. T. Tsong, and Y. E. Strausser, “Correlation of real time spectroellipsometry and atomic force microscopy measurements of surface roughness on amorphous semiconductor thin films,” Appl. Phys. Lett. 69(9), 1297–1299 (1996).
[Crossref]

Vohánka, J.

I. Ohlídal, J. Vohánka, M. Čermák, and D. Franta, “Optical characterization of randomly microrough surfaces covered with very thin overlayers using effective medium approximation and Rayleigh–Rice theory,” Appl. Surf. Sci. 419, 942–956 (2017).
[Crossref]

Wronski, C. R.

J. Koh, Y. W. Lu, C. R. Wronski, Y. L. Kuang, R. W. Collins, T. T. Tsong, and Y. E. Strausser, “Correlation of real time spectroellipsometry and atomic force microscopy measurements of surface roughness on amorphous semiconductor thin films,” Appl. Phys. Lett. 69(9), 1297–1299 (1996).
[Crossref]

Yanguas-Gil, A.

A. Yanguas-Gil, B. A. Sperling, and J. R. Abelson, “Theory of light scattering from self-affine surfaces: Relationship between surface morphology and effective medium roughness,” Phys. Rev. B 84(8), 085402 (2011).
[Crossref]

Appl. Opt. (1)

J. Qiu, D. F. Ran, Y. B. Liu, and L. H. Liu, “Investigation of ellipsometric parameters of 2D microrough surfaces by FDTD,” Appl. Opt. 55(20), 5423–5431 (2016).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

J. Koh, Y. W. Lu, C. R. Wronski, Y. L. Kuang, R. W. Collins, T. T. Tsong, and Y. E. Strausser, “Correlation of real time spectroellipsometry and atomic force microscopy measurements of surface roughness on amorphous semiconductor thin films,” Appl. Phys. Lett. 69(9), 1297–1299 (1996).
[Crossref]

Appl. Surf. Sci. (1)

I. Ohlídal, J. Vohánka, M. Čermák, and D. Franta, “Optical characterization of randomly microrough surfaces covered with very thin overlayers using effective medium approximation and Rayleigh–Rice theory,” Appl. Surf. Sci. 419, 942–956 (2017).
[Crossref]

J. Appl. Phys. (1)

D. Bergström, J. Powell, and A. F. H. Kaplan, “A ray-tracing analysis of the absorption of light by smooth and rough metal surfaces,” J. Appl. Phys. 101(11), 113504 (2007).
[Crossref]

Opt. Commun. (1)

D. Franta and I. Ohlídal, “Comparison of effective medium approximation and Rayleigh–Rice theory concerning ellipsometric characterization of rough surfaces,” Opt. Commun. 248(4–6), 459–467 (2005).
[Crossref]

Phys. Rev. B (2)

A. Yanguas-Gil, B. A. Sperling, and J. R. Abelson, “Theory of light scattering from self-affine surfaces: Relationship between surface morphology and effective medium roughness,” Phys. Rev. B 84(8), 085402 (2011).
[Crossref]

H. Fujiwara, M. Kondo, and A. Matsuda, “Real-time spectroscopic ellipsometry studies of the nucleation and grain growth processes in microcrystalline silicon thin films,” Phys. Rev. B 63(11), 115306 (2001).
[Crossref]

Surf. Sci. (2)

V. Sirtori, L. Magagnin, E. Saglia, and P. L. Cavallotti, “Calculation model of rough gold optical constants,” Surf. Sci. 554(2–3), 119–124 (2004).
[Crossref]

C. A. Fenstermaker and F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
[Crossref]

Thin Solid Films (3)

R. W. Collins, I. An, H. Fujiwara, J. C. Lee, Y. W. Lu, J. Y. Koh, and P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313–314, 18–32 (1998).
[Crossref]

P. Petrik, L. P. Biro, M. Fried, T. Lohner, R. Berger, C. Schneider, J. Gyulai, and H. Ryssel, “Comparative study of surface roughness measured on polysilicon using spectroscopic ellipsometry and atomic force microscopy,” Thin Solid Films 315(1–2), 186–191 (1998).
[Crossref]

H. Arwin and D. E. Aspnes, “Determination of optical properties of thin organic films by spectroellipsometry,” Thin Solid Films 138(2), 195–207 (1986).
[Crossref]

Other (4)

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (John Wiley & Sons, 2007).

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

M. Losurdo and K. Hingerl, Ellipsometry at the Nanoscale (Springer, 2013).

J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, 1995).

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Fig. 1 FDTD simulations of the SE parameters (Ψ, Δ) of Si compared with the EMA model calculations: σ = 0.05 μm; ζ = 0.1 μm, 0.2 μm, 0.4 μm, respectively.

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Fig. 2 Relative mean square error (RMSE) of the SE parameters calculated by the EMA model as a function of the relative roughness: (a) σ = 0.05 μm; σ/ζ = 0.75; (b) σ = 0.05 μm; σ/ζ = 0.125.

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Fig. 3 Relative errors of the Fresnel equations and the EMA model to obtain the complex refractive index as a function of the parameter σ/ζ for different materials in the case of n = 3.4294, σ = 0.05 μm and σ/λ = 0.0125: (a) k = 0; (b) k = 2 and (c) k = 5, respectively.

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Fig. 4 Relative errors of the Fresnel equations and the EMA model to obtain the complex refractive index as a function of the parameter σ/ζ for different materials in the case of k = 2, σ = 0.05 μm and σ/λ = 0.0125: (a) n = 2; (b) n = 3.4294 and (c) n = 5, respectively.

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Fig. 5 Scaling of the EMA model roughness with the parameter σ²/ζ for three materials with different extinction coefficients: (a) k = 0; (b) k = 2 and (c) k = 5, respectively. (n = 3.4294, σ/λ = 0.0125)

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Fig. 6 Scaling of the EMA model roughness with the parameter σ²/ζ for three materials with different refractive indices: (a) n = 2; (b) n = 3.4294 and (c) n = 5, respectively. (k = 2, σ/λ = 0.0125)

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Fig. 7 Scaling of the EMA model roughness with the parameter σ²/ζ corresponding to the cases in Table 2.

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Tables (4)

Table 1 Relationships between the EMA model roughness and morphological parameters of rough surfaces.

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Table 2 Relative errors of the Fresnel equations and the EMA model to obtain the complex refractive indices with different morphologies of rough surfaces. (n = 5.57, k = 0.387).

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Table 3 Thickness of rough layer estimated by the EMA model from SE measurements corresponding to the eighteen cases in Fig. 3 (n = 3.4294, σ = 0.05 μm, σ/λ = 0.0125).

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Table 4 Mathematical relationships between the thickness estimated by the EMA model and the morphological parameters for different materials.

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Equations (6)

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〈 H (\vec{p}) H (\vec{p} + \vec{q}) 〉 = σ^{2} \exp [- (q^{2} / ζ^{2})]

ρ = \tan Ψ \exp (i Δ) = \frac{r_{p}}{r_{s}} = (\frac{E_{r p}}{E_{i p}}) / (\frac{E_{r s}}{E_{i s}})

f \frac{ε_{n} - ε_{e f f}}{ε_{n} + 2 ε_{e f f}} + (1 - f) \frac{ε_{v} - ε_{e f f}}{ε_{v} + 2 ε_{e f f}} = 0

RMSE = \sqrt{\frac{1}{2} [{(\frac{ψ^{F D T D} - ψ^{E M A}}{ψ^{F D T D}})}^{2} + {(\frac{Δ^{F D T D} - Δ^{E M A}}{Δ^{F D T D}})}^{2}]} \times 100 %

n - i k = {{[\frac{(1 - ρ) \sin^{2} θ}{(1 + ρ) \cos θ}]}^{2} + \sin^{2} θ}^{1 / 2}

α = \frac{| N - N^{*} |}{| N^{*} |} \times 100 %

Year	Authors	Materials	Mathematical Relationships
1996	Koh et al. ^[⁷^]	a-Si_1‴_xC_x : H	$d_{E M A} = 1.5 σ + 4 Å$
1998	Petrik et al. ^[⁸^]	Polysilicon	$d_{E M A} = (1.175 \pm 0.285) σ$
2001	Fujiwara et al. ^[⁹^]	μc-Si:H	$d_{E M A} = 0.88 σ + 4.9 Å$
2005	Franta et al. ^[¹⁰^]	Si	$d_{E M A} \propto σ^{2} / ζ$
2011	Yanguas-Gil et al. ^[¹¹^]	a-Si: H	$d_{E M A} = (3.5 \pm 0.1) σ^{2} / ζ^{α} + (3.7 \pm 0.5) Å$

Morphological parameters		Pseudo optical constants	EMA
σ/ζ	σ/λ	α (%)	α (%)
0.33	0.0125	23.51	7.30
	0.025	31.36	10.35
	0.05	45.69	64.30
0.5	0.0125	32.85	12.09
	0.025	45.32	16.89
	0.05	59.26	18.98
0.67	0.0125	38.97	19.67
	0.025	55.67	32.03
	0.05	67.79	43.02

σ/ζ	k	$d_{E M A}$ (μm)
0.75	0	0.101
	2	0.099
	5	0.092
0.625	0	0.096
	2	0.092
	5	0.081
0.5	0	0.081
	2	0.076
	5	0.062
0.375	0	0.060
	2	0.053
	5	0.042
0.25	0	0.044
	2	0.039
	5	0.030
0.125	0	0.018
	2	0.017
	5	0.013

n	k	Mathematical relationships
3.4294	0	$d_{E M A}$ = (2.71 ± 0.25) σ²/ζ + (0.007 ± 0.005)
	2	$d_{E M A}$ = (2.71 ± 0.18) σ²/ζ + (0.003 ± 0.004)
	5	$d_{E M A}$ = (2.60 ± 0.10) σ²/ζ + (−0.004 ± 0.002)
2	2	$d_{E M A}$ = (2.95 ± 0.00) σ²/ζ + (0.221 ± 0.005)
3.4294		$d_{E M A}$ = (2.71 ± 0.00) σ²/ζ + (0.182 ± 0.005)
5		$d_{E M A}$ = (2.72 ± 0.00) σ²/ζ + (0.215 ± 0.005)

Year	Authors	Materials	Mathematical Relationships
1996	Koh et al. ^[⁷^]	a-Si_1‴_xC_x : H	$d_{E M A} = 1.5 σ + 4 Å$
1998	Petrik et al. ^[⁸^]	Polysilicon	$d_{E M A} = (1.175 \pm 0.285) σ$
2001	Fujiwara et al. ^[⁹^]	μc-Si:H	$d_{E M A} = 0.88 σ + 4.9 Å$
2005	Franta et al. ^[¹⁰^]	Si	$d_{E M A} \propto σ^{2} / ζ$
2011	Yanguas-Gil et al. ^[¹¹^]	a-Si: H	$d_{E M A} = (3.5 \pm 0.1) σ^{2} / ζ^{α} + (3.7 \pm 0.5) Å$

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