Optical measurements of transmittance in the far infrared region performed on crystalline silicon wafers exhibit partially coherent interference effects appropriate for the determination of thicknesses of the wafers. The knowledge of accurate spectral and temperature dependencies of the optical constants of crystalline silicon in this spectral region is crucial for the determination of its thickness and vice versa. The recently published temperature dependent dispersion model of crystalline silicon is suitable for this purpose. Because the linear thermal expansion of crystalline silicon is known, the temperatures of the wafers can be determined with high precision from the evolution of the interference patterns at elevated temperatures.
References
1.
J.
Jin
, J. W.
Kim
, C.-S.
Kang
, J.-A.
Kim
, and T. B.
Eom
, “Thickness and refractive index measurement of a silicon wafer based on an optical comb
,” Opt. Express
18
, 18339
–18346
(2010
).2.
S.
Maeng
, J.
Park
, O.
Byungsun
, and J.
Jin
, “Uncertainty improvement of geometrical thickness and refractive index measurement of a silicon wafer using a femtosecond pulse laser
,” Opt. Express
20
, 12184
–12190
(2012
).3.
J.
Jin
, S.
Maeng
, J.
Park
, J.-A.
Kim
, and J. W.
Kim
, “Fizeau-type interferometric probe to measure geometrical thickness of silicon wafers
,” Opt. Express
22
, 23427
–23432
(2014
).4.
H.-J.
Lee
and K.-N.
Joo
, “Optical interferometric approach for measuring the geometrical dimension and refractive index profiles of a double-sided polished undoped Si wafer
,” Meas. Sci. Technol.
25
, 075202
(2014
).5.
J.
Bae
, J.
Park
, H.
Ahn
, and J.
Jin
, “Total physical thickness measurement of a multi-layered wafer using a spectral-domain interferometer with an optical comb
,” Opt. Express
25
, 12689
–12697
(2017
).6.
M. J.
Jansen
, H.
Haitjema
, and P. H. J.
Schellekens
, “Scanning wafer thickness and flatness interferometer
,” Proc. SPIE
5252
, 334
–345
(2004
).7.
D.
Franta
, A.
Dubroka
, C.
Wang
, A.
Giglia
, J.
Vohánka
, P.
Franta
, and I.
Ohlídal
, “Temperature-dependent dispersion model of float zone crystalline silicon
,” Appl. Surf. Sci.
421
, 405
–419
(2017
).8.
H.
Watanabe
, N.
Yamada
, and M.
Okaji
, “Linear thermal expansion coefficient of silicon from 293 to 1000 K
,” Int. J. Thermophys.
25
, 221
–236
(2004
).9.
H. R.
Philipp
and E. A.
Taft
, “Optical constants of silicon in the region 1 to 10 eV
,” Phys. Rev.
120
, 37
–38
(1960
).10.
D. E.
Aspnes
and A. A.
Studna
, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV
,” Phys. Rev. B
27
, 985
–1009
(1983
).11.
B. J.
Frey
, D. B.
Leviton
, and T. J.
Madison
, “Temperature-dependent refractive index of silicon and germanium
,” Proc. SPIE
6273
, 62732J
(2006
).12.
K.
Postava
, M.
Aoyama
, J.
Mistrík
, T.
Yamaguchi
, and K.
Shio
, “Optical measurements of silicon wafer temperature
,” Appl. Surf. Sci.
254
, 416
–419
(2007
).13.
H.
Ibach
, “Thermal expansion of silicon and zinc oxide (I)
,” Phys. Status Solidi
31
, 625
–634
(1969
).14.
L.
Viña
, S.
Logothetidis
, and M.
Cardona
, “Temperature dependence of the dielectric function of gerriianium
,” Phys. Rev. B
30
, 1979
–1991
(1984
).15.
P.
Lautenschlager
, M.
Garriga
, L.
Viña
, and M.
Cardona
, “Temperature dependence of the dielectric function and interband critical points in silicon
,” Phys. Rev. B
36
, 4821
–4830
(1987
).16.
S.
Wei
and M. Y.
Chou
, “Phonon dispersions of silicon and germanium from first-principles calculations
,” Phys. Rev. B
50
, 2221
–2226
(1994
).17.
Z.
Lu
, K.
Munakata
, A.
Kohno
, Y.
Soejima
, and A.
Okazaki
, “Structural relaxation in silicon at low temperatures
,” Mater. Sci. Eng., B
34
, 220
–223
(1995
).18.
D.
Franta
, D.
Nečas
, and L.
Zajíčková
, “Application of Thomas–Reiche–Kuhn sum rule to construction of advanced dispersion models
,” Thin Solid Films
534
, 432
–441
(2013
).19.
D.
Franta
, D.
Nečas
, L.
Zajíčková
, and I.
Ohlídal
, “Utilization of the sum rule for construction of advanced dispersion model of crystalline silicon containing interstitial oxygen
,” Thin Solid Films
571
, 490
–495
(2014
).20.
D.
Franta
, D.
Nečas
, L.
Zajíčková
, and I.
Ohlídal
, “Dispersion model of two-phonon absorption: Application to c-Si
,” Opt. Mater. Express
4
, 1641
–1656
(2014
).21.
D.
Franta
, D.
Nečas
, L.
Zajíčková
, and I.
Ohlídal
, “Broadening of dielectric response and sum rule conservation
,” Thin Solid Films
571
, 496
–501
(2014
).22.
D.
Franta
et al., see http://newad.physics.muni.cz for Software for optical characterization newAD2.23.
W.
Romberg
, “Vereinfachte numerische Integration
,” K. Nor. Vidensk. Selsk. Forh.
28
, 30
–36
(1955
).24.
D.
Franta
, J.
Vohánka
, and M.
Čermák
, “Universal dispersion model for characterization of thin films over wide spectral range
,” in Optical Characterization of Thin Solid Films
, edited by O.
Stenzel
and M.
Ohlídal
(Springer
, 2018
), Vol. 64
, pp. 31
–82
.© 2018 Author(s).
2018
Author(s)
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