We propose a framework for modeling the temperature-dependent infrared optical and radiative properties of metals exhibiting nonideal free-carrier dynamics. In order to do so, we derive a parsimonious model that possesses both multivariate (in temperature and wavelength) and single variate (in wavelength) components. The model is realized as the complex-valued relative permittivity and is applicable to optically smooth media. A procedure is outlined for regressing both components of the model under an appropriate set of physical constraints that preclude superfluous degrees of freedom. The procedure is demonstrated by applying the model to nickel, a transition metal of technical significance that possesses nontrivial valency. The resulting model yields practically accurate results for eight data sets spanning four separate studies, representing the approximate wavelength (λ) bandwidth between 1 and 16μm and the approximate temperature (T) range between 0 and 1400 K. The proposed model framework retains phase information and can, therefore, be directly interfaced with more complex Fresnel frameworks, such as those commonly used for modeling systematically or randomly roughened surfaces.

1.
P.
Drude
, “
Optische Eigenschaften und Elektronentheorie
,”
Ann. Phys.
319
(
10
),
936
961
(
1904
).
2.
H.
Kocer
,
S.
Butun
,
Z.
Li
, and
K.
Aydin
, “
Reduced near-infrared absorption using ultra-thin lossy metals in Fabry-Perot cavities
,”
Sci. Rep.
5
,
8157
(
2015
).
3.
D. J.
Shelton
,
T.
Sun
,
J. C.
Ginn
,
K. R.
Coffey
, and
G. D.
Boreman
, “
Relaxation time effects on dynamic conductivity of alloyed metallic thin films in the infrared band
,”
J. Appl. Phys.
104
(
10
),
103514
(
2008
).
4.
D. K.
Edwards
and
N.
Bayard De Volo
, “Useful approximations for the spectral and total emissivity of smooth bare metals,” in Advances in Thermophysical Properties at Extreme Temperatures and Pressures, edited by S. Gratch (ASME, New York, 1965), pp. 174–188.
5.
C.-D.
Wen
and
I.
Mudawar
, “
Modeling the effects of surface roughness on the emissivity of aluminum alloys
,”
Int. J. Heat Mass Transfer
49
(
23
),
4279
4289
(
2006
).
6.
G.
Teodorescu
,
P. D.
Jones
,
R. A.
Overfelt
, and
B.
Guo
, “
Normal emissivity of high-purity nickel at temperatures between 1440 and 1605 K
,”
J. Phys. Chem. Solids
69
(
1
),
133
138
(
2008
).
7.
S.
Roberts
, “
Interpretation of the optical properties of metal surfaces
,”
Phys. Rev.
100
(
6
),
1667
1671
(
1955
).
8.
S. R.
Nagel
and
S. E.
Schnatterly
, “
Frequency dependence of the Drude relaxation time in metal films
,”
Phys. Rev. B
9
(
4
),
1299
1303
(
1974
).
9.
H.
Schulz-Baldes
, “
Anomalous Drude model
,”
Phys. Rev. Lett.
78
(
11
),
2176
2179
(
1997
).
10.
H. U.
Yang
,
J.
D’Archangel
,
M. L.
Sundheimer
,
E.
Tucker
,
G. D.
Boreman
, and
M. B.
Raschke
, “
Optical dielectric function of silver
,”
Phys. Rev. B
91
(
23
),
235137
(
2015
).
11.
J.
Orosco
and
C. F. M.
Coimbra
, “
On a causal dispersion model for the optical properties of metals
,”
Appl. Opt.
57
(
19
),
5333
5347
(
2018
).
12.
J.
Orosco
and
C. F. M.
Coimbra
, “
Anomalous carrier transport model for broadband infrared absorption in metals
,”
Phys. Rev. B
98
(
23
),
235118
(
2018
).
13.
F. E. M.
Silveira
and
J. A. S.
Lima
, “
Attenuation and damping of electromagnetic fields: Influence of inertia and displacement current
,”
J. Phys. A Math. Theor.
42
(
9
),
095402
(
2009
).
14.
N.
Pottier
, Nonequilibrium Statistical Physics: Linear Irreversible Processes, Oxford Graduate Texts (Oxford University Press, Oxford, 2010), ISBN: 978-0-19-955688-5, OCLC: 368051042.
15.
N. W.
Ashcroft
and
N. D.
Mermin
,
Solid State Physics
(
CBS Publishing Asia Ltd.
,
Philadelphia, PA
,
1976
).
16.
I.
Podlubny
, Fractional Differential Equations, Mathematics in Science and Engineering, 1st ed. (Academic Press, San Diego, CA, 1998), Vol. 198, ISBN: 978-0-12-558840-9.
17.
C. F. M.
Coimbra
and
R. H.
Rangel
, “
Spherical particle motion in harmonic Stokes flows
,”
AIAA J.
39
(
9
),
1673
1682
(
2001
).
18.
B.
West
,
M.
Bologna
, and
P.
Grigolini
, Physics of Fractal Operators, Institute for Nonlinear Science (Springer-Verlag, New York, 2003), ISBN: 978-0-387-95554-4.
19.
O. G.
Bakunin
, “
Description of the anomalous diffusion of fast electrons by a kinetic equation with a fractional spatial derivative
,”
Plasma Phys. Rep.
30
(
4
),
338
342
(
2004
).
20.
A.
Jaishankar
and
G. H.
McKinley
,
Power-law rheology in the bulk and at the interface: Quasi-properties and fractional constitutive equations
,”
Proc. R. Soc. A
469
(
2149
),
20120284
(
2013
).
21.
N.
Singh
, Electronic Transport Theories: From Weakly to Strongly Correlated Materials (CRC Press LLC, Boca Raton, FL, 2016), ISBN: 978-1-4987-4360-0, OCLC: 960704909.
22.
I.
Setién-Fernández
,
T.
Echániz
,
L.
González-Fernández
,
R. B.
Pérez-Sáez
, and
M. J.
Tello
, “
Spectral emissivity of copper and nickel in the mid-infrared range between 250 and 900°C
,”
Int. J. Heat Mass Transfer
71
,
549
554
(
2014
).
23.
P.
Drude
, “
Zur Elektronentheorie der Metalle
,”
Ann. Phys.
306
(
3
),
566
613
(
1900
).
24.
P.
Drude
, “
Zur Elektronentheorie der Metalle; II. Teil. Galvanomagnetische und thermomagnetische Effecte
,”
Ann. Phys.
308
(
11
),
369
402
(
1900
).
25.
P.
Drude
, “
Zur Ionentheorie der Metalle
,”
Phys. Z.
1
(
14
),
161
165
(
1900
).
26.
J. W.
Allen
and
J. C.
Mikkelsen
, “
Optical properties of CrSb, MnSb, NiSb, and NiAs
,”
Phy. Rev. B
15
(
6
),
2952
2960
(
1977
).
27.
H. J.
Park
,
B. C.
Park
,
M.-C.
Lee
,
D. W.
Jeong
,
J.
Park
,
J. S.
Kim
,
H. S.
Ji
,
J. H.
Shim
,
K. W.
Kim
,
S. J.
Moon
,
H.-D.
Kim
,
D.-Y.
Cho
, and
T. W.
Noh
, “
Electrodynamic properties of the semimetallic Dirac material SrMnBi2: Two-carrier-model analysis
,”
Phys. Rev. B
96
(
15
),
155139
(
2017
).
28.
K.
Akiba
,
A.
Miyake
,
Y.
Akahama
,
K.
Matsubayashi
,
Y.
Uwatoko
, and
M.
Tokunaga
, “
Two-carrier analyses of the transport properties of black phosphorus under pressure
,”
Phys. Rev. B
95
(
11
),
115126
(
2017
).
29.
J.
Orosco
and
C. F. M.
Coimbra
, “
Variable-order modeling of nonlocal emergence in many-body systems: Application to radiative dispersion
,”
Phys. Rev. E
98
(
3
),
032208
(
2018
).
30.
M.
Dressel
and
G.
Grüner
,
Electrodynamics of Solids: Optical Properties of Electrons in Matter
(
Cambridge University Press
,
Cambridge
,
2002
), ISBN: 978-0-521-59726-5.
31.
S.
Roberts
, “
Optical properties of nickel and tungsten and their interpretation according to Drude’s formula
,”
Phys. Rev.
114
(
1
),
104
115
(
1959
).
32.
M. Q.
Brewster
,
Thermal Radiative Transfer and Properties
(
John Wiley & Sons
,
New York
,
1992
), ISBN: 978-0-471-53982-7.
33.
D. W.
Lynch
,
R.
Rosei
, and
J. H.
Weaver
, “
Infrared and visible optical properties of single crystal Ni at 4 K
,”
Solid State Commun.
9
(
24
),
2195
2199
(
1971
).
34.
D. W.
Lynch
and
W. R.
Hunter
, “Nickel (Ni),” in Handbook of Optical Constants of Solids, edited by E. D. Palik (Academic Press, Burlington, MA, 1997), Vol. 1, pp. 313–323.
35.
J. C.
Phillips
, “
Fermi surface of ferromagnetic nickel
,”
Phys. Rev.
133
(
4A
),
A1020
A1028
(
1964
).
36.
J.
Orosco
and
C. F. M.
Coimbra
, “Thermophysical model for the infrared emissivity of metals,” in AIAA SciTech 2019 Forum (American Institute of Aeronautics and Astronautics, 2019).
37.
G. W.
Autio
and
E.
Scala
, “
The normal spectral emissivity of isotropic and anisotropic materials
,”
Carbon
4
(
1
),
13
28
(
1966
).
38.
K.
Tang
and
R. O.
Buckius
, “
A statistical model of wave scattering from random rough surfaces
,”
Int. J. Heat Mass Transfer
44
(
21
),
4059
4073
(
2001
).
39.
J.
Orosco
and
C. F. M.
Coimbra
, “
Optical response of thin amorphous films to infrared radiation
,”
Phys. Rev. B
97
(
9
),
094301
(
2018
).
40.
S.
Kasap
,
C.
Koughia
,
H.
Ruda
, and
R.
Johanson
, “Electrical conduction in metals and semiconductors,” in Springer Handbook of Electronic and Photonic Materials, Springer Handbooks, edited by S. Kasap and P. Capper (Springer US, Boston, MA, 2007), pp. 19–45, ISBN: 978-0-387-29185-7.
41.
D. K.
Edwards
, “
Radiative transfer characteristics of materials
,”
J. Heat Transfer
91
(
1
),
1
15
(
1969
).
42.
P.
Winsemius
, “
Temperature dependence of the optical properties of Au and Ag
,” Ph.D. thesis (
Leiden University
,
1973
).
43.
C.
Cagran
,
G.
Pottlacher
,
M.
Rink
, and
W.
Bauer
, “
Spectral emissivities and emissivity X-points of pure molybdenum and tungsten
,”
Int. J. Thermophys.
26
(
4
),
1001
1015
(
2005
).
44.
M. A.
Ordal
,
R. J.
Bell
,
R. W.
Alexander
,
L. L.
Long
, and
M. R.
Querry
, “
Optical properties of Au, Ni, and Pb at submillimeter wavelengths
,”
Appl. Opt.
26
(
4
),
744
752
(
1987
).
45.
R.
Kohlrausch
, “
Theorie des elektrischen Rückstandes in der Leidener Flasche
,”
Ann. Phys.
167
(
1
),
56
82
(
1854
).
46.
G.
Williams
and
D. C.
Watts
, “
Non-symmetrical dielectric relaxation behaviour arising from a simple empirical decay function
,”
Trans. Faraday Soc.
66
(
0
),
80
85
(
1970
).
47.
R.
Metzler
and
J.
Klafter
, “
From stretched exponential to inverse power-law: Fractional dynamics, Cole–Cole relaxation processes, and beyond
,”
J. Non-Cryst. Solids
305
(
1
),
81
87
(
2002
).
You do not currently have access to this content.