Overview of the optical properties of
metals, insulators and semiconductors
v. 3
3.5.2021
1
FK020_Electrodynamics of solids
Josef Humlíček, humlicek@physics.muni.cz
Metals
• Characteristics:
large concentration of free electrons; infrared polarizability due to the free charge carriers is typically
very high, decreasing monotonically towards larger frequencies.
• The contribution of free carriers overlaps that of interband electronic transitions.
• Decreasing real part of the dielectric function with decreasing photon energy E is a characteristic
feature of the free-carrier response; in the dc field (E=0), a finite (negative) value is reached. This
behavior is linked to the diverging imaginary part of the dielectric function for E→0.
Optical conductivity remains finite; this might be the reason for using it, in particular in the infrared
range.
2
Real and imaginary part of the complex dielectric function of several elemental metals.
3
0 2 4 6
-300
-200
-100
0
Ag
Ni
Mo
Au
Al
interband
transitions
free carriers
Cr

E (eV)
0 2 4 6
0
20
40
60
80
Ag
Ni
Mo
Au
Al
interband
transitions
free carriers
Cr

E (eV)
Real part of the complex conductivity of Al and Mo
4
0 2 4 6
1000
10000
100000
Al
Mo
1
(1/cm)
E (eV)
• Direct-current conductivity of pure Al and Mo at room temperature is 3.77x105 and 1.87 x105 1/cm,
respectively; in the infrared data of Al, the dc value is almost reached at the lowest photon energy. On the
contrary, the conductivity of Mo has to increase below the lowest photon energy in the shown spectrum.
• The monotonic contribution of free electrons overlaps with that of the interband transitions. In the spectra of Al,
the latter is dominated by a relatively sharp band located at about 1.6 eV; the interband transitions in Mo form a
more complex contribution, including evidently rather low photon energies.
Complex refractive index of Al and Mo
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Optical spectra of Aluminum display a typical “metallic” behavior. The complex refractive index, n+ik, has its real
part (n) noticeably smaller than the real one (k). This is due to the prevailing influence of free electrons`in this
sense, the response of Molybdenum is “less metallic”. In “ideal metal” at positive frequencies, smaller than the
plasma frequency, the dielectric function is real and negative; its square root is therefore pure imaginary.
Complex refractive index is instrumental in dealing with the propagation of light waves in homogeneous material (via
wavelength and decay of the amplitude) and in investigating the phenomena at interfaces (using, e.g., Fresnel
formulae for the amplitude reflectance and transmittance.
The penetration depth of light (1/K) in the optical range is typically rather small, of the order of tens of nanometers.
This is the reason for the opacity of metallic films even of rather small thicknesses. Note also the large values of
the wavelength in Al in the ultraviolet.
0 2 4 6
0.1
1
10
Al
Mo
n
k
refractiveindex
E (eV)
0 2 4 6
10
100
1000
Mo
Mo
Al
Al
vac
/n
(wavelength)
1/K
(penetration depth)
1/K,vac
/n(nm)
E (eV)
Au, problems related to sample quality and spectral measurements
6
Gold: relatively stable surfaces of (poly)crystals, evaporation and sputtering
Aspnes 1980, ellipsometric results and previous measurements
7
Aspnes 1980, ellipsometric measurements, films evaporated at low (LN) and room temperatures of the substrate
(for low temperatures – worse crystallinity of the films, stronger scattering of free electrons).
Au, problems related to sample quality and spectral measurements
8
Au / Si, evaporated at DCMP Brno (2000); ellipsometric data in accord with the assumption of a porous overlayer (~10
nm thick)
1.5 2.0 2.5
-30
-20
-10
0
1.5 2.0 2.5
0
2
4
6
elli Au/Si
80 & 85
o
0.15
0.10
voids (1) in Au (DruLor, fitted to Theye)
f =0
MG, L=0.33 (spheres)
0.05
1
E (eV)
elli Au/Si
80 & 85
o
0.15
0.10
0.05
f =0
MixVM3_MixSpe
2
Au, problems related to sample quality and spectral measurements
9
high-quality massive polycrystals of Au, Ag and Cu: Otter 1961; ellipsometric data on curved samples with no surface
treatment
Au, problems related to sample quality and spectral measurements
Fe: free electrons and interband transitions
10
Large differences between dielectric functions; solid lines – ellipsometric data measured at DCMP (2007);
difficult preparation of good samples.
Drude-like response of free electrons is negligible in the absorptive part (2), however, very well resolved in the real
part (1).
1.0 1.5 2.0 2.5 3.0
-10
-8
-6
-4
-2
0
1.0 1.5 2.0 2.5 3.0
10
20
30
Ordal Yolken
Neuber
Johnson
1
E (eV)
Johnson
Ordal
Neuber
RT
2
Yolken
Example of an alloy: Agx Au1-x
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Alloys usually display properties typical of modifications of their components.
Ag + Au: Peña-Rodríguez (2014), evaporated films (170 – 330 nm on Si, co-deposition by electron-beam evaporation
of Ag and Au, the thickness and composition obtained from RBS), ellipsometric spectra.
Differences seen in the free electron response (eps. in NIR), and also in the onset of the strong interband absorption
(almost 1.5 eV higher in Ag than in Au).
12
Ag + Au: Peña-Rodríguez (2014)
Parametrization of the free electron response – Drude model (JC=“Johnson-Christy” for both endpoints).
Plasma energy almost independent of the composition (Ep ≈ 9 eV).
Typical dependence, x(1-x), of the broadening parameter is due primarily to the electron scattering on the potential of
randomly distributed atoms in the alloy.
Example of an alloy: Agx Au1-x
“Metallic behavior” of TiN
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A number of compounds contains large density of free charge carriers, leading to the optical response similar to
elementary metals and their alloys. TiN is a very hard compound, with the color similar to that of gold.
In IR, the dominant polarizability is that of free electrons, in UV that of the interband transitions; the sum of Drude
contribution and a broad Lorentzian band ( at 3.6 eV) fits very well the measured ellipsometric spectra.
Free electrons in v TiN
14
Parametrization using the Drude model in a narrow
spectral range;
fairly good agreement of the dc resistivity with the
extrapolation to zero frequency (in cm with both
energies in eV):
The overlap of free-electron response with interband transitions in TiN:
plasmon resonance in the visible range:
15
The negative inverse of the dielectric function displays the band of increased absorption of
longitudinal waves at 2.2 eV, in a good agreement with EELS data.
Crystalline semiconductors and insulators
translational symmetry → dispersion of energies (k-dependence) of single-particle states
the quasiparticles are
bosons fermions
16
17
An instructive case study - doped (n-type) crystalline GaAs
low frequencies (FIR-MIR):
• free-electron plasma, small effective mass, large mobility
• polar lattice vibrations, TO 8 THz (34 meV), LO 8.5 THz
larger frequencies (NIR, VIS, UV):
• onset of interband transitions (1.42 eV at RT)
• strong variations of the joint density of states above gap, in particular in UV
• strong influence of final-state (excitonic) interaction
Spectra at the onset of absprption near the gap energy, undoped GaAs
(„epiready“ sample, several years in the laboratory atmosphere; ellipsometric measurements)
a strong influence of the excitonic interaction on the 3DM0 direct gap
Eg = 1.42 eV @RT
second derivative of the dielectric function (corrected for the presence of a nonabsorbing
overlayer)
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1.0 1.2 1.4 1.6 1.8 2.0
-200
0
200
d
2
<>/dE
2
(ev
-2
)
Im
Re
GaAs (100) corrected for 3.3nm overlayer
GaAs_bulk_D2_E0
E (eV)
Undoped GaAs – ellipsometric data
Eg = 1.42 eV @RT
the dangerous extrapolations of the spectral dependences
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1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
0
1
2
Eg
(1.42 eV)
GaAs (100) corrected for 3.3nm overlayer
GaAs_bulk_eps222
2
2
2
E (eV)
2
2
2
Several insulators and semiconductors, crystalline and amorphous (”glassy”)
materials
LiF
a-SiO2
intrinsic crystalline Si
multiphonon absorption, transparent range
interband transitions
Si:P, metal-insulator transition
20
Lattice vibrations in polar crystals: LiF
FIR (TO) + MIR (LO) ellipsometry, JH, Thin Solid Films 313-314 (1998), 687-691
21
Dielectric function:
(resonance of the IR wave with the TO phonon)
Lorentzian profile of the band
extrapolation of the real part gives the static
permittivity
noticeable broadening at RT
Negative inverse of the dielectric function:
( “LO-resonance“)
nearly Lorentzian lineshape
the width influenced by the presence of
multiple-phonon absorption
Lattice vibrations of a more complex polar crystal: a-SiO2
22
Anisotropic (uniaxial) crystal:
several TO and LO modes identifiable in  and -1/ for ordinary and extraordinary direction;
most of the bands fairly narrow (small broadening parameters G)
23
Ellipsometric data and best-fit Lorentzians, ordinary and extraordinary direction
Interlacing of TO and LO modes in a-SiO2
Transparent range between lattice vibrations (IR) and electronic absorption (UV) in a-SiO2
24
Dielectric function is real (conductivity is imaginary); the dispersion of optical functions can be very
precisely (~10-5) approximated by the expansion
using 2 “phonon” terms (j=-1,-2) and 4 “electron” terms (j=0,...,3)
Multiphonon response of homopolar materials: intrinsic Si
MIR, transmission and reflectivity + KK analysis, JH et al., phys. stat. sol. (a) 92 (1985) 249-255
25
Very small absorption (slabs ~1mm thick
provide easily measurable transmittance
even at the maximum at 605 cm-1;
spectral changes of the real part of refractive
index are smaller than 0.001.
Low-frequency (MIR) part of the transparent
range is described by the following
parametrization of refractive index
(the dispersion is due to the interband
transition above gap wavenumber, ~ 9000
cm-1):
Ellipsometric spectra of Si (rotating analyzer)
Aspnes (Studna, McIntyre), 1983 (c-Si)
Aspnes, Studna, Kinsbron, 1984 (a-Si)
problems in the range of small absorption above the (indirect) gap
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c-Si
a-Si
Optical spectra of silicon in MIR-UV
utilizing SOI structures and homoepitaxial samples
27
28
normal-incidence reflectivity in MIR+NIR (Fourier spectrometer IFS66) , NIR-VIS-UV (fiber spectrometer Avantes 2048)
JH et al., JAP 118 (2015) 195706
Optical spectra of silicon in MIR-UV
utilizing SOI structures and homoepitaxial samples
Elipsometry (rotating compensator) in NIR-VIS-UV, sensitivity to surface overlayers;
utilizint the data obtained from the reflectance (at 3.0 and 3.2 eV)
5.22 5.24 5.26
0.0
0.1
0.2
0.3
0.4
6.05 6.10 6.15 6.20
0.3
0.4
0.5
0.6
0.7
0.8
this work (SOI)
3.5
3.0
2.5
2.0 nm
2.0
0.5 nm
1.5
1.0
HerzingerJellison
Aspnes
<n>
<k>
Si @ 3.0 eV 0.5 nm
1.0
1.5
2.0
2.5
3.0
3.5
this work (SOI)
HerzingerJellison
Aspnes
2.0 nm
Si @ 3.2 eV
29
JH et al., JAP 118 (2015) 195706
Optical spectra of silicon in MIR-UV
utilizing SOI structures and homoepitaxial samples
Donor states and free carriers in n-type semiconductors: Si:P
FIR ellipsometry, JH et al., AIP Conf. Proc. 893 (2007) 33
30
Complex conductivity
(doping very close to the Metal-Insulator
transition)
differs from the Drude lineshape even at RT
complex frequency and temperature
dependences
residues from the absorption lines of
diluted donors (?)
Donor states and free carriers in n-type semiconductors: Si:P
FIR transmission, JH et al., unpubl.
31
Free electrons freeze out with decreasing
temperature
the localized electron states are responsible
for the absorption lines and the
donor - conduction band continuum
Amorphous insulators („glasses“)
Short range order (SRO) preserved, long range order (LRO) missing
→
sharp structures in vibrational and electronic spectra are smeared out,
transparent range between low- and high-frequency absorption remains,
very low attenuation of the light waves is attainable (optical fibers).
32
Vibrations in amorphous polar materials: glassy SiO2
elipsometry in, JH, unpublished
33
Broad vibrational bands (nearly Gaussian profiles) in three ranges of phononic structures of crystalline
SiO2
400 600 800 1000 1200
-5
0
5
10
g-SiO2
etched
Re
Im
MIRElliQ-EpsQ

W (cm
-1
)
Transparent range between vibrational (IR) and electronic (UV) absorption: g-SiO2
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Even powers expansions of the refractive index or dielectric function identifie the vibronic and
electronic contribution to the dispersion.
Data points: several minimum-deviation measurements on prisms (from Palik’s handbook); attaining the high level of
precision requires stringent control of temperature and other conditions.
0 10000 20000 30000 40000
1.30
1.35
1.40
1.45
1.50
1.55
vibrations
electrons
n
W (cm
-1
)
SiO2n-gSiO2n
g-SiO2
Optical glasses
utilize the transparent range between vibrational and electronic absorption
refractive index at selected wavelengths
from visible range
and
dispersive power / Abbe number
35
36
very small attenuation should be observed in heavy (fluoride) glasses
Transparent range between vibrational (IR) and electronic (UV) absorption
Recommended reading
A number of textbooks and monographs dealing with the electrodynamics at
optical frequencies:
Feynman, Landau-Lifshitz, Born-Wolf, Wooten, Dressel-Gruener, Yu-Cardona, …
Layered systems interacting with polarized light are dealt with in detail in
“Handbook of Ellipsometry” (edited by Irene and Tompkins).
Several topics concerning mainly present-day ellipsometry are dealt with in
“Ellipsometry at nanoscale” (edited by Losurdo and Hingerl).
A good deal of discussion of individual materials in the “critiques” of
Palik’s Handbooks.
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