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Optical analysis of doped PbTe samples using UV- VIS and IR ellipsometry
1
Abstract
Lead-tin telluride alloy, Pb1-xSnxTe, is a narrow band gap group IV–VI semiconductor
with NaCl-like crystalline structure. This material has interesting electronic
properties, which makes it suitable for designing infrared photo detectors, diode
lasers, and thermo-photovoltaic energy converters. In this project, we used
spectroscopic ellipsometry in the spectral range of 0.74–6.5 eV to probe the linear
optical response of Pb1−xSnxTe alloys in terms of the complex dielectric function ε(ω)
= ε1(ω) + iε2(ω). A strong optical response in the range of 0.7-2.0 eV arising from
optical absorption was found. We studied eleven different samples of Pb1−xSnxTe of ptype origin with x values in the range (0 ≤ x ≤ 1). They were prepared by means of
molecular beam epitaxy (MBE) on BaF2 substrates with 15mm2 area.
2
Acknowledgement
The completion of this book has been through the effort and support of many people.
First of all I will like to thank my supervisor Prof. Hans Arwin for given me the
opportunity to work in this field. You were all the time there for me when I needed
you. You taught me a lot of things and you were patience is just all I needed to reach
this far.
I am indebted to my wife Martha and my son Egil for their support in prayer, love and
confidence. I know it has not been very easy for you in the last six months but thanks
to God for the strength.
I am also indebted to my colleagues here at IFM such as Mohammed Ibrahim, Nazir,
Khalid all of Sudan. To all my Thai and Pakistani friends who were part of this
course.
I also thank all friends and loved ones that I met here during this short period of
studies.
The last and not the least to my entire family back home in Ghana and in Germany. I
say a big thank to you all. I dedicate this book to you all.
3
Table of Contents
Chapter
1.
2.
3.
4.
5.
Page
Introduction
1.1
PbTe as an important material for electronic application
1.2
The Measuring Parameter
5
6
Optical properties
2.1
The complex dielectric function ε
2.2
The complex index of refraction
2.3
The absorption coefficient α
2.4
Polarization
6
7
8
8
Reflection
3.1
The Fresnel’s equations as important parameters
3.2
Optical modeling
3.3
The Cauchy model
10
11
11
Ellipsometry
4.1
Definition and theory of ellipsometry
4.2
Available ellipsometers
i.
VASE-2
ii.
VASE-IR
4.3
Sample preparation
4.4
BaF2 as a Substrate
4.5
Experimental
4.6
Fourier Transform Infrared ellipsometry
4.7
Electronic Energy Band Structure
11
13
13
14
15
15
16
16
17
Results and discussion
5.1
Comparing results of SE and Infra-red ellipsometry
5.2
Conclusion
18
23
25
References
27
Appendix I
28
Appendix II
32
4
1
Introduction
1.1
PbTe as an important material for electronic application
PbTe is a narrow-gap semiconducting compound and is often alloyed with tin and
used as an infra-red detector material. It has low thermal conductivity and thus has a
good performance as a thermoelectric material. It occurs naturally as the mineral
altaite. It possesses high dielectric constant and high mobility and has a band gap of
about 0.3 eV. PbTe, which has bcc lattice, exhibits properties that are unusually
unique and somehow related to other semiconductors [1].
The photon energy (E) depending dielectric function ε(E) of a semiconductor is
known to be closely related to its electronic band structure which can easily be traced
from the critical points in the optical spectra. The Kramers-Kronig analysis of
reflectivity data can be used to obtain the optical properties of PbTe, which enables
one to understand the energy band structure.
The real (ε1) and imaginary (ε2) parts of the dielectric function, ε(E) =ε1(E)+iε2(E), of
PbTe have been measured by using spectroscopic ellipsometry (SE) in the 1.15-5.4
eV photon-energy range at room temperature in 1993 by Norihiro Suzuki and Sadao
Adachi. The measured SE spectra showed distinct structures at energies of E1, E2 and
E3 critical points. Two theoretical models, namely, the model dielectric function
(MDF) and standard critical-point (SCP) model were used to analyze the data. The
SCP model gave a satisfactory fit to the second derivatives of ε1(E) spectra
[d2ε1(E)/dE2], but could not yield good fits to ε2(E) spectra. The MDF, on the other
hand, gave results that were in excellent agreement with the experimental data for
both ε(E) spectra and their derivatives [1].
Other dielectric-related functions, such as the complex refractive index (N = n +ik),
the absorption coefficient (α) and the normal-incidence reflectivity, of PbTe can also
be determined.
It is ascertained that various calculations and experiments that have been performed,
have given much information about the electronic energy-band structure as well as the
optical properties of PbTe in both the visible and ultraviolet spectra regions [1].
5
1.2
The Measuring Parameter
With spectroscopy ellipsometry we can determine the complex reflectance ratio as a
function of photon energy in the IR-visible-UV optical spectral range. Both the real
and imaginary parts of the dielectric function ( ε = ε 1 + iε 2 ) of a thin film (PbTe) or
substrate (BaF2) can be determined from the complex reflectance ratio. Ellipsometry
works best for film characterization when the film thickness is not too much smaller
or larger than the wavelength of the light used for the measurement.
The real (ε1) and imaginary (ε2) parts of the dielectric function, ε(E) = ε1(E)+iε2(E), of
PbTe can measured by spectroscopic ellipsometry (SE) in a low photon-energy range
at room temperature. The measured SE spectra can reveal distinct structures at
energies of E1, E2 and E3 critical points. These can be easily analyzed using two
theoretical models, as discussed in the introduction [1].
2
Optical Properties
2.1
The complex dielectric function ε
Maxwell’s equations are the fundamental rules governing the electromagnetic
phenomena and for harmonic monochromatic fields with frequency ω, such that
(d dt ) = −iω , the equations are:
→
r
∇• D = ρ
→
→
∇• B = 0
→
r r
r
∇× H = J − iωD
→ →
r
∇× E = iωB
(2.1)
(2.2)
(2.3)
(2.4)
r
r
where E and H are the electric and magnetic fields, which describe the
electromagnetic field. The sources of these fields are the electric charge density ρ and
r
r
r
the current density J . The effects of E and H on matter can be described by the
→
r
electric displacement D and the magnetic induction B , respectively. The properties
of matter enter into equations (2.1) and (2.4) through the constitutive equations [2]:
r
r
B = µµ 0 H
(2.5)
r
r
D = ε~ε 0 E
(2.6)
6
r
r
J = σ~E
(2.7)
From the equations above, (2.5), (2.6) and (2.7), we define the relative magnetic
permeability µ, the relative dielectric function ε and the conductivity σ[2] .
At optical frequencies we always have µ = 1 since magnetic dipoles cannot follow the
rapid oscillations of the magnetic field. It is also difficult to differentiate between ε
and σ and normally one defines an effective dielectric function ε eff that includes the
contribution from σ. By inserting equation (2.6) and (2.7) into equation (2.3) we
obtain equation (2.8) which condenses the optical properties of solids into one
parameter, ε eff , which can be measured by optical instruments as well as be
calculated from the first principles.
r r
⎛ σ~
∇ × H = - iω ⎜⎜ i
+ ε~
ωε
0
⎝
r
⎞ r
⎟⎟ε 0 E =- iωε eff ε 0 E
⎠
(2.8)
r
In the general case, the displacement D will not inevitably follow the time variation
of the electric field as stated in equation (2.6) and such a material has losses and
r
r
absorbs energy. The difference between D and E can be mathematically described
by introducing an imaginary part of ε given by [2]
ε=ε 1 + i ε 2
(2.9)
This shows that the dielectric function in general is a complex number.
2.2
The complex index of refraction
Fresnel’s equations are used to describe the reflection and transmission of light by a
plane interface between two media with different refractive indices. The assumptions
made are that, the incident light is described by a monochromatic plane wave and that
the media are homogeneous and isotropic, such that the optical properties of the
material can be described as scalar functions. The media can then be described by the
complex refractive index [2].
N = n + ik
(2.10)
where n is the index of refraction and k is the extinction coefficient. The complex
refractive is related to ε through
7
2.3
ε = ε 1 + iε 2 = N 2
(2.11)
ε1 = n2 − k 2
(2.12)
ε 2 = 2nk
(2.13)
The absorption coefficient α
The light attenuation during propagation in a medium is expressed in terms of the
intensity variation and this depends on the absorption coefficient α. This quantity is
calculated from Beer-Lambert’s law
I = I 0 e −αd
(2.14)
where I and I0 are the transmitted and incident intensity, respectively and d is the path
length of light which is propagating through the medium. The relationship between α
and k is given by
α=
2.4
4πk
(2.15)
λ
Polarization
The state of polarization is an important property of a plane wave and here, the plane
wave can be described by its propagation direction as well as temporal and spatial
dependencies of the electric field E, presented in a complex form as
∧
∧
E = E x x+ E y y
(2.16)
E x and E y are the complex field amplitudes and they represent the projection of the
field along the x-and y-axes of the coordinate system.
The plane wave is said to be unpolarized if E x and E y are totally uncorrelated and
totally polarized if they are completely correlated. On the other hand, we obtain
partially polarized light for partial correlation between E x and E y .
In view of the possible polarization of values of amplitude and phase, four different
types of polarization are recognized and these are illustrated in Fig.1 together with
their Jones vectors with the components Ex and Ey of an electromagnetic plane waves
[2].
8
a
c
b
d
Fig.1. The Jones vectors describing some states of polarization
Assume that an electromagnetic wave is traveling along the z-direction with phase
difference between E x and E y being equal to zero or an integral multiple of π. This
gives linearly polarized light in the x-direction, which is illustrated in Fig. 1a above or
in the y-direction. When linearly polarized light is passed through a quarter-wave
plate at an angle of 45o to the optical axis of the quarter-wave plate which is made-up
of two plane waves of equal amplitude but differs in phase by π/2, it leads to
circularly polarized light as seen in Fig. 1b. If the Jones vector in Fig. 1a which
represents linear polarized waves whose electric vector oscillating along the xdirection has its angle inclined at α to the x- axis, we have left to right circularly
polarized waves as shown in Fig. 1c. One obtains elliptically polarized light when the
phase shift between E x and E y is different from 0 ± nπ or π/2 ± nπ . This is shown
by the Jones vector in Fig. 1d [2].
9
3
Reflection
3.1
The Fresnel’s equations as important parameters
The effect of light at the interfaces between media of different index of refraction can
be described by the Fresnel equations, which were deduced by Augustin-Jeans
Fresnel.
This gives the relation between angle of incidence and that of refraction from Snell’s
law as [2, 3]
n0 sin θ 0 = n1 sin θ 1
(3.1)
where n0 and n1 are the refractive indices of the two media.
When the s-polarized light is considered, the Fresnel equation is given by
rs =
n0 cosθ 0 − n1 cosθ 1
n0 cosθ 0 + n1 cosθ 1
(3.2)
When the p- polarization is considered, the electric and magnetic fields of incident,
reflected and refractive wave are defined as shown in Fig. 3 and the corresponding
Fresnel expression is given by [2, 3]
rp =
n1 cos θ 0 − n0 cos θ 1
n1 cos θ 0 + n0 cosθ 1
(3.3)
10
3.2
Optical modeling
Ellipsometry is an indirect method of measurement in that the measured ψ and ∆
cannot be converted directly to the optical properties of the sample. Hence a desired
model such as the Cauchy model is used.
3.3
The Cauchy model
Cauchy’s dispersion relation is expressed in a polynomial form as shown below. This
analysis is usually used for organic and other thin film layers that are weakly
absorbing dielectric [2].
n(λ ) = A +
B
λ
2
+
C
λ4
+ ...
(3.4)
Here n is refractive index and coefficients A, B and C are adjustable parameters [2].
4
Ellipsometry
4.1
Definition and theory of ellipsometry
Ellipsometry is an optical technique used for surface, thin film and multilayer
characterization. The measured quantity is the change of polarization of light which is
reflected upon or transmitted through the sample under investigation. The schematic
diagram showing the principle of ellipsometry is shown in Fig. 4. Ellipsometry was
11
introduced by Drude in 1889. Ellipsometry was first used to demonstrate the presence
of surface layers and their characterizations. Ellipsometry can be divided into classes
depending on the wavelength range, sample scanning area and detection mechanism
[2].
When the ellipsometer was introduced it became possible to measure thickness of thin
films in a more accurate and objective way. Two factors that make ellipsometry
especially attractive for surface analysis are the fact that it is non-destructive and that
it is possible to use for in situ studies in a non-vacuum ambient. It can even be used at
the solid/liquid interface. Also it is remarkably sensitive for small surface changes and
sub-nanometer resolution in such a way that thin film thickness can readily be
obtained. It is known that, measurements from ellipsometry are usually performed to
describe an “optical system” that modifies the polarization state of a beam of light.
For thin film sample analysis, the “optical system” is due to the reflection of light
from the sample.
The ratio of amplitude changes and the difference in phase changes for the p- and spolarized components are often denoted with tanψ and ∆ , respectively. These
parameters are defined by introducing the complex reflectance ratio ρ according to
the relation
ρ=
Rp
Rs
= tan(ψ )e i∆
(4.1)
where R p and Rs are the complex-valued reflection coefficients for the p- and spolarization, respectively [2, 4, 5].
The two parameters ∆ and ψ are called the ellipsometric angles and are measured by
determining the change of polarization of light reflected at a surface. The reflected
light is in general elliptically polarized; hence the technique is referred to as
ellipsometry. These ellipsometric angles are simple to measure very accurately.
Ellipsometry measures a ratio of two values and it can be highly accurate and very
reproducible. Because the ratio is a complex number, it also contains “phase”
information (∆), which makes the measurement very sensitive [5].
12
Fig. 4. Schematic set up for an ellipsometer of the rotating analyzer type
4.2
Available Ellipsometers
The J. A. Woollam Company, which was founded in 1987 in response to commercial
demand for ellipsometry technology, made the instruments used in this investigation.
The high precision ellipsometers from the VASE series and VASE-2 is designed to
work in a higher energy range than VASE-IR [6].
i.
VASE-2
This instrument is connected through an optical fibre to a source of light, which is a
high-pressure xenon lamp that is used in the light energy range from 0.74 eV to 6.5
eV. In optical measurement, a xenon lamp is often used because of its wide energy
spectrum and light intensity that is almost equal in a given range.
VASE-2 has a large advantage over the second ellipsometer in that it can measure
samples in a given energy range and thus does not need to measure in the full
available range.
13
Fig .5 VASE-2 Spectroscopic ellipsometer
ii.
VASE-IR
Fig. 6 “VASE-IR” spectroscopic ellipsometer
Here the light source is a hot SiC element whose spectrum can be well explained by
blackbody emission theory. The ellipsometer parameters were set at several
wavelengths with angles ranging from 60º to 70º.
The IR-VASE is a highly accurate spectroscopic technique able to cover a wide
spectral range from 2 to 33 microns (333 to 5000 wavenumbers) and automated angle
of the incidence from 250 to 900. Characterization of both thin films and bulk
materials can be done and various properties can be determined from this
measurement such as thickness, optical constants (n and k, ε1 and ε2), material
composition-molecular vibration, chemical bonding, doping concentration and more.
14
IR-VASE has many advantages such as the rotating compensatory configuration,
which produces excellent data from both transparent and opaque samples. And also
this commercially designed and built instrument is extremely stable, which it can even
be non-stop operated a week. Moreover the instrument can handle variable
temperature of the sample and in the future anisotropic samples of arbitrary
orientation can be studied [7].
4.3
Samples preparation
The samples used in this experiment were prepared at Departamento de Ciências
Exatas, Área de Informática, Universidade Estadual de Feira de Santana, Feira de
Santana, Bahia, Brazil. The main crystals structure for the alloy Pb1−xSnxTe which is
under investigation is the PbTe. There are 11 samples of this type as stated earlier and
their optical properties placed under investigation.
4.4
BaF2 as a Substrate
The layers are deposited on BaF2 (111) substrates which are transparent in wide a
spectrum band. BaF2 is a crystal, which is relatively hard but is highly sensitive to
thermal shock. Its transmission range is 0.2 µm - 11 µm. It has a refractive index of
1.46 at 3.2 µm and a reflection loss of 6.8% at 3.0 µm (for 2 surfaces). Some physical
properties are melting point at 1280° C, hardness of about 82 psi, Young's modulus of
53.07 GPa, modulus of rupture of 3900 psi and cubic structure along (111) cleavage
plane. Chemically, it is soluble in acid and NH4Cl and has solubility of 0.17 g/100g
water at 23° C [8].
BaF2 is used for optical windows, lenses and prisms in UV-IR. It can also be used as
substrate for some applications. In addition, BaF2 is usually used as scintillator for
gamma detection.
Optical crystals, such as CaF2, BaF2, MgF2, LiF, NaCl, KCl, KBr, etc, are widely used
in IR optics and UV optics. High transparency and low loss optical windows, prisms,
lenses, achromatic lenses and parallel planes have been fabricated in these optical
materials. These crystals are widely applied to the entire optical wavelength range,
from vacuum ultraviolet (0.11 µm) to far infrared (40 µm) [9].
15
4.5
Experimental
The ellipsometer used was of the polarizer sample rotating analyzer type with 75W
Xenon lamp as the source of light. The photon energy was set between 0.74 eV and
6.5 eV at room temperature with the angle of incidence set at 60°, 65° and 70°. The
samples were assumed to have some surface layers that are not completely
characterized. However, simple modeling reveals that surface roughness is the main
overlayer, which has been intermixed with some oxide. The total thickness of the
sample differs in the range of 6-12 nm but we assumed an average thickness of 8 nm
for all the samples since it was not possible to obtain the detailed information about
the properties of the overlayers. Using the Bruggeman effective medium
approximation by taking 50% bulk material and 50% air with the said thickness the
complex dielectric function was fitted using all angles of incidence. The dielectric
function which is a quantity derived from the SE data by means of three-phase
(substrate/roughness/ambient) model was found to be somehow equal to the true bulk
dielectric function of a given sample.
4.6
Fourier Transform Infrared Ellipsometry
Spectroscopic ellipsometry is a nondestructive optical characterization technique
mainly used in the field of semiconductor to characterize bare substrates and thin
films. It permits the gathering of information regarding the physical structure of the
sample, such as roughness and film thickness, as well as its optical response. Each
molecule exhibits a characteristic absorption fingerprint in the mid-infrared (IR)
range, which makes this technique chemically selective. In applying this technique for
monitoring semiconductor processing, there is a problem since the substrate surface is
not always a metal. Therefore, we must take reverse-side reflection into account. Even
on a metallic surface, if we measure thin films at the surface, we cannot still neglect
internal multiple reflection and absorption in the bulk film [10].
Fourier transform spectroscopy has two experimental advantages over conventional
techniques: the simultaneous measurement of all observed spectral elements, so-called
multiplex advantage and the large throughput of the interferometer, so-called Jaquinot
advantage. There are other additional advantages such as large resolving power, high
wave number accuracy and vastly reduced stray light problem.
16
The primary results of a measurement are the Fourier transform of the spectrum
(interferogram), so that a Fourier transformation is necessary to obtain the spectrum
under investigation. This calculation is no problem with the modern computer
technique [7, 10].
4.7
Electronic Energy Band Structure
Consider Figs. 7 and 8, which represent the energy band structure of PbTe and SnTe
respectively as calculated by Tung and Cohen [8] with an empirical pseudo-potential
method. The figures indicate the location of several interband transitions. A notation
for double group representations is used to label the electronic states.
The valence band maximum and the conduction band minimum are seen to occur at
the L point indicating that the PbTe has a direct energy gap Ε 0 = Ε 6− − Ε 6+ (≈ 0.3 eV)
We can also see the existence of subsidiary valence-band maximum and the ∑axis.
The most importance fundamental reflectivity features in the energy range are the first
shoulder E1, the highest peak of reflectivity spectrum E2 and the high-energy shoulder
on the main peak E3 in the notation of Cardona and Greenaway [1]. It is also possible
to see from our SE measurement peaks as E4, E5 and E6 occuring at energy higher than
6.0 eV. The only impossible peak is the lowest edge E0 (0.3 eV) which is below the
range of our measurement.
(a)
(b)
Fig.8. Electronic and energy-band structure of (a) PbTe and (b) SnTe (Tung and Cohen)
17
The minimum gap for SnTe is not found at L point but rather it is found to lie in the
hexagonal face of the Brillion Zone, which is somewhat slightly away from L. The
valence band maximum and the conduction band minimum have splitting
L+6 − L−6 = 0.33 eV [11].
5
Results and Discussions
The various measured ellipsometry data are presented below. We shall consider the
first six samples that represent Pb-rich compounds. It was observed that, for Pb-rich
compounds (0.0 ≤ x ≤0.5), there is a shift towards lower photon energies and the
spectra exhibit a “knee” at about 1.0-1.4 eV range in the imaginary part of the
dielectric function Fig. 9b. Also the real part of the dielectric function in Fig. 9a
indicates larger values in the 0.74-1.5 eV range. The straight line parallel to the
photon energy axis in Fig. 9a, represents zero band gap energy which occurs near L
point for x = 0.3 i.e. Eg(Pb1-xSnxTe) ≈ 0 eV. The largest value obtained for PbTe (x=0)
from Fig. 9b is ε2 (E2) ≈ 86 which is much larger than the value reported by Cardona
and Greenaway ( ≈ 20) by means of reflectivity [1].
100
60
Imag(Dielectric Constant), ε 2
Real(Dielectric Constant), ε 1
80
80
pbtex=00
X = 0.0
pbtex=01
X = 0.1
X = 0.2
pbtex=02
X = 0.3
pbtex=03
X = 0.4
pbtex=04
X = 0.5
pbtex=05
40
20
60
40
20
0
-20
-40
0.0
E3
E1
X = 0.0
pbtex=00
X = 0.1
pbtex=01
X = 0.2
pbtex=02
X = 0.3
pbtex=03
X = 0.4
pbtex=04
X = 0.5
pbtex=05
1.0
(a)
2.0
3.0
4.0
Photon Energy (eV)
5.0
E2
0
-20
0.0
1.0
(b)
2.0
3.0
4.0
Photon Energy (eV)
5.0
18
Optical Constants
8
pbtex=00
X = 0.0
pbtex=01
X = 0.1
pbtex=02
X = 0.2
X = 0.3
pbtex=03
X = 0.4
pbtex=04
X = 0.5
pbtex=05
Index of refraction ' n'
8
6
4
2
0
0.0
1.0
(c)
Extinction Coefficient ' k'
10
E1
6
pbtex=00
pbtex=01
pbtex=02
pbtex=03
pbtex=04
pbtex=05
4
2
E2
0
-2
0.0
2.0
3.0
4.0
5.0
Photon Energy (eV)
E3
1.0
(d)
2.0
3.0
4.0
5.0
Photon Energy (eV)
Fig. 9. Observed spectral dependence of (a) real part of dielectric constant ε1, (b) imaginary
part of dielectric constant ε2, (c) index of refraction n, (d) extinction coefficient k for Pb1xSnxTe
(0.0≤x≤0.5) by spectroscopic ellipsometry.
This is due to the fact that ellipsometry deals with intensity independent complex
quantities (ψ and ∆) while reflectance deals with intensities and is a power
measurement which is very sensitive to intensity fluctuations of the source,
temperature drift of electronic components, macroscopic roughness etc. Reflectance
measurement has to be taken very accurate since small difference in reflectivity can
lead to larger errors in ε [1].
Optical Constants
80
Imag(Dielectric Constant), ε 2
Real(Dielectric Constant), ε 1
60
pbtex=06pseu
x = 0.6
x = 0.7
pbtex=07pseu
x = 0.8
pseudox=08
x = 0.9
realmeasx_09
x = 1.0
pb0sn1te_pse
40
20
E1
60
E3
x=06
x=07
x=08
x=09
x=10
40
0
20
-20
-40
0.0
1.0
(a)
2.0
3.0
4.0
5.0
Photon Energy (eV)
0
0.0
E2
1.0
(b)
2.0
3.0
4.0
5.0
Photon Energy (eV)
19
7.0
6
E3
6.0
Extinction Coefficient ' k'
Index of refraction ' n'
8
x=0.6
x=0.7
x=0.8
x=0.9
x=1
5.0
4.0
4
x=0.6
x=0.7
x=0.8
x=0.9
x=1
3.0
2.0
2
E2
1.0
0
0.0
E3
E1
(c)
1.0
2.0
3.0
4.0
Photon Energy (eV)
5.0
0.0
0.0
1.0
(d)
2.0
3.0
4.0
5.0
Photon Energy (eV)
Fig. 10. Observed spectral dependence of (a) real part of dielectric constant ε1, (b) imaginary
part of dielectric constant ε2, (c) index of refraction n and (d) extinction coefficient k for Pb1xSnxTe
(0.6≤x≤1) by spectroscopic ellipsometry.
E3
Fig. 10 above represents the observed spectra dependence of Pb1-xSnxTe for 0.6≤x≤1
for Sn-rich compounds with arrows indicating the positions of each critical point (E1,
E2, E3). We can see from Fig. 10a that, the real part of the dielectric function indicates
that Sn-rich compounds have a larger value in the 0.74-1.5 eV region and this causes
the Sn-rich spectra to shift towards lower energy near E ~ 2 eV. In the above spectra,
we see the optical response in the range 0.8-2.0 eV that is a representation of optical
absorption in the LW region of the Bz. The peak energy that shifts towards lower
energies for Sn-rich compounds signifies that there is a smaller band gap Eg at the Wpoint [1, 12]. The straight line observed in the real part represents zero band gap at or
near W point for x = 0.70.
In general it is seen that the Sn-rich compounds have more flat curvature of the lowest
conduction band along LW-line and hence have stronger response near 2.0 eV. The
observed “knee” in Fig. 10b, which occurs between 1.0-1.4 eV is seen for both Snand Pb-rich compounds. The spectra reveal the E1 at 1.01 eV, E2 at 2.02 eV and E3 at
3.01 eV. The imaginary part is attributed to the lost of energy of the incident radiation
and it is related to the absorption coefficient α.
The real part of the dielectric constant is seen to give the dispersive behavior of the
compound. There is a broad shoulder that is followed by a decrease in the region of
20
energies that correspond to the absorption peak E2 and then followed by an increase to
higher energies. The most vital quantity in the real part of the dielectric function is the
zero energy limit ε1 (0), since it represents the static dielectric constant at that point.
In Fig. 9c, and Fig. 10c, we see the real part of the complex refractive index n (E),
which also represents the actual refractive index of the material. These spectra have
the dispersive form with the broad shoulder descending steeply at E2. Also from Fig.
9d and Fig. 10d, we have the imaginary part of the complex refractive index k (E),
which shows the absorptive behavior of the compound with peaks at energies E1, E2
and E3.
We can compare three main samples PbTe, Pb0.5Sn0.5Te and SnTe, which represents
(x =0.0, 0.5 and 1) respectively and this will enable us to study the trend of the spectra
for the entire eleven samples. The spectra of the dielectric-related optical constant of
Pb1−xSnxTe such as the complex refractive index, absorption coefficient and normalincidence of reflectivity for the selected three samples are presented in Fig. 11a→e. In
Fig. 11b, we observe that, the line shape of the ε2 (E) spectrum is characterized by a
steep low-energy site and a broader high-energy site. The imaginary part ε2 (ω) of the
spectrum increases until the peak E2, which is called the reflectivity peak is formed.
For PbTe, the sharper peak E2 is seen at 2.02 eV as shown in Fig. 11b. The weak peak
in n(E) at ∼3.01 eV in Fig. 11c is related to the E3 transitions. The strongest peak in k
at ~2.1 eV in Fig. 11d, is related to E2.
100
60
PbTe
Pb(0.5)Sn(0.5)
SnTe
40
Imag(Dielectric Constant), ε 2
Real(Dielectric Constant), ε 1
80
80
PbTe
Pb(0.5)Sn(0.5
SnTe
E1
60
20
E1
E3
40
0
20
-20
-40
1.0
1.5
(a)
2.0
2.5
3.0
Photon Energy (eV)
3.5
E2
0
1.0
1.5
(b)
2.0
2.5
3.0
Photon Energy (eV)
3.5
21
7.0
PbTe
Pb(0.5)Sn(0.5
SnTe
8
6
4
6.0
PbTe
Pb(0.5)Sn(0.5
SnTe
5.0
E3
4.0
2
0
1.0
Extinction Coefficient ' k'
Index of refraction ' n'
10
3.0
1.5
(c)
2.0
2.5
3.0
Photon Energy (eV)
2.0
1.0
3.5
E2
1.5
(d)
2.0
2.5
3.0
3.5
Photon Energy (eV)
Absorption Coefficient in 1/cm
1.8E+006
E3
1.5E+006
1.2E+006
E1
9.0E+005
6.0E+005
E2
3.0E+005
0.0E+000
1.0
1.5
(e)
PbTe
Pb(0.5)Sn(0.5)Te
SnTe
2.0
2.5
3.0
Photon Energy (eV)
3.5
Fig. 11. Observed spectral dependence of (a) real part of dielectric constant ε1, (b) imaginary
part of dielectric constant ε2, (c) index of refraction n, (d) extinction coefficient k and (e)
absorption coefficient α for Pb1-xSnxTe by spectroscopic ellipsometry.
We can see from Fig. 11e, which represents the absorption coefficient spectrum of the
compound. This shows an increase value corresponding to E1 and this explains the
fact that at the transition energies, the absorption of the material is increased. The first
energy E0 that corresponds to the step energy of the semiconductor, which is about the
calculated band gap value whose assigned transition, is L (5→6) could not be seen
from the selected spectra as explained in chapter 4, subsession (4.7). The next peak
E1 is located near 1.1 eV from the absorption coefficient spectra. This E1 is the first
shoulder that appears in the spectrum, which is originated from transition ∑ (5→6) (∑
is the direction between Γ and K in the Bz). The absorption coefficient spectra also
show that, there is an energy shift towards lower energies. The shift between PbTe
22
and SnTe is about 0.32 eV for α (E) = 9.0x105/cm i.e. [E (SnTe)- E (PbTe) = 2.001.69 = 0.31 eV]. For PbTe, there is a sharp peak in the range of 1.82-1.90 eV as well
as a sharp growth of the absorption coefficient between E1 and E2 having a maximum
and sharp peak at the later energy. The E2 peak comes from the inter-band transitions
in the direction ∑ (5→7) and ∆ (5→6). From the E2 peak, there is a decrease in the
energy of ε2 (ω) leading to the appearance of peak E3 at energy 2.81-3.01 eV which is
belongs to the inter-band transition ∆ (4→6) and ∑ (4→7) between p-anion valence
states and p-Pb and p-anion conduction states. The absorption coefficient decreases
smoothly between E2 and E3 [1, 12].
It must also be noticed that ellipsometry results are affected by surface artifact at the
measurement level and if identified should be corrected immediately.
5.1
Comparing Results of SE and Infrared ellipsometry
We can compare in the ε (E) spectra obtained from SE to those determined from IR
spectroscopy. High refractive index materials, such as PbTe and SnTe are usually
weakly absorbing semiconductors in the infrared region and as such their optical
constants n, k, and d are strongly dependent on the deposition process.
The graph of the absorption coefficient α of nine of the samples is shown in appendix
(I) and these show-increased values, which correspond to the energy E0.
x = 0.0
ε1
ε2
150
200
100
100
50
0
-100
0.0
0.2
0
0.4
0.6
0.8
1.0
Photon Energy (eV)
x = 0.1
100
80
80
ε1
ε2
60
40
Imag(Dielectric Constant), ε 2
300
Imag(Dielectric Constant), ε 2
Real(Dielectric Constant), ε 1
(b)
200
Real(Dielectric Constant), ε 1
(a)
400
60
40
20
20
0
-20
0.0
0.2
0
0.4
0.6
0.8
1.0
Photon Energy (eV)
23
(c)
300
400
200
200
100
0
0
0.20 0.30 0.40 0.50
Photon Energy (eV)
(e)
Real(Dielectric Constant), ε 1
20
20
10
10
0.20
0
0.30
0.40
0.50
Photon Energy (eV)
(g)
Real(Dielectric Constant), ε 1
60
40
40
20
20
0
0.10
0.20
0
-100
0.00
50
0.10
0
0.20 0.30 0.40 0.50
Photon Energy (eV)
0
0.30 0.40 0.50 0.60
Photon Energy (eV)
x = 0.6
180
100
150
80
ε1
ε2
120
60
90
40
60
20
30
0
0.10
0.20
0
0.30
0.40
0.50
Photon Energy (eV)
x = 0.8
150
80
Imag(Dielectric Constant), ε 2
ε1
ε2
Imag(Dielectric Constant), ε 2
120
60
100
(h) (g)
80
80
100
x = 0.7
140
100
150
Imag(Dielectric Constant), ε 2
30
Imag(Dielectric Constant), ε 2
ε1
ε2
30
200
200
(f)
40
50
0
0.10
ε1
ε2
300
x = 0.5
60
40
250
Real(Dielectric Constant), ε 1
0.10
400
Real(Dielectric Constant), ε 1
-200
0.00
Real(Dielectric Constant), ε 1
ε1
ε2
600
400
300
Imag(Dielectric Constant), ε 2
800
x = 0.4
500
500
Imag(Dielectric Constant), ε 2
Real(Dielectric Constant), ε 1
(d)
x = 0.2
1000
120
ε1
ε2
90
60
60
40
30
0
20
-30
-60
0.0
0.2
0.4
0.6
0.8
Photon Energy (eV)
0
1.0
24
(i)(h)
x = 1.0
150
200
ε1
ε2
150
Imag(Dielectric Constant), ε 2
Real(Dielectric Constant), ε 1
250
120
90
100
60
50
30
0
-50
0.0
0.2
0
0.4
0.6
0.8
1.0
Photon Energy (eV)
Fig. 12a→i. Observed spectral dependence of Real part of dielectric constant ε1 and
imaginary part of dielectric constant ε2 for Pb1-xSnxTe by infrared spectroscopy.
The IR could not reveal this since the range of photon energy was below that of SE.
However we can see E0 at ∼0.3 eV for PbTe. A very important feature of the spectrum
is E2 intensity since it gives a measure of the cleanliness of the surface especially
when using Spectroscopic ellipsometry [1]. The measurements from the IR, which are
limited within the energy range of 0.1-1.0 eV, are so small to reveal the distinct
features as seen with the SE except E0 critical point. However the analysis of the
observed spectra shows that the trend seems to be the same for both SE and IR
spectroscopy.
5.2
Conclusion
The real and imaginary parts of the dielectric function of Pb1−xSnxTe have been
measured the by spectroscopic ellipsometry in the photon energy range of 0.74-6.5 eV
at room temperature. The observed spectra divulge distinct structures at energies of
E1, E2, and E3 critical points at 1.01 eV, 2.02 eV and 3.01 eV respectively. We
observed a strong optical response in the range of 0.8-2.0 eV and a strong peak in the
0.74-1.4 eV regions for both PbTe and SnTe, which shift towards lower energies near
photon energy of 2.0 eV. The estimated band gap was found to be 0.31 eV
25
The determination of the optical constant of such a material with weakly absorbing
property is of great importance in improving the performance of thin film products.
There have also been many theoretical studies of the electronic properties of the
material with different techniques such as the empirical pseudopotential, tight binding
method, the ab initio pseudopotential, the orthogonalized plane-wave, the Green
function method, the augmented- plane- wave and other techniques in which the
results obtained are closely related.
26
References
[1]
Norihiro Suzuki and Sadao Adachi, Optical Properties of PbTe, AA
(Department of Electronic Engineering, Faculty of Engineering, Gunma
University, Kiryu-shi, Gunma 376), Jpn. J. Appl. Phys. Vol. 33 (1994) pp.
193-198, Part 1, No. 1A, January 1994.
[2]
H. Arwin, “Thin film optics”, Linköping University (2007).
[3]
http://en.wikipedia.org/wiki/Fresnel_equations
[4]
T. E. Tiwald, D. W. Thompson, J. A. Woolam, S. V. Pepper, Determination
of the mid-ir optical constants of water and lubricants using ir ellipsometry
combined with an atr cell, Thin Solid Films 313-314 (1998) 718-721.
[5]
J. H. W. G. den Boer. Spectroscopic infrared ellipsometry, components,
calibration, and application. PhD thesis, Eindhoven University of Technology,
The Netherlands, 1995 p. 10-12.
[6]
http://www.jawoollam.com
[7]
J. C. Cigal, A novel spectroscopic ellipsometer in the infrared,
Technische Universiteit Eindhoven, 2002
[8]
http://www.redoptronics.com/BaF2-crystal.html
[9]
http://www.almazoptics.com/BaF2.htm
[10]
http://ao.osa.org/abstract.cfm?id=70648
[11]
Y. W. Tung and M. L. Cohen: Phys. Rev. 180 (1969), p. 823.
[12]
N. Suzuki, K. Sawai and S. Adachi, Jpn. J. Appl. Phys. 77 (1995), p. 1249.
27
Appendix. 1
Fig. 1(a→i) spectra of measured ∆ and ψ of Pb1−xSnxTe from spectroscopy
ellipsometry
x = 0.0
x = 0.0
180
40
150
35
120
90
30
25
60
30
0.0
20
1.0
2.0 3.0 4.0 5.0 6.0
Photon Energy (eV)
15
0.0
x = 0.1
2.0 3.0 4.0 5.0
Photon Energy (eV)
6.0
40
150
35
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
90
Ψ in degrees
∆ in degrees
1.0
x = 0.1
180
120
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
Ψ in degrees
∆ in degrees
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
30
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
25
60
20
30
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Photon Energy (eV)
15
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Photon Energy (eV)
28
x = 0.2
x = 0.2
40
150
35
∆ in degrees
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
120
90
30
25
60
30
0.0
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
Ψ in degrees
180
20
1.0
2.0 3.0 4.0 5.0 6.0
Photon Energy (eV)
15
0.0
1.0
x = 0.3
150
35
∆ in degrees
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
90
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
Ψ in degrees
40
30
25
60
30
0.0
20
1.0
15
0.0
2.0 3.0 4.0 5.0 6.0
Photon Energy (eV)
1.0
x = 0.4
34
140
Ψ in degrees
Model Fit
Exp E 65°
6.0
30
Model Fit
Exp E 65°
∆ in degrees
160
32
2.0 3.0 4.0 5.0
Photon Energy (eV)
x = 0.4
36
120
100
28
26
1.0
6.0
x = 0.3
180
120
2.0 3.0 4.0 5.0
Photon Energy (eV)
80
2.0
3.0 4.0 5.0 6.0 7.0
Photon Energy (eV)
60
1.0
2.0
3.0 4.0 5.0 6.0 7.0
Photon Energy (eV)
29
x = 0.5
x = 0.5
40
150
35
∆ in degrees
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
120
90
30
25
60
30
0.0
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
Ψ in degrees
180
20
1.0
15
0.0
2.0 3.0 4.0 5.0 6.0
Photon Energy (eV)
x = 0.6
1.0
2.0 3.0 4.0 5.0 6.0
Photon Energy (eV)
x = 0.6
180
39
36
∆ in degrees
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
120
33
Ψ in degrees
150
30
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
27
90
24
60
21
30
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Photon Energy (eV)
18
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Photon Energy (eV)
x = 0.7
x = 0.7
39
180
36
150
120
90
Ψ in degrees
∆ in degrees
33
30
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
60
30
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Photon Energy (eV)
27
24
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
21
18
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Photon Energy (eV)
30
x = 0.8
x = 0.8
180
39
36
150
Ψ in degrees
∆ in degrees
33
120
30
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
90
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
27
24
60
21
30
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Photon Energy (eV)
18
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Photon Energy (eV)
x = 0.9
x = 0.9
180
39
150
36
120
33
Ψ in degrees
∆ in degrees
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
30
90
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
27
60
24
30
21
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Photon Energy (eV)
Photon Energy (eV)
x = 1.0
x = 1.0
180
39
150
36
120
90
30
27
60
30
0.0
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
33
Ψ in degrees
∆ in degrees
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
24
1.0
2.0 3.0 4.0 5.0 6.0
Photon Energy (eV)
21
0.0
1.0
2.0
3.0
4.0
Photon Energy (eV)
5.0
31
Appendix 2
Fig. II (a→i) spectra measured ∆ and ψ of Pb1xSnxTe from IR spectroscopy
x = 0.0
x = 0.0
300
100
80
∆ in degrees
Exp E 55°
Exp E 65°
Exp E 75°
100
Exp E 55°
Exp E 65°
Exp E 75°
Ψ in degrees
200
60
40
0
-100
0.0
20
0.2
0
0.0
0.4
0.6
0.8
1.0
Photon Energy (eV)
0.2
x = 0.1
0.4
0.6
0.8
Photon Energy (eV)
1.0
x = 0.1
300
100
80
∆ in degrees
Exp E 55°
Exp E 75°
Exp E 55°
Exp E 75°
Ψ in degrees
200
60
100
40
0
-100
0.0
20
0.2
0.4
0.6
0.8
1.0
Photon Energy (eV)
0
0.0
0.2
x= 0.2
x= 0.2
300
50
∆ in degrees
Exp E 55°
Exp E 57°
Exp E 59°
Exp E 61°
Exp E 63°
Exp E 65°
Exp E 67°
Exp E 69°
Exp E 71°
Exp E 73°
Exp E 75°
100
0
0.10
0.20 0.30 0.40 0.50
Photon Energy (eV)
40
Exp E 55°
Exp E 57°
Exp E 59°
Exp E 61°
Exp E 63°
Exp E 65°
Exp E 67°
Exp E 69°
Exp E 71°
Exp E 73°
Exp E 75°
Ψ in degrees
200
-100
0.00
0.4
0.6
0.8
1.0
Photon Energy (eV)
30
20
10
0
0.00
0.10
0.20 0.30 0.40 0.50
Photon Energy (eV)
32
x = 0.4
x = 0.4
300
50
∆ in degrees
Exp E 59°
Exp E 61°
Exp E 63°
Exp E 65°
Exp E 67°
Exp E 69°
Exp E 71°
Exp E 73°
Exp E 75°
100
0
-100
0.00
40
Exp E 59°
Exp E 61°
Exp E 63°
Exp E 65°
Exp E 67°
Exp E 69°
Exp E 71°
Exp E 73°
Exp E 75°
Ψ in degrees
200
30
20
10
0.10
0
0.00
0.20 0.30 0.40 0.50
Photon Energy (eV)
0.10
x = 0.5
x = 0.5
300
50
∆ in degrees
Exp E 55°
Exp E 57°
Exp E 59°
Exp E 61°
Exp E 63°
Exp E 65°
Exp E 67°
Exp E 69°
Exp E 71°
Exp E 73°
Exp E 75°
100
0
0.20
0.30
0.40
0.50
Photon Energy (eV)
40
Exp E 55°
Exp E 57°
Exp E 59°
Exp E 61°
Exp E 63°
Exp E 65°
Exp E 67°
Exp E 69°
Exp E 71°
Exp E 73°
Exp E 75°
Ψ in degrees
200
-100
0.10
30
20
10
0
0.10
0.20
x = 0.6
0.30
0.40
0.50
Photon Energy (eV)
x = 0.6
300
50
40
∆ in degrees
Exp E 55°
Exp E 57°
Exp E 59°
Exp E 61°
Exp E 63°
Exp E 65°
Exp E 67°
Exp E 69°
Exp E 71°
Exp E 73°
Exp E 75°
100
0
0.20
0.30
0.40
0.50
Photon Energy (eV)
Exp E 55°
Exp E 57°
Exp E 59°
Exp E 61°
Exp E 63°
Exp E 65°
Exp E 67°
Exp E 69°
Exp E 71°
Exp E 73°
Exp E 75°
Ψ in degrees
200
-100
0.10
0.20 0.30 0.40 0.50
Photon Energy (eV)
30
20
10
0
0.10
0.20
0.30
0.40
Photon Energy (eV)
0.50
33
x = 0.7
x = 0.7
250
40
200
50
0
0.20
0.30 0.40 0.50
Photon Energy (eV)
Exp E 57°
Exp E 59°
Exp E 61°
Exp E 63°
Exp E 65°
Exp E 67°
Exp E 69°
Exp E 71°
Exp E 73°
Exp E 75°
Ψ in degrees
∆ in degrees
100
-50
0.10
30
Exp E 57°
Exp E 59°
Exp E 61°
Exp E 63°
Exp E 65°
Exp E 67°
Exp E 69°
Exp E 71°
Exp E 73°
Exp E 75°
150
20
10
0
0.10
0.60
0.20
0.60
x = 0.8
x = 0.8
50
250
40
∆ in degrees
Exp E 55°
Exp E 75°
Exp E 55°
Exp E 75°
Ψ in degrees
200
30
150
20
100
10
50
0
0.0
0.30 0.40 0.50
Photon Energy (eV)
0.2
0
0.0
0.4
0.6
0.8
1.0
Photon Energy (eV)
0.2
0.4
0.6
0.8
1.0
Photon Energy (eV)
x = 1.0
x = 1.0
100
300
80
∆ in degrees
Exp E 50°
Exp E 60°
Exp E 70°
100
Exp E 50°
Exp E 60°
Exp E 70°
Ψ in degrees
200
60
40
0
-100
0.0
20
0.2
0.4
0.6
0.8
1.0
Photon Energy (eV)
0
0.0
0.2
0.4
0.6
0.8
Photon Energy (eV)
1.0
34