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In this chapter we make an attempt to give a comprehensive overview on the optical modeling of layer structures that accommodate or are entirely composed of semiconductor nanocrystals. This research field is huge both in terms of the theories of effective dielectric functions and applications. The dielectric function of single-crystalline semiconductors can be determined on high quality reference materials. The accuracy of the reference data depends mostly on the numerical or experimental elimination of the surface effects like oxides, nanoroughness, contamination, etc.
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Chapter 17
Ellipsometry of Semiconductor Nanocrystals
Peter Petrik and Miklos Fried
In this chapter we make an attempt to give a comprehensive overview on the optical
modeling of layer structures that accommodate or are entirely composed of semi-
conductor nanocrystals. This research field is huge both in terms of the theories
of effective dielectric functions and applications. The dielectric function of single-
crystalline semiconductors can be determined on high quality reference materials.
The accuracy of the reference data depends mostly on the numerical or experimental
elimination of the surface effects like oxides, nanoroughness, contamination, etc.
Most techniques used for the preparation of semiconductor films (e.g. chemical vapor
deposition, evaporation, sputtering, pulsed laser deposition, etc.) result in polycrys-
talline materials with a structure that strongly depends on the parameters of the
sample preparation. Therefore, dielectric function references can only be used in a
parameterized form. The dielectric function of poly- and nanocrystalline1thin films
can be derived from the single-crystalline dielectric functions using parameteriza-
tion of the critical point features or using the effective medium theory by “mixing”
these single-crystalline components with voids, polycrystalline or amorphous com-
ponents. Finally, when the effective dielectric functions are calculated for a given
nanocrystalline structure in a given layer, the vertical change of the composition can
be measured by defining multi-layer models or models with analytical depth profiles.
P. Petrik (B
)·M. Fried
Research Institute for Technical Physics and Materials Science (MTA-MFA),
Hungarian Academy of Sciences, Konkoly Thege Rd. 29-33, 1121 Budapest,
H-1525 Budapest, Hungary
e-mail: petrik.peter@ttk.mta.hu
M. Fried
e-mail: fried.miklos@ttk.mta.hu
1Very recently, a definition of nanomaterial has been given by the European Commision (http://
ec.europa.eu/environment/chemicals/nanotech/index.htm): “A natural, incidental or manufactured
material containing particles, in an unbound state or as an aggregate or as an agglomerate and
where, for 50% or more of the particles in the number size distribution, one or more external
dimensions is in the size range 1–100nm.”
M. Losurdo and K. Hingerl (eds.), Ellipsometry at the Nanoscale, 583
DOI: 10.1007/978-3-642-33956-1_17, © Springer-Verlag Berlin Heidelberg 2013
584 P. Petrik and M. Fried
Fig. 17.1 Real (ε1) and imaginary (ε2) parts of the dielectric functions of single-crystalline silicon
from Refs. [60](line 1,)[14](line 2,)and[63](line 3). Differences for lines 1–2 and 1–3 are also
plotted
17.1 Dielectric Function of Semiconductors
Dielectric functions have been measured and tabulated for a wide range of semi-
conductors (Si, Ge, SiC, SiGe, AlAs, AlGaAs, AlN, AlSb, BN, CdTe, CdS, CIGS,
GaAs, GaN, GaP, GaSb, HgTe, InGaAs, InAs, InN, InP, InSb, PbS, PbSe, PbTe,
ZnS, ZnSe, ZnTe, just to mention some relevant materials, found in a commercial
database). Most of these data have been published [9,14,60,63], are available in
the databases shipped with commercial ellipsometers, or on community websites
[97]. Accurate reference dielectric functions of single-crystalline semiconductors
are crucial when aiming at sub-nanometer precision. Even the dielectric functions
of the purest and most perfect single-crystalline material, the microelectronic grade
silicon shows differences between different authors (Fig. 17.1). Using the various
references of Fig. 17.1 for a native oxide-covered single-crystalline silicon wafer, dif-
ferences of 0.2nm in the oxide thickness are obtained.2The major problem is how
surface imperfections (oxide layer, nanoroughness, contamination) are taken into
account.
2Using the photon energy range of 2–5eV covered by all the references.
17 Ellipsometry of Semiconductor Nanocrystals 585
Fig. 17.2 Real (ε1) and imaginary (ε2) parts of the dielectric functions of single-crystalline silicon
(c-Si, [60]), deposited amorphous silicon (a-Si, [14]) and ion implantation-amorphized silicon
(i-a-Si, [41])
Even amorphous dielectric functions have been tabulated as reference data [14,41],
although their dielectric functions strongly depend on the preparation [41,81,110].
A relevant example is the dielectric function of the ion implantation-amorphized and
relaxed silicon [41]. The dielectric functions of the two kinds of amorphous silicon
differ significantly (Fig. 17.2).
Most of the reference dielectric functions of semiconductors are available in a para-
meterized form as well. The use of analytical functions has several advantages. First
of all, non-idealities, like in the case of polycrystalline materials can be derived
and taken into account using these parameterizations. Furthermore, measurements
at wavelengths other than those given in the data file of the reference dielectric
function can be interpolated and extrapolated with higher accuracy.
There are a range of approaches from using simple Lorentz oscillators through
quantum mechanical considerations (model dielectric function [1], generalized
oscillator model [22]) to empirical formulas like the generalized critical point
model [66].
586 P. Petrik and M. Fried
17.2 Analytical Models
Dielectric responses of all materials follow some relatively simple rules and can be
understood in microscopic terms. Here we consider these rules for both homogeneous
and heterogeneous materials. Examples of generic dielectric functions for single-line
absorption processes, metals and semiconductors are given in Fig. 17.3 and discussed
individually below [53].
17.2.1 Drude
Simple metals are materials whose dominant characteristic is electronic charge that
is able to move more or less freely through the material. In the Drude model, the
Fig. 17.3 Examples of generic dielectric functions for single-line absorption processes (Lorentz,
Tauc-Lorentz and Cody-Lorentz oscillator), Drude metal, semiconductor (GaAs), and insulator
(glass) [Drawn by the WVASE software of the Woollam Co., Inc.]
17 Ellipsometry of Semiconductor Nanocrystals 587
equation of motion of any single carrier in an external field is
F=−eEexp(iωt)=ma =md2x/dt2=−ω2mΔxexp(iωt), (17.1)
where its position is assumed to be given by x(t) =x0+xexp(iωt). Solving
this equation for Δxand substituting the result in Maxwell’s equations leads to the
Drude expression,
ε=14πne2/meω2=1ωp2/ω2,(17.2)
where ωp=4πne2/meis the plasma frequency, nand meare the electron density and
electron mass, respectively. The Drude model works quite well for most metals and
for free-carriers in semiconductors. Note that we derived this dielectric function from
purely classical-mechanical perspective of point charges accelerating in accordance
with Newton’s laws, i.e., the motion in an electric field.
More realistic calculations include electron loss due to scattering, in which case,
ε=1ωp2/ω(ω+i/τ), (17.3)
where τis the lifetime. The lifetime is an important parameter in poly- or nanocrys-
talline metal films of finite grain sizes. It is determined approximately as
1=10+2vf/d,(17.4)
where vfis the Fermi-velocity and dis the diameter of grain. The relevant length
scale is the electron mean free path vfτ0, which in noble metals is on the order of
a few dozen nanometers. Thus for Cu, Au or Ag the dielectric function is affected
when grain sizes drop below these values.
Another important application of the Drude model is to describe the optical contribu-
tion of free-carriers in heavily doped semiconductors. To the lowest order correction
is made simply by adding the Drude expression to the ordinary dielectric function
measured for undoped material. Note the material properties encoded: electron den-
sity and electron mass in the plasma frequency and grain size in the lifetime.
The calculations can be extended by adding a restoring term K(xx0)to the
force equations, where Kis the force constant. The equation then approximately
describes the bound charge that dominates the optical response of dielectrics, oxides
and amorphous semiconductors. In this case εbecomes
ε=1ω2
p/(ω2ω2
0+iω/τ), (17.5)
where ω2
0=K/mis the resonance frequency squared. Resonance leads to strong
absorption in the refractive index at longer wavelengths. The above expression also
applies to excitons, impurity transitions and lattice vibrations.
588 P. Petrik and M. Fried
An important aspect of the above derivations is generality; because the same funda-
mental considerations (mass, restoring forces, loss) apply to all charges, all dielectric
response functions have the same form. Consequently, any dielectric response for any
material can be represented as a superposition of terms of the above form (spectral
representation).
The same principles also apply to semiconductors, although the spectra are now more
complex. Crystalline semiconductors are characterized by relatively sharp struc-
tures, whereas amorphous semiconductors generally have only a single broad peak.
The transition between amorphous and crystalline behavior in semiconductors is ill-
defined, depending in part on the technique used to observe it, but usually considered
to extend approximately from 2 to 50nm. On the other hand, optical excitations in
insulators are usually so well localized that size effects cannot be seen.
A realistic model becomes even more important when one wishes to simulate spectra
from alloys, such as AlxGa1xAs or SixGe1x, which contain critical points in the
optical spectrum that vary continuously with composition x. The best fits can be
obtained when the critical points are modeled with the use of one or more Lorentz
oscillators. For each oscillator, the peak photon-energy, width and amplitude are fit
as a function of x, allowing the composite dielectric function to be calculated as a
function of x. This approach works well near the critical point but breaks down at
small photon energies, where the absorption coefficient becomes small [130].
17.2.2 Lorentz
The Lorentz model is a classical model where a negatively charged electron is bound
to a positively charged atomic nucleus with a spring. Thus single-electron “two-level
atoms” (Ne =Na) are assumed for simplicity.
Each electron is assumed to be bound to its atom with a resonant frequency of ω0in the
near-ultraviolet range; as a result, there is no free-electron or Drude component to the
dielectric function. If light is shone, the AC electric field of the light E=E0exp(iωt)
will induce dielectric polarization in the xdirection. The Lorentz model assumes a
physical model in which the electron oscillates in viscous fluid. In this case, the
position of the atomic nucleus is fixed, since the mass of the atomic nucleus is far
larger than that of the electron. If we use Newtons second law, the physical model is
expressed as:
med2x/dt2=−meΓdx/dt meω2
0xeE0exp(iωt), (17.6)
where meand eshow the mass and charge of the electron, respectively. In Eq. (17.6),
the first term on the right represents the viscous force of the viscous fluid. In general,
the viscous force is proportional to the speed of an object when the speed is slow.
The in Eq. (17.6) represents a proportional constant of the viscous force, known
as the damping coefficient. (It is high in amorphous semiconductors.) The second
17 Ellipsometry of Semiconductor Nanocrystals 589
term on the right expresses that the electron moved by the electric field of light is
restored according to Hooks law F=−KFx, and ω0shows the resonant frequency
of the spring ω2
0=KF/me. The last term on the right shows the electrostatic
force F=qE.Eq.(17.6) represents the forced oscillation of the electron by the
external AC electric field. By this forced oscillation, the electron oscillates at the
same frequency as the AC electric field [i.e., exp (iωt)]. Thus, if we assume that
the solution of Eq. (17.6) is described by the form x(t)=a·exp(iωt), the first and
second derivatives of x(t) are given by dx/dt =iaωexp(iωt)and d2x/dt2=−aω2
exp(iωt), respectively. By substituting these into Eq. (17.6) and rearranging the
terms, we get
a=−(eE0/me)(1/(ω2
0ω2+iω)). (17.7)
On the other hand, if the number of electrons per unit volume is given by Ne, the
dielectric polarization is expressed as P=−eNex(t).Fromx(t)=aexp(iωt),
we obtain P=−eNeaexp(iωt). By substituting P=−eNeaexp(iωt)and E=
E0exp(iωt)into ε=1+P0E=1+χ, where χis the dielectric susceptibility
χP0E, we obtain the dielectric constant as follows:
ε=1+(e2Ne/ε0me)(1/(ω2
0ω2+iω)) (17.8)
This equation represents the Lorentz model. If we multiply by (ω2
0ω2–iω)both
the numerator and the denominator of Eq. (17.8), we get
ε1=1+(e2Ne/ε0me)(ω2
0ω2)/((ω2
0ω2)2+2ω2)), (17.9)
ε2=(e2Ne/ε0me)( ω/((ω2
0ω2)2+2ω2)). (17.10)
In actual data analysis, we commonly express the Lorentz model using the photon
energy Eph:
ε=1+jAj/(Eph0j2Eph2+ijEph)(17.11)
In Eq. (17.11), the dielectric function is described as the sum of different oscillators
and the subscript jdenotes the jth oscillator. In general, A in Eq. (17.11) is called the
oscillator strength.
17.2.3 Tauc-Lorentz and Cody-Lorentz
The Tauc–Lorentz model has been employed to model the dielectric function of
amorphous materials ([65]; for a review see [26]) and of transparent conductive
oxides [52,126]. The shape of ε2peaks calculated from the Lorentz model is com-
pletely symmetric. However the ε2peaks of amorphous materials generally show
asymmetric shapes. In the Tauc-Lorentz model [65], therefore, ε2is modeled from
590 P. Petrik and M. Fried
the product of a unique bandgap of amorphous materials (Tauc gap [134]) and the
Lorentz model.
The Tauc gap Egof amorphous materials is given by the following equation [134]:
ε2=Atauc(Eph Eg)2/E2
ph (17.12)
The ε2of the Tauc-Lorentz model is expressed by multiplying ε2of Eq. (17.11)by
Eq. (17.12)[65]:
ε2=AE0(Eph Eg)2/((Eph2E2
0)2+2Eph2)Eph)(Eph >Eg)(17.13a)
ε2=0(Eph Eg)(17.13b)
The ε1of the Tauc-Lorentz model can be derived by using the Kramers–Kronig rela-
tions [53,65] form Eq. (17.13). Although the equation for ε1is rather complicated,
the dielectric function of Tauc-Lorentz model is expressed from a total of five para-
meters ε1(),A,Γ,En0,Eg.Fig.17.4 shows (a) the dielectric function and (b) the
n-k spectra of an amorphous silicon (a-Si) calculated from the Tauc-Lorentz model
[65]. The values of the analytical parameters in this calculation are A =122 eV,
=2.54eV, En0 =3.45eV, Eg=1.2eV and ε1() = 1.15. It can be seen from
Fig. 17.4a that ε2=0atE
nEgand the ε2peak position is given by En0. The A and
of the Tauc-Lorentz model represent the amplitude and half width of the ε2peak,
respectively, similar to the Lorentz model.
So far, the dielectric function of amorphous materials has also been described using
other models including the Cody–Lorentz model [36], Forouhi–Bloomer model [38],
MDF theory [2], and band model [74].
The Tauc-Lorentz and Cody-Lorentz [36] dispersion types are primarily designed for
modeling amorphous materials. The main difference between the two types is how
they model absorption at photon energies slightly larger than the energy gap. In this
region the Tauc-Lorentz model follows the Tauc law formula while the Cody-Lorentz
follows the Cody formula:
Tauc Absorption Formula: ε2(E) [(E – Eg)2/E2]
Cody Absorption Formula: ε2(E) (E – Eg)2
The Cody-Lorentz type also includes an Urbach absorption term. The fit parameters
for the Tauc-Lorentz are the Amplitude (Amp), Broadening (Br), Center Energy
(Eno), and Band Gap (Eg). The Cody-Lorentz oscillator adds Ep (transition energy
where absorption changes from Lorentzian to Cody), and Et (transition energy where
absorption changes from Cody behavior to Urbach behavior). It has been shown that
the Cody model provides superior fitting to experimental spectra, compared with the
Tauc model [24,25].
17 Ellipsometry of Semiconductor Nanocrystals 591
(a)
(b)
Fig. 17.4 (a) Dielectric function and [36] n k spectra of amorphous silicon (a-Si) calculated from
the Tauc–Lorentz model [53]. Reprinted with permission from Wiley, Fujiwara, Spectroscopic
Ellipsometry: Principles and Applications. Copyright Wiley, New York, 2007
17.3 Dielectric Function of Poly- and Nanocrystalline
Semiconductors
17.3.1 Effective Medium Models
Materials composed of phases much smaller than the radiation wavelength but large
enough to retain their bulk properties can generally be modeled using the effective
medium approximation (EMA) [12,20,72,132,133]. Most poly- and nanocrys-
talline materials fulfill this requirement or at least can properly be modeled using
effective medium theories. The success of this approach is shown by the fact that
the first publication of the most frequently used self-consistent Bruggeman effective
medium approximation (B-EMA) (Ref. [20]) has been cited more than 3,000 times
(note that most articles using the B-EMA don’t cite the original paper any more, so
the influence of this approach is even much greater that the above number of citations
would suggest).
592 P. Petrik and M. Fried
Limitations of both increasing (finite wavelength [12,32,133]) and decreasing (finite
size [35,98,105]) component sizes have been extensively studied. Increasing com-
ponent size leads to scattering- [37], critical dimension- [68,61,62], patterned
wafer- [89] and photonic crystal-based [68,76,95,96] theories and applications,
whereas decreasing component size leads to finite-size effects [35,98], consequently,
to the use of fine- grained reference materials [104] or to the parameterization of the
dielectric function [1,60,94,105,127] (also combined with EMA [112]).
Using effective medium models microstructural information can be obtained by the
volume fraction of the components, e.g. the amorphous/(nano)crystalline ratio as a
function of growth conditions [23,57,58,83,84,91,92], during deposition [26,
56] or annealing [85,106]. Using both nanocrystalline (nc-Si) and single-crystalline
(c-Si) components the nanocrystalline nature (the size of the nanocrystals or the
amount of grain boundaries per unit volume) of polycrystalline silicon films can be
determined [5,105]. The same approach can be used for porous silicon [44,114].
The success of effective medium models is due to their robustness. In most cases,
the samples can be fitted with a relatively small number of parameters, avoiding
parameter correlations, allowing a quick evaluation. The main disadvantage of the
method is that the dielectric function of the components can rarely be described
by bulk references. Furthermore, in most cases isotropic behavior, i.e. not oriented
grain boundaries are assumed, which might not be right in some cases. Usually the
self-consistent Bruggeman method is used, but the type of effective medium model
(i.e. Maxwell-Garnett or B-EMA) may depend on the volume fraction ratios.
Being a robust method, EMA is well suited for in situ ellipsometry [18,26,69,85,
98,106]. This way, the formation of nanocrystals, the transition between amorphous
and crystalline structures can be followed, as well as the evolution of nucleation and
surface roughness layers.
17.3.2 Analytical Function-Based Models
Analytical models can be used when EMA models fail (e.g. the dielectric functions
of the components can not be obtained from bulk or single-phase thin film refer-
ences), when the dielectric function has to be determined as a smooth function of
the wavelength, or when a direct connection with the electronic band properties is
investigated determining the change of band gap, stress, or broadening of critical
points (a recent, sophisticated example on CdTe photovoltaics is given in Ref. [75]).
The analytical functions used for polycrystalline materials can be derived from the
formulas described above for single-crystalline semiconductors.
The Lorentz oscillator is widely used when the EMA models don’t work well enough
[33] or simply to determine a smooth dielectric function that can further be analyzed
using EMA [35]. In this references ion implantation has been used as a way to create
damage and a vanishing long range order in a controlled way. Ion implantation is
17 Ellipsometry of Semiconductor Nanocrystals 593
a versatile and effective method to prepare reference samples for the investigation
of the dielectric function of various disordered lattice structures [33,35,3943,45,
46,7880,94,104,107109,111,112,120,127].
The change of characteristic critical point features can be analyzed using the (second
or third) derivative method [13,116]. In this approach one assumes that the crystal-
related structure is directly connected to the lineshape of the critical point features.
The vanishing long range order (e.g. as a result of ion implantation-caused disorder)
causes the sharp critical point features to broaden, which can be characterized by
second or third derivative amplitudes of the pseudo-dielectric function. The change
of the derivative amplitudes as a function of fluence can be connected to the size of
the single ion tracks. Neglecting a possible surface layer like oxide or nanoroughness
doesn’t cause a significant error in the derivatives, because these layers are practically
non-dispersive in the spectral range around the critical point energies.
The generalized oscillator model uses standard analytical line shapes for the interband
critical point features [11,22,28,73,117]:
ε(E)=Aeiϕ(ECP Ei)μ,(genosc)
where A,ECP,, and φare the amplitude, the critical point energy, the broadening,
and the excitonic phase angle, respectively. Using this equation critical points of
different dimensionalities can be described by adjusting the value of μ. This method
allows the analysis of most important critical point features like the energy positions
or the broadenings as a function of temperature [73], disorder [117] or other process
parameters.
The dielectric function of numerous semiconductors (e.g. Si, Ge, CdS, and CdTe)
have been parameterized by Adachi using the Kramers-Kronig transformation. The
model dielectric function (MDF) calculated using this method has the advantage that
it provides a good fit using a set of dedicated oscillators with a few parameters for
each critical point [1]. This method has been successfully applied for disordered Si
[3,71,127], for III–V semiconductors [34] as well as for nanocrystals in porous
Si [118]. Similar parameterization can be applied for ferroelectric and dielectric
layers [47], but most importantly, it has been demonstrated that it can be applied
for nanocrystals in silicon nitride as well [17]. The fit of the MDF on a fine-grained
polycrystalline silicon reference is shown in Fig. 17.5.
When studying nanocrystalline semiconductors, the dielectric functions range from
the nearly single-crystalline to the nearly amorphous. In the latter case the parame-
trizations are simpler, because the critical point features disappear from the spectrum,
having only one broad peak. The most successful phenomenological model is the
Tauc-Lorentz (TL) model suggested by Jellison, which combines a Tauc gap with
a Lorentzian oscillator [65]. An important feature of the model is that a Kramers-
Kronig consistent analytical formula is given for ε1, which makes the calculation of
the model accurate and fast. This model has later been refined through a more accu-
rate description of the line shape at the gap energy using the Cody parameterization,
594 P. Petrik and M. Fried
Fig. 17.5 Lineshapes of the oscillators of the model dielectric function fitted on a nanocrystalline
polysilicon reference from Ref. [64]. Reprinted with permission from Petrik et al. [118]. Copyright
2009, American Institute of Physics
resulting in the Cody-Lorentz model [36]. The TL models have successfully been
applied for extremely small embedded crystals, carbon based thin films [21,77,113],
nanocrystalline diamond [82] as well as for amorphous SiC [81].
Finally, a general approach that provides the largest flexibility to describe the critical
point features was suggested by Johs et al. [66]. A detailed analysis of the method
termed ‘Generalized critical point model’ can be found in Ref. [28]. In this approach
the line shape of the dielectric function around the critical points are composed of
four Gaussian-broadened polynomials. The fit parameters are the so called control
points of the polynomials, their position, curvature, connection and the broadening
energy, and amplitude values of the critical point. Because of the large number of
fit parameters this model can only be used in a very careful and systematic way
17 Ellipsometry of Semiconductor Nanocrystals 595
to fit each critical point. For nanocrystalline materials coupling or fixing the most
parameters may be needed to avoid cross correlations [119].
17.3.3 Quantum Confinement
Several papers studied silicon nanocrystals embedded in insulator matrix by theo-
retical calculations [101,125] or experimentally [30,31,55,99]. Ögüt et al. [101]
calculated quasiparticle gaps, self-energy corrections, exciton Coulomb energies, and
optical gaps in Si quantum dots from first principles using a real-space pseudopoten-
tial method. The calculations were performed on hydrogen-passivated spherical Si
clusters with diameters up to 2.7nm (800 Si and H atoms). They showed that (i) the
self-energy correction in quantum dots is enhanced substantially compared to bulk,
and is not size independent as implicitly assumed in all semiempirical calculations,
and (ii) quantum confinement and reduced electronic screening result in appreciable
excitonic Coulomb energies. They fitted the calculated data to a power law of the
diameter as da, and found a=0.7. The calculated optical gaps were in very good
agreement with absorption data [55].
Proot et al. [125] calculated the electronic structure of spherical silicon crystallites
containing up to 2058 Si atoms. They predicted a variation of the optical band gap
with respect to the size of the crystallites in very good agreement with available
experimental results [55]. They also calculated the electron-hole recombination time
which is of the order of 104–106s for crystallites with diameters of 2.0–3.0nm.
Their results were applied to porous silicon for which they confirmed that a possible
origin of the luminescence is the quantum confinement. The calculated optical gap
varies from 5 to 1.6eV (size range from 0.8 to 4.3nm) following approximately a
d1.39 law where dis the crystallite diameter. (From a simple effective mass approx-
imation one would expect a d2law. The lower exponent 1.39 shows that the exact
nature of the bands has to be taken into account, in particular for the complex con-
duction band.)
Ding et al. [30,31] studied Si nanocrystals (nc-Si) with different sizes embedded in
SiO2 matrix having been synthesized with various recipes of Si ion implantation. The
influence of nanocrystal size on optical properties, including dielectric functions and
optical constants, of the nc-Si has been investigated with spectroscopic ellipsometry.
The optical properties of the nc-Si are found to be well described by the four-term
Forouhi-Bloomer model [38]. A strong dependence of the dielectric functions and
optical constants on the nc-Si size is observed. For the imaginary part of the dielectric
functions, the magnitude of the main peaks at the transition energies E1and E2
exhibits a large reduction and a significant redshift in E2depending on the nc-Si
size. A band gap expansion is observed when the nc-Si size is reduced. The band gap
expansion with the reduction of nc-Si size is in good agreement with the prediction
of first-principles calculations based on quantum confinement [101]. They fit their
results with the Eg(D)=Eg0+C/Dnformula, where Dis the nanocrystal size in
596 P. Petrik and M. Fried
nm, Eg(D) is the band gap in eV of the nanocrystal, Eg0=1.12 eV is the band gap
of bulk crystalline Si, C=3.9, and n=1.22.
17.4 Modeling of Layer Structure and Composition
17.4.1 Single layers
Even when investigating single layers, in most cases not only the bulk layer is included
in the model, but the surface roughness [10,70] and the nucleation layer [69]isalso
taken into account. The systematic improvement of the fit quality during the intro-
duction of interface layers into the optical model has been investigated by numerous
authors (see Refs. [10,44,121,136], just to mention a few examples). The for-
mation of the nucleation layer and the surface roughness has also been extensively
investigated using in situ spectroscopic ellipsometry [18,19,26,51,69].
The dielectric function of the bulk layer is modeled as described in the previous
chapter. When using single-layer models either the sample is close to perfect (e.g.
prepared by molecular beam epitaxy, atomic layer deposition or other methods that
can create nearly perfect layers), or the non-ideal interfaces are not taken into account.
In the latter case the error caused by not considering the interfaces will be transferred
to the determined layer parameters as errors (to both the layer thickness or the dielec-
tric function). The modeling of the interfaces leads us to the following chapters on
vertically inhomogeneous structures.
17.4.2 Ultrathin nc-Si RTSE Measurements
Nguyen et al. [98100] studied the evolution of thin film Si nanostructure during
growing and etching processes that yields thin layer nanocrystallites. They performed
real-time spectroscopic ellipsometry (RTSE) measurements during the growth of
nanocrystalline silicon (nc-Si:H) or a-Si:H nanoclusters by plasma enhanced chemical-
vapor deposition on chromium at 250C. They focused on the regime when the film
consists of isolated nanocrystallites and intervening void volume. In this regime, the
observed three-dimensional growth behavior allowed to associate the crystallite size
with the physical thickness of the film. The RTSE measurements were self-contained
in that they provided not only microstructural information, including film thickness
and volume fraction, but also the effective optical functions of the film. From this
combination of results, the optical functions the Si crystallites, themselves, could be
deduced by mathematically extracting the influence of the void-volume fraction on
the effective optical functions. A critical-point (CP) analysis of E1transitions visible
near 3.3 eV in the crystallite optical functions provided information on the electronic
properties as a continuous function of crystallite size. Over the physical thickness
17 Ellipsometry of Semiconductor Nanocrystals 597
(a)
(b)
(c)
Fig. 17.6 Optical gap determinations for ultrathin (a) a-Si:H and (b) nc-Si films at 250 C deduced
from RTSE measurements. The optical gaps are obtained by extrapolating observed linear plots to
ε2=0;coptical gaps plotted versus thickness for ultrathin a-Si:H and nc-Si films at 250C, from
aand b. Values for thick a-Si:H and bulk c-Si are provided at the right. Reprinted with permission
from Nguyen et al. [100]. Copyright 1995, The American Physical Society
range in this experiments (max. 25nm), the transition energy and phase deduced in
the CP analysis were constant (at the single-crystal values), while the optical gap
and broadening parameter decreased with increasing thickness. (See Fig. 17.6)This
behavior was consistent with a finite-size effect (quantum confinement).
The etching [99] was performed with thermally generated atomic H in order to avoid
plasma damage. RTSE was applied to characterize the evolution of the thicknesses of
near-surface nc-Si:H and underlying a-Si:H layers, as well as their optical properties.
Using the end-point detection capability of RTSE, they could terminate etching to
obtain an ultrathin (<1.5 nm), single-phase layer of Si nanocrystallites i. e., with no
detectable a-Si:H. These crystallites were densely packed on the substrate compared
to those of conventional, single-phase nc-Si:H prepared by PECVD [98].
17.4.3 Superlattices
In case of superlattices one should assume a periodic layer structure [129]. This can
simplify the optical model because the properties (mainly thicknesses) of the layers
598 P. Petrik and M. Fried
Fig. 17.7 Evolution of sublayer thicknesses in a layer stack of 10×([silicon rich oxide]/SiO2).
Silicon rich oxide (SRO) was modeled using an effective medium mixture of SiO2, a-Si, and voids.
The graph shows the increase of the SiO2sublayer as a function of the annealing temperature.
Reprinted with permission from Agocs et al. [4]. Copyright 2011 Elsevier
within one period only have to be defined once, providing only the number of periods
as a required additional parameter [4]. Equivalently, in case of a fewer number of
layers it can be considered as a coupling of the repeated parameters [128]. Even
if the preparation conditions suggest a periodic structure, the structures are usually
not ideal on the scale of sensitivity of ellipsometry. First, the surface layer usually
has to be fitted independently because its contact with air modifies its properties
(due to oxidation [4] or hydrocarbon contamination [131]). Second, the layer quality
usually deteriorates with each new prepared layer [90] (caused by e.g. the interface
roughness), which results in a vertical inhomogeneity over the whole stack. Of course,
the ideality of the superlattice can be estimated from the fit quality. As sublayers in
superlattice stacks are usually very thin deposited layers, bulk-like references can be
used in very rare cases, and the dielectric function of each stack has to be fitted as
well. Fitting the dielectric function even structural changes can be followed during
e.g. oxidation (Fig. 17.7 from Ref. [4]).
17.4.4 Inhomogeneity in Vertical and Lateral Dimensions
Most methods of thin film preparation result in vertically inhomogeneous layers, at
least on the scale of the precision of ellipsometry. An ideal thin film for ellipsometry
would have atomically sharp interfaces and a vertical refractive index uniformity of
better than about 104. These requirements can rarely be met. Usually, at least an
interface layer of nucleation [28] and a surface roughness layer have to be taken
into account [53]. For example, in case of polysilicon, most layer properties depend
strongly on the layer thickness (Fig. 17.8,[105]).
17 Ellipsometry of Semiconductor Nanocrystals 599
Fig. 17.8 Selected relevant fit
parameters of an optical model
as a function of thickness and
deposition temperature used
to measure low pressure
chemically vapor deposited
polycrystalline silicon. The
optical model consists of
a surface roughness layer,
a polysilicon bulk layer,
and a buried oxide layer
(both the surface roughness
layer and the polysilicon
bulk layer are a Bruggeman
effective medium composition
of single-crystalline Si, fine-
grained polycrystalline Si
and voids). Reprinted with
permission from Petrik et
al. [105]. Copyright 2000,
American Institute of Physics
600 P. Petrik and M. Fried
Fig. 17.9 Optical model for the ellipsometric measurement of the Gaussian damage profile created
by 100keV Ar ions implanted into single-crystal Si. σ1and σ2are the standard deviations of the
coupled half-Gaussian distribution functions. Rpand fare the peak position and the height parameter
of the damage profile, respectively. The fifth parameter of the optical model is the thickness of the
surface oxide layer (Reprinted with permission from Petrik et al. [115]. Copyright 2008, John Wiley
and Sons.)
Neither the amorphous layers created by ion implantation have sharp interfaces.
However, in this case the shape of the profile can be parameterized based on the theory
of ion implantation. This method reduces the number of fit parameters significantly,
because only four key parameters describing the Gaussian profile need to be fitted
[42,107,115] (Fig. 17.9).
Depth profiles of voids in Si created for gettering at a depth as large as 400nm can
sensitively be determined using spectroscopic ellipsometry [54,111]. Although the
penetration depth of light is significantly less than 400nm in a significant part of the
spectrum (at the critical point energies and even below, down to about 2eV), a high
sensitivity on the void profiles has been demonstrated because of the large optical
contrast between the dielectric function of Si and voids. In Ref. [54] a sophisti-
cated parameterization of the depth profiles using a combination of normalized error
functions is suggested and successfully demonstrated.
Because the penetration depth of light at the most sensitive parts of the spectrum (i.e.
at the photon energies of the critical points—in Si about 3.4 and 4.2eV) is only in
the range of 5–10nm, high sensitivity structural characterization using ellipsometry
taking into account the line shape of the dielectric function at the critical points can not
be preformed for deep implanted profiles. However, using wedge etching [16]buried
profiles can be brought to the sample surface allowing sensitive characterizations [48,
17 Ellipsometry of Semiconductor Nanocrystals 601
Fig. 17.10 Imaginary part of the pseudo-dielectric function of Si implanted with 100keV Xe
through a wedge mask created by anodic oxidation [59]. ddenotes the lateral position on the
sample. The oxide thickness decreased from 180 to 0 nm from d=50 to d=5mm. The oxide was
removed after ion implantation, prior to the ellipsometric measurement
Fig. 17.11 Side view of the optical arrangement of the divergentlight source mapping spectroscopic
ellipsometer. (1) point source; (2) polarizer; (3) spherical mirror; (4) non collimated beam; (5)
sample; (6) cylindrical mirror; (7) corrected beam; (8) analyzer; (9) pinhole; (10) beam after pinhole;
(11) corrector-disperser optics; (12) ccd detector; (13) rectangular (narrow) aperture
59]. The bevels are made after ion implantation by pulling the sample out gradually
from the anodization liquid, growing an oxide with laterally increasing thickness. In
the experiment of Ref. [59] the wedge length and maximum height were 50mm and
200nm, respectively. Finally, the oxide is removed by conventional HF etch. It has
to be taken into account that the growth of a unit of SiO2consumes only about 0.45
unit of Si. So in the above case after etching the 200nm SiO2the thickness of the
removed Si is about 90nm, resulting in a slope of about 1×104degree, a slope so
small that is impossible to produce by other techniques. Using this technique even
the plotting of the pseudo-dielectric function as a function of lateral position shows
clearly the locations where the damage peak reaches the surface [115].
602 P. Petrik and M. Fried
The second or third derivative analysis described in chapter “Analytical function-
based models” combined with Rutherford backscattering spectrometry (RBS) and
numerical track modeling can even be used to measure the 3D evolution of the ion
tracks and the remaining nanocrystalline regions during ion implantation [116]. The
broadening of the critical point features measured by ellipsometry as a function of
fluence is proportional to the remaining non-damaged surface and therefore shows
an exponential dependence on the fluence. With RBS we can measure the relative
damage as a function of depth. In the numerical simulation we can define a vertical
distribution of damage and a track size. The dependence of relative damage on the
fluence and depth (obtained from the RBS and SE measurements) can be fitted using
the parameters of the numerical model, so the parameters like the track size and the
depth distribution can be obtained [116].
The complex optical models used for depth profiling often require the fitting of
numerous model parameters. In this case a crucial question is whether the result is
the global minimum? The final step in the fitting procedure is usually a Levenberg-
Marquardt gradient search algorithm that finds the minimum based on the change
of fitting error when changing a parameter locally. Therefore, this method can not
guarantee to find the global minimum. There is a range of techniques developed to
find the global minimum (random global search, genetic algorithms, neural networks,
simulated annealing, hill climbing, etc., see Refs. [122124]). Even a simple global
search and a gradient fit from the best sets of parameters found by the global search
has been proven to be an effective way of avoiding to get into local minima [107].
Lateral mapping is usually performed by moving the samples stage or (in case of
larger substrates) moving the ellipsometer head over the sample [11]. Imaging ellip-
sometry provides a concept of measuring in different lateral positions simultaneously.
Both microscopic [15,137,138] and macroscopic [49,50,67,87,88] concepts have
been demonstrated. The size of the measured area and the resolution ranges from
several microns to several millimeters. For divergent light source non-collimated
beam macroscopic ellipsometric configurations there is theoretically no limit of the
maximum size—the resolution can be defined as the viewing angle of a pixel on the
CCD camera. The concept is shown in Fig. 17.11 [49] demonstrating a new design,
in which all the polarizing optical components can be small—the only component
that scales with the mapping size is the spherical mirror.
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... in general, since the topic is huge. The theory of ellipsometry will also not be included, because it can be found in many books and articles [5][6][7][8][45][46][47][48]. We rather focus on the frequently used measurement configurations of ellipsometry applied for studying the interface between liquids and solids, mainly using examples from our research. ...
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Understanding interface processes has been gaining crucial importance in many applications of biology, chemistry, and physics. The boundaries of those disciplines had been quickly vanishing in the last decade, as metrologies and the knowledge gained based on their use improved and increased rapidly. Optical techniques such as microscopy, waveguide sensing, or ellipsometry are significant and widely used means of studying solid‐liquid interfaces because the applicability of ions, electrons, or X‐ray radiation is strongly limited for this purpose due to the high absorption in aqueous ambient. Light does not only provide access to the interface making the measurement possible, but utilizing the phase information and the large amount of spectroscopic data, the ellipsometric characterization is also highly sensitive and robust. This article focuses on ellipsometry of biomaterials in the visible wavelength range. The authors discuss the main challenges of measuring thickness and optical properties of ultra‐thin films such as biomolecules. The authors give an overview on different kinds of flow cells from conventional through internal reflection to combined methods. They emphasize that surface nanostructures and evaluation strategies are also crucial parts of in situ bioellipsometry and summarize some of the recent trends showing examples mainly from their research.
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Chapter
The research on solar cells based on photonic, plasmonic and various nanostructured materials has been increasing in the recent years. A wide range of nanomaterial approaches are applied from photonic crystals to plasmonics, to trap light and enhance the absorption as well as the efficiency of solar cells. The first part of this chapter presents examples on applications that utilize nanostructured materials for photovoltaics. In the second part, ellipsometry related metrology issues are discussed briefly, dividing the topic in two major parts: effective medium and scatterometry approaches.
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This chapter describes ellipsometric characterization of thin films. The chapter illustrates that the newly activated interest is driven by the demand for rapid, nondestructive analysis of surfaces and thin films—particularly films and surfaces occurring in different device technologies. The fact that ellipsometric measurements can be performed under any ambient conditions is a definite advantage over other surface-science (electron or ion beam) techniques for industrial applications. In ellipsometry, the change in polarization state of a linearly polarized beam of light has to be measured after non-normal reflection from the sample to be studied. The polarization state can be defined by two parameters—for example, the relative phase and relative amplitude of the orthogonal electricfield components of the polarized light wave. The technique of principles of ellipsometry found its first practical use with the development of so-called rotating element ellipsometers and in computers to solve complex equations. These polarimeter-type ellipsometers measure continuously thus, wavelength scanning can be performed.
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