Content uploaded by Peter Petrik

Author content

All content in this area was uploaded by Peter Petrik on Apr 01, 2017

Content may be subject to copyright.

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=1−4πne2/meω2=1−ωp2/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=1−ωp2/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/τ =1/τ0+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(x−x0)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=1−ω2

p/(ω2−ω2

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 AlxGa1−xAs or SixGe1−x, 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

0x−eE0exp(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

0−ω2+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+P/ε0E=1+χ, where χis the dielectric susceptibility

χ≡P/ε0E, we obtain the dielectric constant as follows:

ε=1+(e2Ne/ε0me)(1/(ω2

0−ω2+iω)) (17.8)

This equation represents the Lorentz model. If we multiply by (ω2

0–ω2–iω)both

the numerator and the denominator of Eq. (17.8), we get

ε1=1+(e2Ne/ε0me)(ω2

0−ω2)/((ω2

0−ω2)2+2ω2)), (17.9)

ε2=(e2Ne/ε0me)( ω/((ω2

0−ω2)2+2ω2)). (17.10)

In actual data analysis, we commonly express the Lorentz model using the photon

energy Eph:

ε=1+jAj/(Eph0j2−Eph2+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/((Eph2−E2

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

n≤Egand 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- [6–8,61,62], 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 [33,35,39–43,45,

46,78–80,94,104,107–109,111,112,120,127].

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 −E−i)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 d−a, 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 10−4–10−6s 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

d−1.39 law where dis the crystallite diameter. (From a simple effective mass approx-

imation one would expect a d−2law. 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. [98–100] 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 250◦C. 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 250◦C, 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 10−4. 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 1×10−4degree, 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. [122–124]). 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.

References

1. S. Adachi, Phys. Rev. B 38, 12966 (1988)

2. S. Adachi, Optical Properties of Crystalline and Amorphous Semiconductors: Materials and

Fundamental Principles (Kluwer Academic Publishers, Norwell, 1999)

3. S. Adachi, H. Mori, Phys. Rev. B 62, 10158 (2000)

4. E. Agocs, P. Petrik, S. Milita, L. Vanzetti, S. Gardelis, A.G. Nassiopoulou, G. Pucker, R.

Balboni, M. Fried, Thin Solid Films 519, 3002 (2011)

5. E. Agocs, P. Petrik, M. Fried, A.G. Nassiopoulou, Mater.Res. Soc. Symp. Proc. 1321, 367–372

(2011). doi:10.1557/opl.2011.949

17 Ellipsometry of Semiconductor Nanocrystals 603

6. R. Antos, I. Ohlidal, D. Franta, P. Klapetek, J. Mistrik, T. Yamaguchi, S. Visnovsky, Appl.

Surf. Sci. 244, 221 (2005)

7. R. Antos, J. Pistora, I. Ohlidal, K. Postava, J. Mistrik, T. Yamaguchi, S. Visnovsky, M. Horie,

J. Appl. Phys. 97, 053107 (2005)

8. R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, Y.

Otani, J. Appl. Phys. 100, 054906 (2006)

9. H. Arwin, D.E. Aspnes, J. Vac. Sci. Technol. A 2, 1316 (1984)

10. D.E. Aspnes, J.B. Theeten, F. Hottier, Phys. Rev. B 20, 3292 (1979)

11. D.E. Aspnes, in Handbook on Semiconductors, vol. 2, ed. by M. Balkanski (North-Holland,

Amsterdam, 1980), p. 109

12. D.E. Aspnes, Thin Solid Films 89, 249 (1982)

13. D.E. Aspnes, S.M. Kelso, C.G. Olson, D.W. Lynch, Phys. Rev. Lett. 48, 1863 (1982)

14. D. E. Aspnes, in Handbook of Optical Constants of Solids, ed. by E. D. Palik (Academic,

New York, 1985)

15. Y.M. Bae, B-K Oh, W. Lee, W.H. Lee, J-W Choi, Biosensors and Bioelectronics 20, 895

(2004)

16. A. Balazs, L. Hermann, J. Gyulai, Phys. Stat. Sol. (a) 29, 105 (1975)

17. P. Basa, P. Petrik, M. Fried, L. Dobos, B. Pécz, L. Tóth, Physica E-Low-Dimens. Syst. Nanos-

truct. 38, 76 (2007)

18. A. Bonanni, D. Stifter, A. Montaigne-Ramil, K. Schmidegg, K. Hingerl, H. Sitter, J. Cryst.

Growth 248, 211 (2003)

19. A. Bonanni, K. Schmidegg, A. Montaigne-Ramil, H. Sitter, K. Hingerl, D. Stifter, J. Vac. Sci.

Technol. B 21, 1825 (2003)

20. D.A.G. Bruggeman, Ann. Phys. (Leipzig) 24, 636 (1935)

21. J. Budai, Z. Toth, A. Juhasz, G. Szakacs, E. Szilagyi, M. Veres, M. Koos, J. Appl. Phys. 100,

043501 (2006)

22. M. Cardona, Modulation Spectroscopy, Suppl. 11 of Solid State Physics, ed. by F. Seitz, D.

Turnbull, H. Ehrenreich (Academic, New York, 1969)

23. M.F. Cerqueira, M. Stepikhova, M. Losurdo, M.M. Giangregorio, A. Kozanecki, T. Monteiro,

Opt. Mater. 28, 836 (2006)

24. G.D. Cody, B.D. Brooks, B. Abeles, Sol. Energy Mat. 8, 231–240 (1982)

25. R.W. Collins, K. Vedam, Optical properties of solids, in Encyclopedia of Applied Physics,

vol. 12, ed. by G.L. Trigg (VCH, New York, 1995), p. 285

26. R.W. Collins, Joohyun Koh, H. Fujiwara, P.I. Rovira, A.S. Ferlauto, J.A. Zapien. C.R. Wron-

ski, R. Messier, Appl. Surf. Sci. 154–155, 217–228 (2000)

27. R.W. Collins, Joohyun Koh, A.S. Ferlauto, P.I.Rovira, Yeeheng Lee, R.J. Koval, C.R. Wronski.

Thin Solid Films 364, 129 (2000)

28. R.W. Collins, A.S. Ferlauto, in Handbook of Ellipsometry (William Andrew, Norwich, 2005),

p. 93

29. P. Aryal, J. Chen, Z. Huang, L.R. Dahal, M.N. Sestak, D. Attygalle, R. Jacobs, V. Ranjan, S.

Marsillac, R.W. Collins, 37th IEEE Photovoltaic Specialists Conference (PVSC 2011), Seat-

tle, WA;19–24 June 2011; Category number CFP11PSC-ART; Code 89752, Article number

6186402, pp. 002241–002246

30. L. Ding, T.P. Chen, Y. Liu, C.Y. Ng, S. Fung, Phys. Rev. B 72, 125419 (2005)

31. L. Ding, T.P. Chen, Y. Liu, M. Yang, J.I. Wong, Y.C. Liu, A.D. Trigg, F. R. Zhu M.C. Tan, S.

Fung, J. Appl. Phys. 101, 103525 (2007)

32. W.G. Egan, D.E. Aspnes, Phys. Rev. B 26, 5313 (1982)

33. W.G. Egan, D.E. Aspnes, Phys. Rev. B 26, 5313 (1982)

34. Aleksandra B. Djurisic, Aleksandar D. Rakic, Paul C. K. Kwok, E. Herbert Li, Martin L.

Majewski, J. Appl. Phys. 85, 3638 (1999)

35. G.F. Feng, R. Zallen, Phys. Rev. B 40, 1064 (1989)

36. A.S. Ferlauto, G.M. Ferreira, J.M. Pearce, C.R. Wronski, R.W. Collins, X. Deng, G. Ganguly,

J. Appl. Phys. 92, 2424 (2002)

37. D. Franta, I. Ohlidal, Opt. Commun. 248, 459 (2005)

604 P. Petrik and M. Fried

38. A.R. Forouhi, I. Bloomer, Phys. Rev. B 34, 7018–7026 (1986)

39. M. Fried, T. Lohner, E. Jároli, G. Vizkelethy, G. Mezey, J. Gyulai, M. Somogyi, H. Kerkow,

Thin Solid Films 116, 191 (1984)

40. M. Fried, T. Lohner, J.M.M. de Nijs, A. van Silfhout, L.J. Hanekamp, Z. Laczik, N.Q. Khanh,

J. Gyulai, J. Appl. Phys. 66, 5052 (1989)

41. M. Fried, T. Lohner, W.A.M. Aarnink, L.J. Hanekamp, A. van Silfhout, J. Appl. Phys. 71,

5260 (1992)

42. M. Fried, T. Lohner, W.A.M. Aarnink, L.J. Hanekamp, A. van Silfhout, J. Appl. Phys. 71,

2835 (1992)

43. M. Fried, A. van Silfhout, Phys. Rev. B49, 5699 (1994)

44. M. Fried, T. Lohner, O. Polgar, P. Petrik, E. Vazsonyi, I. Barsony J.P. Piel, J.-L. Stehle, Thin

Solid Films 276, 2223 (1996)

45. M. Fried, T. Lohner, P. Petrik, Ellipsometric Characterization of Thin Films in Handbook

of Surfaces and Interfaces of Materials: "Solid Thin Films and Layers", vol. 4, ed. by H. S.

Nalwa( Academic Press, San Diego, 2001) pp. 335–367

46. M. Fried, P. Petrik, T. Lohner, N.Q. Khánh, O. Polgár, J. Gyulai, Thin Solid Films 455–456,

404 (2004)

47. M. Fried, P. Petrik, Zs. E. Horváth, T. Lohner. Appl. Surf. Sci. 253, 349 (2006)

48. M. Fried, N.Q. Khanh, P. Petrik, Physica Status Solidi C-Curr. Top. Solid State Phys. 5, 1227

(2008)

49. M. Fried, G. Juhász, C. Major, P. Petrik, O. Polgár, Z. Horváth, A. Nutsch, Thin Solid Films

519, 2730 (2011)

50. M. Fried, G. Juhasz, C. Major, A. Nemeth, P. Petrik, O. Polgar, C. Salupo, Lila R. Dahal, R.

W. Collins, Mater. Res. Soc. Symp. Proc. 1323 (2011) doi:10.1557/opl.2011.820

51. H. Fujiwara, J. Koh, C.R. Wronski, R. W. Collins. Appl. Phys. Lett 70, 2151 (1997)

52. H. Fujiwara, M. Kondo, Phys. Rev. B 71 075109–1-10 (2005)

53. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, New York,

2007)

54. W. Fukarek, J.R. Kaschny, J. Appl. Phys. 86, 4160 (1999)

55. S. Furukawa, T. Miyasato, Phys. Rev. B 38, 5726 (1988)

56. M.M. Giangregorio, M. Losurdo, A. Sacchetti, P. Capezzuto, G. Bruno, Thin Solid Films

511–512, 598 (2006)

57. M.M. Giangregorio, M. Losurdo, A. Sacchetti, P. Capezzuto, F. Giorgis, G. Bruno, Appl. Surf.

Sci. 253, 287 (2006)

58. M.M. Giangregorio, M. Losurdo, G.V. Bianco, P. Capezzuto, G. Bruno, Thin Solid Films 519,

2787 (2011)

59. J. Gyulai, G. Battistig, T. Lohner, Z. Hajnal, Nucl. Instrum. Methods Phys. Res. B 266, 1434

(2008)

60. C.M. Herzinger, B. Johs, W.A. McGahan, J.A. Woollam, W. Paulson, J. Appl. Phys. 83, 3323

(1998)

61. H.-T. Huang, W. Kong, F.L. Terry, Jr., Appl. Phys. Lett. 78, 3983 (2001)

62. H.-T. Huang, F.L. Terry Jr., Thin Solid Films 455/456, 828 (2004)

63. G.E. Jellison, Opt. Mater. 1, 41 (1992)

64. G.E. Jellison Jr, M.F. Chisholm, S.M. Gorbatkin, Appl. Phys. Lett. 62, 3348 (1993)

65. G.E. Jellison, F.A. Modine, Appl. Phys. Lett. 69, 371 (1996)

66. B. Johs, C.M. Herzinger, J.H. Dinan, A. Cornfeld, J.D. Benson, Thin Solid Films 313–314,

137 (1998)

67. G. Juhasz, Z. Horvath, C. Major, P. Petrik, O. Polgar, M. Fried, Phys. Status Solidi C-Curr.

Top. Solid State Phys. 5, 1081 (2008)

68. K. Kertész, Z. Bálint, Z. Vértesy, G. Márk, V. Lousse, J.P. Vigneron, M. Rassart, L.P. Biró,

Phys. Rev. E 74, 021922 (2006)

69. J. Koh, Y. Lu , S. Kim, J.S. Burnham, C.R. Wronski, R.W. Collins, Appl. Phys. Lett. 67, 2669

(1995)

17 Ellipsometry of Semiconductor Nanocrystals 605

70. J. Koh, Y. Lu , C.R. Wronski, Y. Kuang, R.W. Collins, T.T Tsong, Y.E. Strausser, Appl. Phys.

Lett. 69, 1297 (1996)

71. K. Kurihara, S. Hikino, S. Adachi, J. Appl. Phys. 96, 3247 (2004)

72. W. Lamb, D.M. Wood, N.W. Ashcroft, Phys. Rev. B 21, 2248 (1980)

73. P. Lautenschlager, M. Garriga, M. Cardona, Phys. Rev. B 36, 4813 (1987)

74. J. Leng, J. Opsal, H. Chu, M. Senko, D.E. Aspnes, Thin Solid Films 313–314, 132–136 (1998)

75. J. Li, J. Chen, M.N. Sestak, R.W. Collins, IEEE J. Photovolt. 1, 187 (2011)

76. C.-H. Lin, H.-L. Chen, W.-C. Chao, C.-I. Hsieh, W.-H. Chang, Microelectron. Eng. 83, 1798

(2006)

77. S. Logothetidis, Diam. Relat. Mater. 12, 141 (2003)

78. T. Lohner, G. Mezey, E. Kótai, F. Pászti, L. Királyhidi, G. Vályi, J. Gyulai, Nucl. Instrum.

Methods 182/183, 591 (1981)

79. T. Lohner, G. Mezey, E. Kótai, F Pászti, A. Manuaba, J. Gyulai, Nucl. Instrum. Methods

209/210, 615 (1983)

80. T. Lohner, M. Fried, J. Gyulai, K. Vedam, N. Nguyen, L.J. Hanekamp, A. van Silfhout, Thin

Solid Films 233, 117 (1993)

81. T. Lohner, Z. Zolnai, P. Petrik, G. Battistig, J.G. Lopez, Y. Morilla, A. Koos, Z. Osvath, M.

Fried, Phys. Status Solidi C-Curr. Top. Solid State Phys. No. 5, 1374 (2008)

82. T. Lohner, P. Csíkvári, N.Q. Khánh, S. Dávid, Z.E. Horváth, P. Petrik, G.Hárs, Thin Solid

Films 519, 2806 (2011)

83. M. Losurdo, G. Bruno, D. Barreca, E. Tondello, Appl. Phys. Lett. 77, 1129 (2000)

84. M. Losurdo, M.F. Cerqueira, M.V. Stepikhova, E. Alves, M.M. Giangregorio, P. Pinto, J.A.

Ferreira, Physica B 308–3010, 374 (2001)

85. M. Losurdo, F. Roca, R. De Rosa, P. Capezzuto, G. Bruno, Thin Solid Films 383, 69 (2001)

86. M. Losurdo, M. Giangregorio, P. Capezzuto, G. Bruno, R. De Rosa, F. Roca, C. Summonte,

J. Pla, R. Rizzoli, J. Vac. Sci. Technol. A 20, 37 (2002)

87. C. Major, G. Juhász, Z. Horvath, O. Polgar, M. Fried, I. Bársony, Physica Status Solidi C-Curr.

Top. Solid State Phys. 5, 1077 (2008)

88. C. Major, G. Juhasz, P. Petrik, Z. Horvath, O. Polgar, M. Fried, Vacuum 84, 119 (2010)

89. H.L. Maynard, N. Layadi, J.T.C. Lee, Thin Solid Films 313/314, 398 (1998)

90. M. Modreanu, M. Gartner, D. Cristea, Mater. Sci. Eng. C 19, 225 (2002)

91. M. Modreanu, M. Gartner, E. Aperathitis, N. Tomozeiu, M. Androulidaki, D. Cristea, Paul

Hurley, Physica E 16, 461 (2003)

92. M. Modreanu, M. Gartner, C. Cobianu, B. O’Looney, F. Murphy, Thin Solid Films 450, 105

(2004)

93. I. Mohacsi, P. Petrik, M. Fried, T. Lohner, J.A. van den Berg, M.A. Reading, D. Giubertoni,

M. Barozzi, A. Parisini, Thin Solid Films 519, 2847 (2011)

94. H. Mori, S. Adachi, M. Takahashi, J. Appl. Phys. 90, 87 (2001)

95. N. Nagy, A.E. Pap, E. Horváth, J. Volk, I. Bársony, A. Deák, Z. Hórvölgyi, Appl. Phys. Lett.

89, 063104 (2006)

96. N. Nagy, A. Deák, Z. Hórvölgyi, M. Fried, A. Agod, I. Bársony, Langmuir 22, 8416 (2006)

97. Public community-driven database on the Nanocharm, http://www.nanocharm.org/index.

php?option=com_content\&task=section\&id=15\&Itemid=122

98. H. V. Nguyen, R.W. Collins, Phys. Rev. B 47, 1911 (1993)

99. H.V. Nguyen, Ilsin An, R.W. Collins, Yiwei Lu, M. Wakagi,C.R. Wronski, Appl. Phys. Lett.

65, 3335 (1994)

100. H.V. Nguyen, Y. Lu, S. Kim, M. Wakagi, R.W. Collins, Phys. Rev. Lett. 74(19), 3880–3883

(1995)

101. S. Ögüt, J.R. Chelikowsky, S.G. Louie, Phys. Rev. Lett. 79, 1770–1773 (1997)

102. E.D. Palik (ed.), Handbook of Optical Constants of Solids II (Academic, New York, 1991)

103. P. Petrik, M. Fried, T. Lohner, R. Berger, L. P. Biró, C. Schneider, J. Gyulai, H. Ryssel, Thin

Solid Films 313/314, 259 (1998)

104. P. Petrik, T. Lohner, M. Fried, N.Q. Khánh, O. Polgár, J. Gyulai, Nucl. Instr. Methods B 147,

84 (1999)

606 P. Petrik and M. Fried

105. P. Petrik, T. Lohner, M. Fried, L.P. Biró, N.Q. Khánh, J. Gyulai, W. Lehnert, C. Schneider, H.

Ryssel, J. Appl. Phys. 87, 1734 (2000)

106. P. Petrik, W. Lehnert, C. Schneider, T. Lohner, M. Fried, J. Gyulai, H. Ryssel, Thin Solid

Films 383, 235 (2001)

107. P. Petrik, O. Polgár, M. Fried, T. Lohner, N.Q. Khánh, J. Gyulai, J. Appl. Phys. 93(5), 1987

(2003)

108. P. Petrik, E.R. Shaaban, T. Lohner, G. Battistig, M. Fried, J. Garcia Lopez, Y. Morilla, O.

Polgár, J. Gyulai, Thin Solid Films 455–456, 239 (2004)

109. P. Petrik, F. Cayrel, M. Fried, O. Polgár, T. Lohner, L. Vincent, D. Alquier, J. Gyulai, Thin

Solid Films 455–456, 344 (2004)

110. P. Petrik, E.R. Shaaban, T. Lohner, G. Battistig, M. Fried, J. Garcia Lopez, Y. Morilla, O.

Polgar, J. Gyulai, Thin Solid Films 455–456, 239 (2004)

111. P. Petrik, M. Fried, T. Lohner, O. Polgár, J. Gyulai, F. Cayrel, D. Alquier, J. Appl. Phys. 97,

123514 (2005)

112. P. Petrik, M. Fried, T. Lohner, N.Q. Khánh, P. Basa, O. Polgár, C. Major, J. Gyulai, F. Cayrel,

D. Alquier, Nucl. Instrum. Methods B 253, 192 (2006)

113. P. Petrik, T. Lohner, L. Égerházi, Zs. Geretovszky. Appl. Surf. Sci. 253, 173 (2006)

114. P. Petrik, M. Fried, É. Vázsonyi, T. Lohner, E. Horváth, O. Polgár, P. Basa, I. Bársony, J.

Gyulai, Appl. Surf. Sci. 253, 200 (2006)

115. P. Petrik, Phys. Status Solidi A 205(4), 732–738 (2008)

116. P. Petrik, N.Q. Khanh, J. Li, J. Chen, R.W. Collins, M. Fried, G.Z. Radnoczi, T. Lohner, J.

Gyulai, Physica Status Solidi C-Curr. Top. Solid State Phys. 5, 1358 (2008)

117. P. Petrik, M. Fried, Z. Zolnai, N.Q. Khánh, J. Li, R.W. Collins, T. Lohner, Mater. Res. Soc.

Symp. Proc. 1123, P05-01 (2009)

118. P. Petrik, M. Fried, E. Vazsonyi, P. Basa, T. Lohner, P. Kozma, Z. Makkai, J. Appl. Phys. 105,

024908 (2009)

119. P. Petrik, S. Milita, G. Pucker, A.G. Nassiopoulou, J.A. van den Berg, M.A. Reading, M.

Fried, T. Lohner, M. Theodoropoulou, S. Gardelis, M. Barozzi, M. Ghulinyan, A. Lui, L.

Vanzetti, A. Picciotto, ECS Trans. 25, 373 (2009)

120. P. Petrik, Z. Zolnai, O. Polgar, M. Fried, Z. Betyak, E. Agocs, T. Lohner, C. Werner, M.

Röppischer, C. Cobet, Thin Solid Films 519, 2791 (2011)

121. P. Pintér, P. Petrik, E. Szilágyi, Sz. Kátai, P. Deák. Diam. Relat. Mater. 6, 1633 (1997)

122. O. Polgár, M. Fried, T. Lohner, I. Bársony, Surf. Sci. 457, 157 (2000)

123. O. Polgár, P. Petrik, T. Lohner, M. Fried, Appl. Surf. Sci. 253, 57 (2006)

124. O. Polgar, M. Fried, N. Khanh, P. Petrik, I. Barsony, Physica Status Solidi C-Curr. Top. Solid

State Phys. 5, 1354–1357 (2008)

125. J.P. Proot, C. Delerue, G. Allan, Appl. Phys. Lett. 61(16), 1948 (1992)

126. P.I. Rovira, R.W. Collins, J. Appl. Phys. 85, 2015–2025 (1999)

127. K. Tsunoda, S. Adachi, M. Takahashi, J. Appl. Phys. 91, 2936 (2002)

128. M. Serényi, T. Lohner, P. Petrik, C. Frigeri, Thin Solid Films 515, 3559 (2007)

129. C. Simbrunner, Tian Li, A. Bonanni, A. Kharchenko, J. Bethke, K. Lischka, H. Sitter. J. Cryst.

Growth 308, 258 (2007)

130. P.G. Snyder, J.A. Woollam, S.A. Alterovitz, B. Johs, J. Appl. Phys. 68, 5925 (1990)

131. E. Strein, D. Allred, Thin Solid Films 517, 1011 (2008)

132. D. Stroud, Phys. Rev. B 12, 3368 (1975)

133. D. Stroud, F.P. Pan, Phys. Rev. B 17, 1602 (1978)

134. J. Tauc, R. Grigorovici, A. Vancu, Phys. Stat. Sol. 15, 627–637 (1966)

135. M. Vaupel, U. Stoberl, Nanocharm Newslett. 3, 4 (2009) http://www.nanocharm.org

136. E. Vazsonyi, E. Szilagyi, P. Petrik, Z.E. Horvath, T. Lohner, M. Fried, G. Jalsovszky, Thin

Solid Films 388, 295 (2001)

137. U. Wurstbauer, C. Röling, U. Wurstbauer, W. Wegscheider, M. Vaupel, Peter H. Thiesen,

Dieter Weiss, Appl. Phys. Lett. 97, 231901 (2010)

138. Y. Zhang, Y. Chen, G. Jin, Appl. Surf. Sci. 257, 9407 (2011)