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Dielectric function spectra and critical-point energies of

from 0.6 to 6.5eV

Name: Ding Yi

Matriculation number: A0091779L

Supervisor:

Assistant Professor. Andrivo Rusydi

Professor. Mark. Breese

National University of Singapore

A thesis submitted to

the Faculty of Science as a Partial Fulfillment of

the Bachelor of Science (Hons.) in Physics

2015

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ACKNOWLEDGEMENTS

I would like to express my deep gratitude to Professor. Andrivo Rusydi and Professor.

Mark. Breese for his enthusiastic encouragement and valuable feedback on this

project. Also, I would also like to offer my special thanks to Dr. Pranjal for his patient

guidance and professional support throughout the project. Last but not least, I would

like to thank my family members for their support and love.

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ABSTRACT

The dielectric function spectra and critical-point energies of

determined by spectroscopic ellipsometry from 0.6 to 6.5eV are presented in this

paper. Ellipsometry data are analyzed using the pseudo-dielectric function and

spectra are extracted. The data exhibit numerous spectral features associated with

critical points, whose energies are obtained by fitting standard line shapes to second

energy derivative of the data using MATLAB. Critical points are reported at 3.33eV,

3.80eV, 4.17eV, 4.79eV, 4.91eV and 6.21eV, and possible origins of the pronounced

critical-point structures are identified. Especially, we report two excitonic critical

points at 3.8eV and 4.7eV.

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CONTENTS

Chapter 1. Introduction ......................................................................................................... - 1 -

1.1 Principles of electromagnetism and optics ................................................................. - 3 -

1.2 Dielectric ...................................................................................................................... - 4 -

1.2.1 Dielectric polarization ........................................................................................... - 4 -

1.2.2 Dielectric function ................................................................................................ - 4 -

1.2.3 Dielectric function of typical materials................................................................. - 6 -

1.3 Fresnel equations ........................................................................................................ - 8 -

1.4 Principles of ellipsometry .......................................................................................... - 10 -

1.5 Pseudo-dielectric function ........................................................................................ - 11 -

1.6 Dielectric function model .......................................................................................... - 12 -

1.6.1 Lorentz model ..................................................................................................... - 13 -

1.7 Kramers-Kronig relation ............................................................................................ - 14 -

1.8 Exciton and excitonic effect....................................................................................... - 15 -

1.9 Strontium Titanate ( or STO) ....................................................................... - 18 -

1.10 Absorption and band gap ........................................................................................ - 22 -

Chapter 2: Experimental set-up and data analysis methodology ....................................... - 24 -

2.1 Experimental set-up .................................................................................................. - 24 -

2.2 Determination of the dielectric function .................................................................. - 25 -

2.3 RefFIT ......................................................................................................................... - 25 -

2.4 Second energy derivative spectra and line shape analysis ....................................... - 26 -

2.4.1 Savitzky-Golay filter used in smoothing ............................................................. - 26 -

2.4.2 Line shape analysis and critical points ................................................................ - 27 -

2.4.3 Curve fitting of the second energy derivative spectra ....................................... - 28 -

Chapter 3: Results and discussion ....................................................................................... - 31 -

3.1 Dielectric spectra of Strontium titanate .................................................................... - 31 -

3.2 Line shape analysis of second energy derivative....................................................... - 34 -

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Chapter4. Conclusion .......................................................................................................... - 50 -

4.1 Conclusion ................................................................................................................. - 50 -

4.2 Future work ............................................................................................................... - 50 -

References ........................................................................................................................... - 51 -

Appendix .............................................................................................................................. - 53 -

A.1 The second energy derivative of esp1 and esp2 at 4K and 300K without and with

smoothing ........................................................................................................................ - 53 -

A.2 data used to fit in MATLAB and samples of trials when fitting ................................. - 57 -

A.3 Samples of MATLAB code.......................................................................................... - 60 -

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Chapter 1. Introduction

Ellipsometry is an optical measurement technique that characterizes light reflection or

transmission from samples. The key feature of ellipsometry is that it measures the

change in polarized light upon light reflection on a sample (or light transmission by a

sample). The name "ellipsometry" comes from the fact that polarized light often

becomes "elliptical" upon light reflection.

Ellipsometry measures the two values . These represent the amplitude ratio

and phase difference between light waves known as p- and s-polarized light waves.

In spectroscopic ellipsometry, spectra are measured by changing the

wavelength of light. In general, the spectroscopic ellipsometry measurement is carried

out in the ultraviolet or visible region.

As shown in Fig. 1, spectroscopic ellipsometry measures spectra for photon

energy or wavelength . The interpretation of measurement results is rather

difficult from the absolute values of . Thus, construction of an optical model is

required for data analysis.

Figure 1: Characterization of physical properties by spectroscopic ellipsometry

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In real-time spectroscopic ellipsometry, spectra are measured continuously

during processing. This technique further allows a number of characterizations

illustrated in Fig. 2

Figure 2: Characterization of thin film structures by real-time spectroscopic

ellipsometry

In this chapter, we will introduce the principles of electromagnetism, optics and

ellipsometry, dielectric function and the Lorentz model, excitons and basics of

Strontium Titanate ( or STO).

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1.1 Principles of electromagnetism and optics

In vacuum, electromagnetic waves always travel at the speed of light, c regardless of

its wavelength and also obey the Maxwell's equations. They are composed of 2

one-dimensional transverse waves of and which are mutually perpendicular.

In general, the propagation of a wave in one-dimension can be expressed in terms of a

general wave expression at by

(1.1)

where A is the wave amplitude, K is the propagation number, ω is the angular

frequency , and δis the initial phase of the wave. This can be brought forward to

define our one-dimensional waves of E and B accordingly

(1.2)

(1.3)

In principle, there are 3 kinds of interactions that can occur as light incidences on a

material surface. Light can be reflected, transmitted or absorbed by the material.

When light advances into optically different media, it gets refracted at the surface due

to a change in its speed. The degree of refraction can be obtained from the definition

of the refractive index of a medium. The refractive index n is defined by

where v is the speed of light in a medium. Accordingly, the propagation of light

waves becomes slower in a medium with high n. The refractive index of air is

and is alomost the same as the refractive index in vacuum .

Where the light advances from medium 1 to another medium 2, the relation between

their indexes of refraction can be described by Snell's law.

(1.5)

where is the angle of incidence and is the angle of refraction.

For transparent media, determines the propagation of electromagnetic waves

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completely. However, for media that show strong light absorption, we need to

introduce another quantity named extinction coefficient to help. Here, we define

the complex refractive index as

The extinction coefficient k is related to the absorption coefficient of a medium as below,

1.2 Dielectric

The dielectric function is determined by the properties of the matter and its interaction

with a light wave propagating through that matter.

1.2.1 Dielectric polarization

When an electric field is applied to a medium, positive and negative charges in the

medium receive electric forces in the opposite direction. In dielectrics, however,

electric charges cannot move freely since atoms, for example, are bound together by

strong chemical bonding. Nevertheless, in the presence of the electric field, the spatial

distributions of positive and negative charges are modified slightly and are separated

into regions that are more electrically positive and negative. This phenomenon is

referred to as dielectric polarization. The most important polarization for

semiconductor characterization is electric polarization. The dielectric polarization P is

defined by a vector whose direction is from the negative charge to the positive charge.

1.2.2 Dielectric function

The magnitude of the polarization generated within a dielectric is expressed by the

permittivity or dielectric constant. In order to define the permittivity physically,

consider a parallel plate capacitor. In this capacitor, a dielectric medium is inserted

between the two electrodes of the capacitor and an ac electric field is applied to the

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capacitor. When the medium between the electrodes is vacuum,

when polarization P is taken into consideration, the electric field within the dielectric

can be written as

In general, the relative permittivity or relative dielectric constant can be expressed

by the following equations,

where is referred to as the dielectric susceptibility.

From Maxwell’s equations for conductors, the complex refractive index is defined as

Combing with the following two equations with Eq.(1.11),

we get

From in Eq.(1.14d), the absorption coefficient can also be obtained using Eq.(1.7).

The study of dielectric properties concerns storage and dissipation of electric and

magnetic energy in materials. Dielectrics are important for explaining various

phenomena in electronics, optics, and solid-state physics.

The quantity represents how much a material becomes polarized when an electric

field is applied due to creation of electric dipoles in the material (see Figure 3). When

the applied field is oscillating between positive and negative such as with a light wave,

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the sign of can be either a positive or negative number depending on whether the

induced dipoles are oscillating in phase or out of phase with the applied field.

Figure 3: In atoms by incident oscillator dipoles induced light wave, resulting in

polarization inside the material.

When the induced dipole oscillations in a material become large it is possible for the

material to start absorbing energy from the applied field. When absorption occurs the

quantity becomes important. When a material is transparent is zero, but

becomes nonzero when absorption begins. Thus represents absorption in a material.

It is important to consider both and together since they affect each other,

meaning the shape of cause corresponding changes in the shape of and

vice-versa. This is known as the Kramers-Kronig relation between the real ( ) and

imaginary ( ) parts of the dielectric function.

In summary,

= volume polarization

= volume absorption

1.2.3 Dielectric function of typical materials

SiO2 Dielectric Function

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Figure 4: Shows an example of a dielectric that is transparent over

the entire spectral region. Notice e1 is positive, but e2=0 indicates a

transparent material.

Aluminum Dielectric Function

Figure 5: Shows an example of a metal material that had e2

absorption due to free carriers over the entire spectral region, causing

e2 to be nonzero over the full spectrum.

GaAs Dielectric Function

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Figure 6: Shows an example of a semiconductor material that has a

band gap near 1.42 eV. Note e2 is zero below the band gap, with

absorption (e2>0) above the band gap.

1.3 Fresnel equations

In the discussion of reflection and refraction of incident light from one medium to

another, we consider 2 kinds of polarization: the p-and s-polarization. When

describing the mathematical relations between the incident light, reflected light and

refractive light, we apply the Fresnel equations.

In case of p-polarized light, the boundary conditions for E and B

combining , we get

using

=

=

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Figure 7: Electric field E and magnetic induction B for (a) p-polarization and

(b)s-polarization. In these figures, B in(a) and E in (b) are perpendicular to the plane

of the paper and are pointing to the reader.

Similarly, the boundary conditions for s-polarized light

we get

=

=

These Fresnel equations still hold if the refractive index n is replaced with the

complex refractive index N.

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1.4 Principles of ellipsometry

Figure 8: measurement principles of ellipsometry

In particular, when a sample structure is simple, the amplitude ratio ψ is characterized

by the refractive index , which Δ represents light absorption described by the

extinction coefficient . In this case, the two values can be determined

directly from the two ellipsometry parameters obtained from a measurement

by applying the Fresnel equations. This is the basic principle of ellipsometry

measurement.

The measured from ellipsometry are defined from the ratio of the amplitude

reflection coefficients for p- and s-polarizations:

(1.22)

When we measured light transmission, instead of light reflection, are defined

as

(1.23)

If we apply the definitions of the amplitude reflection coefficients and

[Equs.(2.57) and (2.61) in Ref[1]], we can write Eq.(4.1) as follows:

≡

(1.24)

As confirmed from Eq.(4.3), and are originally defined by the ratios of

reflected electric fields to incident electric fields, and is defined

further by the ratios of to . In the case of Fig.8, Eq.(1.24) can be simplified to

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since . In Fig.8, therefore, ψ represents the

angle determined from the amplitude ratio between reflected p- and s-polarizations,

while Δ expresses the phase difference between reflected p- and s-polarizations.

1.5 Pseudo-dielectric function

The pseudo-dielectric function (Ψ,Δ) represents a dielectric function obtained directly

from the measured values (Ψ,Δ) and is calculated from an optical model that assumes

a perfectly flat substrate with infinite thickness.

if the dielectric constants of an ambient(air) and a sample are given by and

, respectively, we obtain the pseudo-dielectric function from

Eq.(1.25) as follows:

where and is the incident angle.

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1.6 Dielectric function model

Figure 9: dielectric function models used in ellipsometry data analysis

For dielectric function modeling in a transparent region ( ≈ 0), the Sellmeier or

Cauchy model is used. When there is free-carrier absorption, the data analysis is

generally performed using the Drude model. To express the electric polarization in the

visible/UV region, various models including the Lorentz model, Tauc–Lorentz model,

harmonic oscillator approximation (HOA), and model dielectric function (MDF) have

been used.

In our experiments, RefFIT program is applied to fit the dielectric spectra calculated

from the pseudo-dielectric function and the fit is done with the Lorentz model. Hence,

there will be a more detailed description of the Lorentz model in the following

section.

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1.6.1 Lorentz model

Figure 10: (a) physical model of the Lorentz model and (b) dielectric function from

the Lorentz model[1]

In Lorentz model, a negatively charged electron is bound to a positively charged

atomic nucleus with a spring while the nucleus is fixed. Also, we assume that the

electron oscillates in viscous fluid. If the light is incident, the ac electric field of the

light will induce dielectric polarization in the x direction of Figure. 10(a).

By applying Newton's second law, the physical model can be expressed by

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

Г in Eq. (1.27) represents a proportional constant of the viscous force, known as the

damping coefficient. The second term on the right expresses that the electron moved

by the electric field of light is restored according to Hook’s law , and

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shows the resonant frequency of the spring = . The last term on the

right shows the electrostatic force ( ).

As shown in Fig. 10(a), the direction of the force F applied to the electron is opposite

to that of the electric field, and the restoration force ( ) and viscous force

act in the reverse direction to F. Eq. (1.27) 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.

Using

if the number of electrons per unit volume is given by , the dielectric polarization

is expressed as

we obtain the dielectric constant as follows

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

energy :

1.7 Kramers-Kronig relation

The real and imaginary parts of the dielectric function, and , obey the

Kramers-Kronig relation shown in Eq. (1.34),

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This means that once either of the functions is known for all frequencies, the other can

be immediately known through this Kramers-Kronig relation. It should be pointed out

that the Sellmeier and Cauchy models do not satisfy the Kramers-Kronig relation

because they assume that absorption does not exist in all range.

1.8 Exciton and excitonic effect

In the study of electronic band structures, we may infer insights on the interactions

between charge carriers. One such interaction would be electron-hole interaction . in

the interband transition where photons are absorbed, the electron is excited from the

valence band to the conduction band leaving behind a hole. Excitons may be formed

in such a system where the electron and hole are bound together as a pair by their

Coulomb interaction. Provided that the energy is not too large, the exciton my

continue to reside within the crystal and hence have interesting effects on the optical

properties of the sample as we shall see. If the temperature is high enough, thermally

excited phonons may start to collide with the excitons and provide them with an

energy greater than their binding energy and cause the exiction to dissipate.

In a simple physical picture, it is possible to model an exiction as a hydrogenic system

although it should be taken note that the excition binding energy is usually much

smaller than that for a hydrogen atom because of the screening effects from

neighboring electrons as well as their smaller effective mass.

Excitons can be classified into 2 general kinds: Wannier-Mott excitions which are free

excitions, and Frenkel excitons which are bound excitons.

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Figure 11: The Wannier-Mott exciton and Frenkel exciton

The concepts of excitons plays an extremely important role in the understanding of

the linear and nonlinear optical responses of semiconductor. As the temperature is

increased, the discrete exciton lines become broadened and , in general, the excitonic

contribution to the absorption decreases due to interactions between excitons and

phonons.

The mechanism by which the inclusion of the excitonic effect in the simple model

increases the height of the E1 peak and lowers that of E2 peak, was explained by

Hanke and Sham by invoking the dimensionality of the peaks. Their conclusions were

based on Kramers-Kronig analysis of the step function discontinuity of the E1 peak

and of the inverse square-root singularity of the E2. The Kramers-Kronig analysis had

not included the possibility of bound states below either the E1 or E2 edges.

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Figure 12: Absorption spectra of (a)silicon and (b)germanium and are calculated

values in simple model without interaction and with interaction . Interaction

between states in and those in are included.[6]

The bound exiction state from E2 is in the continuum, of the E1 region , and is

broadened by the interaction between the electrons in the two regions. This interaction

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spreads the strength of the excitonic state, as can be seen in the figure below, where

the resultant resonant state manifests itself as the barely noticeable peaks at 3.51 and

3.78eV in the calculated optical spectra of Si and Ge, respectively.[6]

The crucial point of the approach of finding excitonic effect used in paper[6] is the

line shape analysis which allows for the separation and quantifying of various

contributions.

1.9 Strontium Titanate ( or STO)

At room temperature, Strontium Titanate is a centrosymmetric paraelectric material with

a perovskite structure. Strontium titanate becomes superconducting below 0.35 K

and was the first insulator and oxide discovered to be superconductive. SrTiO3 has an

indirect band gap of 3.25 eV and a direct gap of 3.75 eV. At temperatures lower than

105 K, its cubic structure transforms to tetragonal

The crystal structure of STO is of the perovskite type and consists of alternating

stacks of SrO and TiO2 atomic layers.

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Figure 13：the crystal structure of Strontium titanate

Strontium titanate crystallizes in the cubic perovskite structure. Optical properties are

therefore isotropic, Strontium titanate behaves like a ferroelectric material in the

paraelectric phase. This property is referred to as incipient ferroelectric. the

hypothetical Curie temperature would take place just below the absolute-zero

temperature. As a result, the dielectric constant is strongly dependent on temperature

and increases by a factor as large as 10 when the sample is cooled from room

temperature to liquid-helium temperature. Since the dielectric constant is the

extrapolation down to zero frequency of the dielectric response, which itself is the

square of the complex refractive index, optical constants also vary with temperature,

but mainly in the far infrared. in addition, a phase transition related to alternate tilt of

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oxygen octahedra, which results in the formation of tetragonal structure, takes palace

upon cooling below 105K. The temperature dependence of data related to optical

constants has been studied by many pioneers.[10]

Table 1: Interband transition energies(in eV) for bulk as determined from

spectroscopic ellipsometry, VUV spectroscopy, VEELS, and local density functional

theory(LDFT). Transitions are labeled as in Figs.4 and 5, with p denoting a peak and s

a shoulder.[10]

Table 2: Theoretical calculation of the band gap of STO using different methods[15]

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Figure 14: Schematic energy level diagram for bulk . The transitions from the

valence band into the conduction band assigned in this work are plotted according to

Table 1. Energies are taken from the band structure and DOS calculations and are

calibrated[10]

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Figure 15: the GW band structure along high-symmetry direction. Dotted lines are drawn

according to notable structures in the l-DOS. Peculiar bands have been highlighted

with green dashed and red dashed-dotted lines. [18]

1.10 Absorption and band gap

A common and simple method for determining whether a band gap is direct or

indirect uses absorption spectroscopy. By plotting certain powers of the absorption

coefficient against photon energy, one can normally tell both what value the band gap

has, and whether or not it is direct.

If a plot of versus forms a straight line, it can normally be inferred that there

is a direct band gap, measurable by extrapolating the straight line to the axis.

On the other hand, if a plot of versus forms a straight line, it can normally

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be inferred that there is an indirect band gap, measurable by extrapolating the straight

line to the axis (assuming ).

Direct and indirect band gap energies can be determined by linear fits to the optical

absorption coefficient( α=4πk ), which can be calculated from the ellipsometrically

determined extinction coefficient. Here, k denotes the wave vector and λ the

wavelength. These band gap energies can vary depending on the range of absorption

coefficients used in the linear fit. The direct band gap of of 3.75eV can be

determined as the intercept of a linear fit line to a plot of versus energy E for

the absorption coefficient in the range of . The

indirect band gap of of 3.25eV can be determined as the intercept of the

linear fit line to a plot of versus energy for the absorption coefficient in the

range of .[10]

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Chapter 2: Experimental set-up and data analysis

methodology

In this chapter, we begin with presenting the set-up for ellipsometry used in this

experiment. Then, we will show an sample calculation about how we obtain dielectric

function using pseudo-dielectric formula. Next, a brief introduction of RefFIT

program will be presented. Last but not least, we will discuss the smoothing of data

and curve fitting using line shape analysis to the second energy derivative spectra.

2.1 Experimental set-up

The real set up is shown in the figure below,

Figure 16: the set-up of V-VASE rotating analyzer ellipsometry

The low temperature environment is created in a cryostat with liquid helium and

liquid nitrogen as liquid cryogens at . All the incident angle is set to be 70

degree. The data of , are obtained in the energy range from 0.6eV to 6.5eV

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with a step of 0.02eV at 4K and 300K separately.

SrTiO3 (100) samples procured from Crystec are used for the measurements. All

samples are single side polished of size 10 × 10 × 0.5 mm. AFM measurements show

that the rms roughness for all the samples are less than 5 Å.

2.2 Determination of the dielectric function

In this experiment, the STO samples are measured in air and placed directly on the

substrate. The thickness is larger than ( the maximum penetration depth is no

more than 1 ) and we can consider this bulk infinitely thick. Considering the rms

roughness, we can assume this sample to be flat. Therefore, we can directly apply the

pseudo-dielectric function( Eq.91.26)) to convert the measured , to .

2.3 RefFIT

The primary goal of spectra analysis, that RefFIT does, is to get information about the

material dielectric function on the base of optical spectra. It is done by the fitting of

these spectra using a model of the dielectric function with a set of adjustable

parameters. These parameters are varied in order to obtain the best match between the

experimental and calculated data points.

Usually two ways to model the dielectric function: 1) mathematical formula with a

limited number of parameters, 2)a variational dielectric function. Here, we use the

second one. One sentence summary: every analysis that RefFIT does is a fitting.

The fitting is always a try-and-error activity. The following formula is the Lorentz

model we use in RefFIT,

It describes the optical response of a set of harmonic (damped) oscillators. Here is the

so called ‘high-frequency dielectric constant’, which represents the contribution of all

oscillators at very high frequencies (compared to the frequency range under

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consideration). The parameters , and are the ‘plasma’ frequency, the

transverse frequency (eigenfrequency) and the linewidth (scattering rate) respectively

of the Lorentz oscillator.

In RefFIT, we can see the results of the manipulations with parameters on the graphs

in real time. By adjusting the parameters manually and applying the least square fit

occasionally, we can get a good fit and the parameters featuring a series of Lorentz

oscillators.

2.4 Second energy derivative spectra and line shape

analysis

The second energy derivative spectra is obtained using Igor 6.1, we input the

dielectric function obtained using the pseudo-dielectric function and perform

differentiate to get the first derivative spectra and perform the same process to the

newly-obtained first derivative spectra and get our second energy derivative spectra.

After plotting the second derivative spectra, we find that the spectra is quite noisy in

some range especially the range of energy with 5 eV-6.5 eV. With the purpose of

smoothing the data, we apply the Savitzky-Golay filter contained in Igor. The

Savitzky-Golay filter we use in this part is 19 points, 4th order with End effects

bounce. After that, lines shape analysis have been performed on the smoothed data

using MATLAB.

2.4.1 Savitzky-Golay filter used in smoothing

Savitzky and Golay proposed a method of data smoothing based on local least-squares

polynomial approximation. They showed that fitting a polynomial to a set of input

samples and then evaluating the resulting polynomial at a single point within the

approximation interval to discrete convolution with a fixed impulse response. The

lowpass filters obtained by this method are widely known as Savitzky-Golay filters.

Savitzky and Golay were interested in smoothing noisy data obtained from chemical

- 27 -

spectrum analyzers and they demonstrated that least-squares smoothing reduces noise

while maintaining the shape and height of waveform peaks.[3]

2.4.2 Line shape analysis and critical points

The macroscopic linear optical response of material is represented by its dielectric

function, which is closely related to the electronic band structure of the material. The

structure observed in the spectra are attributed to interband transitions(critical points)

which can be analyzed in terms of standard analytic line shapes:

Г

where a critical point(CP) is described by the amplitude , threshold energy ,

broadening Г, and the excitonic phase angle . The exponent n has the value

for

one-dimensional(1D), 0 [logarithmic, i.e., Г for 2D, and

for 3D

CP's. Discrete excitons are represented by . The information obtained from the

line-shape analysis can be compared with band structure calculations. In this paper,

we also call 1D, 2D and 3D critical points as dimensional CP and when we say 3

dimensional CPs, we refer that these 3 CPs are 1D, 2D or 3D and they may have

different dimensionality. Moreover, the dimensionality is related to the signs and

values of effective mass at x-,y- and z-directions. With different signs, we have

different types of 3D CPs. When the effective mass in one or two dimensions become

very large, we consider the CP to be 1D or 2D. For broadening, it can be considered

as the energy uncertainty in the spectra due to the finite lifetime of electrons in the

presence of phonons.

In order to enhance the structure in the spectra and to perform a line-shape analysis of

the CP, we calculate numerically the second derivative of the dielectric function with

respect to photon energy . The second derivative of the standard analytic

line shapes are shown below:

Г

Г

- 28 -

The fit is performed simultaneously for the real and imaginary parts of

using a least squares process.[7]

Usually, we cannot observe the indirect transitions using line shape analysis because

the indirect transitions are weak second-order processes and can only be observed at

frequencies at which no other stronger processes (say, direct allowed transitions)

occur. However, the indirect transitions, modified by excitonic effects have been seen

and identified in a number of semiconductors for which the lowest energy gap is

indirect. [14]

2.4.3 Curve fitting of the second energy derivative spectra

When performing the fit for the second energy derivative , we use the

MATLAB to help.

MATLAB contains a very powerful curve fitting toolbox. Curve Fitting Toolbox

provides an app and functions for fitting curves and surfaces to data. The toolbox

allows us to perform exploratory data analysis, preprocess and post-process data,

compare candidate models, and remove outliers. This toolbox contains the library of

linear and nonlinear models provided or specify your own custom equations. The

library provides optimized solver parameters and starting conditions to improve the

quality of your fits. The toolbox also supports nonparametric modeling techniques,

such as splines, interpolation, and smoothing. After creating a fit, a variety of

post-processing methods can be applied for plotting, interpolation, and extrapolation;

estimating confidence intervals; and calculating integrals and derivatives.

Though the curve fitting toolbox is very power, it is not convenient for us to do

line-shape analysis because 1) a large number of fitting, in our project, more than 60

fitting are performed for a certain temperature; 2) the fitting formula is quite long and

the number of parameters we use goes beyond 30 sometimes. Hence, we choose to

perform line-shape analysis with the lsqcurvefit.m function.

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Function lsqcurvefit.m

The function lsqcurvefit.m is a function contained in the optimization toolbox and

solves nonlinear curve-fitting problems in least-squares sense. It finds coefficients x

such that the problem

given input data , and the observed output , where and

can be matrices or vectors, and is a matrix-valued or vector-valued

function of the same size as .

In my program, the lsqcurvefit is usually used in the following form,

[a1111,resnorm1111]=lsqcurvefit(@f1111,a,x,y,lb,ub,option);

where

a1111 is the matrix of coefficients solving the problem,

resnorm1111 is the value of the squared 2-norm of the residual at x:

, or SSE at each x,

f1111 is the function we use to perform the least-square curve fitting or the

objective function,

a is the initial point of x and set by me,

x is the input data for objective function,

y is the output data to be matched by objective function,

lb is the vector of lower bounds,

ub is the vector of upper bounds,

option is set using the optimset command and decides the maximum number of

function evaluations allowed or maximum number of iterations allowed.

Moreover, usually, we prefer to use R-square, coefficient of determination, to

evaluate the goodness of a fit, here I will show how to get R-square using the output

of lsqcurvefit.m. SSE, short for the sum of squares due to error, is the sum of the

squared error between the fitted data and the original data. SST, or total sum of

- 30 -

squares, is the sum of squared of difference between original data and their mean.

R-square can be evaluated by this relation

the SST is fixed for a certain data, and we can evaluate the goodness of fitting by

looking at the SSE, which is the resnorm output by the lsqcurvefit.m.

Other contents of function lsqcurvefit.m such as the algorithm and etc can be accessed

in MATLAB help using lsqcurvefit.m as key word to search.

- 31 -

Chapter 3: Results and discussion

3.1 Dielectric spectra of Strontium titanate

By applying the pseudo-dielectric function model, we manage to convert the

measured , to . The values of calculated are attached in the

appendix and the spectra of and at and are plotted in the

Figure. 19,20,21 and 22.

Figure 17:spectrum of esp1 at 4K

Figure 18: spectrum of esp2 at 4K

10

8

6

4

2

esp1

654321Energy(eV)

the esp1 spectra at 4K

8

6

4

2

esp2

654321Energy(eV)

the esp2 spectra at 4K

- 32 -

Figure 19: spectrum of esp1 at 300K

Figure 20: spectrum of esp2 at 300K

10

8

6

4

2

esp1

654321Energy(eV)

esp1 spectra at 300K

6

4

2

0

esp2

654321Energy(eV)

esp2 spectra at 300K

- 33 -

Figure 21: the fitting to 4K spectra using RefFIT

Figure 22: the fitting to 300K spectra using RefFIT

- 34 -

Using RefFIT, we manage to fit the spectra of dielectric function with several

oscillators. The eight oscillators we used are shown in Figure.26. In RefFIT, we

usually use wave number of the photon instead of energy. The relationship between

wave number in and energy in is shown, .

Figure 23: the parameters of Lorentz model used in RefFIT, with 4K left and 300K

right

3.2 Line shape analysis of second energy derivative

In Igor 6.1, we plot the spectra of the second energy derivative and the smoothed

spectra using Savitzky-Golay filter. Figure 27 and 28 shows the second energy

derivative of esp1 at 300K without and with smoothing. It is obvious that the

spectrum becomes less 'noisy' after applying the Savitzky-Golay filter. Moreover, the

original and smoothed second energy derivative of dielectric function at 4K and 300K

are plotted in Appendix.

- 35 -

Figure 24: the second energy derivative of esp1 at 300K without smoothing

Figure 25: the second energy derivative of esp1 at 300K with smoothing

-200

-150

-100

-50

0

50

100

d2e1

/dE

2

654321Energy(eV)

The second energy derivative of esp1 at 300k without smoothing

-200

-150

-100

-50

0

50

100

d2e1

/dE

2

654321Energy(eV)

The second energy derivative of esp1 at 300K with smoothing

- 36 -

We start the line shape analysis by counting the number of local maxima and minima

in the dielectric function, we find 3 local maxima and 2 local minima in the dielectric

spectra both at 4K and 300K. Also, we find 5 critical points by plotting the first

derivative of dielectric function an Figure. 29 shows the first energy derivatives of

eps2 at 300K. From Figure.29, we can get the several x-values with the first energy

derivative of eps2 very close to 0: 4.20, 4.42, 4.76, 6.00 and 6.32eV.

Figure 26: the first energy derivative of esp2 at 300K

Till now, the figures are plotted in usual ways, however, in the following part of our

discussion, the plotting method changes and the detail of my plotting method is

introduced in the following part and the construction of the object function is also

explained.

Plotting method

In line-shape analysis, the fit is performed simultaneously for the real and imaginary

parts of using a least squares process. In order to make the sum of error in

- 37 -

both the real part and imaginary part smallest, I put the real and imaginary parts of

in one diagram with real x and real y coordinates. This diagram can be

divided into two section, section Re with negative x values and section Im with

positive x values. The points in Im section is exactly the imaginary part of ,

while those in the Re section is actually the mirror image of the real part of

with x=0 as the mirror.

For example, when energy=4.56eV, the real part of is -83.244 and the

imaginary part of is 52.0065, then we will get two points in this diagram

with coordinate (-4.56, -83.244) and (4.56, 52.0065). The values of at

300K is plotted below using this plotting method.

Figure 27: the diagram of real and imaginary parts of at 300K

Objective function

Because we plot the real and imaginary parts of in the above way, we need

to rewrite the Eq.(2.2), here we use y as the function value and x as the variable.

when

-8 -6 -4 -2 0 2 4 6 8-150

-100

-50

0

50

100

150300k with smoothing

- 38 -

when

It is obvious that both of Eq.(2.4a) and (2.4b) are not continuous, and the

lsqcurvefit.m requires the objective function to be continuous. Here, we apply another

function to help

Eq.(2.5) is a primary function and obviously continuous. The diagram of Eq.(2.5)

with is plotted below.

Figure 28: diagram of function in Eq.(2.5) with a=10000

Function in (2.5) transits from -1 to 1 rapidly and the width of the transition is

controlled by the value of . When we increase the value of , the width of the

transition becomes smaller. Since the lowest energy in our experiments is 0.6eV, I

choose and in this case the width is 1.2. The width is just twice the value

of our starting energy.

- 39 -

With the help of function (2.5), we can rewrite the piecewise functions in Eq.(2.4a)

and (2.4b) into a whole, and Eq. (2.6) are the object function we use to fit in

MATLAB.

When

When

where a and b are weight factors with expressions below, and their diagrams are

shown in Figure. 18.

Figure 29: diagram of weight factor a and b

- 40 -

Next, we show a sample of 1D, 2D, 3D and excitonic critical points with parameter

(a1,e1,g1,s1) in Figure. 29 and 30, where a1 is the amplitude, e1 is the energy band

gap, g1 is the broadening, and s1 is the phase change.

a1=11.874759342519500;

e1=3.689939007611347;

g1=0.241048162919054;

s1=2.185222673107573;

- 41 -

Figure 30: first derivative diagram of 4 different CPs with the same parameters

- 42 -

Figure 31: second derivative diagrams of 4 different critical points with the same

parameter

By observing the shapes of diagram in Figure. 30 and 31, we can conclude that, each

critical points will produce one shoulder in the Re section of the second energy

derivative and a peak in the Im section of the first energy derivative. In Figure.28,

which is the mirror image of the Re section of the second energy derivative at 300K,

we get 7 peaks and 6 shoulders. In Figure.29, which is the Im section of the first

energy derivative at 300K, we get 5 peaks. From the number of peaks and shoulders,

we include that a minimum of 5 CPs are need to let our fitting share the same number

of peaks and shoulders with the data.

In the process of fitting, I first perform the fitting with unsmoothed data with energy

from 0.6eV to 5.0eV because the noisy falls mainly in the range from 5.0eV to 6.5eV.

The rapid change of the second derivative around 1eV is caused by the missing of

data point around 0.70eV. Figure 32 shows that the unsmoothed and smoothed data in

the range of 0.6eV to 5eV are quite close and differs only a bit. Also we can see that

- 43 -

there only exists three huge peaks in the range of 0.6eV to 5.0eV, and therefore, we

perform the fitting with 3 CPs first. Then a fitting with 4 dimensional CPs are

performed and we find the SSDs of 4 dimensional CPs are nearly twice the other

types of 4CPs. Hence, I decide to constrain maximum of the dimensional CPs to 3.

With no more than 3 dimensional CPs, fittings with 5CPs and 6CPs are performed.

Figure 32: the comparison of smoothed(red) and unsmoothed data(0.6-5.0eV,blue) at

300K

Here goes the details of my fitting loop. The first for loop perform a fitting with 3

1D-CPs and an 1 excitonic CP and the second for loop does a fitting with 3 1D-CPs

and 2 excitonic CPs. In the first for loop, I generate a random matrix with energy

value from 3eV to 8eV and perform a fitting with this random matrix as the starting

point. The if loop acts as a tool to compare and find the fitting with least SSD among

all 60 fittings and the parameters such as amplitude, energy, broadening and phase

change of the best fitting will be recorded in b111e. In the second for loop, a random

matrix is generated and added to the previous fitting results, for example, b1111. The

sum of the random matrix ae and b111e acts as the starting point of following

corresponding fitting, which is a fitting with 3-1D CPs and 2 excitonic CP in the

-8 -6 -4 -2 0 2 4 6 8-150

-100

-50

0

50

100

150unsmoothed(0.6-5.0eV) and smoothed data at 300K

- 44 -

shown code. After all fitting, the sum of squared residuals or SSD will be sorted in

ascending order and the best fittings can be selected.

for i=1:60

a=3+5*rand(4,4);

[a111e,resnorm111e]=lsqcurvefit(@f111e,a,x,y,lb,ub,option);

if resnorm111e<ssd111e

b111e=a111e;

ssd111e=resnorm111e;

end%1

...

end

...

for i=1:10

ae=-4+8*rand(4,4);

[a111ee,resnorm111ee]=lsqcurvefit(@f111ee,[b111e+ae;1 1 1

1],x,y,lbe,ube,option);

if resnorm111ee<ssd111ee

b111ee=a111ee;

ssd111ee=resnorm111ee;

end%1

...

end

Figure 33: fitting(green) of data(blue) from 0.6eV to 5eV at 300K

-5 -4 -3 -2 -1 0 1 2 3 4 5-150

-100

-50

0

50

100

150300K from 0.6eV to 5eV

- 45 -

The three best fittings at 300K are obtained with the same R-square of 0.967(here we

round our results to thousands since our ellipsometry data has 3 decimal places while

for energy values we round them to half of the energy step 0.01eV). The first one is

with the model of 3 2D-CPs and 2 excitonic CPs, and the energy values for 3 2D-CPs

are 3.79eV, 4.15eV and 4.84eV, while the energy values of 2 excitonic-CPs are

3.40eV and 4.63eV. The second one is with 3 3D-CPs and 2 excitonic CPs with

energy values of 4.80eV, 4.52eV, 4.17eV, 3.33eV and 3.80eV. The third one is with 2

3D-CPs and 3 excitonic CPs with energy values of 4.16eV, 4.91eV, 3.33eV, 3.80eV

and 4.77eV.

Types of

fitting

Energy of

1D-CP(eV)

Energy of

2D-CP(eV)

Energy of

3D-CP(eV)

Energy of

excitonic

CP(eV)

222ee*

( =0.967)

3.79,

4.15,

4.84

3.40,

4.63

333ee

( =0.967)

4.80,

4.52,

4.17

3.33,

3.80

33eee

( =0.967)

4.16,

4.91

3.33,

3.80,

4.77

Table 3: energy values of line shape analysis at 300K from 0.6eV to 5.0eV.

*: fitting 222ee means fitting with 3 2D-CPs and 2 excitonic CPs.

Comparing these three results with the theoretical calculations[15] and the

experiments results by pioneers[10], we include that there exists two direct interband

transitions below 5eV, making the fitting with 33eee the best fitting with the respect

of physics. The excitonic effect at 3.33eV illustrates the lowest indirect band of STO,

which is in agreement with results in Ref.[16] and [17]. This excitonic can be

observed because there are no other stronger processes around this energy value. The

dimensional CPs at 4.16eV and 4.91eV agrees with the results in Table.1 from Ref.[10]

and comes from the transition of The excitonic CP at

- 46 -

4.77eV, from Figure.16, can be contributed by transitions.

Considering the line shape analysis we performed on 0.6eV to 5.0eV, we add more

CPs in the range of 5.0eV to 6.5eV to explore the structure. First we construct new

function with 2-3D and 2-excitonic CPs included.(Here I do not take the small

excitonic CP around 3.35eV into consideration since this excitonic CP will not affect

a lot in Figure. 34.) We use the b33eee in Table.3 as starting point and more CPs are

added gradually. With 7 CPs, we report another 1D-CP at 6.21eV and 2D-CP at

6.57eV. Here, the 1D-CP may come from the transition.

After comparing the energy values of CPs we obtain with the data in Table.1, we can

see that a direct band transition of 5.4eV is missing in our analysis. This may be

caused by the relatively high noisy-to-signal ratio in the energy range from 5.0eV to

6.5eV. Also, the CP at 6.57eV is not discussed due to the noise. For this fitting at

300K, the R-square is about 0.970.

Figure 34: the tiny exciton(green) of 3.33eV at 300K

-8 -6 -4 -2 0 2 4 6 8-150

-100

-50

0

50

100

150tiny exciton at 3.33eV at 300K

- 47 -

Figure 35: the fitting(blue) of smoothed data(red) at 300K

While for T=4K, similar process are performed and we report CPs at 3.81eV, 4.18eV,

4.72eV, 4.95eV and 6.13eV. However, for line shape analysis at 4K, many fittings

share the same R-square 0.99, it is so hard to justify the type of CPs using line shape

analysis.

-8 -6 -4 -2 0 2 4 6 8-150

-100

-50

0

50

100

150300K smoothed fit

- 48 -

Figure 36: the fitting(red) in 0.6-5.0eV without data(blue) unsmoothed at 4K

Figure 37: the fitting(red) of smoothed data(blue) at 4K

-5 -4 -3 -2 -1 0 1 2 3 4 5-400

-300

-200

-100

0

100

200

300

-8 -6 -4 -2 0 2 4 6 8-250

-200

-150

-100

-50

0

50

100

150

2004k with smoothing

- 49 -

In summary, the successful application of line shape analysis indicates the

independent-particle frame work is applicable to explain the spectra of STO and STO

is not strongly correlated. The anomalous excitonic effects, a sign of electronic

correlations, are found in STO. Hence, STO is weakly-correlated based on the results

of my project.

- 50 -

Chapter4. Conclusion

4.1 Conclusion

Data for the dielectric function of Strontium Titanate in the 0.6eV-6.5eV range have

been presented both for T=4K and T=300K. A detailed analysis of the critical points

has been performed, and we report critical points at 3.33eV, 3.80eV, 4.17eV, 4.79eV,

4.91eV and 6.21eV. Among them, the critical point at 3.33eV is the exciton near the

lowest indirect band transition. The sharp peaks around 3.80eV and 4.79eV are the

excitonic peaks. The rest of critical points at 4.17eV, 4.91eV and 6.21eV are the direct

band transitions. The existence of excitonic effects in Strontium Titanate has been

verified.

4.2 Future work

The MATLAB function lsqcurvefit.m is very sensitive to the starting point of the

fitting and it returns the results of a 'best' fitting at a local minimum. In order to get a

global minimum and the real best fitting, more starting matrix should be applied. In

my project, I generate 60 random matrix as starting point to get a better fitting and

this process takes about four hours. With more trials or more staring , a better fitting

might be found.

Also, if time permitted, other analysis can be done to the spectroscopic ellipsometry

data at other temperature and the temperature dependence of the critical point energies

can be determined. Finally, the abrupt change of the lowest indirect band gap around

105K may be observed.

- 51 -

References

[1] Fujiwara, H., & Wiley InterScience (Online Service). (2007). Spectroscopic

ellipsometry: Principles and applications. Hoboken, NJ; Chichester, England: John

Wiley & Sons.

[2] Lautenschlager, P., Garriga, M., Logothetidis, S., & Cardona, M. (1987). Interband

critical points of GaAs and their temperature dependence. Physical Review B, 35(17),

9174-9189. doi:10.1103/PhysRevB.35.9174

[3] Blazey, K. W. (1971). Optical absorption edge of SrTiO3 around the 105-K phase

transition. Physical Review Letters, 27(3), 146-148. doi:10.1103/PhysRevLett.27.146

[4] Schafer, R. W. (2011). What is a savitzky-golay filter? [lecture notes]. IEEE Signal

Processing Magazine, 28(4), 111-117. doi:10.1109/MSP.2011.941097

[5] Dejneka, A., Aulika, I., Trepakov, V., Krepelka, J., Jastrabik, L., Hubicka, Z., &

Lynnyk, A. (2009). Spectroscopic ellipsometry applied to phase transitions in solids:

Possibilities and limitations. Optics Express, 17(16), 14322-14338.

[6]del CastilloMussot, M., & Sham, L. J. (1985). Excitonic effect in the optical

spectrum of semiconductors. Physical Review B, 31(4), 2092-2098.

doi:10.1103/PhysRevB.31.2092

[7]Lautenschlager, P., Garriga, M., Vina, L., & Cardona, M. (1987). Temperature

dependence of the dielectric function and interband critical points in silicon. Physical

Review B, 36(9), 4821-4830. doi:10.1103/PhysRevB.36.4821

[8]Persson, C., Repins, I. L., Donohue, A. L., Zhao, H. Y., To, B., Perkins, C. L., . . .

Skolan för industriell teknik och management (ITM). (2012). Dielectric function

spectra and critical-point energies of Cu2ZnSnSe4 from 0.5 to 9.0 eV. Journal of

Applied Physics, 111(3), 033506.

[9]Shokhovets, S., Köhler, K., Ambacher, O., & Gobsch, G. (2009). Observation of

fermi-edge excitons and exciton-phonon complexes in the optical response of heavily

doped n -type wurtzite GaN. Physical Review B, 79(4)

doi:10.1103/PhysRevB.79.045201

- 52 -

[10]van Benthem, K., Elsasser, C., & French, R. H. (2001). Bulk electronic structure

of SrTiO3: Experiment and theory. Journal of Applied Physics, 104(12), 6156-6164.

doi:10.1063/1.1415766

[11]Xu, W., Yang, J., Bai, W., Tang, K., Zhang, Y., & Tang, X. (2013). Oxygen

vacancy induced photoluminescence and ferromagnetism in SrTiO3 thin films by

molecular beam epitaxy. Journal of Applied Physics, 114(15), 154106-154106-6.

doi:10.1063/1.4825257

[12]Yu, P. Y., Cardona, M., & SpringerLink (Online service). (2010). Fundamentals of

semiconductors: Physics and materials properties. Berlin; London: Springer.

[13]Pope, M., & Swenberg, C. E. (1999). Electronic processes in organic crystals and

polymers. New York: Oxford University Press.

[14]Cardona, M. (1969). Modulation spectroscopy. New York: Academic Press.

[15]Piskunov, S. (2004). Bulk properties and electronic structure of SrTiO3, BaTiO3,

PbTiO3 perovskites: An ab initio HF/DFT study. Computational Materials Science,

29(2), 165-178. doi:10.1016/j.commatsci.2003.08.036

[16]M. Cardona, Optical Properties and Band Structure of SrTiO3 and BaTiO3. Phys.

Rev. 140, A651 -A655 (1965).

[17] K. v. Benthem, C. Elsasser, R. H. French, Bulk electronic structure of SrTiO3 :

Experiment and theory. J. Appl. Phys. 90, 6156 - 6164 (2001)

(doi:10.1063/1.1415766).

[18]L. Sponza, V. Veniard, F. Sottile, C. Giorgetti, L. Reining, Role of localized

electrons in electron-hole interaction: The case of SrTiO3. Phys. Rev. B 87, 235102

(2013) (doi:10.1103/PhysRevB.87.235102).

- 53 -

Appendix

A.1 The second energy derivative of esp1 and esp2 at 4K

and 300K without and with smoothing

-200

-150

-100

-50

0

50

100

d2 e1/

dE2

654321Energy(eV)

The second energy derivative of esp1 at 300k without smoothing

-200

-150

-100

-50

0

50

100

d2 e1/d

E2

654321Energy(eV)

The second energy derivative of esp1 at 300K with smoothing

- 54 -

150

100

50

0

-50

-100

d2e2/dE2

654321Energy(eV)

the second energy derivative of esp2 at 300K without smoothing

150

100

50

0

-50

-100

d2e2/dE2

654321Energy(eV)

the second derivative of esp2 at 300k with smoothing

- 55 -

-300

-200

-100

0

100

200

d2e1/dE2

654321Energy(eV)

the second energy derivative of esp1 at 4K without smoothing

-200

-100

0

100

d2e1/dE2

654321Energy(eV)

the second energy derivative of esp1 at 4K with smoothing

300

200

100

0

-100

-200

d2e2/dE2

654321Energy(eV)

the second energy deritive of esp2 at 4K without smoothing

- 56 -

200

100

0

-100

-200

d2e2/dE2

654321Energy(eV)

the second energy derivative of esp2 at 4K with smoothing

- 57 -

A.2 data used to fit in MATLAB and samples of trials

when fitting

When we start our fitting with 3 CPs, however, by plotting the data and our fittings,

-8 -6 -4 -2 0 2 4 6 8-150

-100

-50

0

50

100

150300k with smoothing

-8 -6 -4 -2 0 2 4 6 8-250

-200

-150

-100

-50

0

50

100

150

2004k with smoothing

- 58 -

we discover that it is hard to get a good fitting with 3 CPs because we have 4 peaks in

the real or the imaginary part of the second energy derivative, we cannot get 4 peaks

with only 3 critical points. A sample of the fitting using 3 CPs is shown in Figure.32.

Then, we try to fit our data with 4 CPs and the fitting with 4CPs are quite good.

Samples of fitting with 4CPs are plotted in Figure. 35 and 36. Combining Table.1 in

page 22, from which we can easily see that there are 4 direct band interaction between

0.6 and 6.5eV, hence, we will no longer increase the number of dimensional CPs in

our fitting. After fitting with 4 CPs, we perform the fitting with an extra excitonic CP

based on the previous fitting results of 4CPs. After this, more fitting with increasing

numbers of excitonic CPs are performed in order to find all possible excitonic CPs.

-8 -6 -4 -2 0 2 4 6 8-150

-100

-50

0

50

100

1503CP at 300K

- 59 -

-8 -6 -4 -2 0 2 4 6 8-150

-100

-50

0

50

100

1504CP at 300K

-8 -6 -4 -2 0 2 4 6 8-250

-200

-150

-100

-50

0

50

100

150

2004CPs at 4k with smoothing

- 60 -

A.3 Samples of MATLAB code

lsa300kS.m

data300S=[

-6.5 -87.0401

-6.48 -78.057

-6.46 -59.9936

-6.44 -39.7548

...

6.42 -7.65525

6.44 -4.46428

6.46 -1.2738

6.48 5.5683

6.5 3.4785

];

% part of the input data are not shown, and this function is meant for

showing how we input the data

f123e.m

function f = f123e(a,x )

%1+2+3+ex

%Detailed explanation goes here

a1=a(1,1);

a2=a(2,1);

a3=a(3,1);

a4=a(4,1);

e1=a(1,2);

e2=a(2,2);

e3=a(3,2);

e4=a(4,2);

g1=a(1,3);

g2=a(2,3);

g3=a(3,3);

g4=a(4,3);

- 61 -

s1=a(1,4);

s2=a(2,4);

s3=a(3,4);

s4=a(4,4);

%parameter assignment

a=-1/2*(2/pi*atan(10000*x)-1);

b=1/2*(2/pi*atan(10000*x)+1);

%weight factor

f=a.*real(3/4*a1*exp(1i*s1)*(-x-e1+g1*1i).^(-5/2)+a2*exp(1i*s2)*(-x-e

2+g2*1i).^(-2)+-1/4*a3*exp(1i*s3)*(-x-e3+g3*1i).^(-3/2)+2*a4*exp(1i*s

4)*(-x-e4+g4*1i).^(-3))+...

b.*imag(3/4*a1*exp(1i*s1)*(x-e1+g1*1i).^(-5/2)+a2*exp(1i*s2)*(x-e2+g2

*1i).^(-2)+-1/4*a3*exp(1i*s3)*(x-e3+g3*1i).^(-3/2)+2*a4*exp(1i*s4)*(x

-e4+g4*1i).^(-3));

%object function

end