COMPUTATIONAL MODELING OF AMORPHOUS SiO2 NANOPARTICLES AND

THEIR ELECTRONIC STRUCTURE CALCULATION

A THESIS IN PHYSICS

Presented to the Faculty of the University of Missouri-Kansas City in partial fulfillment of

the requirements for the degree

MASTER IN SCIENCE

by CHANDRA DHAKAL

B.Sc., Tribhuvan University, Biratnagar, Nepal, 2003

M.Sc., Tribhuvan University, Kathmandu, Nepal, 2005

Kansas City, Missouri 2015

© 2015

CHANDRA DHAKAL

ALL RIGHTS RESERVED

iii

COMPUTATIONAL MODELING OF AMORPHOUS SiO2 NANOPARTICLES AND

THEIR ELECTRONIC STRUCTURE CALCULATION

Chandra Dhakal, Candidate for the Master in Science Degree

University of Missouri-Kansas City, 2015

ABSTRACT

The spherical amorphous silica (a-SiO2) nanoparticles (NPs) are constructed from a

previous continuous random network (CRN) model of a-SiO2 with the periodic boundary.

The models of radii 12 Å, 15 Å, 18 Å, 20 Å, 22 Å, 24 Å and 25 Å are built from the CRN

structure. Then, three types of models are constructed. Type I has the surface dangling bonds

not pacified. In type II models, the dangling bonds are pacified by hydrogen atoms. In type

III models, the dangling bonds are pacified by the OH groups. These large models are used to

perform the electronic structure calculation of NPs by using the orthogonalized linear

combination of atomic orbital (OLCAO) method. The results show some trends in band gap

variation for Type I models. The trends in band gap variation for other two types are less

clear.

A series of NP models with a spherical pore in the middle of a solid NP model are

constructed and studied. Spherical pores of radii of 6 Å, 8 Å, 10 Å, 12 Å, 14 Å, 16 Å and 18

Å are introduced within the spherical model of radius 20 Å. After OLCAO calculation, it is

found that the band gap values remain constant (5 eV) up to 21.6% porosity and then

decreases with increased in porosity. The relation with thickness of the porous NP shell and

iv

the surface to volume ratio (S/V) with the calculated band gap are studied in the same

manner and will be discussed.

Key words: Amorphous silica, Continuous random network, Nanoparticles, Porous model,

Band gap, Porosity

v

APPROVAL

The faculty listed below, appointed by the Dean of the College of Arts and Sciences,

have examined a dissertation titled “Computational Modelling of a-SiO2 Nanoparticles and

their Electronic Structure Calculation” presented by Chandra Dhakal, candidate for the

Master in Science degree, and certify that in their opinion, it is worthy of acceptance.

Supervisory Committee

Wai-Yim Ching, Ph.D., Committee Chair

Department of Physics and Astronomy

Da-Ming Zhu, Ph.D.

Department of Physics and Astronomy

Paul Rulis, Ph.D.

Department of Physics and Astronomy

vi

CONTENTS

Page

ABSTRACT ............................................................................................................................ iii

LIST OF ILLUSTRATION .................................................................................................. viii

LIST OF TABLES .................................................................................................................. ix

ACKNOWLEDGEMENTS ..................................................................................................... x

CHAPTER

1. INTRODUCTION ............................................................................................................. 1

1.1 Amorphous silica and nanoparticle .............................................................................. 1

1.2 Nanoporous particles ................................................................................................... 8

1.3 Nanoparticles model .................................................................................................... 9

1.4 Nanoporous model ..................................................................................................... 15

2. THEORETICAL BACKGROUND ................................................................................. 19

2.1 Density functional theory ........................................................................................... 19

2.2 Hohenberg kohn theorem ........................................................................................... 20

2.3 Local density approximation...................................................................................... 23

3. METHOD ........................................................................................................................ 26

3.1 Orthogonalized linear combination of atomic orbitals .............................................. 26

4. RESULTS AND DISCUSSIONS .................................................................................... 33

4.1 Results ........................................................................................................................ 33

4.1.1 Model I ........................................................................................................... 34

4.1.2 Model II ......................................................................................................... 37

4.1.3 Model III ........................................................................................................ 41

vii

4.1.4 Porous model ................................................................................................. 44

4.2 Discussion .................................................................................................................. 50

5. CONCLUSION AND FUTURE WORK ........................................................................ 53

APPENDIX

A. ABBREVIATIONS ................................................................................................... 55

BIBLIOGRAPHY .................................................................................................................. 57

VITA ...................................................................................................................................... 65

viii

LIST OF ILLUSTRATIONS

Figure Page

1.1 Continuous random network of a-SiO2 with periodic boundary condition ....................... 5

1.2 Enlarge model of Continuous random network of a-SiO2 with periodic boundary

condition ............................................................................................................................ 6

1.3 a-SiO2 nanoparticle of radius 12 Å .................................................................................. 12

1.4 a-SiO2 nanoparticle of radius 15 Å .................................................................................. 13

1.5 a-SiO2 nanoparticle of radius 18 Å .................................................................................. 13

1.6 a-SiO2 nanoparticle of radius 20 Å .................................................................................. 14

1.7 a-SiO2 nanoparticle of radius 22 Å .................................................................................. 14

1.8 a-SiO2 nanoparticle of radius 24 Å .................................................................................. 15

1.9 a-SiO2 nanoparticle of radius 25 Å .................................................................................. 15

1.10 a-SiO2 nanoporous model .............................................................................................. 18

4.1 PDOS plots of model I at less accurate potential ............................................................. 35

4.2 PDOS plots of model I at more accurate potential .......................................................... 36

4.3 PDOS plots of model II at less accurate potential ........................................................... 39

4.4 PDOS plots of model II at more accurate potential ......................................................... 40

4.5 PDOS plots of model III at less accurate potential .......................................................... 42

4.6 PDOS plots of model III at more accurate potential ........................................................ 43

4.7 PDOS plots of porous models at less accurate potential.................................................. 46

4.8 PDOS plots of porous models at more accurate potential ............................................... 48

4.9 Other relations with band-gap of porous model .............................................................. 49

ix

LIST OF TABLES

Table Page

1 Introduction of nanoparticle model in tabulated form ............................................... 10

2 Introduction of nanoporous model in tabulated form ................................................ 16

x

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my academic advisor Professor Wai-

Yim Ching for his guidance, support and advice in accomplishing my research thesis. I am

very fortunate to have him as my advisor.

I am grateful to Professor Da-Ming Zhu and Professor Paul Rulis for their

cooperation, encouragement and valuable suggestion for this research and kindly for serving

in my supervisory committee.

I would like to convey my sincere gratitude to Mr. Lokendra Poudel and Mr. Chamila

Dharmawardhana for their advice and assistance to my research work. I am grateful to all

friends (Electronic structure group and computational physics group) who directly or

indirectly helped me by offering their valuable advice and comments towards improving my

research work.

xi

Dedicated to My Family

1

CHAPTER 1

INTRODUCTION

1.1 Amorphous SiO2 and Nanoparticle

Computational Physics is one of the most essential researches in the field of material

sciences, which is not possible without computers. It is the first application of modern

computers from the very beginning. Sometimes, it is considered as an intermediate between

theoretical and experimental Physics. There are different mathematical complex formulas

and the approximations in many body problems where the computational method can work

efficiently to solve the problem. The problem in the theoretical physics can be solved

computationally, which may not possible by analytical calculation. It helps to study and

implement the idea of numerical analysis to solve the problem. In order to unravel any

questions in Physics, the notions of theoretical Physics are the basics tools. The

computational Physics is also essential to construct the model of material such as

nanoparticles (NPs). It is easier to select the small number of atoms while we construct the

model, but difficult to choose the atoms if structures contain several hundred/thousands of

atoms. Therefore, the computers are the required tool for such complex task. The use of

quantum mechanics is also important in theoretical research to determine the electronic

structure calculation. It is helpful to give information about the properties of matter, hence

suitable to describe the electronic structure of NPs.

78% of the earth’s crust consists of Silicon and Oxygen. Silica (SiO2) can be found in

both amorphous and crystalline form. Crystalline silica exists in multiple forms such as -

quartz, -quartz, trydimite and cristobalite. Amorphous silica is divided into natural

2

specimen such as diatomaceous earth, opal, silica glass and artificial products. Amorphous

silica also found in living organisms such as sponges, algae. Silica is one of the most

complex and ubiquitous materials on the earth [1, 2] . With the exception of stishovite and

fibrous silica, the Silicon (Si) atoms in silica show tetrahedral coordination (SiO4) with four

Oxygen atoms surrounding a central Si atom. The amorphous silica, (a- SiO2), also possesses

SiO4 units, i.e. exhibits only short-range ordering of their atoms. Small portion of crystalline

silica is used to observe the whole crystalline form with a non-repeating pattern. It cannot be

done in amorphous silica. Some short range orderliness may exist, but no definite order

extends over a whole range of the a-SiO2 form. Generally, the crystalline silica has ordered

structure and amorphous silica has disordered structure. There are significant differences

between them for the amount and rate of contaminants absorption and desorption. Therefore,

surface ordering can decrease hydrocarbons adsorption, and surface contamination can be

removed more easily with chemical cleaning and lower temperatures [3]. The type, depth

distribution, and density of defects depend on the growth and post-growth processes

experienced by the material. The expectation of the density of unsaturated bonds at the very

surface, Oxygen vacancies in the near selvedge or even across the whole layer as well as the

mismatching (geometrical) defects is related to the amorphous domains size and distribution.

The a-SiO2 is one of the best studied amorphous materials that have broad industrial

applications. There is great literature on such topics.

Nanotechnology is the manipulation of the dimension of material into their atomic,

molecular, and supramolecular scale with their distinct properties. It is going to be a rapidly

emerging artificial stuff in material science. The Mihail Roco of the U. S. national

nanotechnology institute shows the four generations of nanotechnology. The current phase is

3

that of passive nanostructures, which are made to perform one task. The next phase will

introduce active nanostructures for multipurpose use such as actuators, drug delivery, and

sensors. The third generation is projected around 2010 and will feature the nano-systems with

thousands of interacting components. After that, the first integrated nano-system, acting

similar to the animal cells with hierarchical systems within the systems, are expected to

evolve.

Nanoparticles (NPs) are the small objects of size between 1nm and 100nm [4] that act as

a complete unit to study its physical properties and transport phenomena, however, there is

no standard size that exists in this scientific field. The materials at this scale basically show

nanostructure dependent unique physical properties (optical, electrical and magnetic) and

high chemical reactivity. It is well-known that many physicochemical properties of NPs

could affect their biological activity. The NPs size is one of the most important factors in

determining the particular biological behaviors of nanomaterials. The NPs possess specific

large surface area because of their extreme small size and the number of surface atoms

increasing exponentially. It allows a greater proportion of its atoms or molecules to be

displayed on the surface rather than the interior of the material. Therefore, the NPs exhibit

higher chemical and biological reactivity than the fine particles [5]. The NPs are both natural

and artificial. The man made NPs are i) Fullerenes and Carbon Nanotubes, ii) Metals iii)

Ceramics iv) Quantum dots and v) Polymeric [6]. The unique properties of NPs are often

caused their extremely high surface area to volume ratio [7]. The NPs modelling and their

simulation are fascinating due to their size and composition. Many researchers introduced the

NPs differently. The most recent definition is based on the surface area, where they should

have specific area greater than 60 m2/cm3 [8]. The increase in surface area determines the

4

potential number of reactive groups on their surface. This measurement reflects the crucial

importance in different properties of NPs especially in commercial and medical applications

[9].

The a-SiO2 is an inorganic material in simple binary model system. Its initial model

contains three dimensional continuous random networks (CRN) of 432 a-SiO2 molecules

[10] (1296 atoms) as shown in figure 1.1. This model has periodic boundary without any

broken bonds and hence the structure is unique. In general, each Silicon atom is bonded to

four Oxygen atoms and each Oxygen atoms are connected to again two Si atoms. Then the

Silicon and Oxygen are fully coordinated and hence there are no dangling bonds inside the

CRN model. In particular, the CRN model does not allow the defects. Hence, there is the

chance of decreasing the strain in amorphous network. Some works were done on that model

by former co-worker in UMKC [11, 12, 13, 14]. This is the basic structure of a-SiO2 for the

further research on it, such as the a-SiO2 NPs modelling. The CRN of a-SiO2 model doesn’t

have sufficient numbers of atoms as we need to build NPs model. In order to get the different

sizes of it, the size of supercell of CRN model is increased to 2⨯2⨯2 from 1⨯1⨯1 as shown

in figure 1.2. Eventually, the total number of atoms on a-SiO2 models became 10,358. To get

different sizes of a-SiO2 NPs, the data of enlarged CRN model are extracted by choosing a

point inside the structure (figure 2) and distances of each and every atom are obtained from

that point by using simple statistical technique. Then, the spherical structures of nanoparticles

of a-SiO2 of different radii are selected from the same data analysis.

The a-SiO2 NPs have been extremely important in material science and nanophysics

due to their enormous technological importance. Their modelling and simulation are

interesting because of their size and structure. The a-SiO2 NPs have attracted the various

5

Figure 1.1: Continuous random network of a-SiO2 with periodic boundary condition

6

Figure 1.2: Supercell 2⨯2⨯2 model of Continuous random network of a-SiO2 with periodic

boundary condition

scientists and researcher in both experimental and computational sciences [15, 16, 17] and

have been under intensive investigation in recent years. They exhibit the disordered

structures, which are divided into two parts i) core with structural characteristics (size

7

independent) close to that of the corresponding amorphous bulk counterparts [18] and ii)

surface showing the defects such as the pores because of the existing structure [19], higher

concentration of coordinately unsaturated sites and dangling bonds. The interior of the NPs

structure is mentioned in (i), but the surface atoms may not be fully coordinated with each

other. Due to less coordination, the atoms may separate a little to form the vacancies

including unsaturated sites. There is no any rule to determine the surface shell of a-SiO2 NPs.

It is easier to distinguish the shell from the structural point of view of the NPs. The shell can

be assumed that the atoms lying on the surface have no complete coordination to all atomic

pairs. Due to the defects, a-SiO2 NPs have more innovative significance with well-defined

applications such as catalysis [20], chemical reaction [21] and micro-electronic fabrication

[22]. Because of emerging commercialization in the realm of nanotechnology, they are used

in electronic and optoelectronic devices [23, 24], as additives to cosmetics [25], printers [26],

varnishes [27] and food [28]. Moreover, a-SiO2 NPs have been used for the host of

biomedical and biotechnological application such as cancer therapy [29], DNA transfection,

enzyme mobilization [30], gene delivery, drug release control, photoluminescence [31] and

as carriers for indomethacin in solid state dispersion. Also, the common use of silica NPs

generates various sources for potentials in human exposure. It is possible for a-SiO2 NPs to

enter inside the human body through inhalation, ingestion, dermal penetration and injection

[8]. The meticulous study of a-SiO2 NPs surfaces is crucial for addressing the more practical

applications of these ubiquitous materials. Also, the structure and properties of a-SiO2 NPs

are different from their corresponding amorphous bulk counterparts. There are number of

methods for synthesis and characterization of amorphous NPs. Also, the diffraction technique

is used to get the structural characteristics of a-SiO2 NPs, however, the computer simulation

8

is required to get the detailed information of microstructure of the amorphous NPs at atomic

level. Therefore, a-SiO2 NPs become the interesting research material for computer

simulation because of their smaller size [32]. The amorphous a-SiO2 NPs have the other

advanced potential applications in different areas of technology [33].

1.2 Nanoporous Particle

The NPs have holes of varying in size are called nanoporous particle. Usually, the

diameter of holes are less than 100 nm. The porous material are scientifically important

because of the presence of voids or space of dimension at atomic, molecular and nanometer

scales. The most interesting thing is that there is nothing inside the hole. If some interesting

molecules are put inside the pore, their interaction with the NPs surface is detected. The

nanoporous materials are divided into three types by International Union of Pure and Applied

Chemistry (IUPAC). (i) The pore of diameter less than 2 nm are micropores. (ii) Those of

diameter between 2 – 50 nm are termed as mesopores and (iii) the pore diameter greater than

50 nm are macropores. They are found in both biological systems and in natural minerals.

Nanoporous materials in nature are organic-inorganic hybrids. Naturally occurring materials

exhibit synergistic properties. In the past two decades, the field of nanoporous materials has

undergone significant developments. There has been increasing interest and research effort in

the synthesis, characterization, functionalization and designing of nanoporous material. The

main challenges in research include the fundamental understanding of structure-property

relations and tailor-design of nanostructures for specific properties and applications. As these

materials possess high specific surface areas, well-defined pore sizes, and functional sites,

they show a great diversity of applications in many industrial fields. The large number of

unique nanoporous material can be synthesized, varying in chemical composition and

9

topology. Hundreds of such materials have already been synthesized and many of others have

been computationally predicted. The most common task for nanoporous materials in nature is

to make inorganic material much lighter while preserving or improving the high structural

stability of these compounds.

1.3 Nanoparticle Models

In this work, a nearly perfect CRN model of a-SiO2 glass is used as shown in figure 1.1.

This is fully relaxed structure without any broken bonds [34]. It contains 432 a-SiO2

molecules (1296 atoms) and used as the original structure to construct the a-SiO2 NPs. In this

structure, the SiO4 tetrahedral units are linked by bridging Oxygen atoms to form an infinite

array. The above CRN of amorphous SiO2 is not sufficient to obtain the NPs model of

different sizes. The size of structure should be increased by taking supercell 2 2 2 as shown

in figure 1.2. Therefore, the number of atoms in CRN model is increased to 10,358. Here, the

first aim is to make the spherical models of a-SiO2 NPs of different radii. Hence, it is

necessary to cut the increased model in the form of spheres. To obtain such a-SiO2 NPs

model, a point of the increased model is considered at first as a center, but not requires

exactly the center. All the atoms in this increased model are at different distances from that

point. The distances of the atoms from the point are acquired by using the distance formula in

three dimensions, but it is impossible to calculate the distances of such a huge number of

atoms one by one. We have the large data base as we increased the size of model as

mentioned above. Therefore, the simple statistical approach is used to calculate the distances

of all atoms within a few seconds. Then, the atoms lying at different radii from the point are

selected inside the increased model with the help of same approach and taken out one by one

10

Table 1: Introduction of nanoparticle model in tabulated form

in the form of spheres of respective surface areas are shown in above table. Therefore, the

spherical models of a-SiO2 NPs of radii 12 Å, 15 Å, 18 Å, 20 Å, 22 Å, 24 Å and 25 Å are

built as the initial models for our calculation. These models contain the broken bonds on the

surface, either on Silicon or Oxygen or on both, called dangling bonds. As one would

expect, a more disordered surface has more surface defects to interact with contaminants

while an ordered surface with fewer dangling bonds can be more stable, adsorb fewer

hydrocarbons, and desorb them more easily (need to locate position). The dangling bonds on

the surface are saturated by H-atoms. In this process, the silanol (SiOH) and silane like

compounds (SiH, SiH2) are formed on the surface of the a-SiO2 NPs. The silane is the silicon

analogue of methane. Hydrogen in silane is more electronegative than Silicon. So, Hydrogen

has partial negative charge and Silicon has positive charge. It is the gas at normal

temperature and easily combustible in air without external ignition. Silane and silane like

Radius(Å) No. of Si No. of O No. of H Total Area(Å2) a=b=c Volume(Å3)

12 163 327 120 610 1808.64 45 91125

15 308 615 196 1119 2826 50 125000

18 532 1065 278 1875 4069.44 55 166375

20 735 1470 356 2561 5024 62 238328

22 978 1956 410 3344 6079.04 65 274625

24 1272 2535 509 4315 7234.56 65 274625

25 1443 2886 544 4873 7850 70 343000

11

compounds on the surface of NPs are used as coupling agents to adhere fibers to certain the

polymer matrix. They are also applied to couple a bio-inert layer on a titanium implant, water

repellents, masonry protection, control of graffiti [35], manufacturing semiconductor and

sealants. The Si-H bonds are used as reducing agents in organic and organometallic

chemistry [36]. The silanol (OH) group on the surface of a-SiO2 NPs is bounded via the

valence bond with Si atoms on the silica surface. The surface of a-SiO2 is oxide absorbent

and depends on the presence of silanol groups. The surface becomes hydrophilic in presence

of sufficient concentration of these silanols. The OH groups act as the centers of molecular

adsorption by interacting with absorbates capable of forming the Hydrogen bonds with the

OH groups. The adsorption property of NPs surfaces decreases by removing the hydroxyl

groups. Hence, their hydrophobic property increases. These models explained are the

preliminary one. In our improved models, there is the pure a-SiO2 NPs with a fully Oxygen

terminated surface. It means the bonds are broken only on the surface of Oxygen atoms, not

Silicon. Again, the dangling bonds created by Oxygen atoms are saturated by H-atoms and

silanols are formed on the a-SiO2 NPs surface.

To make the a-SiO2 NPs model of radius 12 Å (1.2 nm), a point is taken inside the

enlarged model as mentioned above. Then, the atoms lying at the distance of 12 Å from that

point are selected and taken out. It looks like as we cut the sphere of radius 12 Å from the

enlarged model. During the selection of atoms, the bonds are broken either on Oxygen or

Silicon or both, which takes place only on the surface. These broken bonds are called

dangling bonds. There is no broken bond inside the model. The model consists of outer shell

and inner core. The surface has some defects and dangling bonds as mentioned above. The

models with dangling bonds are unsaturated that can add more other atoms for their

12

(a) (b) (c)

Figure 1.3: a-SiO2 nanoparticle of radius 12 Å

neutrality. Then, H – atoms are added on the unsaturated model so that the dangling bonds

are occupied by these atoms as shown in figure 1.3 (a). Also, before adding H-atoms, each

Silicon atoms on the surface may not be connected with four Oxygen atoms. Therefore, the

extra O-atoms have to be added to Silicon to make the Oxygen coordinated as shown in

figure 1.3(b). The Oxygen atoms on the surface of model have the dangling bonds. This

model is also the unsaturated one. Again, H-atoms are added on models of figure 1.3 (b) in

such a way that these atoms are bonded with oxygen atoms lying on the surface (shell) of

model. Therefore, the H-atoms are connected with O- atoms not the Silicon as shown in

figure 1.3(c).

Similarly, the atoms lying within the distances of 15 Å, 18 Å, 20 Å, 22 Å, 24 Å and

25 Å are carefully selected and taken out in the form of sphere. The atoms lying inside the

spherical models are all bonded, but there are some danggling bonds on the surface of

spherical a-SiO2 NPs. These bonds are occupied after the addition of H-atoms as shown in

figure 1.3(a), 1.4(a), 1.5(a), 1.6(a) and 1.7(a). The Oxygen atoms are added on the Silicon

atoms on the surface of each model, which have the danglling bonds before. Hence, each

13

(a) (b) (c)

Figure 1.4: a-SiO2 nanoparticle of radius 15 Å

Silicon atom on the surface are connected to four Oxygen atoms, not any H-atoms as shown

in figure 1.4(b), 1.5(b), 1.6(b) and 1.7(b). The models contain dangling bonds on the

Oxygen atoms that lie on the surface. Then, the H- atoms are added on this model and the

dangling bonds are saturated by H-atoms. Then, the model is fully saturated as shown in

figure 1.4(c), 1.5(c), 1.6(c) and 1.7(c). During the saturation of structure, SiOH and SiH

compounds are formed on the a-SiO2 NPs surface as in spherical models of varying radii.

(a) (b) (c)

Figure 1.5: a-SiO2 nanoparticle of radius 18 Å

14

(a) (b) (c)

Figure 1.6: a-SiO2 nanoparticle of radius 20 Å

(a) (b) (c)

Figure 1.7: a-SiO2 nanoparticle of radius 22 Å

15

Figure 1.8: a-SiO2 nanoparticle of radius 24 Å

Figure 1.9: a-SiO2 nanoparticle of radius 25 Å

1.4 Nanoporous Model

The pores found on the a-SiO2 NPs model are the smaller holes of varying diameter.

The spherical a-SiO2 NPs model of radius 20 Å is taken as an initial structure without

16

hydrogen. The pores on the model are constructed by removing the certain number of atoms

from the model mentioned. In this process, a point inside the spherical a-SiO2 NPs model of

radius 20 Å is taken. Then distances of all the atoms inside the model from the point is

calculated. After that, the atoms at the distances of 6 Å, 8 Å, 10 Å, 12 Å, 14 Å, 16 Å and 18Å

from the given point of a-SiO2 NPs model are selected and thrown away from the model one

by one. These regions are shown by colors as in figure below. Then these distances become

the radii of the pores inside the a-SiO2 NPs model of radius 20 Å as shown in figure 1.10.

The number of atoms inside the models depends on the radii of the pores. Higher the pores

radii, lesser will be the number of atoms. Also, the atoms on the surface of porous model

contain the broken bonds or dangling bonds. The models with such types of bonds are

unsaturated. These are saturated by adding the H-atoms. The properties of models and atoms

on it are mentioned in below table 2.

Table 2: Introduction of nanoporous model in tabulated form

Radius of

sphere(A0)

Radius of

vacuum(A0)

Thick

ness

(A0)

Si

No.

O

No.

H No. Total a=b=c Volume

of sphere

in Å3

Volume

of

vacuum

in Å3

Porosity

20

10 10 641 1501 438 2580 62 33493.33 4186.67 0.125

12 8 571 1375 466 2412 62 33493.33 7234.56 0.216

14 6 485 1227 514 2226 62 33493.33 11488.21 0.343

16 4 358 1003 574 1935 62 33493.33 17148.59 0.512

18 2 204 686 555 1445 62 33493.33 24416.64 0.729

17

(a) (b) (c)

rp = 6A0 rp = 8A0 rp = 10A0

(d) (e) (f)

rp = 12A0 rp = 14A0 rp = 16A0

18

(g)

rp = 18A0

Figure 1.10: a-SiO2 nanoporous model

19

CHAPTER 2

THEORETICAL BACKGROUND

The Orthogonalized Linear Combination of Atomic Orbitals (OLCAO) [37] is mainly

based on the density functional theory (DFT). DFT is one of the approaches of first principle

calculation, which is relatively new and based upon the quantum mechanical theory from the

1920’s. Here, many electron systems are described in terms of one electron wave functions.

2.1 Density Functional Theory

DFT is used to investigate the electronic structure (principally the ground state) of

many body systems, particularly in atoms, molecules and condensed phases. The functional

is the electron density which is a function of space and time. DFT comes from the use of

functional (functions of another function) of the electron density to reduce a many-electron to

a single-particle problem. The electron density is used in DFT as the fundamental property

unlike Hartree-Fock (HF) theory [38, 39], which deals directly with the many-body wave

function. Using the electron density significantly speeds up the calculation, whereas the

many-body electronic wave function is a function of 3N variables (the coordinates of all N

atoms in the system). It was Hohenberg and Kohn who stated a theorem which tells us that

the electron density is very useful. By focusing on the electron density, it is possible to derive

an effective one-electron-type Schrödinger equation. This is not enough from the structural

view of many solids. Therefore, one electron system is converted into many electrons and the

concept of linear combination of atomic orbital (LCAO) is useful to solve the Schrödinger

equation because it tries to define the structure of solids in terms of its constituents [40].

20

Time-dependent density-functional theory (TDDFT) is an extension of the static ground

state DFT [41]. Despite the failure of DFT, its use continues to increase from the

approximations made for the exchange – correlation contributions, because of its wide

applicability and its favorable scaling with the number of atoms. In the past, the TDDFT [42,

43, 44, 45] has used to do the calculation of electronic excitations of both finite and infinite

system. Currently, it is also used to describe excited state properties of molecules with finite

systems with large numbers of atoms. A wide variety of spectroscopic techniques is being

used to characterize the electronic structure and dynamics of these systems by probing their

excited spectra. The performance of any nano-electronic device, such as a molecular junction

is dominated by its electronic excitations. Here, below is the summary of foundation of DFT

and its connection with the excited state properties and many body perturbation theories.

2.2 Hohenberg-Kohn Theorem

The DFT is based on the H-K theorems [46]. In first theorem, for a system of N

interacting (spinless) electrons, there exists an external potential V (r) (usually the Coulomb

potential of the nuclei). There is only one charge density n(r) for non-degenerate ground

state system that corresponds to a given V (r) without existence of an external magnetic field.

After this, the improvements have been made to accommodate the entire system [47]. Here,

the electron density represents the total energy and the wave function of system. The second

H-K theorem is based on the ground state energy as a minimum versus particle density

variation under normalization condition.

For many-electron system, the Hamiltonian, H with ground state wave function Ψ is

given by

21

H T U V= + + (2.1)

where T is the kinetic energy, U the electron-electron interaction, V the external potential.

The charge density n(r) is defined as

∫= nn drdrrrrrNrn ...............,,()( 2

2

21ψ (2.2)

Consider the different Hamiltonian,

'' VUTH ++= (2.3)

(V and V’ do not differ simply by a constant: V – V’ = const.), with ground state wave

function Ψ’. Suppose ground state charge densities are same. Then,

)()( 'VnVn = (2.4)

Therefore, the following inequality holds

ψψψψψψ VVHHHE −+=<= ''''''' (2.5)

Then, we have

∫ −+< drrnrVrVEE )())]()([( '' (2.6)

Here, Ψ and Ψ’ are different, being the eigenstates of different Hamiltonians. Two different

potentials can’t have same charge density because we obtained absurd result by reversing the

primed and unprimed quantities. From the consequence of the first Hohenberg and Kohn

theorem, the ground state energy, , is also uniquely determined by the ground-state charge

density. In mathematical terms, is a functional [n(r)] of n(r). We can write

∫+=++=++= drrVrnrnFVUTVUTrnE )()()]([)]([ ψψψψψψ (2.7)

where F[n(r)] is unknown but it is the functional of the electron density and [n(r)] is a

universal functional of the charge density n(r) (and not of V (r)). In this way, DFT exactly

reduces the N-body problem to the determination of a 3-dimensional function n(r) which

22

minimizes a functional [n(r)]. In addition, the TDDFT has been used to calculate the excited

states. The energy functional [n(r)] in Kohn and Sham [48] approach of electron effective

potential Veff(r) is given by

)]([)]([)]([)]([)]([ rnErnUrnErnErnE XCieeek +++= −− (2.8)

where [n(r)] is kinetic energy of the electrons, e-e [n(r)] is the electron-electron energy,

e-i[n(r)] is the electron-ion interaction potential and XC[n(r)] is the exchange-correlation

energy resulting from the Pauli exclusion principle and other factors which are not exactly

known. In practice, the kinetic energy and exchange-correlation give the moderate

quantitative agreement with the experimental data [49]. Here,

∑∫=

∇−=N

i

iik drrrrnE1

2* )()(2

1)]([ ψψ (2.9)

∫∫ −=−

'

'

' )()(

2

1)]([ drdr

rr

rnrnrnE ee

(2.10)

∫=− )()()]([ rdrnVrnU extie (2.11)

The second HK theorem tells about the minimization of the total energy through the electron

density. For a fixed number of electrons, N, and the given external potential, the minimizing

[n(r)]- N, is a method of Lagrange multipliers. Then

[ ] 0)()()]([)(

=− ∫ rdrnrnErn

µδ

δ (2.12)

)(

)]([

rn

rnE

δδµ =⇒ (2.13)

where is the Lagrange multiplier with constraint ∫ = Ndrrn )(

From equations (2.8) and (2.13), we get

23

µδ

δδ

δδ

δδ

δ =

+++ −−

)(

)]([

)(

)]([

)(

)]([

)(

)]([

rn

rnE

rn

rnU

rn

rnE

rn

rnE XCieeek (2.14)

By using effective potential eff as

µδ

δ =+ )()(

)]([rV

rn

rnEeff

k (2.15)

where,

∫ +−

+= )()()(

)()( '

'

'

rVrdrr

rnrVrV XCexteff

(2.16)

and

)(

)]([)(

rn

rnErV XC

XC δδ= (2.17)

Also, the one-electron Schrödinger’s equation is given by

iiieff rrV ψεψ =

+∇− )()(2

1 2 (2.18)

From the solutions of above equation, the ground state energy, E0, and the ground state

density 0(r) are determined. The KS approach is an exact in theoretical calculation but the

exchange-correlation functional, XC[n(r)], is unknown in reality.

2.3 Local Density Approximation

Both KS and HF equations are derived from the variational principle and are also self

–consistent equations for one electron wave function. The exchange term in HF equation

appear in the place of the exchange correlation potential of KS equation.

∑∫ =−

−

++∇−

||,

''

'

'*22

2

)()()()(

)()()(2 j

iii

jj

iH rdrrrr

rrerrVrV

m

h ψεψψψ

ψ (2.19)

24

where sum over j extends only to the states with parallel spin. Traditionally, correlation

energy is the difference between the HF and the real energy. The exchange term in HF

equation is the non-local operator – one acting on function as

∫= '' )(),()( drrrrVrV φφ (2.20)

and is quite difficult to compute.

Then,

)()())(( rrVrV x φφ = (2.21)

Also,

3/122

)](3[2

3)( rn

erVx π

π−= (2.22)

Within this approximation, the only difference between KS and HF equation is the form of

exchange correlation. As early as 1965, Kohn and Sham introduced the LDA. They

approximated exchange correlation energy with the function of local density n(r). Then, the

expression for exchange correlation energy functional is given by

∫= drrnrnrnE xcxc )()]([)]([ ε ;

)(])(

)([)]([)(

rnnxc

xcxcxc

dn

ndnnrn

rn

E=+=≡ εεµ

δδ

(2.23)

In this approximation, (n(r)) has same dependence upon the density as for the homogeneous

electron gas. The Latter is unknown except for HF level. The xc is the exchange-correlation

energy of an electron in the homogeneous electron gas of density n(r). The LDA can describe

the various ground state properties and overestimates the exchange part, but underestimates

its correlation resulting from the unexpected accuracy of Exc. The results obtained from the

LDA approximations are partially addressed the systematic error calculation [50]. Hence, it is

25

one of the most famous topics in solid state physics. It has also some limitations and different

methods are subjected for its improvement. In LDA, we generally take the local electron

density at some specific point, say r. The LDA avoids the variable electron density in real

system. Therefore, it is cleared that one can consider the rate of spatial variation of electron

density in exchange-correlation energy functional Exc. Also, highly accurate results from

Quantum Monte-Carlo techniques were found by Ceperley and Alder [51, 52] and

parameterized by Perdew and Zunger [53, 54] with a simple analytical form:

)3334.00529.11(

2846.09164.0)(

sss

xcrrr

n++

−−=ε for rs≥1

= 0960.0023.0ln0622.0ln0040.09164.0 −−++− ssss

s

rrrrr

for rs≤1 (2.24)

Here, rs is the Bohr radius and xc is in Ry. The first term is the HF exchange contribution

and the remaining terms is exchange correlation energy. Other forms of xc can also be found

in other forms of literature. All forms yield very similar results in condensed matter

calculations, which are not surprising, since all parameterizations are very similar in the

range of rs applicable for solid-state phenomena.

26

CHAPTER 3

METHODS

3.1 Orthogonalized Linear Combination of Atomic Orbitals (OLCAO)

The OLCAO package is an extension and the numerous additional modifications of

LCAO method. It is developed entirely at UMKC Electronic Structure Group (ESG) and

Computational Physics Group (CPG). It is an important and efficient for electronic properties

calculations of large complex system, both crystalline and non-crystalline substances [55, 56,

57, 58]. The OLCAO is based on the DFT of all electrons method using atomic orbitals for

the expansion of the bloch wave function. This method is appropriate for complex material

systems containing different types of elements with large number of atoms and materials

with low or no symmetry and so on. From the experience, it is possible to do fully self-

consistence calculation in amorphous structure and materials with defects and

microstructures. The details of the OLCAO method are mentioned in many past publications

especially in Ching (1990) [37] and in Ching and Rulis [59].

There were different methods to solve the Schrödinger’s equation for many electron

systems but the base of the OLCAO method can be seen back to the early days. The atomic

orbitals are very good to describe the atomic electronic configurations, and then it was

obvious procedure to use LCAO to construct the wave function for many electron systems.

The orthogonalized plane wave (OPW) method is developed by Conyers Herring [60] and it

is considered to be the first capable method of band structure calculation. The idea of OPW

method was used to improve LCAO which finally develop to modern OLCAO. It is very

27

useful in the electronic structure calculation and spectroscopic properties of solids. It is

mainly applied to large and complex systems.

In basis expansion, the radial part of atomic orbitals is expanded in the form of Gaussian

type of orbitals (GTO). The OLCAO package is strongly treated as band structure method

and it will succeed to maintain the electronic structure in terms of wave vector . The

expansion of solid state wave function Ψnk(r) in OLCAO method is given by in terms of

bloch sums biϒ(,r) as shown in following equation

Ψ = ∑ ()(, r), (3.1)

where represents nonequivalent atoms in the cell and is the quantum numbers , ! and .

The n marks the band index and is the wave vector. In the expression, the orbital quantum

number contains the principle quantum number, angular momentum quantum number and

spin quantum number.

The bloch sum bi (k, r) in above equation is expressed as linear combination of atom

centered atomic orbitals Ui(r) and given by

= "√$ ∑ %.'((r − t+ − R-)- (3.2)

where t is the position of the th atom in the cell and Rv represents the reciprocal lattice cell.

The atomic orbitals (i(r-t-RV) consist of both radial and angular parts. The radial parts are

expressed as the linear combination of GTO and spherical harmonics Y(θ,Ø) are used to

expand the angular parts. So, the equations are easier and faster to solve.

((.) = /∑ 0.1"%2134'56$0 789:(;, (3.3)

The first term of equation (3.3) is the radial part and is the linear combination of GTO and

second part is the angular part called spherical harmonics. The decaying constant, αj ranging

from αmin to αmax (0.15 to 107) and N has the common value ranging from 16 to 26. The atom

28

having the same wave function has the same set of exponentials (αj) and hence lowers the

computational cost for atoms with large number of orbitals. After the determination of

exponential set [j], the expansion coefficient Cj can be calculated by either the

normalization of eigen-vector coefficient of a single atom problem within the same DFT or

the linear fitting to atomic function in HF method. In the former method, eigen value

problem of an atom is solved self-consistently by using the Gaussian function. The ab initio

atomic calculation [61, 62] is also helpful to obtain the Cj. We need to know the basis

function of specific element and is already fixed in OLCAO method. Therefore, the

calculation of structure containing large number of atoms becomes efficient because it

reduces the number integrals to be evaluated during the process. Also, (i(r) contains the core

orbitals, the occupied valence orbitals and the other additional empty orbitals. Usually, the

minimal basis (MB) is sufficient for the calculation of amorphous structure with large

number of atoms which consists of core orbitals and the occupied or unoccupied orbitals in

the valence shell of the atom. They are used for the calculation of effective charge (Q*) and

bond order (BO) by using Mulliken scheme, but we didn’t do this part in our calculation. In

this method, the effective charge of an atoms and bond order measure the relative strength of

bond between two atoms. Also, the full basis (FB) is much more sufficient for the precise

results. FB has one more unoccupied orbital and is applied to the smaller system in general. It

is used for band structure and density of states (DOS) calculations. The calculations are

significant in presence of more unoccupied orbitals in the basis set. Hence, an additional shell

of the excited atomic basis is added to FB to obtain the extended basis (EB) in spectral

calculations, where unoccupied states at high energy are taken into account. Therefore, the

orthogonalisation of orbitals and the use of basis sets make the OLCAO package very

29

efficient and versatile for electronic and spectroscopic calculations. There are many choices

to select the atomic basis set in order to solve the problem with great accuracy within the

reasonable time interval.

In OLCAO method, the charge density (r) and the one-electron potential Vcry(r) are

also called the atom centered Gaussian function. Therefore, the charge density is

<(.) = ∑ ∑ =0%1>4'5(r − t?)0@"A (3.4)

where B(r) is total charge density at one point due to all atoms in the system. It is significant

and closes to true charge density.

CDE9(r) = ∑ / FG(') %1H'5 − ∑ I0%1>4'5$0@" 7 (r − t+) (3.5)

KL(.) = ∑ ∑ M0%1>4'5(r − t+)$0@" (3.6)

<(.) = ∑ 2L(.) + K(.)6(r − t+) (3.7)

where OB is the total potential at one point because of the all atoms in the system. OP is the

coulomb potential and Ox is the exchange-correlation potential. The coefficients βj are

predefined in database like αj for each atom and the βj values ranging from minimum to

maximum and numbers are arranged in geometric series between them. The term, Q is the

mass number of the atom at the site. All these terms depend on Z and on system used. The

coefficients Bj, Dj and Fj are updated at each self-consistent calculation until the difference of

energy eigen values between the two successive steps drops down to the specified minimum

value, 10-4 /10-5 eV. This convergence is important and easily obtained for non-conducting

system and their band gap is used to differentiate the occupied and unoccupied states. Since

the centered potential function is transferrable, the self-consistent potential obtained from

simple calculations can be used to calculate the properties of complex systems.

30

In OLCAO method, the one of important properties is core-orthogonalisation, where

the core orbitals are removed from the following secular equations

RS,0T(k) − V,0T(k)(k)R = 0 (3.8)

where, Hiγ, jδ (k) and Siγ, jδ (k) are Hamiltonian and overlap matrix respectively and the

overlap matrix is given by

V,0T(k) = X(k)R0T(k)Y (3.9)

Suppose the bloch sum with core orbitals, bcjβ(k,r) and the valence orbitals, bZ α(k,r). Hence,

the orthogonalized valence bloch sum bZ’ α (k,r) is given by

3[\(k, r) = 3[ + ∑ 030, 3C (k, r) (3.10)

where

03 = −X0C (k, r)R3[ (k, r)Y

and

03∗ = −X3[ (k, r)R0>C (k, r)Y are expansion coefficients in equation (3.10).

Also, the core orthogonalisation can be expressed as

X0>C (k, r)R3[\(k, r)Y = ^0>[

\(k, r)R3C (k, r)_ = 0 (3.11)

In OLCAO method, the orbitals of energy less than the oxygen 2s orbital are called

core orbitals. But there is no any separate method to distinguish between the core and valence

orbitals. In this process, the non-diagonal elements of Hamiltonian and overlap matrix

disappear and final matrix consists of the blocks of core and valence orbitals. Therefore,

these parts are solved one by one and reduce the dimensions of secular equations. Finally,

OLCAO method becomes very efficient for large and complex systems.

31

The band structure calculation has played main role during the study of electronic

properties of materials [63]. The OLCAO method is used to calculate the electronic

properties such as band structure, total density of states (TDOS) and partial density of states

(PDOS), effective charge and bond order (BO) and the spectroscopic properties. The band

structure is the plot of energy eigen values as a function of k points in the reciprocal space.

The k points are taken into account along the high symmetry points of the crystal. The

material property like metal, semiconductor and insulator is determined according to the

values of band gap (the space between the topmost valence band and bottom of the

conduction band).

The DOS is the number of states available for electron to occupy at each energy level

in the unit cell. The density of states, (), is given by

( ) ∫Ω=

BZ

dkdE

dEG

32)(

π (3.12)

( ) ∫ ∇Ω=

E

dS32π

where ` is the volume of the unit cell and integral is over the constant energy surface in

Brillion Zone (BZ). From the DOS, the PDOS of atoms and orbitals are determined. In

OLCAO method, the splitting of the TDOS into PDOS of different atomic or orbital

components is natural because of the bloch function. The PDOS is the crucial quantity that

provides the enough about the interactions between atoms or orbitals. The alignments of the

peaks found in the PDOS spectra are due to their interactions. The k is no longer meaningful

for an amorphous material without long range order. Therefore, there is the lacking of the

band structure concept. In our calculation, the models are sufficiently large with periodic

boundary condition and the effects of periodicity can be neglected. The BZ becomes small

32

and we can solve the secular equation at one k-point.

33

CHAPTER 4

RESULTS AND DISCUSSION

4.1 Results

The electronic structure calculation of a-SiO2 NPs is done by using the OLCAO

method. It can also solve the electronic structure calculation of systems containing a large

number of atoms with a higher level of accuracy [59]. Therefore, we can use the OLCAO

package for studying the interaction at the atomistic level of NPs containing a higher number

of atoms. The electronic structure calculation of NPs is done at both less accurate and more

accurate potentials. At less accurate potentials, there is no distinction among the atoms

(Silicon and Oxygen) on the surface and core inside, but there are different types of Silicon,

Oxygen and Hydrogen at more accurate potentials. The Si1 is the Silicon atoms lying inside

the core. The Si2 atoms are concentrated beneath the surface and Si3 atoms are located on the

surface of model. Similarly, O1 and O2 are the Oxygen atoms lying on the surface and core

inside. H1 and H2 are the Hydrogen atoms attached to Silicon and Oxygen on the surface.

Here, the calculated TDOS along with the PDOS of each model are shown in figures below.

The TDOS features of all a-SiO2 NPs models are similar, but there are some differences in

the top of the occupied states close to the top of the valence band (TVB) and the unoccupied

region close to the bottom of the conduction band (BCB). Now, the TDOS of each a-SiO2

NPs model is resolved into atom-resolved PDOS. The upper valence band (UVB) of all a-

SiO2 NPs models come mainly from Oxygen atom and unoccupied CB from Silicon.

34

4.1.1 Model I

The band gap values of a-SiO2 NPs model are calculated by using the OLCAO

package. Their values are 2.3 eV, 4 eV, 2 eV and 1.5 eV for the model of radii 12 Å, 15 Å, 18

Å and 20 Å respectively. These values depend on the methodologies and potentials used. For

a-SiO2 NPs of radii 12 Å, 15 Å, 18 Å and 20 Å, the sharp peaks are found between the energy

ranges of -17.0 eV to 19.0 eV in the lower valence band (LVB). By careful observation, it is

found that these peaks in a-SiO2 NPs models are coming from Oxygen atoms. In lower level

of the upper valence band (UVB), there are two broad and one sharp peak found on all a-

SiO2 NPs model between the energy ranges of -5.0 eV to -10.0 eV. The broad peaks regions

are slightly different, depending upon the model used. These peaks are originating from

Silicon atom. Also, the TDOS has two broad peaks in the upper level of the UVB (0 to -4.5

eV) of all a-SiO2 NPs models. By analyzing the DOS, it is found that the Oxygen atoms are

responsible to develop the peaks in this region. The unoccupied conduction band of all a-

SiO2 NPs model have the initial increases in energy, become maximum and then decreases as

-20 -15 -10 -5 0 5 10 15 20

0

27

54

0

120

240

360

0

52

104

156

0

120

240

360

480

Energy(eV)

H

O

Si

PD

OS

/eV

Ce

ll

TotalR = 12A0

-20 -15 -10 -5 0 5 10 15 20

0

34

68

1020

200

400

600

0

105

210

315

0

220

440

660

Energy(eV)

H

O

Si

R = 15A0

PD

OS

/e

V C

ell

Total

35

Figure 4.1: PDOS plots of model I at less accurate potential

shown in figure 4.1. This part of TDOS is obtained from Silicon atoms in all a-SiO2 NPs

model. The TDOS of a-SiO2 NPs are comparable with the TDOS of α-SiO2 and a-SiO2 [29].

The peaks found in a-SiO2 and a-SiO2 NPs are similar in all energy range of TDOS including

BCB. The peaks found in TDOS of α-SiO2 are relatively sharper than peaks in a-SiO2 NPs.

The gap states are removed by using more accurate potential (level 1), which are

appeared between the VB and CB in presence of less accurate (level 0) potential. Because of

the larger dimension of model of radius 20 Å, it is difficult to perform the electronic structure

calculation by using the recently used version of OLCAO in terms of the more accurate

potential. Also, the TDOS of all a-SiO2 NPs models are similar to each other as shown in

figure 4.2. The peaks are found to be smoother than in previous potential. The band gap

values for a-SiO2 NPs models of radii 12 Å, 15 Å and 18 Å are 4 eV, 5 eV and 4.2 eV

respectively. There is sharp peak found between the energy of -17 eV to -20 eV in the LVB

-20 -15 -10 -5 0 5 10 15 20

0

48

96

144

0

531

1062

1593

0

180

360

540

0

720

1440

2160

Energy(eV)

H

O

Si

R = 18A0

PD

OS

/eV

Cell

Total

-20 -15 -10 -5 0 5 10 15 200

33

66

99

132

0

275

550

825

1100

0

155

310

465

620

0

402

804

1206

Energy (eV)

H

O

Si

R = 20A0

PD

OS

/eV

Cell

Total

36

of each model followed by the broad peaks. These peaks are coming from the O1 atoms in a-

SiO2 NPs mod els of radii 15 Å and 18 Å, but it is from the O2 for 12 Å. In lower level of

Figure 4.2: PDOS plots of model I at more accurate potential

-20 -15 -10 -5 0 5 10 15 20

0

26

52

0

17

34

0

150

3000

43

86

0

3

0

7

14

0

18

36

0

250

500

Energy (eV)

H2

H1

O2

O1

Si3

Si2

Si1

R = 12A0

PD

OS

/e

V C

ell

Total

-20 -15 -10 -5 0 5 10 15 20

0

35

700

35

700

75

150

0

275

5500

14

280

25

500

120

240

0

400

800

Energy(eV)

H2

H1

O2

O1

Si3

Si2

Si1

R = 15A0

PD

OS

/eV

Cell

Total

-20 -15 -10 -5 0 5 10 15 20

0

60

120

0

41

82

0

127

254

0

550

1100

0

17

340

35

70

0

220

440

0

800

1600

Energy (eV)

H2

H1

O2

O1

Si3

Si2

Si1

R = 18A0

PD

OS

/eV

Cell

Total

37

UVB, a sharp peak is noticed between the energy ranges of -5 eV to -10 eV. These peaks are

formed because of the O1 atoms for model of radii 15 Å and 18 Å, but O2 atoms for the model

of radius 12 Å. Also, two broad peaks are seen between the energy ranges of -1 eV to -5 eV

in the upper level of UVB of all a-SiO2 NPs model. The roughness of peak is going to

increase in the model of 12 Å radius than other two. Like others, these peaks in the model of

radii 15 Å and 18 Å are due to the O1 and O2 atoms for the remaining model. The sharp peak

is not seen in unoccupied CB. The energy in CB increases at first and decreases slowly at last

for all a-SiO2 NPs models. The origin of CB up to the maximum energy is from the Si1 for 15

Å and 18 Å and the remaining part of it from the other atoms. The unoccupied CB of 12 Å is

somehow different and is not clearly identified.

4.1.2 Model II

The TDOS and PDOS of a-SiO2 NPs models are in the same energy range as that of

model I. In Oxygen terminated a-SiO2 NPs model, the numbers of sharp peaks along with

some gap are observed between the energy ranges of -14.5 eV to -22 eV in the LVB. These

are coming from the Oxygen atoms for all a-SiO2 NPs models. A sharp peak is obtained

between the energy ranges of -8 eV to -10 eV in the lower level of UVB of all a-SiO2 NPs

model. Also, the multiple sharp peaks are noticed in the upper level of UVB of all NPs

model. The peaks in these two regions are due to the Oxygen atoms. Also, some states are

going to be seen towards the fermi level from upper level of UVB. There is no sharp peak

found in the unoccupied CB and found to be smoother for NPs model of radius 20 Å than

other models. The nature of the increment of energy in unoccupied conduction band is

coming from the Silicon. The band gap values for a-SiO2 NPs model of radii 12 Å, 15 Å, 18

38

Å and 20 Å are 2.5 eV, 2 eV, 1 eV and 1 eV respectively. Hence, band gap values are found

to be decreased with increase in radius of models. Few gap states are found in 18 Å and 20 Å

as shown in figure 4.3.

The peaks in O-terminated model are found to be different at more accurate

potentials. Two sharp peaks are seen in the LVB of NPs model of radii 12 Å and 15 Å

between the energy ranges of -15.5 eV to -20.5 eV, but there is a sharp peak within that

energy range in the model of radius 18 Å. There exist other smaller sharp peaks with some

-20 -15 -10 -5 0 5 10 15 20

0

128

256

384

0

62

124

186

0

160

320

480

Energy (eV)

O

Si

R = 12A0

PD

OS

/e

V C

ell

Total

-20 -15 -10 -5 0 5 10 15 20

0

196

392

588

784

0

92

184

276

368

0

215

430

645

860

Energy (eV)

O

Si

R = 15A0

PD

OS

/e

V C

ell

Total

39

Figure 4.3: PDOS plots of model II at less accurate potential

gap. These peaks in 12 Å are coming from the O1 and O2, but it is from the O1 and O3 in 15 Å

and 18 Å. There is a sharp peak in lower level of UVB of all NPs models between the energy

ranges of -5 eV to -10 eV. The peak in model of radius 12 Å is originating from the O2. In the

models of radii 15 Å and 18 Å, the peaks in the lower level of UVB are coming from the O1.

Also, three peaks are seen in the upper level of UVB of NPs of radii 12 Å and 15 Å between

the energy ranges of 0 eV to -5 eV. The peaks found in 18 Å are different in the same energy

ranges. The peak in this region of the model of radius 12 Å is originating from the O1 and O2.

The peaks are developed from the O1 and O3 for NPs model of radii 15 Å and 18 Å. In

unoccupied CB, the energy increases at first and gain the maximum value. After that, energy

decreases in all a-SiO2 NPs model. The peaks in these regions are not smooth and they are

coming from the Si1. The band gap of a-SiO2 NPs of radius 12 Å, 15 Å and 18 Å are 4.5 eV,

4.5 eV and 1.8 eV. By increasing potentials, the band gap values are found to be increased

-20 -15 -10 -5 0 5 10 15 20

0

425

850

1275

0

180

360

540

0

495

990

1485

Energy (eV)

O

Si

R = 18A0

PD

OS

/eV

Ce

ll

Total

-20 -15 -10 -5 0 5 10 15 20

0

620

1240

1860

0

252

504

756

0

740

1480

2220

Energy (eV)

O

Si

R = 20A0

PD

OS

/e

V C

ell

Total

40

Figure 4.4: PDOS plots of model II at more accurate potential

and some gap states are still present in model of radius 18 Å as shown in figure 4.4. Also, the

results in this potential are found to be better than the previous results.

-20 -15 -10 -5 0 5 10 15 20

0

3

6

0

116

232

0

75

150

0

2

4

0

15

30

0

27

54

0190380570

Energy (eV)

O3

O2

O1

Si3

Si2

Si1

R = 12A0

PD

OS

/eV

Cell

Total

-20 -15 -10 -5 0 5 10 15 20

0

195

390

0

120

240

0

230

460

0

22

44

0

30

60

0

120

240

0

388

776

Energy (eV)

O3

O2

O1

Si3

Si2

Si1

R = 15A0

PD

OS

/e

V C

ell

Total

-20 -15 -10 -5 0 5 10 15 20

0

335

670

0

123

246

0

534

1068

0

11

22

0

50

100

0

230

460

0

682

1364

Energy (eV)

O3

O2

O1

Si3

Si2

Si1

R = 18A0

PD

OS

/e

V C

ell

Total

41

4.1.3 Model III

The electronic structure calculation of the model of radius 20 Å is not included

because of its larger dimension. One sharp peak and other broad peaks are observed between

the energy ranges of -17 eV to -22 eV in the LVB of a-SiO2 NPs model of radii 12 Å, 15 Å

and 18 Å. In all models, the Oxygen atoms are responsible for these peaks. In lower level of

UVB, there is also a sharp peak together with other broad peaks between the energy ranges

of -5 eV to -10 eV in all models. Similarly, two peaks are seen between the energy ranges of

0 eV to -5 eV in the upper level of UVB. These peaks (the lower level of UVB and the upper

level of UVB) are coming from Oxygen. The nature of energy in unoccupied CB is found to

be similar in previous model as shown in figure 4.5. The sharpness of peak is not higher than

other region in unoccupied CB. The peaks in CB are originating from Silicon atoms. The

peaks are smoother in model of radius 18 Å than other two. The band gap values of a-SiO2

NPs models of radii 12 Å, 15 Å and 18 Å are 3.5 eV, 3 eV and 4 eV respectively. Here, some

gap states are found.

At more accurate potential, two sharp peaks are seen between the energy ranges of -

15.5 eV to -20.5 eV in a-SiO2 NPs model of radii 12 Å, 15 Å and 18 Å in LVB and they are

originated from O1 and O2. Also, a sharp peak is observed between the energy ranges of -6

eV to -8 eV in lower level of UVB in model of radii 12 Å, 15 Å and 18 Å. The peaks in model

of radii 15 Å and 18 Å are coming from O1, but in model of radius12 Å, it is from O2. Some

broad peaks are noticed in the upper level of UVB of all a-SiO2 NPs model between the

energy ranges of 0 eV to -5 eV. These are originated from O1 and O2. In unoccupied CB, the

energy increases first and decreases slowly in all a-SiO2 NPs models. The band gap values

are 3 eV, 3 eV and 3.75 eV for a-SiO2 NPs models of radii 12 Å, 15 Å and 18 Å as shown in

42

Figure 4.5: PDOS plots of model III at less accurate potential

-20 -15 -10 -5 0 5 10 15 20

0

28

56

84

0

160

320

480

0

58

116

174

0

205

410

615

Energy (eV)

H

O

Si

R = 12A0

PD

OS

/eV

Cell

Total

-20 -15 -10 -5 0 5 10 15 20

0

40

80

120

0

210

420

630

0

102

204

306

0

280

560

840

Energy(eV)

H

O

Si

PD

OS

/eV

Cell

TotalR = 15A0

-20 -15 -10 -5 0 5 10 15 20

0

56

112

168

0

530

1060

1590

0

184

368

552

0

670

1340

2010

Energy (eV)

H

O

Si

R = 18A0

PD

OS

/eV

Cell

Total

43

Figure 4.6: PDOS plots of model III at more accurate potential

figure 4.6. The unoccupied conduction bands are originated from Si1 in all models.

-20 -15 -10 -5 0 5 10 15 20

0

38

760

126

252

0

97

194

0

1

2

0

10

20

0

24

48

0

275

550

Energy (eV)

H

O2

O1

Si3

Si2

Si1

R = 12A0

PD

OS

/e

V C

ell

Total

-20 -15 -10 -5 0 5 10 15 20

0

60

120

0

155

310

0

270

540

0

17

34

0

26

52

0

114

228

0

400

800

Energy (eV)

H

O2

O1

Si3

Si2

Si1

R = 15A0

PD

OS

/eV

Cell

Total

-20 -15 -10 -5 0 5 10 15 20

0

80

160

0

202

404

0

695

1390

0

3

6

0

35

70

0

215

430

0

802

1604

Energy (eV)

H

O2

O1

Si3

Si2

Si1

R = 18A0

PD

OS

/e

V C

ell

Total

44

4.1.4 Porous Model

The porosity of the model is the ratio of the volume of pore to the effective volume of

NP model. It is a concept related to texture and refers to the pore space in a material. The

pore of different radii is created inside the spherical model of radius 20 Å. The porosity of the

models of pore radii 6 Å, 8 Å, 10 Å, 12 Å, 14 Å, 16 Å and 18 Å are 0.027, 0.064, 0.125, 0.216,

0.343, 0.512 and 0.729 respectively. Therefore, the defect increases with increase in the size

of the pores inside the NPs model. The TDOS of porous models are similar to each other.

There is a sharp peak found in the LVB (-17 eV to 21 eV) of porous models of pore radii 6 Å,

8 Å, 10 Å, 12 Å and 14 Å but it is slightly different in the pore radius 16 Å. These peaks in all

models are coming from Oxygen atoms. Also, a sharp peak along with other smaller broad

peaks are seen in the lower level of the UVB (-5 eV to -10 eV) of model of pore radii 6 Å, 8

Å, 10 Å and 12 Å, but broad peaks are going to be pronounced in the model of pore radii 14 Å

and 16 Å. The Oxygen atoms are responsible to generate the peaks in this region. There are

also some gap states between LVB and the lower level of UVB. These gap states are due to

the Silicon. Two distinct broad peaks are seen in the upper level of the UVB (0 eV to -5 eV)

in the porous model of radii 6 Å, 8 Å and 10 Å. There is a peak in the pore radii 12 Å and 14

Å in the same region. The multiple sharp peaks are observed in the upper level of UVB of

the model of pore radius 16 Å. These peaks in upper level of UVB are due to the Oxygen

atoms. The gap states are seen inside the band gap. The CB of the models of pore radii 6 Å, 8

Å and 10 Å are smoother than the radii 12 Å, 14 Å, 16 Å and 18 Å as shown in figure 4.7. The

variation of energy in CB is similar to the previous result. It is seen that the unoccupied CB

of models originate from Silicon.

45

-20 -15 -10 -5 0 5 10 15 20

0

65

130

195

0

465

930

1395

0

250

500

750

0

590

1180

1770

Energy(eV)

H

O

Si

Rp = 6A

0

PD

OS

/e

V C

ell

Total

-20 -15 -10 -5 0 5 10 15 20

0

67

134

201

0

470

940

1410

0

230

460

690

0

590

1180

1770

Energy(eV)

H

O

Si

Rp = 8A

0

PD

OS

/e

V C

ell

Total

-20 -15 -10 -5 0 5 10 15 20

0

71

142

213

0

385

770

1155

0

210

420

630

0

530

1060

1590

Energy(eV)

H

O

Si

Rp = 10A

0

PD

OS

/e

V

Ce

ll

Total

-20 -15 -10 -5 0 5 10 15 20

0

71

142

213

0

370

740

1110

0

184

368

552

0

490

980

1470

Energy(eV)

H

O

Si

Rp = 12A

0

PD

OS

/e

V C

ell

Total

46

Figure 4.7: PDOS plots of porous models at less accurate potential

Now, more accurate potential is used in the porous model. There is a peak between

the energy ranges of -17 eV to -20 eV in LVB of all porous models, but the peaks in this

region are different for porous model of the pore radii 16 Å and 18 Å. These peaks are

coming from O1 as shown in figure 4.8. The peaks found in lower level of UVB (-5 eV to -

10 eV) are similar to the model of pore radii 6 Å, 8 Å, 10 Å, 12 Å and 14 Å, however, it is

different in the model of pore radii 16A0 and 18A0. Similarly, we can see the similar peaks in

the upper level of UVB (0 eV to -5 eV) for the model of pore radii 6 Å, 8 Å and 10 Å, but the

peaks are completely different for other models as shown in figure 4.8. The peaks in these

two regions are originating from O1 for all porous models except the case of model of pore

radius 18 Å. The energy in unoccupied conduction band regions increases at first and then

decreases. The energy gap of models of pore radii 6 Å, 8 Å, 10 Å, 12 Å, 14 Å, 16 Å and 18 Å

are 5 eV, 5 eV, 5 eV, 5 eV, 4.8 eV, 4.6 eV and 4 eV respectively. Therefore, the band gap

-20 -15 -10 -5 0 5 10 15 20

0

65

130

195

260

0

255

510

765

0

151

302

453

0

360

720

1080

Energy(eV)

H

O

Si

Rp = 14A

0

PD

OS

/e

V C

ell

Total

-20 -15 -10 -5 0 5 10 15 20

0

71

142

213

284

0

205

410

615

0

81

162

243

324

0

290

580

870

Energy (eV)

H

O

Si

Rp = 16A

0

PD

OS

/e

V C

ell

Total

47

can be seen at this potential. Also, the roughness of peaks is increased by increase in the pore

radius as shown in figure 4.8.

-20 -15 -10 -5 0 5 10 15 20

078

1560

65130

0150300

0600

12000

1020

052

1040

400800

01632

0925

1850

Energy (eV)

H2

H1

O2

O1

Si4

Si3

Si2

Si1

Rp=6A

0

PD

OS

/eV

Cell

Total

-20 -15 -10 -5 0 5 10 15 20

084

1680

63126

0150300

0590

118008

160

56112

0352704

01938

0835

1670

Energy (eV)

H2

H1

O2

O1

Si4

Si3

Si2

Si1

Rp=8A

0

PD

OS

/e

V C

ell

Total

-20 -15 -10 -5 0 5 10 15 20

087

1740

73146

0162324

0522

104409

180

295590

055

1100

1938

0835

1670

Energy (eV)

H2

H1

O2

O1

Si4

Si3

Si2

Si1

Rp=10A

0

PD

OS

/e

V C

ell

Total

-20 -15 -10 -5 0 5 10 15 20

088

1760240

77154

0174348

0427854

010200

230460

055

1100

20400

60012001800

Energy (eV)

H3

H2

H1

O2

O1

Si4

Si3

Si2

Si1

Rp=12A

0

PD

OS

/e

V C

ell

Total

48

Figure 4.8: PDOS plots of porous models at more accurate potential

-20 -15 -10 -5 0 5 10 15 20

0106212

0120

94188

0188376

0335670

010200

186372

068

1360

22440

6881376

Energy (eV)

H3

H2

H1

O2

O1

Si4

Si3

Si2

Si1

Rp=14A

0

PD

OS

/e

V C

ell

Total

-20 -15 -10 -5 0 5 10 15 20

01462920.0

2.5

0.0

2.7

0104208

0205410

0190380

01326

0104208

079

1580

2958

0688

1376

Energy (eV)

H4

H3

H2

H1

O2

O1

Si4

Si3

Si2

Si1

PD

OS

/e

V C

ell

TotalRp=16A

0

-20 -15 -10 -5 0 5 10 15 20

0

118

236

0

64

1280

148

296

0

80

160

0

13

26

0

38

76

0

48

96

0255510765

Energy (eV)

H2

H1

O2

O1

Si3

Si2

Si1

Rp=18A

0

PD

OS

/e

V C

ell

Total

49

(a) (b)

(c)

Figure 4.9: Other relations with band-gap of porous model

It is already mentioned that the porosity of the models of pore radii 6 Å, 8 Å, 10 Å, 12 Å,

14 Å, 16 Å and 18 Å are 0.027, 0.064, 0.125, 0.216, 0.343, 0.512 and 0.729 respectively.

There were also a lot of experimental works going on SiO2 on the basis of their thickness. In

selected SiO2 layers, their thickness is found to be a minor importance in their band gap. The

band is increased strongly in presence of Hydrogen atom on the surface of SiO2. Therefore,

the presence of Hydrogen enforces the confinement effect for band gap [64]. In figure 4.9

50

(a), the band gap obtained from OLCAO calculation is initially constant. Hence, band gap

value remains unchanged up to porosity 21.6%. If we increase the diameter of pore, its

porosity is going to be increased. At that time, the band gap value decreases by increasing the

porosity. The thickness of nanoporous material is also crucial for the OLCAO calculation.

The band gap value of nanoporous increases with increase in its thickness. When thickness

approaches the 40% of model, the band-gap values does not change as shown in figure 4.9

(b). The surface to volume (S/V) ratio is other physical parameters in the study of

nanoporous materials that increases with decrease in pore radius. In OLCAO calculation, the

band gap value increases with increase in S/V initially as shown in figure 4.9 (c). The band

gap value remains unchanged from S/V value corresponding to 60% pore size.

4.2 Discussion

The a-SiO2 NPs models are built from CRN model of a-SiO2 with periodic boundary.

To get the spherical a-SiO2 NPs, it is not possible to cut the part of the structure in the form

of sphere directly from the model. Hence, a point inside the model has to select and the

calculations of the distances of all the atoms from the point are mandatory. Three a-SiO2

models are constructed; model I, model II and model III. In the first model, there is dangling

bonds in the atoms that lie on the surface (shell). These bonds are saturated by H-atoms.

Their electronic structure calculation is done by using the OLCAO package. The calculated

band gap values of a-SiO2 NPs models of radii 12 Å, 15 Å, 18 Å and 20 Å are 2.3 eV, 4 eV, 2

eV and 1.5 eV respectively. It shows that the electronic interaction increases with increase in

the size of the model. Also, by increased in the type of potentials, their band gap values are

found to be 4 eV, 5 eV and 4.2 eV for model of radii 12 Å, 15 Å and 18 Å respectively.

Therefore, the electronic interactions are found to be higher in model of radius 12 Å. In

51

model II, all the Silicon atoms on the surface of a-SiO2 NPs model are bonded to four

Oxygen atoms. From OLCAO calculation, it is found that the band gap values for a-SiO2

NPs of radii 12 Å, 15 Å, 18 Å and 20 Å are 2.5 eV, 2 eV, 1 eV and 1 eV respectively. The

band gap values decreases with increase in size of NPs model. Therefore, the electron

interactions are higher in model of radii 18 Å and 20 Å. The peaks in VB regions are sharper

than the VB of model I and CB are comparable. By increase in type of potentials, their band

gap values become 4.5 eV, 4.5 eV and 1.8 eV for the models of radii 12 Å, 15 Å and 18 Å. It

is not easy to get the trends from these three data. It shows that the electron interactions are

higher in the model of radius 18 Å. The structures in model II are saturated by H-atoms to get

the model III. The band gap values of a-SiO2 NPs models of radii 12 Å, 15 Å and 18 Å are 3.5

eV, 3 eV and 4 eV with some gap states. The electron interactions are higher in model of

radius 15 Å. With increase in potential, these values become 3 eV, 3 eV and 3.75 eV. There is

the possibility of getting the trends of band gap values at higher potentials. Hence, the

electron interactions are found to be higher for smaller models. The band gap values of

original sample of a-SiO2 is 5.8 eV [12] but the electronic properties calculation of a-SiO2

nanoparticles are not found by other researchers yet. Some molecular dynamics (MD) a

calculation on it is found by Vo Van Hoang [32], but the calculations are different than here.

The porous a-SiO2 NPs are also made and their porosities are 0.027, 0.064, 0.125, 0.216,

0.343, 0.512 and 0.729 for the pore of radii of 6 Å, 8 Å, 10 Å, 12 Å, 14 Å, 16 Å and 18 Å

respectively. From OLCAO calculation, the gap states are seen between VB and CB. These

gap states are removed by increasing the type of potentials of atoms. Hence, the band gap of

5 eV, 5 eV, 5 eV, 4.75 eV, 4.5 eV and 4 eV are obtained for pore of radii 8 Å, 10 Å, 12 Å, 14

Å, 16 Å and 18 Å. Their band-gap values remains constant up to 21.6% porosity. After that,

52

their band gap decreases with increase in porosities. The electron interactions are equal for

certain value of the porosity. After that, the interactions increase with increase in their

porosity. Also, their band-gap increases with increase in thickness of porous model and

finally, there is no effect of pore thickness on band-gap. The band-gap increases with

increase in (S/V) and finally the band-gap becomes constant. The change in their surfaces by

making the pores plays the major role in their properties.

53

CHAPTER 5

CONCLUSION AND FUTURE WORK

The a-SiO2 NPs models of different sizes are built and perform their electronic

structure calculation. From the result, the number of atoms in a-SiO2 increases with increase

in their size, which is obvious. The PDOS calculation is crucial in electronic structure

calculations. The trends in band gap are found in model II but it is hard to get trends in other

models at lower potential. The electron interactions are higher for the models with smaller

band gap. At higher potentials, there is the lacking of data because of the restriction of

dimension in OLCAO calculation. In porous model, their porosity increases by increasing the

radii. The most important thing in porous model is about nothingness inside the pore, which

is crucial in many applications. The band-gap values remains constant up to 21.6% porosity.

After that, their band-gap decreases with increase in their porosity and hence electron

interactions increases. Also, the band-gap value increases with increase in thickness and

(S/V). The gap states present in NPs model are removed by increasing the type of potentials.

Therefore, the band gap values normally increases. In SiO2 (amorphous, crystalline and NPS)

model, each silicon is attached to four oxygen and the arrangement of atoms inside the

models are similar. Therefore, the TDOS of a-SiO2 NPs are found to be comparable with the

TDOS of a-SiO2 and p-SiO2.

The a-SiO2 NPs contain pore space, where some important molecules or atoms can be

added. The OLCAO is useful to detect the interaction between the molecules and the NPs

surface. Here, the water molecules are inserted inside the pore and the OLCAO jobs are not

completed yet. The alcohol, gold particle, DNA and proteins can be also inserted inside the

54

pores. The better result might be possible, if this model is relaxed and performs the OLCAO

calculation.

55

APPENDIX A

ABBREVIATIONS

a-SiO2: Amorphous Silica

BO: Bond Order

BL: Bond Length

BZ: Brillouin Zone

CRN: Continuous Random Network

DFT: Density Functional Theory

DNA: Deoxyribo Nucleic Acid

EB: Extended Basis

FB: Full Basis

GTO: Gaussian Type Orbitals

HF: Hartee-Fock

HK: Hohenberg-Kohn

IUPAC: International Union of Pure and Applied Chemistry

KS: Kohn-Sham

LCAO: Linear Combination of Atomic Orbital

LDA: Local Density Approximation

LVB: Lower Valence Band

OLCAO: Orthogonalized Linear Combination of Atomic Orbital

MB: Minimal Basis

56

MD: Molecular Dynamics

NPs: Nanoparticles

OPW: Orthogonalized Plane Wave

PDOS: Partial Density of States

Q*: Effective Charge

ROS: Reactive Oxygen Species

Ry: Rydberg

TDDFT: Time Dependent Density Functional Theory

TDOS: Total Density of States

UVB: Upper Valence Band

XC: Exchange Correlation

57

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65

VITA

Chandra Dhakal was born on March 9th, 1981 in Morang, Nepal. He did his schooling

from government school. Mr. Dhakal completed Bachelor degree in Science (B.Sc.) and

Master in Science (M.Sc.) with major in Physics from Tribhuwan University (T.U.),

Kathmandu, Nepal.

Mr. Dhakal came to the United States of America (USA) as a graduate student in the

Department of Physics and Astronomy at University of Missouri-Kansas City (UMKC) in 1st

January 2013. From January 2013, he worked as a Graduate Teaching Assistant (GTA) in

UMKC Physics department. Then, he has joined as a Graduate Research Assistant (GRA) in

Prof. Ching group from August 2014.

Mr. Dhakal is the member of Nepal Physical Society and American Ceramic Society.

He had participated in 57th Mid-West Solid State Conference (MWSSC), Lawrence, Kansas,

2013. Mr. Dhakal had presented the results of his research at 39th International Conference

on Advanced Ceramics and Composites (ICACC), in Orlando, Florida, January 25-30, 2015.