universitas indonesia dynamical mean-field

UNIVERSITAS INDONESIA

DYNAMICAL MEAN-FIELD THEORETICAL APPROACH TO EXPLORE THE MAGNETIC FIELD DEPENDENCE OF MAGNETITE-GRAPHENE OXIDE NANOPARTICLE SYSTEMS

UNDERGRADUATE THESIS

YUSUF WICAKSONO 1206215850

FACULTY OF MATHEMATICS AND NATURAL SCIENCE DEPARTMENT OF PHYSICS DEPOK JUNE 2016

UNIVERSITAS INDONESIA

DYNAMICAL MEAN-FIELD THEORETICAL APPROACH TO EXPLORE THE MAGNETIC FIELD DEPENDENCE OF MAGNETITE-GRAPHENE OXIDE NANOPARTICLE SYSTEMS

UNDERGRADUATE THESIS Proposed in accordance to one of the requirements for the degree of Bachelor of Science

YUSUF WICAKSONO 1206215850

FACULTY OF MATHEMATICS AND NATURAL SCIENCE DEPARTMENT OF PHYSICS DEPOK JUNE 2016

STATEMENT OF ORIGINALITY

I hereby notify that this undergraduate thesisis my original work, and all of the sources that have been cited has been stated clearly.

Name NPM Signature

: Yusuf Wicaksono : 1206215850 :

Defence Date

: 21 June 2016

ii

APPROVAL PAGE

This undergraduate thesis is submitted by : Name NPM Department Title of undergraduate thesis

: Yusuf Wicaksono : 1206215850 : Physics : Dynamical Mean-Field Theoretical Approach to Explore The Magnetic Field Dependence of Magnetite-Graphene Oxide Nanoparticle Systems

Has been defended in front of the Examining Committee and accepted as one of requirements for the degree of Bachelor of Science in the Department of Physics, Faculty Mathematics and Natural Science, Universitas Indonesia.

EXAMINING COMMITTEE

Advisor I

:

Muhammad Aziz Majidi, Ph.D.

(

)

Advisor II

:

Prof. Andrivo Rusydi, Ph.D.

(

)

Examiner I

:

Dr. techn. Djoko Triyono

(

)

Examiner II

:

Efta Yudiarsah, Ph.D.

(

)

Authorized in : Depok Date : 21 June 2016 iii

ACKNOWLEDGEMENT

First and foremost, I would like to say Alhamdulillahirabbil’alamin, all the praises and thanks are to Allah swt., for without Allah’s blessing and mercy this thesis would not be completed. I am also grateful and owe my deep gratitude to the people who have helped and motivated me to finish this thesis. I would like to thank: • Muhammad Aziz Majidi, Ph.D. and Prof. Andrivo Rusydi, Ph.D. for their help, guidance, advice and motivation during my research as an undergraduate student. I am greatly indebted to them. Without advice and motivations from them, maybe I couldnâĂŹt make it this far. • As’ad Saleh Umar, Regi Kusuma Atmadja and Annamaria Bupu for their motivation, discussion, and help during this research. I also need to thanks Angga Dito Fauzi for the unlimited patient to be asked by me and always burn my spirit until the last. • All CM 2014 students, particularly Khalis, Irvan, Hesni, Afi, Christopher and Rizky. Without their motivations, future-talk, and cheers this project would be so much less fun. • All seniors in TCMP, particularly Muhammad Avicenna Naradipa, Choirunnisa Rangkuti and William Yonathan Phan, for always taught me a new things in physis and in computational knowledge. Also for Ardiansyah Taufik and CENS 2013, for accompany me when working overnight at UPP IPD and always open the door of UPP IPD with patient. • Fisika UI 2012 and Sainstek 2013, thank you for those memories we have made together. I hope you all in good condition and may you all succeed in whatever path you have chosen. • Halimah Harfah for her continous support and motivation. I hope you will always support and accompany me until the end of my life. • My beloved elder brother, Mochammad Perbowo, S.K.M., and his wife, for always provide me to bought many books to support my studies. iv

v • Finally, my parents, Dr. Triyono and Sriambarwati, M.Pd., for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this thesis. This accomplishment would not have been possible without them. Thank you.

Depok, 12 July 2016

Yusuf Wicaksono

Universitas Indonesia

HALAMAN PERNYATAAN PERSETUJUAN PUBLIKASI TUGAS AKHIR UNTUK KEPENTINGAN AKADEMIS

Sebagai sivitas akademik Universitas Indonesia, saya yang bertanda tangan di bawah ini: Nama NPM Program Studi Fakultas Jenis Karya

: : : : :

Yusuf Wicaksono 1206215850 Fisika Matematika dan Ilmu Pengetahuan Alam Skripsi

demi pengembangan ilmu pengetahuan, menyetujui untuk memberikan kepada Universitas Indonesia Hak Bebas Royalti Noneksklusif (Non-exclusive Royalty Free Right) atas karya ilmiah saya yang berjudul: Dynamical Mean-Field Theoretical Approach to Explore The Magnetic Field Dependence of Magnetite-Graphene Oxide Nanoparticle Systems beserta perangkat yang ada (jika diperlukan). Dengan Hak Bebas Royalti Noneksklusif ini Universitas Indonesia berhak menyimpan, mengalihmedia/formatkan, mengelola dalam bentuk pangkalan data (database), merawat, dan memublikasikan tugas akhir saya selama tetap mencantumkan nama saya sebagai penulis/pencipta dan sebagai pemilik Hak Cipta. Demikian pernyatan ini saya buat dengan sebenarnya.

Dibuat di : Depok Pada tanggal : 21 June 2016 Yang menyatakan

(Yusuf Wicaksono)

vi

ABSTRAK

Nama : Yusuf Wicaksono Program Studi : Fisika Judul : Pendekatan Dynamical Mean-Field Theory untuk Meneliti Ketergantungan Medan Magnet dari Sistem Nanopartikel Fe3 O4 -Graphene Oxide Kami menunjukkan penelitian secara teoretik pada kenaikan nilai magnetisasi dari sistem nanopartikel Fe3 O4 dengan adanya penambahan reduced Graphene Oxide (rGO). Data eksperimen telah menunjukkan bahwa magnetisasi sistem nanopartikel Fe3 O4 -rGO meningkat dengan peningkatan jumlah rGO sampai sekitar 5 wt%, tetapi menurun kembali dengan bertambah lebih banyaknya jumlah rGO. Kami mengajukkan bahwa kenaikan terjadi dipengaruhi oleh adanya pembalikan spin pada Fe3+ dalam bagian tertrahedral dibantu oleh kekosongan oksigen di perbatasan partikel Fe3 O4 . Kekosongan oksigen diinduksi oleh adanya lapisan rGO yang menarik atom oksigen dari permukaan partikel Fe3 O4 disekitarnya. Untuk memahami peningkatan magnetisasi, kami mengkontruksi model Hamiltonian berdasarkan tight-binding untuk sistem nanopartikel Fe3 O4 dengan konsentrasi kekosongan oksigen dikontrol melalui konten rGO. Kami menghitung magnetisasi sebagai fungsi dari medan magnet eksternal untuk berbagai variasi wt% rGo. Kami menggunakan metode dynamical mean-field theory dan melakukan perhitungan pada temperatur ruangan. Hasil kami untuk ketergantungan rGO wt% dari magnetisasi saturasi menunjukkan hasil yang sangat sesuai dengan data eksperimen dari sistem nanopartikel Fe3 O4 -rGO yang ada. Hasil ini mungkin dapat menkonfirmasi bahwa model kami telah membawa ide paling penting yang dibutuhkan untuk menjelaskan fenomena kenaikan magnetisasi diatas. Kata kunci: reduced graphene oxide, nanopartikel Fe3 O4 , vakansi oksigen, magnetisasi, pembalikan spin, dynamical mean-field theory.

vii

ABSTRACT

Name : Yusuf Wicaksono Program : Physics Title : Dynamical Mean-Field Theoretical Approach to Explore The Magnetic Field Dependence of Magnetite-Graphene Oxide Nanoparticle Systems We present a theoretical study on the enhancement of magnetization of Fe3 O4 nanoparticle system upon addition of reduced Graphene Oxide (rGO). Experimental data have shown that the magnetization of Fe3 O4 -rGO nanoparticle system increases with increasing rGO content up to about 5 wt%, but decreases back as the rGO content increases further. We propose that the enhancement is due to spin-flipping of Fe3+ in the tetrahedral sites assisted by oxygen vacancies at the Fe3 O4 particle boundaries. These oxygen vacancies are induced by the presence of rGO flakes that adsorb oxygen atoms from Fe3 O4 particles around them. To understand the enhancement of the magnetization we construct a tight-binding based model Hamiltonian for the Fe3 O4 nanoparticle system with the concentration of oxygen vacancies being controlled by the rGO content. We calculate the magnetization as a function of the applied magnetic field for various values of rGO wt%. We use the method of dynamical mean-field theory and perform the calculations for a room temperature. Our result for rGO wt% dependence of the saturated magnetization shows a very good agreement with the existing experimental data of the Fe3 O4 -rGO nanoparticle system. This result may confirm that our model already carries the most essential idea needed to explain the above phenomenon of magnetization enhancement. Keywords: reduced graphene oxide, Fe3O4 nanoparticle, oxygen vacancies, magnetization, spin-flipping, dynamical mean-field theory.

viii

TABLE OF CONTENTS

TITLE PAGE

i

STATEMENT OF ORIGINALITY

ii

APPROVAL PAGE

iii

ACKNOWLEDGEMENT

iv

LEMBAR PERSETUJUAN PUBLIKASI ILMIAH

vi

ABSTRACT

vii

Table of Contents

ix

List of Figure

xi

List of Table

xii

1 INTRODUCTION 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Scope of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Research Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 LITERATURE REVIEW 2.1 Many Body Problem and Second Quantization . 2.1.1 Second Quantization . . . . . . . . . . . 2.1.2 Green’s Function . . . . . . . . . . . . . 2.2 Crystal Structure . . . . . . . . . . . . . . . . . 2.3 Hamiltonian in Solid State Physics . . . . . . . 2.3.1 Kinetic Hamiltonian . . . . . . . . . . . 2.3.2 Interaction Hamiltonian . . . . . . . . . 2.3.2.1 Hubbard Model . . . . . . . . . 2.3.2.2 Heisenberg and Heisenberg-Like 2.4 Density of States . . . . . . . . . . . . . . . . . 2.5 Magnetic Order, Magnetization and Hysteresis . 2.6 Dynamical Mean Field Theory . . . . . . . . . . ix

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

1 1 3 3 4 4 4 5 6 7 7 9 9 10 12 14 15

x 3 MODEL 3.1 Model of System . . . . . . . . . . . . . . . . . . . . 3.2 Parameterizing of RGO Contents in the System . . . 3.3 Model Hamiltonian . . . . . . . . . . . . . . . . . . . 3.3.1 Model Hamiltonian without Oxygen Vacancies 3.3.2 Model Hamiltonian with Oxygen Vacancies . . 3.4 Dynamical Mean Field Theory Calculation . . . . . . 3.5 Computational Method . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

4 RESULTS 4.1 Input Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Physical Parameters of Fe3 O4 . . . . . . . . . . . . . . 4.1.2 Numerical Parameter . . . . . . . . . . . . . . . . . . . 4.1.3 Environmental Parameter . . . . . . . . . . . . . . . . 4.2 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Saturated Magnetization of Fe3 O4 with variation RGO Flake Weight Percent . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 External Magnetic Field Dependence Magnetization of Fe3 O4 with rGO Flake Weight Percent Variation . . . . . . . . . . .

. . . . . . .

17 17 19 21 24 27 30 34

. . . . .

35 35 35 37 38 38

. 39 . 42

5 CONCLUSION 45 5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Bibliography

46


LIST OF FIGURE

1.1 1.2

2.1 2.2 2.3

2.4 3.1

3.2 3.3 3.4 3.5

4.1 4.2

4.3 4.4

The model of a nanoparticle cluster. Taken from reference [8] . . The structure of Graphene, Graphene Oxide and Reduced Graphen Oxide. Taken from reference [29] . . . . . . . . . . . .

1

Crystal structure of Fe3 O4 . Taken from reference [32] . . . . . hysteresis curve of ferromagnetic material. Taken from reference [44] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ilustration of mean field theory,(left) the real physics system with interaction between particles and (right) mean-field approximation where particle black feel interaction of others particles as mean density . . . . . . . . . . . . . . . . . . . . . . Ilustration of Vacancy in Crystal. . . . . . . . . . . . . . . . .

6

The system of (a)Fe3 O4 nanoparticle clusters without additional rGO, (b)Fe3 O4 nanoparticle clusters with additional rGO less then 5 wt%, (c)Fe3 O4 nanoparticle clusters with additional rGO around 5 wt% and (d)Fe3 O4 nanoparticle clusters with additional rGO more then 5 wt% . . . . . . . . . . . . . . . . . . . Crystal structure simplification of Fe3 O4 . . . . . . . . . . . . Fe3 O4 unit cell without oxygen vacancy . . . . . . . . . . . . . Fe3 O4 unit cell with oxygen vacancy. . . . . . . . . . . . . . . Flow chart of DMFT calculation in Matsubara domain frequency and real domain frequency . . . . . . . . . . . . . . . Density of States Fe3 O4 unit cell without oxygen vacancy . . . Saturated magnetization of Fe3 O4 +rGO nanoparticle system by vary the wt% of rGO added into system. The calculation was done by using α = 20 and γ = 0.574 . . . . . . . . . . . . . . . External magnetic field dependence magnetization of Fe3 O4 with rGO flake weight percent variation. . . . . . . . . . . . . Temperature dependence magnetization of Fe3 O4 . . . . . . .

xi

.

2

. 14

. 15 . 16

. . . .

17 18 23 27

. 30 . 39

. 40 . 42 . 43

LIST OF TABLE

4.1 4.2 4.3 4.4

Input physical parameters used in calculation of electronic structure and magnetization of Fe3 O4 . . . . . . . . . . . . . . . . Input numerical parameters used in calculation of electronic structure and magnetization of Fe3 O4 . . . . . . . . . . . . . . Input environmental parameters used in calculation of electronic structure and magnetization of Fe3 O4 . . . . . . . . . . . . . . Parameters that used to calculate saturated magnetization versus weight percent of RGO flake . . . . . . . . . . . . . . . . .

xii

. 37 . 38 . 38 . 40

CHAPTER 1 INTRODUCTION 1.1

Background

Magnetite or Fe3 O4 known as magnetic materials with ferrimagnetic ordering. Magnetite is not only important in geophysics and mineralogy, due to half-metallic properties (conductive minority spin channel and semiconducting majority spin channel) with 100% spin polarization at EF [1, 2] and has high Curie temperature (around 851 K), magnetite also becomes prospective material for spintronic devices [3–5]. In form of nanoparticles Fe3 O4 have pretty much the same properties as those of bulk Fe3 O4 . Magnetite (Fe3 O4 ) nanoparticles (NPs), attractive for their biocompatibility and high Curie temperature [6], are important for biomedical technologies such as enhanced MRI contrast imaging, hyperthermia cancer treatment, and tagging. Nevertheless, there are some experiments that have indicated different magnetic properties of the nanoparticles as compared to bulk. Almost universally Fe3 O4 NPs display a reduced saturation magnetization compared to bulk Fe3 O4 , which is worsen for decreased NP size, and suggestive of a surface-related mechanism [7]. Theoretical models indicate that sufficient surface anisotropy could induce a configuration of surface spins pointing radially outward [8–10] that reduces. Surface disordering has also been widely proposed [11–14]. Either in Fe3 O4 bulk or Fe3 O4 NP, surface properties become important to determine the magnetic properties of Fe3 O4 .

Figure 1.1: The model of a nanoparticle cluster. Taken from reference [8]

1

2

Figure 1.2: The structure of Graphene, Graphene Oxide and Reduced Graphen Oxide. Taken from reference [29]

On the other hand, graphene become most interesting materials nowadays. The current interest of Graphene can be ascribed to three main reasons. First, the electronic structure of graphene form dirac cone band structure where the electron transport is described by the Dirac equation that access to quantum electrodynamics [15–19]. Second, The ballistic transport at room temperature combined with chemical and mechanical stability make graphene become a promosing candidate for application in electronic devices [20–24]. This remarkable properties extend to bilayer and few-layer graphene [18–20, 22, 25]. Third, Materials such as graphite, nanotubes, buckyballs and others can be viewed as derivatives of graphene [26]. Another unique property of graphene is the ability of graphene to absorb oxygen to become adatoms on the surface of graphene [27]. However, opposite from Fe3 O4 , graphene is non-magnetic materials According to the experimental data from Computational and Experimental Nano Science (CENS) research group at Department of Physics, Universitas Indonesia, the magnetization of Fe3 O4 nanoparticle system enhance upon addition of reduced graphene oxide (rGO) [31]. The enhancement of Fe3 O4 nanoparticle system increases with increasing rGO flake content up to 5 wt%, at which the highest magnetization enhancement is achieved, but decreases back as the rGO content increases further. The magnetization enhancement of Fe3 O4 nanoparticle system is quite remarkable, as the magnetization saturation enhances up to ≈ 125 emu/gr or increase about 45 emu/gr higher than Fe3 O4 nanoparticle system without rGO flakes added. The reason beyond the enhancement of magnetization is still well understood. However, recent experimental study by a group at National University of Singapore (NUS) suggested that the magnetization enhancement occurs due to oxygen vacancies [30]. Meanwhile, some preliminary result of an on-


3 going research by our group suggest that oxygen vacancies cause a removal of super-exchange (anti-ferromagnetic order) coupling which is replaced by RKKY (ferromagnetic order) coupling. Motivated by the a fore mentioned background, in this thesis we propose a model of Fe3 O4 +rGO nanoparticle system with variation of rGO flake from 5 wt% to 20 wt%. We aim to calculate the magnetization of the combined system and plot it against external magnetic field. Our, computational result are to be compared with experimental data.

1.2

Scope of Problem

In this thesis we hypothesize that the magnetization enhancement of Fe3 O4 + rGO nanoparticle system is due to oxygen vacancies on the surface of Fe3 O4 nanoparticle cluster. The presence of rGO flakes nearby an Fe3 O4 cluster is believed to cause the formation of oxygen vacancies on the Fe3 O4 surface. This argument is based on previous studies [27,28] that have indicated that graphene can easily adsorb oxygens. Our main goal is to calculate magnetization of Fe3 O4 nanoparticle cluster with or without oxygen vacancies and mimic the magnetization of the combined Fe3 O4 +rGO nanoparticle system by connecting the rGO content with the content of oxygen vacancies.

1.3

Research Aim

This research aims to achieve the following: • To model Fe3 O4 nanoparticle with additional reduced Graphene Oxide flake (rGO) and propose the mechanism behind the magnetization enhancement • To theoretically reproduce propose mechanism the plot of magnetization of Fe3 O4 nanoparticle + RGO flake (M) vs external magnetic field (H) and compare it with the experimental data.


CHAPTER 2 LITERATURE REVIEW 2.1

Many Body Problem and Second Quantization

Materials in Condensed Matter Physics accurately described as a manyparticle system that interact each other. We can use quantum mechanics to solve this problem by defining Schroedinger equation for each particles that describes interaction between those particles such as electron-electron and electron-nuclei interaction, then we can obtain the properties of this manybody systems by solving the corresponding Schroedinger equations. However, it is become impractical and requires much computational work to get exact solution for this problems, since the complexity of the many-body wave function (which is superposition of many wave-function) must be solved to explain the properties of the system. Therefore, the strong approximation is necessary to simplify and reformulate the original many-particle Schroedinger equation. Quantum field theory provide us to focus on the part we want to observe in the system or in the other word we can avoid to directly face with the manyparticle wave functions. In Quantum field theory, Green’s function and second quantization are the strongest tools to solve many-body particles problems in condensed matter physics systems. Therefore, we must understand how to simplify and solve our system by using quantum field theory tools.

2.1.1

Second Quantization

Quantum field theory is used in condensed matter physics through second quantization operators. Second quantization operators is incorporate the statistics at each step, which contrast with cumbersome approach of using symmetrized or anti-symmetrized product of single particle wave functions. Second quantization also provides a basic and efficient language to formulate many-particle systems. Therefore, instead of using wave function to describe our systems, we can solve many-body problem more effectively by describing our system using second quantization operators. The term second quantization doesn’t means that we quantize the operators twice. When quantum mechanics were introduced, quantum mechanics successfully convinced people that the particles can behave like waves and they 4

5 obey wave equation known as Schroedinger equation. This idea is known as first quantization. Furthermore, it was realized that the waves can behave like particles and this idea is known as second quantization. For example, electromagnetic waves and lattice vibrations not only behave as waves but they can behave as like-particles where we called that particles as photon and phonon. This idea of second quantization also applied to wave-function of single particle in the system. The wave-function can be transformed into particle-like function or called field operator as the representative of the systems. In our thesis, annihilation and creation operator is used as a representative of field operator. Through annihilation and creation operator, we define our basis for the system and solve the problem by using Green’s Function and Dynamical Mean Field Theory Approximation.

2.1.2

Green’s Function

The many-body systems consisting of strongly interacting real particles can often be described as if they were composed of weakly interacting fictitious particles: quasi particles and collective excitation. To calculate the properties of these fictitious particles, there are various ways of doing this but the most used method to treat many-body problems are played by quantum field theoretical quantities known as Green’s functions or propagators. There are three reason to use Green’s function to calculate many-body problem. First and the most important, they yield in a direct way the most important physical properties of the system. Secondly, they have a simple physical interpretation. Third, they can calculated in a way which is highly systematic and ’automatic’ which appeals to one’s physical intuition. There are two consideration on using the Green’s function to solve manybody problem on physics. If we consider the system is a non-interacting system, then interaction factor in the system, such as electron-electron, electronphonon, or other related interactions, can be neglected. Thus, the Hamiltonian used in the system is only the kinetic part of the Hamiltonian itself. The Green’s function without interacting system can be describe as follow GR 0 (k, z) = lim+ η→0

1 ω + iη − E(k)

(2.1)

However, many of real system is interacting system. To solve many-body problem with an interacting system, then the interaction part of the Hamiltonian cannot be neglected and need to be taken into calculation. The inter-


6

Figure 2.1: Crystal structure of Fe3 O4 . Taken from reference [32]

action part then will be considered as the self-energy (Σ(z)) term. Generally, the Green function of the system can be written as GR (k, z) = lim+ η→0

1 ω + iη − E(k) − Σ(k, z)

(2.2)

Equation 2.2 applied for single particle problem. To generalize the Green’s function equation to many-body problems and to implement our basis which a representation of the system, we transform equation 2.2 into matrix form. The matrix form of Green’s function can be expressed as follow h

2.2

GR (k, z) = [(ω + iη)[I] − [H0 (k)] − [Σ(z)]]−1 i

(2.3)

Crystal Structure

A unit cell of Fe3 O4 consists of 32 oxygen atoms and 24 iron atoms and also crystallizes in a cubic inverse spinel structure. The 24 iron are classified into two groups, eight iron atoms tetrahedrally coordinated with oxygen atoms, and sixteen iron atoms octahedrally coordinated with oxygen atoms [33]. It has a lattice constant of a = 8.396 ˚ A, where oxygen anions form a close-packed facecenter-cubic (fcc) sub-lattice with iron cations located in interstitial sites [34]. Iron ions which are located in tetrahedral and octahedral sites are called FeA and FeB, respectively. FeA ions, which are Fe3+ cations, occupy one-eighth of Universitas Indonesia

7 the tetrahedral sites (A sites) in the unit cell, while FeB, which are 1:1 mixture of Fe2+ and Fe3+ cations, fill half of the octahedral sites (B sites) in the unit cell [35]. Figure 2.1 show the crystal structure of Fe3 O4 . Fe3 O4 belongs to the Fd3m symmetry group [36]. At the B sites, electron hopping between Fe2+ and Fe3+ ions occurs. This results in a high conductivity and an average charge of Fe2.5+ . Fe3+ has 5 d-electrons, whose spins are parallel to one another as dictated by Hund’s rules, forming a filled sub shell. Fe2+ has an additional spin-down electron which can easily hop to a neighboring Fe3+ site if their spins are parallel. In the magnetically ordered state only the spin-down electron can easily move, resulting in spin-polarized electron transport. The electron transport is restricted to the B sites. The spins of Fe2+ and Fe3+ ions in B sites are oriented ferromagnetic because of their mutual anti-ferromagnetic coupling to the spins of Fe3+ ions in A sites. Around 120 K the so-called Verwey transition occurs [37]. At this transition, the structure distorts from cubic symmetry [38] and a charge ordering occurs at the B sites [39] thus reducing the conductivity by two orders of magnitude. The exact transition temperature depends on the purity of the crystal [40,41].

2.3

Hamiltonian in Solid State Physics

In quantum mechanics, the Hamiltonian is the operator corresponding to the total energy of the system. Quantum mechanics in solid state physics hold important role to understand physics behind properties of materials. A system usually consist of two type of Hamiltonian, kinetic Hamiltonian and interaction Hamiltonian. Kinetic Hamiltonian in most case is approached by using tight-binding approximation to describe kinetic energy of electron in solid. Meanwhile, interaction Hamiltonian is vary from one material to another. There are two interaction contribute in our system which is Coulomb repulsive interaction and magnetic exchange interaction.

2.3.1

Kinetic Hamiltonian

In a crystal, an electron is typically bound tightly to one particular atom with some energy E which is less than the potential barrier between atoms. However, since the wave functions of two atoms in the lattice will have some overlap, there is always the possibility that an electron can tunnel through the potential barrier and hop from one atom to another. The tight-binding approximation deals with the case in which the overlap Universitas Indonesia

8 of atomic wave function is enough to require correction to the picture of isolated atoms, but not so much as to render the atomic description completely irrelevant. This approximation is most useful to describe energy band which is arise from partially filled d orbitals of transition metal atoms or to describe insulator in solid. The idea behind the tight-binding model is that the ability to tunnel between atoms is favoured by the electron since the wider space available to it results in its energy being lower. The precise amount by which the energy is lowered depends on the nature of the crystal. If an electron tunnels from crystal lattice site j to site l, its energy changes by an amount −tjl . This tunnelling effect is equivalent (in second quantization language) of annihilating the electron at site j and creating it again at site l, so the portion of the Hamiltonian dealing with tunnelling can be written as ˆ =− H

X

tjl c†l cj

(2.4)

j,l

where c†l and cj are the fermion creation and annihilation operators. The tjl value is decreases along with the separation of two atoms. To simplified the model, assume that electron only hopping to the neighbouring atoms. Thus, tjl = 0 for all other atom pair beside the neighbouring atoms. Then equation above become ˆ =− H

X

tl,l+τ c†l cl+τ

(2.5)

l,τ

where the sum over τ means to sum over atoms close to l. To diagonalize equation 2.5 we can apply Fourier transforms to creation and annihilation operators 1 X ik·rl e ck cl = √ N k

(2.6)

1 X −iq·rl † c†l = √ e cq N q

(2.7)

Therefore equation 2.5 become


9

ˆ =− H

X l,τ

1 X −iq·rl † 1 X ik·(rl +rτ ) e ck tl,l+τ √ e cq √ N q N k

1 XX = −√ tl,l+τ eirl ·(k−q) eik·rτ c†q ck N l,τ k,q

(2.8)

Since X

eirl ·(k−q) = δk,q

(2.9)

l

our equation become ˆ =− H

XX

tl,l+τ eik·rτ c†k ck

(2.10)

l,τ k

Equation 2.10 will be used to calculate kinetic energy of our system. Annihilation and creation operators will be basis orbitals of our model.

2.3.2

Interaction Hamiltonian

The interaction Hamiltonian is depends on the system that will be discovered. For our system there are two interaction Hamiltonian must be included into calculation. The first is Hubbard model and the second is Heisenberg-like model. 2.3.2.1

Hubbard Model

Hubbard model or Coulomb repulsive interaction emerge when Coulomb repulsive energy greater than hopping energy of electron (U > t). Such condition usually occur in material that have half-filled d or f orbital. The Hubbard model explain a system that kinetic interaction alone not enough to describe the electronic structure of materials such as Mott-Insulator systems. The Hubbard model not only complicated to solve but also the complete solution particularly occur in some special case. Let us begin by define Coulomb repulsive Hamiltonian as follow X † † ˆ Coul = 1 cˆ cˆ Vijkl cˆk cˆl H 2 i,j,k,l i j

(2.11)

Recognizing the electron have spin σ that could be | ↑i or | ↓i. Therefore


10 equation 2.12 become X ˆ Coul = 1 H cˆ†i,σ cˆ†j,σ Vijkl cˆk,σ0 cˆl,σ0 2 i,j,k,l,σ,σ0

(2.12)

Assuming there is no spin flipping process occur. Thus, the spin cannot flip from up-to-down or vice versa when hopping. The interaction between electron is correlated with the charge of the electron. Hence, the interaction of electron between opposite spin happen as much as electron with similar spin. In the Hubbard model, Coulomb interaction become assumed to be significant only between two electrons on the same site, Therefore Coulomb interaction potential become U = Viiii . However, the Pauli exclusion principles ensures that should two spin does make a difference. The Hamiltonian of the Hubbard model then ˆU = U H

X

n ˆ i↑ n ˆ i↓

(2.13)

i

The equation looks simple, but the eigenstates tend to complex and highly correlated. Therefore, to solve this problem, some approximation used to solve the Hubbard model. 2.3.2.2

Heisenberg and Heisenberg-Like Model

Exchange interaction or Heisenberg model Hamiltonian describes interaction between two ions spin to form parallel or anti-parallel spin alignment in material. However, exchange interaction Hamiltonian not only used to describes interaction between spin of two ions, this Hamiltonian also applied to describes interaction between ion spin with electron spin which is used in our model. The different is only take place on spin approximation, where for electron we use quantum mechanics approximation, meanwhile for ion we use classical approximation. This Hamiltonian model called Heisenberg-like Hamiltonian. Let us started by understanding exchange interaction between two ions spin and then we carry out into exchange interaction between ion and electron. Consider two electrons with different coordinate where the wave function for the joint state can be written as product of single electron states. Joint state of the system only allowed to make symmetrized or anti-symmetrized product to obtain exchange symmetry. Since our system consist of two electrons, wave function for joint states must be anti-symmetrized so the spin part of wave


11 function consist of two states which is singlet states (S=0) or triplet states (S=1). The wave function for both of the can be expressed as follow

1 ΨS = √ [ψa (r1 )ψa (r2 ) + ψa (r2 )ψa (r1 )]χS 2 1 ΨT = √ [ψa (r1 )ψa (r2 ) − ψa (r2 )ψa (r1 )]χT 2

(2.14) (2.15)

where ΨS is wave function for the singlet case and ΨT is wave function for triplet case. The energy for of the both two possible states are

ES = ET =

Z

ˆ S dr1 dr2 Ψ∗S HΨ

(2.16)

Z

ˆ T dr1 dr2 Ψ∗T HΨ

(2.17)

with the assumption that the spin parts of the wave functions χs and χT are normalized. The difference between two energies of the possible states are ES − ET = 2

Z

ˆ a∗ (r2 )ψb∗ (r1 )dr1 dr2 ψa∗ (r1 )ψb∗ (r2 )Hψ

(2.18)

Recall that the total spin for two electron is S1 +S2 . The hST2 ot i = S21 +S22 + 2S1 · S2 where S1 · S2 can be described as follow for the singlet states −3 4 while 1 for the triplet 4 . Hence Hamiltonian can be written in effective Hamiltonian ˆ = 1 (ES + 3ET ) − (ES − ET )S1 · S2 H (2.19) 4 where consist of a constant term and an a term which depends on spin. The second term constant called exchange constant and can be defined as follow ES − ET Z ∗ ˆ ∗ (r2 )ψ ∗ (r1 )dr1 dr2 = ψa (r1 )ψb∗ (r2 )Hψ a b 2 and then spin-dependent term can be written as follow J=

ˆ spin = −2JS1 · S2 H

(2.20)

(2.21)

If J > 0, ES > ET and the triplet states S=1 is favoured. If J < 0, ES < ET and the singlet states S=) is favoured. This equation motivates the Hamiltonian


12 of Heisenberg model which is ˆ =− H

X

Ji,j S1 · Sj

(2.22)

i,j

The Heisenberg model in equation 2.22 is not the one we will be used in our model. Heisenberg model describe about exchange interaction between ion spins, nevertheless, in our model we used it to calculate exchange interaction between ion spin and electron itinerant. Therefore we define Heisenberg-like model as follow ˆ = −JH H

X

Si · si

(2.23)

i

where JH is Hund’s coupling constant, Si and s are the ion spin and electron spin in the same atom respectively.

2.4

Density of States

Density of States (DOS) is the number of electron states per energy interval which can be obtained by integrating over shell in k space. The purpose of the DOS calculation is to know the electronic properties of the system. Beside of DOS, chemical potential hold important role to identify the electronic structure of the system. Chemical potential is the highest energy of electrons could have in the system. The chemical potential value in DOS describe that the system is insulator, metal or semiconductor. The Density state of the system can be described as delta function with Lorentzian form over all of k-space. Therefore we can expressed as follow DOS(ω) =

1 X δ (ω − (k)) N k

(2.24)

where δ (ω − (k)) function can be represented as follow δ (ω − (k)) =

1 η lim+ π η→0 (ω − E(k))2 + η 2

(2.25)

Recall bare-retarded Green’s function from equation 2.1. To make the denominator become real value, numerator and denominator must multiplied with its conjugate. Therefore, equation 2.1 become


13

1 (ω − E(k)) − iη × (ω − E(k)) + iη (ω − E(k)) − iη (ω − E(k)) − iη = (ω − E(k))2 − η 2 (ω − E(k)) η = −i 2 2 (ω − E(k)) − η (ω − E(k))2 − η 2

GR 0 (k, ω) =

(2.26)

which its imaginary part has similar form with equation 2.25. Thus, equation 2.25 can be expressed as follow δ (ω − (k)) = −

1 1 lim+ = π η→0 (ω − E(k)) + iη

(2.27)

By substitute equation 2.27 into equation 2.24, equation 2.24 become

1 1 X1 lim+ = N k π η→0 (ω − E(k)) + iη 1 1 X 1 =− = π N k (ω − E(k)) + i0+

DOS(ω) = −

(2.28)

Finally, The complete relation between DOS and bare-retarded Green’s function is DOS(ω) = −

1 1 X =GR 0 (k, ω) πN k

(2.29)

or can be expressed as follow 1 DOS(ω) = − =GR (2.30) 0 (k, ω) π where equation 2.30 applied for non-interacting system. For interacting system we must use Green’s function with self-energy (Σ(k, ω)) on it and using same simplification we can get Density of States of interacting system as follow 1 (2.31) DOS(ω) = − =GR (k, ω) π Eventually, to calculate density of states of our system which have more than one basis, we transform equation 2.32 into matrix form. 1 DOS(ω) = − = Tr [GR (k, ω)] π

(2.32)


14

Figure 2.2: hysteresis curve of ferromagnetic material. Taken from reference [44]

2.5

Magnetic Order, Magnetization and Hysteresis

Magnetic ordering is the arrangement of moment magnet in a material. This arrangement of moment magnet in materials describe magnetic properties of material. Various magnetic orderings are possible: ferromagnet is material that the magnetic moments (spins) all line up parallel to one another. antiferromagnet is material that equal magnetic moments (spins) on nearest neighbour sites which tend to line up anti-parallel. More complicated arrangements are possible like a ferrimagnet which has a basic anti-parallel arrangement of the magnetic moments (spins) but the magnitudes of the moments in the two directions are unequal, giving rise to a net magnetization. Unlike paramagnetic material, the magnetic moment of ferromagnetic material (also in ferrimagnetic material) of adjacent atom in this case are aligned in particular direction even in the absence of the applied magnetic field. Thus the a ferro/ferrimagnetic material exhibits a magnetic moment in the absence of a magnetic field. The magnetization existing in a ferro/ferrimagnetic material in the absence of of an applied magnetic field is called the spontaneous magnetization. It exist below a certain temperature critical temperature called the Curie temperature. The alignment of magnetic moments below the Curie temperature is due to the exchange interaction between the magnetic ions. Universitas Indonesia

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Figure 2.3: Ilustration of mean field theory,(left) the real physics system with interaction between particles and (right) mean-field approximation where particle black feel interaction of others particles as mean density

Above the Curie temperature, the thermal effect offset the spin alignment and the ferro/ferrimagnetic substance become paramagnetic. Spontaneous magnetization also yield hysteresis in ferro/ferrimagnetic material as shown in figure 2.2. Magnetization of ferromagnetic material is zero when there is no applied magnetic field applied. If applied magnetic field increase, the magnetization also increase until saturated which is called saturated magnetization. However, the magnetization of ferro/ferrimagnetic material is not vanish when applied magnetic field back to zero. This magnetization is called remanent magnetization. The remanent magnetization itself is spontaneous magnetization of ferro/ferrimagnetic material.

2.6

Dynamical Mean Field Theory

It is well known that theoretical investigations of quantum-mechanical many-body systems are faced with severe technical problems due to complicated dynamics in the system. In the absence of exact methods there is clearly a great need for reliable and controlled approximation schemes. In spite of using complicated method to solve the problem, there are still some rough and simply method but still gives a good physical meaning on it. The method is still including correlation of the system but in the difference way, where all dynamics in the system are represents as the average correlation of the system. This method is called mean field (or mean density) theory. Mean field theory studies the behaviour of large and complex stochastic models by studying a simpler model. Such models consider a large number of small individual components which interact with each other. The effect of all the other individuals on any given individual is approximated by a single averaged effect, thus reUniversitas Indonesia

16

Figure 2.4: Ilustration of Vacancy in Crystal.

ducing a many-body problem to a one-body problem. The general picture of mean field theory shown in figure 2.3. In case of strongly correlated electron system, such as presence of impurity or vacancy, MFT can’t solve the problem alone, therefore, we need modification and advance approach to solve these system. Dynamical mean field theory (DMFT) is one of a method to determine the electronic structure of strongly correlated electron materials. There do exist well-known mean-field approximation schemes, e.g. Hartree-Fock, random-phase approximation, saddlepoint evaluations of path integrals, decoupling of operators. However, these approximations do not provide mean-field theories in the spirit of statistical mechanics, since our model calculate in finite temperature and must be able to give a global description of a given model (including thermodynamics) in the entire range of input parameters. DMFT consists in mapping a many-body lattice problem to a many-body local problem. The mapping in itself doesn’t constitute an approximation. The only approximation made in DMFT is to assume the lattice self-energy to be a momentum independent (local) quantity. The true quantity of lattice selfenergy can be obtain by using iteration in DMFT scheme until self-consistency condition achieve. Since calculation the magnetization of our system in finite temperature, DMFT calculation applied for two domain frequencies which is real frequency domain and Matsubara frequency domain.


CHAPTER 3 MODEL 3.1

Model of System

The enhancement of magnetization of Fe3 O4 nanoparticle system upon addition of reduced graphene oxide (rGO) is due to spin-flipping of Fe3+ in the tetrahedral sites. Oxygen vacancies at the Fe3 O4 nanoparticle clusters assist the existence of spin-flipping, where are the oxygen vacancies cause some superexchange couplings get removed and RKKY coupling arises which change the magnetic moment order of Fe3 O4 from ferrimagnetic ordering transform to ferromagnetic ordering. These oxygen vacancies induced by the presence of rGO flakes that adsorb oxygen from Fe3 O4 clusters around them. Right after oxygen adsorbed by rGO flakes the both system become independent each other. The magnetization of Fe3 O4 -rGO nanoparticle system increase with the increasing of rGO content up to about 5wt%, but decrease back as the rGO content increase further. Figure 3.1 show the ideal model of the system without the addition of rGO and also system with the addition of rGO less then 5wt%, about 5wt% and more than 5wt%. The ideal model of system with additional 5 wt% rGO shows us that the most effective of oxygen adsorption occur when a rGO flake fill the slots (space between two Fe3 O4 nanoparticle cluster). If rGO content less than 5wt%, there are still some slots not filled yet which makes oxygen adsorption become less effective. Therefore, oxygen adsorbed will be

Figure 3.1: The system of (a)Fe3 O4 nanoparticle clusters without additional rGO, (b)Fe3 O4 nanoparticle clusters with additional rGO less than 5 wt%, (c)Fe3 O4 nanoparticle clusters with additional rGO around 5 wt% and (d)Fe3 O4 nanoparticle clusters with additional rGO more than 5 wt%

17

18

Figure 3.2: Crystal structure simplification of Fe3 O4

decreased and the magnetization enhancement not as high as adding rGO about 5wt%. If rGO content more than 5wt%, rGO flakes will be piled up but the active surface to adsorb oxygen is still the same when rGO flakes are not piled up (the surfae between two rGO flakes can’t adsorb oxygen from Fe3 O4 nanoparticle cluster). Since the rGO flake itself is non-magnetic material, the more rGO flake added to Fe3 O4 nanoparticle, the system become less magnetic. Furthermore, the magnetization enhancement of the system will decrease. To convince that our hypothesis is correct, we will calculate the magnetization of the system with the variation of rGO wt%. Calculation of magnetization of the system can be done by modelling the Fe3 O4 nanoparticle cluster as Fe3 O4 bulk. This assumptions is chosen to reduce computational cost but still keep physical properties of the system. Recent studies has been suggested that Fe3 O4 nanoparticle can be modelled as Fe3 O4 bulk [42]. We will use (001) Fe3 O4 surface and only the take first two-layer for our model in order to simplify our system. The half metallic properties of (001) Fe3 O4 surface has been confirmed by using first-principle calculation [42]. However, to preserve the quantum mechanical basis of the system and do the calculation with a small amount of computational time, we clarified further our model as shown as in figure 3.2. Meanwhile, for Fe3 O4 -rGO nanoparticle system where there are oxygen vacancies on the surface of Fe3 O4 clusters has been modelled same as before but without oxygen in site B. However, because in our calculation method, there is no parameter regarding wt% variation, therefore, wt% variation of rGO flake can be performed by parametrizing it to the numerical parameters that will be discussed in section 3.2 and calculate the magnetization using those numerical parameters. Finally, the result of computational calculation will be compared with experimental data. Universitas Indonesia

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3.2

Parameterizing of RGO Contents in the System

The amount of rGO added to Fe3 O4 nanoparticles experimentally measured using weight percent unit (wt%). Weight percent can be defined as the mass ratio of rGO flake added to the base of the system which is Fe3 O4 nanoparticles. Mathematically we can write the equation as follow x=

NG MG , NF MF

(3.1)

where NF and NG respectively are the amount of Fe3 O4 cluster and rGO flake in the system, MF and MG are the mass of Fe3 O4 cluster and rGO flake respectively and x is wt% of rGO flake added to the F e3 O4 nanoparticle. Consider that the mass ratio of Fe3 O4 cluster to rGO flake is a constant called α. Since mass of a Fe3 O4 cluster is much heavier than a rGO flake, so α has a value greater than one. By using definition of α, equation 3.1 become x=

NG NF α

(3.2)

In our model, a slot can be filled by one rGO flake or more. The amount of rGo flake that fill the slots affect the magnetization enhancement of the system. Therefore, The configuration of rGO flakes fill the slots must be included into our model. Let us assume the maximum amount of RGO flakes to fill all of the slots (space between two nanoparticles) without vacant are n. So, if the amount of rGO flakes that fill a slot is still equal or lower than n, there is still probability to find a slot without rGO flake. But if the amount of rGO flakes that fill a slot is already greater than n, there aren’t any slots still unfilled by rGO flakes and all of the slots at least have n rGO flakes. Hence, ratio of the total amount of RGO flakes in the system to maximum amount of RGO flake to fill the slots without vacant is β=

NG NG = NGmax nNF

(3.3)

Thus by substitute this equation to equation 3.2, we get the value of β in the function of x, which is xα (3.4) n Statistically, we can assume the probability to not find any rGO flake between two Fe3 O4 clusters is P0 = (1 − β)n and the probability of finding RGO flake between two Fe3 O4 cluster is 1 − P0 . From experimental data, as we β=


20 know, the saturated magnetization of Fe3 O4 nanoparticles without adding any RGO flake is around 80 emu/gr. Theoretically, this value can be described as follow. M0 =

µ0 ≈ 80emu/gr MF

(3.5)

If we add rGO flakes to the Fe3 O4 nanoparticles, the magnetization of the system is the results of the magnetic moment of Fe3 O4 cluster without oxygen vacancies in the surfaces and magnetic moment of the Fe3 O4 cluster with oxygen vacancies regardless of how many rGO flakes have filled the slots. However in real system, oxygen vacancies does not occur in all unit cells of the Fe3 O4 nanoparticle cluster. As results, magnetization of the system can be expressed as follow M(x) =

P0 NF µ0 + (1 − P0 )NF γµv NF MF + NG MG

(3.6)

where gamma (γ) is a ratio of magnetization enhancement of the system because of oxygen vacancies in some Fe3 O4 cluster surface unit cells to Fe3 O4 cluster magnetization in percent. Thus, if numerator and denominator times with 1/NF and 1/MF , the equation become

P0 NF µ0 + (1 − P0 )NF γµv × M(x) = NF MF + NG MG M(x) =

P0 µ0 + (1 − P0 )γµv 1+

M(x) =

NG NF

MG

×

1 NF 1 NF 1 MF 1 MF

P0 M0 + (1 − P0 )γMv 1+

NG NF

MG MF

(3.7)

where Mv is the magnetization of Fe3 O4 cluster with oxygen vacancies. By substitute equation 3.2 and 3.4 to equation 3.7, equation 3.7 can be simplified as follow

M(x) =

P0 M0 + (1 − P0 )γMv 1 + 2 αβ M(x) =

γMv 1 + 2 αβ

for 0 < β < 1

(3.8)

for β > 1

(3.9)


21

3.3

Model Hamiltonian

In this thesis, tight binding approximation in second quantization form is used to express Hamiltonian kinetic of the electron in our system. On the another hand, F e3 O4 in our model possess two form of Hamiltonian interaction; the first is magnetic interaction term or so-called Heisenberg-like interaction term. This term interprets spin interaction between iron ion which are F e3+ and F e2+ ion in octahedral site and F e3+ ion in tetrahedral site with electron around them. The second is local Coulomb repulsive interaction between two electrons in orbital d of our system or so-called Hubbard term. In our model, Hubbard term takes place on d-orbital, specifically eg orbital, in iron ion. Therefore, the Hamiltonian for our model can be represented as follow H=

X

[H0 (k)] a†k,σ ak,σ − JH

X

Si,σ si,σ + U

i,σ

k,σ

X

n ˆ ↑,i n ˆ ↓,i

(3.10)

i

The kinetic Hamiltonian term construct by using tight-binding approximation as already explained in section 2.3.1. The elements of matrix depend on unit cell and basis orbitals of the system. Meanwhile the second term, magnetic interaction exchange, arise from basis orbitals of iron ions where in this system use eg orbitals (dx2 −y2 ) for each of iron ions. To use this term in DMFT algorithm, we have to transform this term into matrix form. Independent from another interaction Hamiltonian, magnetic interaction exchange can be described as follow [Hspin ] = −JH

X

Sj · sj

(3.11)

j

by operate dot product between iron ions spin and electrons around them, equation above become [Hspin ] = −JH

X

Sx sx + Sy sy + Sz sz j

(3.12)

j

where we approach iron ion spin classically, therefore the iron ion spin can be expressed as follow Sx = S sin θ cos φ

(3.13)

Sy = S sin θ sin φ

(3.14)

Sz = S cos θ

(3.15)


22 on the other hand, electron spin around iron ion describe in quantum approach. The second quantization of spin operator are 













a↑ 0 1 a↑  h ¯ h ¯ sx = (a†↑ a†↓ )[σx ]   = (a†↑ a†↓ )  2 2 a↓ 1 0 a↓ 











(3.16)



a↑ 0 −i a↑  h ¯ h ¯ sy = (a†↑ a†↓ )[σy ]   = (a†↑ a†↓ )  2 2 a↓ i 0 a↓ 











(3.17)



a↑ 1 0  a↑  h ¯ h ¯ sz = (a†↑ a†↓ )[σz ]   = (a†↑ a†↓ )  2 2 a↓ 0 −1 a↓

(3.18)

Finally, by substitute equation 3.13 and 3.16, we get magnetic exchange interaction in matrix form as follow

[Hspin ] = −

X JH Sj h ¯ j

2







cos θj sin θj exp (−iφj ) a↑  (3.19) a↓ sin θj exp (iφj ) − cos θj

(a†↑ a†↓ ) 

Second term of interaction Hamiltonian is Coulomb repulsive interaction between electron on d orbital of iron ions, particularly in eg orbital. In our model, this term is expressed as Hubbard interaction where approached using mean field theory. Our method use Hubbard model in matrix form, therefore we will simplify the model and convert in to matrix form. Recall Hubbard model Hamiltonian as follow [HHubbard ] = U

X

n ˆ i↑ n ˆ i↓

(3.20)

i

where are the counting operators for each spin can be expressed as follow

(3.21)

(3.22)

n ˆ i↑ = hˆ ni↑ i + n ˆ i↑ − hˆ ni↑ i

n ˆ i↓ = hˆ ni↓ i + n ˆ i↓ − hˆ ni↓ i

Consider the second term in equation 3.21 and 3.22 as dynamic fluctuation of the electron, so equation 3.20 become [HHubbard ] = U

X

hˆ ni↑ iˆ ni↓ + hˆ ni↓ iˆ ni↑ − hˆ ni↑ ihˆ ni↓ i

(3.23)

i

In mean field approach, the value of electron fluctuation is small enough to


23

Figure 3.3: Fe3 O4 unit cell without oxygen vacancy

neglect this effect. Therefore, Hubbard model become [HHubbard ] = U

X

hˆ ni↑ iˆ ni↓ + hˆ ni↓ iˆ ni↑ − hˆ ni↑ ihˆ ni↓ i

(3.24)

i

Furthermore, by recall counting operator as annihilation and creation operator the Hubbard model will transform into matrix form as follow



[HHubbard ] = U





hˆ ni↓ i 0  a↑  (a†↑ a†↓ )  − hˆ ni↑ ihˆ ni↓ i a↓ 0 hˆ ni↑ i

X i

(3.25)

The first term will be used as Hubbard interaction in our system, meanwhile the second term is total energy correction and not included in our Hamiltonian model. Finally, the interaction Hamiltonian of our system in matrix form is defined as follow



[Hint ] =

X j

− (a†↑ a†↓ ) 

JH Sj ¯ h cos θj + U hˆ nj↓ i 2 JH Sj ¯ h iφ − 2 sin θj e j

JH Sj ¯ h sin θj e−iφj  a↑  2 JH Sj ¯ h a↓ cos θj + U hˆ nj↑ i 2

−





(3.26) The Hamiltonian is different for two our model, the system without oxygen vacancy and system without oxygen vacancy. Hence, we will explain the Hamiltonian literally for each system.


24

3.3.1

Model Hamiltonian without Oxygen Vacancies

Hamiltonian kinetic term of the system is constructed by using tightbinding approximation as have explained in section 2.3.1. Fe3 O4 unit-cell without oxygen vacancy in our model as shown in figure 3.3 using 11 basis orbitals including px and py orbitals in each of four oxygen atoms and three eg orbitals (dx2 −y2 ) in each of three iron atoms (Fe3+ in octahedral and tetrahedral site; Fe2+ in tetrahedral site). The following basis orbitals are |OA − px i, |OA − py i, |OB − px i, |OB − py i, |OC − px i, |OC − py i, |OD − px i, |OD − py i, |F eA−eg i, |F eB1 −eg i and |F eB2 −eg i. These basis orbitals construct kinetic Hamiltonian matrix with the size 22 × 22 including spin up and spin down. Non-zero elements from kinetic Hamiltonian are

H0 (1, 1) = H0 (2, 2) = OA

(3.27)

H0 (3, 3) = H0 (4, 4) = OB

(3.28)

H0 (5, 5) = H0 (6, 6) = OC

(3.29)

H0 (7, 7) = H0 (8, 8) = OD

(3.30)

H0 (9, 9) = F eA

(3.31)

H0 (10, 10) = H0 (11, 11) = F eB

(3.32) !

kx c 2 ! ky c H0 (1, 6) = H0 (2, 5) = H0 (3, 8) = H0 (4, 7) = −2tO,O(near) cos 2 ! −i(kx + ky )c H0 (1, 7) = H0 (3, 5) = −tO,O(f ar) exp 2 ! −i(kx − ky )c H0 (2, 8) = H0 (4, 6) = −tO,O(f ar) exp 2 ! −iky c H0 (3, 9) = H0 (4, 9) = −tF eA,O exp 4 ! iky c H0 (7, 9) = H0 (8, 9) = −tF eA,O exp 4

H0 (1, 4) = H0 (2, 3) = H0 (5, 8) = H0 (6, 7) = −2tO,O(near) cos

(3.33) (3.34) (3.35) (3.36) (3.37) (3.38)


25

!

i(kx + ky )c H0 (1, 10) = −tF eB1 ,O exp 4 ! i(kx + ky )c H0 (3, 11) = −tF eB2 ,O exp 4 ! −i(ky − kx )c H0 (2, 11) = −tF eB2 ,O exp 4 ! −i(ky − kx )c H0 (4, 10) = −tF eB1 ,O exp 4 ! −i(kx + ky )c H0 (5, 11) = −tF eB2 ,O exp 4 ! −i(kx + ky )c H0 (7, 10) = −tF eB1 ,O exp 4 ! −i(−kx + ky )c H0 (6, 10) = −tF eB1 ,O exp 4 ! −i(−kx + ky )c H0 (8, 11) = −tF eB2 ,O exp 4 ! kx c H0 (10, 11) = −2tF eB,F eB cos 2

(3.39) (3.40) (3.41) (3.42) (3.43) (3.44) (3.45) (3.46) (3.47)

H0 (4, 1) = H0 (3, 2) = H0 (8, 5) = H0 (7, 6) = H0 (1, 4)∗

(3.48)

H0 (6, 1) = H0 (5, 2) = H0 (8, 3) = H0 (7, 4) = H0 (1, 4)∗

(3.49)

H0 (7, 1) = H0 (5, 3) = H0 (1, 7)∗

(3.50)

∗

H0 (8, 2) = H0 (6, 4) = H0 (2, 8)

(3.51)

H0 (9, 3) = H0 (9, 4) = H0 (3, 9)∗

(3.52)

H0 (9, 7) = H0 (9, 8) = H0 (7, 9)∗

(3.53)

H0 (10, 1) = H0 (1, 10)∗

(3.54)

H0 (11, 3) = H0 (3, 11)∗

(3.55)

∗

H0 (11, 2) = H0 (2, 11)

(3.56)

H0 (10, 4) = H0 (4, 10)∗

(3.57)

H0 (11, 5) = H0 (5, 11)∗

(3.58)

H0 (10, 7) = H0 (7, 10)∗

(3.59)

∗

H0 (10, 6) = H0 (6, 10)

(3.60)

H0 (11, 8) = H0 (8, 11)∗

(3.61)

H0 (11, 10) = H0 (10, 11)∗

(3.62)


26

Including the spin degrees of freedom, complete arrangement of matrix [H0 (~k)] is  h

i

H0 ~k

=



H ~k  0 0

↑

0   H0 ~k

(3.63)

↓

Hamiltonian interaction term construct from equation 3.26 and occur only in basis orbitals of iron ion which is |F eA − eg i, |F eB1 − eg i and |F eB2 − eg i including spin orientation. Thus, by substitute following basis orbitals into equation 3.26, non-zero element of Hamiltonian interaction for system without oxygen vacancy are

h ¯ nF eA↓ i Hint (9, 9) = − JH SF eA cos θF eA + U hˆ 2 h ¯ nF eB1 ↓ i Hint (10, 10) = − JH SF eB1 cos θF eB1 + U hˆ 2 h ¯ Hint (11, 11) = − JH SF eB2 cos θF eB2 + U hˆ nF eB2 ↓ i 2 h ¯ Hint (19, 19) = JH SF eA cos θF eA + U hˆ nF eA↑ i 2 h ¯ Hint (20, 20) = JH SF eB1 cos θF eB1 + U hˆ nF eB1 ↑ i 2 h ¯ nF eB2 ↑ i Hint (21, 21) = JH SF eB2 cos θF eB2 + U hˆ 2 h ¯ Hint (9, 20) = − JH SF eA sin θF eA e−iφF eA 2 h ¯ Hint (10, 21) = − JH SF eB1 sin θF eB1 e−iφF eB1 2 h ¯ Hint (11, 22) = − JH SF eB2 sin θF eB2 e−iφF eB2 2 Hint (20, 9) = Hint (9, 10)∗

(3.64) (3.65) (3.66) (3.67) (3.68) (3.69) (3.70) (3.71) (3.72) (3.73)

∗

Hint (21, 10) = Hint (10, 21)

(3.74)

Hint (22, 11) = Hint (11, 22)∗

(3.75)

In section 3.4, kinetic Hamiltonian term is used to construct the retarded Green’s function. On the other hand, interaction Hamiltonian is used to construct Sigma localized. Both of them are the important part in Dynamical Mean Field Theory Algorithm to describe properties of the system.


27

Figure 3.4: Fe3 O4 unit cell with oxygen vacancy.

3.3.2

Model Hamiltonian with Oxygen Vacancies

Hamiltonian kinetic term of this system is constructed by using tightbinding approximation with total 9 basis orbitals as shown in figure 3.4 where Oxygen (also its orbitals) in site B is removed. Thus, the following basis orbitals used in model without oxygen vacancy are i, |OA − py i, |OC − px i, |OC − py i, |OD − px i, |OD − py i, |F eA − eg i, |F eB1 − eg i and |F eB2 − eg i. These basis orbitals construct kinetic Hamiltonian matrix with the size 18 × 18 including spin orientations (spin up and spin down). Non-zero elements from kinetic Hamiltonian are H0 (1, 1) = H0 (2, 2) = OA

(3.76)

H0 (3, 3) = H0 (4, 4) = OB

(3.77)

H0 (5, 5) = H0 (6, 6) = OD

(3.78)

H0 (7, 7) = F eA

(3.79)

H0 (8, 8) = H0 (9, 9) = F eB

(3.80) !

kx c 2 ! ky c H0 (3, 6) = H0 (4, 5) = −2tO,O(near) cos 2

H0 (1, 4) = H0 (2, 3) = −2tO,O(near) cos

(3.81) (3.82)


28

!

−i(kx + ky )c H0 (1, 5) = −tO,O(f ar) exp 2 ! −i(kx − ky )c H0 (2, 6) = −tO,O(f ar) exp 2 ! −iky c H0 (3, 7) = H0 (4, 7) = −tF eA,O exp 4 ! iky c H0 (5, 7) = H0 (6, 7) = −tF eA,O exp 4 ! i(kx + ky )c H0 (1, 8) = −tF eB1 ,O exp 4 ! i(kx + ky )c H0 (3, 9) = −tF eB2 ,O exp 4 ! −i(ky − kx )c H0 (2, 9) = −tF eB2 ,O exp 4 ! −i(ky − kx )c H0 (4, 8) = −tF eB1 ,O exp 4 ! −i(kx + ky )c H0 (5, 8) = −tF eB2 ,O exp 4 ! −i(−kx + ky )c H0 (6, 9) = −tF eB1 ,O exp 4 ! kx c H0 (10, 11) = −2tF eB,F eB cos 2

(3.83) (3.84) (3.85) (3.86) (3.87) (3.88) (3.89) (3.90) (3.91) (3.92) (3.93)

H0 (4, 1) = H0 (3, 2) = H0 (1, 4)∗

(3.94)

H0 (6, 3) = H0 (5, 4) = H0 (3, 6)∗

(3.95)

H0 (5, 1) = H0 (1, 5)∗

(3.96)

H0 (6, 2) = H0 (2, 6)∗

(3.97) ∗

H0 (7, 3) = H0 (7, 4) = H0 (3, 7)

(3.98)

H0 (7, 5) = H0 (7, 6) = H0 (5, 7)∗

(3.99)

H0 (8, 1) = H0 (1, 8)∗

(3.100)

H0 (9, 3) = H0 (3, 9)∗

(3.101)

∗

H0 (9, 2) = H0 (2, 9)

(3.102)

H0 (8, 4) = H0 (4, 8)∗

(3.103)


29

H0 (8, 5) = H0 (5, 8)∗

(3.104)

H0 (9, 6) = H0 (6, 9)∗

(3.105)

H0 (11, 10) = H0 (10, 11)∗

(3.106)

Hamiltonian interaction term construct from equation 3.26 and occur only in basis orbitals of iron ion which is |F eA − eg i, |F eB1 − eg i and |F eB2 − eg i including spin orientation. Thus, by substitute following basis orbitals into equation 3.26, non-zero element of Hamiltonian interaction for system without oxygen vacancy are

h ¯ nF eA↓ i Hint (7, 7) = − JH SF eA cos θF eA + U hˆ 2 h ¯ nF eB1 ↓ i Hint (8, 8) = − JH SF eB1 cos θF eB1 + U hˆ 2 h ¯ Hint (9, 9) = − JH SF eB2 cos θF eB2 + U hˆ nF eB2 ↓ i 2 h ¯ Hint (16, 16) = JH SF eA cos θF eA + U hˆ nF eA↑ i 2 h ¯ Hint (17, 17) = JH SF eB1 cos θF eB1 + U hˆ nF eB1 ↑ i 2 h ¯ nF eB2 ↑ i Hint (18, 18) = JH SF eB2 cos θF eB2 + U hˆ 2 h ¯ Hint (7, 16) = − JH SF eA sin θF eA e−iφF eA 2 h ¯ Hint (8, 17) = − JH SF eB1 sin θF eB1 e−iφF eB1 2 h ¯ Hint (9, 18) = − JH SF eB2 sin θF eB2 e−iφF eB2 2 Hint (16, 7) = Hint (7, 16)∗

(3.107) (3.108) (3.109) (3.110) (3.111) (3.112) (3.113) (3.114) (3.115) (3.116)

∗

Hint (17, 8) = Hint (8, 17)

(3.117)

Hint (18, 9) = Hint (9, 18)∗

(3.118)

Same as before, both of them will be used in DMFT Algorithm and describe the properties of F e3 O4 with oxygen vacancy.


30

3.4

Dynamical Mean Field Theory Calculation Start Input Parameters Process : DMFT Calculation in Matsubara frequency domain

µ

Output : Σ(ωn ), P(θ)

Output : DOS(ω), µ P(θ)

Decision : Is Σ(ωn ) convergent?

no

Process : DMFT Calculation in Real frequency Domain (5 iteration)

yes Output : Magnetization Stop Figure 3.5: Flow chart of DMFT calculation in Matsubara domain frequency and real domain frequency

Dynamical Mean-Field Theory (DMFT) is used to calculate all the substantial value of our systems, such as magnetization, chemical potential and density of states. Since the Fe3 O4 -RGO nanoparticle system is measured at room temperature, the model must be calculated at finite temperature, specifically at room temperature. Unlike system in zero temperature, at finite temperature the system will be statistically distributed over all of its excited levels. As a consequence, DMFT calculation not only perform in real frequency doUniversitas Indonesia

31 main, but also perform in Matsubara frequency domain to include thermal and statistical effect from environment into our system. The mechanism to do DMFT calculation in two domain frequency can be explained as follows : First, DMFT calculation will be carried out in Matsubara frequency domain. From this calculation we will get self-energy of the system. Calculating self-energy must be conducted repeatedly until self-consistency condition fulfilled. In order to get self-consistency condition, Boltzmann weight from this calculation will be used in real frequency domain to perform DMFT calculation. From DMFT calculation in real frequency domain, we can get density of states and chemical potential of the system. The new chemical potential value then used for the next DMFT calculation in Matsubara domain. When self-consistency condition is fulfilled, magnetization calculation can be done. The following flow charts for the mechanism is shown in figure 3.5 Meanwhile, Dynamical Mean Field Theory Calculation begin from define Green’s function matrix from our model. Since our model threat the system in interacting view, we use retarded Green’s function to represent our system. For DMFT approach, retarded Green’s function equation from section 2.3 is modified where the self energy isn’t dependent with reciprocal momentum k. It means that in DMFT approach the interaction in the system can be simplified as the interaction in one specific site which is one unit cell. Therefore, retarded Green’s function from equation 2.3 will become h

GR (k, z) = [z[I] − [H0 (k)] − [Σ(z)]]−1 i

(3.119)

where for real frequency domain z = ω + iη with ω is real frequency and η is positive number approximately zero. Meanwhile, z = iωn + µ for Matsubara frequency domain, where ωn is Matsubara frequency and µ is chemical potential. The value of Matsubara frequency for fermion is ωn = (2n + 1)πT with n is positive integer number and T is temperature. The matrix [H0 (k)] is kinetic Hamiltonian of the system. The elements of matrix has been calculated in section 3.3.1 for system without oxygen vacancy and section 3.3.2 for system with oxygen vacancy. After that we average the retarded Green’s function to all points in the Brilloin zone. The purpose is to make dependencies of all points in Brillouin zone vanish. Therefore, each point in Brilloun zone become independent and


32 there isn’t any interaction each other. i 1 Xh R G (k, z) N k

[GR (z)] =

(3.120)

where N is number of ~k points in the first Brillouin zone. The mean-field function is required to solve the Dynamical Mean Field Approach impurity problem. The mean-field function can be interpreted as local kinetic Green’s function. The following mean field function can be described as follow −1 [GR (z)] + [Σ(z)]

[G(z)]M F =

(3.121)

The next part is to calculate local interacting Green’s function. Hamiltonian of Coulomb repulsive interaction (Hubbard model) and magnetic exchange interaction (Heisenberg-like model) in section 3.3.1 for system without oxygen vacancy and section 3.3.2 for system with oxygen vacancy is substitute to this Green’s function. Therefore, local Green’s function of the system including interacting part and kinetic part is constructed. The following local Green’s function can be expressed as follow h

[G(z, θ1 , θ2 , θ3 )]loc = [G(z)M F ]−1 − [Σ(θ1 , θ2 , θ3 )]loc

i−1

(3.122)

where matrix [Σ(θ1 , θ2 , θ3 )]loc is interaction Hamiltonian which is already calculated in section 3.3.1 and 3.3.2. We simplified our problems in this calculation by consider φ = 0. This Assumption does not change too much the physical meaning of the system, especially in magnetization calculation of the system. Now [G(z, θ1 , θ2 , θ3 )]loc must be averaged over all possible spin orientation at the local site. The average process implemented by using the Boltzmann weight P (θ1 , θ2 , θ3 ) = eSef f

(3.123)

where

Sef f = −

X n

h

i

iωn η ln det [GM F (iωn )][G−1 loc (iωn , θ1 , θ2 , θ3 )] e

(3.124)

to average over the angular distribution of the local spins. The calculation of Boltzmann weight is conducted only in Matsubara domain, because the simplification on doing calculation in finite temperature by using Boltzmann


33 weight factor can be applied in imaginary time frequency. Thus, Average local Green’s function is expressed as follow

1Z Z Z dθ1 dθ2 dθ3 [Gloc (z, θ1 , θ2 , θ3 )] P (θ1 , θ2 , θ3 ) Gave (z) = Z

(3.125)

RRR

where Z = P (θ1 , θ2 , θ3 )dθ1 dθ2 dθ3 . The extra factor of [GM F (iωn )] in equation 3.124 does not change the physics, but it is introduced to aid in convergence. Finally the new value of self energy can be calculated as follow [Σ(z)] = [GM F (z)]−1 − [Gave (z)]−1

(3.126)

The calculation of self-energy in DMFT Algorithm is conducted repeatedly until self-consistency condition is fulfilled. When the self-consistency condition is reached, the self-energy of the system approximately equal to interaction energy in real system. The flowchart of DMFT calculation is shown in figure 3.5 In self-consistency condition, physical value of the system can be extracted by do calculation based on DMFT calculation. Some physical value will be calculated in this thesis are : Density of states calculation can be calculated by using formula in equation 2.32. The calculation of DOS is performed in local site where one local site is representative of whole system. Considering our system is interacting system, equation 2.32 can be modified as follow 1 (3.127) DOS(ω) = − =T r[Gave (ω)] π Chemical potential calculation can be extracted by using the calculation of density of states. Recall that to calculate electron concentration in metal which can be expressed as follow nf illing = −

Z

dωDOS(ω)f (µ, ω, T )

(3.128)

where fermi-dirac function (f (µ, ω, T )) is (e(¯hω−µ)/kT + 1)−1 . To find chemical potential, we must find the root of equation 3.128 by using any numerical method, such as bisection, false position or secant. Chemical potential calculation perform in real frequancy domain. The propose of chemical potential calculation is to use this value in Matsubara domain until the convergence is achieved.


34 Magnetization calculation can be calculated as follow 1Z MZ = dθScos(θ) exp (−[Sef f − βHScos(θ)]) (3.129) Z In our magnetization calculation, the system is triggered by external magnetic field. The external magnetic field decrease slowly in the time of new iteration until external magnetic field final. The purpose of small external magnetic field applied is to break the symmetry along a preferential direction, i.e., the z axis.

3.5

Computational Method

The calculation will be done using a computer cluster with Ubuntu Linux as the operating system (http://www.ubuntu.com/). The programming language of choice is Fortran 90, aided by Linear Algebra Package (http://www.netlib.org/lapack) to do the calculation of higher order matrices and Message Passing Interface (http://www.mpich.org) to utilize parallel computing in computer clusters. The resulting output data is then plotted using xmgrace (http://plasma-gate.weizmann.ac.il/Grace/)


CHAPTER 4 RESULTS 4.1

Input Parameters

Parameters are important thing in our model. It is because our model is not an ab initio model but a model with phenomenological approach to describe the system, so we cannot naturally capture physical properties of the system. Therefore, we parametrizing the physical value we need in calculation to capture possible physics phenomenon within our model. There are three type parameters used in our calculation which is physical parameters, numerical parameters and environmental parameters. These parameters will be explained in following subsection.

4.1.1

Physical Parameters of Fe3 O4

The physical parameters used in calculation of electronic structure and magnetization in chapter 3 shown in table 4.1. Each value in physical parameters has physical meaning and properties of Fe3 O4 . The explanation for each value is described as follow: • The Lattice Parameter of Fe3 O4 has explained in section 2.2 which is 8.396 ˚ A. However, our Fe3 O4 model as shown in figure 3.3 define new lattice constant c to simplified our calculation. Thus, by using simple √ trigonometry we can get the value of c that is a 2/2. • The filling parameter is the total amount of itinerant electron in one unit cell of our model. From section 3.3.1, total basis for system without Oxygen vacancy is 22 basis orbitals and 18 basis orbitals for system with oxygen vacancies including spin degree of freedom. The total itinerant electron from one unit cell system without oxygen vacancies is 20 itinerant electron where each oxygen orbital basis contribute one itinerant electron, FeB1 (Fe3 + on octahedral site) and FeB2 (Fe3 + on tetrahedral site) orbital basis contribute one itinerant electron for each and FeA (Fe2 + on octahedral site) contribute two itinerant electrons. Meanwhile for system with oxygen vacancies, the total of itinerant electron is 16 because of the absence of one oxygen in one unit cell. 35

36 • The spin of iron comes from d-orbitals which is consist of five electron for Fe3+ ion and six electron for Fe2+ ion. Because of one electron of Fe3+ ion is itinerant electron, so the total spin of Fe3+ ion (FeB1 and FeB2) is 2. Meanwhile, Fe2+ (FeA) ion have two itinerant electron, so the total spin is 4, but because of orbital quenching, the total spin of Fe2+ ion become 2. • Our model using phenomenological approach to describe the system, therefore, the exact value of on-site and hopping parameters is no longer important. The most important thing is to determine the value of onsite and hopping parameter in our model while keeping the physical properties of the system. The reason to choose the values as shown in table 4.1 can be explained as follow. – On-site energy of oxygen must be lower than the iron ions, because oxygen’s core electron is fewer than the iron ions’ core electron. Thus, attractive energy from oxygen’s ion core is higher than iron ion (attractive energy is negative value). – On-site energy of electrons in FeA is higher than FeB1 or FeB2 because of their coordinate. FeA coordinated in tetrahedral site, so correction energy from nearest neighbour ion is much smaller than FeB1 or FeB2 which is coordinated in octahedral site. – The value of hopping parameters depend on the distance between ions. When the distance between two ions getting bigger, the value of a hopping parameter becomes smaller and vice versa. • Hund’s coupling constant value is how strong the ionic spin coupled electron itinerant spin. Usually this value is not higher than 1 eV. Because of Fe3 O4 has high temperature Curie, the value of a Hund’s coupling constant equals to 1 eV is chosen. • The Hubbard constant value get from reference [30] that show this value is in agreement with experimental result.


37 Tabel 4.1: Input physical parameters used in calculation of electronic structure and magnetization of Fe3 O4

Parameter Name

a c

Value 8.396˚ A √ a 2/2

nf nf VO SF eA , SF eB1 , SF eB2 εOA , εOA εOB , εOD εF eA εF eB1 , εF eB2 tO−O(near)

20 16 2, 2, 2 -1.5 eV -1.8 eV 0.5 eV 0.4 eV 0.4 eV

Hopping parameter for O-O (far)

tO−O(f ar)

0.3 eV

Hopping parameter for FeA-O Hopping parameter for FeB1-O Hopping parameter for FeB2-O Hopping parameter for FeB1-FeB2 Hopping parameter for FeA-FeB1 Hopping parameter for FeA-FeB2 Hund’s coupling constant Hubbard constant

tF eA−O tF eB1−O tF eB2−O

1.8 eV 0.9 eV 0.9 eV 0.825 eV 1.8 eV 1.8 eV 1 eV 4 eV

Lattice parameter Lattice parameter of the model Filling parameter without oxygen vacancies with oxygen vacancies Spin of iron ion On-site energy for OA and OC On-site energy for OB and OD On-site energy for FeA On-site energy for FeB1 and FeB2 Hopping parameter for O-O (near)

4.1.2

Notation

tF eB1−F eB2 tF eA−F eB1 tF eA−F eB2 JH U

Numerical Parameter

The numerical parameter is chosen to make the result more accurate and precision. But in the same time, the higher numerical parameter then the longer computational time required in calculation. Therefore, the right value of numerical parameters is necessary. In this numerical parameters there is a parameters that must be set with physical consideration. That parameter is ωn discritization parameter (Nωn ). The Nωn value must be high enough to make effective action become convergent, but not to high because computational time will become longer.


38 Tabel 4.2: Input numerical parameters used in calculation of electronic structure and magnetization of Fe3 O4

Parameter Name

Notation

Number of Matsubara iteration Number of real iteration number of interval for k-points ωn discritization parameter k-points discretization parameter ω discritization parameter Artificial broadening parameter

4.1.3

niter nriter nncxint Nωn Nk Nω η

Value 45 5 5 5000 112, 96, 64, 32 16 301 5 × 10−2

Environmental Parameter

Environmental parameters describe environmental condition when experimental measurement is perform. It is necessary because our result will be fitted with experimental result. Tabel 4.3: Input environmental parameters used in calculation of electronic structure and magnetization of Fe3 O4

Parameter Name Temperature External magnetic field initial External magnetic field final Decrement of external magnetic field

4.2

Notation temp Hinitial Hf inal DH

Value 300 K (0.05 − 0.07) eV (0 − 0.02) eV 0.005 eV

Density of States

Density of states is important to describe electronic properties of material. In modelling a system, the density of states of material hold a role to prove that our model is representative of the system or not. In our model, we use Fe3 O4 unit cell to examine magnetic enhancement of Fe3 O4 +RGO nanoparticle. Before we do the magnetization calculation, prove that our model is representative of Fe3 O4 becomes essential. Figure 4.1 is the calculated density of states. The x-axis of the plot is range of energy that already subtracted by chemical potential. Therefore, The zero value of the x-axis represent chemical potential of the system. Meanwhile, the y-axis show the states provided by the system.


39

Figure 4.1: Density of States Fe3 O4 unit cell without oxygen vacancy

The density of states show that our model is half-metallic materials. It is shown that in majority spin channel configuration, chemical potential of the system located inside of d-orbital band. Meanwhile in the minority spin channel, chemical potential located outside the band or in the band gap. The result of calculated density of states as half-metallic material prove that our system is the representative of Fe3 O4 because it is fit with reference [2] that describe Fe3 O4 is semiconductor in minority spin channel and conductor in majority spin channel.

4.3

Saturated Magnetization of Fe3 O4 with variation RGO Flake Weight Percent

Density of states profile able to show that our model is representative of Fe3 O4 system. Therefore, we can calculate saturated magnetization of Fe3 O4 with rGO wt% variation by using equation 3.8 in section 3.2. In calculate saturated magnetization, there are some physical parameters used so the result is in agreement with experimental data. The following parameter is alpha, gamma and n


40

Figure 4.2: Saturated magnetization of Fe3 O4 +rGO nanoparticle system by vary the wt% of rGO added into system. The calculation was done by using α = 20 and γ = 0.574 Tabel 4.4: Parameters that used to calculate saturated magnetization versus weight percent of RGO flake

Parameter Name Mass ratio of Fe3 O4 to RGO flake Percent of Magnetization enhancement The number of RGO flake must filled before stack

Notation

Value

α γ n

20 57.4 % 1, 2, 3

Parameter alpha (α) define the mass ratio of a Fe3 O4 nanoparticle cluster to a RGO flake. Theoretically, the mass of Fe3 O4 nanoparticle cluster obviously is much heavier than RGO flake. However, the exact value of either the mass ratio or mass for Fe3 O4 nanoparticle cluster and rGO flake is not available (experimental data is not provide it). And again, because we are using phenomenological approach, the assumption for the value of α is made without change the fact that the ratio must be greater than 1. Meanwhile, gamma (γ) is ratio of moment magnetic enhancement of Fe3 O4 cluster with some oxygen vacancies in the surface to moment magnetic of Fe3 O4 cluster that was already explained in section 3.2. gamma implicitly show the effectiveness of oxygen adsorption on the surface of Fe3 O4 nanoparticle clusters. If all Fe3 O4 unit cells Universitas Indonesia

41 of our model in the surface of Fe3 O4 nanoparticle cluster have oxygen vacancy, the value of gamma become maximum which is 2 or 200%. The Last, the variation of parameter N is in purpose to know the mechanism of rGO flakes fill the slots. Thus, those parameter fitted until get computational result that resemble the experimental data. Finally, table 4.4 is the most fit parameters which produce computational result most likely with experimental data. The result of calculation by using parameters in table 4.4 is shown in figure 4.2. By using three variation of n, we can see from the result the mechanism of rGO flake fill the slots. For n=1, show us that every additional of rGO flakes, the rGO flakes tend to fill all of the unoccupied slots by one rGO up to ˙ 5 wtThis mechanism make saturated magnetization increase linearly. When 5 wt% rGO added into the system, we can assume all of slots already filled by one rGO flake. Therefore, the saturated magnetization reach the maximum value. Additional of rGO flake more than 5 wt% will make the rGO pill up where the rGO flake tend to fill all of slots that have one rGO flakes first and right after all slots have two rGO flakes, the additional rGO flakes will fill all slot to have three rGO flakes and so on. Therefore, the decrement of saturated magnetization become linear. For others value of n, which is n=2 and n=3, the maximum saturated magnetization shift to the right. It is because there is still probability to find slots unoccupied by rGO flakes when some of slots already filled by two (for n=2) or three (for n=3) rGO flakes. This mechanism also describe that the peak of saturated magnetization for n=2 and n=3 become broader. Figure 4.2 which in agreement with experimental data using γ=0.574. This result show us that not all Fe3 O4 unit cells in the surface have oxygen vacancies. It is because in rGO flake structure some of carbon atoms that adsorbs oxygen from the surface of Fe3 O4 nanoparticle clusters already bonding with oxygen adatoms. Thus, the effective area to adsorb oxygen is reduced. Meanwhile, the value of alpha which is 20 show us that the mass of Fe3 O4 nanoparticle cluster is much heavier that rGO flake which is match with our assumption before.


42

Figure 4.3: External magnetic field dependence magnetization of Fe3 O4 with rGO flake weight percent variation.

4.4

External Magnetic Field Dependence Magnetization of Fe3 O4 with rGO Flake Weight Percent Variation

Dynamcial Mean Field Theory (DMFT) calculation produce external magnetic field dependence of Fe3 O4 unit cell magnetization. Since the result of computational calculation will be compared with experimental data which measure in room temperature, calculation of Fe3 O4 unit cell magnetization perform with T=300 K (room temperature). However, the result of computational calculation is not the hysteresis plot, but only after saturated magnetization achieve and decrease going to remanent magnetization. The result of magnetization calculation using DMFT algorithm for system without oxygen vacancies and with oxygen vacancies shown in figure 4.3. In Fe3 O4 -rGO nanoparticle model, the magnetization of system without adding any rGO flake is represent by black curve. Meanwhile, red curve represent the ideal system which is rGO flakes adsorb oxygen from all Fe3 O4 unit cells in the Universitas Indonesia

43

Figure 4.4: Temperature dependence magnetization of Fe3 O4

surface of Fe3 O4 nanoparticle clusters. By using equation 3.8, these result can be transformed into real system. In using equation 3.8, parameters in table 4.4 is used. The result of external magnetic field dependence of Fe3 O4 -rGO nanoparticle system magnetization is shown in figure 4.3. The results in figure 4.3 show us that the saturated magnetization of our model are in agreement with experimental data. Magnetization of system for variation of rGO flake wt% show the same result with experimental data which is the highest magnetization is come from Fe3 O4 nanoperticle system with adding 5 wt% rGO flake. The magnetization then decrease until the content of rGO flakes in Fe3 O4 nanoparicle is 20 wt%. The lowest magnetization is Fe3 O4 nanoparticle without adding any rGO flakes. Contrast with the saturated magnetization result, all of the remanent magnetization is not fit with experimental data. The computational calculation have remanent magnetization higher than experimental data. Meanwhile, if we find temperature Curie for Fe3 O4 without oxygen vacancy using the model, the temperature Curie of our model as shown in figure 4.4 is much higher than 858 Kelvin. Discrepancy of Curie temperature influence the incompatibility of remanent magnetization. This result arises because we overestimate our model, Universitas Indonesia

44 particularly Hubbard model. The model we used which is the unit cell of bulk Fe3 O4 also not completely represent of Fe3 O4 nanoparticle cluster. Another theoretical study also confirm surface anisotropy occur in Fe3 O4 nanoparticle cluster. Still our model qualitatively confirm magnetization enhancement of Fe3 O4 nanoparticle with additional rGO flake.


CHAPTER 5 CONCLUSION 5.1

Conclusion

The important findings in this research are summarized below. • Our present study support our previous hypothesis that : – The enhancement of magnetization of Fe3 O4 nanoparticle system upon addition of reduced graphene oxide (rGO) is due to spinflipping of Fe3+ in the tetrahedral sites. – Oxygen vacancies at the Fe3 O4 nanoparticle clusters assist the occurrence of spin-flipping. • The presence of rGO flakes nearby Fe3 O4 clusters induces the formation of Oxygen vacancies in the Fe3 O4 clusters, leading to the occurrence of spin-flipping and thus enhancing the magnetization of the system. • Our model for Fe3 O4 nanoparticle cluster can describe the saturated magnetization enhancement of Fe3 O4 nanoparticle system upon addition of reduced graphene oxide (rGO). • Our calculation method overestimates the Curie temperature and the remanent magnetization of Fe3 O4 nanoparticle system. This may be caused by the mean-field approximation used in our method.

45

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